1 Copyright © 2015 by ASME

Proceedings of ASME 2015 DSCC ASME Dynamic Systems and Control Conference

October 28-30, 2015, Columbus, Ohio, USA

ROBUST ADAPTIVE IMPEDANCE CONTROL OF A PROSTHETIC LEG

Vahid Azimi Department of Electrical and

Computer Engineering, Cleveland State University,

Cleveland, OH, USA Email: v.azimi@csuohio.edu

Dan Simon Department of Electrical and

Computer Engineering, Cleveland State University,

Cleveland, OH, USA Email: d.j.simon@csuohio.edu

Hanz Richter Department of Mechanical

Engineering, Cleveland State University,

Cleveland, OH, USA Email: h.richter@csuohio.edu

ABSTRACT

We propose a regressor-based nonlinear robust model

reference adaptive impedance controller for an active prosthetic

leg for a transfemoral amputee. We use an adaptive control term

to estimate the uncertain parameters of the system, and a robust

control term so the system trajectories converge to a sliding

manifold and exhibit robustness to variations of ground reaction

force (GRF). The sliding mode boundary layer not only

compromises between control chattering and tracking

performance, but also bounds the parameter adaptations to

prevent unfavorable parameter drift. We prove the stability of the

closed-loop system with Lyapunov stability theory and the

Barbalat lemma. We use particle swarm optimization (PSO) to

optimize the design parameters of the controller and the

adaptation law. The PSO cost function is comprised of control

signal magnitudes and tracking errors. PSO achieves a 22%

improvement in the objective function. The acceleration-free

regressor form of the system removes the need to measure the

joint accelerations, which would otherwise introduce noise in the

system. Finally, we present simulation results to validate the

effectiveness of the controller. We achieve accurate tracking of

joint displacements and velocities for both nominal and

perturbed values of the system parameters. Variations of ±30%

on the system parameters results in an increase of the cost

function by only 10%whichconfirms the robustness of the

proposed controller.

KEYWORDS Robust control, model reference adaptive control, impedance

control, sliding mode control, prosthetics, particle swarm

optimization

INTRODUCTION

The number of people with limb loss in the United States is

estimated at about two million. Amputation has several main

causes, including accident, cancer, vascular disease, other

disease, birth defect, and paralysis [1]-[2]. Different types of

amputation include transtibial (below the knee), transfemoral

(above the knee), foot amputations, and hip and knee

disarticulations (amputation through the joint). Transfemoral

amputees can use prosthetic legs in an attempt to restore a normal

walking gait.

There are three types of prosthetic legs: passive (no

electronic control), active (motor control), and semi-active

(control without motors). Technology has provided advanced

prosthetic legs for amputees so that they can remain active and

so that they can emulate able-bodied gait. Compared to passive

and semi-active prostheses, active prostheses enable more

natural walking.

The Power Knee was the first commercially available active

transfemoral prosthesis. A combination knee / ankle prosthesis,

both joints of which are active, was recently developed at

Vanderbilt University and is in the process of commercialization.

Many researchers have recently concentrated on the design and

control of these and other active prostheses [3-7]. Recent years

have witnessed numerous advancements in the development of

control and modeling approaches for prosthetic legs [8-10].

A prosthesis can be viewed as a robotic system. A robot’s

environment and the robot itself can be viewed as a mechanical

admittance and impedance respectively. This motivates the

development of impedance control [11]. However, modeling

inaccuracies are unavoidable. Robust controllers can be used to

reduce these effects on the performance and stability of the

system [12-13]. Robust controllers try to achieve a certain level

of performance in the presence of modeling uncertainties,

whereas adaptive controllers try to achieve performance with

learning and adaptation. Adaptive controllers may be preferable

to non-adaptive robust controllers because adaptive controllers

can handle system uncertainties that change with time. Non-

adaptive robust controllers require a priori knowledge of the

bounds of the parameter perturbations, whereas adaptive

approaches do not.

