Robust Adaptive Impedance Control With Application to a … · 2019. 11. 1. · Among adaptive control methods which have already been pub-lished, our work most closely resembles

Vahid AzimiSchool of Electrical and Computer Engineering,

Georgia Institute of Technology,

Atlanta, GA 30332-0250

e-mail: [email protected]

Seyed Abolfazl FakoorianDepartment of Electrical Engineering and

Computer Science,

Cleveland State University,

Cleveland, OH OH 44115


Thang Tien Nguyen1Modeling Evolutionary Algorithms Simulation

and Artificial Intelligence,

Faculty of Electrical & Electronics Engineering,

Ton Duc Thang University,

Ho Chi Minh City, Vietnam


Dan SimonDepartment of Electrical Engineering

and Computer Science,

Cleveland State University,

Cleveland, OH 44115


Robust Adaptive ImpedanceControl With Application to aTransfemoral Prosthesis andTest RobotThis paper presents, compares, and tests two robust model reference adaptive impedancecontrollers for a three degrees-of-freedom (3DOF) powered prosthesis/test robot. We firstpresent a model for a combined system that includes a test robot and a transfemoral pros-thetic leg. We design these two controllers, so the error trajectories of the system con-verge to a boundary layer and the controllers show robustness to ground reaction forces(GRFs) as nonparametric uncertainties and also handle model parameter uncertainties.We prove the stability of the closed-loop systems for both controllers for the prosthesis/test robot in the case of nonscalar boundary layer trajectories using Lyapunov stabilitytheory and Barbalat’s lemma. We design the controllers to imitate the biomechanicalproperties of able-bodied walking and to provide smooth gait. We finally present simula-tion results to confirm the efficacy of the controllers for both nominal and off-nominalsystem model parameters. We achieve good tracking of joint displacements and veloc-ities, and reasonable control and GRF magnitudes for both controllers. We also compareperformance of the controllers in terms of tracking, control effort, and parameter estima-tion for both nominal and off-nominal model parameters. [DOI: 10.1115/1.4040463]

Keywords: : robust adaptive impedance control, transfemoral prosthesis, nonscalarboundary layer trajectories

1 Introduction

Prostheses have become progressively important because thereare about two million people with limb loss in the U.S. as of 2008[1]. Amputation could be due to accidents, cancer, diabetes, vas-cular disease, birth defects, and paralysis. However, the primarycause of lower limb loss is disease—particularly diabetes andother dysvascular etiologies (approximately 75% of all cases)[1,2]. A prosthetic leg can enhance the quality of life and the abil-ity to walk for amputees, so they can regain independence. Ampu-tation could be transtibial (i.e., below knee), transfemoral (i.e.,above knee), at the foot, or disarticulation (i.e., through a joint).Prosthetic legs can be generally classified into three differenttypes: passive prostheses do not include any electronic control,active prostheses include motors, and semi-active prostheses arenot actively driven by motors [3]. Research efforts over the pastfew decades have provided advanced prostheses to closely imitateable-bodied gait and to allow greater levels of activity for ampu-tees. Active prostheses provide gait performance that is more sim-ilar to able-bodied gait than passive or semi-active prostheses.The first commercially available active transfemoral prosthesiswas the Power Knee [3–5]. A combined knee/ankle prosthesis thatincludes active control at both knee and ankle has been developedby Vanderbilt University but has not yet been commercialized [6].Much recent research has focused on the control of these prosthe-ses, along with other prostheses [7–12]. Recent research has pro-vided significant developments in modeling and control forprosthetic legs [13–22], and bipedal robots and rehabilitationrobots [23–25]. Although direct neural integration and electro-myogram signals can be recorded from residual limbs, and theground reaction force (GRF) can be measured from prosthetic

legs to recognize user intent for volitional control of the poweredprosthetic legs, in this paper a pair of “classical” feedback controlstrategies (robust adaptive impedance controllers (RAIC)) are pre-sented to control the robot/prosthesis device using feedback meas-urements of the joints position and velocity and feedback of theGRF model.

An active prosthesis is essentially a robot that interacts with itshuman user. The prosthesis can be controlled to behave as animpedance or admittance [26,27]. The consideration of the inter-action between a robot and its external environment motivated thedevelopment of impedance control [28]. Variable impedance con-troller is one of the most popular approaches to control poweredprosthetic legs, because it can be used in a model independentfashion. However, this control method suffers several shortcom-ings: tedious impedance parameter tuning, lack of feedback, andpassiveness [3,10].

Modeling errors are always present in real-world systems, butrobust control approaches can mitigate the effects of modelingerrors on system performance and stability [29,30]. Robust con-trollers achieve performance in spite of model uncertainty, whileadaptive controllers achieve performance using learning and adap-tation. Nonadaptive controllers generally require prior knowledgeof the parameter variation bounds, while adaptive approaches donot.

The advantages of adaptive control, the availability of able-bodied impedance models, and the uncertainty of robot modelshave motivated the development of impedance model referenceadaptive control [31–33]. However, adaptive control methods cancause instability if disturbances, unmodeled dynamics, or unmod-eled external forces are too large. Robust control can alleviateinstability in such cases [34–39]. Various adaptive and sliding sur-face approaches have also been used for robotic applications[30,40–44].

The contribution of this paper is two robust model referenceadaptive impedance controllers for transfemoral prostheses, thestability analysis of the two controllers, and the investigation of

1Corresponding author.Contributed by the Dynamic Systems Division of ASME for publication in the

JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedJune 30, 2017; final manuscript received May 22, 2018; published online July 2,2018. Editor: Joseph Beaman.

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their performance in simulation. Our control approaches canensure that the system converges to a reference model in the pres-ence of both parametric and nonparametric uncertainties. In thispaper, we present a blending adaptive and nonscalar boundarylayer-based robust control to achieve robustness to GRFs (i.e.,environmental interactions), system uncertainties, and disturban-ces, estimation of the unknown parameters, and a stability proofof the proposed methods.

The first controller comprises a RAIC with a tracking-error-based (TEB) adaptation law, which extracts information about theparameters from only the impedance model tracking error. Thesecond controller comprises a robust composite adaptive imped-ance controller (RCAIC) with bounded-gain forgetting (BGF).Since tracking errors in the joint displacements and predictionerror in the joint torques are influenced by parameter uncertain-ties, RCAIC is designed with TEB/prediction-error-based (PEB)adaptation so that parameter adaptation is driven with both imped-ance model tracking error and prediction error, which, in turn,provides more accurate estimation of system parameters. Moreaccurate estimation of the system parameters results in a moreaccurate model, and in turn, RCAIC can achieve better trackingcompared with RAIC.

Since our goal is that the two closed-loop systems (one withRAIC and the other with RCAIC) match the biomechanics ofable-bodied walking, we use a target impedance model which isbased on able-bodied walking. To balance control chatter and per-formance, we incorporate nonscalar boundary layer trajectories sDin both controllers. We use these trajectories to turn off the TEBadaptation mechanism to prevent unfavorable parameter driftwhen the impedance model tracking errors are small and duemostly to noise and disturbances. We define the trajectories sD sothe error trajectories converge to the boundary layers and the con-trollers show robustness to both parametric and nonparametricuncertainties.

Among adaptive control methods which have already been pub-lished, our work most closely resembles [30] and [42]. In Ref.[30], a direct adaptive controller is proposed whose adaptationmechanism uses joint tracking errors. The control law in Ref. [30]is a combination of a direct adaptive and robust sliding mode con-trol based on a scalar boundary layer to obtain a tradeoff betweencontrol chatter and performance, and to achieve robustness tounmodeled dynamics. Asymptotic stability of the closed-loop sys-tem in the case of a scalar boundary layer is shown.

In Ref. [42], a composite adaptive controller is proposed whoseadaptation law uses tracking errors in the joint motion and errorsin the predicted filtered torque to derive more accurate systemparameters. In addition, a blend of an adaptive feedforward and aproportional–derivative controller is used and exponential stabil-ity of the closed-loop system is proven.

Since a robotic system with more than one degree-of-freedom(1DOF), including the 3DOF prosthesis/controller system in thisresearch, can be considered a nonscalar problem with a couplednature, in this research, we use nonscalar boundary layer trajecto-ries for both control structures.

So, we expand on the work in Ref. [30] by using nonscalarboundary layer trajectories and incorporating impedance control.We prove the asymptotic stability of the system with both control-lers, RAIC and RCAIC, using nonscalar boundary layer trajecto-ries, Barbalat’s lemma, and Lyapunov theory. We also extend thework in Ref. [42] by incorporating nonscalar boundary layer tra-jectories sD and impedance control so that both augmented robustcomposite impedance controllers show robustness to nonparamet-ric model uncertainties and environmental interaction forces(which are GRF variations in our case). We then prove the expo-nential stability of these controllers using nonscalar boundarylayer trajectories.

Simulation results illustrate that both proposed systems havegood tracking performance, strong robustness to system modelparametric and nonparametric uncertainties, and reasonable con-trol signals and GRFs. Furthermore, numerical results show that

the RCAIC demonstrates better parameter estimation and trackingin the presence of system parameter variations. When parametervalues vary by 30% from nominal values, the RCAIC has 9.5%better reference trajectory tracking and 76% better parameter esti-mation, but 9.9% greater control magnitude than RAIC.

The paper is organized as follows: Sec. 2 describes the modelof the transfemoral prosthesis and the robotic test system. Section3 presents the controller structures and proves their stability. Sec-tion 4 presents simulation results. Section 5 presents discussion,concluding remarks, and future work.

2 Prosthetic Leg Model

2.1 Test Robot/Transfemoral Prosthetic Leg. Our systemmodel includes a test robot and a transfemoral prosthesis. The sys-tem includes three links and three degrees-of-freedom. Thisprismatic-revolute-revolute model is shown in Fig. 1. Human hipmotion and thigh motion is emulated by the robot. The knee andshank represent the prosthesis. The vertical motion emulateshuman (or test robot) vertical hip motion, the first axis emulateshuman (or test robot) thigh motion, and the second axis is angularknee (prosthesis) motion [16,45].

Note that the thigh and knee angles are strictly 1DOF in thesagittal walking plane. The transverse DOFs, including adductionand abduction, are also important, but are of secondary considera-tion; the first-order of importance is motion in the direction ofwalking, and so that is what this paper focuses on. The construc-tion of the test robot is detailed in Ref. [45]. Figure 1 also showsthe prototype prosthesis that we developed at Cleveland State Uni-versity, which is detailed in Refs. [46] and [47]. This is a genericlower limb prosthesis that has torque control at the knee andsupercapacitor-based energy regeneration.

