Nonlinear control of mechatronic systems

8/3/2019 Nonlinear control of mechatronic systems

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July 30, 2004

To the Graduate School:

This thesis entitled Nonlinear Control Techniques for Mechatronic Systemsand written by Anup Bajirao Lonkar is presented to the Graduate School ofClemson University. We recommend that it be accepted in partial requirementfor the Master of Science with a major in Electrical Engineering.

Dr. Darren Dawson, Co-Advisor

Dr. Aman Behal, Co-Advisor

We have reviewed this thesisand recommend its acceptance:

Dr. Ian Walker

Dr. Adam Hoover

Accepted for the Graduate School:


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NONLINEAR CONTROL TECHNIQUES FOR

MECHATRONIC SYSTEMS

A Thesis

Presented to

the Graduate School of

Clemson University

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Electrical Engineering

by

Anup Bajirao Lonkar

August 2004

Co-Advisors:

Dr. Darren M. Dawson

Dr. Aman Behal


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ABSTRACT

The thesis describes the application of nonlinear system analysis and control de-

sign techniques for diverse mechatronic systems. Systems considered are a n-DOF

robotic system, a 2-DOF wing-flap aeroelastic system and an underactuated under-

water vehicle system. Experimental results are provided for the robotic system and

simulation results are provided for the remaining two systems.

Two adaptive tracking controllers are developed in the first section that accommo-

date on-line path planning objectives. An example adaptive controller is first modified

to achieve velocity field tracking in the presence of parametric uncertainty in the robot

dynamics. The development aims to relax the typical assumption that the integral of

the velocity field is bounded by incorporating a norm squared gradient term in the

control design so that the boundedness of all signals can be proven. An extension is

then provided that targets the trajectory planning problem where the task objective

can be described as the desire to move to a goal configuration while avoiding known

obstacles. Specifically, an adaptive navigation function based controller is designed

to provide a path from an initial condition inside the free configuration space of the

robot manipulator to the goal configuration. Experimental results for each controller

are provided to illustrate the performance of the approaches.

In the next chapter, a nonlinear 2D wing-flap system operating in an incompress-

ible flowfield is considered. A model free output feedback control law is implemented

and its performance towards suppressing flutter and LCOs as well as reducing vi-

brational level in the subcritical flight speed range is demonstrated. The control

proposed is applicable to minimum phase systems and conditions for the stability

zero dynamics are provided. The control objective is to design a control strategy to

drive the pitch angle to a setpoint while adaptively compensating for uncertainties in

all the aeroelastic model parameters. It is shown that all the states of the closed loop

system are asymptotically stable. Furthermore, an extension is presented in order


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DEDICATION

This thesis is dedicated to my family members who have always supported me in

all my endeavors.


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ACKNOWLEDGEMENTS

I thank Dr. Ian Walker and Dr. Adam Hoover, my committee members for their

technical guidance during my course of study at Clemson. I express deep gratitude to

my thesis advisor, Dr. Darren Dawson for motivating me to achieve higher levels of

excellence in all my work. I thank him for his support, guidance and for the patience

he has shown through my course at Clemson. I thank Dr. Aman Behal, my thesis

co-advisor for his valuable advice and guidance during my work at the controls and

robotics group.

I thank all my colleagues who helped me during my research work: Michael McIn-

tyre, David Braganza, Indrajit Deshmukh, Pradeep Setlur, Abhijit Baviskar, Jian

Chen, Bin Xian, Vilas Chitrakaran, Vishwaprasad Jogurupati, Bryan Jones, Prakash

Chawda and Apoorva Kapadia.

I would also like to thank my friends who made my stay at Clemson a memo-

rable one: Vikesh Handratta, Pramod Shanbhag, Sunil Dsouza, Ankur Pal, Abhijit

Karmokar, Amit Barman, Dedeepya K, Vipul Bhulawala and Naren Mangtani.


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TABLE OF CONTENTS

Page

TITLE PAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

CHAPTER

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Adaptive Tracking Control of Online Path Planners . . . . . . . . . . . . . . . . . . . 1Nonlinear Adaptive Model Free Control of an

Aeroelastic 2D Lifting Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Tracking Control of an Underactuated Unmanned

Underwater Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2. ADAPTIVE TRACKING CONTROL OF ON-LINEPATH PLANNERS: VELOCITY FIELDS ANDNAVIGATION FUNCTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Navigation Function Control Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3. NONLINEAR ADAPTIVE MODEL FREE CONTROLOF AN AEROELASTIC 2D LIFTING SURFACE . . . . . . . . . . . . . . . . . . . . 32

Configuration of the 2-D Wing Structural Model . . . . . . . . . . . . . . . . . . . . . . 32Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Estimation Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Adaptive Control Design and Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . 37Inclusion of Actuator Dynamics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44


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vii

Table of Contents (Continued)

Page

4. TRACKING CONTROL OF AN UNDERACTUATEDUNMANNED UNDERWATER VEHICLE. .. .. .. . . .. . . .. .. .. . .. . . .. . . . 49

System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A. Experiment Velocity Field Selection .. . .. . . .. . . .. .. . . .. . . .. .. . . . .. . . .. .. . 63B. Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114C. Derivative ofu1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 8

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119


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LIST OF FIGURES

Figure Page

2.1. Front view of the experimental setup. .. .. .. . . .. .. .. .. . . .. .. .. . .. . . .. . . . 22

2.2. Experimental and Desired Trajectories . .. .. . . .. .. .. .. . . .. .. .. . .. . . .. . . . 23

2.3. Velocity field tracking errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4. Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5. Control torques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6. Experimental Trajectory for Navigation Function based Controller . . . . 29

2.7. Parameter estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8. Control torque inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1. 2-D wing section aeroelastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2. Time evolution of the closed-loop plunging and pitching deflectionsand flap deflection for U = 20 [ms1]; U = 1.6UFlutter . . . . . . . . . . . . . . . . . . 46

3.3. Time-history of pitching deflection for U =[15 ms1], U =1.25UFlutter . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . 47

3.4. Phase-space of the pitching deflection for U = 15 [ms1], U =1.25UFlutter . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . 48

4.1. Position Tracking Error ep (t) . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 59


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CHAPTER 1

INTRODUCTION

Thesis Organization

The thesis consists of three sections, each of which deals with the control de-

velopment for a specific application. In chapter one, adaptive tracking control of

online path planners is presented. In chapter two, an adaptive output feedback type

controller for suppressing the flutter in a 2 DOF wing flap system is presented. In

chapter three, tracking control of an underactuated unmanned underwater vehicle is

presented. In each of these sections, the system model is presentedfi

rst, followed bythe control design, stability analysis, and finally the simulation/experimental results.

Proofs for theorems used in the control design are provided in the appendix at the

end of these three sections.

Adaptive Tracking Control of Online Path Planners

Traditionally, robot control researchers have focused on the position tracking prob-

lem where the objective is to force the robot to follow a desired time dependent tra-

jectory. Since the objective is encoded in terms of a time dependent trajectory, the

robot may be forced to follow an unknown course to catch up with the desired tra-

jectory in the presence of a large initial error. For example, several researchers (e.g.,

[4], [19]) have reported the so called radial reduction phenomena in which the actual

path followed has a smaller radius than the specified trajectory. In light of this phe-

nomena, the control objective for many robotic tasks are more appropriately encoded

as a contour following problem in which the objective is to force the robot to follow a

state-dependent function that describes the contour. One example of a control strat-

egy aimed at the contour following problem is velocity field control (VFC) where the

desired contour is described by a velocity tangent vector [20]. The advantages of the

VFC approach can be summarized as follows.1

1See [4], [19], and [20] for a more thorough discussion of the advantages and differences of VFCwith respect to traditional trajectory tracking control.


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The velocity field error more effectively penalizes the robot for leaving the de-

sired contour.

The control task can be specified invariant of the task execution speed.

Task coordination and synchronization is more explicit for contour following.

The ability for a velocity field to encode certain contour following tasks has re-

cently prompted researchers to investigate VFC for various applications. For example,

Li and Horowitz utilized a passive VFC approach to control robot manipulators for

contour following applications in [20], and more recently, Dee and Li used VFC to

achieve passive bilateral teleoperation of robot manipulators in [17]. The authors of

[19] utilized a passive VFC approach to develop a force controller for robot manip-

ulator contour following applications. Yamakita et al. investigated the application

of passive VFC to cooperative mobile robots and cooperative robot manipulators in

[30] and [31], respectively. Typically, VFC is based on a nonlinear control approach

where exact model knowledge of the system dynamics are required. Motivated by the

desire to account for uncertainty in the robot dynamics, Cervantes at el. developed a

robust VFC in [4]. Specifically, in [4] a proportional-integral controller was developed

that achieved semiglobal practical stabilization of the velocity field tracking errors

despite uncertainty in the robot dynamics. From a review of VFC literature, it can

also be determined that previous research efforts have focused on ensuring the robot

tracks the velocity field, but no development has been provided to ensure the link

position remains bounded. The result in [4] acknowledged the issue of boundedness

of the robot position; however, the issue is simply addressed by an assumption that

the following norm q(0) + t0

(q())d (1.1)

yields globally bounded trajectories, where q(t) denotes the position, and () denotes

the velocity field.

