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Abstract— We propose two robust model reference adaptive

impedance controllers for a three degree-of-freedom (3-DOF)

active prosthetic leg for transfemoral amputees. In the first

controller we design a robust adaptive impedance controller

(RAIC) with a tracking-error-based (TEB) adaptation law,

whereas in the second controller we propose a robust composite

adaptive impedance controller (RCAIC) with a tracking-error-

based / prediction-error-based (TEB/PEB) adaptation

mechanism and a bounded-gain forgetting (BGF) method. We

present a model for a combined system that includes a test robot

and a transfemoral prosthetic leg. We design these two

controllers so the error trajectories of the joint displacements

converge to boundary layers and the controllers show robustness

to ground reaction forces (GRFs) as non-parametric

uncertainties and also handle model parameter uncertainties.

The boundary layers not only compromise between control signal

chatter and performance, but also stop tracking-error-based

(TEB) adaptation in the boundary layers to prevent unfavorable

parameter estimation drift. We prove the stability of the closed-

loop systems for both controllers for the prosthesis robot in the

case of non-scalar boundary layer trajectories using Lyapunov

stability theory and Barbalat’s lemma. We design the prosthesis

controllers to imitate the biomechanical properties of able-bodied

walking and to provide smooth gait. We finally present

simulation results to confirm the efficacy of the controllers for

both nominal and off-nominal system model parameters. We

achieve good tracking of joint displacements and velocities, and

reasonable control and GRF magnitudes for both controllers. We

also compare performance of the controllers in terms of tracking,

control cost, and parameter estimation for both nominal and off-

nominal cases. Numerical results show that in case of +30%

parameter deviations, RAIC RMS trajectory tracking errors

relative to the reference model are 16 mm for vertical hip

displacement, 0.15 deg for thigh angle, and 0.12 deg for knee

angle. RCAIC RMS trajectory tracking errors are 14 mm for

vertical hip displacement, 0.15 deg for thigh angle, and 0.08 deg

for knee angle. When parameter values vary by 30% from

nominal values, RCAIC achieves 9.5% better trajectory tracking

and 76% better parameter estimation than RAIC, but at the

price of a 9.9% increase in control magnitude.

Index Terms— Non-scalar boundary layer trajectories

Robust adaptive controllers, Target impedance model,

Transfemoral prosthesis

I. INTRODUCTION

ROSTHESES have become progressively important

because there are about two million people with limb loss

in the United States as of 2008 [1].

All authors are with Cleveland State University, Cleveland, OH, USA

† Corresponding author, e-mail: v.azimi@csuohio.edu

Amputation could be due to accidents, cancer, diabetes,

vascular disease, birth defects, and paralysis [1], [2]. A

prosthetic leg can enhance the quality of life and the ability to

walk for amputees so they can regain independence.

Amputation could be transtibial (that is, below knee),

transfemoral (that is, above knee), at the foot, or

disarticulation (that is, through a joint). Prosthetic legs can be

generally classified into three different types: passive

prostheses do not include any electronic control), active

prostheses include motors, and semi-active prostheses include

electronic control but not motors) [3]. Research efforts over

the past few decades have provided advanced prostheses to

closely imitate able-bodied gait and to allow greater levels of

activity for amputees. Active prostheses provide gait

performance that is more similar to able-bodied gait than

passive or semi-active prostheses. The first commercially

available active transfemoral prosthesis was the Power Knee

[3]-[5]. A combined knee / ankle prosthesis that includes

active control at both knee and ankle has been developed by

Vanderbilt University but has not yet been commercialized

[6]. Much recent research has focused on the control of these

prostheses, along with other prostheses [7]-[12]. Recent

research has provided significant developments in modeling

and control for prosthetic legs [13]-[25].

An active prosthesis is essentially a robot that interacts

with its human user. The prosthesis can be controlled to

behave as an impedance or admittance [26], [27]. The

consideration of the interaction between a robot and its

external environment motivated the development of

impedance control [28].

Modeling errors are always present in real-world systems,

but robust control approaches can mitigate the effects of

modeling errors on system performance and stability [29],

[30]. Robust controllers achieve performance in spite of model

uncertainty, while adaptive controllers achieve performance

using learning and adaptation. Non-adaptive controllers

generally require prior knowledge of the parameter variation

bounds, while adaptive approaches do not.

The advantages of adaptive control, the availability of

able-bodied impedance models, and the uncertainty of robot

models, has motivated the development of impedance model

reference adaptive control [31]-[33]. However, adaptive

control methods can cause instability if disturbances,

unmodeled dynamics, or unmodeled external forces are too

large. Robust control can alleviate instability in such cases

[34]-[39]. Various adaptive and sliding surface approaches

have also been used for robotic applications [30], [40]-[44].

The contribution of this paper is two robust model

reference adaptive impedance controllers for transfemoral

Robust Adaptive Impedance Control with

Application to Transfemoral Prostheses

Vahid Azimi†, Seyed Abolfazl Fakoorian, Dan Simon, Hanz Richter

P

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prostheses, the stability analysis of the two controllers, and

investigation of their performance in simulation. Our control

approaches can ensure that the system converges to a

reference model in the presence of both parametric and non-

parametric uncertainties. In this paper, we present a blending

adaptive and non-scalar boundary layer-based robust control

to achieve robustness to GRFs (that is, environmental

interactions), system uncertainties, and disturbances,

estimation of the unknown parameters, and a stability proof of

the proposed methods.

The first controller comprises a robust adaptive

impedance controller (RAIC) with a tracking-error-based

(TEB) adaptation law which extracts information about the

parameters from only the impedance model tracking error. The

second controller comprises a robust composite adaptive

impedance controller (RCAIC) with bounded-gain forgetting

(BGF). Since tracking errors in the joint displacements and

prediction error in the joint torques are influenced by

parameter uncertainties, RCAIC is designed with tracking-

error-based / prediction-error-based (TEB/PEB) adaptation so

that parameter adaptation is driven with both impedance

model tracking error and prediction error, which in turn

provides more accurate estimation of system parameters. More

accurate estimation of the system parameters results in a more

accurate model, and in turn RCAIC can achieve better

tracking compared with RAIC.

Since our goal is that the two closed-loop systems (one

with RAIC and the other with RCAIC) match the

biomechanics of able-bodied walking, we use a target

impedance model with that is based on able-bodied walking.

To balance control chatter and performance, we

incorporate non-scalar boundary layer trajectories 𝑠∆ in both

controllers. We use these trajectories to turn off the TEB

adaptation mechanism to prevent unfavorable parameter drift

when the impedance model tracking errors are small and are

due mostly to noise and disturbances. We define the

trajectories 𝑠∆ so the error trajectories converge to the

boundary layers and the controllers show robustness to both

parametric and non-parametric uncertainties.

Among adaptive control methods which have already

been published, our work most closely resembles [30] and

[42]. In [30], a direct adaptive controller is proposed whose

adaptation mechanism uses joint tracking errors. The control

law in [30] is a combination of a direct adaptive and robust

sliding mode control based on a scalar boundary layer to

obtain a trade-off between control chatter and performance,

and to achieve robustness to unmodeled dynamics.

Asymptotic stability of the closed-loop system in the case of a

scalar boundary layer is shown. In [42], a composite adaptive

controller is proposed whose adaptation law uses tracking

errors in the joint motion and errors in the predicted filtered

torque to derive more accurate system parameters. In addition,

a blend of an adaptive feedforward and a proportional–

derivative (PD) controller is used and exponential stability of

the closed-loop system is proven.

Since a robotic system with more than one DOF,

including the 3-DOF prosthesis/controller system in this

research, can be considered a non-scalar problem with a

coupled nature, in this research we use non-scalar boundary

layer trajectories for both control structures.

So we expand on the work in [30] by using non-scalar

boundary layer trajectories and by incorporating impedance

control. We prove the asymptotic stability of the system with

both controllers, RAIC and RCAIC, using non-scalar

boundary layer trajectories, Barbalat’s lemma, and Lyapunov

theory. We also extend the work in [42] by incorporating

non-scalar boundary layer trajectories 𝑠∆and impedance

control so that both augmented robust composite impedance

controllers show robustness to non-parametric model

uncertainties and environmental interaction forces (which are

GRF variations in our case). We then prove the exponential

stability of these controllers using non-scalar boundary layer

trajectories.

This paper is an extension of our two conference papers

[16] and [17]. In this paper we expand our previous results by

incorporating more complete and comprehensive material on

the model description, the controller design, and the

simulation results.

Simulation results illustrate that both proposed systems

have good tracking performance, strong robustness to system

model parametric and non-parametric uncertainties, and

reasonable control signals and GRFs. Furthermore, numerical

results show that the RCAIC demonstrates better parameter

estimation and tracking in the presence of system parameter

variations. When parameter values vary by 30% from nominal

values, the RCAIC has 9.5% better reference trajectory

tracking and 76% better parameter estimation, but 9.9%

greater control magnitude than RAIC.

Section II describes the model of the transfemoral

prosthesis and the robotic test system. Section III presents the

controller structures and proves their stability. Section IV

presents simulation results. Section V presents discussion,

concluding remarks, and future work.

II. PROSTHETIC LEG MODEL

Our system model includes a test robot and a transfemoral

prosthesis. The system includes three links and three degrees

of freedom. This prismatic-revolute-revolute (PRR) model is

shown in Fig. 1. Human hip motion and thigh motion is

emulated by the robot. The knee and shank represent the

prosthesis. The vertical motion emulates human (or test robot)

vertical hip motion, the first axis emulates human (or test

robot) thigh motion, and the second axis is angular knee

(prosthesis) motion [14]-[17], [45].

The three degree-of-freedom system model can be written

as follows [14]-[17], [45]:

𝑀�̈� + 𝐶�̇� + 𝑔 + 𝑅 = 𝑢 − 𝑇𝑒 (1)

where 𝑞𝑇 = [𝑞1 𝑞2 𝑞3] comprises the generalized

displacements (𝑞1 is vertical displacement, 𝑞2 is thigh angle,

and 𝑞3 is prosthetic knee angle); 𝑀(𝑞) is the inertia matrix;

𝐶(𝑞, �̇�) is the Coriolis and Centripetal matrix; 𝑔(𝑞) is the

gravity vector; 𝑅(𝑞, �̇�) is the nonlinear damping vector; 𝑇𝑒 is

the effect of the combined horizontal (𝐹𝑥) and vertical (𝐹𝑧)

components of the GRF on each joint; u comprises the active

control force at the hip and the active control torques at the

thigh and prosthetic knee.

