Impedance Adaptive Controller for a Prototype of a...

Research ArticleImpedance Adaptive Controller for a Prototype of a WhiplashSyndrome Rehabilitation Device

Isai Guzman-Victoria1 Ivan Salgado-Ramos2 Manuel Mera-Hernandez13

Isaac Chairez14 and Hafiz Ahmed 5

1Unidad Profesional Interdisciplinarıa de Biotecnologıa Instituto Politecnico Nacional Mexico2Centro de Investigacion y Desarrollo de Tecnologıa en Computo Instituto Politecnico Nacional Mexico3Escuela Superior de Ingenierıa Mecanica y Electrica Ticoman Instituto Politecnico Nacional Mexico4Escuela de Ingenieria Instituto Tecnologico de Estudios Superior de Monterrey Campus GuadalajaraMexico5Coventry University Priory Street Coventry CV1 5FB UK

Correspondence should be addressed to Hafiz Ahmed ac7126coventryacuk

Received 15 January 2019 Revised 2 April 2019 Accepted 28 April 2019 Published 28 May 2019

Academic Editor Asier Ibeas

Copyright copy 2019 Isai Guzman-Victoria et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The objective of this study was to design of an output based impedance adaptive controller for a special class of cervical orthoses aclass of biomedical devices for the rehabilitation of neck illnessesThe controller used the adaptive slidingmode theory to enforce thetracking of the reference trajectory if the patient was not resistant to the therapy If the patient rejects the orthosis activity a secondimpedance-based controller governs the orthosis movement allowing the patient to take the leading role in the orthosis sequenceof movements The proposed controller considers a weighted controller combining the tracking and the impedance controls in asingle structureThemonitoring of the external force was evaluated on a novel weighting function defining on-line the role of eachcontroller The proposed orthosis was motivated by the prevalence of whiplash which is a syndrome that is produced by forcedhyperextension and hyperflexion of the neckThis study included the development of a technological prototype of the orthotic typeto support the recovery of patients diagnosed with whiplash The sections that make up the orthotic device are two independentsystems that move the patientrsquos head in the sagittal and frontal planes For this purpose the mechanical structure of the cervicalorthosis was made up of 7 pieces printed in 3D with polylactic acid (PLA) The operation of the cervical orthosis was evaluatedin two sections (a) using a simulation system which consists of a spring with an artificial head and the development of a graphicinterface in Matlab and (b) evaluating the controller on the proposed orthosis With these elements the follow-up of the trajectoryproposed by the actuators was evaluated as well as its performance in the face of the opposition that a patient generates Thesuperiority of the proposed controller was confirmed by comparing the tracking efficiency with proportional-integral-derivativeand first-order sliding variants

1 Introduction

The increasing number of injuries affecting all the articu-lations in human body has motivated the development ofremarkable assistance rehabilitation devices [1] Exoskele-tons lower and upper limbs robotic orthosis and so onappear as new medical tools to help the rehabilitationmedical doctor and physiotherapists in performing moreefficient therapeutic procedures [2] The methodological and

technological approaches aimed at developing such classof rehabilitation devices are now mature and they haveproduced medical devices which are now validated Amongthe illnesses restricting the articulations movements thereare several sicknesses which may affect the mobility of thecervical vertebra Vertebral herniated disk radiculopathymeningitis and vertebral fractures are the most commonpathologies associated with the cervical section of the spinalcord [3ndash5] In general their symptoms include persistent

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 9604179 21 pageshttpsdoiorg10115520199604179

2 Mathematical Problems in Engineering

pain controlled mobilization difficulties of the affected zoneand restricted articulation displacements as well as muscularinflammation [6]

The whiplash syndrome is becoming one of the mostprevalent cervical pathologies This syndrome is the con-sequence of a fast angular movement in either lateral orfrontal anatomical planes [7 8] It is recognized as an inca-pacitate pathology with acute periods of pain and restrictedmobilization of the head This syndrome has been associatedwith violent automobile collisions [9] The hyperextensionof the muscles and ligaments in the cervical section are thephysiological fundamentals of the syndrome The relevanceof this pathology has motivated the introduction of theso-called Quebec classification which contains the clinicalmanifestations and the pathology degree This classificationalso indicates the suggested treatment for each degree of thepathology [10]

In the last few years some technological options appearedas alternative technologies for the treatment of neck vertebralillnesses including the neck hyperextension and hyperflex-ion (whiplash syndrome) [11 12] Most of these devicesmay ensure the complete immobilization of the vertebralarticulation in the cervical area [13 14] The conventionaltreatment consists of immobilizing the cervical region of thespinal cord by wearing therapeutic rigid collars as well aspharmaceutical therapy [15] A cervical collar is an orthosisfitting the patientrsquos neck anywhere from the jaw to the chest Ithas two major functions restricting the cervical movements(flexion extension and lateral tilt) and supporting the headto allow healing the muscles and the ligaments In additionthe collar decreases the muscle spasms and control the painin the neck [16 17]

The rigid or semirigid collars (made of soft material)represent the main assistance technology for the cervicalinjuries treatment Nevertheless there are some drawbacksin using these collars [18] In particular the continuousimmobilization in all planes can lead to sacral pressure pointsheel and elbow breakdown risk of deep venous thrombosismuscle atrophy etc [19]Most of these inconveniences can besaved if a complementary mobilization therapy of isometricexercises is realized It is usual that this complementarytherapy is not recommended because the necessity of aqualified physiotherapist intervention [20]

Nowadays the emerging of rehabilitation technologyincluding the development of active orthosis has motivatedthe introduction of automatized devices aimed to mobilizethe cervical region of patients wearing the rigid collar Themulti-cervical unit is a commercial product (BTE Technolo-gies) aimed at offering pain reduction by restricting the necklateral movements however this device could induce a ver-tical continuous force which may contribute to the treatmentof conical and acute pain vertebral disks compression andsome others This equipment requires the recurrent visits ofthe patient to the specialized clinical facilities [21]

ReSolve Halo is a cervical device with external fixationsystem It has the mechanical sections to immobilize thecervical articulations and it offers complementary regulatedsupport for the head The position of the neck area is fixedin advance and the patient is required to visit the physician

if some adjustment is needed This device offers the weightdischarge of the head by fixing the device to the thorax area[22] The Cervical Thoracic Orthosis provides a conservativetreatment for the cervical pathology It can restrict therotational movement of the neck but it can hold the headsection immobilized This device is attached to the patientrsquosthorax to offer the weight discharge [23]

All the devices proposed for aiding the therapeuticsof the neck illnesses have the disadvantage of forcing theneckrsquos immobilization which is not the best solution for therequested rehabilitationNotice that themobilization realizedby these devices did not consider the potential resistanceof the patient to the orthosis activity This aspect must betaken into account considering the possible pain inducedfor the application of orthosis-based treatment [24] Themain motivation for developing an active orthosis for thewhiplash syndrome is the increasing number of neck illnesseswhich are consequences of car accidents long periods ofmuscle inactivity yielding cervical vertebral inflammationsetc These sicknesses are treated by fixed orthoses which aresupporting the neck by immobilizing the cervical sectionbut not offering an integral rehabilitation The proposedorthosis makes an enhancement in this direction because itis mobilizing the neck section in a controlled way while thepatientrsquos head is still held by the orthosis [25ndash29]

It is usual that active orthosis (mechanized rehabilitationdevices with automatized movements) should be controlledby robust controllers which are not considering the patientresistance This design requirement implies the applicationof a class of human-robot (or patient-orthosis in this case)interaction framework solution In such class of approachesthe tracking of reference trajectories still plays a relevantrole in the design of the automatic controller However thesafe robot operation considering the patient resistance to theorthosis action plays a primary role because the necessity ofensuring a comfortable assisted rehabilitation [30]

The current control designs to solve the human-robotinteraction may not provide sufficient flexibility for collabo-ration tasks (implying a smooth transition between the track-ing and the resistance operation conditions) requiring thatonly two cases are admissible the robot leads or follows thehuman by assessing the performance of human continuously[31] Only a few approaches have considered the possibilityof transiting from the robot leading (orthosis-in-charge)scenario to the human ruling case (patient-in-charge) [32]Within these works diverse studies have provided nonlinearapproaches to regulate the human-robot interaction [33ndash35] inspired by the hybrid control designs Within thehybrid solutions the application of approximate mathemat-ical descriptions to the orthosis model has been used alongthe last years [36ndash38] Such approaches try to produce softinteraction between patient and orthosis Collaborative robotcontrol is also providing new paradigms to control activeorthosis where the patient exerts the rehabilitation procedurebut keeping a predefined safety zone [39ndash41]

Impedanceadmittance-based controllers can solve eachof the proposed scenarios independently [42] Howeverthere is a necessity of changing the desired impedance ineach case This condition may lead to the introduction of

Mathematical Problems in Engineering 3

switched controllers inducing high-frequency oscillationsthat are not justifiable for the rehabilitation of the cervicalregion by the proposed active orthosis [43 44] In additionthe common impedance controllers assume that there is anaccurate mathematical representation of the robotic devicewhich is a strong assumption in many realistic cases Indeedthere is a tread-off between the softness of the automaticcontrol realization ensuring the safe operation of the orthosis(usually solved by adaptive variants of the controller) andthe stipulated robustness against the nonmodeled sections ofthe robotic orthosis Several options of adaptive controllershave dealt with these design problems such as [45ndash47]where relevant approaches handle the thread-off between theposition and the force based forms

A major opportunity to solve the trade-off in the cervicalorthosis development is the implementation of the so-calledadaptive sliding mode theory (ASMT) [48] The recentadvances in the ASMT have led to finding a remarkabledesign strategy that maintains the finite-time convergenceand robustness of the regular sliding mode based controllersbut ensures the reduction of the chattering phenomenon(high-frequency oscillations occurring when the slidingmode is enforced) These combined characteristics motivatethe application of ASMT as a potential solution to developimpedance adaptive sliding mode controllers to regulate thesafe operation of a cervical orthosis This novel controllerrepresents the major contribution of this study

The paper is organized as follows Section 2 details thecervical orthosis proposed for the treatment of the whiplashsyndrome Section 3 defines the impedance adaptive slidingmode controller including the details of the variants of stateand output based approach Section 4 describes the realcervical orthosis developed for testing the proposed adaptivecontroller Section 5 describes the numerical evaluation ofthe proposed controller over a simulated version of theorthosis Section 6 explains the experimental evaluation ofboth the developed cervical orthosis and the applicationof the impedance adaptive controller implemented in anembedded device Section 7 closes the paper with some finalremarks

2 The Cervical Orthosis

The orthosis for the active treatment of the neck disorderassociated with the whiplash syndrome considered a fullyactuated robotic manipulator The design considered a fixedbodymobilized by two independent articulations supportingthe central orthosis body The robotic system included asupportive device allowing the patient to carry the orthosiswithout the necessity of an external support This designagrees with the regular mechanism used to track and movethe articulated section of the injured necks of the selectedpatients

The cervical active orthosis consisted of a low-weighttwo-degree-of-freedom robotic manipulator with a specialend effector which can hold the patientrsquos head The move-ments induced by the suggested robotic manipulator cor-respond to the usual mobilization of the cervical region

in the rehabilitation therapies The orthosis was designedconsidering these main aspects

The dimensions of the mechanical orthosis design werenot prefixed Indeed the orthosis design allowed its config-urations modifications corresponding to averaged anthropo-morphic measures A general view of the computer assisteddesign version of the proposed rehabilitation device appearsin Figure 1

Based on the mechanical structure associated with theneck orthosis let us consider that 119902 isin R2 describes the jointsconfiguration while 119875 isin R3 describes the position of thecenter section in the upper part of the orthosis (Figure 1) inthe task space

The position of point 119875 is related to the join configurationby the nonlinear function Γ (direct kinematics) that is 119875 =Γ(119902) where Γ R2 997888rarr R3

The orthosis structure proposed in this study can bemodeled (considering the joint configuration) by introducinga second order nonlinear mathematical model correspondingto [49]

119872(119902 (119905)) 1198892119902 (119905)1198891199052 + 119866 (119902 (119905)) + 119862(119902 (119905) 119889119902 (119905)119889119905 ) 119889119902 (119905)119889119905+ 120595(119902 (119905) 119889119902 (119905)119889119905 119905) = 120591 (119905) + 120591119903 (119905) (1)

where 119902 = [1199021 1199022]⊤ The matrix 119872 R2 997888rarr R2times2 stands forthe inertia term 119866 R2 997888rarr R2 describes the gravitationaleffect terms and 119862(119902 119902) R2 times R2 997888rarr R2times2 is the so-calledCoriolis matrix The function 119866 must be Lipschitz while theCoriolis matrix must have a bounded consistent matrix Theuncertain term 120595must be continuous with respect to its twofirst arguments By the properties of the inertia matrix thefollowing assumption holds