In addition to VFC, some task objectives are motivated by the need follow a

trajectory to a desired goal configuration while avoiding known obstacles in the con-


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figuration space. For this class of problems, it is more important for the robot to

follow an obstacle free path to the desired goal point than it is to meet a time-based

requirement. Numerous researchers have investigated algorithms to address this mo-

tion control problem. A comprehensive summary of techniques that address the

classic geometric problem of constructing a collision-free path and traditional path

planning algorithms is provided in Section 9, Literature Landmarks, of Chapter 1

of [15]. Since the pioneering work by Khatib in [10], it is clear that the construction

and use of potential functions has continued to be one of the mainstream approaches

to robotic task execution among known obstacles. In short, potential functions pro-

duce a repulsive potential field around the robot workspace boundary and obstacles

and an attractive potentialfi

eld at the goal confi

guration. A comprehensive overviewof research directed at potential functions is provided in [15]. One criticism of the

potential function approach is that local minima can occur that can cause the robot

to get stuck without reaching the goal position. Several researchers have proposed

approaches to address the local minima issue (e.g., see [1], [2], [5], [11], [29]). One

approach to address the local minima issue was provided by Koditschek in [12] for

holonomic systems (see also [13] and [24]) that is based on a special kind of poten-

tial function, coined a navigation function, that has a refi

ned mathematical structurewhich guarantees a unique minimum exists. By leveraging from previous results

directed at classic (holonomic) systems, more recent research has focused on the de-

velopment of potential function-based approaches for nonholonomic systems. For a

review of this literature see [3], [6], [7], [8], [14], [16], [22], [24], [27], and [28].

In chapter two, two adaptive controllers are developed. The first controller focuses

on the VFC problem. Specifically, the benchmark adaptive controller given in [25] is

modifi

ed to yield VFC in the presence of parametric uncertainty. The contributionof the development is that velocity field tracking is achieved by incorporating a norm

squared gradient term in the control design that is used to prove the link positions

are bounded through a Lyapunov-analysis rather than by an assumption. In lieu of

the assumption in (1.1), the VFC development is based on the selection of a velocity

field that is first order differentiable, and that a first order differentiable, nonnegative


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function V(q) R exists such that the following inequality holdsV(q)

q(q) 3(q) + 0 (1.2)

where V(q)

q

denotes the partial derivative of V(q) with respect to q(t), 3()R is a

class K function2, and 0 R is a nonnegative constant. It is interesting to note thatthe velocity field described in the experimental results provided in [4] meets (1.2), (see

Appendix A for proof) . As an extension to the VFC problem, a navigation function

is incorporated with the benchmark adaptive controller in [25] (by also injecting a

gradient term) to track a reference trajectory that yields a collision free path to a

constant goal point in an obstacle cluttered environment with known obstacles.

Nonlinear Adaptive Model Free Control of an Aeroelastic 2D Lifting Surface

Flutter instability can jeopardize aircraft performance and dramatically affect

its survivability. Although flutter boundaries for aircraft are known, due to certain

events occurring during its operational life - such as escape maneuvers - significant

decays of the flutter speed are possible with dramatic implications for its structural in-

tegrity. Passive methods which have been used to address this problem include added

structural stiffness, mass balancing, and speed restrictions [32]. However, all these

attempts to enlarge the operational flight envelope and to enhance the aeroelastic

response result in significant weight penalties, or in unavoidable reduction of nomi-

nal performances. All these facts fully underline the necessity of the implementation

of an active control capability enabling one to fulfil two basic objectives of a) en-

hanced subcritical aeroelastic response, in the sense of suppressing or even alleviating

the severity of the wing oscillations in the shortest possible time, and b) expanded

flight envelope by suppressing flutter instability, thereby, contributing to a significant

increase of the allowable flight speed. The interest toward the development and im-

plementation of active control technology was prompted by the new and sometimes

contradictory requirements imposed on the design of the new generation of the flight

vehicle that mandated increasing structural flexibilities, high maneuverability, and at

2A continuous function : [0,) [0,) is said to belong to class K if it is strictly increasingand (0) = 0 [9].


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the same time, the ability to operate safely in severe environmental conditions. In

the last two decades, the advances of active control technology have rendered the ap-

plications of active flutter suppression and active vibrations control systems feasible

[33]-[40]. A great deal of research activity devoted to the aeroelastic active control

and flutter suppression offlight vehicles has been accomplished. The state-of-the-art

of advances in these areas is presented in [35, 36]. The reader is also referred to a

sequence of articles [39] where a number of recent contributions related to the active

control of aircraft wing are discussed at length.

Conventional methods of examining aeroelastic behavior have relied on a linear

approximation of the governing equations of the flowfield and the structure. How-

ever, aerospace systems inherently contain structural and aerodynamic nonlinearities[32, 58] and it is well known that with these nonlinearities present, an aeroelastic

system may exhibit a variety of responses that are typically associated with nonlinear

regimes of response including Limit Cycle Oscillation (LCO), flutter, and even chaotic

vibrations [59]. These nonlinearities result from unsteady aerodynamic sources, large

deflections, and partial loss of structural or control integrity. Early studies have

shown that the flutter instability can be postponed and consequently the flight en-

velope can be expanded via implementation of a linear feedback control capability.However, the conversion of the catastrophic type of flutter boundary into a benign

one requires the incorporation of a nonlinear feedback capability given a nonlinear

aeroelastic system. In recent years, several active linear and nonlinear control capa-

bilities have been implemented. Digital adaptive control of a linear aeroservoelastic

model [61], -method for robust aeroservoelastic stability analysis [60], gain scheduled

controllers [62], neural and adaptive control of transonic wind-tunnel model [63, 64]

are only few of the latest developed active control methods. Linear control theory,feedback linearizing technique, and adaptive control strategies have been derived to

account for the effect of nonlinear structural stiffness [65]-[71]. A model reference

variable structure adaptive control system for plunge displacement and pitch angle

control has been designed using bounds on uncertain functions, [68]. This approach

yields a high gain feedback discontinuous control system. In [70], an adaptive design


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method for flutter suppression has been adopted while utilizing measurements of both

the pitching and plunging variables.

In the third chapter, an adaptive model free control capability is implemented

in order to rapidly regulate the pitching displacement to a setpoint while utilizing

measurements of only the pitching displacement variable. Specifically, a bank of

filters is designed to estimate the unmeasurable state variables. These filter variables

are then utilized to design a flap deflection control input in tandem with a gradient

based estimation scheme via the use of a Lyapunov function. Since the controller

is designed to be continuously differentiable, we are able to provide an extension in

order to include the effect of manipulator dynamics.

Tracking Control of an Underactuated Unmanned Underwater Vehicle

The autonomous underactuated vehicle control problem offers the challenging

dilemma of developing a suitable control strategy that can simultaneously achieve

a desired objective (position and/or orientation tracking/regulation) while utilizing

a lower number of control inputs than degrees of freedom. For instance, a fully ac-

tuated vehicle would be equipped with three translational force actuators and three

rotational torque actuators where any actuator can independently translate/rotate

the vehicle along/around its respective axis. However, weight of an autonomous

vehicle is an obvious concern when considered in the context of aerial autonomous

vehicles. Underwater vehicles are also plagued with the similar considerations as the

neutral buoyancy depth will be affected by the overall weight and displacement of the

vehicle.

Therefore, more advanced controllers are currently being investigated so that the

tracking objective can be achieved in a satisfactory manner with a reduced number

of actuators. For example, Behal et al. [79] developed a nonlinear controller for

the kinematic model of an underactuated space craft that guaranteed uniform, ulti-

mately bounded tracking provided that initial errors were selected sufficiently small.

The control structure of [79] was motivated by a Lyapunov dynamic oscillator that

originated in the area of wheeled mobile robotics from [81]. Do et al. [82] presented


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an output-feedback controller that guarantees global asymptotic stabilization track-

ing of an underactuated vehicle that operates at a constant depth (omni-directional

intelligent navigator). In [92], a task-space tracking control approach for a redundant

robot manipulator based on quaternion feedback was developed and is applied to

an autonomous underwater vehicle by Xian et al. For further literature concerning

the underactuated control problem please refer to [85], [88], [89], and [90] and the

references within.

Recently, Aguiar et al. [78]utilized an innovative approach to the two and three di-

mensional position tracking problem for underactuated autonomous vehicles. Specif-

ically, a position tracking controller was developed that achieved global stability and

exponential convergence of the position tracking error to within a neighborhood aboutzero which can be made arbitrarily small. Two distinctive techniques were employed

that distinguished [78] from some of the previous work. First, a slightly modified

expression for the dynamics of the rotation matrix was utilized throughout the con-

trol development which represents a shift from previous work. Second, Aguiar et al.

