TP Model Transformation based Control Design for Time-delay Systems
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TP Model Transformation based Control Design for Time-delay Systems: Application in Telemanipulation PhD Dissertation P´ eter Galambos Supervisors: P´ eter Baranyi, DSc (MTA SZTAKI) Guszt´ av Arz, CSc (BME GTT) Budapest, 2013.
Transcript of TP Model Transformation based Control Design for Time-delay Systems
TP Model Transformation based ControlDesign for Time-delay Systems:
Application in Telemanipulation
PhD Dissertation
Peter Galambos
Supervisors:
Peter Baranyi, DSc (MTA SZTAKI)
Gusztav Arz, CSc (BME GTT)
Budapest, 2013.
This dissertation is dedicated to the memory of my father, Dr. Tibor Galambos (1940-1998)
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Acknowledgements
I owe my gratitude to all those people who have made this disserta-tion possible. Most importantly, I would like to express my heartfeltgratitude to my family, as this dissertation would not have been possi-ble without their unconditional love and patience. My mother, Klaraand my wife, Eszter have been a constant source of love, concern andstrength throughout the years.I thoroughly appreciate the work of my supervisors, Dr. Peter Baranyiand Dr. Gusztav Arz, whose support guided me through even the mostdifficult scientific challenges and helped me in finishing this dissertation.I am especially grateful to Dr. Baranyi for his reading various versionsof my manuscripts over and over again, and his detailed comments onhow to make them better.I would also like to express my deep gratitude to all those whohavesupported my research during my PhD studies. I am especiallyindebtedto Prof. Peter Korondi, who has guided me to the research institutewhere I work, allowing me to learn the essence of scientific researchin an inspiring atmosphere. I would like to thank Andras Toth, whoseinfluence has taught me a great deal about R&D projects in robotics.Last but not least, I would like to express my sincere thankfulness to myfather, to whom this dissertation is dedicated. He guided methroughouthis exemplary life and taught me the essence of being an engineer.
The research was supported by HUNOROB project (HU0045), a grantfrom Iceland, Liechtenstein and Norway through the EEA FinancialMechanism and the Hungarian National Development Agency and bythe National Research and Technology Agency, (ERC-HU-09-1-2009-0004 MTASZTAK) (OMFB-01677/2009). Devices and instrumentationof the tele-grasping experiments was supported by the RESCUER (IST-2003-511492) project of the European Commission, that was coordi-nated by the Budapest University of Technology and Economics, De-partment of Manufacturing Science and Technology. The experimentalinvestigations of the CogInfoCom based approach for hapticfeedbackwas supported by the Hungarian National Development Agency, NAPproject,(NAP-1-2005-0021, KCKHA005, OMFB-01137/2008).
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Contents
Page
Preface 1
Goals of the thesis 2
Structure of the dissertation 3
Nomenclature 5
I Introduction 8
1 Preliminaries and scientific background of the research work 9
2 Recent directions in control of time-delay LPV systems 14
3 Impedance control for force reflecting telemanipulation 193.1 Impedance Control with Feedback Delay . . . . . . . . . . . . . . .. . . . . 213.2 Analysis of the Critical Delay . . . . . . . . . . . . . . . . . . . . . .. . . 23
3.2.1 Friction models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 The linear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.3 Non-linear friction . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . .. . 273.4 Comparison of the results . . . . . . . . . . . . . . . . . . . . . . . . . .. 283.5 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31
4 Control structure for stability preservation in impedance control under time-delay 32
5 TP Model Transformation based control design methodology 355.1 Higher Order Singular Value Decomposition of Functions. . . . . . . . . . 35
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5.1.1 Basic concept of tensor algebra . . . . . . . . . . . . . . . . . . .355.1.2 Definition of the HOSVD-based canonical form of TP functions . . 405.1.3 TP model transformation as a numerical reconstruction of the HOSVD-
based canonical form . . . . . . . . . . . . . . . . . . . . . . . . . 425.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.1.5 Convex hull manipulation of TP functions via TP model transformation 465.1.6 Extending the TP model transformation methodology toqLPV models 475.1.7 Definition of the HOSVD-based canonical form of qLPV models . 485.1.8 Convex hull manipulation of qLPV models via TP model transfor-
mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.9 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . 49
6 CogInfoCom in force reflecting telemanipulation 516.1 Conceptual Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
6.1.1 Cognitive Infocommunications . . . . . . . . . . . . . . . . . . .. 526.1.2 Inter-cognitive sensor-bridging in teleoperation .. . . . . . . . . . 526.1.3 CogInfoCom in teleoperation . . . . . . . . . . . . . . . . . . . . 546.1.4 Force feedback in teleoperation . . . . . . . . . . . . . . . . . .. 556.1.5 Haptics in Virtual Environment . . . . . . . . . . . . . . . . . . .566.1.6 Existing solutions for force feedback under the concept of CogInfo-
Com . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
II Theoretical Achievements 58
7 TPτModel Transformation for Dynamical Systems with Feedback Delay 597.1 TPτModel Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.1.1 STEP I: Redefinition-based discretization . . . . . . . . .. . . . . 607.1.2 STEP II: Extracting the TP structure . . . . . . . . . . . . . . .. . . 617.1.3 STEP III: Determination of the weighting functions . .. . . . . . . 62
7.2 A possible way of redefinition . . . . . . . . . . . . . . . . . . . . . . .. 637.3 Some further aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8 Impedance model with feedback delay in TP type polytopic LPV forms 658.1 Specification of the expected LPV reprezentation . . . . . .. . . . . . . . 668.2 The HOSVD based canonical form . . . . . . . . . . . . . . . . . . . . . .66
8.2.1 Components and structure of the exact HOSVD based canonical form 678.2.2 Executing trade-off by TPτmodel transformation . . . . . . . . . . 69
8.3 Manipulation of the convex hull . . . . . . . . . . . . . . . . . . . . .. . 728.3.1 The exact TP model . . . . . . . . . . . . . . . . . . . . . . . . . 738.3.2 Reduced TP model with 5 vertices . . . . . . . . . . . . . . . . . . 768.3.3 Reduced TP model with 4 vertices . . . . . . . . . . . . . . . . . . 788.3.4 Reduced TP model with 3 vertices . . . . . . . . . . . . . . . . . . 80
8.4 Analysis of the convex representation . . . . . . . . . . . . . . .. . . . . . 818.4.1 Constant time-delay . . . . . . . . . . . . . . . . . . . . . . . . . 82
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8.4.2 Varying time-delay . . . . . . . . . . . . . . . . . . . . . . . . . . 838.5 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
9 TPτModel Based Control Design Methodology 879.1 Steps of the proposed control design strategy . . . . . . . . .. . . . . . . . 87
9.1.1 TP type polytopic reconstruction . . . . . . . . . . . . . . . . .. . . 879.1.2 Determination of the controller and observer . . . . . . .. . . . . 889.1.3 Optimization based on convex hull manipulation . . . . .. . . . . 88
10 TPτ transformation based Control Design for Impedance Controlled Robot Grip-per 8910.1 Specification of the control problem . . . . . . . . . . . . . . . .. . . . . 89
10.1.1 Description of the control problem . . . . . . . . . . . . . . .. . . 8910.1.2 Control requirements and constraints . . . . . . . . . . . .. . . . 90
10.2 Execution of the TPτmodel transformation . . . . . . . . . . . . . . . . . . 9010.3 LMI-based multi-objective controller and observer design . . . . . . . . . . . 91
10.3.1 Asymptotically stable controller and observer . . . .. . . . . . . . . 9110.3.2 Constrained control signal . . . . . . . . . . . . . . . . . . . . .. 9210.3.3 Disturbance rejection . . . . . . . . . . . . . . . . . . . . . . . . .93
10.4 Relaxed conservativeness via convex hull manipulation . . . . . . . . . . . 9310.5 Resulted controller and observer gains . . . . . . . . . . . . .. . . . . . . 94
10.5.1 Controller 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9410.5.2 Controller 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9410.5.3 Controller 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
10.6 Evaluation and Validation of the Control Design . . . . . .. . . . . . . . . 9510.7 Carry-over the controller into the unstable domain of the impedance model . 101
III Experimental Validation 102
11 Experimental validation of the results 10311.1 The experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10311.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
11.2.1 Comparative tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 10511.2.2 Further test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
IV CogInfoCom-based approach as alternative solution for force feed-back 121
12 Inter-cognitive Sensor-bridging in Telemanipulation 12212.1 Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
12.1.1 Vibrotactile glove . . . . . . . . . . . . . . . . . . . . . . . . . . . 12312.1.2 Master device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
12.2 The proposed cognitive adapter . . . . . . . . . . . . . . . . . . . .. . . . 125
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12.2.1 Homogeneous linear vibration function . . . . . . . . . . .. . . . 12512.2.2 Inhomogeneous radiating vibration function . . . . . .. . . . . . . 126
12.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12712.3.1 Telemanipulative Grasping . . . . . . . . . . . . . . . . . . . . .. . 12712.3.2 Interaction in Virtual Environment . . . . . . . . . . . . . .. . . . . 127
12.4 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12912.4.1 Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12912.4.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12912.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.5 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
V Conclusion 133
13 Scientific results 13413.1 TPτmodel transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13413.2 Polytopic reconstruction of the mass-damper impedance model with retarded
elastic force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13513.3 Control design for impedance controlled tele-grasping . . . . . . . . . . . . 13513.4 Definition of the force feedback task as inter-cognitive sensor-bridging channel135
14 Concluding remarks and future perspectives 13614.1 Extension of the control design to the unstable region .. . . . . . . . . . . 13614.2 Determination of the momentary time delay . . . . . . . . . . .. . . . . . 13614.3 Concerning the characteristics of time-delay . . . . . . .. . . . . . . . . . . 13714.4 Taking the environmental stiffness into account . . . . .. . . . . . . . . . . 13714.5 Pilot implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 138
15 Theses 140
List of Figures 144
Author’s publications 145
Bibliography 164
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Preface
The motivation of the research concluded in this dissertation dates back to the RESCUER(FP6-IST-2003-511492) project, which was coordinated by the Department of Manufactur-ing Sciences and Technology of BME. The project focused on the development of intelligentinformation and communication technologies and a mobile robot for emergency risk manage-ment, more specifically for improvised explosive device disposal and civil protection rescuemission scenarios.
The RESCUER mobile robot is equipped with two simultaneously working 6-DoF robotarms with teleoperated grippers. In this project, I dealt with the control design of a two-fingered, force feedback capable master-slave gripper thatwas mounted onto the left armof the mobile robot. I faced the challenging problem of forcereflecting telemanipulationwhere the stability of bilateral control and the realistic force sensation (transparency) arecontradicting requirements. Low communication bandwidth, varying time-delay of the com-munication (jitter), nonlinearities in the mechanisms andthe unknown remote environmentcause unstable behaviour and degrade the transparency. Among these causes, time-delay iscrucial because this is an inherent property of distributedcontrol systems. Internet-basedteleoperation is a typical example, where communication delay plays an important role. Inrecent years, several approaches were published addressing the stability problem of closedloop force reflecting telemanipulation over packet-switched network. In the technical partof my research, I have been focusing on the coupled impedancecontrol based approach forbilateral telemanipulation that was also utilized in the RESCUER project.
The theoretical aspects of this work are inspired by the scientific workshop in MTA SZ-TAKI where I have been working since 2009. The TP model transformation based controland the underlying qLPV and LMI theories, which are stronglyrelated to the work of BOKOR
and BARANYI , served as background for the achievements of this dissertation.In this thesis, I endeavour to give a comprehensive view of the results of my research
from the general theoretical discussion to the applicationin a practical engineering problem.
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Goals of the thesis
The goals of this dissertation can be summarized in three main points:
• Control design for time-delay systems often requires case-specific solutions, and rathercomplicated mathematical theories. Thus, there is a gap between the engineering solu-tions and the related scientific literature of control theory. Therefore, my first goal wasto investigate whether it is possible to extend the modern polytopic qLPV and LMIbased control design methodologies already emerging in theengineering solutions toa class of time-delay control design problems. My further goal is to implement thispossible extension in a numerically appealing, routine-like solution.
• The stabilization problem of the mass-spring-damper impedance model based forcefeedback capable bilateral master-slave tele-grasping system is very challenging whenthe pocket switched communication between the master and slave devices introducesvarying time-delay. Since, this is a very up-to-date issue of Internet-based teleopera-tion, my goal was to give a control design method to this tele-grasping problem.
• The Cognitive Infocommunications (CogInfoCom) is an interdisciplinary science thathas been developing via the synergy of cognitive science andinfocommunications andprimarily aimed at engineering solutions. One of the key research topic of CogIn-foCom is dealing with haptic remote sensation based on the plasticity of the humanbrain. Because this is a newly emerging science, my goal is toinvestigate how theCogInfoCom theories may lead to alternative approaches in the field of force feedbackenhanced telemanipulation. Beside the conceptual reformulation of this force feedbackproblem in terms of CogInfoCom, my further goal is to developdifferent experimen-tal testbeds and various evaluation techniques for the CogInfoCom based alternativesolutions.
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Structure of the dissertation
This dissertation is organized into four parts. Part I discusses the preliminaries and the the-oretical background related to the achievements of the dissertation. Part II introduces thenew scientific contributions, while Part III describes the experimental work that serves as avalidation of the results. Part IV introduces an alternative approach to handle the stabilityproblem of haptic devices and telemanipulation with time-delays.
Chapter 1 of Part I gives a brief historical introspection into the related control theorytouching various theoretical aspects and trends of the topic.
Chapter 2 focuses on current directions in control of time-delay LPV systems, which isa newly emerging field in practical system and control science inspite the theory of delayeddifferential equations are quite mature.
In Chapter 3 the impedance control structure and its stability problem is introduced,which served the original motivation for this dissertation. Despite that chapter also discussessome own preliminary results, I decided to include it into the introduction as the content ofit does not directly related to the main achievements ratherserves as additional support forthe motivation.
Chapter 4 introduces a control structure, which is appropriate to handle the control designproblem for delayed impedance model. Applying this structure, the stabilization question isreadily formulated according to the standards of control signal design problem of the moderncontrol theory.
Chapter 5 of the Introduction recalls the theoretical apparatus, namely the Higher Or-der Singular Value Decomposition (HOSVD) of qLPV state-space models and the TensorProduct (TP) model transformation that serves as theoretical basis behind my achievements.However, these two theories can be find in several publications with slightly different nota-tions, for the sake of uniformity and completeness these areincluded in a complete form.
At the end of Part I, chapter 6 introduces the new interdisciplinary scientific field of Cog-nitive Infocommunications that shows alternative perspective in handling the stability prob-lem of haptic and telemanipulation devices by the substitution of the original kinaestheticsensory channel by other sensory input to render force information to the operator.
The new theoretical achievements are discussed in Part II. Chapter 7 introduces theTPτmodel transformation as the extension of the original TP model transformation to dy-namical systems with feedback delays. Chapter 8 applies theTPτmodel transformation onthe specific case of the mass-spring-damper impedance modelwith retarded elastic force andanalyses the resulting TP structure. Chapter 9 introduces the control design methodologybased on the TPτmodel transformation, then in chapter 10 I apply the proposed methodologyto the stabilization problem of the impedance control basedforce reflecting tele-grasping.
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Part III is completely devoted to the experimental validation of the results discussed inPart II. Laboratory experiments on the large-scale prototype of the RESCUER tele-grippersystem are presented.
Finally, Part IV deals with the haptic and telemanipulationforce feedback problem inthe aspect of Cognitive Infocommunications presenting thefundamentals of an alternativesolution and some related experimental results aiming to define a vibrotactile CogInfoComchannel to implement haptic feedback using a lightweight, wireless vibrotactile glove.
Part V concludes the results of this dissertation and presents those in form of theses.
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Nomenclature
A few comments are appropriate on the notation used in this dissertation. To facilitatethe distinction between scalars, vectors, matrices, and higher-order tensors, the type of agiven quantity will be reflected by its representation: scalars are denoted by lower-case let-ters (a, b, . . . ;α, β, . . . ) (italic shaped), vectors are written as bold-face lower case letters(a,b, . . . ), matrices correspond to bold-face capitals(A,B, . . . ), and tensors are written ascalligraphic letters(A,B, . . . ). This notation is consistently used for lower-order parts of agiven structure. For example, the entry with row indexi and column indexj in a matrixA,i.e.,(A)ij, is symbolized byaij (also(a)i = ai and(A)i1i2...iN = ai1i2...iN ); furthermore, theith column vector of a matrixA is denoted asai, i.e.,A = (a1a2 . . . ). To enhance the over-all readability, we have made one exception to this rule: as we frequently use the charactersi, j, r, andn to represent of indices (counters),I, J , R, andN will be reserved to denote theindex upper bounds, unless stated otherwise. The pseudo inverse of matrixA is indicated byA+.
Symbols used in the dissertation
F Vertex system of the state-feedback gainsK Vertex system of the observer gainsS(p(t)) the system matrix of linear parameter-varying state space modelS coefficient tensor of the finite element TP model
constructed from the vertex systemσ singular valueΩ transformation space of the TP modelΘ discretization grid densityu(t) value of the control signalUn matrix representation of the weighting functionsw(p) weighting function of the finite element TP model.x state vector
The superscript of the weighting functionsw(p) shows its type, such aswCANONICAL(p),wSN(p), wNN(p), wNO(p),wCNO(p), wINO(p), andwRNO(p) mean that the type of the weight-ing functions are SN, NN, NO, CNO, INO, or RNO, respectively.The convex weightingfunctions that are at least SN and NN types are indicated aswCO(p).
The multiplen-mode product of a tensor, such asA×1 U1 ×2 U2 ×3 · · · ×N UN can be
shortly denoted asAN
⊠n=1
Un.
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q Vector of generalized coordinatesf External forceM Mass matrixB Damping matrixK Stiffness matrixc Non-linear friction termFh Desired contact force (handling force exerted by the operator) [N ]Fe interaction force with the remote environment [N ]x position of the impedance model [m]m mass in the impedance model [kg]b viscous damping in the impedance model [Ns/m]k stiffness of the environment [N/m]Ff Friction force [N ]Fc Coulomb friction force [N ]Fs Static friction force [N ]Vs Stribeck velocity [m/s]τ Time-delay [s]τcrit Critical (boundary) time-delay [s]M discretization grida, b, . . . scalar valuesa,b, . . . vectorsA,B, . . . matricesA,B, . . . tensorsA×n U n-mode product of a tensor by a matrix
AN
⊠n=1
Un multiple n-mode product as
A×1 U1 ×2 U2 · · · ×N UN
Abbreviations
LPV – Linear Parameter Varying (Used when the context is restricted to the set of LPVmodels.)
qLPV – quasi Linear Parameter Varying (Used, if a statement is valid for the wider set ofqLPV models)
LTI – Linear Time Invariant
LMI – Linear Matrix Inequality
SVD – Singular Value Decomposition
HOSVD – Higher-Order Singular Value Decomposition
CHOSVD – Compact Higher-Order Singular Value Decomposition
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RHOSVD – Reduced Higher-Order Singular Value Decomposition
TP model transformation – Tensor Product model transformation
TPτ – Tensor Product model transformation for time-delay systems
SN – Sum Normalized
NN – Non-Negative
NO – NOrmal
CNO – Close to NOrmal
RNO – Relaxed NOrmal
INO – Inverse NOrmal
IRNO – Inverted and Relaxed NOrmal
PDC – Parallel Distributed Compensation
TS – Takagi-Sugeno
LTF - Linear Fractional Transformation
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Part I
Introduction
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Chapter 1
Preliminaries and scientific backgroundof the research work
The results presented in this Thesis are based on the substantive changes of system and con-trol theory and the underlying field of mathematics that continued in the last decades. In thissection the scientific background of the touched fields and the most important achievementsare introduced.
Multi-objective nonlinear control theory
The quasi Linear Parameter Varying (qLPV) representationsand Linear Matrix Inequal-ity (LMI) based analysis and system control design are some of the topics, which are infocus of modern control theories. qLPV systems appear in theform of Linear Time Invari-ant (LTI) state-space representations where the elements of theS(p(t)) system matrices candepend on an unknown, but at any time instant measurable vector parameterp(t). This pa-rameter can be a function of time or state variables. The parameters may represent constantbut unknown uncertainties or external time signals. These properties show relations to thetheory of uncertain systems with parametric uncertaintiesand to the theory of LTV systems,too. The application of qLPV system representations appeared in relation to aerospace con-trol and it represents a systematic approach to gain scheduling control for nonlinear systems(Shamma and Athans, 1991, [1]). Passivity and H∞ theory have been extended to designrobust controllers for qLPV systems, see e.g. Lim and How (2002), Becker and Packard(1994). Moreover, the study of qLPV systems provides additional insight into some long-standing and sophisticated problems in robust adaptive control (see Athans et. al., 2005 [2]),switching control systems (see Hespanha et. al., 2003) and in intelligent control (see Fengand Ma, 2001, Ravindranathan and Leitch, 1999).
The appearance of Lyapunov-based stability criteria in theform of LMIs made a sig-nificant improvement. From this point on, stability questions were formulated in a newrepresentation, and the feasibility of Lyapunov-based criteria was reinterpreted as a con-vex optimization problem, as well as extended to an extensive model class. The pioneersGahinet, Balas, Chilai, Boyd, and Apkarian were responsible for establishing this new con-cept [3, 4, 5, 6, 7, 8], in Hungarian respective Bokor established the complete geometricalrepresentation of convex optimization. Soon, it was also proved that this new representa-
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tion could be used for the formulation of different control performances beyond the stabil-ity issues together with the optimization problem. Ever since then, the number of papersabout LMIs are increasing drastically in various topics such as LQ optimal control, robustH∞ control /H∞ synthesis,µ-analysis, quadratic stability, Lyapunov-based stability, multi-model and multi-objective state-feedback control of parameter-dependent systems, controlof stochastic systems. Boyd’s paper [7] states that it is true of a wide class of control prob-lems that if the problem is formulated in the form of LMIs, then the problem is practicallysolved.
In parallel, efficient numerical mathematical methods and algorithms were developed forsolving convex optimization problems-thus LMIs (Nesterovand Nemirovski). As a result,with the usage of numerical methods of convex optimization,we consider a large set ofproblems that require the resolution of a huge number of convex algebraic Ricatti-equationssolved today, in spite of the fact that the result of the obtained solution is not a closed (in itsclassical sense) analytical equation.
System modeling and identification theory
In the last decade, various new representations of dynamic models have emerged in thesystems theory. The origins of this paradigm shift can be linked with the famous speechgiven by Hilbert in Paris, in 1900. Hilbert listed 23 conjectures and hypotheses concerningunsolved problems which he believed would prove to be the biggest challenge of the 20thcentury [9]. According to Hilbert’s 13th conjecture, thereexist continuous multi-variablefunctions which cannot be decomposed as the finite superposition of continuous functionsof a smaller number of variables. In 1957, Arnold disproved this hypothesis [10]. Moreover,in the same year, Kolmogorov [11] formulated a general representation theorem, along witha constructive proof, which allows for a decomposition intoone-dimensional functions (seealso Sprecher [12] and Lorentz [13]). This proof justified the existence of ”universal approx-imators”. Based on these results, starting from the 1980s, it has been proved that universalapproximators exist within the categories defined by approximation tools such as biologicallyinspired neural networks and genetic algorithms, as well asfuzzy logic [14, 15]. As a result,these approximators have appeared in the identification models of systems theory, and turnedout to be effective tools even for systems that can hardly be described in an analytical way.Based on the above, we have various effective identificationtechniques today. However, theidentified models obtained as a result of these alternative techniques are described in formsthat are from description point of view quite far form the models given by analytical closedformulas derived via physical considerations engendered by the system under scrutiny.
Tensor algebra
During the last 150 years several mathematicians (Beltrami, Jordan, Sylvester, Schmidtand Weyl to name a few) were responsible for establishing thefoundations of the SingularValue Decomposition (SVD) and for developing its theory, which is one of the most fruitfuldevelopments in linear algebra. A very recent result is the Higher-Order generation of theSVD (HOSVD) to tensors (Lathauwer, 2000, SIAM journal [16]). The Workshop on TensorDecompositions and Applications held in Luminy, Marseille, France, 2005 was the first event
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where the key topic was HOSVD. Its very unique power comes from the fact that it candecompose a givenN-dimensional tensor into a full orthonormal system in a special orderingof higher order singular values, expressing the rank properties of the tensor in the order ofL2-norm. In effect, the HOSVD is capable of extracting the veryclear and unique structureunderlying the given tensor.
HOSVD of continuous functions
HOSVD was applied for fuzzy approximation and fuzzy rule base reduction by Yam[17, 18, 19]. Baranyi gave the concept of the HOSVD of continuous functions and in[20] proposed the Tensor Product model transformation for numerical reconstruction of theHOSVD of continuous functions, which can be scalar, vector or tensor function. Szeidl etal. proved in [21] that the TP model transformation is capable of numerically reconstruct-ing the HOSVD-based canonical form of the functions. TP model transformation is suitableof generating various convex forms of functions besides theHOSVD-based canonical form.Tikk [22, 23, 24] investigated the tradeoff and other approximation capabilities of TP modeltransformation.
Tensor Product model transformation-based control design
We can conclude that on the one hand we have powerful optimization and control designtechniques for polytopic and affine forms of qLPV models, andon the other hand we havea large variety of identification techniques. However, we can hardly link these two aspectsbecause of the lack of a uniform representation. Therefore,there is a great demand for auto-matic and uniform ways to convert various alternative representations into a unique polytopicor affine form.
The TP model transformation was introduced to control by Baranyi in [25] as a methodol-ogy for system control design, based on the theoretical results of tensor algebra. It is capableof transforming given qLPV models into proper polytopic forms, upon which LMI-baseddesign techniques are immediately executable. The qLPV andpolytopic representations ofa given plant are not unique, and different representationsof the same model lead to dif-ferent achievable performance when LMI techniques are usedto design qLPV controllers.Therefore, it is very important to define a unique representation that can be derived from anypolytopic or qLPV representation. Baranyi introduced a definition for the HOSVD canonicalform of the qLPV models and showed that the TP model transformation is capable of numer-ically reconstructing this unique form. Nagy investigatedcomputational relaxed TP modeltransformation of higher dimensional problems in [26], while Petres in [27] by separation ofconstant and non-constant elements.
Various polytopic forms of the same model affect the performance of LMI-based con-trollers (see [28, 29] for more information), thus TP model transformation has to be capableof systematic generation of different convex hulls for tensor functions and qLPV models.This way TP model transformation introduces an additional possibility to multi-objectivecontrol optimization techniques, namely the convex hull manipulation-based optimization,which is the key property of TP model transformation in control design. Control designoptimization can be done in three main steps altogether.
11
i The first step is the correct construction of the system matrix S(p(t)) to avoid numeri-cal anomalies (this is not real optimization, but it does affect the feasibility of LMIs).
ii The second step of optimization is the convex hull manipulation.
iii The third step is the LMI-based convex optimization.
Properties of TP model transformation
The properties of TP model transformation are given below:
• It can be executed uniformly (irrespective of whether the model is given in the formof analytical equations resulting from physical considerations, or as an outcome ofsoft computing-based identification techniques such as neural networks or fuzzy logic-based methods, or as a result of a black-box identification),without analytical interac-tion, within a reasonable amount of time. Thus, the transformation replaces the analyt-ical, and in many cases complex and not obvious, conversionsto numerical, tractable,straightforward operations that can be carried out in a routine fashion.
• It generates the HOSVD-based canonical form of qLPV models.This is a uniquerepresentation. This form extracts the unique structure ofa given qLPV model in thesame sense as the HOSVD does for tensors and matrices, in sucha way that:
– the number of LTI components are minimized;
– the weighting functions are one variable functions of the parameter vector in anorthonormed system for each parameter;
– the LTI components are also in orthogonal position;
– the LTI systems and the weighting functions are ordered according to the higher-order singular values of the parameter vector.
– Unique form
• The TP model transformation was extended to generate different types of convex poly-topic models. This was motivated by the fact that the type of convex hull of the poly-topic models considerably influences the feasibility of theLMI-based design and theresulting performance. This means that instead of developing new LMI equations forfeasible controller design (this is the widely adopted approach) we may rather focuson the systematic (numerical and automatic) modification ofthe convex hull ([28, 29]).It is worth noting that both the TP model transformation and the LMI-based controldesign methods are numerically executable one after the other, and this makes theresolution of a wide class of problems possible in a straightforward and tractable, nu-merical way.
• Based on the higher-order singular values (that express therank properties of the givenqLPV model for each element of the parameter vector inL2 norm), the TP modeltransformation offers a trade-off between the complexity of further LMI design andthe accuracy of the resulting TP model.
12
• The TP model transformation is executed before utilizing the LMI design. This meansthat when we start the LMI design we already have the global weighting functionsand during control we do not need to determine a local weighting of the LTI systemsfor feedback gains to compute the control value at every point of the hyperspace thesystem should go through.
Applications and related works to TP model transformation
One can find several applications and related works to TP model transformation in thetechnical literature, the most important of them without the claim of completeness are givenbelow. These include designing fuzzy controllers for servosystems [30]. [31] refers to TPmodel transformation in dynamic fuzzy controller development and [32] in predictive fuzzycontrollers. [33] refers to TP model transformation in obstacle avoidance trajectory design.TP model transformation-based controller design for gantry crane control system is proposedin [34, 35]. TP model transformation-based experimental validation of iterative feedbacktuning solutions for inverted pendulum crane mode control and stable iterative feedbacktuning-based design of Takagi-Sugeno PI-fuzzy controllers is referred in [36, 37]. [38] refersto TP model transformation in H-infinity control design for delayed discrete fuzzy systems.Tensor product-based real-time control of the liquid levels in a three tank system is applied in[39]. [40] applies TP model transformation for control of a pneumatic cylinder. [41] refersto TP model transformation in applying Takagi-Sugeno fuzzymodel for the estimation ofnonlinear functions.