0 lt 119898minus le inf 119905ge010038171003817100381710038171003817119872minus1 (119902)10038171003817100381710038171003817119865 (2)

with 119898minus being a positive and constant scalar and sdot 119865 aconsistent matrix norm

The function 120595 R2 times R2 times R+ 997888rarr R2 containsall the perturbations and uncertainties affecting the modelof the neck orthosis Such term may include dried frictionnonmodeled internal interconnections between the rotorshaft in the actuator and the corresponding robot joint Theterm 120591 = [1205911 1205912]⊤ represents the input torque vector for bothactuated articulations

The term 120591119903 represents the effect of the constraint forceinduced by the patient that can oppose the orthosis move-ment This opposition may come from the own pathologycharacteristics or the therapy evolution Usually this forcesatisfies the following model 120591119903(119905) = 119869(119902(119905))⊤119891119890119909119905 where119869(119902(119905)) is the Jacobian that relates the terminal velocity ofall arms in the orthosis to the time derivative of generalizedcoordinates 119902 Notice that 119891119890119909119905 is the resistance function ofthe patient against the application of the induced movementby the orthosis

Theorthosis configuration consisting of a two-degree-of-freedom manipulator justifies the Jacobian 119869 being bounded


155cm123cm

35cm 84cm 27cm

56c

m

112

cm

94c

m

118

cm

32c

m

16c

m

42cm 42cm

q1

q2

Y

X

(a) (b)

Figure 1 Computer assisted designmodel of the cervical spinal cord orthosis (a)Dimensions of the elements included in the cervical orthosisas well as the generalized coordinates of movement (b) General view of the cervical orthosis fixation over the supportive collar

according to the result presented in [50] In consequence itis reasonable to introduce the following bound

sup119905ge0

10038171003817100381710038171003817119869 (119902 (119905)) 119869⊤ (119902 (119905))10038171003817100381710038171003817119865 le 119869+ (3)

with 119869+ being a positive scalarNotice that introducing the position coordinates and the

operation space states enforces the necessity of defining thepatient resistance that is the external force which could bemodeled as 119891119890119909119905 = 119870119904119905(119902119904 minus 119902) where 119902119904 isin R3 representsthe position where the distal elements of the orthosis shouldmove back if the external force disappears The matrix 119870119904119905 isinR2times3 defines the patient stiffness and it is usually unknownNotice that the class of admissible external forces satisfies

sup119905ge0

10038171003817100381710038171198911198901199091199051003817100381710038171003817 le 119891+119890119909119905 (4)

where 119891+119890119909119905 is a positive scalarThe definition of the impedance adaptive control requires

an additional part When an external force acts on the setof end effectors (like the resistance enforced by the patient)the objective of the impedance controller is to cause the endeffector to respond according to some defined dynamicsThispart of the controller allows changing the dynamic responseof the orthosis with respect to the patient

For example if the patient should not be forced by theorthosis the desired dynamics can only track the patientsmovements and then reduce the impedance response to zeroin the best case On the contrary if the orthosis must force thepatient to complete a specific therapy the desired dynamics

can be adjusted to keep the resistance in an acceptable rangeThe bounds of this range should be proposed by a medicaldoctor

3 Adaptive Control Design forthe Neck Orthosis Device

Thedesign of the automatic controller requires the definitionof the model aimed at characterizing the human-machineinteraction between the patient and the orthosis This studyconsiders two modes of human-machine interaction

(i) Robot-in-charge The robot plays the leading roleto track the desired trajectory corresponding to theproposed therapy At this condition the human isnot resistant to following the induced moment of therobotic orthosis In this case the interaction force119891119890119909119905is not large

(ii) Human-in-charge The human exerts an increasinginteraction force 119891119890119909119905 on the robotic orthosis totake control actions The robot becomes passive tocomplement humanrsquos intention and movement

Notice that both orthosis-human interaction models aredefining a class of interactions between the patient andthe rehabilitation device but fixing different purposes forthe controller The switch between modes of operation isautomatically defined by estimating the interaction forcebetween the orthosis and the patient

If the robot-in-charge mode is active the working envi-ronment of the orthosis operation is enforcing the patientrsquos


02

0

04

06

08

1

01

0

03

05

06

08

02

04

07

09

1W

eigh

ting

func

tion

w(

ＲＮ)

Second component ＲＮ

05

10

minus10minus5 0

510

minus10minus5

First component ＲＮ

Figure 2 Weighting function 119908(119891119890119909119905) calculated with 119877 = 22 119887 = 30 and 119888 = 20neck movement without resistance In this case the roboticdevice can help the patient to track the desired trajectoryaccurately without supervision from humans In the secondmode (human-in-charge) an external force increases by thepatientrsquos resistance to the robotic orthosis regular movementThen if the force119891119890119909119905 grows over a predefined value then it isexpected that the device stops its movement to avoid forcingthe cervical vertebra beyond a safe condition proposed by thetreating physician

The dual human-robot (patient-orthosis) interactionmodes can also be applied in sensory-feedback robotic ortho-sis to improve the treatment performance without inducingunsafe movement of the cervical region The orthosis func-tions within the robot-in-charge mode when it performs thevertebra manipulation if the external force sensory-feedbackreports a small patientrsquos resistance However if the patienttakes control in the human-in-charge mode because thepatient offers resistance to the execution of the therapy thesensory-feedback reports an external force moving outside apredefined limiting zone

The dual interaction depends on the measurement of theexternal force with certain degree of accuracy In general themeasurement of such force can be obtained directly froma sensor placed at some adequate position in the orthosisstructure An alternative option is using an indirect measure-ment of the external force by introducing a current sensor onthe actuator circuit There are several studies estimating theresistance force of the robot movement by implementing aclass of observers based on the current measurements In thiscase there is the assumption that 119891119890119909119905 can be measured on-line by the corresponding force sensor

The estimation of 119891119890119909119905 named 119891119890 is obtained by directmeasurement of the consumed current at each actuator 119868119894 119894 =1 2 and then the nonlinear transformation Ω defines 119891119890 =

Ω(1198681 1198682) The variation of this force 119891119890 defines the transitionbetween both modes of human-machine interaction

According to [51] it is necessary to introduce a monitor-ing function describing the transition between each interac-tion mode namely ℎ(119891119890) which is proposed as follows

ℎ (119891119890) = 10038171003817100381710038171003817119891119890100381710038171003817100381710038172 minus 1198772 (5)

with 119877 being a positive constant defined by the designerNotice that it may be helpful to define this function if theorthosis could track the reference trajectory without patientrsquosresistance then ℎ(119891119890) lt 0 On the other hand if thepatient starts increasing its resistance against the orthosisperformance then (119891119890) increases from a negative to a positivevalue

Considering the monitoring function (119891119890) a weightingfunction can be suggested to characterize themode transitionnumerically This study used the following vector-valuedweighting function

119908(119891119890) = 1 minus 11 + 119887119890minus119888sdotℎ(119891119890) (6)

In (6) the parameters 119887 and 119888 are positive constant scalarswhich can be adjusted considering the expected transitionbetween interaction modes for each patient Figure 2 showsthe evaluation of function 119908 with respect to the componentsof 119891119890 The smoothness of function (6) is modifiable byadjusting the values of parameters 119887 and 119888

The function 119908 depends on the function 119891119890 which can beestimated by the current measurements on the actuators Thecorresponding relationships can be obtained from adequatecharacterizations which are proposed in the experimentalsection following the general ideas proposed in [52ndash54]


Based on the weighting function it is possible to definethe reference trajectories considering the desired position ofthe reference point 119875119889 The time-dependent sequence of 119875119889defines the movements of the central section for the roboticorthosis In consequence the dynamics of the desired angulardisplacement 119902119894119889 can be estimated as119889119889119905119902119889 (119905) = 119908 (119891119890 (119905)) Γ+ (119902 (119905)) 119889119889119905119875119889 (119905)minus 119908 (119891119890 (119905)) Γ+ (119902 (119905)) Δ119875 (119905) (7)

where Γ+(119902(119905)) is the pseudoinverse of Γ(119902(119905)) and Δ119875 =119875 minus 119875119889 Indeed the desired angular positions come fromthe inverse transformation related to the kinematics relationbetween the position of the end-effect position and theangular variations Notice that the product of Γ+119875119889 is 119902119889Then the only equilibrium point is 0 for Δ119875 which is indeedthe goal of the controller

31 Tracking Error Dynamics Introduce the tracking errordynamics 120575 defined as 120575 = 119902 minus 119902119889 The application of the statevariable theory suggests introducing the states 120575119886 = 120575 and120575119887 = (119889119889119905)120575 and then the following identities hold119889119889119905120575119886 (119905) = 120575119887 (119905)

119889119889119905120575119887 (119905) = 119865 (119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591 (119905)+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905) +119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤sdot 119891119890119909119905 (119905) minus 119889119889119905 [119908 (119891119890 (119905)) Γ+ (119902 (119905)) 119889119889119905119875119889 (119905)minus 119908 (119891119890 (119905)) Γ+ (119902 (119905)) Δ119875 (119905)]

(8)

In (8) the function 119865 R2 times R2 997888rarr R2 corresponds to119865(119902 (119889119889119905)119902) = minus119872minus1(119902)(119866(119902) + 119862(119902 (119889119889119905)119902)(119889119889119905)119902) thefunction 119866 R2 997888rarr R2times2 corresponds to 119866(119902) = minus119872minus1(119902)and the functionΦ R2timesR2timesR2 997888rarr R2 satisfies the follow-ing identity Φ(119902 (119889119889119905)119902 119905) = minus119872minus1(119902)120595(119902(119905) (119889119889119905)119902(119905) 119905)

The function 119865 satisfies the following inequality10038171003817100381710038171003817100381710038171003817119865(119902 119889119889119905119902)10038171003817100381710038171003817100381710038171003817 le 1198910 + 1198911 1003817100381710038171003817119902V1003817100381710038171003817 (9)

where 1198910 and 1198911 are positive constant scalars and 119902V =[119902⊤ (119889119889119905)119902⊤]⊤In equivalent form the uncertainties term Φ satisfies10038171003817100381710038171003817100381710038171003817Φ(119902 119889119889119905119902 119905)10038171003817100381710038171003817100381710038171003817 le 1206010 + 1206011 1003817100381710038171003817119902V1003817100381710038171003817 (10)

where 1206010 and 1206011 are positive constant scalarsThe suggested design of the reference trajectories 119902119889

aggregates both modes of operation The following analysisdemonstrates how each mode operates over the orthosis-patient interaction

(a)When the orthosis is completely ruling the movement(no patient resistance) that is 119908(119891119890(119905)) = 1 the dynamicsof the orthosis-patient dynamics satisfies (with 119891119890119909119905 = 0) thefollowing

119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905)

= 119865(119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591 (119905)+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(11)

The control problem formulated for this case implies thedesign of 120591 = 120591119905119903 in such a way that 120575 = [120575119886 120575119887]⊤ mustbe zero that is the articulation angles track the desiredtrajectory

(b) If the patient is resistant to the orthosis action then119908(119891119890(119905)) = 0 making sure also that 120591 must only compensatethe uncertain term Φ namely 120591 = 120591119888 Then the dynamicsof the orthosis-patient dynamics satisfies (with 119891119890119909119905 = 0) thefollowing

119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865 (119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591119888 (119905)

+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(12)

The control problem in this second case when the patientis ruling the orthosis action (119908(119891119890(119905)) = 1) must providethe fact that 120591119888(119905) = Φ(119902(119905) (119889119889119905)119902(119905) 119905) forall119905 ge 119879119888 Noticethat if the compensation is exact then the orthosis-patientinteraction satisfies the following dynamics

119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865(119902 (119905) 119889119889119905119902 (119905))

+ 119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(13)

The dynamic model presented in (13) is a damping systemin the sense that the patient is taking the control of theorthosis by using 119869(119902(119905))⊤119891119890119909119905(119905) as the external torque