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Fig. 1: Prosthesis test robot: human hip and thigh motions are emulated by a prosthesis test robot where its calf with red cross-hatched lines represents the

prosthesis device with rigid ankle and foot. A treadmill belt serves as the

walking surface. When the foot is in contact with the treadmill belt, the GRF is nonzero.

The prosthesis test robot walks on a treadmill, which we

model as a mechanical stiffness [16], [17], [45]. We model the

vertical component of the GRF (𝐹𝑧) for the foot-treadmill

contact as

𝐹𝑧 = {0 , 𝐿𝑧 < 𝑠𝑧

−𝑘𝑏(𝑠𝑧 − 𝐿𝑧) , 𝐿𝑧 > 𝑠𝑧

(2)

where 𝑘𝑏 is the belt stiffness; 𝑠𝑧 is the treadmill standoff (that

is, the vertical distance from the origin of the world frame (x0,

y0, z0) to 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 = 𝑞1 + 𝑙2 sin(𝑞2) + 𝑙3 sin(𝑞2 + 𝑞3) (3)

where 𝑙2 and 𝑙3 are the length of the thigh and shank

respectively. The horizontal component of the GRF (𝐹𝑥) can

be modeled by an approximation of Coulomb friction as [46]

𝐹𝑥 = −𝛽𝐹𝑧 (1 − 𝑒−𝑣𝑟/𝑣𝑐

1 + 𝑒−𝑣𝑟/𝑣𝑐)

(4)

where 𝛽 is the belt friction coefficient; 𝑣𝑐 is scaling factor; 𝑣𝑟

is the velocity of the foot-treadmill contact relative to the

treadmill, such that

𝑣𝑟 = −�̇�2(𝑙2 sin(𝑞2) + 𝑙3 sin(𝑞2 + 𝑞3))

−�̇�3(𝑙3 sin(𝑞2 + 𝑞3)) − 𝑣 (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 is due to the GRF on each joint and is

given as follows [16], [17], [45]:

𝑇𝑒 = [

𝐹𝑧

𝐹𝑧(𝑙2 cos(𝑞2) + 𝑙3 cos(𝑞2 + 𝑞3)) − 𝐹𝑥(𝑙2 sin(𝑞2) + 𝑙3 sin(𝑞2 + 𝑞3))

𝐹𝑧(𝑙3 cos(𝑞2 + 𝑞3)) − 𝐹𝑥(𝑙3 sin(𝑞2 + 𝑞3)]

(6)

The states and controls are defined as

𝑥𝑇 = [𝑞1 𝑞2 𝑞3 �̇�1 �̇�2 �̇�3]

𝑢𝑇 = [𝑓ℎ𝑖𝑝 𝜏𝑡ℎ𝑖𝑔ℎ 𝜏𝑘𝑛𝑒𝑒] (7)

The left side of Eq. (1) can be written in the following form:

𝑀�̈� + 𝐶�̇� + 𝑔 + 𝑅 = 𝑌ʹ(𝑞, �̇�, �̈�)𝑝 (8)

where 𝑌ʹ(𝑞, �̇�, �̈�) ∈ 𝑅𝑛⤫𝑟 is a regressor matrix that is a

function of joint displacements, velocities, and accelerations; n

is the number of links (n is equal to 3 in this paper; see

Fig. 1); and p ∈ 𝑅𝑟 is the parameter vector. The regressor

𝑌ʹ(𝑞, �̇�, �̈�) and the parameter 𝑝 have many realizations; one

such possibility is

𝑌ʹ(𝑞, �̇�, �̈�) = [�̈�1 − 𝑔 𝑌ʹ12 𝑌ʹ13 0 0 0 0 sgn(�̇�1)

0 𝑌ʹ22 𝑌ʹ23 �̈�2 𝑌ʹ25 �̈�3 �̇�2 00 0 𝑌ʹ33 0 𝑌ʹ35 �̈�2 + �̈�3 0 0

]

𝑌ʹ12 = �̈�2 cos(𝑞2) − �̇�22sin (𝑞2)

𝑌ʹ13 = (�̈�2 + �̈�3)cos (𝑞3 + 𝑞2)

−(2�̇�2�̇�3+�̇�22+�̇�3

2)sin (𝑞3 + 𝑞2)

𝑌ʹ22 = (�̈�1 − g)cos (𝑞2)

𝑌ʹ23 = 𝑌ʹ33 = (�̈�1 − 𝑔) cos(𝑞3 + 𝑞2) 𝑌ʹ25 = (2�̈�2 + �̈�3)cos (𝑞3)−(2�̇�2�̇�3+�̇�3

2)sin (𝑞3) 𝑌ʹ35 = �̈�2 cos(𝑞3) + sin(𝑞3) �̇�2

2 (9)

𝑝 =

[

𝑚1 + 𝑚2 + 𝑚3

𝑚3𝑙2 + 𝑚2𝑙2 + 𝑚2𝑐2𝑚3𝑐3

𝐼2𝑧 + 𝐼3𝑧 + 𝑚2𝑐22 + 𝑚3𝑐3

2 + 𝑚2𝑙22 + 𝑚3𝑙2

2 + 2𝑚2𝑐2𝑙2𝑚3𝑐3𝑙2

𝑚3𝑐32 + 𝐼3𝑧

𝑏𝑓 ]

(10)

III. ROBUST ADAPTIVE IMPEDANCE CONTROL

We design two separate nonlinear robust adaptive

impedance controllers using non-scalar boundary layers and

sliding surfaces to track hip displacement and knee and thigh

angles in spite of parametric and non-parametric uncertainties.

Both controllers use the same control laws, same target

impedance models, and same non-scalar boundary layer

trajectories, but different adaptation laws. In the first

controller, we design a robust adaptive impedance controller

(RAIC) with a TEB adaptation law, which extracts

information about the parameters from the impedance model

tracking error. In the second controller, we propose a robust

composite adaptive impedance controller (RCAIC) with

bounded-gain forgetting (BGF). Since impedance model

tracking errors in the joint displacements and prediction error

in the joint torques are influenced by parameter uncertainties,

in RCAIC we design a TEB/PEB adaptation law which drives

parameter adaptation using both impedance model tracking

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error and prediction error to achieve more accurate estimation

of the system parameters.

A. Target Impedance Model

The robot / prosthesis system interacts with the

environmental admittance, so if we want to have a system that

is well-matched with the mechanical characteristics of the

environment, the closed-loop system should behave as an

impedance. In this way we can achieve a tradeoff between

performance and GRF.

We desire the closed-loop systems with both RAIC and

RCAIC to emulate the biomechanics of able-bodied

walking.We thus define a target impedance model [32] with

characteristics similar to able-bodied walking [16], [47]:

𝑀𝑟(�̈�𝑟 − �̈�𝑑) + 𝐵𝑟(�̇�𝑟 − �̇�𝑑) + 𝐾𝑟(𝑞𝑟 − 𝑞𝑑) = −𝑇𝑒 (11)

The reference mass 𝑀𝑟, damping coefficient 𝐵𝑟 , and spring

stiffness 𝐾𝑟 are positive definite matrices, while 𝑞𝑟 ∈ 𝑅𝑛 is the

state of the reference model and 𝑞𝑑 ∈ 𝑅𝑛 is the reference

trajectory. We assume that the matrices are diagonal:

𝑀𝑟 ∈ 𝑅𝑛⤫𝑛 = diag (𝑀11 𝑀22 … 𝑀𝑛𝑛)

𝐵𝑟 ∈ 𝑅𝑛⤫𝑛 = diag (𝐵11 𝐵22 … 𝐵𝑛𝑛)

𝐾𝑟 ∈ 𝑅𝑛⤫𝑛 = diag(𝐾11 𝐾22 … 𝐾𝑛𝑛) (12)

B. Control Law

In Eq. (9) the regressor depends on acceleration.

However, acceleration measurements are typically noisy, so it

might not be convenient to use 𝑌ʹ(𝑞, �̇�, �̈�) in real time. To

avoid the use of acceleration, we define error vector 𝑠 and

signal vector 𝑣 [40], [41], [48]:

𝑠 = �̇� + 𝜆𝑒 (13)

𝑣 = �̇�𝑟 − 𝜆𝑒 (14)

𝑒 = 𝑞 − 𝑞𝑟 (15)

𝜆 = diag(𝜆1, 𝜆2, … , 𝜆𝑛) , 𝜆𝑖 > 0

We define an acceleration-free controller regressor in place of

the model regressor in Eq. (9):

𝑀�̈� + 𝐶�̇� + 𝑔 + 𝑅 = 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝 (16)

where 𝑌(𝑞, �̇�, 𝑣, �̇�) is a linear function, one realization of

which is given as

𝑌(𝑞, �̇�, 𝑣, �̇�) = [�̇�1 − 𝑔 𝑌12 𝑌13 0 0 0 0 sgn(�̇�1)

0 𝑌22 𝑌23 �̇�2 𝑌25 �̇�3 �̇�2 00 0 𝑌33 0 𝑌35 �̇�2+�̇�3 0 0

]

𝑌12 = �̇�2cos (𝑞2)−𝑣2�̇�2sin (𝑞2)

𝑌13 = (�̇�2 + �̇�3)cos (𝑞3 + 𝑞2) −(𝑣2�̇�3+𝑣2�̇�2+𝑣3�̇�2+𝑣3�̇�3)sin (𝑞3 + 𝑞2)

𝑌22 = (�̇�1 − g)cos (𝑞2)

𝑌23 = 𝑌33 = (�̇�1 − 𝑔) cos(𝑞3 + 𝑞2) 𝑌25 = (2�̇�2 + �̇�3)cos (𝑞3) −(𝑣2�̇�3+𝑣3�̇�3+𝑣3�̇�2)sin (𝑞3)

𝑌35 = �̇�2 cos(𝑞3) + sin(𝑞3) 𝑣2�̇�2 (17)

By substituting Eqs. (13), (14), and (15) in Eq. (1), we rewrite

the model as

𝑀�̇� + 𝐶𝑠 + 𝑔 + 𝑅 + 𝑀�̇� + 𝐶𝑣 = 𝑢 − 𝑇𝑒 (18)

Since Eq. (1) is a second-order system, the error vector of

Eq. (15) can be obtained from the first-order sliding surface

𝑠 = (𝑑

𝑑𝑡+ 𝜆) 𝑒

(19)

where 𝑠 includes n elements. Perfect impedance model

tracking 𝑞 = 𝑞𝑟 (𝑒 = 0) implies that 𝑠 = 0. To reach the

sliding manifold 𝑠 = 0, the following reaching condition must

be satisfied [30]:

sgn(𝑠)�̇� ≤ −𝛾 (20)