The aforementioned advantages of adaptive control, along

with the availability of able-bodied human impedance properties

and uncertain model parameters, has given rise to impedance

model reference adaptive control for robotics [14-16]. Pure

adaptive control approaches may become unstable when

2 Copyright © 2015 by ASME

disturbances, unmodeled dynamics, or external forces affect the

system. Robust control can alleviate instabilities in these cases

[17-19]. Several adaptive control schemes and sliding surface

theories have also been proposed for robotics [20-23].

The contribution of this research is the design of a nonlinear

robust model reference adaptive impedance controller for a

prosthetic leg for transfemoral amputees. We use robust adaptive

control to deal with parameter uncertainties and GRF variations

so that the closed-loop system converges to a target impedance

model. Among related research [14-16], our work has the most

similarity to the controller presented in [15]. New contributions

in this paper include blending adaptive and robust control not

only to reduce the effects of unknown parameters on system

performance and stability, but also to obtain good robustness

against GRF variations (environment interaction). We define a

first-order sliding surface so the system trajectories reach the

sliding manifold 𝑠 = 0 in finite time, given a relevant reaching

condition. We design a control law comprising an adaptive

control term to account for uncertain parameters, and a robust

control term to account for the aforementioned reaching

condition and GRF variations. We extend previous work [15] by

defining a trajectory 𝑠∆ to balance control chattering and tracking

accuracy, and to bound the parameter adaptation. We define a

boundary layer to stop parameter adaptation when tracking

errors reach a satisfactory level. We then prove the stability of

the closed-loop system via Barbalat’s lemma by defining a

suitable Lyapunov function, which leads to a stable adaptation

law. Furthermore, we use PSO to optimize the control design

parameters. The PSO cost function includes control signal

magnitudes and tracking errors, and PSO reduces the cost

function by 22%. Numerical results show that the proposed

system has good robustness to system uncertainties. When we

change the system parameters by −30% and +30%, the total cost

increases by only 7.7% and 10% respectively, and the tracking

performance component of the cost increases by 10.7% and 39%

respectively.

The following section describes the dynamic model of the

prosthetic leg. In the next section we design the controller and

prove its stability. The next section presents simulation results

and robustness results. The final section includes discussion and

concluding remarks.

PROSTHETIC LEG MODEL

We present a model for the prosthetic leg with three rigid

links and three degrees of freedom. The prosthetic component is

modeled as an active transfemoral (above-knee) prosthesis. This

proposed model has a prismatic-revolute-revolute (PRR) joint

structure as illustrated in Fig. 1. The vertical degree of freedom

represents human hip motion, the first rotational axis represents

human angular thigh motion, and the second rotational axis

represents the prosthetic knee angular motion. Human hip and

thigh motion are emulated by a prosthesis test robot [9], [10],

[24].

Fig. 1. Prosthetic leg model with rigid ankle

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

follows [9]:

𝑀�̈� + 𝐶�̇� + 𝑔 + 𝑅 = 𝑢 − 𝑇𝑒 (1)

where 𝑞𝑇 = [𝑞1 𝑞2 𝑞3] is the vector of generalized joint

displacements (𝑞1is the vertical displacement, 𝑞2 is the thigh

angle, and 𝑞3 is the knee angle); 𝑀(𝑞), 𝐶(𝑞, �̇�), 𝑔(𝑞),

and𝑅(𝑞, �̇�) are the inertia matrix, Coriolis matrix, gravity vector,

and nonlinear damping vector respectively; 𝑇𝑒 is the effect of the

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

GRF; u is the control signal that comprises the active control

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

knee.

We use a treadmill as the walking surface of the prosthesis

test robot. We model the treadmill belt as a mechanical stiffness

so the reaction forces from the treadmill are a function of belt

deflection [10]. The effect 𝑇𝑒 of the GRF is given as follows [24]:

𝐿𝑧 = 𝑞1 + 𝑙2 sin(𝑞2) + 𝑙3 sin(𝑞2 + 𝑞3)

(2)

𝐹𝑧 = {0 , 𝐿𝑧 < 𝑠𝑧

𝑘𝑏(𝐿𝑧 − 𝑠𝑧) , 𝐿𝑧 > 𝑠𝑧

(3)

𝐹𝑥 = 𝛽𝐹𝑧 (4)