13
Chapter 2
Recent directions in control of time-delayLPV systems
Control of systems with time-delays is a subject of significant practical and theoretical im-portance, and has been examined extensively in the controlsliterature using both frequencydomain and time domain methods.
Most work has been concentrated on the stability and controlof systems with fixed de-lays or the robust stability and control of systems with uncertain delays. However, in manyengineering applications, the time-delays are variable and are known functions of systemoperating conditions or system parameters that can be measured in real-time. For exam-ple, the transport delay in an internal combustion engine isa known function of the enginespeed. Similarly, parameter dependent time-delays often appear in many manufacturing andchemical processes, biomedical systems and robotic systems where changes in the systemdynamics result in variable delay times. (e.g., see [42, 43,44] and the references therein).
During the last decade, the stability analysis of time-delay systems with LMI based meth-ods have attracted a large number of researchers [45, 46, 47,48, 49, 50]. The criteria devel-oped can be classified into two major categories: the delay independent case and the delaydependent case. The first category assumes the delay unknownand possibly unbounded.The criteria therefore does not depend on the size of the delay. In the second category, mostdelay-dependent results (see [51] and references therein)apply to systems stable withoutdelays and look for the maximal delay that preserves stability. Nevertheless, as it has beenshown in [52, 53, 54] and references therein, there exist systems that are unstable for zerodelay and becomes stable only for some strictly positive values of the delay. It is also of usualpractice to use the Lyapunov-Krasovskii functionals and the Lyapunov-Razumikhin theoryfor delay independent and delay-dependent cases, respectively [55].
The idea of merging time-delay systems and LPV systems is notnew but is a ratherspecific topic. Indeed, works are based on the stability analysis and control synthesis haveappeared only recently and a few researchers deal with the observation and filtering problem.
The most important recent directions in control of time-delayed LPV systems are givenin the following.
Output Feedback RobustH∞ Control of Uncertain Fuzzy Dynamic Systems withTime-Varying Delay
14
Lee et al. in [56] deal with output feedback robustH∞ control problem for a class ofuncertain fuzzy dynamic systems with time-varying delayedstate. The nonlinear systemis represented by a Takagi-Sugeno (T-S)-type fuzzy model; then, the control design is car-ried out on the basis of the fuzzy model via the so-called parallel distributed compensationscheme. Using a single quadratic Lyapunov function, the globally exponential stability anddisturbance attenuation of the closed-loop fuzzy control system is ensured.
LPV Systems with parameter-varying time-delays: analysisand control
Wu et al. in [57, 58] seek to synthesize parameter-varying controllers to stabilize the time-delayed LPV system and to provide disturbance attenuation measured in terms of the inducedL2 norm of the system. The proposed approach utilizes parameter dependent Lyapunovfunctionals to obtain sufficient conditions for stabilization and inducedL2 norm performancein terms of LMIs. Although the single delay case is considered, the results can be easilyextended to treat systems with multiple delays.
L2-L2 and L2-L∞, Output Feedback Control of Time-Delayed LPV Systems
Tan and Grigoraidis examine the analysis and output feedback synthesis problems forlinear parameter-varying (LPV) systems with parameter-varying time-delays in [59]. Thestability is verified using parameter-dependent Lyapunov-Krasovskii functionals [60]. Bothanalysis and synthesis conditions are formulated in terms of linear matrix inequalities (LMIs)that can be solved via efficient interior-point algorithms.They extend previous work on thestate feedback control of time-delayed LPV systems [57] in three directions:
• A main contribution is the extension to the output feedback case.
• Also, theL2 to L∞, gain delayed LPV stabilization and control is addressed, that hasnot been examined before in the literature.
• In addition, a less conservative formulation is developed compared to [57].
In this new formulation both parameter matrices of the Lyapunov-Krasovskii functionalare parameter dependent to provide a wider class of Lyapunovfunctions and reduce con-servatism. The proposed results provide delay-rate dependent stabilization, theL2 gain andL2-to-L∞, gain output feedback synthesis conditions for delay LPV systems.
Stability Analysis of Time-Delay LPV Systems
Zhang et al. provide one of the first attempts to derive computationally tractable criteriafor analyzing the stability of Linear Parameter Varying (LPV) time-delayed systems. Zhanget al. present both delay-independent and delay-dependentstability conditions, which arederived using appropriately selected Lyapunov-Krasovskii functionals. According to thesystem parameter dependence, these functionals can be selected to obtain increasingly non-conservative results. Gridding techniques may be used to cast these tests as Linear MatrixInequalities. To reduce conservatism for the delay-independent stability case [46] introduces
15
parameter-dependent Lyapunov-Krasovskii functionals. All the stability tests are given interms of Linear Matrix Inequalities.
An LMI Approach to Stability of Systems With Severe Time-Delay
Jing et al. in [61] deal with severely time-delayed systems,which may face great chal-lenges in achieving desired stability. New Lyapunov-Krasovskii functionals with a properdistribution of the time-delays are proposed to obtain lessconservative stability conditionsfor a class of uncertain systems with arbitrarily time-varying delay. The results are less con-servative and more generic than the existing ones, however constraints on the time derivativeof the delays are required.
RobustH∞, Filtering for LPV Discrete-Time State-Delayed Systems
Wang et al. in [62, 63] examine the problems of robustH∞ filtering design for linearparameter-varying discrete-time systems with time-varying state delay. [62, 63] extends theprevious works of controller design for time-delayed LPV systems withH∞ filter design.The newH∞ performance criteria depends on the parameters and the delay-varying magni-tude using appropriately selected Lyapunov-Krasovskii functional. The corresponding filterdesign problems are finally cast into convex optimization problems.
Delay-dependent Stability Analysis andH∞ Control for State-delayed LPV System
Zhang et al. in [64] extended the only delay-dependentH∞ control result for LPV systemwith state delays presented in [57]. In [57] the analysis andstate-feedback problems forLPV systems with a parameter-varying time-delay are presented in that work. However, rateinformation for the delay variation has not been used in [57]resulting in conservative results.Zhang et al. in [64] deals with delay-dependent stability and H∞ control of LPV systemswith a rate bounded time-varying state delay. Lyapunov-Krasovskii functionals are used toderive sufficient conditions for stability and inducedL2 norm performance in terms of LMIs.In addition, the variation rate of the time-delay is used to derive the analysis and synthesisconditions.
Stability of Time-Delay Systems with Non-Small Delay
Gouaisbaut and Peaucelle deal with the problem of proving the stability of a linear time-delay system for a given delayτ without assuming the system to be stable forτ = 0.
The concept of Topological Separation [65] is an alternative framework to LyapunovTheory for proving stability of systems. In that framework,stability (or well-posedness) isdefined with respect to some feedback connection and is proved at the expense of findinga separator function. Depending on the feedback connectionmodeling, this separator hasdirect relation with Lyapunov functionals. In case of linear systems, the separator may bechosen as a quadratic function as demonstrated in [66]. These last results where extendedfor descriptor systems in [67] and were applied to time-delay system stability analysis in the[51]. A closely related, alternative to the Lyapunov framework, approach can also be found
16
in [46, 48]. In these, delay-dependent results on time-delay system stability are obtainedapplying small-gain methodology (which is a sub-case of thegeneral quadratic separation)to an artificially modified ”comparison” system that involves not only the Laplace operators and the delay operatore−sτ but also some new combination of these two. However, thesepaper deal only with the usual delay-dependent case for which the system with zero delay hasto be stable. To improve these results and handle the case of non-small delay, the contributionof [68] is based on Taylor series representation of the delayoperator.
Delay-DependentH∞ Filtering for a Class of Time-Delay LPV Systems
Lyapunov-Krasovskii functionals for the delay dependent LPV systems with time-delayare introduced by Mohammadpour in [69, 70], which is concerned with delay-dependentanalysis and design ofH∞ filters for continuous-time LPV systems that include a statedelayin the system. The parameter-dependent Lyapunov-Krasovskii functional is used to deter-mine theH∞ performance criterion that depends on the parameter. Both memoryless andstate-delayed filters are examined and the corresponding filter designs are finally formulatedin the form of convex optimization problems, which can be solved via efficient interior-pointalgorithms.
A Successive Approximation Algorithm to Optimal Feedback Control of time-varyingLPV state-delayed systems
Karimi in [71] gives a solution for finite-time optimal statefeedback control for a classof time-varying linear parameter-varying (LPV) systems with a known delay in the statevector under quadratic cost functional, which is investigated via a successive approximationalgorithm. The method of successive approximation algorithm results an iterative scheme,which successively improves any initial control law ultimately converging to the optimalstate feedback control. On the other hand, by manipulating linear matrix inequalities im-posed by Generalized-Hamiltonian-Jacobi-Bellman methodand the polynomially parameter-dependent quadratic (PPDQ) functions, sufficient conditions with high precision can be givento guarantee asymptotic stability of the time-varying LPV state-delayed systems independentof the time-delay.
Gain-Scheduled Smith PID Controllers for LPV Systems with Time Varying Delay:Application to an Open-flow Canal
Bolea in [72] presents a new approach to design gain scheduled robust linear parame-ter varying (LPV) PID controllers with pole placement constraints (through LMI regions),which is applied for LPV systems with second order structureand time-varying delay. Thecontroller structure includes a Smith predictor, real timeestimated parameters that schedulethe controller (including the known part of the delay) and unstructured dynamic uncertaintywhich covers the unknown portion of the delay.
Parameter dependent state-feedback control of LPV time-delay systems with timevarying delays using a projection approach
17
The stabilization of LPV time-delay systems with time varying delays by parameter de-pendent state-feedback is solved by Briat et al. in [73].
The stability test withH∞ performance is given through a parameter dependent LMI,which is derived from a parameter dependent Lyapunov-Krasovskii functional combinedwith the Jensen’s inequality. From this result Briat derived a state-feedback existence lemmaexpressed through a nonlinear matrix inequality (NMI). TheNMI is then turned into a bilin-ear matrix inequality (BMI) involving a ’slack’ variable. This BMI formulation is shown tobe more flexible than the initial NMI formulation and is more adequate to be solved usingalgorithm such as ’D-K iteration’.
H∞ Filtering of Uncertain LPV Systems with Time-Delays
Preliminary results on filtering time-delayed uncertain LPV systems are provided in[74, 75] where the descriptor model transformation of time-delay systems is considered. In[76, 69, 77] this approach is generalized to LPV time-delay systems. Briat et al. in [78] pro-posed an approach based on a parameter dependent Lyapunov-Krasovskii functional wherewe avoid model transformations/cross-terms and reducing the number of bounding proce-dures to the smallest number. The Lyapunov-Krasovskii functional proposed in this article israther simple but coupled with sufficiently accurate bounding techniques, it leads to promis-ing results. A bounded real lemma is provided for LPV time-delay systems, which has amore useful structure to the filtering problem than the bounded real lemma directly obtainedfrom the parameter dependent Lyapunov- Krasovskii functional. Sufficient conditions (interms of parametrized LMIs) for the existence of memorylessand memoryH∞ robust LPVfilters are derived.
Gain-Scheduled Guaranteed Cost Control of LPV Systems withTime-Varying Stateand Input Delays
Wang in [79] investigates the problem of delay-dependent guaranteed cost control for lin-ear parameter-varying systems with time-varying state andinput delays. Attention is focusedon the design of gain-scheduled guaranteed cost controllersuch that the resulting closed-loopsystem is asymptotically stable and a parameter-dependentcost performance is also satisfied.By parameter-dependent Lyapunov approach, a sufficient condition is proposed for design-ing gain-scheduled state feedback controller, in which thecontroller gain is dependent on thescheduling parameters.
Memory Resilient Gain-scheduled State-Feedback Control of LPV Time-Delay Sys-tems with Time-Varying Delays
Briat et al. in [80] are concerned with the stabilization of LTI/LPV time-delay systemswith time varying delays using LTI and LPV state-feedback controllers. First, a stabilitytest with guaranteed input/outputL2 performance is provided in terms of parameter depen-dent LMIs. The problem of designing both instantaneous and exact-memory state-feedbackcontrol laws is solved. The results are then extended to provide memory-resilient controllersynthesis conditions. Such controllers are guaranteed to stabilize the considered system evenin presence of uncertainty of the delay implemented into thecontroller.
18
Chapter 3
Impedance control for force reflectingtelemanipulation
The goal of this introductory chapter is twofold. First, to give a brief historical overview anda technical discussion on impedance control. Second, to present a pre-study that has beenconducted in order to reveal some basic principles of the stability of impedance controlledrobots with different friction expressions in the impedance model in presence of time-delayin the feedback loop.
As a matter of fact, this chapter contains results from original research, and as such mighthave been included in Part II on Theoretical Achievements. The reason why the chapter isincluded here (in the Introduction) instead is because these results can also be considered aspart of the motivation for later chapters. Thus, while the inclusion of this chapter slightly dis-torts the ratio between the length of the parts, it nevertheless helps to lend a clearer structureto this dissertation.
This chapter introduces a widely applied impedance controlscheme for force reflectingtelemanipulation and then focuses attention on the theoretical limitations of the stability ofsuch systems under time-delay. The stability is analysed interm of the maximum time-delay with the impedance model remains stable. First, it is investigated how the criticaldelay depends on the mass and damping in the impedance model and on the stiffness ofthe environment. Second, it is presented how the non-linearfriction term in the impedancemodel influences the stability boundary. The critical delayvalues are determined numericallyfor linear cases and simulation is used in case of non-linearmodels. The results are validatedby laboratory experiments and compared to other results in the literature.
In recent years, the important role of impedance control hasbeen emerged in severalfield of robotics such as human-robot interaction, telemanipulation or robotic surgery. In thesame time, the upcoming trends in robotics rather turn to logically or spatially distributedrobot control systems (RT-Middleware [81], ROS [82], MRDS [83]), where time-delays areunavoidable and inherent, that leads to unfavourable effect on control performance. Thisphenomenon supported by various practical applications arises a motivation to investigatethe time-delay sensitivity of impedance controlled robots.
Since the extensive work of Hogan [84, 85, 86], wherein the concept of impedance con-trol and its application was formulated, this control strategy became one of the key technol-
19
ogy of modern robot control. Some areas of robotics where impedance control is widelyused are telerobotics, dexterous manipulation [87, 88], and flexible joint robots [89].
Time delay - caused by the sampling time of digital control, or the communication jitterin the computer network between distributed system elements - usually have unfavorable in-fluence in closed loop control systems. Control of time-delay systems (TDS) is a permanentchallenge [90, 91]. Internet based teleoperation is typical area where communication delayplays crucial role [92, 93, 94, 95, 96]. Among several other approaches, impedance controlis an important strategy in bilateral telemanipulation [97, 98, 99, 100, 101]. Stability and per-formance of haptic rendering is also affected by the delay occurs in the control loop [102]. Aseries of papers from DLR’s researchers cover the stabilityof haptic rendering from severalaspects [103, 104, 105].
This chapter focuses on the class of impedance model based robot control that is schemat-ically illustrated in figure 3.1.
Figure 3.1: Schematic structure of impedance control of robots
Figure 3.2 illustrates the operation of such algorithms in teleoperation scenario. Commonproperty of these interaction control systems, that the time-delay whilst the acting force isbeing measured and transmitted to the impedance model, makes the control performanceweaker and over a critical delay, the system become unstable. One important goal of thischapter is to determine the maximum value of time-delay withthe impedance model remainsstable. Because of the complexity of the system, it is not trivial how to investigate thequantitative performance and the stability of a real telemanipulation scenario. Therefore,a simplified, unilateral, one degree of freedom impedance model have been used keepingthe substantial points of the problem. Conventional Mass-Damper virtual impedance withdifferent type of friction models were studied. The critical delay values are determined vianumerical computation in case of linear dissipation model,while simulation is applied innon-linear cases. The resulted stability boundary is validated experimentally using a linearrobotic axis.
Before going into the details let me define some important concepts which are oftenreferred in this Thesis.
Definition 3.1. (Impedance controlled robot): Impedance controlled robotis a manipula-tor that is programmed to act according to a predefined impedance model (usually a virtualmass-spring-damper system). The actual path of the end-effector is resulted by the interac-tion between the robot and its environment, because the dynamic relationship between theinteraction force and the displacement is prescribed. Impedance control of robots is some-times also referred as compliance control.
20
Figure 3.2: Scheme of coupled impedance force reflecting algorithm for bilateral telemanip-ulation
Definition 3.2. (Impedance model): In this dissertation, impedance model is understood asthe dynamic relationship between the force and the resulteddisplacement. Impedance modelis typically given by a virtual mass-spring-damper system.In this chapter, non-linear frictioncomponent (coulomb friction, stiction) is also consideredas part of the impedance model. Ingeneral case the task-space impedance model can be described as
M q+ Bq + Kq+ C(q,q) = f, (3.1)
whereM , B, K are symmetric, positive-definite matrices describing the mass, damping, stiff-ness parameters andC contains non-linear friction terms of the impedance model respec-tively, while F denotes the external forces.
Remark3.1. In some applications the end effector path is prescribed andthe displacementresults from the impedance model is added to the predefined path. In this way the robotmotion became compliant.
Regarding that this work deals with the two-fingered, parallel jaw tele-grasping problem,in the followings a single degree of freedom model is investigated.
3.1 Impedance Control with Feedback Delay
Consider the single degree of freedom mechanical system depicted by Figure 3.3 as theimpedance model. Massm and viscous dampingb are virtual properties defining the desireddynamics of the manipulator, whilek denotes the stiffness of the robot’s environment. Inreal cases, the environment is usually more complicated, but this simplified model is suitableto investigate the effect of time-delay. Virtual parameters have to be chosen according tothe accuracy↔ robustness trade-off [106]: The lower mass and damping result in faster andmore accurate tracking with less robustness against feedback delay and vice versa.
The equation of motion of this system is as follows:
21
Figure 3.3: Mass-Spring-Damper system
x(t) =Fh(t)
m−
b
mx(t)−
Fe(t)
m(3.2)
The non-delayed system can be represented by a standard LTI (ABCD) state space model:
x(t) = Ax(t) +Bu(t) (3.3)
y(t) = Cx(t) +Du(t)
, where the elements are as follows according to (3.2):
x(t) =
[
xx
]
u(t) = Fh(t)
A =
[
− bm
− km
1 0
]
B =
[
1m
0
]
C =[
0 1]
D = 0
Introducing the time-delayτ in the interaction (overall delay of the force monitoring dueto the lag of the signal processing and/or network delays):
x(t) =Fh(t)
m−
b
mx(t)−
Fe(t− τ(t))
m(3.4)
substituting the interaction force (Fe) by the elastic force (kx) in the formula as the simplestmodel of the environment we get the following equation:
x(t) =Fh(t)
m−
b
mx(t)−
k
mx(t− τ(t)). (3.5)
One can see that the resulted equation represents a mass-spring-damper system where theeffect of the spring is delayed byτ(t). Figure 3.4 illustrates the resulted model.
Figure 3.5 shows the effect ofτ feedback delay: By the increasing delay, the step re-sponse of the system getting more susceptible for oscillation. In other words the so-calledpseudo-damping ratio is decreasing until the system becomeunstable.
22
Figure 3.4: Impedance model with feedback delay
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
Time [s]
Pos
ition
[m]
Non−delayed systemTau=0.0329 sTau=0.0561 s
Figure 3.5: Step response of the model with various delay values (model parameters:m =1[kg], b = 100[Ns/m], k = 1000[N/m]). Excitation function defines asFh(t ≦ 0) = 0 ,Fh(t > 0) = 1
3.2 Analysis of the Critical Delay
In this section we are looking for the functionτcrit = f(k, b,m), whereτcrit is the largestτ that the system remain stable with. In the followings, this stability bound for linear andnon-linear friction (dissipation) model is determined.
3.2.1 Friction models
In this investigation a complex friction model is applied that includes coulomb, static andviscous friction (3.6) allowing as to treat the linear and non-linear cases in a common form.This friction model is proposed by Hess and Soom in [107].
Ff(t) = (Fc +Fs − Fc
1 + (x(t)/vs)) · sign(x(t)) + b · x(t) (3.6)
23
Table 3.1: Parameter setsA B C D
Fc 0 10 15 15Fs 0 10 20 8Vs 1 1 0.1 0.01
Figure 3.6 shows the velocity-friction force characteristics of the friction model withdifferent parametrization (b = 100[Ns/m]). Table 3.1 lists each set of parameters that isreferred in this chapter. Parameter set A results the pure viscous damping. Set B impliescoulomb friction and viscous damping, while in sets C and D appears the static and kineticfriction. In parameter D, the static friction force is smaller then the kinetic friction makingthe curve smooth. Presented sets of parameters aim to cover the possible cases by concreteexamples.
−0.2 −0.1 0 0.1 0.2−40
−30
−20
−10
0
10
20
30
40
velocity [m/s]
Fric
tion
forc
e [N
]
param Aparam Bparam Cparam D
Figure 3.6: Friction force as function of model velocity
Including the friction model (3.6) into the equation of motion of the impedance model(3.5) the following second order non-linear delay differential equation is formed:
x(t) =Fh − (Fc +
Fs−Fc
1+(x(t)/vs)) · sign(x(t))
m+
+−b · x(t)− k · x(t− τ(t))
m(3.7)
The stability of equation 3.7 is investigated according to its special cases constructedfrom the listed set of parameters.
When parameter set A is used in the friction expression (3.6), the system (3.7) can be ap-proximated with a linear LTI model and the critical delay canbe derived from the condition
24
of asymptotic stability. In case of non-linear friction, this way is not viable, therefore numer-ical simulation is used to determine the stability bound. The linear and non-linear cases arediscussed separately in the following subsections.
3.2.2 The linear case
The delayed differential equation (3.5) cannot be written in form of LTI state space model,because the exponential transfer function of the pure delayH(s) = e−τs cannot be realizedwith finite number of state variables. The ideal time-delay can be approximated by ratio-nal function. Among many methods (Taylor, Bessel-Thomson etc.) Pade approximation iswidely used for the approximation of dead-time [108]. In this investigation a third order(both the numerator and denominator) Pade approximant is used. With the approximation ofdelay, the following state space model is formed according to the delayed equation (3.5):
x(t) =
xxz1z2z3
u(t) = Fh(t)
A =
− bm
km
− 24mτ
0 −240mτ
1 0 0 0 00 k −12
τ− 60
τ2−120
τ3
0 0 1 0 00 0 0 1 0
B =
1m
0000
C =[
0 1 0 0 0]
D = 0
The so determined linear time invariant (LTI) model is suitable to apply the commonapparatus for analysis of linear systems. The system is asymptotically stable if the real partsof the eigenvalues of matrixA are negative (Re(λi) < 0).
According to the stability condition, the solution is foundnumerically on the intervalb =[0..1000][Ns/m] andk = [2000..16000][N/m]. Stability boundary surface form = 1[kg]shown in Figure 3.7.
From the result of the numerical computation we can draw the following conclusion:The stability margin is mainly depends on the stiffness (k) and viscous damping (b). Depen-dence on mass is not significant however the larger mass movesthe system slightly towardsthe unstable region. The stability boundary can be characterized with the equation (3.8),wherec is a mass dependent dimensionless multiplier. In case ofm = 1kg, c ≈ 1.566 andlimm→∞ c(m) = 1.
τcrit = c(m)b
k(3.8)
25
0
200
400
600
800
10002000
40006000
800010000
1200014000
16000
0
0.2
0.4
0.6
0.8
1
Stiffness (k) [N/m]Damping (b) [Ns/m)
τ crit [s
]
Figure 3.7:τcrit as function of linear damping and stiffness of the environment with parame-ter set A
Table 3.2: Experimental resultsb[Ns/m]
200 400 600 800 1000A D A D A D A D A D
k[N/m]
1921 54 92 275 365 - - - - - -6315 25 41 79 97 129 150 183 200 234 2556898 24 38 68 86 117 134 162 187 200 22814540 12 19 30 38 50 61 72 82 97 107
3.2.3 Non-linear friction
The impedance model implying non-linear friction term (parameter sets B, C and D) can-not be examined with simple stability criterion as it was done in the previous subsection.The stability of the non-linear equation (3.7) is investigated via numerical simulation in Mat-lab/Simulink environment. In the simulation, only the dynamics of the impedance modelwas considered. The dynamics of the controlled axis is neglected. We assumed that thecontrolled axis follows the position (x) of the impedance model without tracking error. Todetermine the critical delay for a given set of parameters, the transient of the system has tobe analysed. In each test, the simulation is started from steady state and perturbed with astep of force input (Fh). The system was judged as stable when after the transient, the outputstays in the±5% environment of the expected terminate position (x∞ = Fh∞
k). This zone is
assigned because of the uncertainty coming from the non-linearities. Figure 3.8 presents thestability boundary that is determined via a series of programmatically executed simulations.
Results shows that for the parameter sets A, B and D (figure 3.8(a)), the simulationprovides stability bounds with similar shape as observed inthe linear case. In the case ofparameter set C (figure 3.8(b)), the characteristics of the surface is changed: The linear-
26
(a) Parameter set A, B and D (b) Parameter set C
Figure 3.8: Stability boundary resulted from the simulation
like relationship betweenτcrit andb cannot be observed. We have noticed, that the stabilitymargin became mass dependent. In the range of higher stiffness, the larger mass effect thetolerable time-delay beneficially.
3.3 Experimental Validation
For the experimental validation of the computed results, a HIRATA (MB-H180-500) DCdrive Cartesian robot was used (Figure 3.9). The first axis ofthe robot was fixed during thetests, while the second axis was connected to the base of the robot (environment) by helicalsprings of different stiffnesses (1921, 6315, 6898, 14540 [N/m]). The interaction force wasmeasured by a Tedea-Huntleight Model 355 load cell mounted between the spring and therobot’s flange. The driving system of the moving axis consisted of a HIRATA HRM-020-100-A DC servo motor connected directly to a ball screw with a20[mm] pitch thread. Therobot was controlled by an MCU-based embedded control unit providing 1000 Hz samplingfrequency for the overall control loop. This controller made it possible to emulate a delay inthe force sensing as integer multiples of1[ms], and to set the control signal by the pulse withmodulation (PWM) of supply voltage of the DC motor. The modalmass and the dampingratio of the robot were experimentally determined:m = 29.57[kg] andb = 1447[Ns/m].The Coulomb friction was measured asC = 16.5[N ]. More details on the experimentalidentification of the system parameters can be found in [109]. The control process wasdeclared unstable if the robot started oscillations for perturbations. Programmatically definedstep of20[N ] in desired force (Fh) was applied as perturbation. During the tests, the delayparameterτ was changed with increments of 10 [ms]. After reaching the unstable region thedelay was decreased in 1 [ms] steps until the process become stable again. The largest stable
27
Figure 3.9: Experimental setup
(a) Parameter set A (b) Parameter set D
Figure 3.10: Comparison of simulated and experimental results
τ value was recorded as stability bound. In most cases, the experimental stability boundarieswere clear and easily recognizable.
Two series of measurements were completed using the friction model with parameterset A and D. The resulted critical delay (τcrit[ms]) values are displayed in Table 3.2. Theexperimental data are illustrated in figure 3.10 together with the stability boundary resultedfrom the simulation. One can see, that the measured points are very close to the theoreticalsurface. Minor differences are coming from the unmodelled dynamics of the robotic axis.
3.4 Comparison of the results
In this section the stability boundary (3.8) that was determined in section 3.2.2 is comparedto the simulation results published in [105] by Gil, Sanchez, Hulin, Preusche and Hirzinger.
28
The simulations have been performed to check the validity ofthe stability condition:
k <b
T2+ τ
, (3.9)
wherek denotes stiffness of a target object, whileb means the overall damping in the system.Since their investigation focused on discrete time systems, the effect of sampling and holdis approximated by a delay of half the sampling periodT
2. Note, that in the investigation
of the stability of haptic systems, usually the critical stiffness of a virtual wall is expressed.However, in teleoperation the overall stiffness of the remote environment is known (boundedby the stiffness of the slave device) so the determination ofthe critical delay makes moresense.
In figure 3.11, simulation results of [105] and the computed boundary (τcrit) accordingto (3.8) are illustrated together in the diagrams. Figure 3.11(a) shows an overview of the in-vestigated range of damping and stiffness, while figures 3.11(b), 3.11(c), 3.11(d) and 3.11(e)show sections along the stiffness axis at damping values0.2, 0.3, 0.4 and0.5 [Ns/m] re-spectively. On theτ axis in case of simulated results the overall delay values are shown(T2+ τ ).One can see, that the simulated points are very close to the computed boundary. In con-
clusion, we can say that the validity of stability condition(3.9) - which is derived for discrete-time haptic systems - can be extended to the stability problem of impedance controlled robotsunder time-delay. It is also shown that the stability criterion (3.8) - that is determined fromthe continuous time model of time-delayed impedance control using the Pade approxima-tion of time-delay - gives practically identical result. For further details about the physicalbackground behind the stability of haptic devices please bereferred to [103, 104, 105].