The three degrees-of-freedom system model can be written asfollows:

M€q þ C _q þ gþ R ¼ u� Te (1)

where qT ¼ q1 q2 q3� �

comprises the generalized displace-ments (q1 is vertical displacement, q2 is thigh angle, and q3 isprosthetic knee angle); MðqÞ is the inertia matrix; Cðq; _qÞ is theCoriolis and Centripetal matrix; gðqÞ is the gravity vector; Rðq; _qÞis the nonlinear damping vector;

Te ¼ JTF is the effect of the combined horizontal (Fx) and ver-tical (Fz) components of the GRF on each joint, where J is

the Jacobian matrix and F ¼ Fx Fz� �T

is the GRF vector; ucomprises the active control force at the hip and the active controltorques at the thigh and prosthetic knee. It should be noted that“thigh torque” is also known as “hip torque” in the biomechanicalliterature.

We assume that the positions and velocities of each joint aremeasured accurately with encoders and differentiators. More detailsabout the sensors are provided in Ref. [45]. However, there is noneed to measure joint accelerations for our proposed controllers.

Although in a real-world prosthesis application, the hip forceand thigh torque are controlled by the human amputee, in thisresearch, they are control inputs of the test robot shown in Fig. 1,which emulates human hip and thigh motions. Also, real-timemeasurements of q1 and q2 are challenging in amputees, but inthis paper, they can easily be measured by incremental encoders.

2.2 Ground Reaction Forces Model. The prosthesis testrobot walks on a treadmill, which we model as a mechanical stiff-ness [16]. We model the vertical component of the GRF (Fz) forthe foot-treadmill contact as

Fz ¼0; Lz < sz�kbðsz � LzÞ; Lz > sz

�(2)

where kb is the belt stiffness; sz is the treadmill standoff (i.e., thevertical distance from the origin of the world frame (x0, y0, z0) to

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the belt); Lz is the vertical position of bottom of the foot in the world frame, which is given as follows (see Fig. 1):

Lz ¼ q1 þ l2sin q2ð Þ þ l3sin q2 þ q3ð Þ (3)

where l2 and l3 are the length of the thigh and shank, respectively. There is much less slack in the x-direction than in the z-direction, sowe consider slack only in z-direction [46]. The horizontal component of the GRF (Fx) can be modeled by an approximation of Coulombfriction as [48]

Fx ¼ �bFz1� e�vr=vc1þ e�vr=vc

� �(4)

where b is the belt friction coefficient;vc is scaling factor; vr is the velocity of the foot-treadmill contact relative to the treadmill, suchthat

vr ¼� _q2 l2sin q2ð Þ þ l3sin q2 þ q3ð Þ� �

� _q3 l3sin q2 þ q3ð Þð Þ � vt (5)

where vt is the treadmill speed. Based on Eq. (2), we divide one stride into two phases: swing phase, where Lz < sz; and stance phase,where Lz > sz. Therefore, we have zero Fz and zero GRF in the swing phase, and when the point foot hits the ground (stance phase),GRF appears as the belt stiffness times the belt deflection. Te ¼ JTF is due to the effect of GRF on each joint and is given as follows[16,17]:

Te ¼Fz

Fz l2cos q2ð Þ þ l3cos q2 þ q3ð Þ� �

� Fx l2sin q2ð Þ þ l3sin q2 þ q3ð Þ� �

Fz l3cos q2 þ q3ð Þð Þ � Fxðl3sin q2 þ q3ð Þ

264375

J ¼0 �l2sinðq2Þ � l3sinðq2 þ q3Þ �l3sinðq2 þ q3Þ1 l2cosðq2Þ þ l3cosðq2 þ q3Þ l3cosðq2 þ q3Þ

" # (6)

2.3 Model Regressor. The states and controls are defined as

xT ¼ q1 q2 q3 _q1 _q2 _q3� �

uT ¼ fhip sthigh sknee� �

(7)

Robot dynamics can be linearly parameterized by a model regressor Y0 q; _q; €qð Þ 2 Rn�r and parameter vector p 2 Rr , so the right side ofEq. (1) can be written in the following form:

M€q þ C _q þ gþ R ¼ Y0 q; _q; €qð Þp (8)

where Y0 q; _q; €qð Þ is a function of joint displacements, velocities, and accelerations; n is the number of links (n¼ 3 in this paper; see Fig.1); r is the number of parameter vector elements (r¼ 8 in this paper as shown below). The regressor Y0 q; _q; €qð Þ and the parameter phave many realizations; one such possibility is

Y0 q; _q; €qð Þ ¼€q1 � g Y012 Y013 0 0 0 0 sgn _q1ð Þ

0 Y022 Y023 €q2 Y

025 €q3 _q2 0

0 0 Y033 0 Y035 €q2 þ €q3 0 0

24 35Y012 ¼ €q2cos q2ð Þ � _q22sinðq2ÞY013 ¼ ð€q2 þ €q3Þcosðq3 þ q2Þ�ð2 _q2 _q3þ _q22þ _q23Þsinðq3 þ q2ÞY022 ¼ ð€q1 � gÞcosðq2ÞY023 ¼ Y033 ¼ ð€q1 � gÞcos q3 þ q2ð ÞY025 ¼ ð2€q2 þ €q3Þcosðq3Þ�ð2 _q2 _q3þ _q23Þsinðq3ÞY035 ¼ €q2cos q3ð Þ þ sin q3ð Þ _q22

(9)

p ¼

m1 þ m2 þ m3m3l2 þ m2l2 þ m2c2

m3c3I2z þ I3z þ m2c22 þ m3c32 þ m2l22 þ m3l22 þ 2m2c2l2

m3c3l2m3c3

2 þ I3zbf

266666666664

377777777775(10)

3 Robust Adaptive Impedance Control

We design two separate nonlinear robust adaptive impedance controllers using nonscalar boundary layers and sliding surfaces to trackhip displacement and knee and thigh angles in spite of parametric and nonparametric uncertainties. Both controllers use the same

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control laws, same target impedance models, and same nonscalarboundary layer trajectories but different adaptation laws. In thefirst controller, we design a RAIC with a TEB adaptation law,which extracts information about the parameters from the imped-ance model tracking error. In the second controller, we propose aRCAIC with BGF. Since impedance model tracking errors in thejoint displacements and prediction error in the joint torques areinfluenced by parameter uncertainties, in RCAIC we design aTEB/PEB adaptation law which drives parameter adaptation usingboth impedance model tracking error and prediction error toachieve more accurate estimation of the system parameters.

3.1 Target Impedance Model. The robot/prosthesis systeminteracts with the environmental admittance, so if we want to havea system that is well-matched with the mechanical characteristicsof the environment, the closed-loop system should behave as animpedance. In this way, we can achieve a tradeoff between per-formance and GRF.

We desire the closed-loop systems with both RAIC and RCAICto emulate the biomechanics of able-bodied walking. We thusdefine a target impedance model [32] with characteristics similarto able-bodied walking [16,49]

Mr €qr � €qdð Þ þ Br _qr � _qdð Þ þ Kr qr � qdð Þ ¼ �Te (11)

The reference mass Mr , damping coefficient Br , and spring stiff-ness Kr are positive definite matrices, while qr 2 Rn is the state ofthe reference model and qd 2 Rn is the reference trajectory. Weassume that the matrices are diagonal

Mr2 Rn�n ¼ diag M11 M22 … Mnn� �

Br2 Rn�n ¼ diag B11 B22 … Bnn� �

Kr 2 Rn�n ¼ diag K11 K22 … Knn� � (12)

3.2 Control Law. In Eq. (9), the regressor depends on accel-eration. However, acceleration measurements are typically noisy,so it might not be convenient to use Y0 q; _q; €qð Þ in real time. Toavoid the use of acceleration, we define error vector s and signalvector v [40,41,50]

s ¼ _e þ ke (13)

v ¼ _qr � ke (14)

e ¼ q� qr (15)

k ¼ diag k1; k2;…; knð Þ; ki > 0

where ki is a positive scalar tuned by the user. The joint accelera-tion measurements can be very noisy, so an acceleration-dependent model regressor as presented in Eq. (9) might not beconvenient for control design implementation. Consequently, toavoid the need to measure the joint accelerations, we define anacceleration-free controller regressor using signal vector v in Eq.(14) in place of the model regressor in Eq. (9)

M€q þ C _q þ gþ R ¼ Y q; _q; v; _vð Þp (16)

where Y q; _q; v; _vð Þ is a linear function, one realization of which isgiven as

Y q; _q;v; _vð Þ ¼_v1� g Y12 Y13 0 0 0 0 sgn _q1ð Þ

0 Y22 Y23 _v2 Y25 _v3 _q2 00 0 Y33 0 Y35 _v2þ _v3 0 0

24 35Y12 ¼ _v2cosðq2Þ�v2 _q2sinðq2ÞY13 ¼ ð _v2 þ _v3Þcosðq3 þ q2Þ

�ðv2 _q3þv2 _q2þv3 _q2þv3 _q3Þsinðq3 þ q2ÞY22 ¼ ð _v1 � gÞcosðq2ÞY23 ¼ Y33 ¼ ð _v1 � gÞcos q3 þ q2ð ÞY25 ¼ ð2 _v2 þ _v3Þcosðq3Þ

�ðv2 _q3þv3 _q3þv3 _q2Þsinðq3ÞY35 ¼ _v2cos q3ð Þ þ sin q3ð Þv2 _q2

(17)

By substituting Eqs. (13)–(15) in Eq. (1), we rewrite the model as

M _s þ Csþ gþ RþM _v þ Cv ¼ u� Te (18)

Fig. 1 The left figure shows the test robot/transfemoral prosthetic leg with a passive ankle;the right figure shows the 3DOF unified model with a point prosthetic foot. Human hip andthigh motions are emulated by a prosthesis test robot where the calf represents the prosthesisdevice with rigid ankle and foot. A treadmill belt serves as the walking surface. When the footis in contact with the treadmill belt, the GRF is nonzero.