[78] augmented the filtered tracking error signal with a constant design vector. At

first glance, this addition of a constant vector within a tracking error signal seems

counterproductive; however, it is this additional design constant which facilitates theLyapunov-based control synthesis. Unfortunately, the control development requires

an input related matrix to be full rank. As mentioned in [78], it is not clear whether

this full rank condition is always satisfied.

These two distinct approaches coupled with our previous research efforts in similar

underactuated applications prompted us to investigate the possibility of relaxing the

full rank condition for the tracking control problem. That is, the objective of this

research is the development of a position tracking controller for an underactuatedautonomous vehicle equipped with one translational actuator and three rotational

actuators. To achieve this objective, the control design must be crafted in such a

manner such that rotational torque can be transmitted to the translational system

via coupling terms in order to influence position tracking. In addition, the controller

must account for the fact that uncertainty exists within select parameter values.


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Therefore, a robust structure will be embedded within the control design. A stability

analysis of the resulting tracking error signals demonstrates that the developed control

strategy drives the position tracking error signal into a neighborhood about zero with

an arbitrarily small radius set through the adjustment of design parameters. The

approach of the proposed controller follows closely to that of [78]; however, the full

rank condition set forth in [78] has been removed through a modified translational

velocity error signal and through careful crafting of the error systems based on the

stability analysis.


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CHAPTER 2

ADAPTIVE TRACKING CONTROL OF ON-LINE

PATH PLANNERS: VELOCITY

FIELDS AND NAVIGATION

FUNCTIONS.

System Model

The mathematical model for an n-DOF robotic manipulator is assumed to have

the following form

M(q)q+ Vm(q,q)

q+ G(q) = . (2.1)

In (2.1), q(t), q(t), q(t) Rn denote the link position, velocity, and acceleration,respectively, M(q) Rnn represents the positive-definite, symmetric inertia matrix,Vm(q, q) Rnn represents the centripetal-Coriolis terms, G(q) Rn represents theknown gravitational vector, and (t) Rn represents the torque input vector. Wewill assume that q(t) and q(t) are measurable. The dynamic model in (2.1), exhibits

the following properties that are utilized in the subsequent control development and

stability analysis.

Property 1: The inertia matrix can be upper and lower bounded by the following

inequalities [18]

m1 2 TM(q) m2(q) 2 Rn (2.2)

where m1 is a positive constant, m2() is a positive function, and denotesthe Euclidean norm.

Property 2: The inertia and the centripetal-Coriolis matrices satisfy the following

relationship [18]

T

1

2M(q) Vm(q, q)

= 0 Rn (2.3)

where M(q) represents the time derivative of the inertia matrix.


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Property 3: The robot dynamics given in (2.1) can be linearly parameterized as

follows [18]

Y(q, q, q) M(q)q+ Vm(q, q)q+ G(q) (2.4)

where Rp contains constant system parameters, and Y(q, q, q) Rnpdenotesa regression matrix composed ofq(t), q(t), and q(t).

Adaptive VFC Control Objective

As described previously, many robotic tasks can be effectively encapsulated as

a velocity field. That is, the velocity field control objective can be described as

commanding the robot manipulator to track a velocity field that is defined as a

function of the current link position. To quantify this objective, a velocity field

tracking error, denoted by 1(t) Rn, is defined as follows

1(t) q(t) (q) (2.5)

where () Rn denotes the velocity field. To achieve the control objective, thesubsequent development is based on the assumption that q(t) and q(t) are measurable,

and that (q) and its partial derivative (q)q

Rn, are assumed to be boundedprovided q(t) L.

Benchmark Control Modification

To develop the open-loop error dynamics for 1(t), we take the time derivative of

(2.5) and premultiply the resulting expression by the inertia matrix as follows

M(q)1 = Vm(q, q)q G(q) + + Vm(q, q)(q) (2.6)

Vm(q,

q)

(q) M(q)(q)

q

q

where (2.1) was utilized. From (2.5), the expression in (2.6) can be rewritten as

follows

M(q)1 = Vm(q, q)1 Y1(q, q) + (2.7)


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where was introduced in (2.4) and Y1(q, q) Rnp denotes a measurable regressionmatrix that is defined as follows

Y1(q, q) M(q)(q)

qq+ Vm(q, q)(q) + G(q). (2.8)

Based on the open-loop error system in (2.7), a number of control designs could be

utilized to ensure velocity field tracking (i.e., 1(t) 0) given the assumption in(1.1). Motivated by the desire to eliminate the assumption in (1.1), a norm squared

gradient term is incorporated in an adaptive controller introduced in [25] as follows

(t)

K+

V(q)

q

21 + Y1(q, q)1 (2.9)

where K Rnn is a constant, positive definite diagonal matrix, and V(q)q

was

introduced in (1.2). In (2.9), (t) Rp denotes a parameter estimate that is generatedby the following gradient update law

.

1 (t) = 1YT1 (q, q)1 (2.10)

where 1 Rpp is a constant, positive definite diagonal matrix. After substituting(2.9) into (2.7), the following closed-loop error system can be obtained

M(q)1 = Vm(q, q)1 Y1(q, q)1

K+

V(q)q21 (2.11)

where the parameter estimation error signal 1(t) Rp is defined as follows

1(t) 1. (2.12)

Remark 1 While the control development is based on a modification of the adaptive

controller introduced in [25], the norm squared gradient term could also be incorporated

in other benchmark controllers to yield similar results (e.g., sliding mode controllers).

Stability Analysis

To facilitate the subsequent stability analysis, the following preliminary theorem

is utilized.


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Theorem 1 LetV(t) R denote the following nonnegative, continuous differentiablefunction

V(t) V(q) + P(t) (2.13)

where V(q) R denotes a nonnegative, continuous differentiable function that satis-fies (1.2) and the following inequalities

0 1(q) V(q) 2(q) (2.14)

where1(), 2() are class K functions, and P(t) R denotes the following nonneg-ative, continuous differentiable function

P(t)

t

t0

2()d (2.15)

where R is a positive constant, and (t) R is defined as follows

V(q)q 1 . (2.16)

If (t) is a square integrable function, wherett0

2()d , (2.17)

and if after utilizing (2.5), the time derivative of V(t) satisfies the following inequality

.

V (t) 3(q) + 0 (2.18)

where3(q) is the class K function introduced in (1.2), and 0 R denotes a positiveconstant, then q(t) is global uniformly bounded.

Proof: The time derivative of V(t) can be expressed as follows

.

V (t) =V(q)

q(q) +

V(q)

q1 2(t)

where (2.5) and (2.15) were utilized. By exploiting the inequality introduced in (1.2)

and the definition for (t) provided in (2.16), the following inequality can be obtained

.

V (t) 3(q) + 0 +(t) 2(t) . (2.19)


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After completing the squares on the bracketed terms in (2.19), the inequality intro-

duced in (2.18) is obtained where

0 0 +1

4.

Hence, if(t) L2, then Lemma 5 in the appendix can be used to prove that q(t) isGUB.

In the following analysis, we first prove that (t) L2. Based on the conclusionthat (t) L2, the result from Theorem 1 is utilized to ensure that q(t) is boundedunder the proposed adaptive controller given in (2.9) and (2.10).

Theorem 2 The adaptive VFC given in (2.9) and (2.10) yields global velocity field

tracking in the sense that

1(t) 0. (2.20)

Proof: Let V1(t) R denote the following nonnegative function

V1 1

2T1 M1 +

1

2T1

11 1. (2.21)

After taking the time derivative of (2.21) the following expression can be obtained

V1 = T

Y1(q, q)1 +

K+

V(q)

q

21

T1 11

.

1 (2.22)

where (2.3) and (2.11) were utilized. After utilizing the parameter update law given

in (2.10), the expression given in (2.22) can be rewritten as follows

V1 = T1

K+

V(q)q21. (2.23)

The expressions given in (2.16), (2.21), and (2.23) can be used to conclude that 1(t),

1(t) L and 1(t), (t) L2. From the fact that 1(t) L, (2.12) can be used toprove that 1(t) L. Based on the fact that (t) L2, the results from Theorem 1

can be used to prove that q(t) L. Based on the fact that q(t), 1(t) L and theassumption that (q), V(q)

q L provided q(t) L, the expressions in (2.5) and

(2.16) can be used to prove that q(t), (t) L. Based on these facts, the expressionsgiven in (2.8)-(2.11) can be used to prove that Y1(q, q), (t),

.

1 (t), 1(t) L. Giventhat 1(t), 1(t) L and 1(t) L2, Barbalats Lemma [25] can now be utilized toprove (2.20).


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14

Navigation Function Control Extension

Control Objective

The objective in this extension is to navigate a robots end-effector along a collision-

free path to a constant goal point, denoted by q D, where the set D denotes afree configuration space that is a subset of the whole configuration space with all

configurations removed that involve a collision with an obstacle, and q Rn denotesthe constant goal point in the interior of D. Mathematically, the primary control

objective can be stated as the desire to ensure that

q(t) q as t (2.24)

where the secondary control is to ensure that q(t) D. To achieve these two controlobjectives, we define (q) R1 as a function (q) : D [0, 1] that is assumed tosatisfy the following properties:

P1) The function (q) is first order and second order differentiable (i.e., q (q)and

q

q (q)

exist on D).