29
(a) Surface overview
0 200 400 600 800 10000
1
2
3
4
5
6x 10
−3
Stiffness (k) [N/m]
τ crit [s
]
Computed τ
crit (b=0.2 [Ns/m])
Gil et al. 2009 (b=0.2 [Ns/m])
(b) b = 0.2[Ns/m]
0 200 400 600 800 10000
1
2
3
4
5
6
7
8x 10
−3
Stiffness (k) [N/m]
τ crit [s
]
Computed τ
crit (b=0.3 [Ns/m])
Gil et al. 2009 (b=0.3 [Ns/m])
(c) b = 0.3[Ns/m]
0 200 400 600 800 10000
0.002
0.004
0.006
0.008
0.01
0.012
Stiffness (k) [N/m]
τ crit [s
]
Computed τ
crit (b=0.4 [Ns/m])
Gil et al. 2009 (b=0.4 [Ns/m])
(d) b = 0.4[Ns/m]
0 200 400 600 800 10000
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Stiffness (k) [N/m]
τ crit [s
]
Computed τ
crit (b=0.5 [Ns/m])
Gil et al. 2009 (b=0.5 [Ns/m])
(e) b = 0.5[Ns/m]
Figure 3.11: Comparison of the computed stability boundarywith the results of Gil et al.
30
3.5 Summary of the chapter
In this chapter, the critical time-delay in the force sensing of impedance control is investi-gated. This research is motivated by the destabilizing effect of time-delay in bilateral controlloops for telemanipulation and in interaction control of robots where the sensor, actuatorand control elements are connected via packet-switched network. Stability bounds weredetermined for mass-spring-damper impedance models with different type of linear and non-linear friction characteristics. Stability bounds are characterized with surfaces formed bythe critical delay values (τcrit) over the stiffness-damping plane. The result shows that thestability bound depends on the stiffness and damping but notdepends significantly on themass in the impedance model. Using friction models containing non-linear elements suchas coulomb friction and stiction increases the critical delay. The theoretical and simulatedresults were validated experimentally and compared to other results in the literature. Theresults shows the theoretical similarities of control problems induced by time-delay in hapticdevices and in impedance controlled robots.
31
Chapter 4
Control structure for stabilitypreservation in impedance control undertime-delay
This chapter, introduces the most current approaches for stability preservation in impedancecontrol-based bilateral telemanipulation, and describe acontrol structure which is appropri-ate for the impedance model in the telemanipulation scenario described in Chapter 3. In thiscontrol structure, the impedance model - under feedback delay - can be embedded in sucha way that its stabilization design problem readily leads tothe class of general control the-ories developed for control signal design. This approach slightly reinterprets the previouslyapplied stabilization techniques which are based on the adaptive tuning of the impedancemodel’s parameters. Hence this structure, actually, extends the class of control design theo-ries applicable for stable impedance control design.
In the past two decades a large variety of approaches has beenintroduced addressing thestability issues of telemanipulators in presence of time-delay. Hokayem and Spong publisheda comprehensive survey [110] that introduces most of the different approaches that can be cat-egorized in passivity based, prediction based, sliding mode and other techniques [111]. Thewidest group contains the methods which are based on energy related considerations, namelythe passivity theory. To be more specific, they are directly or indirectly manipulating the so-called energy tanks of the dynamic system in order to guarantee its stability. Among thesedirections adaptive tuning of the applied impedance parameters has a special significance(Figure 4.1). Dubey et al. [112] published a variable damping impedance control methodto enhance the quality of master-slave force reflecting telemanipulation. Wen-Hong Zhuand Salcudean introduced an adaptive controller in [113] wherein the master-slave systembehaves essentially as a linearly damped free-floating massin which the mass and dampingparameters are changing according to the estimated dynamics of the environment. In 2004Love and Book published an other adaptive impedance controlalgorithm [114] which setsthe master impedance based on the estimated time-varying, position-dependent representa-tion of the remote environment. In their solution, the environment estimation and impedanceadaptation are executed simultaneously and in real time.
32
A general problem of the before mentioned directions that there are no simple method tofind the appropriate dissipation model that makes the systemstable but transparent enoughfor comfortable use. Thus, these methods are usually very conservatives making the overallteleoperator system more dissipative (less transparent) than it would be really required.
Figure 4.1: Stabilization of impedance control based forcereflecting telemanipulation byparameter tuning
An other approach is to apply a structure that manipulates the dissipative characteristicsof the system indirectly by an external damper force that is additional to the damping whichis included in the impedance model itself. This structure isillustrated by Figure 4.2.
Based on this structure, we can formulate the following DDE as the equation of motionof the impedance model which to be stabilized by the appropriate design ofFc(t).
x(t) =Fh(t)
m+
Fc(t)
m−
b
mx(t)−
k
mx(t− τ(t)) (4.1)
The theoretical contribution of this work is focused on a design methodology that is ap-propriate to find the delay dependent control law that maintain the stability of the impedancemodel without unnecessary degradation of the transparency.
33
Figure 4.2: Control scheme for the stabilization of force reflecting telemanipulation undertime-delay
34
Chapter 5
TP Model Transformation based controldesign methodology
This chapter introduces the Higher Order Singular Value Decomposition of qLPV state-spacemodels and the control design methodology based on TP Model Transformation as the math-ematical background of the theoretical results of this Thesis. Even though one can find thesemethods in the literature, for the sake of completeness, I recall them based on the work ofTakarics [115].
5.1 Higher Order Singular Value Decomposition of Func-tions
5.1.1 Basic concept of tensor algebra
First of all, let us view a brief introduction to the HOSVD of tensors. This introduction isbased mainly on Lathauwer’s work [16], which proposes SVD model forN th-order tensors.To facilitate the concept, we first recall the matrix SVD as follows:
Theorem 5.1(Matrix SVD). Every real(I1 × I2)-matrixA can be written as the product
A = U1 · S ·UT2 , (5.1)
in which
i U1 = (u1,1u1,2 · · ·u1,I1) is a unitary(I1 × I1)-matrix,
ii U2 = (u2,1u2,2 · · ·u2,I2) is a unitary(I2 × I2)-matrix,
iii S is an(I1 × I2)-matrix with the properties of
(a) pseudodiagonality:
S = diag(
σ1, σ2, . . . , σmin(I1,I2)
)
, (5.2)
35
(b) ordering:σ1 ≥ σ2 ≥ · · · ≥ σmin(I1,I2) ≥ 0. (5.3)
Theσi are the singular values ofA and the vectorsu1,i andu2,i are, respectively, anithleft and anith right singular vector.
Remark5.1. The number of non-zero singular valuesσi equals to the rank of matrixA.
Remark5.2. Note that the SVD uniquely determines matrixS and matricesU1 andU2 to theextent of the signs (these can be systematically switched) if there are no equivalent singularvalues (σi 6= σi+1). The determination of matricesU1 andU2 may not be unique if there areequivalent singular values (σi = σi+1).
Remark5.3. The singular values of matrixA define the Frobenius norm (see Definition 5.3)of matrixA as:
‖A‖ =
min(I1,I2)∑
i=1
√
σ2i .
Before extending the matrix SVD to HOSVD of tensors we extendthe well-known defi-nitions of scalar product, orthogonality, and Frobenius-norm to tensors as:
Definition 5.1(Scalar product). The scalar product〈A,B〉 of two tensorsA,B ∈ RI1×I2×···×IN
is defined as〈A,B〉
def=∑
i1
∑
i2
· · ·∑
iN
bi1i2...iNai1i2...iN .
Definition 5.2 (Orthogonality). Arrays of which the scalar product equals0 are orthogonal.
Definition 5.3 (Frobenius-norm). The Frobenius-norm of a tensorA is given by
‖A‖def=√
〈A,A〉.
In order to define the Higher Order Singular Value Decomposition we should define then-mode rank of tensors. There are major differences between matrices and higher-ordertensors when rank properties are concerned and two approaches exist. The general methodto define the rank of a tensor is the parallel factorization (PARAFAC) and the other approachis to define the rank of each dimension of the tensor. We chose the latter approach, since itfits well to the HOSVD.
Definition 5.4 (n-mode matrix of tensorA). Assume anN th-order tensorA ∈ RI1×···×IN .
The matrixA(n) ∈ RIn×(In+1In+2···INI1I2···In−1) contains the elementai1,i2,...,iN at the position
with row numberin and column number equal to:
(in+1 − 1)In+2In+3 · · · INI1I2 · · · In−1 + (in+2 − 1)In+3In+4 · · · INI1I2 · · · In−1 + · · ·+
+ (iN − 1)I1I2 · · · In−1 + (i1 − 1)I2I3 · · · In−1 + (i2 − 1)I3I4 · · · In−1 + · · ·+ in−1.
Remark5.4. The ordering of the column vectors can be arbitrarily determined. The onlyimportant thing is that in all cases the same ordering and reordering must be used system-atically later on. In general, therth column ofn-mode matrixA(n) is equivalent to theI1, I2, . . . , In−1, In+1, . . . , IN -th vector of dimensionn, where
r = ordering(i1, i2, . . . , in−1, in+1, . . . , iN).
36
I 1
I 2
I 3
A
⇒
I 2 I 2 I 2
I 1
A (1)
I 1
I 2
I 3
A
⇒
I 3 I 3 I 3 I 3
I 2
A (2)
I 1
I 2
I 3
A
⇒
I 1 I 1 I 1 I 1 I 1
I 3
A (3)
Figure 5.1: Illustration of3-mode matrices of a 3rd-order tensorA
r is a linear index equivalent of array indexesi1...in. Figure 5.1 shows an example forthen-mode matrix of a 3rd-order tensor.
Definition 5.5 (n-mode rank of tensorA). Then-mode rank of tensorA, denoted byRn =rankn(A), is the dimension of the vector space spanned by then-mode matrix of tensorA.
Remark5.5. From a computational point of view, it directly follows thatrankn(A) =rank
(
A(n)
)
.
Definition 5.6 (n-mode product of a tensor by a matrix). Then-mode product of a tensorA ∈ R
I1×I2×···×IN by a matrixU ∈ RJn×In , denoted byA×nU, is an(I1×I2×· · ·×In−1×
Jn × In+1 × · · · × IN)-tensor of which the entries are given by
(A×n U)i1,i2,...,in−1,jn,in+1,...,iNdef=∑
in
ai1,i2,...,in−1,in,in+1,...,iNujn,in.
The multiplen-mode product of a tensor, such asA ×1 U1 ×2 U2 ×3 · · · ×N UN can beshortly denoted as
AN
⊠n=1
Un.
From a computational point of view, then-mode product of a tensor by a matrixA =B ×n U can be defined asA(n) = UB(n).
Then-mode product satisfies the following properties.
37
Property 5.1. Given the tensorA ∈ RI1×I2×···×IN and the matricesF ∈ R
Jn×In,G ∈R
Jm×Im , (n 6= m), one has
(A×n F)×m G = (A×m G)×n F = A×n F×m G.
Property 5.2. Given the tensorA ∈ RI1×I2×···×IN and the matricesF ∈ R
Jn×In,G ∈R
Kn×Jn, one has(A×n F)×n G = A×n (G · F).
Now we state the generalization of matrix SVD:
Theorem 5.2(Higher Order SVD, HOSVD). Every real (I1 × I2 × · · · × IN )-tensorA canbe written as the product
A = S ×1 U1 ×2 U2 ×3 · · · ×N UN = SN
⊠n=1
Un, (5.4)
in which
i Un = (un,1un,2 · · ·un,In), n = 1 . . . N is a unitary(In × In)-matrix,
ii S is a real(I1×I2×· · ·×IN )-tensor of which the subtensorsSin=α obtained by fixingthenth index toα have the properties of
(a) all-orthogonality: two subtensorsSin=α andSin=β are orthogonal for all possi-ble values ofn, α andβ subject toα 6= β:
〈Sin=α,Sin=β〉 = 0, whenα 6= β, (5.5)
(b) ordering:‖Sin=1‖ ≥ ‖Sin=2‖ ≥ · · · ≥ ‖Sin=In‖ ≥ 0, (5.6)
for all possible values ofn.
The Frobenius-norms‖Sin=i‖, symbolized byσ(n)i , aren-mode singular values ofA and the
vectorun,i is anith n-mode singular vector.
The decomposition is visualized for third-order tensors inFigure 5.2.
Remark5.6. Note that the HOSVD uniquely determines tensorS, matricesUn only to theextent of the signs (these can be systematically switched).The determination of matricesUn may not be unique if there are equivalent singular values at least in one dimension (σi =σi+1).
Remark5.7. Then-mode singular matrixUn (and then-mode singular values) can directlybe found as the left singular matrix (and the singular values) of ann-mode matrix of tensorA. Hence computing the HOSVD of anN th-order tensor leads to the computation ofNdifferent matrix SVDs of matrices with size(In × I1I2 . . . In−1In+1 . . . IN ), (1 ≤ n ≤ N):
A(n) = UnΘn (Vn)T .
Afterwards, the core tensorS can be computed by bringing the matrices of singular vectorsto the left side of (5.4):
S = A×1 UT1 ×2 U
T2 ×3 · · · ×N UT
N = AN
⊠n=1
UTn .
38
AI 1
I 2I 3
=
S
I 1
I 2I 3
U (1)
I 1
I 1 U (2)
I 2
I 2
U (3)
I 3I 3
Figure 5.2: Visualization of the HOSVD for a third-order tensor
In the TP model transformation we use a minimal or compact form of HOSVD as:
Definition 5.7 (Compact HOSVD (CHOSVD)). If we discard the zero singular values andthe related singular vectorsuRn+1...In,n, whereRn = rankn(A) during the SVD computationof each dimension then we obtain the Minimized form of HOSVD as
A = DN
⊠n=1
Un, (5.7)
which has all the properties as in Theorem 5.2 except the sizeof Un andD. HereUn hasthe size ofIn ×Rn andD has the size ofR1 × · · · ×RN .
Definition 5.8 (Reduced HOSVD (RHOSVD)). If we discard non-zero singular values (notonly the zero ones) and the corresponding singular vectors then the decomposition results inan approximation of tensorA with the following property.
Property 5.3. Assume the HOSVD of tensorA is given according to Theorem 5.2 asA =
DN
⊠n=1
Un, and then-mode rank ofA isRn (1 ≤ n ≤ N). Let us defineA by discarding the
elements of singular valuesσ(n)I′n+1, σ
(n)I′n+2, . . . σ
(n)Rn
of tensorD, for a givenI ′n < Rn, and thecorresponding vectors of the singular matricesUn. In this case
γ = ‖A − A‖2 ≤R1∑
i1=I′1+1
(
σ(1)i1
)2
+
R2∑
i2=I′2+1
(
σ(2)i2
)2
+ · · ·+RN∑
iN=I′N+1
(
σ(N)iN
)2
. (5.8)
This property is theN th-order generalization of the connection between the singularvalue decomposition of a matrix and its best, lower ranked matrix approximation (in thesense of least square). In the higher order case, the discarding of singular values in eachdimension results in a lower rank along each dimension, while, contrarily to the singularvalue decomposition for matrices, the resulting tensorA, having a lowered rank along eachof its dimensions, is not the best approximation to the giventensorA. Irrespective of thisfact, the descending order of the singular values indicatesthe increasing error during their
39
discard. The upper limit of this error can be given by (5.8). The best approximation for agiven rank reduction can be achieved by the proper modification of the elements of tensorA. For instance [116, 117] define HOOI (Higher-Order Orthogonal Iteration) for best rank-1approximation.
5.1.2 Definition of the HOSVD-based canonical form of TP functions
Definition 5.9 (Tensor Product (TP) function). Consider the following function:
y = f(x) (5.9)
wherey is scalar andx ∈ RN .
For brevity we call (5.9) TP function, if it can be given in thefollowing form:
y = BN
⊠n=1
wn(xn), (5.10)
whereB ∈ RI1×...×IN is bounded and the row vectorwn (xn) ∈ [0, 1] contains one
variable and continuous weighting functionswn,in(xn) ,(in = 1 . . . IN).
Remark5.8. Note that not all functions can be given in form (5.10) with bounded size oftensorB. The comprehensive analysis on this subject is given in detail by [23, 118]. TIKK in[22] proves that the TP functions are nowhere dense in the space of approximation functionif the size of tensorB is bounded. However, practically, this TP function is widely acceptedas a good approximator since the size ofB can be enlarged, for instance B-spline, fuzzytransfer functions etc.
If a function is not a TP function it can still be reprensentedin form (5.10) with boundedsize ofB as approximation.
Consider the given TP functiony = f(x). In the present case let’s assume that its TPform is known as:
y = f(x) = BN
⊠n=1
wn(xn), (5.11)
wherex ∈ Ω : [a1, b1]× [a2, b2]× · · · × [aN , bN ].For this TP function, we can assume that the functionswn,in(xn) ,in = 1...In, n = 1...N
are linearly independent (in the means ofL2[an, bn]) over the intervals[an, bn]. In oppositecase we can choose linearly independent functions fromwn,in(pn),in = 1...In and we canexpress the remaining functions via the linear combinationof the independent ones.
The linearly independent functionswn,in(xn) are determinable by the linear combinationsof orthonormal functions (for instance by Gram–Schmidt-type orthogonalization method):thus, one can determine such a system of orthonormal functions for all n = 1...N asϕn,in(xn), in = 1...In from the linear combination of weighting functionswn,in(xn). Inthis manner, one can always derive an equivalent orthonormal system form as:
y = f(x) = CN
⊠n=1
ϕn(xn), (5.12)
40
whereC ∈ RI1×···×IN and row vectorsϕn(xn) consists of elementsϕn,in(xn) which form an
orthonormal system as:
∀n :
∫ bn
an
ϕn,i(xn)ϕn,j(xn)dxn = δi,j, 1 ≤ i, j ≤ In,
whereδi,j is the Kronecker-function (δij = 1, if i = j andδij = 0, if i 6= j).Now we execute CHOSVD (Definition 5.7) on the system tensorC ∈ R
I1×···×IN :
C = DN
⊠n=1
Un, (5.13)
whereUn has the size ofIn × Rn, whereRn = rankn(B). Substituting (5.13) in (5.12)we have:
y = f(x) = [DN
⊠n=1
Un]N
⊠n=1
ϕn(xn). (5.14)
That leads to
y = f(x) = DN
⊠n=1
(ϕn(xn)Un) = DN
⊠n=1
wn(xn), (5.15)
where
wn(xn) = ϕn(xn)Un. (5.16)
Observe thatUn are orthonormal matrices and functionsϕn,in(xn) form an orthonormalsystem (see above) for alln = 1..N . Therefore, the components of the functionwn(xn) alsoform an orthonormal system. Based on the above we obtain the following theorem:
Theorem 5.3 (HOSVD-based canonical form of TP functions). Let’s assume a function,which can be given in form (5.10):
y = f(x) = BN
⊠n=1
wn(xn) (5.17)
Based on the above and executing CHOSVD via equations (5.11-5.15) one can determinethe HOSVD-based canonical form of (5.17) as
y = DN
⊠n=1
wn(xn) (5.18)
where tensorD has the size ofR1 × ...× RN .Properties:1. The weighting functionswn,in(xn), in = 1..Rn (termed asinth singular function on
dimensionn = 1..N) contained in vectorwn(xn) form an orthonormal system:
∀n :
∫ bn
an
wn,i(xn)wn,j(xn)dxn = δi,j, 1 ≤ i, j ≤ In,
whereδi,j is the Kronecker-function (δij = 1, if i = j andδij = 0, if i 6= j).
41
2. SubtensorsDin=i have the properties of(i) all-orthogonality: two subtensorsDin=i and Din=j are orthogonal for all possible
values ofn, i andj : 〈Din=i,Din=j〉 = 0 wheni 6= j,(ii) ordering: ‖Din=1‖ ≥ ‖Din=2‖ ≥ . . . ≥ ‖Din=rn‖ > 0 for all possible values of
n = 1, . . . , N .3. Based on the analogy of tensor algebra terms we call the Frobenius-norm‖Din=i‖,
symbolized byσ(n)i , then-mode singular values of the TP functiony = f(x).
4. D is termed core tensor containing the vertex points.5. Based on the analogy of tensor algebra terms we callR1, R2 ... RN the rank of the TP
function in each dimension, which determines the minimal number of linearly independentweighting functions in each dimension respectively.
Remark5.9. If there are equal singular values on any dimensions when CHOSVD is executedthen the canonical form is not unique. In this case then-mode singular vectors correspondingto the samen-mode singular value can be replaced by any orthonormal linear combination(see [16]). The decomposition is unique to the extent of the signs of wn, which can besystematically switched.
5.1.3 TP model transformation as a numerical reconstruction of theHOSVD-based canonical form
This section shows that the TP model transformation, namelythe discretisation of the givenfunction and executing HOSVD on the discretised system, leads to a numerical reconstruc-tion of the HOSVD-based canonical form. When we increase thediscretisation density, theresult of the HOSVD converges to the canonical form. This section is based on the works of[20, 21]
Notation 5.1 (Transformation spaceΩ). Ω is a bounded hyper rectangular space, wherex ∈ Ω : [a1, b1]× [a2, b2]× · · · × [aN , bN ].
Notation 5.2 (Discretisation grid M). M denotes a hyper rectangular discretisation grid de-fined inΩ. Mn (n = 1..N) denotes the number of grid points on then-th dimension andan ≤ gn,mn
≤ bn (mn = 1..Mn) denotes the location of the grid in dimensionn. In general,the grid can arbitrarily located in the intervals, for instance one can define an equidistantlocation:gn,mn
= an +bn−anMn−1
(mn − 1) if no special information is given about the system inthis regard. The grid points are defined by vector
gm1,..,mN=
g1,m1
...gN,mN
.
. (5.19)
Definition 5.10 (The discretised form of a given function). TensorFD(Ω,M) defines the dis-cretised form of functiony = f(x) in the transformation spaceΩ over the discretisation gridM . SuperscriptD stands for the discretised form.
Namely, functiony = f(x) is defined for all grid points ofM in Ω as:
42
hD(Ω,M)m1,...,mn
= f(gm1,..,mN).
TensorFD(Ω,M) ∈ RM1×...×MN is constructed from elements ofhD(Ω,M)
m1,...,mn.If y = f(x) is a TP function given in the form of (5.10),FD(Ω,M) can simply be given by
discretising the weighting functions. Thus if we have:
y = f(x) = BN
⊠n=1
wn(xn), (5.20)
then its discretised form is:
FD(Ω,M) = BN
⊠n=1
wD(Ω,M)n , (5.21)
wherewD(Ω,M)n are the discretised weighting functions of the dimensions as:
wD(Ω,M)n = [w
D(Ω,M)n,1 ,w
D(Ω,M)n,2 , ...,w
D(Ω,M)n,N ] (5.22)
wherewD(Ω,M)n,1 is the discretised column vector of functionwn,1(xn) :
wD(Ω,M)n,1 = w
D([an,bn],Mn)n,1 . (5.23)
In conclusion:
WD(Ω,M)n =
wn,1(gn,1) wn,2(gn,1) · · · wn,rn(gn,1)wn,1(gn,2) wn,2(gn,2) · · · wn,rn(gn,2)
.... . .
...wn,1(gn,Mn
) wn,2(gn,Mn) · · · wn,rn(gn,Mn
)
(5.24)
The goal of TP model transformation is to transform a given function (5.9) into finiteelement TP function form (5.17) (see Definition 5.9) in a given transformation spaceΩ.The key idea is that the function is not reconstructible onlyat the grid points of the hyper-rectangular gridM , but also continuously between these points.
TP model transformation has three main steps.
STEP I: Discretisation
The goal of this step is to represent the given function by itsdiscretised tensorFD(Ω,M) thatis ready to find the tensor product structure of the function.
First of all we define the transformation spaceΩ (see Definition 5.1) in which we ex-pect the TP function to be relevant. Secondly, we define the hyper-rectangular gridM (seeDefinition 5.2).
As a result of discretising the functiony = f(x) over the grid points we haveFD(Ω,M) ∈R
M1×...×MN .As a matter of fact, we define as dense a discretisation grid aspossible to guarantee that
the discretised tensor describes the function as close as possible inΩ.
43
STEP II: Extracting the discretised TP function
The goal of this step is to reveal the TP structure of the givenfunction. We use HOSVD tofind the TP structure of the function.
As a result of STEP 1, we have the tensor of the discretised function FD(Ω,M). TheCHOSVD results inFD(Ω,M):
FD(Ω,M) = DN
⊠n=1
Un = DN
⊠n=1
wD(Ω,M)n , (5.25)
where the size ofD is R1 × R2 × . . . × RN . SinceRn = rankn(FD(Ω,M)), Rn ≤ Mn,
for all n = 1..N . TensorD contains the constant componentsDi1,i2,...,iN , in = 1..In. Basedon the previous chapter, we haveUn = w
D(Ω,M)n .
STEP III: Reconstruction of the continuous TP function
The weighting functions can be determined over any points ofintervals[an, bn] by the helpof the giveny = f(x). In order to determine the weighting functions in vectorwd(xd), letxn be fixed to selected grid-lines as:
xn = gn,in n = 1 . . .N, n 6= d, in ∈ 1 . . . In,
wherein can be chosen arbitrarily. Then forxd:
wd(xd) = (f(x))(3)
(
(
BN
⊠n 6=d
un,in
)
(d)
)+
, (5.26)
where vectorx consists of elementsxn andxd asx =(
g1,i1 g2,i2 . . . pd . . . gN,in
)
,and superscript “+” denotes pseudo inverse andun,in is the first row vector ofUn. The third-mode matrix(f(x))(3) of functionf(x) is understood such that functionf(x) is consideredas a three-dimensional tensor, where the length of the third-dimension is one. This in practicemeans that the functionf(x) is stored in a one-row vector.
Remark5.10. (5.26) is the solution to the linear equation
y = f(x) = BN
⊠n=1
wn(xn)
where the only unknowns are the elements of thewd(xd). (We assume that the originalsystem is available soy = f(x) is known and forn 6= d the weightings are known as wellwn(xn) = un,in). The equation system can be rewritten in a linear form as
(f(x))(3) = wd(xd)
(
BN
⊠n=1,n 6=d
un,in
)
(d)
which usually can be solved by (5.26), but it can also be under-determined or numericallyill-conditioned.
This problem can be solved by using more equations by querying the model at additionalx parameter points using different grid lines (ie. differentin index combinationsn = 1 . . . N ,
44
n 6= d). It is hard to tell in advance which grid points will result in a solvable equation, butthe number of examined parameter points can be incrementally increased until the solutionis acceptable.
5.1.4 Discussion
Assume that functiony = f(x) belongs to the class of TP functions. Szeidl proves that TPmodel transformation is surely able to find the rank of the function by increasing the numberof sampling points in gridM , thus it is possible to obtain the TP representation of functiony = f(x). Szeidl also proves that the reconstruction of the TP function will converge to thecanonical form of functiony = f(x) beyond a certain grid density. The proof and additionaldetails can be found in [20].
The number of selected discretisation grid points can be examined form two points ofview:
• The aim is to find the canonical form of the function;
• The aim is to refine the weighting functions.
In the first case as a general recommendation we can say that itis necessary to increase thenumber of grid points until the rank of the HOSVD is increasing. The computational capacityrequired to execute the HOSVD raises at least to the power of the number of discretisationpoints, which sets a theoretical limit for the maximal number of sampling points. However,in the field of engineering applications the computational time for executing the HOSVDdecreased from several hours to several minutes in the latter years, which means that thelarge number of sampling points does not usually lead to a computational problem.
In the second case the number of points of the weighting functions can be increased afterthe HOSVD, thus the number of sampling grid points does not directly affect this option.More information about the mathematical background can be find in [119]
Concerning approximation with TP model transformation we should discuss two caseshere, one is the case when the function is a non-TP function, the other case is when thereconstructed function is a TP-function, but we want to reduce complexity (number of theweighting functionswn).
Approximation in non-TP function class
If function y = f(x) does not belong to the TP class, there is no finite rank, which meansthere will be no zero singular values after executing HOSVD no matter how dense the sam-pling gridM is. Similarly to Taylor series expansion, we have to discardsome of the non-zero singular values. Since the singular values are in descending order, we can alreadyestimate the approximation error based on the values of the discarded singular values.
Approximation in TP function class - Complexity reduction
On the other hand, if functiony = f(x) = BN
⊠n=1
wn(xn) belongs to the TP class, one
can still choose RHOSVD for complexity reduction of the TP representation to decrease the
45
number of weighting functionswn at the price of approximating the original function. TheTP function will be an approximation of the original function as:
f(x) ≈ DN
⊠n=1
wn(xn) (5.27)
The approximation errorγ is defined by (5.8) in Definition 5.8. Since the discardedsingular values assign the error and then-mode rank, namely the number of constant com-ponents, we can readily perform trade-off here between the error and the number of constantcomponents and weighting functionswn.
Remark5.11. The error of the approximation can already be estimated fromthe discardedsingular values, but beyond this it can be numerically calculated and evaluated at any instanceon a large number of random points.
5.1.5 Convex hull manipulation of TP functions via TP model transfor-mation
An important property of TP model transformation is that in step II Un can be changed andaccordingly,D can be recalculated keeping the equality in 5.25. Thus, matrix Un can betransformed in order to define various convex hulls. This means that we can define differenthull types of the TP function besides the canonical form.
Un = U′nT, (5.28)
whereT is the transformation matrix. We can give the discretised system tensorFD(Ω,M)
as:
FD(Ω,M) = DN
⊠n=1
Un = DN
⊠n=1
(U′nT) = (D
N
⊠n=1
T)N
⊠n=1
U′n = D′
N
⊠n=1
U′n. (5.29)
Definition 5.11. The TP function is convex if the weighting functions satisfythe followingcriteria:
∀n, i, xn(t) : wn,i(xn(t)) ∈ [0, 1]; (5.30)
∀n, xn(t) :
In∑
i=1
wn,i(xn(t)) = 1. (5.31)
Convex hulls have a special role, for instance in LMI-based control theory, thus we definesome basic convex hull types:
Definition 5.12. (SN type TP function): The convex TP function is SN (Sum Normalized)if the sum of the weighting functions for allx ∈ Ω is 1.