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Since Eq. (1) is a second-order system, the error vector of Eq. (15)can be obtained from the first-order sliding surface

s ¼ ddtþ k

� �e (19)

where s includes n elements. Perfect impedance model trackingq ¼ qr (e ¼ 0) implies that s ¼ 0. To reach the sliding manifolds ¼ 0, the following reaching condition must be satisfied [30]:

sgn sð Þ _s � �c (20)

This vector inequality is taken one element at a time, and c is an

n-element vector denoted as c ¼ c1 c2 … cn� �T

where ci >0 is a design parameter. Eq. (20) shows that in the worst case,sgn sð Þ _s ¼ �c, so we calculate the worst-case reaching time of thetracking error trajectory asð0

sð0ÞsgnðsÞds ¼ �c

ðT0

dt! s 0ð Þ sgn sð Þ ¼ cT T ¼ s 0ð Þ

c(21)

This equation gives n different reaching times, s 0ð Þ is the error atthe initial time, and the quotient s 0ð Þ=c is defined one element at atime. We can see from Eq. (21) that a larger c gives smaller reach-ing times T. The system parameters are not known, so we use acontroller [30] to handle parameter uncertainty and to satisfy thecondition of Eq. (20)

u ¼ bM _v þ bCvþ bg þ bR þ bT e � KdsgnðsÞ (22)where bM; bC; bg; bR; and bTe are estimates of M;C; g;R, and Te, andKd is a tuning matrix denoted as Kd ¼ diag Kd1;Kd2;ð…;KdnÞ;where Kdi > 0. Note that sgnðsÞ is discontinuous, whichmeans that it would result in control chattering; therefore, wereplace it with the saturation function satðs=diagðuÞÞ (see Fig. 2).The division and saturation operations in satðs=diagðuÞÞ are takenone element at a time. The term diagðuÞ is an n-element vector.This all results in a modification of the controller of Eq. (22)

u ¼ bM _v þ bCvþ bg þ bR þ bTe � Kdsatðs=diagðuÞÞ (23)The diagonal elements of u are the widths of the saturation func-tion. The control law of Eq. (23) includes two parts. The first part,bM _v þ bCvþ bg þ bR, is an adaptive term that handles uncertainparameters. The second part, bT e � Kdsatðs=diagðuÞÞ, is a robust-ness term that satisfies Eq. (20) and the variations of the externalinputs Te as nonparametric uncertainties. We substitute Eq. (23)

into Eq. (18) and define ~M ¼ bM �M, ~C ¼ bC � C, ~g ¼ bg � g,~R ¼ bR � R, and ~p ¼ bp � p, to derive the closed-loop systemM _s þ Csþ Kdsat s=diagðuÞð Þ þ Te � bT e� � ¼ ð ~M _v þ ~Cvþ ~g þ ~RÞ

(24)

where bp is the estimate of p.We separate the right side of Eq. (24) into two parts: the regres-

sor Y q; _q; v; _vð Þ and the parameter estimation error ~p. We can,thus, write Eq. (24) in the following regressor (linear parametric)form:

M _s þ Csþ Kdsat s=diagðuÞð Þ þ Te � bT e� � ¼ Y q; _q; v; _vð Þ~p (25)3.3 Nonscalar Boundary Layer Trajectories. One of the

challenges with adaptive control is that in the presence of non-parametric uncertainties such as noise and disturbances, and alsoin the presence of large adaptation gains and reference trajecto-ries, the estimated parameters are prone to oscillate and growwithout bound because of instability in the control system. Thisphenomenon is known as parameter drift. However, if the model

regressor Y0 q; _q; €qð Þ satisfies persistent excitation (PE) conditions,the adaptive control scheme exhibits robustness against nonpara-metric uncertainties and unmodeled dynamics, and parameter driftcan be avoided [30,50].

To turn off the TEB adaptation mechanism to prevent unfavora-ble parameter drift when the impedance model tracking errors aresmall and due mostly to noise and disturbances, we incorporatenonscalar boundary layer trajectories sD into both controllersRAIC and RCAIC. We define these trajectories to balance controlchatter and performance. Furthermore, we define the trajectoriessD so the error trajectories converge to the boundary layers andboth proposed controllers show robustness to nonparametricuncertainties. We define these boundary trajectories sD as follows[30]:

sD ¼0; sj j � diagðuÞs� usatðs=diagðuÞÞ; sj j > diagðuÞ

�(26)

Note that sD is an n-element vector. We call the region sj j �diagðuÞ the boundary layer, where the inequality is taken one ele-ment at a time. Note that the diagonal elements of u comprise thethickness values of the boundary layer and are denoted as u ¼diag u1;u2;…;unð Þ; where tunable ui > 0. We illustrate sD andsatðs=diagðuÞÞ for a single dimension in Fig. 2.

3.4 Robust Adaptive Impedance Controller. RAIC uses thecontrol law in Eq. (23), nonscalar boundary layer trajectories inEq. (26), and the TEB adaptation law, so the prosthesis/RAICcombination converges to the target impedance model in Eq. (11).The TEB adaptation law can be presented as

_bp ¼ �l�1YT q; _q; v; _vð ÞsD (27)where l 2 Rr�r is a positive definite matrix with diagonal ele-ments, which is adjusted by the user.

THEOREM 1. Consider the following scalar positive definite Lya-punov function [50]

V sD; ~pð Þ ¼1

2sD

TMsD� �

þ 12

~pTl~p� �

(28)

where l is a design parameter such that l ¼ diag l1; l2;ð…;lrÞ;with li > 0. The closed-loop system using RAIC resultsin _V sD; ~pð Þ ! 0 as t!1. That is, the closed-loop systems areasymptotically stable. The error vector s converges to the bound-ary layer, which implies convergence of the closed-loop system tothe target impedance model.

Proof of Theorem 1: See Appendix A.

3.5 Robust Composite Adaptive Impedance Controller.The RCAIC uses the same control law in Eq. (23) and nonscalarboundary layer trajectories in Eq. (26) as the RAIC uses but uses a

Fig. 2 Saturation function and sD in one dimension


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different adaptation law, i.e., the TEB/PEB mechanism, so theprosthesis/RCAIC combination converges to the target impedancemodel in Eq. (11). In the TEB adaptive controller (RAIC), theadaptation law extracts information about the parameters onlyfrom the impedance model tracking error. However, the trackingerror is not the only source of parameter information; predictionerror also contains parameter information. Therefore, by using acombination of the impedance model tracking and predictionerrors, the performance of the adaptive controller can beimproved. For the RCAIC, a TEB/PEB adaptation law is intro-duced as follows [30,50]:

_bp ¼ �P tð Þ½YT q; _q; v; _vð ÞsD þWTRep� (29)where R ¼ dIn�n is a positive definite diagonal weighting matrixthat indicates how much the adaptation law uses the predictionerror (d is a positive constant); P tð Þ is time-varying adaptationgain; W is a filtered version of the model regressor matrixY0 q; _q; €qð Þ given in Eq. (9), where this filtering is introduced toavoid the need for joint acceleration in the regressor [50]; and epis the prediction error and is calculated from W q; _qð Þ~p (detailswill be presented later in this section). The filtering can be donewith a first-order stable filter as follows:

W q; _qð Þ ¼ csþ c Y

0 q; _q; €qð Þ (30)

where c > 0. To filter in the time domain, we convolve both sidesof Eq. (1) with the impulse response of c=ðsþ cÞ(thatis;w tð Þ ¼ ce�ct):ðt

0

w t� hð Þ M€q þ C _q þ gþ R½ �dh ¼ðt

0

w t� hð Þ½u� Te�dh (31)

The first part of Eq. (31),Ð t

0w t� hð ÞM€qdh, can be written as

follows:ðt0

w t� hð ÞM€qdh ¼ w t� hð ÞM _qjt0

�ðt

0

d

dhw t� hð ÞMð Þ _qdh

¼ w 0ð ÞM _q � w tð ÞM q 0ð Þ� �

_q 0ð Þ

�ðt

0

½w t� hð Þ _M _q þ ddh

w t� hð Þð ÞM _q�dh

(32)

That is, convolving the left-hand side of Eq. (31) can be inter-preted as filtering that side and is equal to W q; _qð Þp, so that

y tð Þ ¼ W q; _qð Þp ¼ w 0ð ÞM _q � w tð ÞM q 0ð Þ� �

_q 0ð Þ

�ðt

0

½w t� hð Þ _M _q þ ddh

w t� hð Þð ÞM _q�dh

þðt

0

w t� hð Þ½C _q þ gþ R�dh

(33)

where y tð Þ is the filtered version of the right side of Eq. (1) and isgiven as follows:

y tð Þ ¼ðt

0

w t� hð Þ½u� Te�dh (34)

The estimated value of y tð Þ can be written as follows:

by tð Þ ¼ W q; _qð Þbp (35)Therefore, the prediction error ep is derived as

ep ¼ by tð Þ � y tð Þ ¼ W q; _qð Þ~p (36)It is important to note that past data are generated from pastparameter values, and the algorithm should therefore pay lessattention to past data when generating current parameter esti-mates. Therefore, exponential data forgetting is advisable for esti-mating time-varying parameters. The composite adaptation law inEq. (29) can benefit from an exponentially forgetting least-squaresgain update for P tð Þ as follows [42,50]:

d

dtP�1ð Þ ¼ �# tð ÞP�1 þWT tð ÞW tð Þ (37)

where # tð Þ � 0 denotes the time-varying forgetting factor. Tobenefit from data forgetting and to avoid unboundedness in P tð Þ,the BGF method can be used to tune the time-varying forgettingfactor # tð Þ as follows [50]:

# tð Þ ¼ #0 1�Pk kK0

� �(38)

where #0 is the maximum forgetting rate; K0 is the upper boundof P tð Þ; and Pð0Þ must be smaller than K0I. The second part of theTEB/PEB adaptation law in Eq. (29) can be written as

_~p ¼ �P tð ÞWTRW~p (39)

Solving Eq. (39) gives

~pðtÞ ¼ ~pð0Þexpðt

0

�P tð ÞWT tð ÞRW tð Þdt !

(40)

Therefore, ~p ¼ bp � p will exponentially converge to zero ifW q; _qð Þ is PE. The speed of convergence can be heavily depend-ent on the magnitude of the adaptation gain. W q; _qð Þ must satisfythe following PE condition:

limt!1

ðt0

�WT tð ÞWðtÞdt ¼ 1 (41)

Therefore, ~p will exponentially converge to zero for nonzero andconstant W. It is interesting to note that when W is not PE, ~p can-not converge to zero, even if there are no nonparametric uncer-tainties, and the robustness property cannot be guaranteed. In thisprocedure, the time-varying forgetting factor is tuned so that dataforgetting is active when WðtÞ is PE and it is off when WðtÞ is notPE. From Eq. (38), Pk k shows the level of PE of WðtÞ so that if

Pk k decreases, WðtÞ is strongly PE (# tð Þ ¼ #0), and if Pk kincreases, WðtÞ is weakly PE. In the BGF composite controller, ~pand P tð Þ are upper bounded, and if WðtÞ is strongly PE, then ~pexponentially converges to zero, P tð Þ is upper and lower boundedby positive numbers, and # tð Þ > # > 0.

THEOREM 2. Consider the scalar positive definite Lyapunovfunction

V sD; ~pð Þ ¼1

2sD

TMsD� �

þ 12

~pTP�1 ~p� �

(42)

The controller of Eq. (23), when used in conjunction with theupdate law of Eq. (29) and applied to the system of Eq. (1), resultsin _V sD; ~pð Þ ! 0 as t!1, which means the prosthesis/RCAICcombination is globally exponentially stable. The error vector sconverges to the boundary layer, indicating perfect estimation ofthe system parameters and convergence of the closed-loop systemto the target impedance model.

Proof of Theorem 2: See Appendix B.