P2) The function (q) obtains its maximum value on the boundary of D.

P3) The function (q) has unique global minimum at q(t) = q.

P4) If q (q) = 0 then q(t) = q.

Based on (2.24) and the above definition, an auxiliary tracking error signal, de-

noted by 2(t) Rn, can be defined as follows to quantify the control objective

2(t) q(t) +(q) (2.25)

where (q) = q (q) denotes the gradient vector of (q) defined as follows

(q)

q1

q2...

qn

T. (2.26)


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Remark 2 As discussed in [24], the construction of the function (q), coined a

navigation function, that satisfies all of the above properties for a general obstacle

avoidance problem is nontrivial. Indeed, for a typical obstacle avoidance, it does not

seem possible to construct (q) such that q (q) = 0 only at q(t) = q. That is, as

discussed in [24], the appearance of interior saddle points (i.e., unstable equilibria)

seems to be unavoidable; however, these unstable equilibria do not really cause any

difficulty in practice. That is, (q) can be constructed as shown in [24] such that only

a few initial conditions will actually get stuck on the unstable equilibria.

Benchmark Control Modification

To develop the open-loop error dynamics for 2(t), we take the time derivative of

(2.25) and premultiply the resulting expression by the inertia matrix as follows

M2 = Vm(q, q)2 + Y2(q, q) + . (2.27)

where (2.1) and (2.25) were utilized. In (2.27), the linear parameterization Y2(q, q)

is defined as follows

Y2(q, q) M(q)f(q, q) + Vm(q, q) (q) G(q) (2.28)

where Y2(q, q) Rnm denotes a measurable regression matrix, Rm was intro-duced in (2.4), and the auxiliary signal f(q, q) Rn is defined as

f(q, q) d

dt((q))

= H(q)q (2.29)

where the Hessian matrix H(q) Rnn is defined as follows

H(q)

2

q21

2

q1q2

2

q1qn2

q2q1

2

q22

2

q2qn 2

qnq1

2

q2n

. (2.30)


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Based on (2.27) and the subsequent stability analysis, the following adaptive controller

introduced in [25] can be utilized

k2 Y2(q, q)2 (2.31)

where k R is a positive constant gain, and 2(t) Rn denotes a parameter updatelaw that is generated from the following expression

.

2 (t) 2YT2 (q, q)2 (2.32)

where 2 Rmm is a positive definite, diagonal gain matrix. Note that the trajectoryplanning is incorporated in the controller through the gradient terms included in

(2.28) and (2.29). After substituting (2.31) into (2.27) the following closed loop error

systems can be obtained

M2 = Vm(q, q)2 k2 + Y2(q, q)2 (2.33)

where 2(t) Rp is defined as follows

2(t) 2. (2.34)

Stability Analysis

Theorem 3 The adaptive controller given in (2.31) and (2.32) ensures that the robot

manipulator tracks an obstacle free path to the unique goal configuration in sense that

q(t) q as t (2.35)

provided the control gain k introduced in (2.31) is selected to be sufficiently large.

Proof: Let V2(q, 2, 2) R denote the following nonnegative function

V2 (q) +

1

2T2 M2 +

T2

12 2

. (2.36)

where R is an adjustable, positive constant. After taking the time derivative of(2.36) the following expression can be obtained

V2 = [(q)]T q+ T2k2 + Y2(q, q)2

T2 12

.

2 (2.37)


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17

where (2.3), (2.26), (2.33), and (2.34) were utilized. By utilizing (2.25), (2.32), and

the triangle inequality, the following expression can be obtained

V2 12

(q)2 (k 2) 22 . (2.38)

Provided k is selected sufficiently large to satisfy

k >2

, (2.39)

it is clear from (2.2), (2.36), and (2.38) that

0 (q(t)) + (q, t) (q(0)) + (q(0), 0) (2.40)

where (q, t) R is defined as

(q, t)

m2(q)

22(t)2 + max{12 }

2(t)2

. (2.41)

From (2.34), (2.40), and (2.41) it is clear that 2(t), (q), 2(t), 2(t) L. Let theregion D0 be defined as follows

D0 { q(t)| 0 (q(t)) (q(0)) + (q(0), 0)} . (2.42)

Hence (2.36), (2.38) and (2.40) can be utilized to show that q(t) D0 providedq(0) D0 (i.e., q(t) D0 q(0) D0). Based on property P1 given above, we nowknow that (q) L q(0) D0. Since 2(t), (q) L q(0) D0, (2.25) canbe used to conclude that q(t) L q(0) D0; hence, property P1) and (2.29) can beused to conclude that f(q, q) L q(0) D0. Based on the previous boundednessstatements, (2.28) can be used to show that all signals are bounded q(0) D0.Since all signals are bounded, it easy to show that 2(t),

ddt

(

(q))

L

q(0)

D0,

and hence, (q), 2(t) are uniformly continuous q(0) D0. From (2.38), it canalso be determined that (q), 2(t) L2 q(0) D0. From these facts, Barbalatslemma [25] can be used to show that (q), 2(t) 0 as t q(0) D0. Since(q) 0 , property P4) given above can be used to prove that q(t) q as t q(0) D0. To ensure that q(t) will remain in a collision-free region, we must account


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for the effects of the (q(0), 0) term introduced in the definition of the region D0

given in (2.42). To this end, we first define the region D1 as follows

D1 {q(t)| 0 (q(t)) < 1} (2.43)

where D1 denotes the largest collision-free region. It is now clear from (2.42) and

(2.43) that if the weighting constant is selected sufficiently small to satisfy

(q(0), 0) + (q(0)) < 1, (2.44)

then D0 D1, and hence, the robot manipulator tracks an obstacle free path.

Experimental Verification

To investigate the behavior of the proposed trajectory planning controllers, both

the Adaptive VFC and Navigation Function Control Extension approaches were im-

plemented on two links of the Barrett Whole Arm Manipulator (WAM).

Experimental Setup

The WAM is a highly dexterous backdriveable manipulator. The biggest challenge

faced during both experimental trials was not knowing the actual dynamics of the

WAM. To overcome this obstacle, 5 links of the robot were locked at a fixed, specified

angle during the experiment and the remaining links of the manipulator were used as

a 2-link planer robot manipulator (see Figure 2.1). In this configuration, the dynamics

of the robot can be expressed in the following form [26]

=

M11 M12M21 M22

q1q2

+

Vm11 Vm12Vm21 Vm22

q1q2

+

fd1 00 fd2

q1q2

(2.45)

the entries of the inertia matrix are given as

M11 = p1 + 2p2cos(q2)M12 = p3 +p2cos(q2)M21 = p3 +p2cos(q2)M22 = p3

(2.46)

where p1, p2, p3 denote the mass parameters. The entries of the centripetal-Coriolis

matrix are formed to satisfy the skew-symmetric property between the inertia matrix


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and the centripetal-Coriolis matrix and are given as follows

VM11 = p2sin(q2)q2VM12 = p2sin(q2)q1 p2sin(q2)q2VM21 = p2sin(q2)q1

VM22 = 0.

(2.47)

The dynamic friction coefficients were set equal to the following values: fd1 = 6.8 and

fd2 = 3.8. The gravitational effects do not appear in (2.45) because of the selection

of locked and actuated joints. The constant parameter vector defined in (2.4) was

constructed as follows

=

p1 p2 p3T

. (2.48)

All links of the WAM manipulator are driven by brushless motors supplied with

sinusoidal electronic commutation. Each axis has an encoder located at the motor

shaft for link position measurements. Since no tachometers are present for velocity

measurements, link velocity signals are calculated via a filtered backwards difference

algorithm. An AMD Athlon 1.2GHz PC running QNX 6.1 RTP (Real Time Plat-

form), a real-time micro-kernel based operating system, hosts the control algorithms

which were written in C++. To facilitate real time graphing, data logging and

on-line gain tuning, the in-house graphical user interface, Qmotor 3.0 [21], was used.

Data acquisition and control implementation were performed at a frequency of 1.0kHz

using the ServoToGo I/O board.

Remark 3 The joint angles were measured using encoders having a resolution of

4096 counts per revolutions which may be considered inadequate for high precision

maneuvers. The WAM, noted for its light weight and speed, also imposes limitations

on the bandwidth of the control. Hence the constraints in the physical systems used

for testing, prevented the further increase of control gains even though additional

torque capacity was available for control. As the main intention in conducting these

experiments was merely to show that the control could be implemented on a physical

system, we believe the magnitude of the errors are not a reflection of the performance

of these controllers.


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Remark 4 For the Navigation Function Control Extension experiment, a work-space

with obstacles was defined in the WAMs task space. Four circular obstacles were

placed in the work-space and sized such that a path could be found around the obstacles.