Definition 5.13 (NN type TP function). The convex TP function is NN (Non-Negative) ifthe values of the weighting functions for allx ∈ Ω are non-negative.
46
Definition 5.14 (NO/CNO, NOrmal type TP function). The convex TP function is a NO(Normal) type model if itsw(p) weighting functions are Normal, that is, if it satisfies (5.30)and (5.31), and the largest value of all weighting functionsis 1. Also, it is CNO (close tonormal), if it is satisfies (5.30) and (5.31) and the largest value of all weighting functions is1 or close to 1.
Definition 5.15 (RNO type TP function). The convex TP function is Relaxed NO (RNO)type, if the largest values of all weighting functions are the same (if the matrix is SN and NNtype, then this value is always between0 and1).
Definition 5.16 (INO type TP function). The convex TP function is Inverse NO (INO) type,if the smallest value of all columns is0.
Definition 5.17 (IRNO, Inverted and Relaxed NOrmal type TP function). The TP functionis IRNO type if the smallest values of all weighting functions are 0, and the largest values ofall weighting functions are the same.
5.1.6 Extending the TP model transformation methodology toqLPVmodels
Definition 5.18 (qLPV model). Consider the Linear Parameter Varying State Space model:
x(t) = A(p(t))x(t) +B(p(t))u(t) (5.32)
y(t) = C(p(t))x(t) +D(p(t))u(t),
with inputu(t) ∈ Rm, outputy(t) ∈ R
l and state vectorx(t) ∈ Rk. The system matrix
S(p(t)) =
(
A(p(t)) B(p(t))C(p(t)) D(p(t))
)
(5.33)
is a parameter-varying object, wherep(t) ∈ Ω is time varyingN−dimensional parametervector, whereΩ = [a1, b1] × [a2, b2] × .. × [aN , bN ] ∈ R
N is a closed hypercube.p(t) canalso include some elements ofx(t), in this case (5.33) is termed as quasi LPV (qLPV) model.Therefore, this type of model is considered to belong to the class of non-linear models. Let’sassume that the size of the system matrixS(p(t)) isO timesI.
Definition 5.19 (Finite element polytopic model).
S(p(t)) =
R∑
r=1
wr(p(t))Sr. (5.34)
wherep(t) ∈ Ω. S(p(t)) is given for any parameter vectorp(t) as the parameter varyingcombinations of LTI system matricesSr ∈ R
O×I called LTI vertex systems. The combina-tion is defined by the weighting functionswr(p(t)) ∈ [0, 1]. By finite we mean thatR isbounded.
47
Definition 5.20(Finite element TP type polytopic model). S(p(t)) in (5.34) is given for anyparameter as the parameter-varying combination of LTI system matricesSr ∈ R
O×I .
S(p(t)) =
I1∑
i1=1
I2∑
i2=1
..
IN∑
iN=1
N∏
n=1
wn,in(pn(t))Si1,i2,..,iN , (5.35)
applying the compact notation based on the previous chapters we have:
S(p(t)) = SN
⊠n=1
w (pn (t)) (5.36)
where the(N +2) dimensional coefficient tensorS ∈ RI1×I2×···×IN×O×I is constructed from
the LTI vertex systemsSi1,i2,...,iN (5.35) and the row vectorwn (pn (t)) ∈ [0, 1] contains onevariable and continuous weighting functionswn,in(pn(t)) ,(in = 1 . . . IN) .
Remark5.12. : TP model (5.36) is a special class of polytopic models (5.34), where theweighting functions are decomposed to the Tensor Product ofone variable functions.
5.1.7 Definition of the HOSVD-based canonical form of qLPV models
The HOSVD-based canonical form for the qLPV model given in Definition 5.18 can bedefined similarly to the definition of the HOSVD-based canonical form of TP functions. Themain difference here is that the system matrix (5.33) has dimensionO timesI, while functiony = f(x) is scalar.
Notation 5.3. State variable vectorx(t), output vectory(t) and input vectoru(t) are alldependent on timet. For simpler notification we do not emphasize this dependency throughthis document from now on. Notationx, y andu is considered equivalent tox(t), y(t) andu(t). On the other hand, we would like to keep the emphasis that parameter vectorp(t) andthe weighting functionw (pn (t)) are time-varying.
Theorem 5.4 (HOSVD-based canonical form of qLPV models). Assume that the Gram-Schmidt-type orthogonalization is already carried out andthat the TP model can be given inthe form:
(
x
y
)
=
(
SN
⊠n=1
wn(pn(t))
)(
x
u
)
. (5.37)
Executing the three steps of TP model transformation on discretised tensorS one candetermine
(
x
y
)
=
(
DN
⊠n=1
wn(pn(t))
)(
x
u
)
, (5.38)
where tensorD has the size ofR1 × ...× RN ×O × I.CHOSVD is executed only on the firstN dimension of the qLPV model, which is the
parameter space.
48
The weighting functionswn,in(pn(t)), in = 1..Rn (termed asinth singular function ondimensionn = 1..N) contained in vectorwn(pn(t)) form an orthonormal system for the firstN dimension:
∀n :
∫ bn
an
wn,i(pn(t))wn,j(pn(t))dpn = δi,j, 1 ≤ i, j ≤ In,
whereδi,j is the Kronecker-function (δij = 1, if i = j andδij = 0, if i 6= j).
The Frobenius-norm‖Din=i‖, symbolized byσ(n)i , aren-mode singular values ofD.
D is termed core tensor consisting the LTI systems.
Remark5.13. The steps of TP model transformation of qLPV are the same as ofthe TPmodel transformation of continuous functions with the onlydifference that the CHOSVD inthe second step is executed only for the firstN dimension.
Discussion
As it is the case with TP functions, the same holds for qLPV models, namely some qLPVmodels belong to the TP model class and some do not. Based on this classification we discusstwo cases, the first is when the qLPV model is a non-TP model andthe second is when thereconstructed qLPV model is a TP-qLPV model, but we want to reduce complexity (numberof the weighting functionswn). For both of the cases we can use RHOSVD (see Definition5.8 for further details). The error of the approximation canalready be estimated from thediscarded singular values, but beyond this, it can be numerically calculated at any instanceon a huge number of random points.
5.1.8 Convex hull manipulation of qLPV models via TP model trans-formation
Definition 5.21. The TP model is convex if the weighting functions satisfy thefollowingcriteria:
∀n, i, pn(t) : wn,i(pn(t)) ∈ [0, 1]; (5.39)
∀n, pn(t) :In∑
i=1
wn,i(pn(t)) = 1. (5.40)
The same convex hull types can be defined for qLPV models as forTP functions. Thebasic ones are: SN, NN, NO/CNO, NOrmal, RNO, INO and IRNO types, for further details,see previous sections.
5.1.9 Summary of the chapter
This chapter introduced the SVD of matrices, which was extended to HOSVD of tensorsbased on Lathauwer’s work, the definitions of HOSVD, CHOSVD and RHOSVD were given.The HOSVD-based canonical form of TP functions was presented, that can be obtainedexecuting the steps of TP model transformation. Convex hullmanipulation via TP model
49
transformation was introduced and various convex hulls of TP functions are defined. At theend of this chapter the TP model transformation methodologywas extended to qLPV models.These concepts and methodologies compose the mathematicalbackground of this Thesis.
50
Chapter 6
CogInfoCom in force reflectingtelemanipulation
In telemanipulation and 3D virtual interactions it is important to transmit force sensationfrom the remote or virtual environment to the human operator. Due to the weak points (con-trol issues, robustness, cost) of real force feedback devices, methods where force is renderedon non-native sensory channels have grounds. In this chapter, a survey of the related lit-erature is presented and the concept of sensor-bridging type cognitive infocommunicationsbased force reflecting schemes is discussed. A complete experimental infrastructure withhardware and software components is built providing a background for the investigation ofthe proposed methods from practical usability aspects. This environment has been utilizedin a pilot experiment with human participants providing substantial observations on the us-ability of sensor-bridging type vibrotactile force feedback methods. The test confirms thatvibrotactile glove equipped with shaftless vibration motors can be successfully applied astactile/haptic feedback device in immersive virtual reality applications.
One of the key problem of telemanipulation is to pass sensoryinformation from the re-mote site to the human operator. Performance of the task can be improved by the complexityand quality of transmitted sensory channels. In most cases,visual and haptic informationare the most important, however e.g. in robot assisted surgery it is the most challenging toprovide transparent force feedback for the operator. The difficulties of real haptic feedbackare caused by the diverse complexity of the problem. Issues can be sorted in three main cate-gories: Control, Infocommunication and Cognitive aspects. From the control theory perspec-tive, time-delay, parameter uncertainties and nonlinearities all influence the force feedbackcontrol performance in negative way. Infocommunication isalso a crucial point because thevarying time-delay on the packet-switched networks causesunmanageable problem for thecontrol algorithms. The most complex problems rise up around the cognitive process fromthe sensation trough the understanding of the incoming multi-modal information during theremote or virtual interaction. Complexity of the connection between the human sensory sys-tems and the technical devices forms a challenging researchtask namely, the developmentof devices that can be applied to convey all the sensory percepts from the remote (real orvirtual) environment to the human operator. The goal is to develop and optimize methods
51
and tools in order to transmit the remotely measured data to the human brain via the sensoryorgans.
6.1 Conceptual Background
The first part of this section gives a summary of the goals and perspectives of CognitiveInfocommunications (CogInfoCom) based on [120], then a survey of the concerned topicsis provided. CogInfoCom is a newly emerging multi-disciplinary field which is concernedwith the analysis of existing and the synthesis of novel forms of communication betweenhumans and electronic devices with various levels of cognitive capabilities (also referred toas artificially cognitive devices). Towards the end of the section, it will become clear thatwhen force feedback in teleoperation is viewed as a channel of communication between theteleoperation process and the human operator, research directions motivated by the philoso-phy of CogInfoCom arises novel and useful tools to tackle theproblem of providing effectiveforce feedback under various circumstances.
6.1.1 Cognitive Infocommunications
Cognitive Infocommunications (CogInfoCom) investigatesthe links between the research ar-eas of infocommunications, informatics and cognitive sciences, as well as the various fieldswhich have emerged as a combination of this sciences. The field of CogInfoCom is sectionedalong two dimensions, the mode and the type of the communication. Mode of communica-tion can be intra-cognitive or inter-cognitive according to the level of cognitive capabilitiesof the endpoints participating in the communication process. Type of communication refersto the type of information that is conveyed between the nodesand the way in which this isdone. The communication is sensor-sharing type when the sensory information is merelytransferred on the infocommunication line thus the same sensory modality is used on bothends to perceive the information. The type of communicationis sensor-bridging when thesensory information obtained or experienced is not only transferred to the other end of theline, but also reallocated and transferred to an appropriate sensory modality on the receiverend. Terminology of CogInfoCom is often used within this chapter. The definition of CogIn-foCom and further introspection can be found in [120].
6.1.2 Inter-cognitive sensor-bridging in teleoperation
Human brain is able to interpret sensory information even ifit is not presented on the natu-rally coupled sensory modality. It practically means that ablind person might have real visualexperience getting tactile stimulation of her skin. This phenomena namely the cross-modalplasticity of the human brain is the basis of the sensory substitution as Paul Bach-y-Ritapublished in the late ’60s [121, 122, 123, 124]. Since the early times, the idea of sensorysubstitution has been tested in various research areas mainly in rehabilitation of persons withspinal cord injuries. The first working implementation was the tactile vision substitution sys-tem (TVSS) introduced by Bach-y-Rita: A video camera signalwas converted into a tactile
52
image and projected onto the blind subject’s back. In [125] Bach-y-Rita reviews the theoret-ical aspects and many applications from the beginning to 1999. In the past decade furtherresults were published: Bach-y-Rita, Kaczmarek and Tyler introduced the tongue based manmachine interface [126]. Danilov et al. using the tongue based interface got promising resultsin rehabilitation of patients with balance disorders [127,128]. In the teleamnipulation casethe motivation is not the recovery of a lost sensory capability, but the exploiting unloadedsensory skills of human. In conventional telemanipulatorsthe force/position input and theposition/force display is realized on the same actuated part of a master device (Figure 6.1).
Figure 6.1: Conventional force-position bilateral control of telemanipulation
This structure - supposed that the bilateral control ensurethe stable operation - providephysical constraint that prevent the unwanted destructionof the remote environment andprovide the operator with realistic force feedback in a natural way. Based on the conceptof sensory substitution, authors aim to develop inter-cognitive sensor-bridging methods forforce feedback. The proposed architecture is showed in Figure 6.2 supposing a telamanipu-lative grasping task.
Figure 6.2: Telemanipulation with sensory substitution
In this scheme, the master device acts as the position input of the system. The mea-sured master positionxM is the reference signal of the slave side position controller. The
53
interaction between the slave gripper and the remote environment is monitored via forcemeasurement. In the bilateral control, the measured slave forcefs is rendered on the masterdevice (Figure 6.1), while in the proposed sensor-bridgingtechnique the interaction force ispresented in the form of non native sensory stimulus (vibrotactile, visual, audio) applyingan appropriate coupling algorithm. The separation of the force/position input and outputchannels opens up the joint closed loops of the traditional bilateral control guarantying thestability of the telemanipulation system.
6.1.3 CogInfoCom in teleoperation
CogInfoCom appears from various points of view – including sensory substitution and multi-modal interaction – in existing teleoperation applications. The key motivation for using suchapproaches is multifold:
• Contradictory goals in terms of situation awareness and unencumberment
Situation awareness (also commonly referred to as telepresence) is a measure of thedegree to which the user feels present in the remote or virtual environment [129, 130].In an early work on the subject, Sheridan outlines3 key components of telepresence:the extent of sensory information, the control of relation of sensors to the environment,and the ability to modify the physical environment [131]. Other researchers have con-jectured that situation awareness is less quantitative, and that it has both subjective andobjective aspects [132].
Encumberment is a term used often in the literature to describe the extent to which theuser is burdened with having to wear various kinds of sensorsin order to interact with asystem [133, 134]. It is natural to try to reduce encumberment in virtual environments,however, doing this conflicts the goal of increased situation awareness.
In order to resolve these conflicting goals, a widely adoptedand natural direction forteleoperation research was to try to use alternative forms of feedback (e.g., [135, 136,137, 138, 139]).
• The usefulness of corroboration
There is extensive proof in the literature that different sensory channels are not inde-pendent of each other. While contradicting information from various senses can causeconfusion, simulation sickness or other discomfort, illusions in which stimulation inone sensory channel leads to the illusion of stimulation in another sensory channel canbe very powerful in virtual and/or remote teloperation [140].
The ability of human cognition to integrate experience fromvarious sensory channelsis referred to as intermodal (or intersensory) integration. According to [141], inter-modal integration may be a ”key psychological mechanism contributing to a sense ofpresence in virtual environments”. It was consistently proven that information sharedby various senses can reinforce each other to create powerful representations of inter-nally consistent worlds [142, 141]
54
• Alleviating the negative effects of reduced resolution
There are several results in the literature which underlinethe fact that the use of sen-sory substitution can be extremely beneficial in cases wherethe signals which canbe provided to the natural sensory modality are reduced in resolution or degrees offreedom.
For instance, according to Verner and Okamura, providing force feedback that is re-duced in degrees of freedom can result in the destabilization of the teleoperation sys-tem [143]. In specific applications, such as remote surgicalknot-tying in telesurgery, itwas shown that the forces applied by the telesurgeon were closer to the normal, manualcase when auditory and graphical displays were used insteadof direct force feedback[138].
It is important to note that the challenge behind sensory substitution lies not only in theeffective mapping of representations from one sensory modality to another, but also in con-sidering the cross-effects between sensory modalities. Researchers have long ago discoveredthat the impression that different sensory modalities are independent of each other is ”moreillusory than real” [144]. Thus, when designing feedback strategies in teleoperation systems,care must be taken so that the operator is not overloaded withsensory information. Althoughmulti-sensory information can help in many cases, its effects can also be counterproductiveif the user is burdened with too much information [135, 145].The question as to whethermulti-sensory feedback is productive or not has much to do with the degree of redundancyin the information that is presented [135, 146]. However, Biocca et al in [140, 141] alsosuggests that it is possible for one sensory modality to yield realistic sensations normallyperceived through another modality, while another sensorymodality gives no contribution torealistic sensations, but significantly increases the user’s sense of telepresence.
Besides sensory overload, another key point of interest when designing multi-modalinterfaces is how the various sensory modalities relate to one another in terms of impor-tance to human cognition. This is referred to as the questionof sensory dominance. Therehave been a number of studies which show that vision dominates haptic touch and audition[131, 147, 148, 149], but it was also shown that relationships of dominance can become morecomplex if more than two modalities are under stimulation atthe same time [149].
6.1.4 Force feedback in teleoperation
In force feedback capable haptic or telemanipulator devices the stability of bilateral controland the realistic force sensation (transparency) are contradicting requirements. Low commu-nication bandwidth, varying time-delay of the communication (jitter), nonlinearities in themechanisms and the unknown remote environment cause the unstable behavior or degradethe transparency. Among this causes, time-delay is crucialbecause this is an inherent prop-erty of distributed control systems. Internet-based teleoperation is a typical example, wherecommunication delay plays important role [92, 96]. In internet-based telemanipulation, thevarying and unbounded time-delays represents the main problem, even though the averagecommunication delay is far less than the reaction time of thehuman operator. Transparencyand stability of bilateral teleoperation was studied by Lawrence [150]. In the past decades
55
several approaches were published addressing the stability problem of closed loop force re-flecting telemanipulation. A comprehensive survey can be found in [110].
Several researchers attempted to solve the problem of time-delays by creating predictivemodels based on the application. In modalities other than force feedback, two basic ap-proaches – also known as predictor displays – were proposed:one which involves a Taylor-series based extrapolation based on current state variables and their derivatives, and a secondone which involves running a predictive model with time constants that are faster than thoseof the actual process [151]. The first approach has been shownto yield good results for short-term predictions, while the second approach can be useful inaddressing problems caused bynonlinear dynamic properties such as saturation [151].
Such predictor displays have proven extremely useful for tasks in which visual feedbackis needed [151]. However, due to operator-induced instabilities and problems with closed-loop control (as first demonstrated in [152]), the price of erroneous predictions is muchhigher in the case of force feedback than in the case of visualfeedback. The reason for thisis that unexpected disturbances can arise from both the natural inertia of the system, as wellas the feedback of erroneous force feedback on the same hand that is operating the masterdevice [153, 154]. Thus, although efforts were made to use predictive force feedback (e.g.,[155]), the importance of alternative approaches also became clear. Such approaches includesensory substitution using visual, audio and vibrotactilefeedback [135, 156, 138, 157, 139],wave dissipation and transformation techniques to spread out the negative effects of time-delays through time [158, 159], and supervisory control using high-level commands.
6.1.5 Haptics in Virtual Environment
Importance of haptic rendering in virtual reality applications was continuously growing inthe last twenty years. Haptic rendering provides the user with force and tactile feedbackduring manipulation in virtual environment. A general survey on haptic rendering is pub-lished by Salisbury et al. in [160]. The role of haptics in multimedia is reviewed in [161].Application of haptics in virtual reality ranges from entertainment to rehabilitation via 3Ddesign, virtual collaboration and different training purposes. Coles et al. investigated therole of haptic feedback in simulators for medical training [162]. Kammerl, Chaudhari andSteinbach introduced a method for efficient haptic data communication for networked virtualenvironments [163]. Considering, that control of haptic devices meets the same difficultiesas the bilateral control of telemanipulation [164], the prosed sensor-bridging method helpsto overcome the control issues in a sort of applications.
6.1.6 Existing solutions for force feedback under the concept of CogIn-foCom
The idea to use sensory substitution in order to convey forcefeedback has been under in-vestigation for several decades [165, 166, 167]. However, the first detailed and conclusiveexperiments on the subject were carried out by Massimino [135, 168, 156]. In his PhD thesis,Massimino drew the following conclusions regarding sensory substitution for force feedback[135]:
56
• In the presence of a time-delay, sensory substitution may beused, as it does not intro-duce instabilities
• Auditory displays are useful for representing accurate force direction information
Massimino’s results sparked active interest in using sensory substitution through variousmodalities in order to compensate for compromised force feedback due to lack of equipmentand/or time-delays.
Through the course of this research, it was found that the useof sensory substitutioncan be both superfluous and extremely valuable, depending onthe application. In an all-encompassing review of the topic, Kaczmarek demonstrated that already in the early 1990s,the use of electrotactile and vibrotactile displays was prevalent in the feedback of variouskinds of information, including tactile and force information [136]. At the end of the pa-per, Kaczmarek concluded that in order to make progress, researchers would have to designmore accurate stimulation waveforms for electrotactile feedback, develop smaller, less noisyand less power-consuming vibrator arrays, as well as betterunderstand the correlation be-tween standardized measures such as the just-noticeable-difference (JND) and number ofdiscernible levels.
Today, the most popular forms of sensory substitution for force feedback occur throughvisual, vibrotactile, and to a lesser extent auditory stimuli [169, 170, 171, 139]. It was shownin [169] and [170] that the use of multi-channel vibrotactile displays result in reduced meanerrors and reduced peak forces when users have to trace the outline of a shape at a fixedforce. Such results are especially encouraging for the design of remote telesurgical devices,of which even the most prominent are still completely lacking in force feedback [171, 139].
57
Part II
Theoretical Achievements
58
Chapter 7
TPτModel Transformation forDynamical Systems with Feedback Delay
The goal of this chapter is to extend the TP model transformation to a set of control problemshaving time-delay. The motivation behind this extension isto develop a compact, reliable andnumerically appealing method to present time-delay systems in qLPV model form – wherethe time-delay is transformed to be a parameter only – with polytopic structure where uponthe modern multi-objective LMI-based control design theories can immediately be applied.Actually, this means that the explicit time-delay will be represented implicitly as an externalparameter in the system description. This conceptually leads to the extension of the moderncontrol theories developed for non-delayed systems represented in qLPV form to a set ofcontrol problems containing time-delay. The key point of this extension to the TPτmodeltransformation is based on the modification of the first and the third step of the original TPmodel transformation. The essence of the modification is that the simple sampling-based dis-cretization is replaced by a redefinition-based discretization in the first step, and accordinglywe introduce the same redefinition for the system matrixS(p) when the continuous weight-ing functions are determined in the third step of the original TP model transformation. Theredefinition-based discretization means that we identify or approximate the delayed system– where the delay is given byτ – by a non-delayed system over every discretization pointsalong dimensionτ . When the system is more complex and depends on more parametersthanτ , then the discretization grid point is defined by the elements of the parameter vectorandτ , so as the discretization results in a parameter independent, non-delayed, simple LTIsystem over each grid point. Consequently, this results in the same form as the first step ofthe original TP model transformation, where a simple sampling is applied.
We can find a number of identification theories utilizable forthis redefinition idea in therelated literature. Due to the limited extent of this dissertation, only one of the appropriatemethods (which is utilized later in this dissertation) is discussed in this chapter as example.In the following sections the steps of TPτare described. For the sake of completeness, eachsteps are included in this chapter in its full form.
59
7.1 TPτModel Transformation
Assume that the time-delay qLPV model is given in the form of the following delay-differentialalgebraic equation [172]
x(t) = A(p)x(t) +B1(p)u(t) +B2(p)w(t) (7.1)
y(t) = C1(p)x(t) +D11(p)u(t) +D12(p)w(t)
z(t) = C2(p)x(t) +D21(p)u(t) +D22(p)w(t),
wherew(t) = [z1(t− τ), ..., zN(t− τ)]T . (7.2)
To maintain the generality in the followings it is assumed that all elements of the systemmatrices can be parameter dependent whether it has physicalmeaning or not. The equationcan be rewritten in compact matrix form such as
x(t)y(t)z(t)
= S(p)
x(t)u(t)w(t)
, (7.3)
where
S(p) =
A(p) B1(p) B2(p)C1(p) D21(p) D12(p)C2(p) D21(p) D22(p)
. (7.4)
The goal of the TPτ is to transform the above delayed system (7.1,7.2) into[
x(t)y(t)
]
= S(p′(t))
[
x(t)u(t)
]
, (7.5)
where
S(p′(t)) = SN
⊠n=1
w (p′n (t))) (7.6)
and
p′ = [p, τ ]T ,p′ ∈ Ω ⊂ RN . (7.7)
Thus,τ is transformed to be a simple element of the parameter vectorin a non-delayedpolytopic qLPV form.
7.1.1 STEP I: Redefinition-based discretization
The goal of this step is to represent the system (7.1,7.2) by its redefined and discretized tensorSD(Ω,M) that is ready to find the tensor product structure.
First of all we define the transformation spaceΩ (see Notation 5.1) in which we expectthe TP function to be relevant. The first important difference from the original TP model
60
transformation is that the dimensions ofΩ are assigned to the dimensions ofp′, namelyto the dimensions of the parameter vectorp and τ . (As a matter of fact,τ can be multi-dimensional if we consider systems with multiple delay-dependency. Without the loss ofgenerality we assume a scalarτ later on.) Secondly, we define the hyper-rectangular gridMby its grid pointsgm1
, .., gmNin Ω, see Notation 5.2.
Let the elementHm1,m2,m3,...,mNof SD(Ω,M) be the redefined LTI model of the delayed
system (7.1,7.2) over a given grid pointgm1..gmN
.
Hm1,m2,m3,...,mN= redef
([
xy
]
= f(gm1, gm2
, gm3, ..., gmN
, u(t))
)
, (7.8)
whereredef denotes the identification that redefines the original delayed and parameterdependent system by an LTI state space model without delay asdiscussed in the beginningof this chapter.
As a matter of fact, we define as dense discretization grid as possible to guarantee thatthe discretized tensor describes the system as close as possible inΩ.
As result of STEP I, we haveSD(Ω,M) ∈ RM1×...×MN .
7.1.2 STEP II: Extracting the TP structure
- i - The goal of this step is to reveal (from the results of STEP1) the TP structure of thegiven system model. We use HOSVD to find the TP structure of themodel:
As a result of STEP I, we have the discretized system tensorSD(Ω,M). ExecutingCHOSVD 5.7 on the firstN dimension ofSD(Ω,M) results in:
SD(Ω,M) = DN
⊠n=1
Un = DN
⊠n=1
wD(Ω,M)n , (7.9)
where the size ofD in the firstN dimension isR1 × R2 × . . . × RN , whereRn =rankn(S
D(Ω,M)), Rn ≤ Mn, for all n = 1..N . TensorD contains the LTI vertexcomponents. If we perform further complexity trade-off we execute RHOSVD – wediscard non-zero singular values that means the elimination of LTI vertex componentshaving the smallest contribution – the size of tensorD becomes even smaller. To bespecific, if we keepIn singular values along each dimension (n = 1..N) then the sizeof D in the firstN dimension isI1 × I2 × . . .× IN , whereIn ≤ Rn, for all n = 1..N .
- ii - In this step of the TPτmodel transformation we are capable of generating variouskinds of convex hulls (Definitions 5.12, 5.13, 5.14, 5.15, 5.16, 5.15, 5.17) in the sameway as in the original TP model transformation. Thus the following TP forms can beformulated:
SD(Ω,M) = DN
⊠n=1
WCNO,IRNO,SNNNn . (7.10)
61
7.1.3 STEP III: Determination of the weighting functions
STEP II generates the values of the weighting functions overthe discrete points defined byM along the dimensions ofΩ. First it is shown that matricesUn defines the discretizedweighting functions. Then it is presented how to reconstruct the continuous weighting func-tions over any arbitraryp′.
Discretized weighting functions
un,in of matrix Un ∈ RMn×In determineswD
n that is the discretization ofwn,in(p′n) (n =
1..N) overM . Thus, the valuesun,mn,in of column vectorun,in define the values of theweighting functionwn,in(p
′n) over the grid-pointsgn,mn
:
wn,in(gn,mn) = un,mn,in,
because of
Hm1,m2,m3,...,mN= S
N
⊠n=1
wDn (p
′n), (7.11)
for all grid-point
p′ = g =
g1,m1
...gN,mN
. (7.12)
Reconstruction of the continuous weighting functions
In the case of original TP model transformation the weighting functions are determined overany points of intervals[an, bn] by sampling the givenS(p). Obviously, in the case of TPτ thesame redefinition is necessary as in the STEP I.
In order to determine the weighting functions in vectorwd(p′d), letp′n be fixed to selected
grid-lines as:
p′n = gn,in n = 1 . . . N, n 6= d, in ∈ 1 . . . In,
wherein can be chosen arbitrarily. Then forp′d:
wd(p′d) = (H)(3)
(
(
SN
⊠n=1,n 6=d
un,in
)
(d)
)+
, (7.13)
whereH denotes the redefined delayed system resulted from
H = redef(f(p′,u(t))) (7.14)
and vectorp′ consists of elementsp′n andpd asp′ =(
g1,i1 g2,i2 . . . p′d . . . gN,in
)
,and superscript “+” denotes pseudo inverse andun,in is the first row vector ofUn. Thethird-mode matrix(H)(3) of matrixH is understood such that matrixH is considered as a
62
three-dimensional tensor, where the length of the third-dimension is one. This practicallymeans that the matrixH is stored in a one-row vector by placing the rows ofH next to eachother, respectively.