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

an n-element vector denoted as 𝛾 = [𝛾1 𝛾2 … 𝛾𝑛]𝑇 where

𝛾𝑖 > 0. Eq. (20) shows that in the worst case, sgn(𝑠)�̇� = −𝛾,

so we calculate the worst-case reaching time of the tracking

error trajectory as

∫ sgn(𝑠)𝑑𝑠 = −𝛾0

𝑠(0)

∫ 𝑑𝑡𝑇

0

→ |𝑠(0)|sgn(𝑠) = 𝛾 𝑇

𝑇 =𝑠(0)

𝛾

(21)

This equation gives n different reaching times, 𝑠(0) is the

error at the initial time, and the quotient 𝑠(0)/𝛾 is defined one

element at a time. We can see from Eq. (21) that a larger 𝛾

gives smaller reaching times T. The system parameters are

note known, so we use a controller [30] to handle parameter

uncertainty and to satisfy the condition of Eq. (20):

𝑢 = �̂��̇� + �̂�𝑣 + �̂� + �̂� + �̂�𝑒 − 𝐾𝑑sgn(𝑠) (22)

where �̂�, �̂�, �̂�, �̂�, and �̂�𝑒 are estimates of 𝑀,𝐶, 𝑔, 𝑅, and 𝑇𝑒,

and 𝐾𝑑 is a tuning matrix denoted as 𝐾𝑑 =

diag(𝐾𝑑1, 𝐾𝑑2

, … , 𝐾𝑑𝑛) , where 𝐾𝑑𝑖

> 0. Note that sgn(𝑠) is

discontinuous, which means that it would result in control

chattering; therefore, we replace it with the saturation function

sat(𝑠/diag(𝜑)) (see Fig. 2). The division and saturation

operations in sat(𝑠/diag(𝜑)) are taken one element at a time.

The term diag(𝜑) is an n-element vector. This all results in a

modification of the controller of Eq. (22):

𝑢 = �̂��̇� + �̂�𝑣 + �̂� + �̂� + �̂�𝑒 − 𝐾𝑑 sat(𝑠/diag(𝜑) ) (23)

The diagonal elements of 𝜑 are the widths of the saturation

function. The control law of Eq. (23) includes two parts. The

first part, �̂��̇� + �̂�𝑣 + 𝑔 + �̂�, is an adaptive term that handles

uncertain parameters. The second part, �̂�𝑒 − 𝐾𝑑 sat(𝑠/diag(𝜑)), is a robustness term that satisfies Eq. (20) and the

variations of the external inputs 𝑇𝑒 as non-parametric

uncertainties. We substitute Eq. (23) into Eq. (18) and define

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�̃� = �̂� − 𝑀, �̃� = �̂� − 𝐶, �̃� = �̂� − 𝑔, �̃� = �̂� − 𝑅, and

𝑝 = �̂� − 𝑝, to derive the closed-loop system

𝑀�̇� + 𝐶𝑠 + 𝐾𝑑sat(𝑠/diag(𝜑)) + ( 𝑇𝑒 − �̂�𝑒) =

(�̃��̇� + �̃�𝑣 + �̃� + �̃�)

(24)

We separate the right side of Eq. (24) into two parts: the

regressor 𝑌(𝑞, �̇�, 𝑣, �̇�) and the parameter estimation error 𝑝.

We can thus write Eq. (24) in the following regressor (linear

parametric) form:

𝑀�̇� + 𝐶𝑠 + 𝐾𝑑sat(𝑠/diag(𝜑)) + ( 𝑇𝑒 − �̂�𝑒) =

𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝

(25)

C. Non-Scalar 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 also in the presence of large adaptation gains

and reference trajectories, the estimated parameters are prone

to oscillate and grow without bound because of instability in

the control system. This phenomenon is known as parameter

drift. However, if the model regressor 𝑌ʹ(𝑞, �̇�, �̈�) satisfies

persistent excitation (PE) conditions, the adaptive control

scheme exhibits robustness against non-parametric

uncertainties and unmodeled dynamics, and parameter drift

can be avoided [30], [48].

To turn off the TEB adaptation mechanism to prevent

unfavorable parameter drift when the impedance model

tracking errors are small and are due mostly to noise and

disturbances, we incorporate non-scalar boundary layer

trajectories 𝑠∆ into both controllers RAIC and RCAIC. We

define these trajectories to balance control chatter and

performance. Furthermore, we define the trajectories 𝑠∆ so the

error trajectories converge to the boundary layers and both

proposed controllers show robustness to non-parametric

uncertainties. We define these boundary trajectories 𝑠∆ as

follows [30]:

𝑠∆ = {0 , |𝑠| ≤ diag(𝜑)

𝑠 − 𝜑 sat(𝑠/diag(𝜑)) , |𝑠| > diag(𝜑)

(26)

Note that 𝑠∆ is an n-element vector. We call the region |𝑠| ≤diag(𝜑) the boundary layer, where the inequality is taken one

element at a time. Note that the diagonal elements of 𝜑

comprise the thickness values of the boundary layer and are

denoted as 𝜑 = diag(𝜑1, 𝜑2, … , 𝜑𝑛) , where 𝜑𝑖 > 0. We

illustrate 𝑠∆ and sat(𝑠/diag(𝜑)) for a single dimension in

Fig. 2.

Fig. 2: Saturation function and 𝑠∆ in one dimension

D. Robust Adaptive Impedance Controller (RAIC)

RAIC uses the control law in Eq. (23), non-scalar

boundary layer trajectories in Eq. (26), and the TEB

adaptation law, so the prosthesis/RAIC combination converges

to the target impedance model in Eq. (11). The TEB

adaptation law can be presented as

�̇̂� = −𝜇−1𝑌𝑇(𝑞, �̇�, 𝑣, �̇�)𝑠∆ (27)

Theorem 1: Consider the following scalar positive

definite Lyapunov function [48]:

𝑉(𝑠∆, 𝑝) =1

2(𝑠∆

𝑇𝑀 𝑠∆) +1

2(𝑝𝑇𝜇 𝑝)

(28)

where 𝜇 is a design parameter such that 𝜇 =diag(𝜇1, 𝜇2, … , 𝜇𝑟) , with 𝜇𝑖 > 0. The closed-loop system

using RAIC results in �̇�(𝑠∆, 𝑝) → 0 as 𝑡 → ∞. That is, the

closed-loop systems is asymptotically stable. The error vector

𝑠 converges to the boundary layer, which implies convergence

of the closed-loop system to the target impedance model.

Proof of Theorem 1: See Appendix 1.

E. Robust Composite Adaptive Impedance Controller

(RCAIC)

The RCAIC uses the same control law in Eq. (23) and

non-scalar boundary layer trajectories in Eq. (26) as the RAIC

uses, but uses a different adaptation law, i.e., the TEB/PEB

mechanism, so the prosthesis/RCAIC combination converges

to the target impedance model in Eq. (11). In the TEB

adaptive controller (RAIC), the adaptation law extracts

information about the parameters only from the impedance

model tracking error. However, the tracking error is not the

only source of parameter information; prediction error also

contains parameter information. Therefore, by using a

combination of the impedance model tracking and prediction

errors, the performance of the adaptive controller can be

improved. For the RCAIC, a TEB/PEB adaptation law is

introduced as follows [30], [48]:

�̇̂� = −𝑃(𝑡)[𝑌𝑇(𝑞, �̇�, 𝑣, �̇�)𝑠∆ + 𝑊𝑇𝑅𝑒𝑝] (29)

where 𝑅 = 𝑑𝐼n⤫n is a positive definite diagonal weighting

matrix that indicates how much the adaptation law uses the

prediction error (d is a positive constant); 𝑃(t) is time-varying

adaptation gain; W is a filtered version of the model regressor

matrix 𝑌ʹ(𝑞, �̇�, �̈�) given in Eq. (9), where this filtering is

introduced to avoid the need for joint acceleration in the

regressor [48]; and 𝑒𝑝 is the prediction error and is calculated

from 𝑊(𝑞, �̇�)𝑝 (details will be presented later in this section).

The filtering can be done with a first order stable filter as

follows:

𝑊(𝑞, �̇�) =𝑐

𝑠 + 𝑐𝑌ʹ(𝑞, �̇�, �̈�)

(30)

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6

where 𝑐 > 0. To filter in the time domain, we convolve both

sides of Eq. (1) with the impulse response of 𝑐/(𝑠 + 𝑐)

(that is, 𝑤(𝑡) = 𝑐𝑒−𝑐𝑡):

∫ 𝑤(𝑡 − ℎ)[𝑀�̈� + 𝐶�̇� + 𝑔 + 𝑅]𝑑ℎ =𝑡

0

∫ 𝑤(𝑡 − ℎ)[𝑢 − 𝑇𝑒]𝑑ℎ𝑡

0

(31)

The first part of Eq. (31), ∫ 𝑤(𝑡 − ℎ)𝑡

0𝑀�̈� 𝑑ℎ, can be written

as follows:

∫ 𝑤(𝑡 − ℎ)𝑡

0

𝑀�̈�𝑑ℎ = 𝑤(𝑡 − ℎ)𝑀�̇�|0𝑡

−∫

𝑑

𝑑ℎ

𝑡

0

(𝑤(𝑡 − ℎ)𝑀)�̇� 𝑑ℎ

= 𝑤(0)𝑀�̇� − 𝑤(𝑡)𝑀(𝑞(0))�̇�(0)

−∫ [𝑤(𝑡 − ℎ)�̇��̇� +

𝑡

0

𝑑

𝑑ℎ(𝑤(𝑡 − ℎ))𝑀�̇�] 𝑑ℎ

(32)

That is, convolving the left hand side of Eq. (31) can be

interpreted as filtering that side, and is equal to 𝑊(𝑞, �̇�)𝑝 so

that

𝑦(𝑡) = 𝑊(𝑞, �̇�)𝑝 = 𝑤(0)𝑀�̇� − 𝑤(𝑡)𝑀(𝑞(0))�̇�(0)

−∫ [𝑤(𝑡 − ℎ)�̇��̇� +

𝑡

0

𝑑

𝑑ℎ(𝑤(𝑡 − ℎ))𝑀�̇�] 𝑑ℎ

+∫ 𝑤(𝑡 − ℎ)[𝐶�̇� + 𝑔 + 𝑅]

𝑡

0

𝑑ℎ

(33)

where 𝑦(𝑡) is the filtered version of the right side of Eq. (1)

and is given as follows:

𝑦(𝑡) = ∫ 𝑤(𝑡 − ℎ)[𝑢 − 𝑇𝑒]𝑡

0𝑑ℎ (34)

The estimated value of 𝑦(𝑡) can be written as follows:

�̂�(𝑡) = 𝑊(𝑞, �̇�)�̂� (35)

Therefore, the prediction error 𝑒𝑝 is derived as

𝑒𝑝 = �̂�(𝑡) − 𝑦(𝑡) = 𝑊(𝑞, �̇�)𝑝 (36)

It is important to note that past data are generated from

past parameter values, and the algorithm should therefore pay

less attention to past data when generating current parameter

estimates. Therefore, exponential data forgetting is advisable

for estimating time-varying parameters. The composite

adaptation law in Eq. (29) can benefit from an exponentially

forgetting least-squares gain update for 𝑃(t) as follows [42],

[48]:

𝑑

𝑑𝑡(𝑃−1) = −𝜗(𝑡)𝑃−1 + 𝑊𝑇(𝑡)𝑊(𝑡)

(37)

where 𝜗(𝑡) ≥ 0 denotes the time-varying forgetting factor. To

benefit from data forgetting and to avoid unboundedness in

𝑃(𝑡), the bounded-gain forgetting (BGF) method can be used

to tune the time-varying forgetting factor 𝜗(𝑡) as follows [48]:

𝜗(𝑡) = 𝜗0 (1 −‖𝑃‖

𝐾0

)

(38)

where 𝜗0 is the maximum forgetting rate; 𝐾0 is the upper

bound of 𝑃(𝑡); and 𝑃(0) must be smaller than 𝐾0𝐼. The

second part of the TEB/PEB adaptation law in Eq. (29) can be

written as

�̇� = −𝑃(𝑡)𝑊𝑇𝑅𝑊𝑝 (39)

Solving Eq. (39) gives

𝑝(𝑡) = 𝑝(0)exp (∫ −𝑃(t)WT(t)R𝑊(𝑡)𝑑𝑡t

0

)

(40)

Therefore, 𝑝 = �̂� − 𝑝 will exponentially converge to zero

if 𝑊(𝑞, �̇�) is PE. The speed of convergence can be heavily

dependent on the magnitude of the adaptation gain. 𝑊(𝑞, �̇�)

must satisfy the following PE condition:

lim𝑡→∞

∫ −𝑊𝑇(𝑡)𝑊(𝑡)𝑑𝑡𝑡

0

= ∞

(41)

Therefore, 𝑝 will exponentially converge to zero for non-

zero and constant 𝑊. It is interesting to note that when 𝑊 is

not PE, 𝑝 cannot converge to zero, even if there are no non-

parametric uncertainties, and the robustness property cannot

be guaranteed. In this procedure, the time-varying forgetting

factor is tuned so that data forgetting is active when 𝑊(𝑡) is

PE and it is off when 𝑊(𝑡) is not PE. From Eq. (38), ‖𝑃‖

shows the level of PE of 𝑊(𝑡) so that if ‖𝑃‖ decreases, 𝑊(𝑡)

is strongly PE (𝜗(𝑡) = 𝜗0), and if ‖𝑃‖ increases, 𝑊(𝑡) is

weakly PE. In the BGF composite controller, 𝑝 and 𝑃(𝑡) are

upper bounded, and if 𝑊(𝑡) is strongly PE, then 𝑝

exponentially converges to zero, 𝑃(𝑡) is upper and lower

bounded by positive numbers, and 𝜗(𝑡) > 𝜗 > 0.

Theorem 2: Consider the scalar positive definite

Lyapunov function

𝑉(𝑠∆, 𝑝) =1

2(𝑠∆

𝑇𝑀 𝑠∆) +1

2(𝑝𝑇𝑃−1 𝑝)

(42)

The controller of Eq. (23), when using in conjunction with the

update law of Eq. (29) and applied to the system of Eq. (1),

results in �̇�(𝑠∆, 𝑝) → 0 as 𝑡 → ∞, which means the

prosthesis/RCAIC combination is globally exponentially

stable. The error vector 𝑠 converges to the boundary layer,

indicating perfect estimation of the system parameters and

convergence of the closed-loop system to the target impedance

model.

Proof of Theorem 2: See Appendix 2.

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7

To get a feeling for the RAIC/ RCAIC structure, consider

the general pattern loop of Fig. 3. To show that both proposed

controller structures RAIC and RCAIC result in closed-loop

systems that converge to the target impedance model, we use

Eqs. (13)-(15) to write the closed-loop system as

�̃��̈�𝑟 + (�̃� + �̃�𝜆)�̇�𝑟 + �̃�𝜆𝑞𝑟 =

�̃�𝜆�̇� + �̃�𝜆𝑞 − �̃� − �̃� + 𝑀�̇� + 𝐶𝑠

+𝐾𝑑sat(𝑠/diag(𝜑)) + ( 𝑇𝑒 − �̂�𝑒) (43)

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

𝑀𝑟�̈�𝑟 + 𝐵𝑟�̇�𝑟 + 𝐾𝑟𝑞𝑟 = 𝑀𝑟�̈�𝑑 + 𝐵𝑟�̇�𝑑 + 𝐾𝑟𝑞𝑑 − 𝑇𝑒

(44) From Theorems 1 and 2, 𝑠∆ → 0 as 𝑡 → ∞ so the

trajectories of 𝑠 are bounded in the boundary layers. Now

since 𝑠 is bounded, 𝑒 and �̇� are bounded. The boundedness of

𝑞𝑟, �̇�𝑟, 𝑒, and �̇� implies that 𝑞 and �̇� are bounded, which in

turn implies that the right side of Eq. (43) is bounded, just as

the right side of Eq. (44) is bounded.

It is seen that the closed-loop system in Eq. (43) has the

same structure as the impedance model of Eq. (44), which

means both proposed controllers result in closed-loop systems

that converge to the target impedance model of Eq. (44);

where comparing Eq. (43) with Eq. (44) gives 𝑀𝑟 = �̃�, 𝐵𝑟 =�̃� + �̃�𝜆, and 𝐾𝑟 = �̃�𝜆. We see that the proposed control law

in Eq. (23) for both RAIC and RCAIC drives the closed-loop

system in Eq. (25) to match the impedance model in Eq. (11).

IV. SIMULATION RESULTS

The reference trajectory is obtained from the Motion Studies

Lab (MSL) of the Cleveland Veterans Affairs Medical Center

(VAMC) [11]. Here we demonstrate the performance of RAIC

and RCAIC with simulation.

A. Prosthesis Test Robot Model, Controllers, and Target

Impedance Model Parameters

In the prosthesis test model considered here, we have 𝑞 ∈𝑅3, so target impedance model coefficients presented in Eq.

(12) can be written as 𝑀𝑟 = diag(𝑀11 𝑀22 𝑀33), 𝐵𝑟 =diag(𝐵11 𝐵22 𝐵33), and 𝐾𝑟 = diag(𝐾11 𝐾22 𝐾33). To

obtain critically damped responses (two equal roots for each

joint displacement) in the reference impedance model of

Eq. (11), we set 𝐵𝑖𝑖 = 2√𝐾𝑖𝑖𝑀𝑖𝑖 and the two roots are both

equal to −√𝐾𝑖𝑖/𝑀𝑖𝑖 (𝑖 = 1, 2, 3). To obtain two different real

roots, 𝐵𝑖𝑖 > 2√𝐾𝑖𝑖𝑀𝑖𝑖. Here we use a reference impedance

model with roots −11 and −88 for the thigh, −5 and −94 for

the knee, and −3 and −497 for the hip. These values provide a

reference impedance model that is stable, that performs

similarly to able-bodied walking, that provides good reference

model tracking, and that provides control signals and GRFs

with the same order of magnitude as able-bodied ones. We

thus obtain the reference impedance matrices

𝑀𝑟 = diag (10, 10, 10)

𝐾𝑟 = diag (15000, 10000, 5000)

𝐵𝑟 = 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 prosthesis test robot parameters are partly unknown

and can vary by up to 30% from their nominal values [16].

The nominal system parameters are shown in Table 2. The

initial state is 𝑥0𝑇 = [0.019 1.13 0.09 0.09 0 1.6].

After some experimentation, we achieve good performance for

RAIC and RCAIC with the design parameters in Table 3. As

seen from Table 3, the controller design parameters are round

numbers, which means that the controllers are relatively easy

to tune and do not require a tedious gain-tuning process.

Note that the performance of RCAIC is slightly

influenced by each 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 the filter is larger than the system bandwidth

and smaller than the noise frequency. We choose 𝑑 = 2 so

that the adaptation law in Eq. (29) weights the prediction error

twice as much as the impedance model tracking error. The

value of 𝜗0 in Eq. (38) influences the speed of forgetting and

determines the compromise between parameter tracking speed

and oscillation of the estimated parameters. 𝐾0 in Eq. (38)

represents a tradeoff between parameter update speed and the

disturbance effect on the prediction error. 𝑃(0) represents a

tradeoff between parameter error value and the degree of

stability. However, we should choose 𝑃(0) as high as the

noise sensitivity allows to achieve the lowest parameter error

value; to avoid unbounded 𝑃(𝑡), 𝑃(0) must be smaller than

𝐾0𝐼. The value of 𝜑 for both proposed controllers provides a

trade-off between chattering on the control signal and tracking

error bound, adjusts the robustness of the system to non-

parametric uncertainties, and adjusts the sensitivity of the

controllers to parameter drift.