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)]

(5)

3 Copyright © 2015 by ASME

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

𝐿𝑧 is the vertical position of bottom of the foot in the world frame

(x0, y0, z0); 𝑠𝑧 is the treadmill standoff (vertical distance between

the origin of the world frame and the belt); 𝑘𝑏 is the belt stiffness;

and 𝛽 is a friction coefficient. See Fig. 1 for details. The states

and control inputs are defined as

𝑥𝑇 = [𝑞1 𝑞2 𝑞3 �̇�1 �̇�2 �̇�3]

𝑢𝑇 = [𝑓ℎ𝑖𝑝 𝜏𝑡ℎ𝑖𝑔ℎ 𝜏𝑘𝑛𝑒𝑒] (6)

We convert the left hand side of Eq. (1) into the following

parameterized form [25]-[26]:

𝑀�̈� + 𝐶�̇� + 𝑔 + 𝑅 = 𝑌ʹ(𝑞, �̇�, �̈�)𝑝ʹ (7)

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

of the joint displacements, velocities, and accelerations; n is the

number of rigid link and 2n is the dimension of the state-space

system (n is equal to 3 in our case; see Eq. (6)); and pʹ ∈ 𝑅𝑟is a

parameter vector.

ROBUST ADAPTIVE IMPEDANCE CONTROL

The main contribution of this research is the design of a

nonlinear robust adaptive impedance controller using a boundary

layer and a sliding surface to track hip displacement, knee and

thigh angles, and their velocities, in the presence of parameter

uncertainties. We desire the closed-loop system to imitate the

biomechanical propertiesof able-bodied walking and thus

provide near-normal gait for amputees. Therefore, we definea

target impedance model with characteristics that are similar to

those of able-bodied walking [26]:

𝑀𝑟(�̈�𝑟 − �̈�𝑑) + 𝐵𝑟(�̇�𝑟 − �̇�𝑑) + 𝐾𝑟(𝑞𝑟 − 𝑞𝑑) = −𝑇𝑒 (8)

where the desired mass 𝑀𝑟, the damping coefficient 𝐵𝑟 , and the

spring stiffness 𝐾𝑟 are the positive definite matrices of the target

model. For the sake of simplicity, we suppose these matrices are

diagonal:

𝑀𝑟 ∈ 𝑅𝑛⤫𝑛 = diag (𝑀11 𝑀22 … 𝑀𝑛𝑛)

𝐵𝑟 ∈ 𝑅𝑛⤫𝑛 = diag (𝐵11 𝐵22 … 𝐵𝑛𝑛)

𝐾𝑟 ∈ 𝑅𝑛⤫𝑛 = diag(𝐾11 𝐾22 … 𝐾𝑛𝑛)

where 𝑞𝑟 ∈ 𝑅𝑛 and 𝑞𝑑 ∈ 𝑅𝑛 are the state vectors of the reference

model and the desired trajectory respectively.

In the model presented in Eq. (7), the regressor matrix

depends on the joint position, velocity, and acceleration. In

practice the joint acceleration measurements can be very noisy,

so 𝑌ʹ(𝑞, �̇�, �̈�) might not be convenient for real time

implementation. Consequently, to avoid measuring the joint

accelerations, we define error and signal vectors 𝑠 and 𝑣

respectively, based on Slotine and Li’s approach [22], [23], [25],

[26]:

𝑠 = �̇� + 𝜆𝑒 (9)

𝑣 = �̇�𝑟 − 𝜆𝑒 (10)

𝑒 = 𝑞 − 𝑞𝑟 (11)

𝜆 = diag(𝜆1, 𝜆2, … , 𝜆𝑛) , 𝜆𝑖 > 0 (12)

In place of the regressor model of Eq. (7), we define an

acceleration-free regressor model as follows:

𝑀�̈� + 𝐶�̇� + 𝑔 + 𝑅 = 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝 (13)

𝑌(𝑞, �̇�, 𝑣, �̇�) is a linear combination of 𝑞, �̇�, 𝑣, and �̇�. One

realization of the regressor matrix 𝑌(𝑞, �̇�, 𝑣, �̇�) and the

associated parameter vector 𝑝 is given as follows:

𝑝 =

[

𝑚1 + 𝑚2 + 𝑚3

𝑚3𝑙2 + 𝑚2𝑙2 + 𝑚2𝑐2𝑚3𝑐3

𝐼2𝑧 + 𝐼3𝑧 + 𝑚2𝑐22 + 𝑚3𝑐3

2 + 𝑚2𝑙22 + 𝑚3𝑙2

2 + 2𝑚2𝑐2𝑙2𝑚3𝑐3𝑙2

𝑚3𝑐32 + 𝐼3𝑧

𝑏𝑓 ]

(14)

𝑌(𝑞, �̇�, 𝑣, �̇�) = [�̇�1 − 𝑔 𝑌12 𝑌13 0 0 0 0 sign(�̇�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 (15)

By substituting Eqs. (9), (10), (11) and (12) in Eq. (1), we rewrite

the model in the following form:

𝑀�̇� + 𝐶𝑠 + 𝑔 + 𝑅 + 𝑀�̇� + 𝐶𝑣 = 𝑢 − 𝑇𝑒 (16)

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

the error vector of Eq. (9) is derived from the following first-

order sliding surface:

𝑠 = (𝑑

𝑑𝑡+ 𝜆) 𝑒

(17)

where 𝑠 is n-element vector. Perfect tracking 𝑞 = 𝑞𝑟 (𝑒 = 0) is

equivalent to 𝑠 = 0. In order to reach the sliding manifold 𝑠 = 0

in finite time, the following reaching condition must be attained

[20]:

sgn(𝑠)�̇� ≤ −𝛾 (18)

𝛾 > 0

4 Copyright © 2015 by ASME

where the inequality is interpreted element-wise. From Eq. (18)

we see that in the worst case,sgn(𝑠)�̇� = −𝛾, so we can calculate

the worst-case reaching time to 𝑠 = 0 of the tracking error

trajectories as follows:

∫ sgn(𝑠)𝑑𝑠 = −𝛾0

𝑠(0)

∫ 𝑑𝑡𝑇

0

→ |𝑠(0)|sgn(𝑠) = 𝛾 𝑇

𝑇 =𝑠(0)

𝛾

(19)

where this component has n different reaching time and 𝑠(0) is

the initial error. It is seen from Eq. (19) that increasing 𝛾 results

in a smaller reaching time T.

Since the parameters of system are unknown, we use a

control law [19] to not only consider parameter uncertainties but

also to satisfy the reaching condition of Eq. (18):

𝑢 = �̂��̇� + �̂�𝑣 + �̂� + �̂� + �̂�𝑒 − 𝐾𝑑sgn(𝑠) (20)

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

respectively; 𝐾𝑑 is a robust control design coefficient with 𝐾𝑑 =

diag(𝐾𝑑1, 𝐾𝑑2

, … , 𝐾𝑑𝑛) , and𝐾𝑑𝑖

> 0. Since the function sgn(𝑠)

is discontinuous and causes control chattering, the saturation

function sat(𝑠/𝜑) (see Fig.2) promises to provide better

performance than the sign function. So we modify the control

law of Eq. (20) as follows:

𝑢 = �̂��̇� + �̂�𝑣 + �̂� + �̂� + �̂�𝑒 − 𝐾𝑑 sat(𝑠/𝜑) (21)

where 𝜑 is the width of the saturation function.

The control law of Eq. (21) comprises two different parts.

The first part, �̂��̇� + �̂�𝑣 + �̂� + �̂�, is an adaptive control term that

is responsible for handling the uncertain parameters. The second

part,�̂�𝑒 − 𝐾𝑑sat(𝑠/𝜑), is a robust control term that is responsible

for dealing with the condition of Eq. (18) and the variations of

the external inputs 𝑇𝑒. Substituting Eq. (21) into Eq. (16) and

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

and 𝑝 = 𝑝 − �̂�, we derive the closed-loop system as follows:

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

(22)