The patientrsquos impedance against the robotic orthosismay vary from a comparatively large to a small value Theimpedance variation forces the transition from one model


to the other one (from the orthosis-in-charge to the patient-in-charge mode) Therefore there is a smooth combinationof the situation when the human can guide the orthosismovement and the case when the robotic device is playingthe assistance forces The transition scenario 0 lt 119908(119891119890(119905)) lt1 defines the so-called shared-control scenarios combiningthe patient control actions and the orthosis robot controlledrealization The proportion of transition stage along thistransition stage can be adjusted by setting the parameters inthe weighting function The variation of the radius 119877 definesspecific cases for different patients If the radius is set largeenough then the orthosis plays a more active role along thetransition stage but if the patient should be allowed to takethe device control the119877must be decreasedThismodificationdepends on the rehabilitation stage for each patient In thecase of fully paralysed vertebral movement the orthosis musttake the leading role to control the stability of the neckrsquosspinal cord section Otherwise if the patient is recovering themovement abilities then the 119877 value can be decreased

32 Design of the Reference Trajectories for the PatientrsquosTherapy The definition of the reference trajectory for thecentral point in the orthosis structure that is 119875119889 requires aparticular technique Usually this part of the process receivesthe name of planning trajectory which can be solved byimplementing the so-called Bezier polynomials In this studythis planning process considered a suitable interpolationmethod based on nonlinear differentiable functions In thiscase this part of the process also uses sigmoid functions

The efficient design of the reference trajectories impliesthe explicit solution of suitable algebraic systems of equationswhich defines remarkable sections of the planning process(defined by the time instants when the reference trajectoryis inflected) Such solution is time consuming with highcomputational complexity

The simpler composition of sigmoid functions may over-come these two drawbacks Such composition uses the initialand final values of the sigmoid functions as well as theso-called transition time for each inflection section 119905119896 Thesigmoid function considered in this part of the study obeys

119904 (119905) = 11990111 + 1199012119890(119905minus119905119896) + 1199013 forall119905 isin [119905119896 119905119896+1] (14)

where the parameters 1199011 1199012 and 1199013 are positive constantscalars The procedure aimed at constructing a referencetrajectory when a particular element must move from aninitial position 1205770 to a final position 120577119891 in a time period of119889119905 = 119905119896+1 minus 119905119896 seconds can be explained as follows

(i) Fix the value 1199011(1 + 1199012) + 119888 = 1205770(ii) Fix the value 1199011(1 + 1199012119890119889119905) + 1199013 = 120577119891(iii) Fix the value 1199011(1 + 119901211989005119889119905) + 1199013 = 05(120577119891 minus 1205770)The solution of this algebraic system yields the third-

order polynomial for the parameter 119901211990132 + 119908211990122 + 11990811199012 + 1199080 = 0 (15)

where 1199082 1199081 and 1199080 are straightforwardly estimatedAccording to [55] if 12058821 minus 412058830 ge 0 (1205880 = 11990822 minus 31199081 1205881 =211990832 minus 911990811199082 + 271199080) then 1199012 can be explicitly calculated as

1199012 = minus1199082 + 1198621 + 119862231198621 = (1205881 minus radic12058821 minus 4120588302 )

13

1198622 = (1205881 + radic12058821 minus 4120588302 )13 (16)

which is the real root of cubic equation (15) Based on theresult for 1199012 the value of 1199011 is as follows

1199011 = 05 (120577119891 minus 1205770) (1 + 119887) (1 + 119887119890119889119905)119887 (1 minus 119890119889119905) (17)

The parameter 1199013 is as follows1199013 = 1205770 minus 1198861 + 119887 (18)

The time derivative of the reference trajectory can bedirectly calculated from the sigmoid function In conse-quence 119875119889 and (119889119889119905)119875119889 can be used in the controller design

33TheAdaptive Controller The controller structure to solvethe orthosis-patient interaction process satisfies the followingmixed structure

120591 (119905) = 119866minus1 (119902 (119905))sdot 119867minus1 (119908 (119891119890 (119905)) 120591119905119903 (119905) + [1 minus 119908 (119891119890 (119905))] 120591119888 (119905)) (19)

To complete the control design let us define the slidingvector 119878 as

119878 (119905) = 120575119886 (119905) + 119867120575119887 (119905) (20)

with119867 isin R2times2 being a positive definite matrixThe dynamics of 119878 satisfies the following

119889119889119905119878 (119905) = 120575119887 (119905) + 119867 119889119889119905120575119887 (119905) (21)

Here (119889119889119905)120575119887(119905) comes from model (8) Notice that thedynamics of 119878 depends on themode executed by the orthosis-patient interaction

The proposed controller uses the following adaptivesliding mode theory

120591119905119903 (119905) = minus119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905) minus Π119905119903120591119888 (119905) = minus119870119888 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905) minus Π119888Π119905119903 = 120575119887 (119905) minus 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

Π119888 = 120575119887 (119905) + 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(22)


Rotary sensors

DC actuators+

+-

1

+

fext

xlowasttr

c

Ktr

Kc

xa[In 0n]

x

5 100minus5minus10

50

minus5 minus10

Figure 3 Simplified diagram of the proposed adaptive controller

where 120582 ge 0 120598 gt 0 and 119870119905119903 isin R2times2 and 119870119888 isin R2times2 are theadaptive controller matrix gains satisfying

119889119889119905119870119905119903 (119905) = (1205740 + 1205741 1003817100381710038171003817119902V1003817100381710038171003817)119870119905119903 (119905) sign119890 (119878 (119905))minus 119872 [119870119905119903 (119905) minus 119870+119905119903]++119872[120583119905119903 minus 119870119905119903 (119905)]+119889119889119905119870119888 (119905) = (1205740 + 1205741 1003817100381710038171003817119902V1003817100381710038171003817)119870119888 (119905) sign119890 (119878 (119905))minus 119872 [119870119888 (119905) minus 119870+119888 ]++119872[120583119888 minus 119870119888 (119905)]+

(23)

with sign119890(119878(119905)) = |119878(119905)119878(119905)| minus 120572V where sign119890(119878(119905))is the vector formed with the components of the filteredsignal associated with 119878119894119878(119905) 119894 = 1 2 Figure 3 showsa simplified diagram of the proposed control including theestimation of the external force 119891119890119909119905 and the simultaneouscalculus of the time-varying gains The state dependenceof the gains (23) motivates labeling the controller (22) asan adaptive structure Notice that this description tries tohighlight the difference between the proposed controller andthe classical ones where a fixed value of gain may counteractthe perturbationsuncertainties effect without taking care ofthe operation mode

This study considers the application of a direct estimationof the external force119891119890119909119905 depending on the currents measuredat the DC actuators Therefore this study did not consider adirect estimation of a second ordermodel for the119891119890119909119905 Clearlythis is a potential future trend on our orthosis device but thisstudy considered a simpler form which can be solved withthe additional instrumentation of the current sensors as wellas the previous characterization of the relationship betweenexternal forces and actuators currents

The motivation of using adaptive form of sliding modecontroller is the necessity of having an accurate estimation ofthe uncertainties affecting the orthosis-patient system Evenmore some recent works are showing that some adaptiveforms of sliding controllers offer robustness against some

kind of perturbations (bounded with restricted derivative)as well as the finite-time convergence Originally the intro-duction of adaptive gains motivated the application of dis-continuous controllers (sliding mode type) with the smallerattainable chattering effect The application of the equivalentcontrol concept also offers the advantage of recovering anaccurate estimation of the uncertainties affecting the con-trolled system Such estimation can be further used as partof the control as a compensation term

This section presents the main result of this study in thefollowing lemma

Lemma 1 Consider the closed-loop dynamics of the activeneck orthosis under the assumption of a human-robot inter-action given in (8) satisfying restrictions (9) and (10) with theadaptive controller (19) adjusted with the time-dependent gains119870119905119903 and119870119888 given in (23) If the tuning parameters 120583119905119903 and 120582 ofthe gains119870119905119903 and119870119888 given in (23) satisfy120583119905119903 = 119867119865 1205780 + 120588119905119903

120582 = 1205781120583119905119903120583119888 = 120583119905119903(24)

with 1205780 = 1198910 + 1206010 1205781 = 1198911 + 1206011 and 120588119905119903 gt 0 then the origin isa finite-time stable equilibrium point for the tracking error 120575Proof The proof of this lemma is presented in three sectionsthe first considers the case when the robotic orthosis ishandling the neck without patient resistance the secondconsiders the case when the patient is taking control ofthe human-robot interaction and the third analyzes thetransition phase between the two previous cases

Orthosis Control Mode Based on the definition of the pro-posed controller the time derivative of the sliding vectorsatisfies the following119889119889119905119878 (119905) = 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905))

+ (119908 (119891119890 (119905)) 120591119905119903 (119905) + [1 minus 119908 (119891119890 (119905))] 120591119888 (119905))


+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+ 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)minus 119867 119889119889119905 [119908 (119891119890 (119905)) Γ+ (119902 (119905)) 119889119889119905119875119889 (119905)minus 119908 (119891119890 (119905)) Γ+ (119902 (119905)) Δ119875 (119905)]

(25)

Let us consider the case when the orthosis is ruling theneckrsquos movement then 119908(119891119890(119905)) = 1 and then (25) yields thefollowing119889119889119905119878 (119905)

= 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119905119903 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(26)

The substitution of 120591119905119903 leads to the following119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)

(27)

Now let us propose the energetic (Lyapunov-like) func-tion

119881119905119903 (119878) = 05 1198782 (28)

The function 119881119905119903 is positive definite 119881119905119903(0) = 0 and radi-ally unbounded The Lie-derivative of1198811 over the trajectoriesof (27) is as follows119889119889119905119881119905119903 (119878 (119905)) = 119878⊤ (119905) [119867119865(119902 (119905) 119889119889119905119902 (119905))

+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)]

(29)

Using inequalities (10) and (9) one gets the following119889119889119905119881119905119903 (119878 (119905)) le 119867119865 (1205780 + 1205781 1003817100381710038171003817119902V1003817100381710038171003817) 119878 (119905)minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905) (30)

Notice the following

119867119865 (1205780 + 1205781 1003817100381710038171003817119902V1003817100381710038171003817) minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172]lt 119867119865 (1205780 + 1205781 1003817100381710038171003817119902V1003817100381710038171003817) minus 120583119905119903 [1 + 120582 1003817100381710038171003817119902V1003817100381710038171003817] (31)

In view of the parameter selection in (24) then(119889119889119905)119881119905119903(119878(119905)) le 0 and then

119889119889119905119881119905119903 (119878 (119905)) le minusradic2120588119905119903radic119881119905119903 (119878 (119905)) (32)

The standard procedure can be used to prove that119881119905119903(119878(119905)) = 0 forall119905 ge 119879119905119903 = 120588minus1119905119903 radic119881119905119903(119878(0))Patient Control Mode Let us consider the case when119908(119891119890(119905)) = 0 then (25) yields the following

119889119889119905119878 (119905) = 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119888 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+ 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(33)

The application of the compensating controller 120591119888 yieldsthe following

119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119888 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)

(34)

Considering the value of 120583119888 and following a similarprocedure to the one shown in the case when 119908(119891119890(119905)) = 1then (119889119889119905)119881119888(119878(119905)) le 0 with 119881119888(119878) = 051198782 and then


The standard procedure can be used to prove that119881119888(119878(119905)) = 0 forall119905 ge 119879119888 = 120588minus1119888 radic119881119888(119878(0))Transition Phase If the transition process is considered andboth controllers 120591119905119903 and 120591119888 are applied then (25) yields thefollowing


119889119889119905119878 (119905) = 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 119908 (119891119890 (119905))sdot (minus119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905) minus 120575119887 (119905)+ 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)])+ [1 minus 119908 (119891119890 (119905))]sdot (minus119870119888 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905) minus 120575119887 (119905)minus 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905) + 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤sdot 119891119890119909119905 (119905) minus 119867 119889119889119905 [119908 (119891119890 (119905)) Γ+ (119902 (119905)) 119889119889119905119875119889 (119905)minus 119908 (119891119890 (119905)) Γ+ (119902 (119905)) Δ119875 (119905)]

(36)

The reorganization of (36) justifies the following differen-tial equation

119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905) + minus (119908 (119891119890 (119905))119870119905119903 (119905)+ [1 minus 119908 (119891119890 (119905))]119870119888 (119905)) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172]sdot 119878 (119905)119878 (119905) + 119908 (119891119890 (119905))sdot 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905) + 119908 (119891119890 (119905))sdot 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]minus 119867 119889119889119905 [119908 (119891119890 (119905)) Γ+ (119902 (119905)) 119889119889119905119875119889 (119905)minus 119908 (119891119890 (119905)) Γ+ (119902 (119905)) Δ119875 (119905)]