3.6 System Convergence. To get a feeling for the RAIC/RCAIC structure, consider the general structure of Fig. 3. To


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show that both proposed controller structures RAIC and RCAICresult in closed-loop systems that converge to the target imped-ance model, we use Eqs. (13)–(15) to write the closed-loop systemas

~M €qr þ ~C þ ~Mkð Þ _qr þ ~Ckqr ¼ ~Mk _q þ ~Ckq� ~g � ~R þM _s þ CsþKdsat s=diagðuÞð Þ þ Te � bT e� �

(43)

From Eq. (11) we have the target impedance model

Mr €qr þ Br _qr þ Krqr ¼ Mr €qd þ Br _qd þ Krqd � Te (44)

From Theorems 1 and 2, sD ! 0 as t!1 so the trajectories of sare bounded in the boundary layers. Now since s is bounded, eand _e are bounded. The boundedness of qr , _qr , e, and _e impliesthat q and _q are bounded, which in turn implies that the right sideof Eq. (43) is bounded, just as the right side of Eq. (44) isbounded.

It is seen that the closed-loop system in Eq. (43) has the samestructure as the impedance model of Eq. (44), which means bothproposed controllers result in closed-loop systems that convergeto the target impedance model of Eq. (44); where comparing

Eq. (43) with Eq. (44) gives Mr ¼ ~M;Br ¼ ~C þ ~Mk, andKr ¼ ~Ck. We see that the proposed control law in Eq. (23) forboth RAIC and RCAIC drives the closed-loop system in Eq. (25)to match the impedance model in Eq. (11).

4 Simulation Results

4.1 Experimental Reference Trajectory. The reference tra-jectory is obtained from the motion studies lab (MSL) of theCleveland Veterans Affairs Medical Center (VAMC) [11]. Inorder to calculate three-dimensional joint angles, a three-dimensional model was constructed from 47 reflective markersplaced on the research participants. The research participants werevolunteers in this study, which was approved by the InstitutionalReview Board of the Cleveland VAMC. Data were collected at aspecific walking speed. The research participants walked on atreadmill for 10–30 s trials while kinematic and kinetic data were

collected at their preferred walking speed. This speed was deter-mined using previous methods [51], which allowed for acclimat-ing to the treadmill. Moreover, all of the research participants hadprevious treadmill experience. The kinematic data were collectedat 100 Hz via a 16 camera passive marker motion capture system(Vicon, Oxford Metrics, UK) with the markers mounted accordingto the Human Body Model (Motek, Amsterdam, The Nether-lands). In addition, GRFs were collected at 1000 Hz via two forceplates within the treadmill (ADAL3DM-F-COP-Mz, Tecmachine,France). For data processing, 100 frames were taken from a stand-ing trial for initialization of the subject-specific model comprised18 body segments and 46 kinematic degrees-of-freedom. Thekinematics and the ground reaction forces were the input for aninverse dynamic analysis after low pass filtering at 6 Hz with a

Fig. 3 RAIC/RCAIC structure. Note that this flowchart includes both the hip model and theprosthesis model.

Table 1 Nominal system parameter values

Parameter Description Value Units

sz Treadmill standoff (Eq. (2)) 0.905 mkb Belt stiffness (Eq. (2)) 37000 N/mb Belt friction coefficient (Eq. (4)) 0.2 —vc Scaling factor (Eq. (4)) 0.05 m/svt Treadmill speed (Eq. (4)) �1.25 m/s

Table 2 Nominal values of model parameters

Parameter Description Nominal value Units

m1 Mass of link 1 40.5969 kgm2 Mass of link 2 8.5731 kgm3 Mass of link 3 2.29 kgl2 Thigh length 0.425 ml3 Length of knee joint to bottom of shoe 0.527 mc2 Center of mass on thigh 0.09 mc3 Center of mass on shank 0.32 mf Sliding friction in link 1 83.33 Nb Rotary actuator damping 9.75 N m sI2z Rotary inertia of link 2 0.138 kg/m

2

I3z Rotary inertia of link 3 0.0618 kg/m2

g Acceleration of gravity 9.81 m/s2


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second-order low pass Butterworth filter. The data were finallyprocessed to solve the skeletal motion and to compute the inversedynamics for each model.

4.2 Prosthesis Test Robot Model, Controllers, and TargetImpedance Model Parameters. Here we demonstrate the per-formance of RAIC and RCAIC with simulation. In the prosthesis

test model considered here, we have q 2 R3, so target impedancemodel coefficients presented in Eq. (12) can be written as

Mr ¼ diag M11 M22 M33� �

, Br ¼ diag B11 B22 B33� �

, and

Kr ¼ diag K11 K22 K33� �

. To obtain critically damped

responses (two equal roots for each joint displacement) in the ref-

erence impedance model of Eq. (11), we set Bii¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiKiiMiip

and

the two roots are both equal to �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiKii=Mii

p(i ¼ 1; 2; 3). To obtain

two different real roots, Bii > 2ffiffiffiffiffiffiffiffiffiffiffiffiKiiMiip

. Here, we use a referenceimpedance model with roots �11 and �88 for the thigh, �5 and�94 for the knee, and �3 and �497 for the hip. These values pro-vide a reference impedance model that is stable, performs similarto able-bodied walking, provides good reference model tracking,and provides control signals and GRFs with the same order ofmagnitude as able-bodied ones. We, thus, obtain the followingreference impedance matrices:

Mr ¼ diag 10; 10; 10� �

Kr ¼ diag 15000; 10000; 5000� �

Br ¼ diag 5000; 1000; 1000� �

We assume that the treadmill parameters (i.e., GRFs parameters)are constant and listed in Table 1. We suppose that the prosthesistest robot parameters are partly unknown and can vary by up to30% from their nominal values [16]. The nominal systemparameters are shown in Table 2. The initial state isx 0T ¼ 0:019 1:13 0:09 0:09 0 1:6

� �. After some experi-

mentation, we achieve good performance for RAIC and RCAICwith the design parameters in Table 3. As seen from Table 3, thecontroller design parameters are round numbers, which meansthat the controllers are relatively easy to tune and do not require atedious gain-tuning process.

Note that the performance of RCAIC is slightly influenced byeach of its design parameters. From Eq. (30), 1=c is the steady-state gain of the filter and should be tuned so the bandwidth of thefilter is larger than the system bandwidth and smaller than thenoise frequency. We choose d ¼ 2; so that the adaptation law inEq. (29) weights the prediction error twice as much as the imped-ance model tracking error. The value of #0 in Eq. (38) influencesthe speed of forgetting and determines the compromise betweenparameter tracking speed and oscillation of the estimated parame-ters. K0 in Eq. (38) represents a tradeoff between parameterupdate speed and the disturbance effect on the prediction error.

Pð0Þ represents a tradeoff between parameter error value and thedegree of stability. However, we should choose Pð0Þ as high asthe noise sensitivity allows to achieve the lowest parameter errorvalue; to avoid unbounded P tð Þ;Pð0Þ must be smaller than K0I.The value of u for both proposed controllers provides a trade-offbetween chattering on the control signal and tracking error bound,adjusts the robustness of the system to nonparametric uncertain-ties, and tunes the sensitivity of the controllers to parameter drift.

4.3 Controller Performance Evaluation. We define a costfunction to evaluate the performance of RAIC and RCAIC, wherethe tracking error part of the cost and the control part of the costare defined as

RMSEi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

T

ðT0

ðxi � rdiÞ2dt

s; i ¼ 1;…; 6 (45)

RMSUj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

T

ðT0

uj2dt

s; j ¼ 1;…; 3 (46)

where T is the simulation time period, and x, r, and u are given as

xT ¼ q1 q2 q3 _q1 _q2 _q3� �

rT ¼ rd1 rd2 rd3 rd4 rd5 rd6� �¼ qd1 qd2 qd3 _qd1 _qd2 _qd3� �

uT ¼ fhip sthigh sknee� �

(47)

The components of the normalized cost function are defined as

CostEi ¼RMSEi

maxt2½0;T� rdij jcostUj ¼

RMSUj

maxt2½0;T� uabij j(48)

where uabi indicates the ith able-bodied control signal (able-bod-ied control comprises hip force, thigh torque, and knee torque).The total desired trajectory tracking cost and the total control costare defined as follows:

CostE ¼X6i¼1

CostEi (49)

CostU ¼X3j¼1

CostUi (50)

The total cost is a combination of the total desired trajectorytracking cost in Eq. (49), and the total control cost in Eq. (50):

Table 3 Controllers design parameters

Controller type Parameter Description Value

u Boundary layer thicknesses (Eq. (26)) 0.5IRAIC Kd Robust term coefficients (Eq. (22)) 100I

l Adaptation rate (Eq. (27)) 0.01Ik Sliding term coefficients (Eq. (13)) 100Iu Boundary layer thicknesses (Eq. (26)) 0.5IKd Robust term coefficients (Eq. (22)) 100Ik Sliding term coefficients (Eq. (13)) 100I

RCAIC #0 Maximum forgetting rate (Eq. (38)) 5K0 Upper bound of the adaptation gain (Eq. (38)) 400

Pð0Þ Initial condition of the adaptation gain (Eq. (37)) 100IC Filter constant (Eq. (30)) 1D Weighting constant (Eq. (29)) 2


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Cost ¼ CostE þ CostU (51)

We define a cost function to evaluate the estimation of the param-eter vector p 2 R8 presented in Eq. (10) as

RMSPk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

T

ðT0

ðbpk � p0kÞ2dts

; k ¼ 1;…; 8 (52)

where bpk and p0k are kth elements of the estimated parameter vec-tor and the true parameter vector, respectively. The normalizedand total estimation costs are given as follows:

CostPk ¼RMSPk

maxt2½0;T�

p0kj j(53)

CostP ¼X8i¼1

CostPk (54)

4.4 Simulation Results

4.4.1 Tracking Performance. Figure 4 compares the states ofthe system with RAIC and RCAIC and the reference trajectories(qd) when all system parameters are equally varied by 30% fromtheir nominal values (Table 2).

It should be noted that to consider the tracking sensitivity to thesystem parameters, a sensitivity analysis is required in which theparameters should be perturbed one by one not altogether. Figure4 shows that both controllers demonstrate robustness and alsowalking behavior of the prosthesis is similar to human-like walk-ing. Although, Theorems 1 and 2 guarantee that the states of thesystem accurately track the model reference trajectory (qr), Fig. 4compares the states of the system with the reference trajectories

Fig. 4 Tracking performance with 130% parameter deviations:desired trajectory (dotted line), response with RAIC (dashedline), and response with RCAIC (solid line)

Fig. 5 Control signals for 30% parameter deviations: RAIC(dashed line) and RCAIC (solid line)

Fig. 6 GRFs for 30% parameter deviations: RAIC (dashed line)and RCAIC (solid line). The circles on the x-axis of the right plotshow the foot strikes on the treadmill.


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(qd). The reason is to show the flexibility of the prosthesis walkingin following the desired trajectories provided by the impedancemodel of Eq. (11). In this paper, the actual hip position should notperfectly follow its desired value and if it does, the walking losesthe flexibility in the presence of the GRF impacts and other exter-nal effects.