From (2.40) and (2.41), the selection of the starting and goal points affect the actual

size of the work-space and obstacles, in the task space. This change in size is attributed

to the addition of(q(0), 0). Because the actual dynamics are unknown for the WAM,

it is impossible to calculate(q(0), 0), therefore the actual size of the obstacles and the

work-space were found experimentally and can be see in Figure 2.6. In Figure 2.6, a

small increase in the physical obstacles along with a small decrease in the size of the

work-space were determined.

Experimental Results

Adaptive VFC Approach

An experiment using two links of the WAM was utilized to demonstrate the perfor-

mance of the adaptive VFC given in (2.9) and (2.10). For the experiment, a velocity

field for a planar, circular task-space contour was proven to meet the qualifying con-

dition given in (1.2), (see Appendix for proof) . Specifically, the following contour

was utilized [4]

(x) = K(x)f(x)

2(x1 xc1)2(x2 xc2)

+ c(x)

2(x2 xc2)2(x1 xc1)

. (2.49)

In (2.49), xc1 = 0.54155[m] and xc2 = 0.044075[m] denote the circle center, and the

functions f(x), K(x), and c(x) R are defined as follows

f(x) = (x1 xc1)2 + (x2 xc2)2 r2o (2.50)K(x) =

k0f2(x)

f(x)x + c(x) =

c0 exp

f2(x)

f(x)x

where ro = 0.2[m] denotes the circle diameter; = 0.005[m3] and = 20[m1] are

auxiliary parameters; and k0 = 0.25[ms1] and c0 = 0.25[ms

1] are tracking speed


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21

parameters. Values for the above parameters were chosen based on the velocity field

presented in [4].

The following forward kinematics for the WAM,

x1x2

=1 cos(q1) + 2 cos(q1 + q2)1 sin(q1) + 2 sin(q1 + q2)

(2.51)

and manipulator Jacobian

J(q) =

1 sin(q1) 2 sin(q1 + q2) 2 sin(q1 + q2)1 cos(q1) + 2 cos(q2 + q1) 2 cos(q2 + q1)

(2.52)

where utilized to implement this control problem. In (2.51) and (2.52), the robot

link lengths denoted by 1 = 0.558, 2 = 0.291[m]. From (2.49)-(2.52) the joint-space

velocity field can be determined as follows (q) = J1

(q)(x). Upon starting eachexperimental run, a PD controller was utilized to move the WAM to the following

initial position

q =

0 90 90 60 90 20 0 T , all in degrees (2.53)then links 2, 3, 5, 6 and 7 were locked (the desired set-point of the PD controller was

set to the initial values) while links 1 and 4 were driven by the proposed controller.

The control gains were adjusted by trial and error, with the best performance found

with the following values

K = diag(25, 15) = diag(3, 1, 5)

where diag() denotes a diagonal matrix with the arguments as the diagonal entries.

Figures 2.2 - 2.5 depicts the trajectories of the WAMs endeffector, the velocity field

tracking errors, the parameter estimates, and the control torque inputs, respectively.

Navigation Function Control Extension

To illustrate the performance of the controller given in (2.31) and (2.32), an ex-

periment using two links of the WAM was also performed which navigated the robots

end effector from an initial position to the task-space goal point, denoted by x [x1,


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Figure 2.1 Front view of the experimental setup.


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Figure 2.2 Experimental and Desired Trajectories


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Figure 2.4 Parameter Estimates


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Figure 2.5 Control torques


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x2]T R2. The same forward kinematic and Jacobian as defined in (2.51) and (2.52)

were utilized. A task-space potential function (x) was chosen as follows

(x) =x x2

x x2 + 012341/(2.54)

where x(t) [x1(t), x2(t)]T R2. In (2.54), R denotes some positive integer. If

is selected large enough, it is proven in [12] that (x, x) is a navigation function for

x(t). In (2.54), the boundary function 0(x) R and the obstacle functions 1(x),2(x), 3(x), 4(x) R are defined as follows

0 = r20 (x1 x1r0)2 (x2 x2r0)2 (2.55)

1 = (x1 x1r1)2

+ (x2 x2r1)2

r2

1

2 = (x1 x1r2)2 + (x2 x2r2)2 r223 = (x1 x1r3)2 + (x2 x2r3)2 r234 = (x1 x1r4)2 + (x2 x2r4)2 r24.

In (2.55), (x1 x1ri) and (x2 x2ri) where i = 0, 1, 2, 3, 4 are the centers of theboundary and obstacles respectively, r0, r1, r2, r3, r4 R are the radii of the boundary

and obstacles respectively. From (2.54) and (2.55) it is clear that the model space isa circle that excludes four smaller circles described by the obstacle functions 1(x),

2(x), 3(x), 4(x). For this experiment the model-space configuration is selected as

followsx1r0 = 0.5064 x2r0 = 0.0275 r0 = 0.28x1r1 = 0.63703 x2r1 = 0.11342 r1 = 0.03x1r2 = 0.4011 x2r2 = 0.0735 r2 = 0.03x1r3 = 0.3788 x2r3 = 0.1529 r3 = 0.03x1r4 = 0.6336 x2r4 = 0.12689 r4 = 0.03

where all values are in meters.The control gains were adjusted to the following values which yielded the best

performance

= 14 k = 45

2 = diag (0.02, 0.01, 0.01) .


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Upon starting each experimental run, a PD controller was utilized to move the WAM

to the following initial position

q = 58.84 90 90 140.72 11.5 84.5 0 T

, all in degrees (2.56)

then links 2, 3, 5, 6 and 7 were locked (the desired set-point of the PD controller was

set to the initial values) while links 1 and 4 were driven by the proposed controller.

The actual trajectory of the WAM robots end effector can be seen in Figure 2.6.

From Remark 4, the outer circle pairs in Figure 2.6 depicts the physical and actual

boundaries for the work space. The four inner circle pairs depict the physical and

actual boundaries for the obstacles. Figure 2.6 illustrates that the robots end effector

avoids the actual obstacles as it moves to the goal point. The parameter estimatesand control torque inputs are provided in Figures 2.7 and 2.8, respectively.


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Figure 2.6 Experimental Trajectory for Navigation Function based Controller


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Figure 2.7 Parameter estimates


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Figure 2.8 Control torque inputs.


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CHAPTER 3

NONLINEAR ADAPTIVE MODEL FREE CONTROL

OF AN AEROELASTIC 2D LIFTING

SURFACE

Configuration of the 2-D Wing Structural Model

Figure 3.1 shows a schematic for a plunging-pitching typical wing-flap section that

is considered in the present analysis. This model has been widely used in aeroelastic

analysis [46],[47]. The plunging (h) and pitching () displacements are restrained by

a pair of springs attached to the elastic axis of the airfoil (EA) with spring constants

kh and k (), respectively. Here, k () denotes a continuous, linear parameterizable

nonlinearity, i.e., the aeroelastic system has a continuous nonlinear restoring moment

in the pitch degree of freedom. Such continuous nonlinear models for stiffness result

from a thin wing or propeller being subjected to large torsional amplitudes [32, 58].

Similar models [65, 66, 68, 69] have been examined and provide a basis for comparison.

As can be clearly seen in (3.1), the aeroelastic system permits two degree-of-freedom

motion, whereas an aerodynamically unbalanced control surface is attached to the

trailing edge to suppress instabilities. h denotes the plunge displacement (positive

downward), the pitch angle (measured from the horizontal at the elastic axis of

the airfoil, positive nose-up) and is the aileron deflection (measured from the axis

created by the airfoil at the control flap hinge, positive flap-down). The governing

equations of motion for the aeroelastic system under consideration are given by [65]-

[66]m mxbmxb I

h

+

ch 00 c

h

+

kh 00 k ()

h

=

LM

(3.1)

where k () denotes the pitch spring constant a particular choice for which is pre-

sented in Section 5. The quasi-steady aerodynamic lift (L) and moment (M) can be


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Figure 3.1 2-D wing section aeroelastic model


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34

modeled as [72]

L = U2bcl

+

h

U+

1

2 a

b

U

+ U2bcl

M = U2b2cm +

h

U +1

2 a

b

U

+ U2b2cm

(3.2)

One can transform the governing EOM of (3.1) into the following equivalent state

space form as follows [66]z = f(z) + g(z)y = z2

(3.3)

where z(t) =

z1 (t) z2 (t) z3 (t) z4 (t)T R4 is a vector of system states that

is defined as follows

z =

h h

T (3.4)

(t) R1 is a flap deflection control input, y (t) R1 denotes the designated output,while f(z), g (z) R1 assume the following form

f(z) =

z3z4k1z1 (k2U2 +p (z2)) z2 c1z3 c2z4k3z1 (k4U2 + q(z2)) z2 c3z3 c4z4

g(z) =

00g3U

2

g4U2

, g4 = 0

(3.5)

where p (z2) , q(z2) R1 are continuous, linear parameterizable nonlinearities in theoutput variable resulting from the nonlinear pitch spring constant k(). The auxil-

iary constants ki, ci i = 1...4 as well as g3, g4 are explicitly defined in Appendix B.Motivated by our desire to rewrite the equations of (3.3) in a form that is amenable

to output feedback control design, we apply Kalmans observability test to the pairA, C

where A R44, C R14 are explicitly defined as

A =

0 1 0 0k4U2 c4 k3 c30 0 0 1k2U2 c2 k1 c1

, C =

1000

. (3.6)