(37)

Considering that the function 119908(119891119890(119905)) is explicitly deter-mined in time then (38) is equivalent to the following

119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))

sdot [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905) + 119908 (119891119890 (119905))sdot 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(38)

Now let us propose the energetic (Lyapunov-like) func-tion 119881119898119905 (119878) = 05 1198782 (39)

This function is also positive definite 119881119898119905(0) = 0and radially unbounded The Lie-derivative of 119881119898119905 over thetrajectories of (38) corresponds to the following119889119889119905119881119898119905 (119878 (119905)) = 119878⊤ (119905)

sdot [119867119865(119902 (119905) 119889119889119905119902 (119905)) + 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)]+ 119878⊤ (119905)sdot [119908 (119891119890 (119905))119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)]+ (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))sdot [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)

(40)

Using inequalities (10) and (9) as well as the assumptionsregarding the bounds for the Jacobian 119869 (3) and the 119891119890119909119905 (4)one gets the following119889119889119905119881119905119903 (119878 (119905)) le 119867119865 (1205780 + 119898minus (119869+)12 119891+119890119909119905 + 1205781 1003817100381710038171003817119902V1003817100381710038171003817)

sdot 119878 (119905)minus (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))sdot [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)

(41)



Then in view of the values for 120583119905119903 and 120582 (119889119889119905)119881119905119903(119878(119905)) le0 yielding the following119889119889119905119881119905119903 (119878 (119905)) le minusradic2120588119905119903radic119881119905119903 (119878 (119905)) (43)

The standard procedure can be used to prove that119881119905119903(119878(119905)) = 0 forall119905 ge 119879119905119903 = 120588minus1119905119903 radic119881119905119903(119878(0))Considering that all the three phases (orthosis control

patient control and transition) admits a Lyapunov functionand all three functions served to prove that the origin is arobust finite-time stable equilibrium point then it is possibleto argument that for all possible variants of 119908 this stabilitycharacteristic holds


34 The Output Feedback Version of the Adaptive ControllerThe adaptive controller proposed above can be implementedunder a technical restriction and the angular displacementand the angular velocity must be measured simultaneouslyyielding the estimation of the tracking error and its derivativeon-line that is (119889119889119905)119890(119905) and 119890(119905) can be applied directly inthe calculus of 119878

However these variables are rarely measured at the sametime unless important resources investment is realized Analternative is using a simple and effective robust and exactobserver or differentiator aimed at estimating a suitable aswell as accurate approximation of the tracking error timederivative [56]

In opposition to other sliding modes algorithms (secondorder) the super twisting (ST) can be applied in the caseof systems with relative degree equal to one with respect tothe chosen output [57] The ST algorithm application as arobust differentiator is described as follows Consider that1199081(119905) = 119903(119905) where 119903(119905) isin R must be differentiated andalso consider that 1199082(119905) = (119889119889119905)119903(119905) corresponds to its timederivative Based on the assumption of |(11988921198891199052)119903(119905)| le 119903+then the following auxiliary equations hold

1198891198891199051199081 (119905) = 1199082 (119905)1198891198891199051199082 (119905) = 11988921198891199052 119903 (119905)

(44)

The differential equations (44) correspond to the staterepresentation of 119903(119905) The ST algorithm used to obtain thederivative of 119903(119905) satisfies the following

1198891198891199051199081 (119905) = 1199082 (119905) minus 1205821 10038161003816100381610038161199081 (119905)100381610038161003816100381612 sign (1199081 (119905))1198891198891199051199082 (119905) = minus1205822sign (1199081 (119905))

1199081 = 1199081 minus 1199081119889 (119905) = 1198891198891199051199081 (119905)

(45)

Here 1199081 and 1199082 define the ST algorithm states and1205821 1205822 gt 0 are the algorithm gains [56] The variable 119889(119905)defines the differentiator result given in (45) Here

sign (]) = 1 119894119891 ] gt 0minus1 119894119891 ] lt 0 (46)

Notice that the sign function used here agrees with theformulation given by [49] which is not defining the multi-valued set for sign(0) corresponding to [minus1 +1] Notice thateither definition yields the slidingmode equationwithout anyambiguousness

The application of the STA as robust differentiator impliesthat the orthosis system (11) should be represented as thecomposition of the following couple of second order systems119889119889119905120575119886119894 (119905) = 120575119887119894 (119905)

119889119889119905120575119887119894 (119905) = 119891119894 (120575119886 (119905) 120575119887 (119905)) + 2sum119895=1

119892119894119895 (120575119886 (119905)) 119906119895 (119905)+ 120578119894 (120575119886 (119905) 120575119887 (119905) 119905)

(47)

where 120575119886119894 and 120575119887119894 are the 119894 minus 119905ℎ and (119899 + 119894) minus 119905ℎ states of (11)respectively The nonlinear functions 119891119894(sdot) and 119892119894(sdot) are thefunctions associated with the states 120575119886119894 and 120575119887119894 Similarly 120578119894is the corresponding uncertainty to the same subsystem

Using the STA the impedance adaptive control onlyrequires that the surface 119878 should be calculated as follows119878 (119905) = 120575119886 + 119867120575119887 (48)

where 120575119887 is the estimated velocity based only on the infor-mation provided by 120575119886 The forward complete characteristicof the orthosis (based on its robotic manipulator configura-tion) allows substituting the original surface value 119878 by theestimated value using the output of the STA

The study proposed in [51] provides the fundamentals fordesigning the soft-switch function between the human-in-control and the robot-in-control operation modes The men-tioned study offers just the manner to propose the weightingfunction which gives a continuous variation between thehuman-in-control and the robot-in-control modes and viceversa The rest of this study appears as a contribution in thesense that it offers a novel manner to consider the resistanceeffect of the patient against the orthosis action Also theexplicit estimation of the control adaptive gains is given inthis study considering the desired reference position Theseadaptive gains also establish the gain restrictions which yieldthe design of a bounded feedback controller Moreover theintroduction of an on-line robust and exact differentiatorintroduces the possibility of designing the output feedbackvariant of the suggested controller

4 The Orthosis Device

A simplified orthosis prototype served to evaluate thesuggested composite controller The SolidWorks computerassisted design software verified the proposed mechanicaldesign of the robotic orthosis The design tools included inthis software contributed to the selection of the constructionmaterial the machining processes and the actuators char-acteristics A finite-element numerical analysis defined thestatic and dynamic characteristics of the cervical orthosisdesign

The orthosis design included a support section aimed atcarrying up the mobile section of the orthosis This supportstructure consisted of an EVS-r4 collar which is madeof carbon fibers The complete weight of this section was0635 119896119892 The original objective of this collar was preventingwhiplash-like injuries for sportsmen


Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation

Micro-motor reducer Positionsensor

Figure 4 Cervical orthosis electronic instrumentation scheme depicting all the components needed to perform the proposed therapeuticprocedures The rotatory potentiometers are used to measure the angular position at each of the two articulations The H-bridge electronicboards as well as the micro DC motors are the electronic devices enforcing the movement of each orthosis coordinateThe optical couplingdevices isolated the high-power phase of the digital micro-controller (Tiva C 1294 Series)

A three-dimensional printing technique constructed themobile section of the orthosis The polymer acrylonitrilebutadiene styrene was the building material for the orthosisA 119885 minus 18 replicator printing machine executed the diversemachining processes in the orthosis device The printingcharacteristics were as follows resolution = 002119898119898 velocity= 1119888119898119898119894119899 etc

The instrumentation of the orthosis and its integrationwith the mechanical design and other steps are representedin Figure 4

The selection of each joint actuator considered the torquein the corresponding junction a set of two DC motors wasselected An electromechanical analysis was realized in orderto obtain the torque corresponding to each joint of theorthosis device

The set of sensors was composed of five linear precisionsingle turn potentiometers used for measuring the positionfor each joint The maximum resistance of the potentiometerwas 10 119896Ω and it is designed to be 33 119881 compatible theminimum range is 03∘ (Phidgets Slider 60 USA) The

actuators were DC motors (2501 HPCB 12V Pololu USA)with 105 g weight high-power carbon brushes operationvoltage = 12 volts angular velocity of 130 RPM torque = 30Kgcm The driving of actuators implemented the VNH5019(Pololu EVAL-VNH5019-P2 USA) Motor Driver Carrier(motor driver IC working with a voltage range of 55 to 24V continuous current of 12 A 25 to 5 V logic levels input20 kHz of PWM) The current sensor at the input of theactuator was the 20Amp Current Sensor ACDC with 50000v isolation (Phidgets RB-Phi-72 USA)

To implement the control algorithm a low-cost devel-opment platform was used This platform works with ahigh speed Texas Instrument ARM Cortex-M4F micro-controller The platform is the Tiva C Series TM4C1294and has the following features 120 MHz 32 bits CPU 1MB memory flash 256 KB SRAM 6KB EEPROM andserial communication interfaces The analog-digital con-verter (ADC) channels read the angular position sensorsignals The signals are processed individually Because themovements of all other joints are considered as perturbations


RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)

Figure 5 Comparison of the tracking trajectories associated with the movements executed by the simulated orthosis regulated by theproposed impedance adaptive sliding mode controller the PID and the regular first-order sliding mode variantThe simulation was realizedin the MatlabSimulink environment using the SimMechanics Toolbox

this led to the development of a novel scheme to manipulateeach of the movable sections of the orthosis the derivativeand the control for each actuator are computed Outputsignals are sent through the general purpose input-output(GPIO) pins as pulse-width modulation (PWM) signals Theoutput signals of the platform pass through an optocoupler toisolate the power stage and the controller board The powerstage for the actuators was handled by an H-bridge board inthe case of the micro metal gear motors a VNH5019 devicefrom Pololu was used This device is a dual H-bridge andit allows two motors to be driven at the same time for theother three metal gear motors a Pair Motor Driver CarrierVNH3SP30 of Pololu was used These drivers support thetechnical specification for the actuators estimated for theorthosis operation

5 Simulated Evaluation of theSuggested Controller

The first part of results corresponded to the class of exercisesdeveloped by the AO These numerical evaluations wereselected according to the regular treatment methods usedin actual physiotherapies of the whiplash syndrome Thesesequences of movements were included in Figure 5 andthey correspond to some exercises moving both articulationstogether These movements were obtained as solution ofthe numerical implementation (using the so-called SimMe-chanics Toolbox) of the AO in Matlab This orthosis wassimulated to evaluate the performance of the distributedstrategy of adaptive control The simulated model in Matlabwas evaluated with the actual mass and dimensions for eachmechanical section included in the AO This strategy servedto evaluate the controller and define the gains that must beused for tuning both differentiators as well as controllers

The trajectories shown in Figure 5 correspond to thesimultaneous lateral and frontal movements of the neck Inthis case the study evaluated the tracking of the reference

trajectories but considering that the patient is not offeringany resistance to the orthosis action

Figure 5 shows the comparison of trajectories obtainedby the application of three different controllers PID (blueline) first-order (green line) and the adaptive sliding mode(black line) variantsThePID controller structure satisfied thefollowing

120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)

The simulation of this controller used the following gains

119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)

The second controller used for comparison purposes wasa first-order sliding mode variant satisfying

120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)

with 119870119878119872 being the gain of this controller and119867 the positivedefinite matrix given by the following

119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM

minus11minus108minus106minus104minus102

Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)

Figure 6 Amplified view of the trajectories obtained with the implementation of the different controllers corresponding to referencetrajectories describing movements executed by the simulated orthosis The simulation was realized in the MatlabSimulink environmentusing the SimMechanics Toolbox

The proposed adaptive controller (22) was simulatedusing the following parameters

119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)

A set of diverse reference trajectories evaluated the abilityof the cervical orthosis to realize the movements correspond-ing to distinct therapies These different trajectories weredeveloped by implementing the composite sigmoid func-tions Figure 5 demonstrates the corresponding movementswhich were produced by the controller in simulation atdifferent stages of the therapy This figure emphasizes theeffect of tracking controller on both articulations leading tothe general movement of the orthosis Notice that in thisfigure the generated trajectories of the three controllers seemto be similar Nevertheless one may notice that there aresignificant differences between them Figure 6 demonstratesthe trajectories produced by all the three evaluated con-trollers in a time range of 633-654 seconds This amplifiedview exhibits the improvements in the tracking task of theadaptive controller (without taking into account the effect ofthe patient resistance) The adaptive gains included in the

controller reduced the regular high-frequency oscillationsproduced as a consequence of introducing the traditional dis-continuous sliding mode controller Moreover the proposedcontroller tracked the reference trajectory with smaller track-ing error than the other two controllers This characteristicwas observed for both articulations in the cervical orthosis