Figure 5 shows the control signals of RAIC and RCAIC (thecontrol force for the hip, and the control torques for the thigh andknee) with 30% parameter deviations. The control magnitudes forthe off-nominal case have similar magnitudes as able-bodied aver-aged hip force (–800 to 200 N), thigh torque (–50 to 100 N�m),and knee torque (–50 to 50 N�m) [52–54]. Note that the hip forceand thigh torque represent able-bodied walking, and the kneetorque acts on the prosthesis, which has the same magnitude asable-body knee torque. This indicates a strong potential for theproposed controllers to be useful in real-world prosthesis applica-tions. In addition, the results demonstrate that the controllers candeal with parameter variations without large increases in the con-trol magnitudes.

For both controllers, high gains in the reference impedancemodel not only provide better tracking, particularly for hip dis-placement, but also increase the control effort. Figure 6 depictsthe GRFs when the system parameters vary by 30% from nominal.We see that the generated forces are similar to able-bodied aver-aged horizontal GRF (�150 to 150 N) and vertical GRF (0–800N) [52–54], again indicating strong potential for real-world appli-cation. As can be observed from Fig. 6, we have no GRF in swingphase, and after the point foot hits the ground (circles on thex-axis), horizontal and vertical GRFs become nonzero.

4.4.2 Parameter Identification. Figure 7 shows the estimatedparameter vector p (presented in Eq. (10)) for RAIC and RCAICwhen the system parameters vary by 30%. As expected, the RAICparameter estimates do not match the true parameter values.

Fig. 7 True parameter values (dotted lines) and estimatedparameter values for 30% parameter deviations: RAIC (dashedline) and RCAIC (solid line)

Fig. 8 Trajectories of sD and s for the RAIC (dashed line), andthe RCAIC (solid line) with 130% parameter deviations

Fig. 9 (a) Norm of P, (b) time-varying forgetting factor, and (c)joint prediction errors (epi; i 5 1;2;3); all plots represent the sit-uation of 30% parameter uncertainty


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However, RCAIC using the BGF composite adaptation law per-forms noticeably better regarding parameter estimation comparedto RAIC. The estimated parameter vector of the proposed control-ler RCAIC perfectly matches the true value except for the fourthelement P4. P4 is the most complex parameter in terms of its con-stituent elements (see Eq. (10)), so errors in the constituent ele-ments of P4 can cause cumulative errors in P4.

Figure 8 compares the trajectories of s and sD as described inEqs. (13) and (24), respectively for RAIC and RCAIC. Based onthe values of u1;u2, u3, and the definition of sD, it is seen that theTEB adaptation mechanism in RAIC is active only when s is out-side its boundary layer (i.e., sD is nonzero). When sD is zero, theparameter adaptation of the RAIC (Eq. (27)) stops and its esti-mated parameters remain constant, whereas sD ¼ 0 only turns offTEB adaptation part of the RCAIC (the first part of the Eq. (29)).

It is observed that none of the s trajectories for the RCAICexceed the boundary layer (the area between ui ¼ �0:5 andui ¼ þ0:5) and in turn all sD trajectories are zero. This shows thatthe RCAIC only uses prediction errors, which appear in the PEBadaptation, and the TEB adaptation mechanism is turned off.

On the other hand, all s trajectories for the RAIC exceed theboundary layer. From Fig. 8, we can see that the s trajectories ofthe RAIC for the hip, thigh, and knee exceed the boundary layerfour, three, and two times, respectively, and in turn, the sD trajec-tories are nonzero.

4.4.3 Persistent Excitation Verification. Figure 9 shows thenorm of the adaptation gain P, the time-varying forgetting factor# tð Þ, and the joint prediction errors (epi; i ¼ 1; 2; 3) for the RCAICwith þ30% uncertainty on the system parameters. Figure 9(a)illustrates that P tð Þ is upper and lower bounded by two positivenumbers (P tð Þ is upper bounded by K0 ¼ 400 and lower boundedby Pð0Þ ¼ 100). Figure 9(b) shows that the forgetting factor satis-fies the condition # tð Þ > # > 0. These observations imply thatWðtÞ is PE. Since WðtÞ is PE, ~p and epi exponentially converge tozero as shown in Fig. 9(c).

4.4.4 Numerical Evaluation. Table 4 summarizes the desiredtrajectory tracking, parameter estimation, and control performancefor RAIC and RCAIC for the nominal system parameter valuesand also when the parameter values vary 630% relative tonominal.

Table 4 lists total desired trajectory tracking cost CostE, totalcontrol cost CostU , total estimation cost CostP, and total cost Cost

(which is sum of the desired trajectory tracking and control costs)for both controllers. Table 4 shows that for the nominal case,RAIC has better performance for the control cost and estimation,while tracking performance maintains the same level as RCAIC,and in turn RAIC slightly improves the total cost by 1.2%. Whenthe system parameter values vary �30%, RCAIC has a smallimprovement in control cost, but an improvement in estimation by40% in comparison with the RAIC, while tracking performance ofthe RCAIC slightly deteriorates. In general, in the case of �30%parameter uncertainty, the total cost of the RCAIC decreases by4%.

Table 4 shows that when the parameter values vary 30% fromnominal, RCAIC has a remarkable superiority to the RAIC interms of the estimation and tracking performances. This superior-ity is because by more accurately estimating the system parame-ters, the RCAIC includes a more accurate model (~p and epexponentially converge to zero) and in this way achieves bettertracking. Table 4 shows that desired trajectory tracking perform-ance (CostE) and estimation performance (CostP) of the proposedRCAIC considerably improves by 9.5% and 76%, respectively,whereas the control signal magnitude (CostU) and total cost (Cost)increases by 9.9% and 3.6%, respectively compared with RAIC.As it is seen from Table 4, the most notable difference betweenthe controllers is their parameter estimation performances whiletheir tracking performances are not noticeably different from eachother. This is because the RCAIC uses a different adaptation lawto improve the parameter estimation accuracy, whereas both con-trollers use the same control law.

Table 5 shows the maximum control effort (Umax), maximumtracking error (Emax), and maximum estimation error (Pmax) forthe nominal and off-nominal cases. The table shows that RCAICresults in smaller Umax for the nominal and þ30% cases, whereasRAIC performs better for the –30% case. Although RCAIC out-performs RAIC in terms of average parameter estimation for theoff-nominal cases ( CostP in Table 4), their Pmax values are withinapproximately 1% of each other. Table 5 also shows that Emax isabout the same for both controllers for the nominal and off-nominal cases.

5 Conclusions and Future Work

We designed two robust adaptive impedance controllers, RAICand RCAIC, for a combined test robot and transfemoral prosthesisdevice. The controllers estimate the system parameters and alsodriving joint tracking errors to boundary layers while compensat-ing for the variations of GRFs and nonparametric uncertainties.We defined the boundary layers to make a tradeoff between con-trol signal chatter and performance, and also to stop TEB adapta-tion mechanism in these layers to prevent unfavorable parameterdrift.

We designed both controllers to imitate the characteristics ofnatural walking and to provide flexible, smooth, gait. We thusdefined a reference model with impedance similar to that of able-bodied gait. We also proved closed-loop system stability for bothRAIC and RCAIC based on nonscalar boundary layers using Bar-balat’s lemma and Lyapunov theory.

Table 4 Controller performance

Controller type Controller performance

Parameter uncertainty CostE (Eq. (49)) CostU (Eq. (50)) CostP (Eq. (54)) Cost (Eq. (51))

Nominal RAIC 0.96 2.40 0.00 3.36RCAIC 0.96 2.44 0.80 3.40

–30% RAIC 0.93 2.65 4.42 3.58RCAIC 0.96 2.48 2.66 3.44

þ30% RAIC 1.05 2.22 14.62 3.27RCAIC 0.95 2.44 3.46 3.39

Table 5 Controller performance

Controller performance

Parameter uncertainty Controller type Umax Emax Pmax

Nominal RAIC 741.60 0.78 4.36RCAIC 737.38 0.78 4.54

–30% RAIC 700.17 0.77 25.00RCAIC 737.37 0.78 25.44

þ30% RAIC 868.75 0.80 25.48RCAIC 849.14 0.78 25.21


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We performed simulations for both proposed controllers with30% parameter errors, and we showed that trajectory trackingremained good, which demonstrated robustness of the proposedcontrollers. We demonstrated good transient responses with nomi-nal system parameter values and also with system parameter valuedeviations of up to 630%. When we used the first controllerRAIC for the 30% parameter deviations, desired trajectory track-ing errors were 16 mm for vertical hip position, 0.15 deg for thighangle, and 0.12 deg for prosthetic knee angle. When we used thesecond controller, RCAIC, with 30% parameter uncertainties, tra-jectory tracking errors were 14 mm for vertical hip position,0.15 deg for thigh angle, and 0.08 deg for knee angle.

Therefore, numerical results showed that when the systemparameter values varied by 30% from nominal, the proposed con-troller RCAIC had better tracking performance by 9.5% in com-parison to RAIC, while resulting in more control cost by 9.9%.Furthermore, RCAIC using the BGF composite adaptation lawachieved much better parameter estimation by 76% compared tothe RAIC. We also achieved reasonable control signals and GRFsfor the both controller structures. Note that, however, RCAIC, ingeneral, performed better than RAIC; RCAIC has larger computa-tional time and higher programming complexity.

For future work, we will incorporate rotary and linear actuatordynamics in the system model to obtain motor voltage control sig-nals. We will also apply the controllers to a prosthesis prototypethat has been developed at Cleveland State University. We willalso include an active ankle joint to the system model to extendthe controllers to a 4-DOF robot/prosthesis model. We will testthe proposed prosthesis model and controllers on a human-prosthesis hybrid system [22]. We will also implement the pro-posed controllers experimentally on a powered transfemoral pros-thesis, AMPRO3 (AMBER Prosthetic) [24]. In this paper, allsystem parameters are perturbed equally and at the same time, soa more in-depth sensitivity analysis of the system performance tothe parameter variation would be interesting. The results of thispaper are obtained for one experimental reference trajectory(Sec. 4.1). So, understanding the effects of varying speed, stridefrequency, and other gait kinematic parameters on the overall per-formance will be an interesting problem.

The results in this paper can be reproduced with the MATLABcode that is available at the website link.2

Acknowledgment

The authors are grateful to Jean-Jacques Slotine, Elizabeth C.Hardin, Antonie van den Bogert, and and Mojtaba Sharifi for sug-gestions that improved this paper.

Funding Data

� National Science Foundation (Grant No. 1344954).