TABLE 1: NOMINAL SYSTEM PARAMETER VALUES

Parameter Description Value Units

𝑠𝑧 Treadmill standoff (Eq. (2)) 0.905 m

𝑘𝑏 Belt stiffness (Eq. (2)) 37000 N/m

𝛽 Belt friction coefficient (Eq. (4)) 0.2 -

𝑣𝑐 Scaling factor (Eq. (4)) 0.05 m/s

𝑣𝑡 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 kg

𝑚2 Mass of link 2 8.5731 kg

𝑚3 Mass of link 3 2.29 kg

𝑙2 Thigh length 0.425 m

𝑙3 Length of knee joint to bottom of shoe 0.527 m

𝑐2 Center of mass on thigh 0.09 m

𝑐3 Center of mass on shank 0.32 m

𝑓 sliding friction in link 1 83.33 N

𝑏 Rotary actuator damping 9.75 N-m-s

𝐼2𝑧 Rotary inertia of link 2 0.138 kg-m^2

𝐼3𝑧 Rotary inertia of link 3 0.0618 kg-m^2

𝑔 Acceleration of gravity 9.81 m/s^2

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8

Fig. 3. RAIC/RCAIC structure

TABLE 3: CONTROLLERS DESIGN PARAMETERS

Controller

Type

Parameter Description Value

𝜑 Boundary layer thicknesses (Eq. (26)) 0.5I

RAIC 𝐾𝑑 Robust term coefficients (Eq. (22)) 100I

𝜇 Adaptation rate (Eq. (27)) 0.01I

𝜆 Sliding term coefficients (Eq. (13)) 100I

𝜑 Boundary layer thicknesses (Eq. (26)) 0.5I

𝐾𝑑 Robust term coefficients (Eq. (22)) 100I

𝜆 Sliding term coefficients (Eq. (13)) 100I

RCAIC 𝜗0 Maximum forgetting rate (Eq. (38)) 5

𝐾0 Upper bound of the adaptation gain (Eq. (38))

400

𝑃(0) Initial condition of the adaptation gain

(Eq. (37))

100I

c Filter constant (Eq. (30)) 1

d Weighting constant (Eq. (29)) 2

B. Controller Performance Evaluation

We define a cost function to evaluate the performance of

RAIC and RCAIC, where the tracking error part of the cost,

and the control part of the cost, are defined as

𝑅𝑀𝑆𝐸𝑖 = √1

𝑇∫ (𝑥𝑖 − 𝑟𝑑𝑖

)2 𝑑𝑡

𝑇

0

, 𝑖 = 1, … , 6

(45)

𝑅𝑀𝑆𝑈𝑗 = √1

𝑇∫ 𝑢𝑗

2 𝑑𝑡

𝑇

0

, 𝑗 = 1, … ,3

(46)

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

as

𝑥𝑇 = [𝑞1𝑞

2𝑞

3�̇�

1�̇�

2�̇�

3]

𝑟𝑇 = [𝑟𝑑1𝑟𝑑2

𝑟𝑑3𝑟𝑑4

𝑟𝑑5𝑟𝑑6]

= [𝑞𝑑1𝑞𝑑2

𝑞𝑑3�̇�𝑑1

�̇�𝑑2�̇�𝑑3]

𝑢𝑇 = [𝑓ℎ𝑖𝑝𝜏𝑡ℎ𝑖𝑔ℎ 𝜏𝑘𝑛𝑒𝑒] (47)

The components of the normalized cost function are defined as

𝐶𝑜𝑠𝑡𝐸𝑖 =𝑅𝑀𝑆𝐸𝑖

maxt∈[0,T]

|𝑟𝑑𝑖|

𝐶𝑜𝑠𝑡𝑈𝑗 =𝑅𝑀𝑆𝑈𝑗

maxt∈[0,T]

|𝑢𝑎𝑏𝑖|

(48)

where 𝑢𝑎𝑏𝑖 indicates the ith able-bodied control signal (able-

bodied control comprises hip force, thigh torque, and knee

torque). The total desired trajectory tracking cost and the total

control cost are defined as follows.

𝐶𝑜𝑠𝑡𝐸 = ∑ 𝐶𝑜𝑠𝑡𝐸𝑖

6

𝑖=1

(49)

𝐶𝑜𝑠𝑡𝑈 = ∑𝐶𝑜𝑠𝑡𝑈𝑖

3

𝑗=1

(50)

The total cost is a combination of the total desired trajectory

tracking cost in Eq. (49), and the total control cost in Eq. (50):

𝐶𝑜𝑠𝑡 = 𝐶𝑜𝑠𝑡𝐸 + 𝐶𝑜𝑠𝑡𝑈 (51)

We define a cost function to evaluate the estimation of the

parameter vector p ∈ 𝑅8 presented in Eq. (10) as

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9

𝑅𝑀𝑆𝑃𝑘 = √1

𝑇∫ (�̂�

𝑘− 𝑝

0𝑘)2𝑑𝑡

𝑇

0

, 𝑘 = 1, … , 8

(52)

where �̂�𝑘 and 𝑝0𝑘 are 𝑘th elements of the estimated parameter

vector and the true parameter vector respectively. The

normalized and total estimation costs are given as follows

𝐶𝑜𝑠𝑡𝑃𝑘 =𝑅𝑀𝑆𝑃𝑘

maxt∈[0,T]

|𝑝0𝑘

|

(53)

𝐶𝑜𝑠𝑡𝑃 = ∑𝐶𝑜𝑠𝑡𝑃𝑘

8

𝑖=1

(54)

C. Simulation Results

Fig. 4 compares the states of the system with RAIC and

RCAIC and the reference trajectories when the system

parameters are varied 30% from nominal (Table 2). Fig. 4

shows that both controllers demonstrate robustness. Fig. 4

shows that the walking behavior of the prosthesis is similar to

human-like walking.

Fig. 5 shows the control signals of RAIC and RCAIC (the

control force for the hip, and the control torques for the thigh

and knee) with 30% parameter deviations. The control

magnitudes for the off-nominal case have similar magnitudes

as able-bodied averaged hip force (800 to 200 N), thigh

torque (50 to 100 N.m), and knee torque (50 to 50 N.m)

[49]-[51]. Note that the hip force and thigh torque represent

able-bodied walking, and the knee torque acts on the

prosthesis, which has the same magnitude as able-body knee

torque. This indicates a strong potential for the proposed

controllers to be useful in real-world prosthesis applications.

In addition, the results demonstrate that the controllers can

deal with parameter variations without large increases in the

control magnitudes. For both controllers, high gains in the

reference impedance model provide better tracking,

particularly for hip displacement, but also increase the control

effort.

Fig. 6 depicts the GRFs when the system parameters vary

by 30%from nominal. We see that the generated forces are

similar to able-bodied averaged horizontal GRF (-150 to 150

N) and vertical GRF (0 to 800 N) [49]-[51], again indicating

strong potential for real-world application. As can be observed

from Fig. 6, we have no GRF in swing phase, and after the

point foot hits the ground (circles on the x-axis), horizontal

and vertical GRFs become nonzero.

Fig. 7 shows the estimated parameter vector 𝑝 (presented

in Eq. (10)) for RAIC and RCAIC when the system

parameters vary by 30%. As expected, the RAIC parameter

estimates do not match the true parameter values.

However, RCAIC using the BGF composite adaptation

law performs significantly better regarding parameter

estimation compared to RAIC. The estimated parameter vector

of the proposed controller RCAIC perfectly matches the true

value except for the fourth element P4.

Fig. 4: Tracking performance with +30% parameter deviations: desired

trajectory (magenta dotted line), response with RAIC (red dashed line), and response with RCAIC (blue solid line)

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

RCAIC (blue solid)

0 2 4-0.04

-0.02

0

0.02

0.04

Time (s)

Hip

dis

pla

ce

men

t (m

)

0 2 4-0.4

-0.2

0

0.2

0.4

Time (s)

Hip

velo

city (

m/s

)

0 2 40.5

1

1.5

2

Time (s)

Th

igh

an

gle

(ra

d)

0 2 4-4

-2

0

2

4

Time (s)

Thig

h v

elo

city (

rad

/s)

0 2 4-0.5

0

0.5

1

1.5

Time (s)

Kne

e a

ngle

(ra

d)

0 2 4-10

-5

0

5

10

Time (s)

Kn

ee

ve

locity (

rad

/s)

0 2 4

-800

-600

-400

-200

0

200

Time (s)

Hip

forc

e (

N)

0 2 4

-100

0

100

200

Time (s)

Thig

h torq

ue (

N.m

)

0 2 4

-50

0

50

Time (s)

Kn

ee

to

rqu

e (

N.m

)

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

10

P4 is the most complex parameter in terms of its constituent

elements (see Eq. (10)), so errors in the constituent elements

of P4 can cause cumulative errors in P4.

Fig. 8 compares the trajectories of 𝑠 and 𝑠∆ as described in

Eqs. (13) and (24) respectively for RAIC and RCAIC. Based

on the values of 𝜑1, 𝜑2, 𝜑3, and the definition of 𝑠∆, it is seen

that the TEB adaptation mechanism in RAIC is active only

when 𝑠 is outside its boundary layer (that is, 𝑠∆ is nonzero).

Fig. 6. GRFs for 30% parameter deviations: RAIC (red dashed line) and

RCAIC (blue solid line). The circles on the x-axis of the right plot show the

foot strikes on the treadmill.

Fig. 7. True parameter values (magenta dotted lines) and estimated parameter

values for 30% parameter deviations: RAIC (red dashed line) and RCAIC

(blue solid line)

Fig. 8: Trajectories of 𝑠∆ and 𝑠 for the RAIC (red dashed line), and the RCAIC (blue solid line) with +30% parameter deviations

When 𝑠∆ is zero, the parameter adaptation of the RAIC (Eq.

(27)) stops and its estimated parameters remain constant,

whereas 𝑠∆ = 0 only turns off TEB adaptation part of the

RCAIC (the first part of the Eq. (29)). It is observed that none

of the 𝑠 trajectories for the RCAIC exceed the boundary layer

(the area between 𝜑𝑖 = −0.5 and 𝜑𝑖 = +0.5) and in turn all

s∆ trajectories are zero. This shows that the RCAIC only uses

prediction errors, which appear in the PEB adaptation, and the

TEB adaptation mechanism is turned off.

On the other hand, all 𝑠 trajectories for the RAIC exceed

the boundary layer. From Fig. 8, we can see that the 𝑠

trajectories of the RAIC for the hip, thigh, and knee exceed the

boundary layer four, three, and two times respectively, and in

turn the 𝑠∆ trajectories are nonzero.

Fig. 9 shows the norm of the adaptation gain 𝑃, the time-

varying forgetting factor 𝜗(𝑡), and the joint prediction errors

(𝑒𝑝𝑖 , 𝑖 = 1, 2, 3) for the RCAIC with +30% uncertainty on the

system parameters. Fig. 9(a) illustrates that 𝑃(t) is upper and

lower bounded by two positive numbers (𝑃(t) is upper

bounded by 𝐾0 = 400 and lower bounded by 𝑃(0) = 100).

Fig. 9(b) shows that the forgetting factor satisfies the

condition 𝜗(𝑡) > 𝜗 > 0. These observations imply that 𝑊(𝑡)

is PE. Since 𝑊(𝑡) is PE, 𝑝 and 𝑒𝑝𝑖 exponentially converge to

zero as shown in Fig. 9(c).