The control signals obtained in the numerical evaluationof trajectory tracking for the cervical orthosis demonstratedthat the adaptive sliding controller proposed in this studyreduced the energy needed to complete the reference track(Figure 7) The behavior of the control action demonstratesthat less energy is needed to complete the trajectory problemthan that in the case when PID or the nonadaptive slidingmode controllers were considered This can be confirmedby the range where the control action evolves during thenumerical simulations Moreover the adaptive controllershowed a similar response to the PID with better trackingquality and it has smaller high-frequency oscillations thanthe regular sliding mode version

The diminution in the vibrations exhibited in the controlsignals plays a relevant role in the orthosis-human interac-tion These vibrations may cause long-term injuries on eitherthe vertebra or the muscles needed to mobilize the patientrsquoscervical section This aspect represents a serious issue inthe application of active orthosis controlled by sliding modecontroller Nevertheless the accurate gain tuning based onthe adaptive form as well as the force compensation reducedthe oscillations on the tracking trajectory as demonstrated inthis study

The second set of numerical evaluations consisted indescribing the effect of the external force 119891119890119909119905 and how itmodifies the trajectories produced by the orthosis Figure 8demonstrates the components (a and b) of the simulatedexternal force which represents the patient resistance to theorthosis action These components correspond to gradualand fast patientrsquos resistance which ismaintained for 5 seconds


minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)

Figure 7 Comparison of the time evolution for the control signals associated with the movements executed by the simulated orthosisregulated by the PID and the classical first-order and the impedance adaptive sliding mode controller

10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)

Figure 8 Characterization of the simulated external force 119891119890119909119905 used in the numerical evaluation of the impedance adaptive sliding modecontroller (a) First component of 119891119890119909119905 (b) second component of 119891119890119909119905 (c) magnitude of 119891119890119909119905 and (d) evaluation of the weighing function interms of the simulated 119891119890119909119905and then released The magnitude of this external force isdepicted in Figure 8(c) The time variation of this magnitudeevaluated on the weighting function 119908(119891119890119909119905) appears inFigure 8(d) The variation of this weighting defines therelative contribution of both sections of the controller 120591119905119903 and120591119888

The presence of the external force119891119890119909119905modifies the trajec-tory tracking of the reference signal in the cervical orthosisNotice that if the external force produces the modificationof the weighting function and then the tracking quality is

reduced leading to the loss of the tracking as noticed in Fig-ure 9 Once the external force disappears then the controllersenforce the tracking of the reference trajectory This factconfirmed in two cases as shown in Figure 9 demonstratesthe benefits of introducing the weighted controller proposedin this study

Notice that Figure 9 contains the comparison of theadaptive sliding mode without and with the impedanceevaluation effect This comparison demonstrates that if theimpedance effect is not considered the adaptive sliding mode


05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance

Figure 9 Comparison of the trajectory tracking associatedwith themovements executed by the simulated orthosis regulated by the proposedimpedance adaptive sliding mode controller with and without taking the patient resistance effect

controller keeps tracking the reference without consideringthe resistance of the patient to the orthosis action This isa remarkable advance of the suggested controller because itmay ensure a safe operation of the patient-orthosis processfor the rehabilitation of the whiplash syndrome Moreoverthe resistance of the patient to the orthosis ensures that theorthosis keeps a constant position (shown in the figure at thesame time when the external force appears) This conditionensures that the patient is ruling the orthosis activity and thetracking problem is not considered in this case

6 Experimental Results

The functioning evaluation of the actual cervical orthosisused the adaptive controller implemented in the TIVA1294device The explicit discretization algorithm defined theimplementation of the adaptive controller The controllerused the Code Composer v80 software

The numerical integration implemented in the TIVA129board was the first-order Euler ODE algorithm The prede-fined sampling period was 001 s (this value was confirmedexperimentally)

The experimental evaluation of the orthosis functionconsidered a dummy phantom to characterize the trackingtrajectory effectiveness The dummy cannot move freelybecause it is fixed on a supporting spring which is repre-senting the force exerted by the neck muscles Then theforce applied by the spring on the robot could be regulatedby holding the spring tight to the base In this way thedummy served to prove the efficiency of the suggestedcontroller Moreover the modification of the holding forcebetween the basement and the spring represents differentscenarios of impedance responses between the patient andthe proposed orthosis An extended polystyrene head-likedummy phantom attached to the constructed orthosis served

as experimental platform The neck resistance and elasticitywere represented by a metallic spring (Hooke constant of48119873119898) which was attached to the bottom part of the head-like phantom (Figure 10)

The implemented impedance adaptive sliding mode con-troller used the following gains to adjust the orthosis move-ment

119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)

Figure 11 depicts the trajectories of the measured anglesfrom the potentiometric sensors placed at each articulation inthe orthosis The reference angular movement correspondedto a basic lateral movement which is considered one of thefirst therapies in the rehabilitation of the whiplash syndromeThe selected reference trajectories for each articulation con-firmed the capacity of the designed cervical orthosis to realizethe proposed movements corresponding to a simple lateraloscillatory movement one of the first therapy proposed byspecialists Notice that the suggested experimental resulthas the presence of the resistance enforced by the spring


(a) Frontal view (b) Lateral view

Figure 10 Frontal and lateral projections of the developed cervical orthosis and the attached head-like dummy phantomThis phantom hasthe spring attached at the bottom section which represented the neck resistance and elasticity

0

05

1

15

Ref ExpASM-Impedance

minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)

Figure 11 Experimental tracking trajectories associated with the movements executed by the real orthosis regulated by the proposedimpedance adaptive sliding mode controller

device This is the reason of the deviation observed along theintervals of [284 315]119904 and [463 545]119904 Such experimentalresults confirmed the effectiveness of the weighted controllersuggested in this study

Figure 12 shows a sequence of photos taken alongthe proposed trajectory shown in Figure 11 The referencetrajectories were proposed to mobilize the head in thelateral plane The orthosis movements were produced by the


Figure 12 Sequence of photos showing the movements executed by the real orthosis regulated by the proposed impedance adaptive slidingmode controller considering the resistance of the spring attached to the head-like dummy phantom

controller and tracked the proposed trajectory despite thepresence of the resistance induced by the spring Similarevaluationswere realized for three different trajectories whereindividual articulations or both articulations simultaneouslywere mobilized

7 Conclusions

This study detailed the design numerical evaluation andexperimental validation of an impedance adaptive slidingmode controller aimed at regulating the movements of an


active cervical orthosis The controller considered the effectof the patient resistance to the orthosis action as well as thetrajectory tracking problem considering the design of refer-ence trajectories corresponding to regular therapy sequenceof movements A novel weighting function introduces adynamic rule to characterize the activity of both controllersthe tracking oriented one and the force dependent realiza-tionThe specific rules to select the parameters of the adaptivecontroller were proposed The numerical evaluations ofthe controller confirmed the superior performance of theproposed controller with respect to the classical PID andfirst-order sliding mode variant Additionally the weighting rulefor the components (tracking and impedance) was evaluatedin the case when the patient resisted the orthosis actionThe controller responded avoiding the mobilization of thecervical section of the patient if the resistance is present Thesuggested controller did not have the usual switched structureproposed in diverse impedance base control designs whichsimplified the theoretical analysis of the trajectory trackingerror convergence The suggested controller evaluated onan own-developed cervical orthosis was able to completethe tracking trajectory corresponding to a basic cervicalrehabilitation therapy suggested for patients with whiplashsyndrome This study considered the application of a slidingmode controller with state dependent gains which pro-vided a solution for a robotic orthosis Notice that novelmethods including parametric and nonparametric adaptiveestimations of uncertainties and perturbations may offer adistinctive manner of solving the tracking trajectory problemfor rehabilitation devices such as the one considered in thisstudy [38 58 59]

Data Availability

The simulation and experimental data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] J M Matuszak J McVige J McPherson B Willer and JLeddy ldquoA practical concussion physical examination toolboxevidence-based physical examination for concussionrdquo SportsHealth vol 8 no 3 pp 260ndash269 2015

[2] D C Tong and J Breeze ldquoDamage control surgery and combat-relatedmaxillofacial and cervical injuries A systematic reviewrdquoBritish Journal of Oral and Maxillofacial Surgery vol 54 no 1pp 8ndash12 2016

[3] R Kanwar B E Delasobera K Hudson and W FrohnaldquoEmergency department evaluation and treatment of cervicalspine injuriesrdquo Emergency Medicine Clinics of North Americavol 33 no 2 pp 241ndash282 2015

[4] P Huang A Anissipour W McGee and L Lemak ldquoReturn-to-play recommendations after cervical thoracic and lumbarspine injuries a comprehensive reviewrdquo Sports Health vol 8no 1 pp 19ndash25 2016

[5] N Khezri T Ailon and B K Kwon ldquoTreatment of facet injuriesin the cervical spinerdquo Neurosurgery Clinics of North Americavol 28 no 1 pp 125ndash137 2017

[6] C Hrysomallis ldquoNeck muscular strength training perfor-mance and sport injury risk a reviewrdquo Sports Medicine vol 46no 8 pp 1111ndash1124 2016

[7] A Cesarani D Alpini R Boniver et al Whiplash InjuriesDiagnosis and Treatment Springer Science amp Business Media2013

[8] J J Wong P Cote H M Shearer et al ldquoClinical practiceguidelines for the management of conditions related to trafficcollisions A systematic review by the OPTIMa CollaborationrdquoDisability and Rehabilitation vol 37 no 6 pp 471ndash489 2015

[9] C Ritchie J Hendrikz G Jull J Elliott and M SterlingldquoExternal validation of a clinical prediction rule to predictfull recovery and ongoing moderatesevere disability followingacutewhiplash injuryrdquo Journal of Orthopaedic amp Sports PhysicalTherapy vol 45 no 4 pp 242ndash250 2015

[10] P Sarrami E Armstrong J M Naylor and I A HarrisldquoFactors predicting outcome in whiplash injury a systematicmeta-review of prognostic factorsrdquo Journal of Orthopaedics andTraumatology vol 18 no 1 pp 9ndash16 2017

[11] K Nas L Yazmalar V Sah A Aydin and K Ones ldquoRehabilita-tion of spinal cord injuriesrdquo World Journal of Orthopedics vol6 no 1 pp 8ndash16 2015

[12] S Mehta A McIntyre S Janzen E Loh and R TeasellldquoSystematic review of pharmacologic treatments of pain afterspinal cord injury an updaterdquoArchives of Physical Medicine andRehabilitation vol 97 no 8 pp 1381ndash1391 2016

[13] J J Leddy J G Baker and B Willer ldquoActive rehabilitation ofconcussion and post-concussion syndromerdquo Physical Medicineand Rehabilitation Clinics of North America vol 27 no 2 pp437ndash454 2016

[14] G E Francisco N Yozbatiran J Berliner et al ldquoRobot-assistedtraining of arm andhandmovement shows functional improve-ments for incomplete cervical spinal cord injuryrdquo Americanjournal of physical medicine amp rehabilitation vol 96 no 10 ppS171ndashS177 2017

[15] A S Burns R J Marino S Kalsi-Ryan et al ldquoType andtiming of rehabilitation following acute and subacute spinalcord injury a systematic reviewrdquo Global Spine Journal vol 7no 3 pp 175ndash194 2017

[16] C Thompson J F Gonsalves and D Welsh ldquoHyperextensioninjury of the cervical spine with central cord syndromerdquoEuropean Spine Journal vol 24 no 1 pp 195ndash202 2014

[17] S Mateo A Roby-Brami K T Reilly Y Rossetti C Colletand G Rode ldquoUpper limb kinematics after cervical spinal cordinjury a reviewrdquo Journal of NeuroEngineering and Rehabilita-tion vol 12 no 1 9 pages 2015

[18] K Barati M Arazpour R Vameghi A Abdoli and F FarmanildquoThe effect of soft and rigid cervical collars on head and neckimmobilization in healthy subjectsrdquoAsian Spine Journal vol 11no 3 pp 390ndash395 2017

[19] L Rogers ldquoNo place for the rigid cervical collar in pre-hospitalcarerdquo International Paramedic Practice vol 7 no 1 pp 12ndash152017