Appendix A

A.1 Stability Analysis of the Robust AdaptiveImpedance Controllers

Proof of Theorem 1: Even though sD is not differentiable every-where, V is differentiable because it is a quadratic function of sD.The derivative of the Lyapunov function of Eq. (28) is given asfollows:

_V sD; ~pð Þ ¼1

2_sD

TMsD þ sDTM _sD� �

þ 12

sDT _MsD

� �þ 1

2_~p

Tl~p þ ~pTl _~p

� �¼ sDTM _sD þ

1

2sD

T _MsD� �

þ _~p Tl~p

(A1)

Note that inside the boundary layer in Eq. (26), _sD ¼ 0, and out-side the boundary layer _sD ¼ _s, so using the closed-loop form inEq. (25) gives

_V sD; ~pð Þ ¼ sDTð�Cs� Kdsatðs=diagðuÞÞ þ bTe � Te� �

þY q; _q; v; _vð Þ~pÞ þ 12

sDT _MsD

� �þ _~p Tl~p

¼ �sDTCsþ1

2sD

T _MsD� �

�sDTKdsat s=diag uð Þ� �

þsDT bTe � Te� �þ sDTY q; _q; v; _vð Þ~p þ _~p Tl~p(A2)

To derive the adaptation law, we constrain _~pTl~p þ

sDTY q; _q; v; _vð Þ~p to zero, which gives the update law _bp ¼�l�1YT q; _q; v; _vð ÞsD as already presented in Eq. (27). As seenfrom Eq. (27), the adaptation law extracts information about theparameters from only the tracking error (i.e., TEB). Therefore,_V sD; ~pð Þ can be written as follows:

_V sD; ~pð Þ ¼ �sDTCsþ 1

2sD

T _MsD� �

�sDTKdsatðs=diagðuÞÞ þ sDT bT e � Te� � (A3)We see from Eq. (26) that if sj j � diagðuÞ, then sD ¼ 0 and_V sD; ~pð Þ converges to zero inside the boundary layer. Conversely,

if sj j > diagðuÞ, then sD is defined by the second part of Eq. (26),in which case s ¼ sD þ usatðs=diagðuÞÞ outside the boundarylayer. If we substitute s ¼ sD þ usatðs=diagðuÞÞ in Eq. (57), weobtain _V sD; ~pð Þ outside the boundary layer as follows:

_V sD; ~pð Þ ¼1

2sD

T _M � 2Cð ÞsD�sDTCusat s=diag uð Þ� �

�sDTKdsatðs=diagðuÞ þ sDT bT e � Te� � (A4)Matrix _ðM � 2CÞ is skew symmetric, so sDT _M � 2Cð ÞsD ¼ 0 andwe simplify _V sD; ~pð Þ as

_V sD; ~pð Þ ¼ �sDT Cuþ Kdð Þsatðs=diagðuÞÞ þ sDT bT e � Te� �(A5)

We choose Kd and u as tuning parameters to keep Cuþ Kdbounded from below by the KmI, where Km is a positive scalar.We can see that Cuþ Kd � KmI ensures that Cuþ Kd is positivedefinite. We use Eq. (A5) to write

_V sD; ~pð Þ � �KmsDTsatðs=diagðuÞÞ þ sDT bT e � Te� � (A6)We note that sD

Tsatðs=diagðuÞÞ is the one-norm of sD, so we writeEq. (A6) as

_V sD; ~pð Þ � �Km sDk k1 þ sDT bT e � Te� � (A7)

We now define Km ¼ Fm þ cm, where bT ei � Tei � Fi � Fm,Fm ¼ maxðFiÞ, and cm ¼ maxðciÞ. We can then write Eq. (A7) asfollows:

_V sD; ~pð Þ � �cm sDk k1 � Fm sDk k1 þ sDT bT e � Te� � (A8)

Noting that bT ei � Tei � Fi � Fm and sDi � sDij j, we see thatsD

T bT e � Te� � in Eq. (A8) is bounded from above by Fm sDk k1, so_V sD; ~pð Þ � �cm sDk k1 (A9)

This indicates that outside the boundary layer (the second condi-tion of Eq. (26)), the Lyapunov derivative is negative2http://embeddedlab.csuohio.edu/prosthetics/research/robust-adaptive.html


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semidefinite, so we can prove the stability of the closed-loop sys-tem with Barbalat’s lemma [50].

Barbalat’s Lemma: If a Lyapunov function V ¼ Vðt; xÞ satisfiesthe following conditions:

ið Þ Vðt; xÞ is lower bounded, andðiiÞ _Vðt; xÞ is negative semi-definite, andðiiiÞ €V t; xð Þ is boundedthen _Vðt; xÞ ! 0 as t!1; that is, the closed-loop system is

asymptotically stable. Now we state an intermediate lemma thatwe will need to complete the proof of Theorem 1.

LEMMA 1. The derivative of the Lyapunov function of Eq. (A9)converges to zero, which guarantees that the system converges tothe boundary layer.

Proof of Lemma 1: Conditions I and II in Barbalat’s Lemma areconfirmed from Eqs. (28) and (A9), which means that V isbounded. This implies that all of the terms in V in Eq. (28) arebounded, including sD and ~p. Since p is constant, this means thatbp is bounded. Since sD is bounded, this means that s is bounded.The second derivative of V is bounded as follows: €V sD; ~pð Þ ��cm ddt sDk k1. In the worst case (i.e., at the upper bound), we have

€V sD; ~pð Þ¼�cmd

dtsDk k1¼�cm

XsD _sDsDj j¼6cm

X_sD¼6cm

X_s

(A10)

where sD 6¼ 0 outside the boundary layer. We substitute _s fromEq. (25) into Eq. (A10) to obtain

€V sD; ~pð Þ ¼ 6cmX

M�1ð�Cs� Kdsatðs=diagðuÞÞ

þ bTe � Te� �� Y q; _q; v; _vð Þ~pÞ (A11)Recall that ~p and s are bounded. The boundedness of s implies theboundedness of e and _e, as seen from Eq. (13). Since qr , _qr , and €qrare bounded, we know that q, _q, v, and _v are also bounded. There-fore, in Eq. (A11), M;C;u; ~p; Y; cm; s; and Kd are bounded.bTe � Te is upper bounded by Fm, so we can conclude that €V isbounded. Therefore, as conditions I, II, and III from Barbalat’s

Lemma hold, we can conclude that _V sD; ~pð Þ ! 0 as t!1. Thismeans that �cm sDk k1 in Eq. (A9) is equal to zero, which meansthat Eq. (A9) can be written as the equality _V sD; ~pð Þ ¼ �cm sDk k1.We, therefore, have _V sD; ~pð Þ ! 0 ) �cm sDk k1 ! 0 ) sD ! 0.This indicates that the control ensures that s converges to theboundary layer.

A.2 QED (Lemma 1). The RAIC mitigates system uncertain-ties more than a standard adaptive controller but also has a largertracking error. RAIC drives the system to the boundary layer andresults in robustness to GRF as a nonparametric uncertainty.Inside the boundary layer, Eqs. (A4)–(A11) can be reformulatedfor sD ¼ 0, in which case s remains in the boundary layer, whichstops adaptation, and the estimated parameters remain constant.Therefore, the system with the RAIC converges to the referenceimpedance model.

A.3 QED (Theorem 1). It should be noted that the aforemen-tioned proof of asymptotic closed-loop system stability impliesthat Kd should be bounded byFm � Cu� that is;Kd � FmI3�3 � Cuþ cmI3�3.

Appendix B

B.1 Stability Analysis of the Robust Composite AdaptiveImpedance Controller. Proof of Theorem 2: Note that V in Eq.(42), which is a quadratic function of sD, is continuously differen-tiable. The derivative of the Lyapunov function is given as

_V sD; ~pð Þ ¼1

2_sD

TMsD þ sDTM _sD� �

þ 12

sDT _MsD

� �þ 1

2_~p

TP�1 ~p þ ~pTP�1 _~p

� �þ 1

2~pT

d

dtP�1ð Þ~p

� �¼ sDTM _sD þ

1

2sD

T _MsD� �

þ ~pTP�1 _~p þ 12

~pTd

dtP�1ð Þ~p

� �(B1)

Now we want to prove global exponential stability of the closed-loop system both outside and inside the boundary layer defined inEq. (26). Inside the boundary layer _sD ¼ 0, and outside the bound-ary layer _sD ¼ _s, so Eq. (B1) can be written as

_V sD; ~pð Þ ¼ sDTð�Cs� Kdsatðs=diagðuÞÞ þ bTe � Te� �þY q; _q; v; _vð Þ~pÞ þ 1

2sD

T _MsD� �

þ ~pTP�1 _~p

þ 12

~pTd

dtP�1ð Þ~p

� �¼ �sDTCsþ

1

2sD

T _MsD� �

þsDTY q; _q; v; _vð Þ~p�sDTKdsatðs=diagðuÞÞ

þsDT bT e � Te� �þ ~pTP�1 _~p þ 12

~pTd

dtP�1ð Þ~p

� �(B2)

Outside the boundary layer we see that if sj j > diagðuÞ, then sDcomes from the second condition of Eq. (26), in which casewe have s ¼ sD þ usatðs=diagðuÞÞ. Substituting s ¼ sD þusatðs=diagðuÞÞ in the first term of Eq. (B2), we write _V sD; ~pð Þoutside the boundary layer as

_V sD; ~pð Þ ¼1

2sD

T _M � 2Cð ÞsD�sDTCusat s=diag uð Þ� �

þsDTY q; _q; v; _vð Þ~p�sDTKdsatðs=diagðuÞÞ

þsDT bT e � Te� �þ ~pTP�1 _~p þ 12

~pTd

dtP�1ð Þ~p

� �(B3)

_ðM � 2CÞ is skew-symmetric, so sDT _M � 2Cð ÞsD ¼ 0 and we cansimplify _V sD; ~pð Þ as

_V sD; ~pð Þ ¼ �sDT Cuþ Kdð Þsatðs=diagðuÞÞþsDTY q; _q; v; _vð Þ~p þ sDT bT e � Te� �þ~pTP�1 _~p þ 1

2~pT

d

dtP�1ð Þ~p

� � (B4)We tune the design parameters Kd and u so that Cuþ Kd � KmI,which guarantees the positive definiteness of Cuþ Kd , where Kmis a positive scalar. We then use Eq. (B4) to write

_V sD; ~pð Þ � �KmsDTsatðs=diagðuÞÞ þ sDTY q; _q; v; _vð Þ~p

þsDT bTe � Te� �þ ~pTP�1 _~p þ 12

~pTd

dtP�1ð Þ~p

� �(B5)

We replace sDTsatðs=diagðuÞÞ with the one-norm of sD and then

we write Eq. (B5) as

_V sD; ~pð Þ � �Km sDk k1 þ sDTY q; _q; v; _vð Þ~p þ sDT bTe � Te� �

þ~pTP�1 _~p þ 12

~pTd

dtP�1ð Þ~p

� �(B6)

We define Km ¼ Fm þ cm, where bT ei � Tei � Fi � Fm,Fm ¼ maxðFiÞ, and cm ¼ maxðciÞ with i ¼ 1; 2; 3. Noting thatsDi � sDij j, we see that sDT bT e � Te� � in Eq. (B6) is bounded fromabove by Fm sDk k1, so


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_V sD; ~pð Þ � �cm sDk k1 þ sDTY q; _q; v; _vð Þ~p

þ~pTP�1 _~p þ 12

~pTd

dtP�1ð Þ~p

� �(B7)

Since ~p ¼ bp � p, we can substitute Eq. (29) and Eq. (36) intoEq. (B7), and _V sD; ~pð Þ can be written as

_V sD; ~pð Þ � �cm sDk k1 þ sDTY q; _q; v; _vð Þ~p

þ 12

~pTd

dtP�1ð Þ~p

� �� ~pTYT q; _q; v; _vð ÞsD

�~pTWTRW~p ¼ �cm sDk k1 þ1

2~pT

d

dtP�1ð Þ~p

� ��~pTWTRW~p (B8)

By substituting Eq. (37) into Eq. (B8), we can write

_V sD; ~pð Þ � �cm sDk k1 �1

2~pT# tð ÞP�1 ~p � ~pTWT dI � 1

2I

� �W~p

(B9)

where cm > 0, # tð Þ � 0, and P is positive definite, so by choosingd > 1

2, we can see that outside the boundary layer (i.e., the second

condition of Eq. (26)), the derivative of the Lyapunov function isnegative semidefinite. This in turn means that we can use Barba-lat’s lemma to prove global exponential stability. If Vðt; xÞ satis-fies the Barbalat’s Lemma conditions, then _Vðt; xÞ ! 0 as t!1,which means that RCAIC results in a closed-loop system that isglobally exponentially stable.