Table 4 summarizes the desired trajectory tracking,

parameter estimation, and control performance for RAIC and

RCAIC for the nominal system parameter values and also

when the parameter values vary ±30% relative to nominal.

0 2 4-100

0

100

200

Time (s)

Fx (

N)

0 2 40

200

400

600

800

Time (s)

Fz (

N)

0 2 440

50

60

70

p1

(kg

)

Time (s)0 2 4

5

10

15

p2

(kg

-m)

Time (s)

0 2 4-2

0

2

4

6

p3

(kg-m

)

Time (s)0 2 4

0

5

10

15

p4

(kg

-m2)

Time (s)

0 2 4-1

-0.5

0

0.5

1

p5

(kg-m

2)

Time (s)

0 2 4-1

0

1

2

3

p6

(kg

-m2)

Time (s)

0 2 49

10

11

12

13

p7

(N

-m-s

)

Time (s)0 2 4

80

90

100

110

p8

(N

)

Time (s)

0 2 4-0.6

-0.4

-0.2

0

0.2

Time (s)

s fo

r h

ip

0 2 4-0.06

-0.04

-0.02

0

Time (s)

s for

hip

0 2 4-1

-0.5

0

0.5

1

Time (s)

s fo

r th

igh

0 2 4-0.1

-0.05

0

0.05

0.1

Time (s)

s fo

r th

igh

0 2 4-0.5

0

0.5

1

Time (s)

s fo

r kn

ee

0 2 40

0.02

0.04

0.06

Time (s)

s for

kne

e

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

11

Table 4 lists total desired trajectory tracking cost 𝐶𝑜𝑠𝑡𝐸,

total control 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%. When the system parameter values vary

−30%, RCAIC has a small improvement in control cost, but a

significant improvement in estimation by 40% in comparison

with the RAIC, while tracking performance of the RCAIC

slightly deteriorates.

In general, in the case of −30% parameter uncertainty, the

total cost of the RCAIC decreases by 4%.

Table 4 shows that when the parameter values vary 30%

from nominal, RCAIC has a remarkable superiority to the

RAIC in terms of the estimation and tracking performances.

This superiority is because by more accurately estimating the

system parameters, the RCAIC includes a more accurate

model (𝑝 and 𝑒𝑝 exponentially converge to zero) and in this

way achieves better tracking. Table 4 shows that desired

trajectory tracking performance (𝐶𝑜𝑠𝑡𝐸) and estimation

performance (𝐶𝑜𝑠𝑡𝑃) of the proposed RCAIC 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.

Fig. 9. (a) Norm of P, (b) time-varying forgetting factor, (c) joint prediction

errors (𝑒𝑝𝑖 , 𝑖 = 1, 2 ,3); all plots represent the situation of 30% parameter

uncertainty.

TABLE 4: CONTROLLER PERFORMANCE

Parameter Uncertainty

Controller Type

Controller Performance

𝐶𝑜𝑠𝑡𝐸 (Eq.(49))

𝐶𝑜𝑠𝑡𝑈 (Eq.(50))

𝐶𝑜𝑠𝑡𝑃 (Eq.(54))

𝐶𝑜𝑠𝑡 (Eq.(51))

Nominal RAIC 0.96 2.40 0.00 3.36

RCAIC 0.96 2.44 0.80 3.40

30% RAIC 0.93 2.65 4.42 3.58

RCAIC 0.96 2.48 2.66 3.44

+30% RAIC 1.05 2.22 14.62 3.27

RCAIC 0.95 2.44 3.46 3.39

V. CONCLUSIONS AND FUTURE WORK

We designed two robust adaptive impedance controllers,

RAIC and RCAIC, for a combined test robot and transfemoral

prosthesis device. The controllers estimate the system

parameters and also driving joint tracking errors to boundary

layers while compensating for the variations of GRFs and non-

parametric uncertainties. We defined the boundary layers to

make a tradeoff between control signal chatter and

performance, and also to stop TEB adaptation mechanism in

these layers to prevent unfavorable parameter drift.

We designed the both controllers to imitate the

characteristics of natural walking and to provide flexible,

smooth, gait. We thus defined a reference model with

impedance similar to that of able-bodied gait. We also proved

closed-loop system stability for both RAIC and RCAIC based

on non-scalar boundary layers using Barbalat’s lemma and

Lyapunov theory.

We performed simulations for both proposed controllers

with 30% parameter errors, and we showed that trajectory

tracking remained good, which demonstrated robustness of the

proposed controllers. We demonstrated good transient

responses with nominal system parameter values and also with

system parameter value deviations of up to ±30%. When we

used the first controller RAIC for the 30% parameter

deviations, desired trajectory desired trajectory tracking errors

were 16 mm for vertical hip position, 0.15 deg for thigh angle,

and 0.12 deg for prosthetic knee angle. When we used the

second controller, RCAIC, with 30% parameter uncertainties,

trajectory 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 system

parameter values varied by 30% from nominal, the proposed

controller RCAIC had better tracking performance by 9.5% in

comparison to RAIC, while resulted in more control cost by

9.9%. Furthermore, RCAIC using the BGF composite

adaptation law performed significantly better parameter

estimation by 76% compared to the RAIC. We also achieved

reasonable control signals and GRFs for the both controller

structures. Note that, however RCAIC in general performed

better to RAIC, RCAIC has larger computational time and

higher programming complexity.

For future work, we will incorporate rotary and linear

actuator dynamics in the system model to obtain motor voltage

control signals. We will also apply the controllers to a

prosthesis prototype that has been developed at Cleveland

State University. We will also include an active ankle joint to

0 2 4100

200

300

400

Time (s)

No

rm o

f P

0 2 40

1

2

3

4

Time (s)

Fo

rge

ttin

g fa

cto

r

0 2 4-6

-4

-2

0

2

Time (s)

Jo

ints

pre

dic

tio

n e

rro

r

e

p1

ep2

ep3

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

12

the system model to extend the controllers to a 4-DOF robot /

prosthesis model.

We will test the proposed prosthesis model and

controllers on a human-prosthesis hybrid system [22]. We will

also implement the proposed controllers experimentally on a

powered transfemoral prosthesis, AMPRO3 (AMBER

Prosthetic) [24].

The results in this paper can be reproduced with the

Matlab code that is available at

http://embeddedlab.csuohio.edu/prosthetics/research/robust-

adaptive.html.

APPENDIX 1

Stability Analysis of the RAIC

Proof of Theorem 1: Even though 𝑠∆ is not differentiable

everywhere, V is differentiable because it is a quadratic

function of 𝑠∆. The derivative of the Lyapunov function of Eq.

(28) is given as follows:

�̇�(𝑠∆, 𝑝) =1

2(�̇�∆

𝑇𝑀 𝑠∆ + 𝑠∆𝑇𝑀 �̇�∆) +

1

2(𝑠∆

𝑇�̇�𝑠∆)

+

1

2(�̇�𝑇𝜇 𝑝 + 𝑝𝑇𝜇 �̇�)

= 𝑠∆

𝑇𝑀 �̇�∆ +1

2(𝑠∆

𝑇�̇�𝑠∆) + �̇�𝑇𝜇 𝑝

(55)

Note that inside the boundary layer in Eq. (26), �̇�∆ = 0, and

outside the boundary layer �̇�∆ = �̇�, so using the closed-loop

form in Eq. (25) gives

�̇�(𝑠∆, 𝑝) = 𝑠∆𝑇(−𝐶𝑠 − 𝐾𝑑sat(𝑠/diag(𝜑)) + (�̂�𝑒 − 𝑇𝑒)

+𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝) +

1

2(𝑠∆

𝑇�̇�𝑠∆) + �̇�𝑇𝜇 𝑝 =

−𝑠∆

𝑇𝐶𝑠 +1

2(𝑠∆

𝑇�̇�𝑠∆)−𝑠∆𝑇𝐾𝑑sat(𝑠/diag(𝜑))

+𝑠∆𝑇(�̂�𝑒 − 𝑇𝑒) + 𝑠∆

𝑇𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝 + �̇�𝑇𝜇 𝑝 (56)

To derive the adaptation law, we constrain �̇�𝑇𝜇 𝑝 +

𝑠∆𝑇𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝 to zero, which gives the update law �̇̂� =

−𝜇−1𝑌𝑇(𝑞, �̇�, 𝑣, �̇�)𝑠∆ as already presented in Eq. (27). As seen

from Eq. (27), the adaptation law extracts information about

the parameters from only the tracking error (i.e., TEB).

Therefore, �̇�(𝑠∆, 𝑝) can be written as follows:

�̇�(𝑠∆, 𝑝) = −𝑠∆𝑇𝐶𝑠 +

1

2(𝑠∆

𝑇�̇�𝑠∆)

−𝑠∆𝑇𝐾𝑑sat(𝑠/diag(𝜑)) + 𝑠∆

𝑇(�̂�𝑒 − 𝑇𝑒) (57)

We see from Eq. (26) that if |𝑠| ≤ diag(𝜑), then 𝑠∆ = 0 and

�̇�(𝑠∆, 𝑝) converges to zero inside the boundary layer.