[20] M C Makhni E C Makhni E F Swart and C S Day ldquoInitialcervical spine managementrdquo in Orthopedic Emergencies pp 7ndash9 Springer International Publishing 2017

[21] A F Burnett F L Naumann R S Price and R H SandersldquoA comparison of training methods to increase neck musclestrengthrdquoWork vol 25 no 3 pp 205ndash210 2005


[22] P C Ivancic and C J Telles ldquoNeck motion due to the halo-vest in prone and supine positionsrdquo in Proceedings of the 6thWorld Congress of Biomechanics (WCB rsquo10) vol 31 of IFMBEProceedings pp 569ndash572 Springer Berlin Germany 2010

[23] T M Gavin G Carandang R Havey P Flanagan AGhanayem and A G Patwardhan ldquoBiomechanical analysisof cervical orthoses in flexion and extension A comparisonof cervical collars and cervical thoracic orthosesrdquo Journal ofRehabilitation Research and Development vol 40 no 6 pp527ndash537 2003

[24] R Jimenez-Fabian and O Verlinden ldquoReview of controlalgorithms for robotic ankle systems in lower-limb orthosesprostheses and exoskeletonsrdquo Medical Engineering amp Physicsvol 34 no 4 pp 397ndash408 2012

[25] J F Cusick F A Pintar and N Yoganandan ldquoWhiplashsyndrome Kinematic factors influencing pain patternsrdquo TheSpine Journal vol 26 no 11 pp 1252ndash1258 2001

[26] R Ferrari and H Schrader ldquoThe late whiplash syndrome abiopsychosocial approachrdquo Journal of Neurology Neurosurgeryamp Psychiatry vol 70 no 6 pp 722ndash726 2001

[27] M Koelbaek Johansen T Graven-Nielsen A Schou Olesenand L Arendt-Nielsen ldquoGeneralised muscular hyperalgesia inchronic whiplash syndromerdquo Pain vol 83 no 2 pp 229ndash2341999

[28] L Provinciali M Baroni L Illuminati and M G CeravololdquoMultimodal treatment to prevent the late whiplash syndromerdquoJournal of Rehabilitation Medicine vol 28 no 2 pp 105ndash1111996

[29] M Sterling ldquoTreatment of patients with whiplash associateddisordersrdquo Psychological Approaches to Pain Management APractitionerrsquos Handbook p 425 2018

[30] A J del-Ama A D Koutsou J C Moreno A de-los-Reyes AGil-Agudo and J L Pons ldquoReview of hybrid exoskeletons torestore gait following spinal cord injuryrdquo Journal of Rehabilita-tion Research andDevelopment vol 49 no 4 pp 497ndash514 2012

[31] M R Tucker J Olivier A Pagel et al ldquoControl strategiesfor active lower extremity prosthetics and orthotics A reviewrdquoJournal of NeuroEngineering and Rehabilitation vol 12 no 12015

[32] L Marchal-Crespo and D J Reinkensmeyer ldquoReview of con-trol strategies for robotic movement training after neurologicinjuryrdquo Journal of NeuroEngineering and Rehabilitation vol 6no 1 article 20 2009

[33] X Li Y Pan G Chen and H Yu ldquoMulti-modal control schemefor rehabilitation robotic exoskeletonsrdquo International Journal ofRobotics Research vol 36 no 5-7 pp 759ndash777 2017

[34] H Yu S Huang G Chen Y Pan and Z Guo ldquoHumanndashrobotinteraction control of rehabilitation robots with series elasticactuatorsrdquo IEEE Transactions on Robotics vol 31 no 5 pp1089ndash1100 2015

[35] J Zhang and C C Cheah ldquoPassivity and stability of human-robot interaction control for upper-limb rehabilitation robotsrdquoIEEE Transactions on Robotics vol 31 no 2 pp 233ndash245 2015

[36] M-S Ju C-C K Lin D-H Lin I-S Hwang and S-MChen ldquoA rehabilitation robot with force-position hybrid fuzzycontroller Hybrid fuzzy control of rehabilitation robotrdquo IEEETransactions on Neural Systems and Rehabilitation Engineeringvol 13 no 3 pp 349ndash358 2005

[37] W He S S Ge Y Li E Chew and Y S Ng ldquoNeural networkcontrol of a rehabilitation robot by state and output feedbackrdquoJournal of Intelligent amp Robotic Systems vol 80 no 1 pp 15ndash312015

[38] K Guo Y Pan and H Yu ldquoComposite learning robot controlwith friction compensation a neural network-based approachrdquoIEEE Transactions on Industrial Electronics 2018

[39] M Bernhardt M Frey G Colombo and R Riener ldquoHybridforce-position control yields cooperative behaviour of the reha-bilitation robot lokomatrdquo in Proceedings of the 9th InternationalConference on Rehabilitation Robotics ICORR rsquo05 pp 536ndash539IEEE Chicago Ill USA 2005

[40] P K Jamwal S Hussain M H Ghayesh and S V RogozinaldquoImpedance control of an intrinsically compliant parallel anklerehabilitation robotrdquo IEEE Transactions on Industrial Electron-ics vol 63 no 6 pp 3638ndash3647 2016

[41] WMeng Q Liu Z ZhouQ Ai B Sheng and S S Xie ldquoRecentdevelopment of mechanisms and control strategies for robot-assisted lower limb rehabilitationrdquo Mechatronics vol 31 pp132ndash145 2015

[42] P Song Y Yu and X Zhang ldquoA tutorial survey and comparisonof impedance control on robotic manipulationrdquo Robotica vol37 no 5 pp 801ndash836 2019

[43] J A Blaya and H Herr ldquoAdaptive control of a variable-impedance ankle-foot orthosis to assist drop-foot gaitrdquo IEEETransactions on Neural Systems and Rehabilitation Engineeringvol 12 no 1 pp 24ndash31 2004

[44] S Hussain S Q Xie and P K Jamwal ldquoAdaptive impedancecontrol of a robotic orthosis for gait rehabilitationrdquo IEEETransactions on Cybernetics vol 43 no 3 pp 1025ndash1034 2013

[45] I Ranatunga F L Lewis D O Popa and SM Tousif ldquoAdaptiveadmittance control for humanndashrobot interaction using modelreference design and adaptive inverse filteringrdquo IEEE Transac-tions on Control Systems Technology vol 25 no 1 pp 278ndash2852017

[46] M Sharifi S Behzadipour and G Vossoughi ldquoNonlinearmodel reference adaptive impedance control for human-robotinteractionsrdquo Control Engineering Practice vol 32 pp 9ndash272014

[47] H Modares I Ranatunga F L Lewis and D O Popa ldquoOpti-mized assistive human-robot interaction using reinforcementlearningrdquo IEEE Transactions on Cybernetics vol 46 no 3 pp655ndash667 2016

[48] V I Utkin and A S Poznyak ldquoAdaptive sliding mode controlwith application to super-twist algorithm equivalent controlmethodrdquo Automatica vol 49 no 1 pp 39ndash47 2013

[49] V Utkin J Guldner and J Shi Sliding Mode Control in Electro-Mechanical Systems Automation and Control Engineering2009

[50] C C Cheah S Kawamura and S Arimoto ldquoFeedback controlfor robotic manipulator with an uncertain jacobian matrixrdquoJournal of Robotic Systems vol 16 no 2 pp 119ndash134 1999

[51] X Li Y Pan G Chen and H Yu ldquoAdaptive human-robotinteraction control for robots driven by series elastic actuatorsrdquoIEEE Transactions on Robotics vol 33 no 1 pp 169ndash182 2017

[52] M Bertoluzzo G Buja and R Menis ldquoDirect torque controlof an induction motor using a single current sensorrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 778ndash7842006

[53] M E Haque L Zhong andM F Rahman ldquoA sensorless initialrotor position estimation scheme for a direct torque controlledinterior permanent magnet synchronous motor driverdquo IEEETransactions on Power Electronics vol 18 no 6 pp 1376ndash13832003


[54] J Maes and J A Melkebeek ldquoSpeed-sensorless direct torquecontrol of induction motors using an adaptive flux observerrdquoIEEE Transactions on Industry Applications vol 36 no 3 pp778ndash785 2000

[55] A Polyakov and I Chairez ldquoA new homogeneous quasi-continuous second order sliding mode controlrdquo in XVI Con-greso Latinoamericano de Control Automatico 2014

[56] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[57] J A Moreno and M Osorio ldquoStrict lyapunov funtions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[58] L A Castaneda A Luviano-Juarez and I Chairez ldquoRobusttrajectory tracking of a delta robot through adaptive activedisturbance rejection controlrdquo IEEE Transactions on ControlSystems Technology vol 23 no 4 pp 1387ndash1398 2015

[59] N Martınez-Fonseca L A Castaneda A Uranga A Luviano-Juarez and IChairez ldquoRobust disturbance rejection control of abiped robotic system using high-order extended state observerrdquoISA Transactions vol 62 pp 276ndash286 2016

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pain controlled mobilization difficulties of the affected zoneand restricted articulation displacements as well as muscularinflammation [6]

The whiplash syndrome is becoming one of the mostprevalent cervical pathologies This syndrome is the con-sequence of a fast angular movement in either lateral orfrontal anatomical planes [7 8] It is recognized as an inca-pacitate pathology with acute periods of pain and restrictedmobilization of the head This syndrome has been associatedwith violent automobile collisions [9] The hyperextensionof the muscles and ligaments in the cervical section are thephysiological fundamentals of the syndrome The relevanceof this pathology has motivated the introduction of theso-called Quebec classification which contains the clinicalmanifestations and the pathology degree This classificationalso indicates the suggested treatment for each degree of thepathology [10]

In the last few years some technological options appearedas alternative technologies for the treatment of neck vertebralillnesses including the neck hyperextension and hyperflex-ion (whiplash syndrome) [11 12] Most of these devicesmay ensure the complete immobilization of the vertebralarticulation in the cervical area [13 14] The conventionaltreatment consists of immobilizing the cervical region of thespinal cord by wearing therapeutic rigid collars as well aspharmaceutical therapy [15] A cervical collar is an orthosisfitting the patientrsquos neck anywhere from the jaw to the chest Ithas two major functions restricting the cervical movements(flexion extension and lateral tilt) and supporting the headto allow healing the muscles and the ligaments In additionthe collar decreases the muscle spasms and control the painin the neck [16 17]

The rigid or semirigid collars (made of soft material)represent the main assistance technology for the cervicalinjuries treatment Nevertheless there are some drawbacksin using these collars [18] In particular the continuousimmobilization in all planes can lead to sacral pressure pointsheel and elbow breakdown risk of deep venous thrombosismuscle atrophy etc [19]Most of these inconveniences can besaved if a complementary mobilization therapy of isometricexercises is realized It is usual that this complementarytherapy is not recommended because the necessity of aqualified physiotherapist intervention [20]

Nowadays the emerging of rehabilitation technologyincluding the development of active orthosis has motivatedthe introduction of automatized devices aimed to mobilizethe cervical region of patients wearing the rigid collar Themulti-cervical unit is a commercial product (BTE Technolo-gies) aimed at offering pain reduction by restricting the necklateral movements however this device could induce a ver-tical continuous force which may contribute to the treatmentof conical and acute pain vertebral disks compression andsome others This equipment requires the recurrent visits ofthe patient to the specialized clinical facilities [21]

ReSolve Halo is a cervical device with external fixationsystem It has the mechanical sections to immobilize thecervical articulations and it offers complementary regulatedsupport for the head The position of the neck area is fixedin advance and the patient is required to visit the physician

if some adjustment is needed This device offers the weightdischarge of the head by fixing the device to the thorax area[22] The Cervical Thoracic Orthosis provides a conservativetreatment for the cervical pathology It can restrict therotational movement of the neck but it can hold the headsection immobilized This device is attached to the patientrsquosthorax to offer the weight discharge [23]

All the devices proposed for aiding the therapeuticsof the neck illnesses have the disadvantage of forcing theneckrsquos immobilization which is not the best solution for therequested rehabilitationNotice that themobilization realizedby these devices did not consider the potential resistanceof the patient to the orthosis activity This aspect must betaken into account considering the possible pain inducedfor the application of orthosis-based treatment [24] Themain motivation for developing an active orthosis for thewhiplash syndrome is the increasing number of neck illnesseswhich are consequences of car accidents long periods ofmuscle inactivity yielding cervical vertebral inflammationsetc These sicknesses are treated by fixed orthoses which aresupporting the neck by immobilizing the cervical sectionbut not offering an integral rehabilitation The proposedorthosis makes an enhancement in this direction because itis mobilizing the neck section in a controlled way while thepatientrsquos head is still held by the orthosis [25ndash29]