Remark7.1. (7.13) is the solution to the linear equation
redef(f(p′,u(t))) = H = SN
⊠n=1
wn(p′n),
where only the elements ofwd(p′d) are unknown. (We assume that the original system is
available soH is redefinable and forn 6= d the weightings are known as wellwn(p′n) =
un,in). The equation system can be rewritten in a linear form as
(H)(3) = wd(pd)
(
SN
⊠n=1,n 6=d
un,in
)
(d)
which usually can be solved by (5.26), but it can also be underdetermined or numerically ill-conditioned. This problem can be solved by using more equations by querying the model atadditionalp′ parameter points using different gridlines (i.e. different in index combinationsn = 1 . . . N , n 6= d). It is hard to tell in advance which grid points will result solvableequation, but the number of examined parameter points can beincreased incrementally untilthe solution is acceptable.
7.2 A possible way of redefinition
In this section we introduce a system identification method that is selected from the literatureas an example for the reidentification in the TPτmodel transformation. We emphasize hereagain, that a huge variety of different identification techniques are available in the relatedliterature. The reason why we briefly introduce only one method below is that this will beutilized in the further parts of the dissertation when TPτmodel transformation is applied.
Assume that the delayed qLPV model is given in form of (7.1,7.2). The goal is to identifyS(p′), when the parameter vectorp′ andτ are given byp′.
Output-Error (OE) Model Black-box identification
Equation Error (EE) methods and Output Error (OE) approaches can be differentiated amongsystem identification methods. In this section, the Output error method is described briefly.
The general structure of the output-error model is
y(t) =B(s)
F (s)u(t− nk) + e(t) (7.15)
B(s) = b1 + b2q−1 + · · ·+ bnbq
−nb+1 (7.16)
F (s) = 1 + f1q−1 + · · ·+ fnfq
−nf (7.17)
wherenb andnf are the orders of the output-error model.
63
OE methods are minimizing a usually quadratic objective function of output error. Thisis the error between the original system and the estimated model. In case of continuous timeidentification the resulted model is given in form of continuous time transfer function
G(s) =B(s)
F (s)=
bnbsnb−1 + bnb−1s
nb−2 + · · ·+ b1snf + fnfsnf−1 + · · ·+ f1
(7.18)
The coefficients of the polynomials are estimated using a prediction error/maximum like-lihood method. An extensive work on the identification of continuous time systems waspublished by Garnier and Wang [173].
Once we have identifiedG(s) in (7.18), we can readily generate its state space represen-tation denoted byH in 7.8.
7.3 Some further aspects
TPτ is basically introduced for delayed systems, however due tothe redefinition in STEP Iit is appropriate for such non-delayed systems where the sampling-based discretization isnot possible. Thus, the application field of TP Model Transformation has been extended.By the extension, TP Model Transformation will be applicable in those cases, when theinitial model is not given in form of state space model, but ingeneralx = f(x(t),u(t))form, because the redefinition brings this in state space form. Conclusively, when the systemmodel can be identified over the points of the discretizationgrid, the introduced method isapplicable. However, the described process is viable this Thesis does not deal with its furtherinvestigation. Study of the limitations of this method willbe the subject of future work.
64
Chapter 8
Impedance model with feedback delay inTP type polytopic LPV forms
The aim of this chapter is to create and manipulate the different convex polytopic structuresof the impedance model with feedback delay. Furthermore, toperform a trade-off betweenthe complexity and the accuracy of the resulted TP model. Further considerations on theeffectiveness of the convex polytopic model based LMI design techniques and the conserva-tiveness of the resulting controller is investigated in Chapter 10.
The literature of modern control theory shows that the representation of a given planthas a considerable effect on the usability of the proper controller design method and on theachievable control performance. For instance in case of theqLPV state-space model givenin a polytop representation and the LMI based design techniques we can observe that the dis-position layout of the system matrix elements at the very beginning modelling phase alreadydetermine the set of achievable control performance. Furthermore the resulting controlleris really depends on the applied LMIs that is the reason why the majority of the related lit-erature discusses how to manipulate LMIs in order to optimize for multi-objective controlperformance. In the same time, one of the key trends in moderncontrol -H∞ based method-ologies - bases the optimization of the required control constraints on integrating weightingfunctions into the system model before determining the polytop representation and construct-ing the LMI-based synthesis.
Nevertheless, it was not emphasized as much that the LMIs arevery sensitive for thepolytop structure. Since the LMI in that sense can be considered as a non-linear transfor-mation, a little modification of the convex hull may leads to considerable deviation of theresulting controller. Therefore one may arise the questionwhether the convex hull manipu-lation plays an important role in the optimization of the control performance. Actually thiswas one of the key motivations to develop the TP model transformation that readily capableof manipulating the convex hull of the convex polytop representation of a given model.
Applying the TP model transformation Grof at al. [174] presented a deep investigationhow the convex hull manipulation influences the effectiveness of the LMIs or even morehow the improper selection of the convex hull may leads to infeasible LMIs. With this in-vestigation she has shown that the manipulation of the convex hull is as much important asthe selection of the LMIs to reach the best control performance. TP model transformation
65
give us good manipulation tool so thus it is important to investigate whether TPτmodel trans-formation is capable of performing similar optimization ondelayed systems and to checkif TPτmodel transformation holds all the advantages for delayed system as the original TPmodel transformation does.
In this chapter, different convex hulls of the impedance model are generated for the LMIbased control design described in the further part of the dissertation. This actually validatesthe applicability of TPτmodel transformation. While the convex hull is created we examinetwo types of manipulation techniques. One type performs complexity trade-off on the num-ber of the LTI vertex models, while the other focuses on the manipulation of the convex hull.Thus, we create the HOSVD based canonical form of the impedance model with approxi-mation trade-off, and generate different convex hulls using different convex transformationsatisfying various constraints.
8.1 Specification of the expected LPV reprezentation
The aim of this chapter is to find a representation of the investigated impedance model in TPtype polytopic form:
S(p′(t)) = SN
⊠n=1
w (p′n (t))) (8.1)
that meets the following requirements:
i Fulfils the specifications of HOSVD based canonical form by finding minimum num-ber of LTI components that represent the original system in polytopic structure (8.1).
ii Complexity trade-off capability by mean of approximation.
iii Eligible for LMI based multi-objective control design.
iv The generated convex hull indirectly supports the feasibility of optimal control perfor-mance under the LMI based design concept.
8.2 The HOSVD based canonical form
In this section we utilize TPτmodel transformation to determine the so called HOSVD basedcanonical form (Theorem 5.4) of the investigated impedancemodel (3.5), which is minimumand unique TP type polytopic representation. Along this chapter, the impedance model withfeedback delay is regarded as the subject of the investigation. As the TPτmodel transfor-mation is a fully numerical method, the chapter goes througha typical numerical example,wherein the following model parameters are considered:
It is important to note, that the main properties of the polytop structure are not influencedsignificantly by the model parameters in a wide range with practical relevancy, supposed thatthe TPτmodel transformation is executed with the reidentificationtechnique described in thesection 7.1.
66
Table 8.1: Parameters of the impedance modelDescription Parameter Value UnitsMass m 1 kgviscous damping b 100 Ns/mStiffness of the environment k 2000 N/mDelay interval τ 0..0.07 s
8.2.1 Components and structure of the exact HOSVD based canonicalform
After the execution of the TPτmodel transformation on the impedance model, we get theminimum size LPV representation composed by 6 LTI vertex models, as the HOSVD leadsto 6 non-zero singular values in the second step of the TPτmodel transformation. Singularvalues are as follow:
σ1 = 2.3414× 104
σ2 = 3.5305× 102
σ3 = 1.0331
σ4 = 2.2164× 10−2
σ5 = 1.2964× 10−3
σ6 = 6.9808× 10−5
Note again that the different model parameters have no substantial effect on the resultedsingular values, on the rank of the model and the underlying polytopic structure, so this ex-ample properly shows the uniqueness of the representation.The consecutive singular valuesdecrease exponentially by a factor of two orders of magnitude, which suggests a balancedcontribution of vertices. Figure 8.1 displays the formation of the 6 singular values.
In the followings, the components of the HOSVD based canonical form of the impedancemodel are introduced. The system is represented by the vertex models and the weightingfunctions. Let us partition the LTI vertices (Scan
r ) as follows:
Scanr =
[
A B
C D
]
(8.2)
As the reidentification (7.1.1) results inC = [1 0] andD = 0 for all τ ∈ Ω, onlyA andB are written in the list below:
67
1 2 3 4 5 610
−5
10−4
10−3
10−2
10−1
100
101
102
103
104
105
N−th singular value
Figure 8.1: Singular values of the HOSVD based canonical form
[AB]can1 =
[
−1.1515× 104 −1.1896× 104 −6.39441.1521× 104 1.1890× 104 6.3906
]
[AB]can2 =
[
−1.8082× 102 1.7522× 102 3.04961.7781× 102 −1.7209× 102 −3.0457
]
[AB]can3 =
[
2.8404× 10−1 −2.7537× 10−1 −6.0693× 10−1
2.9928× 10−1 −2.9109× 10−1 6.0668× 10−1
]
[AB]can4 =
[
8.7316× 10−3 −9.6607× 10−3 8.6616× 10−3
8.7412× 10−3 −9.6704× 10−3 −8.7600× 10−3
]
[AB]can5 =
[
6.6570× 10−4 6.2594× 10−4 9.5164× 10−5
6.6535× 10−4 6.2629× 10−4 3.9904× 10−5
]
[AB]can6 =
[
−2.5581× 10−6 −2.5893× 10−6 4.9197× 10−5
−2.5570× 10−6 −2.5904× 10−6 4.9258× 10−5
]
Figure 8.2 shows the weighting functionsw(τ) over the range ofΩ. The smoothnessof the weighting functions shows that the applied reidentification method is stable alongthe investigated range ofτ . This means that the applied identification algorithm does notalternate between different local solutions (local minimums). It is worth mentioning thatif the identification method is switching between differentsolutions, additional ranks couldappear in the HOSVD canonical form. By neglecting the extra singular values, HOSVDis able to (smoothly) approximate the ruggedness in least-square sense in a way similar tohow SVD can be used for noise filtering in digital signal processing [175]. However, if thefluctuation is large, such approximation should be applied with circumspection.
68
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
−0.3
−0.2
−0.1
0
0.1
0.2
feedback delay [s]
wei
ghts
Figure 8.2: Weighting functions of the HOSVD based canonical form
8.2.2 Executing trade-off by TPτmodel transformation
As it was mentioned before, a trade-off can be determined on the complexity and the accuracyof the TP type polytopic model. The goal of this section is to reveal the correlation betweenthe accuracy and the number of utilized vertices. Regardingthat the LMI based designprocess is very sensitive for the complexity of the polytop model, it is very important to findthe minimum complexity that reach the accuracy threshold ofthe given engineering problem.
Even the state-of-the-art computational solutions as LMI toolbox for MATLAB utilizesthe so-called interior point LMI solvers [176] which are much more efficient then the classi-cal convex optimization algorithms, the computational complexity of the problem explodesexponentially by the number of vertex models. Over a certainnumber of vertices the LMIsolvers may not be able to provide the solution. Even the solution exists the solver mightreport infeasibility. Considering that the significance ofthe vertex models decreasing uni-formly (Figure 8.1), there is no theoretically appealing point from where to cut the less sig-nificant vertices to reduce the complexity of the model. However, a systematically executedtrade-off could help to find the reasonable complexity. The HOSVD based canonical form isreadily support a kind of principal component analysis of the investigated dynamical systemmodel. In this analysis the model accuracy is measured by themodelling errorǫr defined as:
ǫr =∥
∥
∥SD(Ω,M) − S
D(Ω,M)Approxr
∥
∥
∥
L2
, (8.3)
whereSD(Ω,M) can be computed as it is written in (7.9).SD(Ω,M)Approxr
is computed analo-gously but considering only the vertex models according to the firstr singular values. Mod-eling errors results as follows:
69
1 2 3 4 5 610
−12
10−10
10−8
10−6
10−4
10−2
100
102
104
Keeping N singular value
Mod
el e
rror
Figure 8.3: Accuracy-complexity trade-off: Modelling error (ǫr) in function of dimensional-ity of HOSVD based canonical form
ǫ1 = 3.53× 102 ≤ 3.541× 102
ǫ2 = 1.0333 ≤ 1.0566
ǫ3 = 2.22× 10−2 ≤ 2.35× 10−2
ǫ4 = 1.3× 10−3 ≤ 1.4× 10−3
ǫ5 = 6.9808× 10−5 ≤ 6.9808× 10−5
ǫ6 = 1.1272× 10−11 ≈ 0 (numerically zero)
As matrix [AB]r contains element in the order of magnitude103, due to the definitionof ǫr, ǫ6 is much larger then10−15 what is typically considered as numerically zero if allthe matrix elements are in the range of101. ǫr is upper bounded by the sum of the singularvalues of the neglected vertices. The upper bounds are also exposed in the above list.
Figure 8.3 displays the modelling errors. One can see that the modelling error decreasesbetweenǫ6 andǫ5 much larger then in case of the other reduction steps.
This measure describes the model accuracy only over the discrete delay values definedby M , does not give information about the correctness between the discrete points that havebeen used in the first step of the TPτmodel transformation. To follow out a more extensiveinvestigation and to ensure that the resulted TP model is notundersampled let us define thefollowing measure:
Definition 8.1 (ǫRND1000r ).
ǫRND1000r =∥
∥
∥SD(Ω,M ′) − S
D(Ω,M ′)Approxr
∥
∥
∥
L2
, (8.4)
70
1 2 3 4 5 610
−1
100
101
102
103
Keeping N singular value
Mod
ellin
g er
ror
εRN
D10
00N
Figure 8.4: Accuracy-complexity trade-off: Modelling error (ǫRND1000r ) in function of di-
mensionality of hosvd-based canonical form
whereM ′ denotes a discretization grid with1000 randomly generated grid points overΩ.Grid M ′ is not equidistant andM ′
⋂
M = ∅.
The measureǫRND1000r compares the reidentified and the approximated systems in1000randomly generated points considering the firstr vertices of the HOSVD based model.ǫRND1000r shows the model accuracy better in real situations where arbitrary varying delaysoccur. The resultedǫRND1000r s are listed below:
ǫRND10001 = 9.6325× 102
ǫRND10002 = 2.7698
ǫRND10003 = 1.2099× 10−1
ǫRND10004 = 1.0519× 10−1
ǫRND10005 = 1.0514× 10−1
ǫRND10006 = 1.0514× 10−1
The values forr = 4, r = 5, r = 6 are almost the same and start to increase only atr = 3. Figure 8.4 displays theǫRND1000r data in logarithmic scale and the big step byr = 3 isevident.
These results support the hypothesis that the number of non-zero singular values do notincrease, even when the density ofM is increased without bounds. Thus, results show thatthe representation is minimal and exact. It can be also concluded that the applied discretiza-tion is not under-sampled hence complexity reduction by neglecting the less significant ver-
71
tices is well established. Beyond this pure numerical comparison the dynamic accuracy ofthe TP model has also been investigated in section 8.4.
8.3 Manipulation of the convex hull
As it was already emphasized, LMIs are very sensitive for theshape of the convex hullthat defines the polytopic qLPV representation together with the weighting functions. In thissection different type of convex hulls of the delayed impedance model are generated utilizingthe hull manipulation capabilities of TPτmodel transformation. For the sake of completeness,the exact TP type polytopic model and the non-exact model with 5 to 3 vertices are examined.In the following subsections, IRNO, SNNN and CNO type convexhulls are given by thevertices and the weighting functions. Even using the most developed optimization strategiesit is not possible to generate NO type convex hulls. From engineering aspect this hypothesiscan be accepted as CNO type convex hulls fulfil the requirements of control synthesis. Thedefinitions of different type of convex hulls can be found in section 5.
The convex hulls are given in the following structure:
• The exact TP model
– SNNN type convex hull (Figure 8.5)
– IRNO type convex hull (Figure 8.6)
– CNO type convex hull (Figure 8.7)
• Reduced TP model with 5 vertices
– SNNN type convex hull (Figure 8.8)
– IRNO type convex hull (Figure 8.9)
– CNO type convex hull (Figure 8.10)
• Reduced TP model with 4 vertices
– SNNN type convex hull (Figure 8.11)
– IRNO type convex hull (Figure 8.12)
– CNO type convex hull (Figure 8.13)
• Reduced TP model with 3 vertices
– SNNN type convex hull (Figure 8.14)
– IRNO type convex hull (Figure 8.15)
– CNO type convex hull (Figure 8.16)
72
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.1
0.2
0.3
0.4
0.5
feedback delay [s]
wei
ghts
Figure 8.5: Weighting functions of SNNN type convex hull of the exact TP model
8.3.1 The exact TP model
SNNN type convex hull
[AB]snnn1 =
[
9.8260× 102 1.0206× 103 −6.1076× 10−1
−9.8207× 102 −1.0211× 103 6.1092× 10−1
]
[AB]snnn2 =
[
9.6385× 102 1.0388× 103 2.7141× 10−1
−9.6410× 102 −1.0385× 103 −2.7127× 10−1
]
[AB]snnn3 =
[
9.0316× 102 1.0976× 103 2.4409−9.0555× 102 −1.0952× 103 −2.4383
]
[AB]snnn4 =
[
8.7268× 102 1.1271× 103 3.3613−8.7601× 102 −1.1238× 103 −3.3583
]
[AB]snnn5 =
[
1.4307× 103 5.5785× 102 −6.8959−1.4239× 103 −5.6496× 102 6.8865
]
[AB]snnn6 =
[
9.2741× 102 1.0741× 103 1.5495−9.2893× 102 −1.0725× 103 −1.5480
]
73
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.05
0.1
0.15
0.2
0.25
0.3
feedback delay [s]
wei
ghts
Figure 8.6: Weighting functions of IRNO type convex hull of the exact TP model
IRNO type convex hull
[AB]irno1 =
[
9.8112× 102 1.0197× 103 3.3566× 10−1−9.8148× 102 −1.0193× 103 −3.3532× 10−1
]
[AB]irno2 =
[
9.5875× 102 1.0435× 103 4.9710× 10−1−9.5923× 102 −1.0430× 103 −4.9670× 10−1
]
[AB]irno3 =
[
9.4525× 102 1.0554× 103 1.3775−9.466× 102 −1.0540× 103 −1.3761
]
[AB]irno4 =
[
9.3838× 102 1.0623× 103 1.4845−9.3985× 102 −1.0608× 103 −1.4832
]
[AB]irno5 =
[
1.1100× 103 8.8698× 102 −1.5891−1.1084× 103 −8.8863× 102 1.5866
]
[AB]irno6 =
[
8.9945× 102 1.1029× 103 1.9874−9.0140× 102 −1.1009× 103 −1.9853
]
74
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
feedback delay [s]
wei
ghts
Figure 8.7: Weighting functions of CNO type convex hull of the exact TP model
CNO type convex hull
[AB]cno1 =
[
9.5246× 102 1.0486× 103 1.0530−9.5350× 102 −1.0475× 103 −1.0520
]
[AB]cno2 =
[
9.8994× 102 1.0101× 103 4.7248× 10−1
−9.9040× 102 −1.0096× 103 −4.7217× 10−1
]
[AB]cno3 =
[
1.0169× 103 9.8246× 102 1.2070× 10−2
−1.0169× 103 −9.8248× 102 −1.2565× 10−2
]
[AB]cno4 =
[
1.0049× 103 9.9497× 102 8.5882× 10−2
−1.0050× 103 −9.9488× 102 −8.6082× 10−2
]
[AB]cno5 =
[
9.5135× 102 1.0492× 103 1.1945−9.5253× 102 −1.0480× 103 −1.1934
]
[AB]cno6 =
[
9.5001× 102 1.0513× 103 1.0027−9.5099× 102 −1.0503× 103 −1.0017
]
75
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
feedback delay [s]
wei
ghts
Figure 8.8: Weighting functions of SNNN type convex hull of the reduced TP model with 5vertices
8.3.2 Reduced TP model with 5 vertices
SNNN type convex hull
[AB]snnn51 =
[
9.4975× 102 1.0516× 103 1.0061−9.5074× 102 −1.0506× 103 −1.0051
]
[AB]snnn52 =
[
9.5019× 102 1.0509× 103 1.0864−9.5126× 102 −1.0498× 103 −1.0853
]
[AB]snnn53 =
[
9.7368× 102 1.0262× 103 8.6877× 10−1
−9.7454× 102 −1.0254× 103 −8.6819× 10−1
]
[AB]snnn54 =
[
1.0024× 103 9.9751× 102 1.5039× 10−1
−1.0025× 103 −9.9736× 102 −1.5050× 10−1
]
[AB]snnn55 =
[
1.0140× 103 9.8548× 102 4.6375× 10−2
−1.0140× 103 −9.8546× 102 −4.6693× 10−2
]
IRNO type convex hull
[AB]irno51 =
[
9.7734× 102 1.0230× 103 6.0370× 10−1
−9.7795× 102 −1.0223× 103 −6.0321× 10−1
]
[AB]irno52 =
[
9.6627× 102 1.0353× 103 5.0519× 10−1
−9.6678× 102 −1.0347× 103 −5.0454× 10−1
]
[AB]irno53 =
[
9.5116× 102 1.0504× 103 9.2294× 10−1
−9.5206× 102 −1.0495× 103 −9.2220× 10−1
]
[AB]irno54 =
[
1.0828× 103 9.1455× 102 −1.0172−1.0818× 103 −9.1562× 102 1.0154
]
[AB]irno55 =
[
9.0530× 102 1.0962× 103 2.1220−9.0738× 102 −1.0940× 103 −2.1198
]
76
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.05
0.1
0.15
0.2
0.25
0.3
0.35
feedback delay [s]
wei
ghts
Figure 8.9: Weighting functions of IRNO type convex hull of the reduced TP model with 5vertices
CNO type convex hull
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
feedback delay [s]
wei
ghts
Figure 8.10: Weighting functions of CNO type convex hull of the reduced TP model with 5vertices
[AB]cno51 =
[
9.4984× 102 1.0515× 103 1.0050−9.5083× 102 −1.0505× 103 −1.0040
]
[AB]cno52 =
[
9.7541× 102 1.0245× 103 8.3214× 10−1
−9.7623× 102 −1.0236× 103 −8.3163× 10−1
]
[AB]cno53 =
[
9.5280× 102 1.0482× 103 1.0526−9.5384× 102 −1.0471× 103 −1.0515
]
[AB]cno54 =
[
1.0021× 103 9.9773× 102 1.5382× 10−1
−1.0023× 103 −9.9758× 102 −1.5393× 10−1
]
[AB]cno55 =
[
1.0028× 103 9.9689× 102 2.3357× 10−1
−1.0030× 103 −9.9668× 102 −2.3363× 10−1
]
77
8.3.3 Reduced TP model with 4 vertices
SNNN type convex hull
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
feedback delay [s]
wei
ghts
Figure 8.11: Weighting functions of SNNN type convex hull ofthe reduced TP model with4 vertices
[AB]snnn41 =
[
1.0009× 103 9.9880× 102 2.9652× 10−1
−1.0011× 103 −9.9852× 102 −2.9658× 10−1
]
[AB]snnn42 =
[
1.0005× 103 9.9947× 102 1.8125× 10−1
−1.0006× 103 −9.9928× 102 −1.8130× 10−1
]
[AB]snnn43 =
[
9.6070× 102 1.0397× 103 1.0297−9.6172× 102 −1.0387× 103 −1.0288
]
[AB]snnn44 =
[
9.5016× 102 1.0512× 103 1.0004−9.5114× 102 −1.0502× 103 −9.9936× 10−1
]
IRNO type convex hull
[AB]irno41 =
[
9.4288× 102 1.0592× 103 9.7691× 10−1
−9.4384× 102 −1.0582× 103 −9.7584× 10−1
]
[AB]irno42 =
[
9.7528× 102 1.0245× 103 9.1178× 10−1
−9.7616× 102 −1.0236× 103 −9.1119× 10−1
]
[AB]irno43 =
[
1.0524× 103 9.4607× 102 −6.4437× 10−1
−1.0517× 103 −9.4674× 102 6.4320× 10−1
]
[AB]irno44 =
[
9.4577× 102 1.0555× 103 1.1012−9.4686× 102 −1.0544× 103 −1.1001
]
78
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
feedback delay [s]
wei
ghts
Figure 8.12: Weighting functions of IRNO type convex hull ofthe reduced TP model with 4vertices
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
feedback delay [s]
wei
ghts
Figure 8.13: Weighting functions of CNO type convex hull of the reduced TP model with 4vertices
CNO type convex hull
[AB]cno41 =
[
1.0018× 103 9.9785× 102 2.8284× 10−1
−1.0021× 103 −9.9758× 102 −2.8292× 10−1
]
[AB]cno42 =
[
1.0004× 103 9.9948× 102 1.8152× 10−1
−1.0006× 103 −9.9930× 102 −1.8157× 10−1
]
[AB]cno43 =
[
9.6003× 102 1.0404× 103 1.0393−9.6105× 102 −1.0393× 103 −1.0384
]
[AB]cno44 =
[
9.5018× 102 1.0512× 103 1.0000−9.5116× 102 −1.0502× 103 −9.9904× 10−1
]
79
8.3.4 Reduced TP model with 3 vertices
SNNN type convex hull
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
feedback delay [s]
wei
ghts
Figure 8.14: Weighting functions of SNNN type convex hull ofthe reduced TP model with3 vertices
[AB]snnn31 =
[
9.7873× 102 1.0209× 103 8.6643× 10−1
−9.7958× 102 −1.0200× 103 −8.6591× 10−1
]
[AB]snnn32 =
[
9.4943× 102 1.0519× 103 1.0101−9.5043× 102 −1.0509× 103 −1.0091
]
[AB]snnn33 =
[
1.0005× 103 9.9947× 102 1.8248× 10−1
−1.0007× 103 −9.9929× 102 −1.8255× 10−1
]
IRNO type convex hull
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.1
0.2
0.3
0.4
0.5
feedback delay [s]
wei
ghts
Figure 8.15: Weighting functions of IRNO type convex hull ofthe reduced TP model with 3vertices
80
[AB]irno31 =
[
9.6225× 102 1.0392× 103 6.5354× 10−1
−9.6290× 102 −1.0385× 103 −6.5284× 10−1
]
[AB]irno32 =
[
1.0288× 103 9.7002× 102 −1.6177× 10−1
−1.0287× 103 −9.7021× 102 1.6113× 10−1
]
[AB]irno33 =
[
9.3329× 102 1.0679× 103 1.4790−9.3475× 102 −1.0664× 103 −1.4776
]
CNO type convex hull
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
feedback delay [s]
wei
ghts
Figure 8.16: Weighting functions of CNO type convex hull of the reduced TP model with 3vertices
[AB]cno31 =
[
9.7873× 102 1.0209× 103 8.6644× 10−1
−9.7957× 102 −1.0200× 103 −8.6591× 10−1
]
[AB]cno32 =
[
9.4943× 102 1.0519× 103 1.0101−9.5043× 102 −1.0509× 103 −1.0091
]
[AB]cno33 =
[
1.0005× 103 9.9947× 102 1.8247× 10−1
−1.0007× 103 −9.9929× 102 −1.8254× 10−1
]
8.4 Analysis of the convex representation
The goal of this section is to illustrate the dynamical accuracy of the different TP models.Here the model accuracy is investigated by mean of the difference between the step responseof the polytopic model and the original delayed model. Due tothe limited extent of thisdissertation, only a set of practically interesting validation cases have been imparted. Thecomparison is broken into two parts according to the constant delay and varying delay cases.
81
8.4.1 Constant time-delay
Firstly, the dynamic accuracy of the HOSVD based canonical forms of the delayed impedancemodel with different complexity is examined. Figure 8.17 shows the step responses of thecompared models at an arbitrarily chosen constant delay value (τ = 0.05567). As input sig-nal, a1[N ] force step was used at0.1s in the simulation. Subfigures 8.17/a-f shows the stepresponses of TP models with different number of neglected less significant vertices. The timeplots confirm the result of the modelling error analysis was done in 8.2.2. As the values ofǫRND1000r suggested, the TP models show similarly good accuracy with 6, 5 and 4 vertices andthe model accuracy begins to relapse with 3 vertices. TP model with 2 and only 1 verticescannot describe the dynamics of the original delayed systemproperly.
For the purpose of confidence, the same simulations have beenexecuted on CNO typeTP models with 5, 4 and 3 vertices (Figure 8.18). As it was expected, the time plots showthe same result as the HOSVD based canonical type with equivalent complexities.
We can conclude that the CNO type TP models with 5,4 and 3 vertices give very similarresponses independently from the complexity. Convex hull of the investigated polytopicmodel cannot be formed with less then three vertices becausethe resulted domain of LPVmodels are not on the hyperline that can be defined by the convex combinations of two vertexsystems. The results suggest that the accuracy of CNO type convex TP model with 3 verticesprovide sufficient accuracy for controller design purposes.
For the quantitative comparison, theL2 norm of the position error (the square root of sumof squares) and the maximum error is computed at four arbitrarily chosenτ values (neitherof them are on the gridM) considering the1s long execution of the previously discussedsimulation scenario. Results are displayed in Table 8.2.