Conversely, if |𝑠| > diag(𝜑), then 𝑠∆ is defined by the second

part of Eq. (26), in which case 𝑠 = 𝑠∆ + 𝜑 sat(𝑠/diag(𝜑))

outside the boundary layer. If we substitute 𝑠 = 𝑠∆ +

𝜑 sat(𝑠/diag(𝜑)) in Eq. (57), we obtain �̇�(𝑠∆, 𝑝) outside the

boundary layer as follows:

�̇�(𝑠∆, 𝑝) =1

2𝑠∆

𝑇(�̇� − 2𝐶)𝑠∆−𝑠∆𝑇𝐶𝜑 sat(𝑠/diag(𝜑))

−𝑠∆𝑇𝐾𝑑 sat(𝑠/diag(𝜑) + 𝑠∆

𝑇(�̂�𝑒 − 𝑇𝑒) (58)

Matrix (�̇� − 2𝐶) is skew symmetric, so 𝑠∆𝑇(�̇� − 2𝐶)𝑠∆ = 0

and we simplify �̇�(𝑠∆, 𝑝) as

�̇�(𝑠∆, 𝑝) = −𝑠∆𝑇(𝐶𝜑 + 𝐾𝑑) sat(𝑠/diag(𝜑)) + 𝑠∆

𝑇(�̂�𝑒 − 𝑇𝑒)

(59)

We choose 𝐾𝑑 and 𝜑 as tuning parameters to keep 𝐶𝜑 + 𝐾𝑑

bounded from below by the 𝐾𝑚𝐼, where 𝐾𝑚 is a positive

scalar. We can see that 𝐶𝜑 + 𝐾𝑑 ≥ 𝐾𝑚𝐼 ensures that 𝐶𝜑 + 𝐾𝑑

is positive definite. We use Eq. (59) to write

�̇�(𝑠∆, 𝑝) ≤ −𝐾𝑚𝑠∆𝑇sat(𝑠/diag(𝜑)) + 𝑠∆

𝑇(�̂�𝑒 − 𝑇𝑒) (60)

We note that 𝑠∆𝑇sat(𝑠/diag(𝜑)) is the one-norm of 𝑠∆, so we

write Eq. (60) as

�̇�(𝑠∆, 𝑝) ≤ −𝐾𝑚‖𝑠∆‖1 + 𝑠∆𝑇(�̂�𝑒 − 𝑇𝑒) (61)

We now define 𝐾𝑚 = 𝐹𝑚 + 𝛾𝑚, where |�̂�𝑒𝑖− 𝑇𝑒𝑖

| ≤ 𝐹𝑖 ≤ 𝐹𝑚,

𝐹𝑚 = max (𝐹𝑖), and 𝛾𝑚 = max (𝛾𝑖). We can then write

Eq. (61) as follows:

�̇�(𝑠∆, 𝑝) ≤ −𝛾𝑚‖𝑠∆‖1 − 𝐹𝑚‖𝑠∆‖1 + 𝑠∆𝑇(�̂�𝑒 − 𝑇𝑒) (62)

Noting that |�̂�𝑒𝑖− 𝑇𝑒𝑖

| ≤ 𝐹𝑖 ≤ 𝐹𝑚 and 𝑠∆𝑖≤ |𝑠∆𝑖

|, we see that

𝑠∆𝑇(�̂�𝑒 − 𝑇𝑒) in Eq. (62) is bounded from above by 𝐹𝑚‖𝑠∆‖1,

so

�̇�(𝑠∆, 𝑝) ≤ −𝛾𝑚‖𝑠∆‖1 (63)

This indicates that outside the boundary layer (the second

condition of Eq. (26)), the Lyapunov derivative is negative

semidefinite, so we can prove the stability of the closed-loop

system with Barbalat’s lemma [48].

Barbalat’s Lemma: If a Lyapunov function 𝑉 = 𝑉(𝑡, 𝑥)

satisfies the following conditions:

I. 𝑉(𝑡, 𝑥) is lower bounded, and

II. �̇�(𝑡, 𝑥) is negative semi-definite, and

III. �̈�(𝑡, 𝑥) is bounded

then �̇�(𝑡, 𝑥) → 0 as 𝑡 → ∞; that is, the closed-loop system is

asymptotically stable. Now we state an intermediate lemma

that we will need to complete the proof of the theorem 1.

Lemma 1: The derivative of the Lyapunov function of Eq.

(63) converges to zero, which guarantees that the system

converges to the boundary layer.

Proof of Lemma 1: Conditions I and II in Barbalat’s

Lemma are confirmed from Eqs. (28) and (63), which means

that V is bounded. This implies that all of the terms in V in

Eq. (28) are bounded, including 𝑠∆ and 𝑝. Since 𝑝 is constant,

this means that �̂� is bounded. Since 𝑠∆ is bounded, this means

that 𝑠 is bounded. The second derivative of V is bounded as

follows: �̈�(𝑠∆, 𝑝) ≤ −𝛾𝑚𝑑

𝑑𝑡‖𝑠∆‖1. In the worst case (that is, at

the upper bound) we have

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

13

�̈�(𝑠∆, 𝑝) = −𝛾𝑚𝑑

𝑑𝑡‖𝑠∆‖1 = −𝛾𝑚 ∑

𝑠∆�̇�∆

|𝑠∆|=

± 𝛾𝑚 ∑ �̇�∆ = ± 𝛾𝑚 ∑ �̇�

(64)

where 𝑠∆ ≠ 0 outside the boundary layer. We substitute �̇�

from Eq. (25) into Eq. (64) to obtain

�̈�(𝑠∆, 𝑝) = ± 𝛾𝑚 ∑𝑀−1(−𝐶𝑠 − 𝐾𝑑sat(𝑠/diag(𝜑))

+(�̂�𝑒 − 𝑇𝑒) − 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝) (65)

Recall that 𝑝 and 𝑠 are bounded. The boundedness of 𝑠

implies the boundedness of 𝑒 and �̇�, as seen from Eq. (13).

Since 𝑞𝑟, �̇�𝑟, and �̈�𝑟 are bounded, we know that 𝑞, �̇�, 𝑣, and �̇�

are also bounded. Therefore, in Eq. (65), 𝑀,𝐶, 𝜑, 𝑝, 𝑌, 𝛾𝑚 , 𝑠,

and 𝐾𝑑 are bounded. |�̂�𝑒 − 𝑇𝑒| is upper bounded by 𝐹𝑚, so we

can conclude that �̈� is bounded. Therefore, as conditions I, II,

and III from Barbalat’s Lemma hold, we can conclude that

�̇�(𝑠∆, 𝑝) → 0 as 𝑡 → ∞. This means that −𝛾𝑚‖𝑠∆‖1 in Eq. (63)

is equal to zero, which means that Eq. (63) can be written as

the equality �̇�(𝑠∆, 𝑝) = −𝛾𝑚‖𝑠∆‖1. We therefore have

�̇�(𝑠∆, 𝑝) → 0 ⟾ −𝛾𝑚‖𝑠∆‖1 → 0 ⟾ 𝑠∆ → 0. This indicates

that the control ensures that 𝑠 converges to the boundary layer.

QED (Lemma 1)

The RAIC mitigates system uncertainties more than a

standard adaptive controller, but also has a larger tracking

error. RAIC drives the system to the boundary layer and

results in robustness to GRF as a non-parametric uncertainty.

Inside the boundary layer, Eqs. (58)(65) can be reformulated

for 𝑠∆ = 0, in which case 𝑠 remains in the boundary layer,

which stops adaptation, and the estimated parameters remain

constant. Therefore, the system with the RAIC converges to

the reference impedance model.

QED (Theorem 1)

It should be noted that the above proof of asymptotic

closed-loop system stability implies that 𝐾𝑑 should be

bounded by 𝐹𝑚 − 𝐶𝜑 − that is, 𝐾𝑑 ≥ 𝐹𝑚 − 𝐶𝜑 + 𝛾𝑚.

APPENDIX 2

Stability Analysis of the RCAIC

Proof of Theorem 2: Note that V in Eq. (42), which is a

quadratic function of 𝑠∆, is continuously differentiable. The

derivative of the Lyapunov function is given as

�̇�(𝑠∆, 𝑝) =1

2(�̇�∆

𝑇𝑀 𝑠∆ + 𝑠∆𝑇𝑀 �̇�∆) +

1

2(𝑠∆

𝑇�̇�𝑠∆)

+

1

2(�̇�𝑇𝑃−1 𝑝 + 𝑝𝑇𝑃−1 �̇�) +

1

2(𝑝𝑇

𝑑

𝑑𝑡(𝑃−1) 𝑝) =

𝑠∆𝑇𝑀 �̇�∆ +

1

2(𝑠∆

𝑇�̇�𝑠∆) + 𝑝𝑇𝑃−1 �̇� +1

2(𝑝𝑇 𝑑

𝑑𝑡(𝑃−1) 𝑝)

(66)

Now we want to prove global exponential stability of the

closed-loop system both outside and inside the boundary layer

defined in Eq. (26). Inside the boundary layer �̇�∆ = 0, and

outside the boundary layer �̇�∆ = �̇�, so Eq. (66) can be written

as

�̇�(𝑠∆, 𝑝) = 𝑠∆𝑇(−𝐶𝑠 − 𝐾𝑑𝑠𝑎𝑡(𝑠/diag(𝜑)) + (�̂�𝑒 − 𝑇𝑒)

+𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝) +

1

2(𝑠∆

𝑇�̇�𝑠∆) + 𝑝𝑇𝑃−1 �̇� +

1

2(𝑝𝑇

𝑑

𝑑𝑡(𝑃−1) 𝑝) = −𝑠∆

𝑇𝐶𝑠 +1

2(𝑠∆

𝑇�̇�𝑠∆) +

𝑠∆𝑇 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝−𝑠∆

𝑇𝐾𝑑𝑠𝑎𝑡(𝑠/diag(𝜑)) +

𝑠∆𝑇(�̂�𝑒 − 𝑇𝑒) + 𝑝𝑇𝑃−1 �̇� +

1

2(𝑝𝑇

𝑑

𝑑𝑡(𝑃−1) 𝑝)

(67)

Outside the boundary layer we see that if |𝑠| > diag(𝜑),

then 𝑠∆ comes from the second condition of Eq. (26), in which

case we have 𝑠 = 𝑠∆ + 𝜑 sat(𝑠/diag(𝜑)). Substituting 𝑠 =𝑠∆ + 𝜑 sat(𝑠/diag(𝜑)) in the first term of Eq. (67), we write

�̇�(𝑠∆, 𝑝) outside the boundary layer as

�̇�(𝑠∆, 𝑝) =1

2𝑠∆

𝑇(�̇� − 2𝐶)𝑠∆−𝑠∆𝑇𝐶𝜑 sat(𝑠/diag(𝜑)) +

𝑠∆𝑇 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝−𝑠∆

𝑇𝐾𝑑𝑠𝑎𝑡(𝑠/diag(𝜑)) +

𝑠∆𝑇(�̂�𝑒 − 𝑇𝑒) + 𝑝𝑇𝑃−1 �̇� +

1

2(𝑝𝑇

𝑑

𝑑𝑡(𝑃−1) 𝑝)

(68)

(�̇� − 2𝐶) is skew-symmetric, so 𝑠∆𝑇(�̇� − 2𝐶)𝑠∆ = 0 and we

can simplify �̇�(𝑠∆, 𝑝) as

�̇�(𝑠∆, 𝑝) = −𝑠∆𝑇(𝐶𝜑 + 𝐾𝑑) sat(𝑠/diag(𝜑)) +

𝑠∆𝑇 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝 + 𝑠∆

𝑇(�̂�𝑒 − 𝑇𝑒) +

𝑝𝑇𝑃−1 �̇� +

1

2(𝑝𝑇

𝑑

𝑑𝑡(𝑃−1) 𝑝)

(69)