Now we state an intermediate lemma that we will need to com-plete the proof of Theorem 2.

LEMMA 2: The derivative of the Lyapunov function of Eq. (B9)globally exponentially converges to zero, which guarantees con-vergence to the boundary layer (sD ! 0). Also, the predictionerror in Eq. (36) of the proposed RCAIC converges to zero, whichimplies perfect estimation of the system parameters.

Proof of Lemma 2: Conditions I and II in Barbalat’s Lemma aresatisfied from Eqs. (42) and (B9) and we, therefore, conclude thatV is bounded, which means that all terms in V (including sD, and~p) are bounded. Since p is constant bp is bounded, and since sD isbounded s is bounded. From Eq. (11), since qd is bounded, _qd , €qd ,qr , _qr , and €qr are bounded. From Eqs. (13)–(15), since s isbounded, we see that e and _e are both bounded. These facts implythat q, _q, €q, v, and _v are bounded as well. So, Y q; _q; v; _vð Þ,Y0 q; _q; €qð Þ, and W q; _qð Þ are bounded.

By taking the derivative of _V sD; ~pð Þ at its upper bound, weobtain


_sD � ~pTWT 2dI � Ið ÞW _~p�~pTWT 2dI � Ið Þ _W ~p � ~pT# tð ÞP�1 _~p

� 12

~pT _# tð ÞP�1 ~p � 12

~pT# tð Þ ddt

P�1ð Þ~p

(B10)

Substituting _~p , ddt P�1ð Þ, and _s from Eqs. (29), (37), and (25),

respectively, into Eq. (B10), €V sD; ~pð Þ can be written as follows:


M�1ð�Cs� Kdsatðs=diagðuÞÞ

þ bTe � Te� �þ Y q; _q; v; _vð Þ~pÞ þ ~pTWT 2dI � Ið ÞWP tð ÞYTsD þ ~pT# tð ÞYTsD þ ~pTWT 2dI � Ið ÞWP tð ÞWT dIð ÞW~p � ~pTWT 2dI � Ið Þ _W ~p þ ~pT

# tð ÞWT dIð ÞW~p � 12

~pT _# tð ÞP�1 ~p

þ 12

~pT#2 tð ÞP�1 ~p � 12

~pT# tð ÞWTW~p

(B11)

Since P tð Þ is bounded and its norm is bounded by K0, then fromEq. (38), # tð Þ and _# tð Þ are bounded. Moreover, since M, C,s, Y,W, _W , ~p, and sD are bounded and bT e � Te � Fm, we see that€V sD; ~pð Þ is bounded. Since we have verified all conditions in Bar-balat’s Lemma, we know that _V sD; ~pð Þ ! 0 as t!1)cm sDk k1 ! 0) sD ! 0 as t!1, which means that outsidethe boundary layer, RCIAC guarantees convergence of s to theboundary layer. Furthermore, _V sD; ~pð Þ ! 0 means that~pT# tð ÞP�1 ~p ! 0 and since P�1 tð Þ � 1K0 I, then if W is PE,# tð Þ > # > 0, so we have

# tð Þ~pTP�1 ~p � # ~pT ~p=K0 (B12)

Therefore, ~pT# tð ÞP�1 ~p ! 0 means that ~p ! 0. To achieve fasterexponential convergence of s to the boundary layer and conver-gence of the prediction error to zero, we define a strictly positiveconstant X0, where X0 ¼ minð2#0; #Þ, and we write [50]:

_V tð Þ þX0V tð Þ � 0; VðtÞ � Vð0Þe�X0t (B13)

On the other hand, inside the boundary layer, wheresj j � diagðuÞ, Eqs. (B3)–(B13) can be rewritten for sD ¼ 0. For

this condition, s remains inside the boundary layer and the predic-tion error exponentially converges to zero. Outside the boundarylayer, both sD and ep exponentially converge to zero, which meansthat we achieve perfect parameter estimation and guarantee con-vergence of s to the boundary layer.

B.2 QED (Lemma 2). We see from the above that the closed-loop system with the proposed RCAIC converges to the targetimpedance model. Therefore, the controller drives the system tothe boundary layer, achieves perfect parameter estimation, andachieves robustness against GRFs.

B.3 QED (Theorem 2). Note that the exponential stabilityproof implies that Kd must be bounded from below by Fm � Cu(that is,

Kd � FmI3�3 � Cuþ cmI3�3Þ:

References[1] Ziegler-Graham, K., 2008, “Estimating the Prevalence of Limb Loss in

the United States: 2005 to 2050,” Arch. Phys. Med. Rehabil., 89(3), pp.422–429.

[2] Robbins, J. M., Strauss, G., Aron, D., Long, J., Kuba, J., and Kaplan, Y., 2008,“Mortality Rates and Diabetic Foot Ulcers,” J. Am. Podiatric Med. Assoc.,98(6), pp. 489–493.

[3] Sup, F., Varol, H. A., and Goldfarb, M., 2010, “Upslope Walking With a Pow-ered Knee and Ankle Prosthesis: Initial Results With an Amputee Subject,”IEEE Trans. Neural Syst. Rehabil. Eng., 19(1), pp. 71–78.

[4] Lawson, B. E., Varol, H. A., and Goldfarb, M., 2011, “Standing StabilityEnhancement With an Intelligent Powered Transfemoral Prosthesis,” IEEETrans. Biomed. Eng., 58(9), pp. 2617–2624.

[5] Hoover, C. D., Fulk, G. D., and Fite, K. B., 2013, “Stair Ascent With a PoweredTransfemoral Prosthesis Under Direct Myoelectric Control,” IEEE/ASMETrans. Mechatronics, 18(3), pp. 1191–1200.

[6] Sup, F., Varol, H. A., Mitchell, J., Withrow, T. J., and Goldfarb, M.,2009, “Preliminary Evaluations of a Self-Contained AnthropomorphicTransfemoral Prosthesis,” IEEE/ASME Trans. Mechatronics, 14(6), pp.667–676.

[7] Fite, K., Mitchell, J., Sup, F., and Goldfarb, M., 2007, “Design and Control ofan Electrically Powered Knee Prosthesis,” IEEE Tenth International Confer-ence on Rehabilitation Robotics, Noordwijk, The Netherlands, June 13–15, pp.902–905.

[8] Popovic, D., Oguztoreli, M. N., and Stein, R. B., 1991, “Optimal Controlfor the Active above-Knee Prosthesis,” Ann. Biomed. Eng., 19(2), pp. 131–150.

[9] Popovic, D., Tomovic, R., Tepavac, D., and Schwirtlich, L., 1991, “ControlAspects of Active above-Knee Prosthesis,” Int. J. Man-Mach. Stud., 35(6), pp.751–767.

[10] Sup, F., Bohara, A., and Goldfarb, M., 2008, “Design and Control of a PoweredTransfemoral Prosthesis,” Int. J. Rob. Res., 27(2), pp. 263–273.


Dow





ovember 2019

http://dx.doi.org/10.1016/j.apmr.2007.11.005http://dx.doi.org/10.7547/0980489http://dx.doi.org/10.1109/TBME.2011.2160173http://dx.doi.org/10.1109/TBME.2011.2160173http://dx.doi.org/10.1109/TMECH.2012.2200498http://dx.doi.org/10.1109/TMECH.2012.2200498http://dx.doi.org/10.1109/TMECH.2009.2032688http://dx.doi.org/10.1109/ICORR.2007.4428531http://dx.doi.org/10.1007/BF02368465http://dx.doi.org/10.1016/S0020-7373(05)80159-2http://dx.doi.org/10.1177/0278364907084588

[11] Khademi, G., Mohammadi, H., Hardin, E. C., and Simon, D., 2015,“Evolutionary Optimization of User Intent Recognition for TransfemoralAmputees,” Biomedical Circuits and Systems Conference (BioCAS), Atlanta,GA, Oct. 22–24, pp. 1–4.

[12] Gregg, R. D., Lenzi, T., Hargrove, L. J., and Sensinger, J. W., 2014, “VirtualConstraint Control of a Powered Prosthetic Leg: From Simulation to Experi-ments With Transfemoral Amputees,” IEEE Trans. Rob., 30(6), pp. 1455–1471.

[13] Gregg, R. D., and Sensinger, J. W., 2014, “Towards Biomimetic Virtual Con-straint Control of a Powered Prosthetic Leg,” IEEE Trans. Control Syst. Tech-nol., 22(1), pp. 246–254.

[14] Liu, M., Wang, D., and Huang, H., 2016, “Development of an Environment-Aware Locomotion Mode Recognition System for Powered Lower Limb Pros-theses,” IEEE Trans. Neural Syst. Rehabil. Eng., 24(4), pp. 434–443.

[15] Plauch�e, A., Villarreal, D., and Gregg, R. D., 2016, “A Haptic Feedback Systemfor Phase-Based Sensory Restoration in above-Knee Prosthetic Leg Users,”IEEE Trans. Haptics, 9(3), pp. 421–426.

[16] Azimi, V., Simon, D., and Richter, H., 2015, “Stable Robust Adaptive Imped-ance Control of a Prosthetic Leg,” ASME Paper No. DSCC2015-9794.

[17] Azimi, V., Simon, D., Richter, H., and Fakoorian, S., 2016, “Robust CompositeAdaptive Transfemoral Prosthesis Control With Non-Scalar Boundary LayerTrajectories,” American Control Conference (ACC), Boston, MA, July 6–8, pp.3002–3007.