A sufficient condition for the system of (3.3) to be observable from the pitch angle

output (t) is given as

dobs = k23 k3c3c1 + c23k1 = 0


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which is easily satisfied by the nominal model parameters of (3.49) presented in Sec-

tion 5. After a series of coordinate transformations, we can then express the governing

equations for the aeroelastic model in the following convenient state space form

x = Ax +(y) + By = CTx =

1 0 0 0

x

(3.7)

where = U2 is an auxiliary control input, x (t) =

x1 (t) x2 (t) x3 (t) x4 (t)T

R4 is a new vector of system states, A R44, B R4 are explicitly defined as

A =

0 1 0 00 0 1 00 0 0 10 0 0 0

, B =

0234

(3.8)

where i i = 2, 3, 4 are constants that are explicitly defined in Appendix B. In (3.7)above, (y) =

1(y) 2(y) 3(y) 4(y)

T R4 is a smooth vector field thatcan be linearly parameterized as follows

(y) = W(y)=

pj=1

jWj(y) (3.9)

where Rp is a vector of constant unknowns, W(y) R4p is a measurable,

nonlinear regression matrix while the notation Wj () R4

denotes the j

th

columnof the regression matrix W () j = 1...p. For a particular choice of the pitchspring nonlinearity k(), explicit expressions for in terms of model parameters are

provided in Appendix B.

Remark 5 The proposed control strategy is predicated on the assumption that the

system of (3.7) is minimum phase. In Appendix B, we study the zero dynamics of the

system and provide conditions for stability of the zero dynamics.


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Control Objective

Provided the structures of both the aeroelastic model and the pitch spring constant

nonlinearity is known, our control objective is to design a control strategy to drive

the pitch angle to a setpoint while adaptively compensating for uncertainties in allparameters of the model and the nonlinearity. We assume that the only measurements

available are those of the output variable y = while we compensate for the remaining

states via use of state estimators. Toward these goals, we begin by first defining the

pitch angle setpoint error e1 (t) R1 as follows

e1 = y yd (3.10)

We also defi

ne a state estimation error x (t) R4

as follows

x = x x (3.11)

where x (t) is a state vector previously defined in (3.7) while x (t) R4 is yet to bedesigned. Motivated by our subsequent control design, we define parameter estimation

error signals 0 (t) Rp1 and 1 (t) Rp2 as follows

0 = 0 0 1 = 1 1 (3.12)

where 0 Rp1, 1 Rp2 are unknown constant vectors that will be subsequentlyexplicitly defined while 0 (t) , 1 (t) are their corresponding dynamic estimates that

will be generated as part of a Lyapunov function based control design scheme. We

note here that p1, p2 are non-negative integers whose values are determined through

an explicit choice for the nonlinearity in the aerodynamic model.

Estimation Strategy

In this section, we design the state estimation vector x (t) . Given the fact that

both (y) and B of (3.7) contain unmeasurable terms, we take a modular approach

[74] to the state estimation problem. Ignoring the last two terms in the first equation

of (3.7), we design the standard observation signal 0 (t) R4 as follows

0 = A0 + kCT(x 0) (3.13)


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where k R4 is a gain vector chosen so that A0 = A kCT is Hurwitz. Giventhat A and CTx = y are measurable, one can see that the design of (3.13) is readily

implementable. Moving on to the second term in the first equation of (3.7), one can

now exploit the linear parameterization of (3.9) and the fact that A0

is Hurwitz in

order to design a set of implementable observers j (t) R4 for Wj(y) as follows

j = A0j + Wj(y) j = 1....p (3.14)

Moving on to the last term in the first equation of (3.7), the following facts are readily

noticeable: (a) B is unmeasurable, and (b) B(t) can be conveniently rewritten as

B=4

j=2

jej (3.15)

where the notation ej represents the jth standard basis vector on R4. Motivated by

the observation design of (3.14), it is easy to design the following implementable set

of observers for (t) as follows

i = A0i + ei i = 2, 3, 4 (3.16)

Given (3.13),(3.14), and (3.16), we are now in a position to patch together an unmea-

surable state estimate x (t) as follows

x = 0 +

pj=1

jj +4

j=2

jj (3.17)

By taking the derivative of (3.11) and substituting for the dynamics of (3.7) and

(3.17), we can obtain the following exponentially stable estimation error system

.

x= A0x (3.18)

Although the state estimate is unmeasurable, the worth of the design above lies in

the linear parameterizability of the right hand side of the expression of (3.17)as well

as the exponential stability of the estimation error as will be amply evident in Section

4.


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Adaptive Control Design and Stability Analysis

After taking the derivative of the regulation error e1 (t) and substituting (3.7) for

the system dynamics, we obtain

e1 = x2 +p

j=1

jWj,1 (y) (3.19)

Since x2 (t) is unmeasurable, we add and subtract (3.17) to the right hand side of the

above equation in order to obtain the open loop error dynamics as follows

e1 = 0,2 +

pj=1

j [j,2 + Wj,1 (y)] +4

j=2

jj,2 + x2 (3.20)

where (3.11) has been utilized where the notation j,i denotes the ith

element in the jth

column vector of. From (3.16), it is easy to notice that 2,2 (t) is only one derivative

removed from the control input (t) and is a suitable candidate for a virtual control

input. By applying the backstepping methodology in [74], we define an auxiliary error

signal e2 (t) R1 as followse2 = 2,2 2,2,d (3.21)

where 2,2d (t) R1 is a yet to be designed backstepping signal. After adding andsubtracting 2,2d (t) as well as a feedback term (c11 d11) e1 to the right hand sideof (3.20) and rearranging the terms in a manner amenable to the pursuant analysis,

one obtains

e1 = 2

12 (c11 + d11) e1 +

12 02 +

pj=1

12 j [j,2 + Wj,1 (y)] +

4j=3

12 jj,2

+ 22,2d (c11 + d11) e1 + 2e2 + x2

(3.22)

where c11, d11 are positive gain constants. It is easy to see that the terms inside

the brackets in the above equation can be linearly parameterized as T1 0 where

0,1 (t) Rp1 denote, respectively, a vector of unknown constants and a measurableregression vector and are explicitly written as follows

T0 =12

12 1

12 2 .....

12 p

12 3

12 4

(3.23)


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T1 =

c11e1 + d11e1 + 0,2 1,2 + W1,1 (y) 2,2 + W2,1 (y) ..... p,2 + Wp,1 (y) 3,2 4,2

(3.24)

One can now succinctly rewrite the expression of (3.22) as follows

e1 = 2T1 0 + 2,2d

(c11 + d11) e1 + 2e2 + x2 (3.25)Given the structure of (3.25) above, we design the desired backstepping signal 2,2d (t)

as follows

2,2d = T1 0 (3.26)

After substituting this control input into the open loop expression of (3.25), one can

obtain the following expression

e1 = 2T1 0 (c11 + d11) e1 + 2e2 + x2 (3.27)

where the definition of (3.12) has been utilized. At this stage, one can define a

Lyapunov like function to perform a preliminary stability analysis at this stage as

follows

V1 =1

2e21 +

|2|

2T0 0 + d

111 x

TPx (3.28)

where P R44 is a positive-definite symmetric matrix such that P A0 + AT0 P = I.

After taking the derivative ofV1 (t) along the system trajectories of (3.18) and (3.27)

and rearranging the terms, we obtain

V1 = c11e21 + |2|T0 1e1sign(2) T0

.

0

+2e1e2 d11e21 + e1x2 d11xTx (3.29)

From (3.29), the choice for a dynamic adaptive update law is made as follows

.