We tune the design parameters 𝐾𝑑 and 𝜑 so that 𝐶𝜑 + 𝐾𝑑 ≥𝐾𝑚𝐼, which guarantees the positive definiteness of 𝐶𝜑 + 𝐾𝑑,

where 𝐾𝑚 is a positive scalar. We then use Eq. (69) to write

�̇�(𝑠∆, 𝑝) ≤ −𝐾𝑚𝑠∆𝑇sat(𝑠/diag(𝜑)) + 𝑠∆

𝑇 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝 +

𝑠∆𝑇(�̂�𝑒 − 𝑇𝑒) + 𝑝𝑇𝑃−1 �̇� +

1

2(𝑝𝑇

𝑑

𝑑𝑡(𝑃−1) 𝑝)

(70)

We replace 𝑠∆𝑇sat(𝑠/diag(𝜑)) with the one-norm of 𝑠∆ and

then we write Eq. (70) as

�̇�(𝑠∆, 𝑝) ≤ −𝐾𝑚‖𝑠∆‖1 + 𝑠∆𝑇 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝 + 𝑠∆

𝑇(�̂�𝑒 − 𝑇𝑒)

+𝑝𝑇𝑃−1 �̇� +

1

2(𝑝𝑇

𝑑

𝑑𝑡(𝑃−1) 𝑝)

(71)

We define 𝐾𝑚 = 𝐹𝑚 + 𝛾𝑚, where |�̂�𝑒𝑖− 𝑇𝑒𝑖

| ≤ 𝐹𝑖 ≤ 𝐹𝑚,

𝐹𝑚 = max (𝐹𝑖), and 𝛾𝑚 = max (𝛾𝑖) with 𝑖 = 1, 2, 3. Noting

that 𝑠∆𝑖≤ |𝑠∆𝑖

|, we see that 𝑠∆𝑇(�̂�𝑒 − 𝑇𝑒) in Eq. (71) is

bounded from above by 𝐹𝑚‖𝑠∆‖1, so

�̇�(𝑠∆, 𝑝) ≤ −𝛾𝑚‖𝑠∆‖1 + 𝑠∆𝑇 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝 +

𝑝𝑇𝑃−1 �̇� +

1

2(𝑝𝑇

𝑑

𝑑𝑡(𝑃−1) 𝑝)

(72)

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

14

Since 𝑝 = �̂� − 𝑝, we can substitute Eq. (29) and Eq. (36) into

Eq. (72), and �̇�(𝑠∆, 𝑝) can be written as

�̇�(𝑠∆, 𝑝) ≤ −𝛾𝑚‖𝑠∆‖1 + 𝑠∆𝑇𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝

+

1

2(𝑝𝑇

𝑑

𝑑𝑡(𝑃−1) 𝑝) − 𝑝𝑇 𝑌𝑇(𝑞, �̇�, 𝑣, �̇�)𝑠∆ −

𝑝𝑇𝑊𝑇𝑅𝑊𝑝 = −𝛾𝑚‖𝑠∆‖1 +

1

2(𝑝𝑇

𝑑

𝑑𝑡(𝑃−1) 𝑝)

−𝑝𝑇𝑊𝑇𝑅𝑊𝑝 (73)

By substituting Eq. (37) into Eq. (73), we can write

�̇�(𝑠∆, 𝑝) ≤ −𝛾𝑚‖𝑠∆‖1 −1

2𝑝𝑇𝜗(𝑡)𝑃−1𝑝

− 𝑝𝑇𝑊𝑇(𝑑𝐼 −

1

2𝐼)𝑊𝑝

(74)

𝛾𝑚 > 0, 𝜗(𝑡) ≥ 0, and 𝑃 is positive definite, so by

choosing 𝑑 >1

2, we can see that outside the boundary layer

(that is, the second condition of Eq. (26)), the derivative of the

Lyapunov function is negative semidefinite. This in turn

means that we can use Barbalat’s lemma to prove global

exponential stability. If 𝑉(𝑡, 𝑥) satisfies the Barbalat’s

Lemma conditions, then �̇�(𝑡, 𝑥) → 0 as 𝑡 → ∞, which means

that RCAIC results in a closed-loop system that is globally

exponentially stable.

Now we state an intermediate lemma that we will need to

complete the proof of Theorem 2.

Lemma 2: The derivative of the Lyapunov function of Eq.

(74) globally exponentially converges to zero, which

guarantees convergence to the boundary layer (𝑠∆ → 0). Also,

the prediction error in Eq. (36) of the proposed RCAIC

converges to zero, which implies perfect estimation of the

system parameters.

Proof of Lemma 2: Conditions I and II in Barbalat’s

Lemma are satisfied from Eqs. (42) and (74) and we therefore

conclude that V is bounded, which means that all terms in V

(including 𝑠∆, and 𝑝) are bounded. Since 𝑝 is constant �̂� is

bounded, and since 𝑠∆ is bounded 𝑠 is bounded. From

Eq. (11), since 𝑞𝑑 is bounded, �̇�𝑑, �̈�𝑑, 𝑞𝑟, �̇�𝑟 , and �̈�𝑟 are

bounded. From Eqs. (13)(15), since 𝑠 is bounded, we see

that 𝑒 and �̇� are both bounded. These facts imply that 𝑞, �̇�, �̈�,

𝑣, and �̇� are bounded as well. So 𝑌(𝑞, �̇�, 𝑣, �̇�), 𝑌ʹ(𝑞, �̇�, �̈�), and

𝑊(𝑞, �̇�) are bounded.

By taking the derivative of �̇�(𝑠∆, 𝑝) at its upper bound

we obtain

�̈�(𝑠∆, 𝑝) = ± 𝛾𝑚 ∑�̇�∆ − 𝑝𝑇𝑊𝑇(2𝑑𝐼 − 𝐼)𝑊�̇� −

𝑝𝑇𝑊𝑇(2𝑑𝐼 − 𝐼)�̇�𝑝 − 𝑝𝑇𝜗(𝑡)𝑃−1�̇�

−1

2𝑝𝑇�̇�(𝑡)𝑃−1𝑝 −

1

2𝑝𝑇𝜗(𝑡)

𝑑

𝑑𝑡(𝑃−1)𝑝

(75)

Substituting �̇�, 𝑑

𝑑𝑡(𝑃−1), and �̇� from Eqs. (29), (37), and (25)

respectively into Eq. (75), �̈�(𝑠∆, 𝑝) can be written as follows:

�̈�(𝑠∆, 𝑝) = ± 𝛾𝑚 ∑𝑀−1(−𝐶𝑠 − 𝐾𝑑sat(𝑠/diag(𝜑))

+(�̂�𝑒 − 𝑇𝑒) + 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝) + 𝑝𝑇𝑊𝑇(2𝑑𝐼 − 𝐼)

𝑊𝑃(t)𝑌𝑇𝑠∆ + 𝑝𝑇𝜗(𝑡)𝑌𝑇𝑠∆ + 𝑝𝑇𝑊𝑇(2𝑑𝐼 − 𝐼) 𝑊𝑃(t)𝑊𝑇(𝑑𝐼)𝑊𝑝 − 𝑝𝑇𝑊𝑇(2𝑑𝐼 − 𝐼)�̇�𝑝 + 𝑝𝑇

𝜗(𝑡)𝑊𝑇(𝑑𝐼)𝑊𝑝 −

1

2𝑝𝑇�̇�(𝑡)𝑃−1𝑝

+

1

2𝑝𝑇𝜗2(𝑡)𝑃−1𝑝 −

1

2𝑝𝑇𝜗(𝑡)𝑊𝑇𝑊𝑝

(76)

Since 𝑃(t) is bounded and its norm is bounded by 𝐾0, then

from Eq. (38), 𝜗(𝑡) and �̇�(𝑡) are bounded. Moreover, since 𝑀,

𝐶, 𝑠, 𝑌, 𝑊, �̇�, 𝑝, and 𝑠∆ are bounded and |�̂�𝑒 − 𝑇𝑒| ≤ 𝐹𝑚, we

see that �̈�(𝑠∆, 𝑝) is bounded. Since we have verified all

conditions in Barbalat’s Lemma, we know that �̇�(𝑠∆, 𝑝) → 0

as 𝑡 → ∞ ⟾𝛾𝑚‖𝑠∆‖1 → 0⟾ 𝑠∆ → 0 as 𝑡 → ∞, which means

that outside the boundary layer, RCIAC guarantees

convergence of 𝑠 to the boundary layer. Furthermore,

�̇�(𝑠∆, 𝑝) → 0 means that 𝑝𝑇𝜗(𝑡)𝑃−1𝑝 → 0 and since

𝑃−1(𝑡) ≥1

𝐾0𝐼, then if 𝑊 is PE, 𝜗(𝑡) > 𝜗 > 0, so we have

𝜗(𝑡)𝑝𝑇𝑃−1𝑝 ≥ 𝜗𝑝𝑇𝑝/𝐾0 (77)

Therefore, 𝑝𝑇𝜗(𝑡)𝑃−1𝑝 → 0 means that 𝑝 → 0. To achieve

faster exponential convergence of 𝑠 to the boundary layer and

convergence of the prediction error to zero, we define a

strictly positive constant Ч0, where Ч0 = min (2𝜗0, 𝜗), and we

write [48]:

�̇�(𝑡) + Ч0𝑉(𝑡) ≤ 0 , 𝑉(𝑡) ≤ 𝑉(0)𝑒−Ч0𝑡 (78)

On the other hand, inside the boundary layer, where |𝑠| ≤diag(𝜑), Eqs. (68)(78) can be rewritten for 𝑠∆ = 0. For this

condition, 𝑠 remains inside the boundary layer and the

prediction error exponentially converges to zero. Outside the

boundary layer, both 𝑠∆ and 𝑒𝑝 exponentially converge to zero,

which means that we achieve perfect parameter estimation and

guarantee convergence of th 𝑠 to the boundary layer.

QED (Lemma 2)

We see from the above that the closed-loop system with

the proposed RCAIC converges to the target impedance

model. Therefore, the controller drives the system to the

boundary layer, achieves perfect parameter estimation, and

achieves robustness against GRFs.

QED (Theorem 2)

Note that the exponential stability proof implies that 𝐾𝑑

must be bounded from below by 𝐹𝑚 − 𝐶𝜑 (that is,𝐾𝑑 ≥ 𝐹𝑚 −𝐶𝜑 + 𝛾𝑚).

ACKNOWLEDGMENT

This research is supported by NSF Grant 1344954. The

authors are grateful to Jean-Jacques Slotine, Mojtaba Sharifi,

Antonie van den Bogert, and Thang Tien Nguyen for

suggestions that improved this paper.

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <

15

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