We can separate the right side of Eq. (22) into two different parts:

the regressor matrix 𝑌(𝑞, �̇�, 𝑣, �̇�) and the parameter estimation

error vector𝑝. Therefore, we can present Eq. (22) in the

following regressor (linear parametric) form:

𝑀�̇� + 𝐶𝑠 + 𝐾𝑑sat(𝑠/𝜑) + ( 𝑇𝑒 − �̂�𝑒) = −𝑌(𝑞, �̇�, 𝑣, �̇�)�̃� (23)

Next, to trade off control chattering and tracking accuracy, and

to create an adaptation dead zone to prevent unfavorable

parameter drift, we define a trajectory 𝑠∆ as follows [20]-[25]:

𝑠∆ = {0 , |𝑠| ≤ 𝜑

𝑠 − 𝜑 sat(𝑠/𝜑) , |𝑠| > 𝜑

(24)

where the region |𝑠| ≤ 𝜑 is the boundary layer; and 𝜑 is the

boundary layer thickness and the width of the saturation

function. We depict the trajectory 𝑠∆ and the function sat(𝑠/𝜑)

in Fig. 2.

Fig. 2. Saturation function and trajectory 𝒔∆

To drive a stable adaptation law based on the trajectory 𝑠∆,

we present a continuously differentiable scalar positive definite

Lyapunov function as follows [25]:

𝑉(𝑠∆, 𝑝) =1

2(𝑠∆

𝑇𝑀 𝑠∆) +1

2(𝑝𝑇𝜇 𝑝)

(25)

where 𝜇 is a design parameter such that 𝜇 =diag(𝜇1, 𝜇2, … , 𝜇𝑟) , with 𝜇𝑖 > 0. We find the derivative of the

Lyapunov function 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

Inside the boundary layer �̇�∆ = 0, and outside of it �̇�∆ = �̇�, so

�̇�(𝑠∆, 𝑝) = 𝑠∆𝑇(−𝐶𝑠 − 𝐾𝑑𝑠𝑎𝑡 (

𝑠

𝜑) + (�̂�𝑒 − 𝑇𝑒) −

𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝) +1

2(𝑠∆

𝑇�̇�𝑠∆) + �̇�𝑇𝜇 𝑝

=1

2𝑠∆

𝑇(�̇� − 2𝐶)𝑠∆ − 𝑠∆𝑇 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝 −

𝑠∆𝑇𝐾𝑑sat(𝑠/𝜑) + 𝑠∆

𝑇(�̂�𝑒 − 𝑇𝑒) + �̇�𝑇𝜇 𝑝

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

Also, 𝑠∆𝑇sat(𝑠/𝜑) is equal to |𝑠∆

𝑇|, so we can simplify the

derivative as follows:

�̇�(𝑠∆, 𝑝) = −𝐾𝑑|𝑠∆| + 𝑠∆𝑇(�̂�𝑒 − 𝑇𝑒) +

�̇�𝑇𝜇 𝑝 − 𝑠∆𝑇𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝

(26)

5 Copyright © 2015 by ASME

In order to ensure semi-negative definiteness for�̇�(𝑠∆, 𝑝),

we constrain the term �̇�𝑇𝜇 𝑝 − 𝑠∆𝑇𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝 to zero in

Eq. (26), which allows us to derive a stable update law as

follows:

�̇̂� = −𝜇−1𝑌𝑇(𝑞, �̇�, 𝑣, �̇�)𝑠∆ (27)

By defining 𝐾𝑑𝑖= 𝐹 + 𝛾 for some n-element vectors F and 𝛾,

which comes from the inequality |�̂�𝑒 − 𝑇𝑒| ≤ 𝐹 (that is, the

difference between �̂�𝑒 and 𝑇𝑒 is bounded element-wise) and

from the inequality sgn(𝑠)�̇� ≤ −𝛾, we obtain the final form of

�̇�(𝑠∆, 𝑝) as follows:

�̇�(𝑠∆, 𝑝) ≤ −𝛾|𝑠∆| ≤ 0 (28)

It is seen that the derivative of the proposed Lyapunov

function is negative semi-definite, so we utilize Barbalat’s

lemma [25] to prove the asymptotic stability of the closed-loop

system.