It is usual that active orthosis (mechanized rehabilitationdevices with automatized movements) should be controlledby robust controllers which are not considering the patientresistance This design requirement implies the applicationof a class of human-robot (or patient-orthosis in this case)interaction framework solution In such class of approachesthe tracking of reference trajectories still plays a relevantrole in the design of the automatic controller However thesafe robot operation considering the patient resistance to theorthosis action plays a primary role because the necessity ofensuring a comfortable assisted rehabilitation [30]

The current control designs to solve the human-robotinteraction may not provide sufficient flexibility for collabo-ration tasks (implying a smooth transition between the track-ing and the resistance operation conditions) requiring thatonly two cases are admissible the robot leads or follows thehuman by assessing the performance of human continuously[31] Only a few approaches have considered the possibilityof transiting from the robot leading (orthosis-in-charge)scenario to the human ruling case (patient-in-charge) [32]Within these works diverse studies have provided nonlinearapproaches to regulate the human-robot interaction [33ndash35] inspired by the hybrid control designs Within thehybrid solutions the application of approximate mathemat-ical descriptions to the orthosis model has been used alongthe last years [36ndash38] Such approaches try to produce softinteraction between patient and orthosis Collaborative robotcontrol is also providing new paradigms to control activeorthosis where the patient exerts the rehabilitation procedurebut keeping a predefined safety zone [39ndash41]

Impedanceadmittance-based controllers can solve eachof the proposed scenarios independently [42] Howeverthere is a necessity of changing the desired impedance ineach case This condition may lead to the introduction of













119872(119902 (119905)) 1198892119902 (119905)1198891199052 + 119866 (119902 (119905)) + 119862(119902 (119905) 119889119902 (119905)119889119905 ) 119889119902 (119905)119889119905+ 120595(119902 (119905) 119889119902 (119905)119889119905 119905) = 120591 (119905) + 120591119903 (119905) (1)








155cm123cm

35cm 84cm 27cm

56c

m

112

cm

94c

m

118

cm

32c

m

16c

m

42cm 42cm

q1

q2

Y

X

(a) (b)



sup119905ge0

10038171003817100381710038171003817119869 (119902 (119905)) 119869⊤ (119902 (119905))10038171003817100381710038171003817119865 le 119869+ (3)



sup119905ge0

10038171003817100381710038171198911198901199091199051003817100381710038171003817 le 119891+119890119909119905 (4)












02

0

04

06

08

1

01

0

03

05

06

08

02

04

07

09

1W

eigh

ting

func

tion

w(

ＲＮ)


05

10

minus10minus5 0

510

minus10minus5








ℎ (119891119890) = 10038171003817100381710038171003817119891119890100381710038171003817100381710038172 minus 1198772 (5)











(8)







119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905)

= 119865(119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591 (119905)+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(11)



119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865 (119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591119888 (119905)

+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(12)


119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865(119902 (119905) 119889119889119905119902 (119905))

+ 119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(13)














13

1198622 = (1205881 + radic12058821 minus 4120588302 )13 (16)


1199011 = 05 (120577119891 minus 1205770) (1 + 119887) (1 + 119887119890119889119905)119887 (1 minus 119890119889119905) (17)






119878 (119905) = 120575119886 (119905) + 119867120575119887 (119905) (20)


119889119889119905119878 (119905) = 120575119887 (119905) + 119867 119889119889119905120575119887 (119905) (21)




Π119888 = 120575119887 (119905) + 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(22)


Rotary sensors

DC actuators+

+-

1

+

fext

xlowasttr

c

Ktr

Kc

xa[In 0n]

x

5 100minus5minus10

50

minus5 minus10




(23)







120582 = 1205781120583119905119903120583119888 = 120583119905119903(24)



+ (119908 (119891119890 (119905)) 120591119905119903 (119905) + [1 minus 119908 (119891119890 (119905))] 120591119888 (119905))



(25)


= 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119905119903 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(26)


(27)


119881119905119903 (119878) = 05 1198782 (28)


+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)]

(29)







119889119889119905119878 (119905) = 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119888 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+ 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(33)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119888 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)

(34)






(36)



(37)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))


(38)




(40)



(41)










1198891198891199051199081 (119905) = 1199082 (119905)1198891198891199051199082 (119905) = 11988921198891199052 119903 (119905)

(44)



1199081 = 1199081 minus 1199081119889 (119905) = 1198891198891199051199081 (119905)

(45)


sign (]) = 1 119894119891 ] gt 0minus1 119894119891 ] lt 0 (46)



119889119889119905120575119887119894 (119905) = 119891119894 (120575119886 (119905) 120575119887 (119905)) + 2sum119895=1

119892119894119895 (120575119886 (119905)) 119906119895 (119905)+ 120578119894 (120575119886 (119905) 120575119887 (119905) 119905)

(47)









Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation










RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018




















119872(119902 (119905)) 1198892119902 (119905)1198891199052 + 119866 (119902 (119905)) + 119862(119902 (119905) 119889119902 (119905)119889119905 ) 119889119902 (119905)119889119905+ 120595(119902 (119905) 119889119902 (119905)119889119905 119905) = 120591 (119905) + 120591119903 (119905) (1)








155cm123cm

35cm 84cm 27cm

56c

m

112

cm

94c

m

118

cm

32c

m

16c

m

42cm 42cm

q1

q2

Y

X

(a) (b)



sup119905ge0

10038171003817100381710038171003817119869 (119902 (119905)) 119869⊤ (119902 (119905))10038171003817100381710038171003817119865 le 119869+ (3)



sup119905ge0

10038171003817100381710038171198911198901199091199051003817100381710038171003817 le 119891+119890119909119905 (4)












02

0

04

06

08

1

01

0

03

05

06

08

02

04

07

09

1W

eigh

ting

func

tion

w(

ＲＮ)


05

10

minus10minus5 0

510

minus10minus5








ℎ (119891119890) = 10038171003817100381710038171003817119891119890100381710038171003817100381710038172 minus 1198772 (5)











(8)







119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905)

= 119865(119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591 (119905)+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(11)



119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865 (119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591119888 (119905)

+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(12)


119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865(119902 (119905) 119889119889119905119902 (119905))

+ 119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(13)














13

1198622 = (1205881 + radic12058821 minus 4120588302 )13 (16)


1199011 = 05 (120577119891 minus 1205770) (1 + 119887) (1 + 119887119890119889119905)119887 (1 minus 119890119889119905) (17)






119878 (119905) = 120575119886 (119905) + 119867120575119887 (119905) (20)


119889119889119905119878 (119905) = 120575119887 (119905) + 119867 119889119889119905120575119887 (119905) (21)




Π119888 = 120575119887 (119905) + 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(22)


Rotary sensors

DC actuators+

+-

1

+

fext

xlowasttr

c

Ktr

Kc

xa[In 0n]

x

5 100minus5minus10

50

minus5 minus10




(23)







120582 = 1205781120583119905119903120583119888 = 120583119905119903(24)



+ (119908 (119891119890 (119905)) 120591119905119903 (119905) + [1 minus 119908 (119891119890 (119905))] 120591119888 (119905))



(25)


= 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119905119903 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(26)


(27)


119881119905119903 (119878) = 05 1198782 (28)


+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)]

(29)







119889119889119905119878 (119905) = 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119888 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+ 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(33)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119888 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)

(34)






(36)



(37)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))


(38)




(40)



(41)










1198891198891199051199081 (119905) = 1199082 (119905)1198891198891199051199082 (119905) = 11988921198891199052 119903 (119905)

(44)



1199081 = 1199081 minus 1199081119889 (119905) = 1198891198891199051199081 (119905)

(45)


sign (]) = 1 119894119891 ] gt 0minus1 119894119891 ] lt 0 (46)



119889119889119905120575119887119894 (119905) = 119891119894 (120575119886 (119905) 120575119887 (119905)) + 2sum119895=1

119892119894119895 (120575119886 (119905)) 119906119895 (119905)+ 120578119894 (120575119886 (119905) 120575119887 (119905) 119905)

(47)









Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation










RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018









155cm123cm

35cm 84cm 27cm

56c

m

112

cm

94c

m

118

cm

32c

m

16c

m

42cm 42cm

q1

q2

Y

X

(a) (b)



sup119905ge0

10038171003817100381710038171003817119869 (119902 (119905)) 119869⊤ (119902 (119905))10038171003817100381710038171003817119865 le 119869+ (3)



sup119905ge0

10038171003817100381710038171198911198901199091199051003817100381710038171003817 le 119891+119890119909119905 (4)












02

0

04

06

08

1

01

0

03

05

06

08

02

04

07

09

1W

eigh

ting

func

tion

w(

ＲＮ)


05

10

minus10minus5 0

510

minus10minus5








ℎ (119891119890) = 10038171003817100381710038171003817119891119890100381710038171003817100381710038172 minus 1198772 (5)











(8)







119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905)

= 119865(119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591 (119905)+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(11)



119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865 (119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591119888 (119905)

+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(12)


119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865(119902 (119905) 119889119889119905119902 (119905))

+ 119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(13)














13

1198622 = (1205881 + radic12058821 minus 4120588302 )13 (16)


1199011 = 05 (120577119891 minus 1205770) (1 + 119887) (1 + 119887119890119889119905)119887 (1 minus 119890119889119905) (17)






119878 (119905) = 120575119886 (119905) + 119867120575119887 (119905) (20)


119889119889119905119878 (119905) = 120575119887 (119905) + 119867 119889119889119905120575119887 (119905) (21)




Π119888 = 120575119887 (119905) + 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(22)


Rotary sensors

DC actuators+

+-

1

+

fext

xlowasttr

c

Ktr

Kc

xa[In 0n]

x

5 100minus5minus10

50

minus5 minus10




(23)







120582 = 1205781120583119905119903120583119888 = 120583119905119903(24)



+ (119908 (119891119890 (119905)) 120591119905119903 (119905) + [1 minus 119908 (119891119890 (119905))] 120591119888 (119905))



(25)


= 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119905119903 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(26)


(27)


119881119905119903 (119878) = 05 1198782 (28)


+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)]

(29)







119889119889119905119878 (119905) = 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119888 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+ 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(33)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119888 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)

(34)






(36)



(37)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))


(38)




(40)



(41)










1198891198891199051199081 (119905) = 1199082 (119905)1198891198891199051199082 (119905) = 11988921198891199052 119903 (119905)

(44)



1199081 = 1199081 minus 1199081119889 (119905) = 1198891198891199051199081 (119905)

(45)


sign (]) = 1 119894119891 ] gt 0minus1 119894119891 ] lt 0 (46)



119889119889119905120575119887119894 (119905) = 119891119894 (120575119886 (119905) 120575119887 (119905)) + 2sum119895=1

119892119894119895 (120575119886 (119905)) 119906119895 (119905)+ 120578119894 (120575119886 (119905) 120575119887 (119905) 119905)

(47)









Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation










RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018









02

0

04

06

08

1

01

0

03

05

06

08

02

04

07

09

1W

eigh

ting

func

tion

w(

ＲＮ)


05

10

minus10minus5 0

510

minus10minus5








ℎ (119891119890) = 10038171003817100381710038171003817119891119890100381710038171003817100381710038172 minus 1198772 (5)











(8)







119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905)

= 119865(119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591 (119905)+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(11)



119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865 (119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591119888 (119905)

+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(12)


119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865(119902 (119905) 119889119889119905119902 (119905))

+ 119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(13)














13

1198622 = (1205881 + radic12058821 minus 4120588302 )13 (16)


1199011 = 05 (120577119891 minus 1205770) (1 + 119887) (1 + 119887119890119889119905)119887 (1 minus 119890119889119905) (17)






119878 (119905) = 120575119886 (119905) + 119867120575119887 (119905) (20)


119889119889119905119878 (119905) = 120575119887 (119905) + 119867 119889119889119905120575119887 (119905) (21)




Π119888 = 120575119887 (119905) + 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(22)


Rotary sensors

DC actuators+

+-

1

+

fext

xlowasttr

c

Ktr

Kc

xa[In 0n]

x

5 100minus5minus10

50

minus5 minus10




(23)







120582 = 1205781120583119905119903120583119888 = 120583119905119903(24)



+ (119908 (119891119890 (119905)) 120591119905119903 (119905) + [1 minus 119908 (119891119890 (119905))] 120591119888 (119905))