0= sign (2)1e1 (3.30)

After substituting (3.30) into (3.29) and performing some algebraic manipulation, anupperbound for V1 (t) can be obtained in the following manner

V1 c11e21 3

4d111 x

Tx + 2e1e2 (3.31)

where the first two terms are negative definite terms and the last term is an indefinite

interconnection term which is dealt with in the next stage of control design. Our next


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step involves design of our actual control input (t) which is easily reachable through

differentiation of (3.21) along the dynamics of (3.16) as follows

e2 = k22,1 + 2,3 + 2,2d (3.32)

From (3.26) and (3.24), it is obvious that 2,2d () can be represented in the following

manner

2,2d = 2,2d (y, yd, 0,2, 1,2, 2,2,..., p,2,3,2, 4,2, 0) (3.33)

Since 3,2 (t) ,4,2 (t) are more than one derivative away from the control, potential

singularities in the control design are avoided when the desired control signal 2,2d ()

is differentiated as follows

2,2d =2,2d

y

02 +

pj=1

j [j,2 + Wj,1 (y)] +4

j=2

jj,2 + x2

+

2,2d

0[A00 + ky]

+p

j=1

2,2d

j[A0j + Wj (y)] +

4j=3

2,2d

jA0j +

2,2d

0[sign (2)1e1]

(3.34)

It is now possible to separate out (3.34) into its measurable and immeasurable com-

ponents as follows

2,2d = 2,2dm + 2,2du +2,2d

yx2 (3.35)

where 2,2dm (t) R1 denotes the measurable components that is matched with thecontrol input (t) and can be done away via direct cancellation, 2,2du (t) R1

denotes unmeasurable components that are linearly parameterizable and dealt with

via the design of an adaptive update law, while the last term can be damped out

owing to the exponential stability of (3.18) (For explicit expressions for 2,2dm (t) and

2,2du (t), we refer the reader to the Appendix B). We are now in a position to rewrite

the open-loop dynamics for e2 (t) in the following manner

e2 = k22,1 + 2,3 + 2,2dm 2,2dy

x2 2e1 + [2e1 2,2du] (3.36)

where we have added and subtracted 2e1 to the right hand side of (3.32) and (3.35)

has been utilized. It is important to note that the bracketed term in the above

expression can be parameterized as T2 1 where 1,2 (t) Rp1 denote, respectively,


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a vector of unknown constants and a measurable regression vector and are explicitly

written as follows

T1 =

1 2 ..... p 2 3 4

(3.37)

T2 = 2,2dy {1,2 + W1,1 (y)} ... 2,2dy {p,2 + Wp,1 (y)} 2,2dy 2,2 + e1 3,2 4,2

(3.38)

Motivated by our preliminary stability analysis and the structure of (3.36), we design

our control input (t) as follows

= c22e2 d222,2d

y

2e2 2,3 + k22,1 + 2,2dm T2 1 (3.39)

where c22, d22 are positive gain constants. After substituting this control input into

(3.36), we obtain the closed loop dynamics for e2 (t) in the following manner

e2 = c22e2 d222,2d

y

2e2 2,2d

yx2 2e1 + T2 1 (3.40)

Motivated by the subsequent stability analysis, we design an update law for 1 (t) as

follows.

1= 2e2 (3.41)

Toward a final stability analysis, we augment the function of (3.28) as follows

V2 = V1 +1

2e22 +

T1 1 + d

122 x

TPx (3.42)

The time derivative of V2 (t) along the closed-loop trajectories of (3.40) and (3.41)

yields the following upperbound

V2 c11e21 c22e22 3

4

d111 + d

122

xTx (3.43)

where we utilized the nonlinear damping argument [75]. Since

V2 (t) is negative semi-definite and V2 (t) is positive definite, we can conclude that V2 (t) L. Therefore,from (3.28) and (3.42), it is easy to see that 0 (t) , 1 (t) L and e1 (t) , e2 (t) , x (t) LL2. From the apriori boundedness ofyd, 0,1, we can say that y (t) , 0 (t) , 1 (t) L. The boundedness ofy (t) guarantees that i (t) L i = 0....p. From (3.7) andthe assumption of stable zero dynamics, the boundedness of 2,1 (t) ,3,1 (t) , 3,2 (t) ,


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4,1 (t) , 4,2 (t) , 4,3 (t) can be readily established. Hence, 1 (t) , 2,2d (t) L from(3.24) and (3.26) which implies from the definition of (3.21) that 2,2 (t) L. Fromthis and the preceding assumptions, one can state that 2 (t) , (t) , (t) L. From(3.16), one can guarantee the boundedness of all

i i = 2, 3, 4. Thus, all the states of

the system are bounded in closed-loop operation. From (3.27) and (3.40), one can see

that e1 (t) , e2 (t) L. Given all the preceding facts, we can now apply BarbalatsLemma [76] to state that lim

te1 (t) , e2 (t) = 0. From (3.18), the estimation error x (t)

is exponentially regulated to the origin.

Inclusion of Actuator Dynamics

In order to model the control surface dynamics along with the quasi-steady equa-

tions of (3.1), we follow the approach of [77] and assume a second-order system as

follows

+p1+p2= p2u p2 = 0 (3.44)

where (t) R1 denotes the actual control surface deflection while u (t) is the flapactuator output and the de facto control input signal. In this Section, we will discuss

3 different strategies to include the effect of the dynamics of the flap.

Ifp1 and p2 are chosen such that the dynamics of (3.44) are faster than the dynam-ics of (3.1), then, similar to a cascaded control structure, we can neglect the coupling

of the plunge and pitch motion of the wing section. The signal (t) introduced in

(3.1) and designed subsequently in (3.39) can be treated as a desired flap deflection

d (t) and the control input u (t) can be simply chosen to be

u (t) = d +p1d +p2d

such that the closed-lop system for (3.44) becomes

+p1 +p2 = 0 (3.45)

which is an exponentially stable system and where = d denotes the error be-tween the desired and actual flap deflection. Thus, (t) tracks d (t) exponentially


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43

fast. However, if the actuator has slow dynamics, then one must consider the inter-

connection between the dynamics of (3.1) and (3.44). One must then rewrite (3.36)

as follows

e2 = k22,1 + 2,3 + U2d 2,2dm 2,2dy

x2 2e1 + T2 1 + U2 (3.46)

and design d (t) as follows

d = U2

c22e2 d22

2,2d

y

2e2 2,3 + k22,1 + 2,2dm T2 1 + U2

(3.47)

A substitution of the above desired flap deflection signal into (3.46) leaves a U2r

mismatched indefinite term in the closed-loop dynamics for e2 (t) where r (t) (t) +

(t) is a filtered error variable. This mismatch must be compensated for in the control

design for u (t) . We begin by writing the open-loop dynamics for the actuator as

follows

r = + = d p1p2+ +p2u

Motivated by the structure of our dynamics, we can design the control input signal

as follows

u (t) = p12 d +p1+p2

kur

U2e2 (3.48)

which yields the actuator dynamics closed-loop form of

r = kur U2e2

We can now augment the function of (3.42) as V3 = V2 +1

2r2 which yields after

differentiation along the system closed-loop trajectories the following upperbound

V3 c11e21 c22e22 3

4 d111 + d122 xTx kur2Again, a signal chasing argument would show that lim

te1 (t) , e2 (t) , x (t) , r (t) =

0. It is to be noted here that the control redesign of (3.47) and (3.48) does not

affect either the state estimation strategy or the parameter update laws. However,

we needed to utilize measurements of the state variables (t) , (t) as well as the

model parameters p1 and p2. In order to mitigate this dependence and design a true


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44

output feedback control strategy with measurements of only the pitch displacement

as well as adaptively compensate for uncertainty in p1and p2, one would need to

comprehensively redesign the estimation, adaptation, and control strategies. The

methodology followed in this research would still be valid but the estimation strategy

would have to include observers for the new states, i.e., 1 (t) and its time derivative.

Additionally, the parameter update laws would need to be augmented to compensate

for the effects of the actuator model in terms of p1 and p2.

Simulation Results

The model described in (3.3) was simulated using the output feedback observation,

control, and estimation algorithm of (3.13), (3.14), (3.16), (3.30), (3.39), and (3.41).

The values for the model parameters were taken from [65] and are listed below

b = 0.135 [m] kh = 2844.4 [Nm1] ch = 27.43 [Nm

1s1]c = 0.036 [Ns] = 1.225 [kgm3] cl = 6.28cl = 3.358 cm = (0.5 + a) cl cm = 0.635m = 12.387 [kg] I = 0.065 [kgm2] x = [0.0873 (b + ab)] /ba = 0.4

(3.49)

The nonlinear pitch spring stiffness was represented by a quintic polynomial as follows

ky(y) =

5i=1

iyi1 (3.50)

where the unknown i extracted from experimental data [66] are given as

{i} =

2.8 62.3 3709.7 24195.6 48756.9 TThe first simulation was run with freestream velocity U = 20 [ms1] while the initial

conditions for the system states, observed variables, and estimates were chosen as

follows (0) = 0.1 [rad] (0) = 0 [rads1] h (0) = 0 [m]

h (0) = 0 [ms1] 0 (0) = 0 1 (0) = 00 (0) = 0 j (0) = 0 j (0) = 0

(3.51)

Figure 3.2 shows the closed-loop plunging, pitching, and control surface deflection

time-histories for the case when the actuation is on at time zero. In the presence

of active control, the system states are regulated to the origin fairly rapidly. It is


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interesting to note that the adaptive control strategy is able to quickly regulate the

plunging and pitching response in the presence of large uncertainty in the parameters.