Table 8.2: Quantitative comparison of the original delayedmodel and the CNO typeTPτmodel with 3 vertices
L2 error Max errorτ = 0.01375s 2.6279× 10−5 9.8521× 10−7
τ = 0.02941s 4.0380× 10−5 5.9765× 10−6
τ = 0.04752s 4.3281× 10−5 1.0500× 10−5
τ = 0.06393s 1.0851× 10−4 1.3048× 10−5
It is a noteworthy observation that the investigated TP models have a minor steady-state error (see Figure 8.17 and 8.18), which often cause problems in control design. Theimpedance model is understood as the dynamical relationship between the force and the re-sulted velocity. The applied reidentification method oftenresults in models where the resid-ual velocity is not0[m/s] but some small value. In the figures, the position is displayed asthe integral of the model output, and thus, the non-zero residual velocity results a drift in theposition value. As it can be seen in Chapter 10, this type of model inaccuracy does not causethe failure of the control design. In our case, due to the observer-based control approachthe TP model does not appear directly in the controller, but the state observer receives thevelocity from the original system. In other applications, where the model is directly used inthe control algorithm, a dead zone filter around zero can be a handy solution. The residual
82
error is investigated more deeply by my student Jozsef Kutiin [177, 178], where he proposedan enhancement for the reidentification that solves this issue.
8.4.2 Varying time-delay
The models have been compared under varying delay as well. The value ofτ(t) was variedas a sine function of timeτ(t) = 0.03 + sin(tπ)0.025. The input signal was a square wavewith the frequency of2[Hz] and amplitude of1[N ]. Figure 8.19 shows the result of thesimulation.
Figure 8.19: Comparison under varying delay
One can observe that the different system behaviours aroundthe two terminal positions.The reason of the difference is that the square wave and the sine function in this example hasthe same period, and thus, the outer terminate position happens at larger momentary delays.A thorough investigation of varying parameter cases can be found in [179, 177, 178].
83
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
time [s]
posi
tion
[m]
Original delayed modelTP model
(a) Canonical form with 6 vertices (exact model)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
time [s]
posi
tion
[m]
Original delayed modelTP model
(b) Canonical form with 5 vertices
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
time [s]
posi
tion
[m]
Original delayed modelTP model
(c) Canonical form with 4 vertices
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
time [s]
posi
tion
[m]
Original delayed modelTP model
(d) Canonical form with 3 vertices
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
time [s]
posi
tion
[m]
Original delayed modelTP model
(e) Canonical form with 2 vertices
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
time [s]
posi
tion
[m]
Original delayed modelTP model
(f) Canonical form with 1 vertices
Figure 8.17: Comparison of the original delayed model and the HOSVD-based canonicalform of the TPτmodel with different complexity
84
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
time [s]
posi
tion
[m]
(a) CNO5
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
time [s]
posi
tion
[m]
(b) CNO4
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
time [s]
posi
tion
[m]
(c) CNO3
Figure 8.18: Comparison of the original delayed model and the CNO type TPτmodel withdifferent complexity
85
8.5 Summary of the chapter
• I deduced the HOSVD based canonical form of the impedance model then I performeda complexity trade off supported by TPτ transformation to determine non-exact TPmodels neglecting vertex systems with the less contribution.
• I proved that the TPτ transformation is capable of manipulating the convex hull of themodel, where variableτ appears as an external parameter.
• I proved that a convex polytop structure requires 6 vertex models for exact representa-tion of the impedance model for anyτ ∈ ω.
• I presented the correlation between the number of vertex models and the number ofsingular values of the HOSVD based canonical form andL2 norm based error of thepolytopic structure over the transformation spaceΩ.
• In order to satisfy the basic requirements of LMI based design and further convex hullmanipulation based optimization of the control design, I generated the SNNN, INO,IRNO, CNO type convex TP models for the reduced 3 vertex modelbased convexhull.
86
Chapter 9
TPτModel Based Control DesignMethodology
The original TP model transformation is based on the direct sampling of the given model,therefore, it is capable of transforming the functions thatdescribe the dynamics of the modelinto tensor product form. However, for the same reason, the TP model transformation isnot capable of transforming the original representation inthe ”dynamical sense” - i.e., whileit is capable of transforming parameter-varying functionsappearing in a state-space qLPVstructure into TP functions, it has no effect on the dynamic structure of the state space.
In contrast, the TPτ transformation performs a dynamic re-identification of theoriginalmodel instead of relying on direct sampling. As a result, theTPτ transformation can be usedto diverge from the dynamic structure of the original model,and to re-identify it in a differentstate space form. For example, it is even possible to use the TPτ transformation to re-identifya dynamical model originally described in non-qLPV form andto transform it to qLPVform. A concrete application based on this notion was described in Chapter 8, in which adynamical system that contained a time-delay was transformed into a qLPV representationwithout a time-delay.
Based on the above, it is clear that the TPτ transformation has significant advantagesover the original TP model transformation, and can be used toaddress a broader range ofproblems. The goal of this chapter is to re-formulate the complete TP-model transformationbased design methodology in a more general form, so as to incorporate the possibility to usethe proposed TPτ transformation.
9.1 Steps of the proposed control design strategy
The control design strategy has the following three key steps:
9.1.1 TP type polytopic reconstruction
Assume that we have the following initial model:
87
[
xy
]
= f(x,u,p, τ), (9.1)
where the time-delay is represented byτ . The application of the TPτmodel transformationresults in the following form:
(
x
y
)
=
[(
SZ
⊠n=1
wn(pn)
)
N
⊠n=Z+1
wn(τn)
](
x
u
)
=
(
SN
⊠n=1
wn(p′n)
)(
x
u
)
, (9.2)
wherep′ ∈ Ω andp′ = [p′1 = p1, . . . , p′Z = pZ , p
′Z+1 = τ1, . . . , p
′N = τN−Z ] supposing
thatp ∈ RZ andτ ∈ R
N−Z .In other words, the vector of delaysτ become a concatenated sequence at the end ofp′.
Remark9.1. If there is no information on which elements of the state vector have non-linearmapping effect on the system, we do not know which state variables belong to the parametervectorp. It is worth to assume that each state variables are concerned, and by the repetitionand validation of the transformation this dependency can bedetermined.
9.1.2 Determination of the controller and observer
Controller is searched in the following TP form:
u = −
(
FN
⊠n=1
wn(pn)
)
x. (9.3)
When the state variables are not accessible, observer basedcontrol can be applied. Inthis case, the state observer is searched in the form below:
ˆx = A(p)x+B(p)u+
(
KN
⊠n=1
wn(pn)
)
(y − y) . (9.4)
The LTI feedback gainsFi1,i2,...,iN and observer gainsKi1,i2,...,iN are stored in tensorFandK respectively and they are calledvertex feedback gainsandvertex observer gains.
Remark9.2. As a matter of fact, there are many ways to derive the feedbackgains of thecontroller from the LTI vertex systems of the model. In orderto achieve multi-objectiveoptimization we typically use LMI’s. Thus, we select the LMIcontrol design theoremsaccording to the desired control performance and the solution of the LMI’s will determinethe feedback and observer gains:
S → LMI+optimization parameters→ F ,K (9.5)
9.1.3 Optimization based on convex hull manipulation
According to the work of Grof et al. [174] the shape ofw(p) (that defines the shape of theconvex hull) significantly influences the resulting controlperformance. In the second stepof the TPτmodel transformation, the convex hull manipulation based control performanceoptimization can be performed in the same way as in the original TP model transformation.
88
Chapter 10
TPτ transformation based Control Designfor Impedance Controlled Robot Gripper
Previously, in Chapter 7 the complete apparatus of TPτmodel transformation has been intro-duced for qLPV modelling of systems with feedback delay. In Chapter 8 the transformationmethodology was applied on the impedance controlled robot interaction under feedback de-lay. Chapter 9 proposes a complete design method based on theTPτmodel transformationthat can be applied in a routine-like fashion. In this chapter, summarizing all the previ-ously proposed techniques, I apply this design methodologywith multi-objective LMI basedsynthesis. The herein described observer and controller design process and its validation ac-tually serves as a common usability proof of the previously introduced theorems in a currentengineering problem.
10.1 Specification of the control problem
For the sake of clarity, the whole control problem is resumedhere briefly. In the first subsec-tion, the control problem is recalled, then the required performance of the controller to besynthesized is described. The detailed description of the delayed impedance model can befound in section 3.
10.1.1 Description of the control problem
Let us recall the equation of motion of the impedance model that is embedded in the controlstructure proposed in section 4.
x(t) =Fh(t)
m+
Fc(t)
m−
b
mx(t)−
k
mx(t− τ(t)) (10.1)
Task of the stability preserving controller is to provide theFc(t) control signal that satis-fies the below defined performance requirements of the impedance model during the teleop-erated grasping.
89
10.1.2 Control requirements and constraints
The controller should satisfies the following criteria:
i The supervised impedance model must be stable, sox(t) → 0 ast → ∞, if Fh(t) →const.
ii Stability must be guaranteed with constrained control signal (‖Fc(t)‖2 < µ).
iii Relax the conservativeness of the solution.
Accordingly, we search the state observer and state feedback gains that fulfil the abovelisted requirements.
10.2 Execution of the TPτmodel transformation
For the purpose of completeness, execution of TPτmodel transformation is evoked here, asproposed in Chapter 7.
I tested a wide range of different TP type polytopic models – created by TPτmodel trans-formation – in LMI based control synthesis, but due to the limited extent of this work, onlythe reduced (3 vertices), CNO type TP model is recalled here from Chapter 8, which wasresulted in the best control performance during my investigations. In the rest of this chapter,this type of TP model is applied for controller and observer synthesis.
Let me call back the CNO type reduced TP model of the delayed impedance modelderived in Chapter 8:
[AB]cno31 =
[
9.7873× 102 1.0209× 103 8.6644× 10−1
−9.7957× 102 −1.0200× 103 −8.6591× 10−1
]
(10.2)
[AB]cno32 =
[
9.4943× 102 1.0519× 103 1.0101−9.5043× 102 −1.0509× 103 −1.0091
]
[AB]cno33 =
[
1.0005× 103 9.9947× 102 1.8247× 10−1
−1.0007× 103 −9.9929× 102 −1.8254× 10−1
]
Figure 10.1 shows the weighting functions according to the vertex systems described by(10.2).
90
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
feedback delay [s]
wei
ghts
Figure 10.1: Weighting functions of CNO type convex hull of the reduced TP model with 3vertices
10.3 LMI-based multi-objective controller and observer de-sign
LMI based controller and observer design techniques are readily applicable on the convexTP type polytopic model that is introduced in the previous section. Due to the very activeresearch of this field, the appropriate LMIs can be find in the related literature for the desiredcontrol requirements. In this section, a set of LMI theoremsare introduced which are appliedin the controller and observer design later on. These theorems are quoted from [180].
10.3.1 Asymptotically stable controller and observer
As the state variables of the delayed impedance model are notreadily available for measure-ment, we need to resort to output feedback control that is based on the simultaneous design ofan observer capable of estimating the unavailable state values from the output of the systemand of a state-feedback controller.
In this chapter, I am going to use the observer and controllerstructure as it was introducedin the TPτbased control methodology in chapter 9. In this concrete case, the controller andthe observer are formulated as follows:
91
Fc(t) = u(t) = −
(
3∑
r=1
wr(p)Fr
)
x(t)
ˆx =3∑
r=1
wr(p)Arx+3∑
r=1
wr(p)Bru+
+3∑
r=1
wr(p)Kr(y− y) (10.3)
y =3∑
r=1
wr(p)Crx.
The goal of the controller and observer design is to determine gainsFr andKr (r = 1..3)in such a way that the stability of the overall output-feedback control structure is guaranteed.According to this stability requirement, I selected the following LMIs derived in chapter 4of the book [180]:
Theorem 10.1(Globally and asymptotically stable observer and controller). Assume thepolytopic model (9.2) with controller (9.3) and observer structure (9.4). This output-feedbackcontrol structure is globally and asymptotically stable ifthere exists suchP1 > 0,P2 > 0
andM1,r,N2,r (r = 1, . . . , R) satisfying equations
(10.4)
P1ATr −MT
1,rBTr +ArP1 −BrM1,r < 0,
ATr P2 −CT
r NT2,r +P2Ar −N2,rCr < 0,
P1ATr −MT
1,sBTr +AsP1 −BrM1,s +P1A
Ts −MT
1,rBTs +AsP1 −BsM1,r < 0,
ATr P2 −CT
s NT2,r +P2Ar −N2,rCs +AT
s P2 −CTr N
T2,s +P2As −N2,sCr < 0
for r < s ≤ R, except the pairs(r, s) such that∀p(t) : wr(p(t))ws(p(t)) = 0, and whereM1,r = FrP1 andN2,r = P2Kr. The feedback gains and the observer gains can then beobtained from the solution of the above LMIs asFr = M1,rP
−11 andKr = P−1
2 N2,r.
10.3.2 Constrained control signal
According to the 2nd requirement defined in section 10.1.2 , Iselected the following theoremderived in chapter 3 of the book [180]:
Theorem 10.2(Constraint on the control value). Assume that‖x(0)‖ ≤ φ, wherex(0) isunknown, but the upper boundφ is known. The constraint‖u(t)‖2 ≤ µ is enforced at alltimest ≥ 0 if the LMIs
φ2I ≤ X (10.5)(
X MTr
Mr µ2I
)
≥ 0
hold.
92
Remark10.1. Note, that if LMIs of theorem 10.1 and 10.2 are used simultaneously,X inequation 10.5 is identical toP1 in equation 10.4.
10.3.3 Disturbance rejection
Assume, that the disturbance appears in the model as follows:
x(t) = A(p)x(t) +B(p)u(t) + E(p)v(t), (10.6)
wherev(t) is the disturbance. In the specific case of the impedance model the disturbanceis affect the states through the same input gains as the control signalu(t), thusE(p) = B(p).The disturbance rejection can be realized by minimizingγ subject to
sup‖v(t)‖2 6=0
‖y(t)‖2‖v(t)‖2
≤ γ. (10.7)
According to the simultaneous fulfilment of the requirements defined in section 10.1.2, Iselected the following theorem derived in chapter 3 of the book [180]:
Theorem 10.3(Disturbance rejection). The feedback gainsFr that stabilizes the system andminimizeγ in (10.7) can be obtained by solving the following minimization problem basedon LMIs.
minimizeX,M1,...,Mr
γ2
(
−12XAT
i −MTj B
Ti +AiX−BiMj
+XATj −MT
i BTj +AjX−BjMi
)
−12(Ei + Ej)
12X(Ci +Cj)
T
−12(Ei + Ej)
T γ2I 012(Ci +Cj)X 0 I
> 0 (10.8)
Remark10.2. Note, that if LMIs of theorem 10.1 and 10.3 are used simultaneously,X inequation 10.8 is identical toP1 in equation 10.4.
Remark10.3. Because of the limited length of this Thesis, I restrict thistreatment to thedisclosure of the pure theorem, however this LMI significantly contributes to the tractabilityof the design process.
10.4 Relaxed conservativeness via convex hull manipula-tion
I have derived and tested different controllers and observers to a large variety of convex rep-resentations. I concluded that the CNO type hull resulted innoticeably less conservativesolution than any other type of convex hulls. Therefore, hereafter, only these controller andobserver vertex gains will be discussed. In [181] Grof et al. (including me as co-author) per-formed a more sophisticated convex hull manipulation basedinvestigation, and we demon-strated that further improvements can be achieved on the control performance. This chapter
93
will not go further in details, since this manipulation was aheuristic investigation of theconservativeness only, and rather focused on the theoretical aspects of the convex hull ma-nipulation instead of giving solution to the present control design problem.
10.5 Resulted controller and observer gains
Making the numerical example complete, the controller and observer gains are summarizedhere, which are resulted from the solution of the previouslyintroduced LMIs in theorems10.1, 10.2 and 10.3. In order to satisfy the given performance requirements, all the LMIsof the applied theorems must be solved simultaneously. Thissection shows the solutions ofsuch combinations of LMIs:
• Controller 1 is resulted from 10.1 providing asymptotically stable observer and con-troller.
• Controller 2 is resulted from 10.1 and 10.2 providing asymptotically stable observerand controller with constrained control signal.
• Controller 3 is resulted from 10.1, 10.2 and 10.3 providing asymptotically stable ob-server and controller with constrained control signal and prescribed disturbance rejec-tion performance.
10.5.1 Controller 1
Fcno31 =
[
−4.693366697674861× 101
−7.568178739709092× 101
]
(10.9)
Fcno32 =
[
9.305971161607453× 102
8.498685679030033× 102
]
Fcno33 =
[
−1.473244940427044× 102
−1.406297436375449× 102
]
Kcno31 =
[
−4.038966693651662× 101
2.773932874373593× 103
]
(10.10)
Kcno32 =
[
−4.088474064644633× 101
2.689376377428563× 103
]
Kcno33 =
[
−7.930535944776077× 101
2.866274487959281× 103
]
10.5.2 Controller 2
µ < 200
94
Fcno31 =
[
−1.083065587303841× 102
−1.382708449040355× 102
]
(10.11)
Fcno32 =
[
1.161938393154183× 102
8.602599797438528× 101
]
Fcno33 =
[
−3.882266623580870× 102
−4.064172428857575× 102
]
Kcno31 =
[
4.202738085358277× 102
−3.706627491954388× 102
]
(10.12)
Kcno32 =
[
4.513238288314137× 102
−3.980516180904803× 102
]
Kcno33 =
[
3.913248140080873× 102
−3.451262331158127× 102
]
10.5.3 Controller 3
µ < 200 E = B
Fcno31 =
[
−5.528369213968272× 101
−7.731154196175747× 101
]
(10.13)
Fcno32 =
[
2.347047967210873× 102
2.129777441712156× 102
]
Fcno33 =
[
−2.274116741892123× 102
−2.405310852966929× 102
]
Kcno31 =
[
−3.632338488681152× 101
3.995608638767787× 102
]
(10.14)
Kcno32 =
[
1.1766149247803543.556439446795744× 102
]
Kcno33 =
[
−7.536285829290905× 101
4.457147943312266× 102
]
10.6 Evaluation and Validation of the Control Design
In the previous parts of this chapter, the whole design process was applied on a concretedelayed impedance model. In this section, the control performances with and without theTP type polytopic controller are compared via numerical simulations. Part III validates thewhole design methodology in real experimental environment, investigating the same opera-tion cases which are simulated in this section. In the courseof the validation process a large
95
variety of impedance models and realistic operational situations were tested. In this section,some selected characteristic cases are discussed. Table 10.6 contains two parameter sets thatare used in this section. The stiffness values are chosen according to a sort of real helicalsprings in order to make the experimental validation (in chapter 11) possible under the sameconditions as the simulations executed here.
Table 10.1: Parameter sets applied in the validationDescription Parameter SET A SET B UnitsMass m 1 1 kgviscous damping b 80 120 Ns/mStiffness of the environment k 1921 6315 N/mDelay interval τ 0..0.04 0..0.019 s
For the sake of clarity, the investigated operation conditions and some important conceptsare defined here:
Grasping task Within the simulation a typical grasping process is imitated using pre-recordedoperator inputFh(t).
Environment The mechanical environment of the slave device, which is in physical inter-action with the device during the simulated grasping task. In the simulations andexperiments, helical sprigs were applied as environment.
Fixed contact It means that the slave device is attached to the environmentin such a waythat both positive (press) and negative (pull) forces can arise.
Free space motionIf the slave device is not in fixed contact with the environment, it makesfree space motion before touching the environment in the very first part of the graspingtask.
Constant delay The time-delay is not changing during the grasping task (τ(t) = const.).
Varying delay The time-delay is randomly changing during the grasping task, emulating thereal network delay. This type of emulated delay is often usedfor similar investigationse.g. in [182].
In the following part of this section, a sort of simulation results that show the fulfilmentof the performance requirements (section 10.1.2) are discussed:
• Fixed contact
– Constant time-delay (Figure 10.2)
– Varying time-delay (Figure 10.3)
• Contact after free space motion
– Constant time-delay (Figure 10.4)
96
– Varying time-delay (Figure 10.5)
Each figure contains three sub-plots: The uppermost diagramshows the position responseof the impedance model as the function of time with (continuous line) and without (dashedline) the TPτcontroller. The graph in the middle displays the time-delayin the feedback loop(τ(t)), while the third diagram shows the intervention of the human operator (Fh(t)), theinteraction force in the remote environment (Fe(t)) and the control signal (u(t)). In thesesimulations the action of the human operator (Fh(t)) was emulated using a pre-defined forcecurve.
Figure 10.2 illustrates a case when feedback delay is constant during the emulated grasp-ing action. One can see that when the TPτcontroller is switched off, unfavourable oscillationsoccur around the maximum deformation with both parameter sets. Switching the TPτbasedstability preserving controller on this oscillations disappear. The same behaviour can beobserved in the Figure 10.3 in which the feedback delay is varying.
0 0.5 1 1.5 2 2.5 3 3.5 4−2
0
2
4
6
8x 10
−3
time [s]
posi
tion
[m]
Simulation result: m=1 Kg, k=1921 N/m, b=80 Ns/m
without TPτ controller
with TPτ controller
0 0.5 1 1.5 2 2.5 3 3.5 40
0.01
0.02
0.03
0.04
0.05delay profile
time [s]
Tau
(t)
[s]
0 0.5 1 1.5 2 2.5 3 3.5 4−15
−10
−5
0
5
10
15
time [s]
[N]
u(t)F
h(t)
Fe(t)
(a) parameter set A
0 0.5 1 1.5 2 2.5 3 3.5 4−5
0
5
10
15
20x 10
−4
time [s]
posi
tion
[m]
Simulation result: m=1 Kg, k=6315 N/m, b=120 Ns/m
without TPτ controller
with TPτ controller
0 0.5 1 1.5 2 2.5 3 3.5 40
0.005
0.01
0.015
0.02delay profile
time [s]
Tau
(t)
[s]
0 0.5 1 1.5 2 2.5 3 3.5 4−15
−10
−5
0
5
10
15
time [s]
[N]
u(t)F
h(t)
Fe(t)
(b) parameter set B
Figure 10.2: Simulation with fixed contact under constant time-delay
Figure 10.4 and 10.5 show a more realistic grasping situation, where the jaw of the grip-per is not fastened to the remote object. In this case, the jawcan accelerate during the freespace motion until it touches the grasped object. The highervelocity (the higher kinetic en-ergy) causes longer and oscillations with larger amplitudewhen the controller is switchedoff, but it does not occur with the TPτcontroller.
97
0 0.5 1 1.5 2 2.5 3 3.5 4−2
0
2
4
6
8x 10
−3
time [s]
posi
tion
[m]
Simulation result: m=1 Kg, k=1921 N/m, b=80 Ns/m
without TPτ controller
with TPτ controller
0 0.5 1 1.5 2 2.5 3 3.5 40
0.01
0.02
0.03
0.04
0.05delay profile
time [s]
Tau
(t)
[s]
0 0.5 1 1.5 2 2.5 3 3.5 4−15
−10
−5
0
5
10
15
time [s]
[N]
u(t)F
h(t)
Fe(t)
(a) parameter set A
0 0.5 1 1.5 2 2.5 3 3.5 4−5
0
5
10
15
20x 10
−4
time [s]
posi
tion
[m]
Simulation result: m=1 Kg, k=6315 N/m, b=120 Ns/m
without TPτ controller
with TPτ controller
0 0.5 1 1.5 2 2.5 3 3.5 40
0.01
0.02
0.03
0.04delay profile
time [s]
Tau
(t)
[s]
0 0.5 1 1.5 2 2.5 3 3.5 4−15
−10
−5
0
5
10
15
time [s]
[N]
u(t)F
h(t)
Fe(t)
(b) parameter set B
Figure 10.3: Simulation with fixed contact under varying time-delay
These figures properly illustrate the fulfilment of the performance requirements. Furtherconclusion will be drawn in chapter 11 wherein the before discussed cases are executed inreal laboratory experiments.
98
0 0.5 1 1.5 2 2.5 3 3.5 40
0.005
0.01
0.015
time [s]
posi
tion
[m]
Simulation result: m=1 Kg, k=1921 N/m, b=80 Ns/m
without TPτ controller
with TPτ controller
0 0.5 1 1.5 2 2.5 3 3.5 40
0.01
0.02
0.03
0.04delay profile
time [s]
Tau
(t)
[s]
0 0.5 1 1.5 2 2.5 3 3.5 4−15
−10
−5
0
5
10
15
time [s]
[N]
u(t)F
h(t)
Fe(t)
(a) parameter set A
0 0.5 1 1.5 2 2.5 3 3.5 40
2
4
6
8x 10
−3
time [s]
posi
tion
[m]
Simulation result: m=1 Kg, k=6315 N/m, b=120 Ns/m
without TPτ controller
with TPτ controller
0 0.5 1 1.5 2 2.5 3 3.5 40
0.005
0.01
0.015
0.02delay profile
time [s]
Tau
(t)
[s]
0 0.5 1 1.5 2 2.5 3 3.5 4−15
−10
−5
0
5
10
15
time [s]
[N]
u(t)F
h(t)
Fe(t)
(b) parameter set B
Figure 10.4: Simulation with contact after free space motion under constant time-delay
99
0 0.5 1 1.5 2 2.5 3 3.5 40
0.002
0.004
0.006
0.008
0.01
0.012
time [s]
posi
tion
[m]
Simulation result: m=1 Kg, k=1921 N/m, b=80 Ns/m
without TPτ controller
with TPτ controller
0 0.5 1 1.5 2 2.5 3 3.5 40
0.02
0.04
0.06
0.08delay profile
time [s]
Tau
(t)
[s]
0 0.5 1 1.5 2 2.5 3 3.5 4−15
−10
−5
0
5
10
15
time [s]
[N]
u(t)F
h(t)
Fe(t)
(a) parameter set A
0 0.5 1 1.5 2 2.5 3 3.5 40
2
4
6
8x 10
−3
time [s]
posi
tion
[m]
Simulation result: m=1 Kg, k=6315 N/m, b=120 Ns/m
without TPτ controller
with TPτ controller
0 0.5 1 1.5 2 2.5 3 3.5 40
0.01
0.02
0.03
0.04delay profile
time [s]
Tau
(t)
[s]
0 0.5 1 1.5 2 2.5 3 3.5 4−15
−10
−5
0
5
10
15
time [s]
[N]
u(t)F
h(t)
Fe(t)
(b) parameter set B
Figure 10.5: Simulation with contact after free space motion under varying time-delay
100
10.7 Carry-over the controller into the unstable domain ofthe impedance model
The proposed design methodology gives mathematically provable control performance onlyin the range of delay within the TPτ transformation was applied. As the system identificationmethod which was used in this example is appropriate for stable systems only, the resultedcontroller is strictly satisfies the design requirements (that is actually guaranteed by the ap-plied LMI theorems) up till the critical delay, where the uncontrolled impedance model goesinto an unstable limit-cycle oscillation as it was discussed in the Introduction (Chapter 3). Inthis section, it is shown, how the controller works out of thedesigned domain ofτ (τ > τcrit,τcrit ≅
bk). Figure 10.6 shows a constant (10.6(b)) and a varying (10.6(b)) delay case. The
emulated delay was higher than the critical (0.019s in this case), in both simulations. Theplots show, that the proposed controller keep the impedancemodel stable, while the uncon-trolled system behaves unstable.
0 0.5 1 1.5 2 2.5 3 3.5 4−5
0
5
10
15x 10
−3
time [s]
posi
tion
[m]
Simulation result: m=1 Kg, k=6315 N/m, b=120 Ns/m
without TPτ controller
with TPτ controller
0 0.5 1 1.5 2 2.5 3 3.5 40
0.01
0.02
0.03
0.04
0.05
0.06delay profile
time [s]
Tau
(t)
[s]
0 0.5 1 1.5 2 2.5 3 3.5 4−15
−10
−5
0
5
10
15
time [s]
[N]
u(t)F
h(t)
Fe(t)
(a) Constant delay
0 0.5 1 1.5 2 2.5 3 3.5 40
0.002
0.004
0.006
0.008
0.01
0.012
time [s]
posi
tion
[m]
Simulation result: m=1 Kg, k=6315 N/m, b=120 Ns/m
without TPτ controller
with TPτ controller
0 0.5 1 1.5 2 2.5 3 3.5 40
0.01
0.02
0.03
0.04delay profile
time [s]
Tau
(t)
[s]
0 0.5 1 1.5 2 2.5 3 3.5 4−15
−10
−5
0
5
10
15
time [s]
[N]
u(t)F
h(t)
Fe(t)
(b) Varying delay
Figure 10.6: Control performance in the unstable domain of the impedance model
101
Part III
Experimental Validation
102
Chapter 11
Experimental validation of the results
In this chapter, an experimental proof is given for the theoretical results achieved in thisresearch. The experiments has been conducted on the prototype of a master-slave grippersystem [101] developed in the RESCUER (IST-2003-511492) project of the European Com-mission.
11.1 The experimental setup
The proposed TPτ transformation based control was implemented on a pair of HIRATA (MB-H180-500) DC drive robots at the BME Department of Manufacturing Science and Technol-ogy (Figure 11.1). These robots has two linear axes: The firstaxis (the y-direction) was fixedduring the experiments, while the second axis (the x-direction) was used for the master-slavegrasping. The master robot is equipped with a handle that is used by the operator whilethe slave robot is modified in such a way that the robot’s flangecan be connected to thebase of the robot by a helical springs of various stiffness. The interaction force between themaster robot and the human operator and the contact force on the slave side was measuredby a Tedea–Huntleight Model 355 load cell mounted on both robot’s flange. The drivingsystem of the moving axis consisted of a HIRATA HRM-020-100-A dc servo motor con-nected directly to a ball screw with a20[mm] pitch thread. The robots were controlled by amicrocontroller-based control unit providing the sampling frequency of1[kHz] for the over-all control loop. The control action can be set by the pulse-width modulation of the supplyvoltage of the DC motors.