(25)


= 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119905119903 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(26)


(27)


119881119905119903 (119878) = 05 1198782 (28)


+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)]

(29)







119889119889119905119878 (119905) = 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119888 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+ 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(33)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119888 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)

(34)






(36)



(37)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))


(38)




(40)



(41)










1198891198891199051199081 (119905) = 1199082 (119905)1198891198891199051199082 (119905) = 11988921198891199052 119903 (119905)

(44)



1199081 = 1199081 minus 1199081119889 (119905) = 1198891198891199051199081 (119905)

(45)


sign (]) = 1 119894119891 ] gt 0minus1 119894119891 ] lt 0 (46)



119889119889119905120575119887119894 (119905) = 119891119894 (120575119886 (119905) 120575119887 (119905)) + 2sum119895=1

119892119894119895 (120575119886 (119905)) 119906119895 (119905)+ 120578119894 (120575119886 (119905) 120575119887 (119905) 119905)

(47)









Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation










RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018













(8)







119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905)

= 119865(119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591 (119905)+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(11)



119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865 (119902 (119905) 119889119889119905119902 (119905)) + 119866 (119902 (119905)) 120591119888 (119905)

+ Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(12)


119889119889119905120575119886 (119905) = 120575119887 (119905)119889119889119905120575119887 (119905) = 119865(119902 (119905) 119889119889119905119902 (119905))

+ 119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(13)














13

1198622 = (1205881 + radic12058821 minus 4120588302 )13 (16)


1199011 = 05 (120577119891 minus 1205770) (1 + 119887) (1 + 119887119890119889119905)119887 (1 minus 119890119889119905) (17)






119878 (119905) = 120575119886 (119905) + 119867120575119887 (119905) (20)


119889119889119905119878 (119905) = 120575119887 (119905) + 119867 119889119889119905120575119887 (119905) (21)




Π119888 = 120575119887 (119905) + 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(22)


Rotary sensors

DC actuators+

+-

1

+

fext

xlowasttr

c

Ktr

Kc

xa[In 0n]

x

5 100minus5minus10

50

minus5 minus10




(23)







120582 = 1205781120583119905119903120583119888 = 120583119905119903(24)



+ (119908 (119891119890 (119905)) 120591119905119903 (119905) + [1 minus 119908 (119891119890 (119905))] 120591119888 (119905))



(25)


= 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119905119903 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(26)


(27)


119881119905119903 (119878) = 05 1198782 (28)


+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)]

(29)







119889119889119905119878 (119905) = 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119888 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+ 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(33)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119888 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)

(34)






(36)



(37)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))


(38)




(40)



(41)










1198891198891199051199081 (119905) = 1199082 (119905)1198891198891199051199082 (119905) = 11988921198891199052 119903 (119905)

(44)



1199081 = 1199081 minus 1199081119889 (119905) = 1198891198891199051199081 (119905)

(45)


sign (]) = 1 119894119891 ] gt 0minus1 119894119891 ] lt 0 (46)



119889119889119905120575119887119894 (119905) = 119891119894 (120575119886 (119905) 120575119887 (119905)) + 2sum119895=1

119892119894119895 (120575119886 (119905)) 119906119895 (119905)+ 120578119894 (120575119886 (119905) 120575119887 (119905) 119905)

(47)









Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation










RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018



















13

1198622 = (1205881 + radic12058821 minus 4120588302 )13 (16)


1199011 = 05 (120577119891 minus 1205770) (1 + 119887) (1 + 119887119890119889119905)119887 (1 minus 119890119889119905) (17)






119878 (119905) = 120575119886 (119905) + 119867120575119887 (119905) (20)


119889119889119905119878 (119905) = 120575119887 (119905) + 119867 119889119889119905120575119887 (119905) (21)




Π119888 = 120575119887 (119905) + 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)(22)


Rotary sensors

DC actuators+

+-

1

+

fext

xlowasttr

c

Ktr

Kc

xa[In 0n]

x

5 100minus5minus10

50

minus5 minus10




(23)







120582 = 1205781120583119905119903120583119888 = 120583119905119903(24)



+ (119908 (119891119890 (119905)) 120591119905119903 (119905) + [1 minus 119908 (119891119890 (119905))] 120591119888 (119905))



(25)


= 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119905119903 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(26)


(27)


119881119905119903 (119878) = 05 1198782 (28)


+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)]

(29)







119889119889119905119878 (119905) = 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119888 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+ 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(33)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119888 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)

(34)






(36)



(37)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))


(38)




(40)



(41)










1198891198891199051199081 (119905) = 1199082 (119905)1198891198891199051199082 (119905) = 11988921198891199052 119903 (119905)

(44)



1199081 = 1199081 minus 1199081119889 (119905) = 1198891198891199051199081 (119905)

(45)


sign (]) = 1 119894119891 ] gt 0minus1 119894119891 ] lt 0 (46)



119889119889119905120575119887119894 (119905) = 119891119894 (120575119886 (119905) 120575119887 (119905)) + 2sum119895=1

119892119894119895 (120575119886 (119905)) 119906119895 (119905)+ 120578119894 (120575119886 (119905) 120575119887 (119905) 119905)

(47)









Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation










RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018









Rotary sensors

DC actuators+

+-

1

+

fext

xlowasttr

c

Ktr

Kc

xa[In 0n]

x

5 100minus5minus10

50

minus5 minus10




(23)







120582 = 1205781120583119905119903120583119888 = 120583119905119903(24)



+ (119908 (119891119890 (119905)) 120591119905119903 (119905) + [1 minus 119908 (119891119890 (119905))] 120591119888 (119905))



(25)


= 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119905119903 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(26)


(27)


119881119905119903 (119878) = 05 1198782 (28)


+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)]

(29)







119889119889119905119878 (119905) = 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119888 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+ 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(33)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119888 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)

(34)






(36)



(37)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))


(38)




(40)



(41)










1198891198891199051199081 (119905) = 1199082 (119905)1198891198891199051199082 (119905) = 11988921198891199052 119903 (119905)

(44)



1199081 = 1199081 minus 1199081119889 (119905) = 1198891198891199051199081 (119905)

(45)


sign (]) = 1 119894119891 ] gt 0minus1 119894119891 ] lt 0 (46)



119889119889119905120575119887119894 (119905) = 119891119894 (120575119886 (119905) 120575119887 (119905)) + 2sum119895=1

119892119894119895 (120575119886 (119905)) 119906119895 (119905)+ 120578119894 (120575119886 (119905) 120575119887 (119905) 119905)

(47)









Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation










RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018










(25)


= 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119905119903 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119867 119889119889119905 [Γ+ (119902 (119905)) 119889119889119905119875119889 (119905) minus Γ+ (119902 (119905)) Δ119875 (119905)]

(26)


(27)


119881119905119903 (119878) = 05 1198782 (28)


+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119905119903 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)]

(29)







119889119889119905119878 (119905) = 120575119887 (119905) + 119867119865(119902 (119905) 119889119889119905119902 (119905)) + 120591119888 (119905)+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)+ 119867119872minus1 (119902 (119905)) 119869 (119902 (119905))⊤ 119891119890119909119905 (119905)

(33)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus 119870119888 (119905) [1 + 120582radic120598 + 1003817100381710038171003817119902V10038171003817100381710038172] 119878 (119905)119878 (119905)

(34)






(36)



(37)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))


(38)




(40)



(41)










1198891198891199051199081 (119905) = 1199082 (119905)1198891198891199051199082 (119905) = 11988921198891199052 119903 (119905)

(44)



1199081 = 1199081 minus 1199081119889 (119905) = 1198891198891199051199081 (119905)

(45)


sign (]) = 1 119894119891 ] gt 0minus1 119894119891 ] lt 0 (46)



119889119889119905120575119887119894 (119905) = 119891119894 (120575119886 (119905) 120575119887 (119905)) + 2sum119895=1

119892119894119895 (120575119886 (119905)) 119906119895 (119905)+ 120578119894 (120575119886 (119905) 120575119887 (119905) 119905)

(47)









Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation










RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018










(36)



(37)


119889119889119905119878 (119905) = 119867119865(119902 (119905) 119889119889119905119902 (119905))+ 119867Φ(119902 (119905) 119889119889119905119902 (119905) 119905)minus (119908 (119891119890 (119905))119870119905119903 (119905) + [1 minus 119908 (119891119890 (119905))]119870119888 (119905))


(38)




(40)



(41)










1198891198891199051199081 (119905) = 1199082 (119905)1198891198891199051199082 (119905) = 11988921198891199052 119903 (119905)

(44)



1199081 = 1199081 minus 1199081119889 (119905) = 1198891198891199051199081 (119905)

(45)


sign (]) = 1 119894119891 ] gt 0minus1 119894119891 ] lt 0 (46)



119889119889119905120575119887119894 (119905) = 119891119894 (120575119886 (119905) 120575119887 (119905)) + 2sum119895=1

119892119894119895 (120575119886 (119905)) 119906119895 (119905)+ 120578119894 (120575119886 (119905) 120575119887 (119905) 119905)

(47)









Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation










RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018












1198891198891199051199081 (119905) = 1199082 (119905)1198891198891199051199082 (119905) = 11988921198891199052 119903 (119905)

(44)



1199081 = 1199081 minus 1199081119889 (119905) = 1198891198891199051199081 (119905)

(45)


sign (]) = 1 119894119891 ] gt 0minus1 119894119891 ] lt 0 (46)



119889119889119905120575119887119894 (119905) = 119891119894 (120575119886 (119905) 120575119887 (119905)) + 2sum119895=1

119892119894119895 (120575119886 (119905)) 119906119895 (119905)+ 120578119894 (120575119886 (119905) 120575119887 (119905) 119905)

(47)









Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation










RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018









Micro-motor reducer

Positionsensor

H bridge

Matlabinterface

NE555LM7805

Powersource 5V

Firstarticulation

Secondarticulation










RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018









RefPID

SMASM

RefPID

SMASM

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

005

115

Ｋ1

(rad

)








120591 (119905) = 119870119901120575119886 (119905) + 119870119889120575119887 (119905) + 119870119894 int119905119904=0

120575119886 (119904) 119889119904 (49)


119870119901 = [500 0000 500]119870119889 = [20 0000 20]119870119894 = [05 0000 05]

(50)


120591 (119905) = 119870119878119872 (120575119886 (119905) + 119867120575119887 (119905)) (51)


119867 = [001 0000 001]119870119878119872 = [400 0000 400]

(52)


RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018









RefPID

SMASM

RefPID

SMASM


Ｋ2

(rad

)

636 638 64 642 644 646 648 65 652 654634Time (s)

636 638 64 642 644 646 648 65 652 654634Time (s)

132134136138

Ｋ1

(rad

)



119870119905119903 (0) = [500 0000 550]119870119888 (0) = [200 0000 250]

119872 = [450 0000 200]120598 = 005120582 = 0201205740 = 251205741 = 42

(53)







minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018









minus2

0

2

PIDSMASM

PIDSMASM

minus2

0

2Ｏ2

Ｏ1(

lowastＧ)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

(lowastＧ)


10 20 30 400Time (s)

0

1

2

3

Firs

t com

pone

nt ＲＮ (

N)

(a)

10 20 30 400Time (s)

0

05

1

15Se

cond

com

pone

nt ＲＮ (

N)

(b)

10 20 30 400Time (s)

0

1

2

3

4

Mag

nitu

de o

fＲ

Ｎ (N

)

(c)

10 20 30 400Time (s)

02

04

06

08

1

Wei

ghtin

g fu

nctio

n W

( ＲＮ)

(d)






05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018









05

1

15

Ｋ1

(rad

)

minus25minus2

minus15minus1

minus05Ｋ2

(rad

)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 400Time (s)

PIDASMASM-Impedance

PIDASMASM-Impedance









119870119905119903 (0) = [15 0000 25]119870119888 (0) = [12 0000 9 ]

119872 = [12 0000 12]120598 = 005120582 = 0201205740 = 251205741 = 22

(54)





0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018











0

05

1

15


minus25minus2

minus15minus1

minus05


10 20 30 40 50 60 700Time (s)

10 20 30 40 50 60 700Time (s)

Expe

rimen

talＫ

2(r

ad)

Expe

rimen

talＫ

1(r

ad)







7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018











7 Conclusions




Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018










Data Availability




References





































































Journal of












Journal of







Volume 2018






Volume 2018























































Journal of












Journal of







Volume 2018






Volume 2018








Impedance Adaptive Controller for a Prototype of a...

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