The second simulation was run with freestream velocity U = 15 [ms1] with the same

initial conditions as (3.51). However, in this case, the system is allowed to evolve

open-loop (i.e., (t) 0) for 15 [s] to observe the development of an LCO. Duringthe first 15 seconds, the estimation strategy is allowed to evolve according to the

dynamics of (3.13), (3.14), (3.16). This is possible because the estimation strategy

is designed to be stable independent of the control methodology used. However, the

estimation strategy of (3.30) and (3.41) are turned off because of their dependence

on the control strategy of (3.39). At t = 15 [s], the control is turned on and the

immediate attenuation of oscillations can be seen in Figures 3.3 and 3.4. Again, theresponse is seen to be quick in the presence of large model uncertainty.

Remark 6 It is worth remarking that the methodology presented here can be ex-

tended to the compressibleflight speed regimes. In this sense, pertinent aerodynamics

for the compressible subsonic, supersonic and hypersonicflight speed regimes have to

be applied [73]. However, the goal of this research was restricted to the issue of the

illustration of the methodology and to that of highlighting the importance of the im-

plementation of the nonlinear adaptive control on the lifting surfaces equipped with a

flap.


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Figure 3.2 Time evolution of the closed-loop plunging and pitching deflections andflap deflection for U = 20 [ms1]; U = 1.6UFlutter


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Figure 3.3 Time-history of pitching deflection for U =[15 ms1], U = 1.25UF lutter


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Figure 3.4 Phase-space of the pitching deflection for U = 15 [ms1], U = 1.25UFlutter


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CHAPTER 4

TRACKING CONTROL OF AN UNDERACTUATED

UNMANNED UNDERWATER

VEHICLE.

System Model

The kinematics of the UUV attitude can be represented by the unit quaternion

[86] q(t)

qo(t) qv(t) R R3, which describes the orientation of an orthogonal

coordinate frame B attached to the UUV center of mass with respect to an inertial

reference frame I expressed in I and is governed by the following differential equation

[80]

qv =1

2(qv + qo)

qo = 12

qTv (4.1)

where (t) R3 is the angular velocity of B with respect to I expressed in B.The

rotation matrix that translates vehicle coordinates into inertial coordinates is denoted

by R (q) R33

and is calculated from the following formula [87]

RT (q) =

q2o qTv qv

I3 + 2qvqTv 2qoS(qv) (4.2)

where I3 R33 is the 3 3 identity matrix, and S() R33 is a skew-symmetricmatrix with the following mapping characteristics

S(y) =

0 y3 y2y3 0 y1

y2 y1 0

y R3 . (4.3)

The translational kinematic relationship for the UUV is given by the following ex-

pression [84]

p = Rv (4.4)

where p (t) R3 represents the position of the origin of B expressed in I, v (t) R3

denotes the translational velocity of the UUV with respect to I expressed in B.


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The translational and rotational equations of motion for the UUV are given by the

following [78]Mv = S() Mv + h (v) + B1u1Jw = S(v) M v S() J + g () + B2u2 (4.5)

where M, J R33 denote the positive definite, constant mass and inertia matrices,respectively, h (v) R31 captures the hydrodynamic damping interactions, B1 =

1 0 0T R3, u1 (t) R1 represents the translational force input, g () R31

is the matrix of gravitational and buoyant effects, B2 = I3 R33, and u2 R3

denotes the rotation vector input.

Remark 7 The following properties of the rotation matrix for R (q) and the skew-

symmetric matrix S() will be utilized in the subsequent control development[78]

I) R = RS()II) RTR = I3III) ST (y) = S(y) y R3

(4.6)

As previously mentioned, Property I of (4.6) represents a slight shift in the structure

of the dynamics for R (q) [78] as compared to previous work.

Problem Formulation

Our main objective is to design the translational force input u1 (t) and the rota-

tional torque input u2 (t) such that p (t) tracks a sufficiently smooth desired position

trajectory pd (t) R3 (i.e., pd (t), pd (t), pd (t),...pd (t) L). The position tracking

objective is complicated due to the fact that the UUV is equipped with only one

translational actuator aligned along one axis of B. Therefore, the translational con-

trol input must be designed in conjunction with the rotational torque input such that

the vehicle tracks pd (t) . To this end, the position tracking error signal ep (t) R3

isdefined in the following manner

ep = RT (p pd) . (4.7)

In addition to the underactuation constraint, the control design must compensate for

uncertainty in parameter values of the inertia matrix J, the hydrodynamic damping


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coefficients of h (v), and the coefficients of the gravitational matrix g (). In lieu of

a dynamic estimation approach, a constant parameter estimate will be utilized in

a feedforward term with the resulting parameter mismatch effects compensated for

through the injection of a bounding function within the control input expressions for

u1 (t) and u2 (t). The controller is designed under the assumption that the inertia

matrix M is known, and the translational position p (t), the translational velocity

v (t), and the angular velocity (t) are measurable.

Controller Design

After taking the time derivative of (4.7) and substituting in the translational

kinematics of (4.4), the open-loop tracking error dynamics for ep (t) are given by the

following expression

ep = S() RT (p pd) + v RT pd= S() ep + M1ev + (M1 I3) RT pd (4.8)

where the properties of (4.6) have been utilized, the term M1RT pd has been added

and subtracted to the right hand side of (4.8), and the translational velocity tracking

error signal ev (t) R3 is defined in the following manner

ev = Mv RT pd. (4.9)

After taking the time derivative ofev (t) and substituting in the translational dynam-

ics of (4.5), the open-loop linear velocity tracking error dynamics are given by the

following

ev = S() ev +

h (v) RTpd

+ B1u1. (4.10)

where the definition of (4.9) has been utilized.

In order to simplify the control design, thefi

ltered tracking error signal r (t) R3

is defined in the following manner [78]

r = ev + ep + (4.11)

where R1 denotes a positive, scalar constant, = 1 0 0 T R3, and 1 R1represents a positive design constant. The open-loop filtered tracking error dynamics


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where the auxiliary matrix B R33 and the auxiliary vector (t) R3 are definedby the following

B = 1 0 00 0

1

0 1 0 = u1 2 3

T. (4.17)

At this point, the scalar control input u1 (t) is designed in the following manner

u1 =

1 0 0

u1 + 1 (4.18)

where u1 (t) R3 represents an auxiliary control input vector. After substituting(4.18) into (4.16), the open-loop filtered tracking error dynamics for r (t) can be

expressed as follows

r =

S() r + Y11

M1ep+M1ev RT pd + (M1 I3) RT pd

+M1ep + [Bu1 + Bz](4.19)

where the auxiliary vector signal z(t) R3 is defined by the following

z = Bzu1, Bz = 0 0 00 1 0

0 0 1

. (4.20)

Based on the structure of (4.19), the auxiliary control input u1 (t) is designed in the

following manner

u1 = B1

M1ev + RTpd (M1 I3) RT pdM1ep Y11 krr

21 (vs) r1 (vm) rm + 1

(4.21)where kr R1 denotes a positive scalar control gain, 1 R1 represents a smallpositive constant, and the functions s and m are defined in the following manner

ys =

yTy +

ym = yTy + = ys

y R3 (4.22)

where R1 denotes a small positive scalar constant. After substituting (4.21) intothe open-loop dynamics of (4.19), the closed-loop filtered tracking error dynamics for

r (t) are given by the following expression

r = krr S() r M1ep + Bz+

Y1 (v) 1

21 (vs) r

1 (vm) rm + 1

. (4.23)


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Remark 8 Based on the definitions of (4.22) and the non-decreasing characteristic

of1 (v (t)), the following inequality will be used in the subsequent stability to boundthe effects of the parameter mismatch term Y1 (v) 1

1 (vs) 1 (v) > 1 (vm) . (4.24)

Remark 9 The functions given in (4.22) are utilized in lieu of the standard Euclidean

norm in u1 (t) so as to ensure that the time derivative of 1 (v (t)) is well-definedwhen backstepping on the signal z(t). That is, the time derivative of s and mcan be calculated from the following expression

d

dtys =

d

dtym =

yTy

(yTy + )y R3 (4.25)

The remainder of the control design involves the development of the torque input

u2 (t) such that the auxiliary variable z(t) is regulated to zero. That is if z(t) = 0,

then the auxiliary control input signal u1 (t) is injected into the filtered position

tracking error dynamics (4.19) via the angular velocity vector (t) in order to promote

position tracking.

After taking the time derivative ofz(t), multiplying both sides of the resulting ex-

pression by the unknown inertia matrix J, and substituting in the rotational dynamicsof (4.5), the open-loop dynamics for z(t) are given by the following expression

Jz = S(v) Mv +

g () S() J JBz.u1

+ B2u2 (4.26)

where the explicit elements of.u1 (t) are provided in the Appendix. At this point, the

structure of g () is assumed to be of a form to allow the bracketed term of (4.26) to

be linearly parameterized into the following form

Y2 (p,, v) 2 = g () S() J JBz.

u1 (4.27)

where Y2 (p,, v) R3n denotes a known regression vector, and 2 Rn is the vectorof unknown constants. In addition, it is also assumed that the following bounding

relationship exists Y2 (p,v, ) 2 2 () (4.28)

8/3/2019 Nonlinear control of mechatronic

Nonlinear control of mechatronic systems

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