The so called impedance control based bilateral force-reflecting structure (introduced inchapter 3) extended with the proposed TPτbased stability preserving control is implementedon the controller PC that provides the control signal for both robotic axis. The communica-tion delay between the master and slave robots is emulated bythe controller software in sucha way that allow us to access the delay value in runtime.
103
Figure 11.1: Experimental setup composed by two linear servo axes.
11.2 Experiments
The protocol of the experimental investigation were designed in order to check the validity ofthe simulation results (see 10.6) and to proof the real-world usability of the proposed controlstrategy in the interactive tele-grasping scenario. To fulfil this requirements, the followingtwo types of experiments were conducted:
Grasping with predefined master force: Similarly to the simulations in 10.6, – where atypical grasping process was imitated using pre-recorded operator inputFh(t) –, thesame predefined force curve is used in this type of experiment.
Interactive grasping: In contrast to the previous case, real and interactive tele-grasping isperformed where the master robot is handled by a human operator.
The definition of the different testing conditions (constant delay; varying delay; fixedcontact; free space motion) given in 10.6 are also valid in this chapter.
In addition to the parameter sets discussed in 10.6, a third case is introduced here in table11.2. According to these three different cases, controllerA, B and C are defined respectively.
Table 11.1: Parameter sets applied in the experimental validationDescription Parameter SET A SET B SET C UnitsMass m 1 1 1 kgviscous damping b 80 120 120 Ns/mStiffness of the environment k 1921 6315 8757 N/mDelay interval τ 0..0.04 0..0.019 0..0.013 s
104
Controller A
Fcno31 =
[
−4.1451410002046451× 101
2.7234055824589132× 103
]
Fcno32 =
[
−5.33016774461802092.6401576644398883× 103
]
Fcno33 =
[
−7.9284541740421076× 101
2.8121855760002345× 103
]
Kcno31 =
[
7.7030214525720702−1.6564005429782792× 101
]
Kcno32 =
[
9.5738208561972510× 102
8.8434760605421877× 102
]
Kcno33 =
[
−9.0132987464574626× 101
−8.0043771298835026× 101
]
Controller B
Fcno31 =
[
−6.0199168240745941× 101
4.8966987105197704× 103
]
Fcno32 =
[
−1.1909357552483412× 102
5.0990439389898320× 103
]
Fcno33 =
[
−6.11498326779037444.7169302770939648× 103
]
Kcno31 =
[
−4.3226382410828357× 101
−1.8146815819387905× 10+2
]
Kcno32 =
[
−1.2167326920378878× 102
−2.0294205316014754× 102
]
Kcno33 =
[
6.3662978426229574× 102
4.0284085174146776× 102
]
Controller C
Fcno31 =
[
−1.1879958960215724× 102
5.4247530233644793× 103
]
Fcno32 =
[
−8.69237730489954795.0350985330205722× 103
]
Fcno33 =
[
−6.1861261742423892× 101
5.2199451241994575× 103
]
Kcno31 =
[
7.3708893779761570× 102
7.9053676698540028× 102
]
Kcno32 =
[
1.3935349936580853× 103
1.3305320155723475× 103
]
Kcno33 =
[
8.1318866711222074× 102
8.1417100181479282× 102
]
In the rest of this chapter the experimental results are introduced. Each figures containsthree sub-plots: The uppermost diagram shows the position of the master (xm) and slave(xs) robot as the function of time with dashed lines. In the plotswhere free space motionoccurs black dash-dot line denotes the bound where the gripper jaw touches the environment.The graph in the middle displays the time delay (τ(t)) profile (same for each figures), whilethe third diagram shows the handling force (Fh(t)), the interaction force with the remoteenvironment (Fe(t)) and the control signal (Fc(t)).
11.2.1 Comparative tests
In this subsection, comparative test cases are included regarding the parameter set B. Thedifferent situations reveal the effect of the stabilizing controller and show the performance ofthe supervised system. In these tests, a varying delay profile is used emulating the packagedelay in real network control systems.
105
The first case (Figure 11.2), shows a grasping process with predefined handling forceand fixed contact. In Figure 11.2(a) the stabilising controlis switched off and as one cansee the impedance model, and thus, the master and slave axis immediately begin to oscillatewhen the predefined master force starts to increase. Figure 11.2(b) shows the same case withthe proposed controller that preserves the stability of thewhole system in presence of thevarying communication delay.
0 0.5 1 1.5 2−0.04
−0.02
0
0.02
0.04Measurement: k=6315 N/m b=120 Ns/m
time [s]
posi
tion
[m]
x
m
xs
0 0.5 1 1.5 20
0.01
0.02
0.03
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2−200
−100
0
100
200
time [s]
forc
e [N
]
F
c(t)
Fh(t)
Fe(t)
(a) Without TP controller
0 0.5 1 1.5 20
1
2
3x 10
−3Measurement: k=6315 N/m b=120 Ns/m
time [s]
posi
tion
[m]
x
m
xs
0 0.5 1 1.5 20
0.01
0.02
0.03
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2−20
−10
0
10
20
time [s]
forc
e [N
]
F
c(t)
Fh(t)
Fe(t)
(b) With TP controller
Figure 11.2: Grasping with predefined handling force in fixedcontact
Figure 11.3 represents a similar comparison but with non-fixed contact at the slave side.Without the TP controller the slave jaw rebounds when touching the environment which isa typical unstable behaviour of telemanipulators in contact with stiff environment. Similarlyto the fixed contact case, the TP controller managed to maintain the stability of the systemwhen the slave jaw contacts the environment after free spacemotion.
106
0 0.5 1 1.5 2−0.04
−0.02
0
0.02
0.04Measurement: k=6315 N/m b=120 Ns/m
time [s]
posi
tion
[m]
x
m
xs
0 0.5 1 1.5 20
0.01
0.02
0.03
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2−50
0
50
100
150
time [s]
forc
e [N
]
F
c(t)
Fh(t)
Fe(t)
(a) Without TP controller
0 0.5 1 1.5 2−20
−15
−10
−5
0
5x 10
−3Measurement: k=6315 N/m b=120 Ns/m
time [s]
posi
tion
[m]
x
m
xs
0 0.5 1 1.5 20
0.01
0.02
0.03
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2−10
−5
0
5
10
time [s]
forc
e [N
]
F
c(t)
Fh(t)
Fe(t)
(b) With TP controller
Figure 11.3: Grasping with predefined handling force and free space motion before contact
In figure 11.4, the results are shown that obtained for interactive grasping. Figure 11.4(a)shows the situation where the slave jaw is not fixed to the helical spring, while Figure 11.4(a)depicts the fixed contact case. Like in the previous test situations, the system remains stablein both cases all along the registered process although the large jump in the delay profile at2s that causes some minor oscillations.
107
0 0.5 1 1.5 2 2.5−0.04
−0.02
0
0.02
0.04Measurement: k=6315 N/m b=120 Ns/m
time [s]
posi
tion
[m]
x
m
xs
0 0.5 1 1.5 2 2.50
0.01
0.02
0.03
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−200
−100
0
100
200
time [s]
forc
e [N
]
F
c(t)
Fh(t)
Fe(t)
(a) With freespace motion
0 0.5 1 1.5 2 2.5−0.03
−0.02
−0.01
0
0.01Measurement: k=6315 N/m b=120 Ns/m
time [s]
posi
tion
[m]
x
m
xs
0 0.5 1 1.5 2 2.50
0.01
0.02
0.03
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−200
−100
0
100
200
time [s]
forc
e [N
]
F
c(t)
Fh(t)
Fe(t)
(b) Fixed contact
Figure 11.4: Interactive grasping
11.2.2 Further test cases
In this subsection, different test cases are introduced regarding the A, B and C parametersets highlighting various operating conditions. Commentsof the comparative tests fromthe previous subsections can be analogously interpreted here as well, therefore, these arepresented as supplement without further explanation.
The figures are organized in the following structure:
• Behaviour of the system without the proposed stability preserving controller
• Grasping with predefined master force
• Interactive grasping
Remark11.1. If the TPτbased controller is switched off and the feedback delay is larger thenthe critical value (τ > τcrit) (Figure 11.5), the system behaves unstable even without anyinteraction with the user or the environment.
108
0 0.5 1 1.5 2 2.5−0.01
0
0.01
0.02Measurement: k=1921 N/m b=80 Ns/m
time [s]
posi
tion
[m]
x
m
xs
0 0.5 1 1.5 2 2.50
0.02
0.04
0.06
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−40
−20
0
20
time [s]
forc
e [N
]
Fc(t)
Fh(t)
Fe(t)
Figure 11.5: Unstable behaviour of the tele-grasping system without the TPτcontroller
Grasping with predefined master force
109
0 0.5 1 1.5 2 2.5−2
0
2
4
6
8x 10
−3Measurement: k=1921 N/m b=80 Ns/m
time [s]
posi
tion
[m]
xm
xs
0 0.5 1 1.5 2 2.50
0.01
0.02
0.03
0.04
0.05τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−20
−10
0
10
20
time [s]
forc
e [N
]
Fc(t)
Fh(t)
Fe(t)
Figure 11.6: Experimental validation: parameter set A, constant delay (τ = 40ms), prede-finedFh(t)
110
0 0.5 1 1.5 2 2.5−2
0
2
4
6
8x 10
−3Measurement: k=1921 N/m b=80 Ns/m
time [s]
posi
tion
[m]
xm
xs
0 0.5 1 1.5 2 2.50
0.02
0.04
0.06
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−20
−10
0
10
20
time [s]
forc
e [N
]
Fc(t)
Fh(t)
Fe(t)
Figure 11.7: Experimental validation: parameter set A, constant delay (τ = 60ms), prede-finedFh(t)
111
0 0.5 1 1.5 2 2.5−2
0
2
4
6
8x 10
−3Measurement: k=1921 N/m b=80 Ns/m
time [s]
posi
tion
[m]
xm
xs
0 0.5 1 1.5 2 2.50
0.02
0.04
0.06
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−20
−10
0
10
20
time [s]
forc
e [N
]
Fc(t)
Fh(t)
Fe(t)
Figure 11.8: Experimental validation: parameter set A, varying delay (τ = 40ms), prede-finedFh(t)
112
0 0.5 1 1.5 2 2.5−5
0
5
10
15
20x 10
−4Measurement: k=6315 N/m b=120 Ns/m
time [s]
posi
tion
[m]
xm
xs
0 0.5 1 1.5 2 2.50
0.01
0.02
0.03
0.04τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−20
−10
0
10
20
time [s]
forc
e [N
]
Fc(t)
Fh(t)
Fe(t)
Figure 11.9: Experimental validation: parameter set B, constant delay (τ = 30ms), prede-finedFh(t)
113
0 0.5 1 1.5 2 2.5−1
0
1
2
3x 10
−3Measurement: k=6315 N/m b=120 Ns/m
time [s]
posi
tion
[m]
xm
xs
0 0.5 1 1.5 2 2.50
0.01
0.02
0.03
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−20
−10
0
10
20
time [s]
forc
e [N
]
Fc(t)
Fh(t)
Fe(t)
Figure 11.10: Experimental validation: parameter set B, varying delay (τ = 20ms), prede-finedFh(t)
114
0 0.5 1 1.5 2 2.5−5
0
5
10
15x 10
−4Measurement: k=8757 N/m b=120 Ns/m
time [s]
posi
tion
[m]
xm
xs
0 0.5 1 1.5 2 2.50
0.01
0.02
0.03τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−20
−10
0
10
20
time [s]
forc
e [N
]
Fc(t)
Fh(t)
Fe(t)
Figure 11.11: Experimental validation: parameter set C, constant delay (τ = 20ms), prede-finedFh(t)
115
0 0.5 1 1.5 2 2.5−5
0
5
10
15
20x 10
−4Measurement: k=8757 N/m b=120 Ns/m
time [s]
posi
tion
[m]
xm
xs
0 0.5 1 1.5 2 2.50
0.005
0.01
0.015
0.02
0.025
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−20
−10
0
10
20
time [s]
forc
e [N
]
Fc(t)
Fh(t)
Fe(t)
Figure 11.12: Experimental validation: parameter set C, varying delay (τ = 10ms), prede-finedFh(t)
116
Interactive grasping
0 0.5 1 1.5 2 2.5−20
−15
−10
−5
0
5x 10
−3Measurement: k=1921 N/m b=80 Ns/m
time [s]
posi
tion
[m]
xm
xs
0 0.5 1 1.5 2 2.50
0.01
0.02
0.03
0.04
0.05τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−40
−20
0
20
40
time [s]
forc
e [N
]
Fc(t)
Fh(t)
Fe(t)
Figure 11.13: Experimental validation: parameter set A, constant delay (τ = 40ms), inter-active grasping
117
0 0.5 1 1.5 2 2.5−20
−15
−10
−5
0
5x 10
−3Measurement: k=6315 N/m b=120 Ns/m
time [s]
posi
tion
[m]
xm
xs
0 0.5 1 1.5 2 2.50
0.01
0.02
0.03
0.04τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−100
−50
0
50
100
time [s]
forc
e [N
]
Fc(t)
Fh(t)
Fe(t)
Figure 11.14: Experimental validation: parameter set B, constant delay (τ = 30ms), interac-tive grasping
118
0 0.5 1 1.5 2 2.5−15
−10
−5
0
5x 10
−3Measurement: k=6315 N/m b=120 Ns/m
time [s]
posi
tion
[m]
xm
xs
0 0.5 1 1.5 2 2.50
0.01
0.02
0.03
τ profile
time [s]
τ (t
) [s
]
0 0.5 1 1.5 2 2.5−100
−50
0
50
100
time [s]
forc
e [N
]
Fc(t)
Fh(t)
Fe(t)
Figure 11.15: Experimental validation: parameter set B, varying delay (τ = 15ms), interac-tive grasping
119
The here exposed experimental results show the fulfilment ofthe stability requirement inthe presence of varying time delay. The observation, that the tele-grasping system remainedstable in each tested situations even when the delay profile exceeds the maximum of the con-sidered interval shows the robustness of the controller design and reconfirms the statementsof section 10.7.
120
Part IV
CogInfoCom-based approach asalternative solution for force feedback
121
Chapter 12
Inter-cognitive Sensor-bridging inTelemanipulation
This chapter aims to develop an experimental apparatus for the investigation of CogInfoComtheories in haptics and force feedback enhanced tele-grasping tasks and to conduct pilotexperiments using the newly developed test environments. The motivation of this researchwas discussed in Chapter 6, where the problem of force feedback was reformulated underthe concepts of CogInfoCom.
In this chapter, vibrotactile feedback strategies are considered as a possible way to trans-mit the percept of tactile and grasp forces in human-machineinteraction. Two experimentalapplications will be introduced: The first one is a simulatedtelemanipulation scenario wherethe operator can grasp virtual remote objects using a masterdevice and receive visual feed-back on a screen, and grasp force feedback via a vibrotactileglove. For telemanipulation,many types of devices are available on the market providing force feedback in its nativemechanical way however, the time-delay in the infocommunication channels limits their ap-plicability due to the control stability issues.
In the second application, a human operator can interact in the virtual space as an avatarmoved by skeleton-based motion capture device while she/heis provided with vibrotactilestimulation. A cognitive adapter computes the vibrotactile stimulus according to the inter-action forces between the operators hand and the manipulated objects. Sensory informationmust be generated synthetically from the physical model of the virtual environment. Thepresented experimental software is implemented on the VirCA1 platform [183].
The proposed experimental setups provide a platform to develop and investigate cognitiveinfocommunications based solutions [120]. From the aspectof Cognitive Infocommunica-tions - tactile, force and haptic feedback strategies wherethe original sensory information ismapped to alternative sensory channels - it is calledsensor-bridging. In this abstraction, thecommunication takes place between the manipulation process and the human operator so thelevels of intelligence are different. Accordingly, the communication is inter-cognitive.
1VirCA - Vir tual CollaborationArena, developed in MTA SZTAKI (Computer and Automation ResearchInstitute, Hungarian Academy of Sciences). VirCA is free touse for academic purposes and available online atwww.virca.hu.
122
Regarding the similarities of application areas discussedin this chapter, the existence ofthe inter-cognitive sensor-bridging type cognitive adapter is investigated that gives solutionfor both application.
12.1 Devices
In this section, a vibrotactile glove and a master device fortelemanipulative grasping areintroduced. Both devices have been developed by the author and his associates in MTASZTAKI.
12.1.1 Vibrotactile glove
For the sensor-bridging purpose, a vibrotactile glove has been developed (Figure 12.1). Inthis glove, one shaftless vibration motor is placed on each finger endings on the nail sideleaving the fingertips free and let the user to grasp the master device comfortably.
Figure 12.1: Wireless vibrotactile glove with five shaftless vibration motors
The glove is equipped with a controller board capable of wireless communication witha host PC or embedded system using bluetooth serial connection. Each vibration motorscan be controlled independently with pulse width modulated(PWM) signal. The PWM dutycycle can be set in 256 steps with serial commands from the host device. A C# API and anRT-Middleware component are available for integrating of the glove into newly developed orexisting systems.
12.1.2 Master device
The master gripper was developed from a commercially available (SCHUNK PG-70) servo-electrical, two-fingered, parallel jaw robot gripper (Figure 12.2). The below listed propertiesof the gripper make it well suited to this application.
123
Stroke per finger 35[mm]Encoder resolution 0.001[mm]Max. gripping force 200[N ]Communication RS-232, CAN-Bus, Profibus DP
The implementation of the real haptic feedback requires relatively high sampling frequency,so CAN-Bus communication at1MBaud was selected. The gripper internally providesclosed-loop velocity and position control and open-loop direct PWM control. The imple-mented communication is running at approximately500Hz of command frequency. Thesampling period is varying due to the non-real-time characteristics of the MS Windows oper-ating system, but during the experiments no malfunction wasobserved. Because of the largegear-ratio, the gripper is non back-drivable. It means thatwithout active control the operatorcannot move the jaws of the master device, so it is consideredas admittance display [184].To make this gripper applicable as a master device, additional force sensors are necessary.Binocular type load cells are integrated with the fingers of the gripper. Load cells are drivenby a commercial instrumentation amplifier circuit providing 12 bit A/D conversion and com-munication via RS-232 line at 115200 baud rate (electronicswas made by Tenzi Ltd.). Rawforce data is sent to the host PC at the rate of2000Hz.
Figure 12.2: The force feedback capable master gripper
The master device has two operation modes: force feedback and the a position inputmode without force reflection. Details about the implemented bilateral control scheme canbe found in [101]. In position input mode, the master device is controlled in a feed-forwardmanner. Due to the non-backdrivability of the gripper, its embedded drive system is operatingin open-loop control mode and gets PWM duty value from the host computer according tothe following equation:
PWM duty = min(max(0,f1 + f2
2G), 1)), (12.1)
wheref1 and f2 refers to the measured force on the two fingers respectively and G is aproportional gain. The value ofG is tuned so that the master device does not show highresistance against the operator hands, but does not allow sudden or accidental movements.
124
m1
m2
m3
(b) inhomogeneous linear modulation
f
m1
mn
(a) homogeneous linear modulation
(c) homogeneous pulse frequency modulation
m1
m2
mn
(e) homogeneous pulse strength modulation
m1
m2
mn
(f) inhomogeneous pulse strength modulation
m1
m2
m3
m1
m2
mn
(d) homogeneous pulse width modulation
Figure 12.3: Various stimulation strategies
12.2 The proposed cognitive adapter
The cognitive adapter transforms a force-like quantity into a vibration pattern that is exe-cuted on the vibrotactile glove. In our investigation, grasping force in telemanipulation andelastic force in virtual manipulation are considered as input data. A vibration pattern meansthe way of control of the vibration motors according to a given force value. In simple cases,such a rule can be formulated as a scalar-vector functionp = g(f) , where the elementsof p are real numbers between zero and one, representing the vibration intensity of eachmotors. More complicated methods can be formed, if the motors are controlled using peri-odic signals because the characterization of these patterns requires more parameters and thusmore complex stimulus can be generated. Supposed that the vibrotactile stimulator deviceis equipped with vibro-actuators in discrete positions (e.g. at the fingertips), different typeof stimulation strategies can be distinguished. Figure 12.3 schematically illustrates variouspossible control approaches: The upper plot of diagram (a) shows the measured or computedforce f as the function of time. Other plots in the figure displays themomentary intensityof the vibro-actuators (m1, m2, m3,..., mn) according to thef signal. In the experiments,homogeneous and inhomogeneous amplitude modulated strategies (Figure 12.3/a,b) are im-plemented, which are introduced in the following subsections.
12.2.1 Homogeneous linear vibration function
As an obvious approach, the homogeneous linear mapping was implemented (Figure 12.3/a).In this case, all five vibro-actuators are controlled in the same way in a linear function of theinput force.
125
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
Mot
or P
WM
dut
y cy
cle
[%]
Force [N]
Figure 12.4: Homogeneous linear vibration function
Figure 12.4 shows the functionp(f) = g · f whereg is a properly selected proportionalgain. Asp saturates at the force valuef = 1/g, the linear section should be arranged intothe typical force range of the application.
12.2.2 Inhomogeneous radiating vibration function
More sophisticated mapping can be defined with the inhomogeneous stimulation (Figure12.3/b). In this mode, the vibration is radiating from the thumb to the little finger by theincreasing force value. Figure 12.5 shows how the motors arecontrolled in the function
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
Mot
or P
WM
dut
y cy
cle
[%]
Force [N]
Thumb & Index fingerMiddle fingerRing fingerLittle finger
Figure 12.5: Inhomogeneous radiating vibration function
of input force. The thumb and index fingers stimulated in the same way and after this twomotors reach the100% PWM duty, further fingers come in to action. In this way the dynamicrange of the stimulus is much higher then in the homogeneous case.
126
12.3 Applications
12.3.1 Telemanipulative Grasping
In this application, inter-cognitive sensor-bridging is applied to display force information tothe user in a telemanipulative grasping task. The testbed incorporates the force feedbackcapable master device, a simulated remote environment and the vibrotactile glove.
In the course of experiment, human subject uses the master device to grasp a simulatedremote object with a certain force. During the task, subjectis provided with visual feedbackregarding the vision as primary sensory modality. Optionally, real haptic feedback, sensor-bridging type feedback or both can be enabled.
Figure 12.6: Telemanipulative grasping experiment to investigate sensor-bridging type feed-back methods
Figure 12.6 shows the experimental setup. A PC is serving as the controller of the ac-tive devices (master gripper, vibrotactile glove) and provides the simulated virtual remoteenvironment. Two type of remote objects can be selected in the software: a helical springwith adjustable linear stiffness and a breathing chicken. The chicken’s size is changing pe-riodically due to a breathing motion which has time varying periodicity in order to make itmore realistic. The nonrigid character of the chicken is defined by its stiffness while otherdetails of mechanical behavior like viscous damping and stress relaxation were neglected.During the tests, several system parameters like grasping force and gripper finger positioncan be recorded. This setup makes possible the comparative investigation of native andsensor-bridging type force feedback in telemanipulative grasping tasks. An earlier versionof this setup was applied for the experimental investigation of peripheral vision-based grasp-ing force feedback [185].
12.3.2 Interaction in Virtual Environment
In this experimental scenario, a human operator representshimself in an immersive virtualenvironment and manipulates objects using his humanoid avatar. The operator can receivevibrotactile feedback according to the interaction between the avatar and the manipulatedobjects. VirCA and the immersive 3D visualisation laboratory in MTA SZTAKI is appliedas background infrastructure for this setup.
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A sort of motion capture devices (MoCap suites, MS KinectTM) can be used to track themotion of the operator nearly in real-time. A 3 by 3 meters area is available for the user tomanage his movements but the virtual room can be any size. Three walls of the room is 3Dprojected hence the users have to wear passive 3D glasses. Operator’s motion is reproducedby a humanoid avatar in the virtual space thus the user can observe his own activity from anout-of-body perspective [186]. The avatar has the look and skeleton structure of the widelyused NAO humanoid robot. An abstract contact sensor device is attached onto the robot’shand in order to detect the contact with other objects. This sensor is capable of providing thedescription of the contacted object and the contact state. Feedback to the user is computed bythe cognitive adapter according to this description. User receives the vibrotactile feedbackvia the vibrotactile glove on his right hand. Figure 12.7 shows the operator and his avatar inthe virtual arena.
Figure 12.7: Manipulation in the immersive 3D Virtual reality
Technically, the application is implemented as an assemblyof RT-Middleware compo-nents (RTCs) involving VirCA. Components and connections are shown in Figure 12.7.VirCA core component serves as the central management module and provides the 3D virtualenvironment. Other components are stringed to VirCA via RT-Middleware connections.
The first branch consist of an RTC for the motion capture device (Measurand) that isconnected to an RTC which performs transformation between the MoCap suit data type andthe input format of the robot and a Cyber Device that represents the NAO humanoid robotin VirCA. In the next branch, the abstract sensor device is connected directly to VirCA.The vibrotactile glove gets force information from the sensor. Environmental objects canbe connected to the virtual room. They are able to react with the contact sensor and handlesurface and stiffness information characterizing their physical behavior. Over the vibrotactileforce feedback, the cognitive adapter can transform tactile information into auditory icons[187]. This scenario illustrates a possible application ofinter-cognitive sensor-bridging toachieve effective haptic feedback avoiding control instability using low-cost and lightweightdevices.
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Figure 12.8: VirCA assembly for manipulation in a shared virtual space
12.4 Experimental Study
A pilot experiment has been done to investigate the vibrotactile feedback in interaction withvirtual objects in immersive virtual reality. Motivation of the experimental study is twofold.Firstly, to gain initial experience in this quite new area ofhuman computer interaction. Andto reveal, how many grade of hardness/stiffness can be distinguished based on the vibrotactilestimulus that displays the interaction force.
12.4.1 Participants
Ten people has been participated in the experiment. Their age varied between 24-42 yearsand three of them were female. All the participants are well experienced computer usersand half of them also play 3D video games. The vibrotactile glove that was used in theexperiment is only available in one size thus it was not fit equally for everyone. Participantswith smaller palm had problems with improper alignment of vibro-actuators on the fingers.
12.4.2 Method
The experimental environment was arranged similarly as it is described in subsection 12.3.2.The only difference is that the motion of the participants was tracked by a KinectTMsensor sothey did not have to wear any disturbing gadgets. Three type of stimulation strategies havebeen examined: first one was the linear homogeneous vibration function (Figure 12.4). Thesecond strategy was the inhomogeneous radiating vibrationfunction (Figure 12.5). The thirdwas a modified version of the second wherein all fingers treated individually.
Each subject passed two tests with each vibration modes. In both tests, stiffness/hardnessof visually uniform objects should have been discriminated. In the first test, a given stiffnessrange was equally sectioned in five discrete values, while inthe second one only three gradeswere assigned on the same range. This means, that the first (softest) and the last (hardest)grades meant the same hardness, while the medium (2) on the second scale was equal to thethird grade of the first scale. The two scales are illustratedin fugure 12.9.
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Figure 12.9: Stiffness grades in the first and second tests
The gain parameter of the first strategy was set so that the covered stiffness range tobe between the lower and upper bound of the linear part of the vibration function (Figure12.4). The second strategy (Figure 12.5) was also tuned in such a way that the interactionforces vary in the valid input range. In the third strategy, the parameters and the stiffnessof the objects adjusted to set the saturation point of the fingers coinciding with the totaloverlapping of the test object and the virtual touch sensor.
The test process was the following: In the first part of the test - yet before the actualprobes -, participants practised 5-10 minutes on sample objects, which were five uniformboxes in a row with increasing stiffness in five grades. Afterthe practice, participants had tojudge the stiffness of an unknown object (test object) five times consecutively. Subjects wereasked to tell the number of the stiffness grade (1-5) that they felt. In the five consecutive test,the stiffness of the test object was selected randomly in order to exclude any possibility toeduce the subsequent answer. In the second part, the same process (5-10 minutes practice,5 probes) was executed with three grades. Figure 12.10 showsa moment when a subjectpractices on the sample objects.
Figure 12.10: Experimental setup in the immersive virtual reality
12.4.3 Results
The before described test has been completed by the ten participants thus overall 300 testevents are registered. In this subsection, these events areevaluated using standard statistical
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methods. Results are presented in figure 12.11 by box plots for each stimulation strategies,separately for the 3-grade and 5-grade cases. In the diagrams, six numerical quantities arepresented: The thick horizontal line shows the median, the lower and upper bound of theboxes means the lower and upper quartile respectively, horizontal lines connected to theboxes with dashed lines shows the minimum and maximum valuesof guesses , while thesmall crosses shows the judges that are considered as outliers. Numbers in the boxes meanthe percentage of correct guesses.
1
1.5
2
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3
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user
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ction
94%
45%
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(a) Strategy 1 / three-grade test
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(c) Strategy 2 / three-grade test
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(f) Strategy 3 / five-grade test
Figure 12.11: Experimental comparison of three stimulation strategies
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Diagrams show, that the ratio of correct guesses and the variance is improving in thesecond and also in the third stimulation strategy compared to the first one. In the tests withthe first strategy, discrimination of three grades (Figure 12.11/a) were already ambiguouseven though the first grade was recognized in 94%. In the five-grade test of the first strategy(Figure 12.11/b), the uncertainty was even larger as the median values illustrate. The pictureis much better in case of the second stimulation strategy (Figure 12.11/c,d) as the mediansare lying on the expected grade. We got the best results with the third vibration function(Figure 12.11/e,f). Both 3-grade and 5-grade tests passed with low number of mistakes. Byeach strategy, discrimination at lower stiffnesses (1,2) was better than at higher (3,4,5) gradeswhat shows that the discrimination is better at lower vibration intensities (and frequencies).In conclusion we can state, that the linear homogeneous vibration function (strategy 1) isnot applicable to display multi-grade information precisely, using vibrotactile gloves withshaftless vibration motors where the amplitude and the frequency of the vibration are coupled.Strategies that implement different stimulation for each fingers such as strategy 2 and 3 aremore applicable to transmit such information to the user. The investigation show that with aproperly tuned inhomogeneous radiating vibration function, five different stiffness grade canbe precisely discriminated.