[18] Yuan, K., Wang, Q., and Wang, L., 2017, “Energy-Efficient Braking TorqueControl of Robotic Transtibial Prosthesis,” IEEE/ASME Trans. Mechatronics,22(1), pp. 149–160.

[19] Lawson, B. E., Ledoux, E. D., and Goldfarb, M., 2017, “A Robotic Lower LimbProsthesis for Efficient Bicycling,” IEEE Trans. Rob., 33(2), pp. 432–445.

[20] Zhao, H., and Ames, A. D., 2014, “Quadratic Program Based Control of Fully-Actuated Transfemoral Prosthesis for Flat-Ground and Up-Slope Locomotion,”American Control Conference (ACC), Portland, OR, June 4–6, pp. 4101–4107.

[21] Zhao, H., Horn, J., Reher, J., Paredes, V., and Ames, A. D., 2017, “First StepsToward Translating Robotic Walking to Prostheses: A Nonlinear OptimizationBased Control Approach,” Auton. Robots, 41(3), pp. 725–742.

[22] Zhao, H., Horn, J., Reher, J., Paredes, V., and Ames, A. D., 2016, “MulticontactLocomotion on Transfemoral Prostheses Via Hybrid System Models andOptimization-Based Control,” IEEE Trans. Autom. Sci. Eng., 13(2), pp. 502–513.

[23] Ames, A. D., 2014, “Human-Inspired Control of Bipedal Walking Robots,”IEEE Trans. Autom. Control, 59(5), pp. 1115–1130.

[24] A. D., Ames, 2012, “First Steps Toward Automatically Generating BipedalRobotic Walking From Human Data,” Robot Motion and Control, Vol. 422,Springer, New York, pp. 89–116.

[25] E., Westervelt, J., Grizzle, and D. E., Koditschek, 2003, “Hybrid Zero Dynam-ics of Planar Biped Walkers,” IEEE Trans. Autom. Control, 48(7), pp. 42–56.

[26] Ott, C., Mukherjee, R., and Nakamura, Y., 2010, “Unified Impedance andAdmittance Control,” IEEE International Conference on Robotics and Automa-tion (ICRA), Anchorage, AK, May 3–7, pp. 554–561.

[27] Abdossalami, A., and Sirouspour, S., 2009, “Adaptive Control for ImprovedTransparency in Haptic Simulations,” IEEE Trans. Haptics, 2(1), pp. 2–14.

[28] Hogan, N., 1985, “Impedance Control: An Approach to Manipulation: Part I,Part II and Part III,” ASME J. Dyn. Syst., Meas. Control, 107(1), pp. 1–24.

[29] S. P., Chan, B., Yao, W. B., Gao, M., and Cheng, 1991, “Robust ImpedanceControl of Robot Manipulators,” Int. J. Rob. Autom., 6(4), pp. 220–227.

[30] Slotine, J.-J. E., and Coetsee, J. A., 1986, “Adaptive Sliding Controller Synthe-sis for Non-Linear Systems,” Int. J. Control, 43(6), pp. 1631–1651.

[31] Hussain, S., Xie, S. Q., and Jamwal, P. K., 2013, “Adaptive Impedance Controlof a Robotic Orthosis for Gait Rehabilitation,” IEEE Trans. Cybern., 43(3), pp.1025–1034.

[32] Sharifi, M., Behzadipour, S., and Vossoughi, G., 2014, “Nonlinear Model Ref-erence Adaptive Impedance Control for Human–Robot Interactions,” ControlEng. Pract., 32(8), pp. 9–27.

[33] Park, H., and Lee, J., 2004, “Adaptive Impedance Control of a Haptic Inter-face,” Mechatronics, 14(3), pp. 237–253.

[34] Zhijun, L., and Ge, S. S., 2013, “Adaptive Robust Controls of Biped Robots,”Control Theory Appl., IET, 7(2), pp. 161–175.

[35] Tomei, P., 2000, “Robust Adaptive Friction Compensation for Tracking Controlof Robot Manipulators,” IEEE Trans. Autom. Control, 45(11), pp. 2164–2169.

[36] Huh, S.-H., and Bien, Z., 2007, “Robust Sliding Mode Control of a RobotManipulator Based on Variable Structure-Model Reference Adaptive ControlApproach,” IET Control Theory Appl., 1(5), pp. 1355–1363.

[37] Salehi, A., and Rashidian, M., 2013, “Intelligent Robust Feed-Forward FuzzyFeedback Linearization Estimation of PID Control With Application to Contin-uum Robot,” Int. J. Inf. Eng. Electron. Bus., 5(1), pp. 1–16.

[38] Liu, Y. H., 2017, “Saturated Robust Adaptive Control for Uncertain Non-Linear Systems Using a New Approximate Model,” IET Control Theory Appl.,11(6), pp. 870–876.

[39] Deng, M., Bi, S., and Inoue, A., 2009, “Robust Non-Linear Control and Track-ing Design for Multi-Input Multi-Output Non-Linear Perturbed Plants,” IETControl Theory Appl., 3(9), pp. 1237–1248.

[40] Slotine, J.-J. E., and Li, W., 1987, “Adaptive Strategy in Constrained Manipula-tors,” IEEE International Conference on Robotics and Automation, Raleigh,NC, pp. 1–16.

[41] J.-J. E., Slotine, and W., Li, 1988, “Adaptive Manipulator Control: A StudyCase,” IEEE Trans. Autom. Control, 33(11), pp. 995–1003.

[42] Slotine, J.-J. E., and Coetsee, J. A., 1989, “Composite Adaptive Controller ofRobot Manipulators,” Automatica, 25(4), pp. 509–519.

[43] Yuan, J., and Stepanenko, Y., 1993, “Composite Adaptive Controller of Flexi-ble Joint Robots,” Automatica, 29(3), pp. 609–619.

[44] Sanner, R. M., and Slotine, J.-J. E., 1992, “Gaussian Networks for DirectAdaptive Control,” IEEE Trans. Neural Networks, 3(6), pp. 837–863.

[45] Richter, H., and Simon, D., 2014, “Robust Tracking Control of a ProsthesisTest Robot,” ASME J. Dyn. Syst., Meas., Control, 136(3), p. 3.

[46] Warner, H., 2015, “Optimal Design and Control of a Lower-Limb ProsthesisWith Energy Regeneration,” Master’s thesis, Cleveland State University,Cleveland, OH.

[47] Khademi, G., Mohammadi, H., Richter, H., and Simon, D., 2018, “Optimal MixedTracking\Impedance Control With Application to Transfemoral Prostheses WithEnergy Regeneration,” IEEE Trans. Biomed. Eng., 65(4), pp. 894–910.

[48] Ackermann, M., and van den Bogert, A. J., 2010, “Optimality Principles forModel-Based Prediction of Human Gait,” J. Biomech., 43(6), pp. 1055–1060.

[49] Huang, A.-C., and Chien, M.-C., 2010, Adaptive Control of Robot Manipula-tors, World Scientific Publishing, Singapore.

[50] Slotine, J.-J. E., and Li, W., 1991, Applied Nonlinear Control, Prentice Hall,Upper Saddle River, NJ.

[51] Dingwell, J. B., and Marin, L. C., 2006, “Kinematic Variability and LocalDynamic Stability of Upper Body Motions When Walking at Different Speeds,”J. Biomech., 39(3), pp. 444–452.

[52] Winter, D. A., 1984, “Kinematic and Kinetic Patterns in Human Gait: Variabil-ity and Compensating Effects,” Human Movement Sci., 3(1–2), pp. 51–76.

[53] Winter, D. A., 2009, Biomechanics and Motor Control of Human Movement,Wiley, Hoboken, NJ.

[54] van den Bogert, A. J., Geijtenbeek, T., Even Zohar, O., Steenbrink, F., and Har-din, E. C., 2013, “A Real-Time System for Biomechanical Analysis of HumanMovement and Muscle Function,” Med. Biol. Eng. Comput., 51(10),pp. 1069–1077.


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http://dx.doi.org/10.1109/BioCAS.2015.7348280http://dx.doi.org/10.1109/TRO.2014.2361937http://dx.doi.org/10.1109/TCST.2012.2236840http://dx.doi.org/10.1109/TCST.2012.2236840http://dx.doi.org/10.1109/TNSRE.2015.2420539http://dx.doi.org/10.1109/TOH.2016.2580507http://dx.doi.org/10.1115/DSCC2015-9794http://dx.doi.org/10.1109/ACC.2016.7525376http://dx.doi.org/10.1109/TMECH.2016.2620166http://dx.doi.org/10.1109/TRO.2016.2636844http://dx.doi.org/10.1109/ACC.2014.6859014http://dx.doi.org/10.1007/s10514-016-9565-1http://dx.doi.org/10.1109/TASE.2016.2524528http://dx.doi.org/10.1109/TAC.2014.2299342http://dx.doi.org/10.1109/TAC.2002.806653http://dx.doi.org/10.1109/ROBOT.2010.5509861http://dx.doi.org/10.1109/TOH.2008.18http://dx.doi.org/10.1115/1.3140702http://dx.doi.org/10.1080/00207178608933564http://dx.doi.org/10.1109/TSMCB.2012.2222374http://dx.doi.org/10.1016/j.conengprac.2014.07.001http://dx.doi.org/10.1016/j.conengprac.2014.07.001http://dx.doi.org/10.1016/S0957-4158(03)00040-0http://dx.doi.org/10.1049/iet-cta.2012.0066http://dx.doi.org/10.1109/9.887661http://dx.doi.org/10.1049/iet-cta:20060440http://dx.doi.org/10.5815/ijieeb.2013.01.01http://dx.doi.org/10.1049/iet-cta.2016.0979http://dx.doi.org/10.1049/iet-cta.2008.0218http://dx.doi.org/10.1049/iet-cta.2008.0218https://slidept.net/document/intelligent-robust-feed-forward-fuzzy-feedback-linearization-estimation-of-pid-control-with-application-to-continuum-robothttp://dx.doi.org/10.1109/9.14411http://dx.doi.org/10.1016/0005-1098(89)90094-0http://dx.doi.org/10.1016/0005-1098(93)90058-2http://dx.doi.org/10.1109/72.165588http://dx.doi.org/10.1115/1.4026342https://engagedscholarship.csuohio.edu/cgi/viewcontent.cgi?article=1678&context=etdarchivehttp://dx.doi.org/10.1109/TBME.2017.2725740http://dx.doi.org/10.1016/j.jbiomech.2009.12.012http://dx.doi.org/10.1016/j.jbiomech.2004.12.014http://dx.doi.org/10.1016/0167-9457(84)90005-8http://dx.doi.org/10.1007/s11517-013-1076-z

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Robust Adaptive Impedance Control With Application to a … · 2019. 11. 1. · Among adaptive control methods which have already been pub-lished, our work most closely resembles