12.5 Summary of the chapter
In this chapter, sensor-bridging type force feedback scheme has been discussed with theconceptual background of cognitive infocommunications. Acomplete experimental infras-tructure is developed for the investigation of vibrotactile force feedback strategies in twocurrent applications: The first setup provides a test environment for telemanipulative grasp-ing where the operator can grasp a simulated remote object using a master device and receiveforce feedback via a vibrotactile glove. The second setup implements vibrotactile feedbackfor immersive 3D environments serving the user with the sense of interaction forces. Initialtests show promising perspective in both application field however the improvement of thetele-grasping setup with real remote side is reasonable. This chapter introduces a pilot exper-iment in immersive 3D virtual world conducted with ten participants and reports substantialobservations on the usability of different vibrotactile feedback strategies. As the main conclu-sion of the experiment, it is shown that five stiffness gradescan be distinctly discriminatedwith properly tuned vibrotactile feedback produced by a vibrotactile glove equipped withshaftless vibration motors.
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Part V
Conclusion
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Chapter 13
Scientific results
In this chapter, I summarize my scientific results in light ofthe goals laid down at the begin-ning of the research work.
13.1 TPτmodel transformation
I developed the theoretical foundations and the numerical implementation of the TPτmodeltransformation. I proved that the TPτmodel transformation is capable of transforming time-delay systems into non time-delay TP-type polytopic qLPV forms, where time-delay appearsas an external parameter only. I also proved that the TPτ transformation preserves severaladvantageous properties of the original TP model transformation, notably:
i Enables a directly controllable trade-off between the accuracy and complexity of theresulting model.
ii The ability to reconstruct the HOSVD-based canonical form.
iii The possibility to manipulate the properties of the convex hull defined by the resultingpolytop through matrix operations executed on the core tensor of the TP structure andon the single-variable weighting functions.
iv The LMI-based control methods are directly applicable to the resulting TP-type poly-topic qLPV form.
Thus, in a way analogous to the original TP model transformation based control method-ology, I proposed the TPτmodel transformation based control design methodology that isconceptually consistent with the modern polytopic qLPV andLMI-based control theory andhence extends its applicability onto the class of control problems comprising so-called inter-nal time delays that occurs in the following cases:
• Time Delay take place in the feedback loop
• Systems with input/output delay are connected together
• Transforming a MIMO system with transfer delays into state space model
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13.2 Polytopic reconstruction of the mass-damper impedancemodel with retarded elastic force
I derived the non time-delayed TP type polytopic LPV forms ofthe mass-spring-damper me-chanical impedance model with retarded elastic force component. Using the TPτmodel trans-formation I derived an exact HOSVD-based canonical form of this model, which consistedof a 6-component vertex systems, and several further characteristic convex representations(i.e. CNO, IRNO, SNNN). In order to obtain practically viable trade-offs between the com-plexity and accuracy of these models, I determined the relationship between higher-ordersingular values of the system and the different types of convex representations, and moreimportantly how these factors quantitatively affect modelaccuracy. Based on these results,I derived various reduced TP type polytopic forms of the original, time-delay impedancemodel. I demonstrated via numerical simulations and laboratory measurements that for prac-tical stabilization purposes, a reduced model consisting of 3 vertices can be used to obtain acontroller and observer of adequate performance.
13.3 Control design for impedance controlled tele-grasping
As a practical specialization of the TPτmodel transformation based control design methodol-ogy, I proposed a numerical, non-heuristic control design method for a common impedancemodel based bilateral telemanipulation task with known, varying time-delay. I applied theproposed control design method to stabilize a 1-DoF servo-electric master-slave telemanip-ulation device with parallel jaws for force feedback enhanced remote grasping tasks. Ivalidated the effectiveness of the control design method under laboratory conditions on atele-grasping system through a series of experiments with varying stiffness in the remoteenvironment, and varying time-delay profiles.
13.4 Definition of the force feedback task as inter-cognitivesensor-bridging channel
Based on the conceptual framework and characteristic methodologies of cognitive infocom-munications (a newly emerging inter-disciplinary research field), I reformulated the forcefeedback problem as an inter-cognitive and sensor-bridging application in which vibrotac-tile channels are used to convey force feedback perception.This replacement of the forcerendering by vibration leads to the decoupling of the position and force feedback, hence,opening the closed control loop and considerably relaxes the stabilization problem of forcefeedback enhanced tele-grasping. Based on this work, I re-contextualized the problem offorce feedback with time-delay in a way that is consistent with the cognitive infocommuni-cations perspective.
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Chapter 14
Concluding remarks and futureperspectives
In this chapter, I briefly revisit some issues related to the industrial applicability of the resultsand show further practical and theoretical perspectives ofthe proposed methods that supplyappealing research topics for the coming years.
14.1 Extension of the control design to the unstable region
The reidentification-based discretisation is a key point inthe TPτmodel transformation baseddesign methodology. The applied system identification method determines the achievableparameter range. Open loop identification techniques enable to investigate stable systemsonly. There also exist closed loop identification methods capable of handling unstable plantstogether with a suitable control algorithm that makes the closed loop stable. Using suchan approach, it is possible to form a recursive identification process that can be applied togather good model of a given system in a wide range of operation parameters even in theregion where the plant is unstable.
A mathematically less precise, but promising engineering approach is based on the extrap-olation of the results gained from the reidentification in the stable domain. This regressiveapproach should be used with a certain circumspection, because the dynamic behaviour ofthe so determined model can considerably differ from the real system and thus might resultsin ineligible controller.
14.2 Determination of the momentary time delay
In the so called bounding box, or convex hull design approaches – like the LMI design basedon the TPτmodel transformation –, the resulted gain scheduling controller guarantees thestability over the whole parameter space. However, the precise knowledge of the actualvalues of the scheduling variables helps to assure a less conservative operation, or on theopposite, helps to increase the robustness against parameter uncertainties and hence providea better overall performance.
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In the concrete example of the time-delayed bilateral telemanipulation, the monitoringof the varying package delay is essential to attain the best possible transparency. Systemintegrators have several viable approaches to measure the time elapsed between the sendingand the receiving of a data package.
One option is the token-based communication, where the master sends an initial packagewith his own time stamp and a unique package ID then the slave send back the package withthe measured force data. When the package get back to the master, the round-trip time (RTT)can be computed [188]. This approach assumes symmetric delay.
An other possibility of measuring the delay is to synchronize the master and slave con-troller’s clock. There exist several time synchronisationprotocols (e.g. IEEE 1588) whichenable the precise synchronisation of the computer’s internal clock via the Internet [189, 190].The GPS service also can be utilized to obtain the exact time with suitable precision at boththe master and slave site [191, 192]. These strategies allowus to estimate the momentary de-lay with a1− 10[ms] accuracy which is suitable for the discussed tele-graspingapplication.
14.3 Concerning the characteristics of time-delay
In the design methodologies, where the model of a plant is determined experimentally via sys-tem identification, the derivatives of a given variable (rate of change) is often neglected. Forinstance, in the identification of aeroelastic wing dynamics via wind-tunnel experiments, thefree stream velocity (relative wind speed) is set to discrete constant values, while the accel-eration is not taken into account [193]. Similarly, I have not considered theτ(t)′ and higherderivatives in the examples of this Thesis and the experimental validation approved the suit-ability of this simplification. Nevertheless, the TPτ transformation-based design frameworkallows to take further dynamical properties of the time-delay into consideration at the ex-pense of the model complexity. In turn, one should balance along a certain trade-off in theindividual engineering problems. It should be addressed infurther investigation, how theprobabilistic properties of the varying time-delay [194] can be involved into the schedulingvector.
14.4 Taking the environmental stiffness into account
For the sake of compactness, the experiments in this Thesis are focused on the rather tractable,one variable cases, when only the time-delay are consideredas scheduling parameter but theTPτdesign framework allows to involve multiple parameters. Ina tele-grasping scenario,when the mechanical properties of the remote environment isunknown (but estimable), itis an evident extension to add the apparent stiffness to the scheduling variables as the mostimportant model parameter. For this extension, we need to obtain the representation of thedelayed dynamical system in TP type polytopic form, whereinthe time delayτ and the en-vironmental stiffnessk are the model parameters. Based on the TPτmodel transformation,my student Jozsef Kuti has developed the extended model in his MSc thesis [178] in thefollowing form:
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S(k, ϑ) = SD2
⊠n=1
wn(pn) =
Rk∑
ik=1
Rϑ∑
iϑ=1
wik(k)wiϑ(ϑ)Sik,iϑ (14.1)
,wherek andϑ denotes the stiffness and dimensionless time-delay respectively, whileRk andRϑ equals the ranks of theSD tensor. Using the two-parameter model, it has been managedto design a time-delay, and environmental stiffness dependent TP type gain scheduling con-troller which were evaluated experimentally with convincing results. The real-time stiffnessestimation method proposed by Grioli and Bicchi [195] were successfully applied in the ex-periments. The comprehensive documentation of the design and the experimental evaluationcan be found in [179, 177, 178].
14.5 Pilot implementation
Concerning the industrial-grade justification of the theoretical results, a complete bilateral,Internet-based teleoperated robot system is under implementation in the course of a project(has been started in 2013) that is inspired by the scientific results of this Thesis and fundedby MTA SZTAKI. In this system, the operator and the remote workplace will be locatedat different sites of the Institute. In the planned master arrangement, the human operatorcan remote control an industrial robot arm while receiving multi-modal feedback from theremote environment. The arm motion will be controlled utilizing a 6 DoF input device,while the remote grasping function will be realized in a single DoF force reflecting master-slave gripper. Besides the auditory, visual and the kinesthetic force displays, the VirCA(Virtual Collaboration Arena) platform and the 3D CAVE of MTA SZTAKI will be involvedto provide an interactive and immersive replication of the remote environment to enhancethe quality of telepresence. This experimental arrangement will also serve as demonstrationplatform fostering the industrial dissemination of the results. A conceptual plan for our futurework is illustrated in Figure 14.1.
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Figure 14.1: Illustration of the teleoperated workplace for experimental and demonstrationpurposes
The extensive engineering application of the proposed theories induces further researchquestions that should be investigated as a follow-up of thiswork. In a large set of dynamicalsystems, certain parameters can not be chosen arbitrarily because of physical constraints (e.g.stable/unstable regions in the parameter space). With thisconstrains, a given closed and com-pact hyper-rectangular domain of the parameter space will contains sub-regions where thephysical system is infeasible. Development of suitable methods to handle irregular shaped(frustum) parameter domains via TP model transformation isa subject of further research.
Further practical difficulties may come forward when dynamic estimators are used toserve the gain scheduling controller with actual system parameters and state variables. Ro-bustness of the closed loop control strongly depends on the conservativeness or relaxednessof the observer and controller design. Investigation of this topic is also a possible focus ofthe future.
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Chapter 15
Theses
My new scientific results based on this dissertation are summarized by the following theses:
Thesis I
The TPτmodel transformation extends the original TP model transformation to dynamicalsystems with time-delay. The TPτmodel transformation is capable of transforming time-delay systems into non time-delay TP-type polytopic qLPV forms, where time-delay appearsas an external parameter only. The TPτmodel transformation preserves several advantageousproperties of the original TP model transformation, notably:
i Enables a directly controllable trade-off between the accuracy and complexity of theresulting model.
ii The ability to reconstruct the HOSVD-based canonical form.
iii The possibility to manipulate the properties of the convex hull defined by the resultingpolytop through matrix operations executed on the core tensor of the TP structure andon the single-variable weighting functions.
iv The LMI-based control methods are directly applicable to the resulting TP-type poly-topic qLPV form.
Analogously to the original TP model transformation based control methodology, theTPτmodel transformation based control design methodology is conceptually consistent withthe modern polytopic qLPV and LMI-based control theory and hence extends its applicabilityonto the large class of control problems comprising internal time-delay.
Related publications: [J-2], [J-5], [C-1], [C-5], [C-7], [C-11], [C-13]
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Thesis II
The mass-spring-damper mechanical impedance model with retarded elastic force compo-nent can be transformed into non time-delayed TP type polytopic LPV form using theTPτmodel transformation. The HOSVD-based canonical form of this model consists 6-component vertex systems. By the trade-off between the model accuracy and complexityit is shown that a reduced model consisting 3 vertices can be used to obtain a controller andobserver of adequate performance. These results are validated via numerical simulations andlaboratory experiments.
Related publications: [J-2], [J-5], [C-1], [C-5], [C-7], [C-11], [C-13]
Thesis III
The TPτmodel transformation based control design methodology is appropriate to be special-ized as a numerical, non-heuristic control design method for a common impedance modelbased bilateral telemanipulation task with known, varyingtime-delay. This method can beapplied to stabilize a 1-DoF servo-electric master-slave telemanipulator device with paralleljaws for force feedback enhanced remote grasping tasks. Theeffectiveness of the control de-sign method has been proofed under laboratory conditions ona tele-grasping system througha series of experiments with various stiffness in the remoteenvironment, and various time-delay profiles.
Related publications: [J-1], [C-1], [C-11]
Thesis IV
Based on the conceptual framework and characteristic methodologies of cognitive infocom-munications (a newly emerging inter-disciplinary research field), the force feedback problemcan be reformulated as an inter-cognitive and sensor-bridging application in which vibrotac-tile channels are used to convey force feedback perception.This replacement of the forcerendering by vibration leads to the decoupling of the position and force feedback, hence,opening the closed control loop and considerably relaxes the stabilization problem of forcefeedback enhanced tele-grasping. Based on this work, the problem of force feedback withtime-delay is re-contextualized in a way that is consistentwith the cognitive infocommuni-cations perspective.
Related publications: [J-3], [C-1], [C-6], [C-8], [C-9], [C-18]
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List of Figures
3.1 Schematic structure of impedance control of robots . . . .. . . . . . . . . 203.2 Scheme of coupled impedance force reflecting algorithm for bilateral telema-
nipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Mass-Spring-Damper system . . . . . . . . . . . . . . . . . . . . . . . .. 223.4 Impedance model with feedback delay . . . . . . . . . . . . . . . . .. . . 233.5 Step response of the model with various delay values (model parameters:
m = 1[kg], b = 100[Ns/m], k = 1000[N/m]). Excitation function definesasFh(t ≦ 0) = 0 , Fh(t > 0) = 1 . . . . . . . . . . . . . . . . . . . . . . 23
3.6 Friction force as function of model velocity . . . . . . . . . .. . . . . . . 243.7 τcrit as function of linear damping and stiffness of the environment with pa-
rameter set A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.8 Stability boundary resulted from the simulation . . . . . .. . . . . . . . . . 273.9 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.10 Comparison of simulated and experimental results . . . .. . . . . . . . . . 283.11 Comparison of the computed stability boundary with theresults of Gil et al. 30
4.1 Stabilization of impedance control based force reflecting telemanipulationby parameter tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Control scheme for the stabilization of force reflectingtelemanipulation un-der time-delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.1 Illustration of3-mode matrices of a 3rd-order tensorA . . . . . . . . . . . . 375.2 Visualization of the HOSVD for a third-order tensor . . . .. . . . . . . . . 39
6.1 Conventional force-position bilateral control of telemanipulation . . . . . . 536.2 Telemanipulation with sensory substitution . . . . . . . . .. . . . . . . . 53
8.1 Singular values of the HOSVD based canonical form . . . . . .. . . . . . 688.2 Weighting functions of the HOSVD based canonical form . .. . . . . . . . 698.3 Accuracy-complexity trade-off: Modelling error (ǫr) in function of dimen-
sionality of HOSVD based canonical form . . . . . . . . . . . . . . . . .. 708.4 Accuracy-complexity trade-off: Modelling error (ǫRND1000
r ) in function ofdimensionality of hosvd-based canonical form . . . . . . . . . . .. . . . . . 71
8.5 Weighting functions of SNNN type convex hull of the exactTP model . . . 738.6 Weighting functions of IRNO type convex hull of the exactTP model . . . 748.7 Weighting functions of CNO type convex hull of the exact TP model . . . . 75
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8.8 Weighting functions of SNNN type convex hull of the reduced TP modelwith 5 vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.9 Weighting functions of IRNO type convex hull of the reduced TP model with5 vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8.10 Weighting functions of CNO type convex hull of the reduced TP model with5 vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8.11 Weighting functions of SNNN type convex hull of the reduced TP modelwith 4 vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8.12 Weighting functions of IRNO type convex hull of the reduced TP model with4 vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.13 Weighting functions of CNO type convex hull of the reduced TP model with4 vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.14 Weighting functions of SNNN type convex hull of the reduced TP modelwith 3 vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.15 Weighting functions of IRNO type convex hull of the reduced TP model with3 vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.16 Weighting functions of CNO type convex hull of the reduced TP model with3 vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.19 Comparison under varying delay . . . . . . . . . . . . . . . . . . . .. . . 838.17 Comparison of the original delayed model and the HOSVD-based canonical
form of the TPτmodel with different complexity . . . . . . . . . . . . . . . 848.18 Comparison of the original delayed model and the CNO type TPτmodel with
different complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.1 Weighting functions of CNO type convex hull of the reduced TP model with3 vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
10.2 Simulation with fixed contact under constant time-delay . . . . . . . . . . . . 9710.3 Simulation with fixed contact under varying time-delay. . . . . . . . . . . 9810.4 Simulation with contact after free space motion under constant time-delay . 9910.5 Simulation with contact after free space motion under varying time-delay . 10010.6 Control performance in the unstable domain of the impedance model . . . . . 101
11.1 Experimental setup composed by two linear servo axes. .. . . . . . . . . . 10411.2 Grasping with predefined handling force in fixed contact. . . . . . . . . . 10611.3 Grasping with predefined handling force and free space motion before contact 10711.4 Interactive grasping . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 10811.5 Unstable behaviour of the tele-grasping system without the TPτcontroller . 10911.6 Experimental validation: parameter set A, constant delay (τ = 40ms), pre-
definedFh(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11011.7 Experimental validation: parameter set A, constant delay (τ = 60ms), pre-
definedFh(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11111.8 Experimental validation: parameter set A, varying delay (τ = 40ms), prede-
finedFh(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11211.9 Experimental validation: parameter set B, constant delay (τ = 30ms), pre-
definedFh(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
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11.10Experimental validation: parameter set B, varying delay (τ = 20ms), prede-finedFh(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
11.11Experimental validation: parameter set C, constant delay (τ = 20ms), pre-definedFh(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
11.12Experimental validation: parameter set C, varying delay (τ = 10ms), prede-finedFh(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
11.13Experimental validation: parameter set A, constant delay (τ = 40ms), inter-active grasping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
11.14Experimental validation: parameter set B, constant delay (τ = 30ms), inter-active grasping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
11.15Experimental validation: parameter set B, varying delay (τ = 15ms), inter-active grasping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
12.1 Wireless vibrotactile glove with five shaftless vibration motors . . . . . . . 12312.2 The force feedback capable master gripper . . . . . . . . . . .. . . . . . . 12412.3 Various stimulation strategies . . . . . . . . . . . . . . . . . . .. . . . . . 12512.4 Homogeneous linear vibration function . . . . . . . . . . . . .. . . . . . 12612.5 Inhomogeneous radiating vibration function . . . . . . . .. . . . . . . . . 12612.6 Telemanipulative grasping experiment to investigatesensor-bridging type
feedback methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12712.7 Manipulation in the immersive 3D Virtual reality . . . . .. . . . . . . . . 12812.8 VirCA assembly for manipulation in a shared virtual space . . . . . . . . . 12912.9 Stiffness grades in the first and second tests . . . . . . . . .. . . . . . . . 13012.10Experimental setup in the immersive virtual reality .. . . . . . . . . . . . 13012.11Experimental comparison of three stimulation strategies . . . . . . . . . . . . 131
14.1 Illustration of the teleoperated workplace for experimental and demonstra-tion purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
144
Author’s publications
Book chapters
[B-1] L. L. Kovacs, P. Galambos, A. Juhasz, and G. Stepan, “Experiments on the stabilityof digital force control of robots,” inModeling, Simulation and Control of NonlinearEngineering Dynamical Systems, 2009.
[B-2] T. Insperger, L. L. Kovacs, P. Galambos, and G. Step´an, “Act-and-Wait control con-cept for a force control process with delayed feedback,” inMotion and Vibration Con-trol, 2009.
[B-3] I. Fulop, P. Galambos, and P. Baranyi, “Semantic intelligent space for ambient assistedliving,” in Information and Communication Technologies, ser. Lecture Notes in Com-puter Science, R. Szabo and A. Vidacs, Eds. Springer Berlin Heidelberg, 2012, vol.7479, pp. 396–407.
Journal papers
[J-1] P. Galambos, P. Baranyi, and G. Arz, “TP model transformation based control designfor force reflecting tele-grasping under time delay,”Proceedings of the Institution ofMechanical Engineers, Part C: Journal of Mechanical Engineering Science, Condi-tionally accepted.
[J-2] P. Galambos and P. Baranyi, “Representing the model ofimpedance controlled robot in-teraction with feedback delay in polytopic LPV form: TP model transformation basedapproach,”Acta Polytechnica Hungarica, vol. 10, no. 1, pp. 139–157, 2013.
impact factor: 0.385
[J-3] P. Galambos, “Vibrotactile feedback for haptics and telemanipulation: Survey, conceptand experiment,”Acta Polytechnica Hungarica, vol. 9, no. 1, pp. 41–65, 2012.
impact factor: 0.385independent citations: 3
[J-4] P. Galambos, I. M. Fulop, and P. Baranyi, “Virtual collaboration arena, platform forresearch, development and education,”Acta Technica Jauriensis, vol. 4, no. 1, pp. 145–155, 2011.
145
independent citations: 1
[J-5] P. Galambos, P. Baranyi, and P. Korondi, “Extended TP model transformation for poly-topic representation of impedance model with feedback delay,” WSEAS Transactionson Systems and Control, vol. 5, no. 9, pp. 701–710, 2010.
[J-6] T. Insperger, L. L. Kovacs, P. Galambos, and G. Step´an, “Increasing the accu-racy of digital force control process using the Act-and-Wait concept,”Mechatronics,IEEE/ASME Transactions on, vol. 15, no. 2, pp. 291–298, Apr. 2010.
impact factor: 2.577independent citations: 7
Conference papers
[C-1] P. Baranyi, P. Galambos, A. Csapo, and P. Varlaki, “Stabilization and synchroniza-tion of dynamicons through CogInfoCom channels,” in2012 IEEE 3rd InternationalConference on Cognitive Infocommunications (CogInfoCom), Dec. 2012, pp. 33–36.
[C-2] P. Galambos, C. Weidig, P. Baranyi, J. C. Aurich, B. Hamann, and O. Kreylos,“VirCA NET: a case study for collaboration in shared virtualspace,” in2012 IEEE3rd International Conference on Cognitive Infocommunications (CogInfoCom), Dec.2012, pp. 273–277.
[C-3] P. Galambos, C. Weidig, Z. Peter, A. Csapo, P. Baranyi, J. C. Aurich, B. Hammann,and O. Kreylos, “VirCA NET: a collaborative use case scenario on factory layoutplanning.” Kosice, Slovakia: IEEE, Dec. 2012, pp. 467–468.
[C-4] H. Steiner, Z. Kertesz, P. Baranyi, P. Galambos, P. Kovacs, and N. Juhasz, “Motioncapture ariel performance analysis system and virtual collaboration arena,” Kosice,Slovakia, Dec. 2012, pp. 127–128.
[C-5] J. Kuti, P. Galambos, and P. Baranyi, “Delay and stiffness dependent polytopic LPVmodel of impedance controlled robot interaction,” inFifth Gyor Symposium and FirstHungarian-Polish Joint Conference on Computational Intelligence, 2012, pp. 174–175.
[C-6] P. Galambos and P. Baranyi, “New Directions in immersive 3D virtual reality re-search from the aspect of CogInfoCom,” inCollaborative Conference on 3D Re-search (CC3DR) 2012, Seoul, South Korea, Jun. 2012.
[C-7] P. Galambos, “Idokeseses dinamikai rendszerek qLPV modellezese,” inFiatalMuszakiak TudomanyosUlesszaka XVII. Kolozsvar: Erdelyi Muzeum Egyesulet,2012, pp. 135–138.
[C-8] P. Galambos and J. Kocsis, “Experimental testbed for the investigation of sensor-bridging feedback in tele-grasping,” inProceedings of the 10th IEEE Jubilee In-ternational Symposium on Applied Machine Intelligence andInformatics, Herl’any,Slovakia, 2012, pp. 489–494.
146
[C-9] P. Galambos and P. Baranyi, “Vibrotactile force feedback for telemanipulation: Con-cept and applications,” in2011 2nd International Conference on Cognitive Infocom-munications (CogInfoCom). IEEE, Jul. 2011, pp. 1–6.
[C-10] P. Galambos and P. Baranyi, “Virca as virtual intelligent space for RT-Middleware,”in Advanced Intelligent Mechatronics (AIM), 2011 IEEE/ASME International Con-ference on, july 2011, pp. 140 –145.
independent citations: 7
[C-11] P. Galambos, P. Baranyi, and P. Korondi, “Control design for impedance model withfeedback delay,” inRobotics in Alpe-Adria-Danube Region (RAAD), 2010 IEEE 19thInternational Workshop on, 2010, pp. 475–480.
[C-12] P. Galambos, A. Roka, G. Soros, and P. Korondi, “Visual feedback techniques fortelemanipulation and system status sensualization,” inApplied Machine Intelligenceand Informatics (SAMI), 2010 IEEE 8th International Symposium on, 2010, pp. 145–151.
[C-13] P. Galambos, P. Baranyi, and P. Korondi, “Polytopic representation of impedancemodel with feedback delay,” inProceedings of the 12th WSEAS International Con-ference on AUTOMATIC CONTROL, MODELLING & SIMULATION, Catania, Italy,May 2010, pp. 390–396.
[C-14] P. Galambos, P. Baranyi, and P. Korondi, “Extending the concept of tensor productmodelling for delayed systems,” in8th IEEE International Symposium on IntelligentSystems and Informatics, Subotica, Serbia, Sep. 2010, pp. 541–546.
[C-15] P. Galambos, B. Resko, and P. Baranyi, “Introduction of virtual collaboration arena(VirCA),” in The 7th International Conference on Ubiquitous Robots and AmbientIntelligence, Busan, Korea, Nov. 2010, pp. 575–576.
independent citations: 4
[C-16] P. Galambos, B. Resko, I. M. Fulop, and P. Baranyi,“Virtual collaboration arena(VirCA),” in The 7th International Conference on Ubiquitous Robots and AmbientIntelligence, Busan, Korea, Nov. 2010, p. 651.
[C-17] P. Galambos, A. Roka, P. Baranyi, and P. Korondi, “Contrast vision-based graspforce feedback in telemanipulation,” inAdvanced Intelligent Mechatronics, 2010IEEE/ASME International Conference on, Montreal, Canada, Jun. 2010.
independent citations: 1
[C-18] P. Galambos, A. Roka, and P. Korondi, “Ergonomic vibrotactile feedback designfor HMI applications,” in11th International Carpathian Control Conference, Eger,HUNGARY, May 2010, pp. 31–34.
independent citations: 1
147
[C-19] P. Galambos, A. Roka, G. Soros, and P. Korondi, “Periferiass latasonn alapulo infor-maciomegjelenites vizsgalata,” inXXIV. microCAD International Scientific Confer-ence, Miskolc, HUNGARY, Mar. 2010, pp. 71–78.
[C-20] A. Roka, P. Galambos, and P. Baranyi, “Contrast sensitivity model of the human eye,”in 2009 4th International Symposium on Computational Intelligence and IntelligentInformatics, Luxor, Egypt, Oct. 2009, pp. 93–99.
[C-21] T. Insperger, L. L. Kovacs, P. Galambos, and G. Stepan, “Act-and-wait concept fordigital force control of robots,” inProceedings of RoManSy’08 : the 17th CISM-IFToMM Symposium on Robot Design, Dynamics, and Control, Tokyo, Japan, Jul.2008.
[C-22] L. L. Kovacs, P. Galambos, A. Juhasz, and G. Stepan, “Experiments on the stabil-ity of digital force control of robots,” inProceedings of the 9th CONFERENCE onDYNAMICAL SYSTEMS THEORY AND APPLICATIONS, Lodz, Poland, Dec. 2007.
[C-23] C. Beltran-Gonzalez, A. Gasteratos, A. Amanatiadis, D. Chrysostomou, R. Guzman,A. Toth, L. Szollosi, A. Juhasz, and P. Galambos, “Methods and techniques for in-telligent navigation and manipulation for bomb disposal and rescue operations,” inSafety, Security and Rescue Robotics, 2007. SSRR 2007. IEEEInternational Work-shop on, 2007, pp. 1–6.
[C-24] P. Galambos, M. Boleraczki, A. Juhasz, A. Toth, J. Puspoki, and G. Arz, “Develop-ment of the prototype of a master-slave servoelectrical gripper with forcefeedbackfor telemanipulation tasks,” inProceedings of the Fifth Conference on MechanicalEngineering, K. L. Penninger A, Ed. BME OMIKK, May 2006.
148
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