Barbalat’s lemma: If a candidate Lyapunov function 𝑽 =𝑽(𝒕, 𝒙) satisfies the following conditions:

I. 𝑽(𝒕, 𝒙) is lower-bounded.

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

III. �̈�(𝒕, 𝒙) is bounded (�̇�(𝒕, 𝒙)is uniformly

continuous)

then �̇�(𝑡, 𝑥) → 0 as 𝑡 → ∞, which means that the closed-loop

system is asymptotically stable.

Theorem1: The tracking errors defined in Eq. (11)

asymptotically converge to zero, which in turn results in

asymptotically perfect tracking (𝑞 → 𝑞𝑟).

Proof: Items I and II in Barbalat’s lemma can easily be

shown from Eqs. (25) and (26) respectively. We thus conclude

that V is bounded; therefore, all terms in V in Eq. (25), namely,

𝑠∆ and 𝑝, are bounded. Since𝑝 is constant, �̂� is bounded, and

since𝑠∆ is bounded, 𝑠 is bounded. �̈�(𝑠∆, 𝑝) ≤ −𝛾�̇� and in the

worst case we have

�̈�(𝑠∆, 𝑝) = −𝛾�̇�

= −𝛾𝑀−1(−𝐶𝑠 − 𝐾𝑑sat(𝑠/𝜑) +

(�̂�𝑒 − 𝑇𝑒) − 𝑌(𝑞, �̇�, 𝑣, �̇�)𝑝)

(29)

Since the model and controller parameters (𝑀,𝐶, 𝑌, 𝛾, and

𝐾𝑑) are bounded, |�̂�𝑒 − 𝑇𝑒| ≤ 𝐹 and 𝑝 and 𝑠 are bounded, we

conclude that �̈� is bounded. Consequently, since we confirm

premises I, II, and III in Barbalat’s lemma, we conclude that

�̇�(𝑠∆, 𝑝) → 0 as 𝑡 → ∞. This implies that −𝛾|𝑠∆| in Eq. (28)

converges to 0, and we easily conclude that the inequality of

Eq. (28) can be written as �̇�(𝑠∆, 𝑝) = −𝛾|𝑠∆|. �̇�(𝑠∆, 𝑝) →0⟾𝛾|𝑠∆| → 0⟾ 𝑠 → 0 ⟾𝑒 and �̇� → 0 and eventually 𝑞

converges to 𝑞𝑟 to attain perfect tracking. ■

By using the Lyapunov function of Eq. (25), and the

adaptation law of Eq. (27), and by considering 𝐾𝑑𝑖> 𝐹, the

proposed system is asymptotically stable and the controller is

robust to the effects of GRF. The robust model reference adaptive

impedance controller structure is summarized in Fig.3.

Fig. 3. Robust model reference adaptive impedance

controller structure

Fig. 3 shows that PSO tunes the optimal design parameters to

minimize tracking errors between the knee, thigh, and hip

trajectories, and the desired trajectories.

Theorem 1 proved that the errors of Eq. (11) between the

states of the target impedance model (�̇�𝑟𝑖 and 𝑞𝑟𝑖) and the states

of the system (�̇�𝑖 and 𝑞𝑖) converge to zero. PSO optimizes the

controller design parameters (𝜆, 𝜇, and 𝐾𝑑) so that the errors

between the states of the system and the desired trajectories (�̇�𝑑𝑖

and 𝑞𝑑𝑖) converge as quickly as possible to zero, while

preventing large magnitudes in the control signals. To achieve

these goals, we define a PSO cost function. First we define the

tracking error portion of the cost function, and the control portion

of the cost function, as follows:

𝑅𝑀𝑆𝐸𝑖 = √1

𝑇∫ (𝑥𝑖 − 𝑟𝑑𝑖

)2 𝑑𝑡𝑇

0

, 𝑖 = 1, … , 6

(30)

𝑅𝑀𝑆𝑈𝑗 = √1

𝑇∫ 𝑢𝑗

2 𝑑𝑡𝑇

0

, 𝑗 = 1, … ,3

(31)

where T is the time period (one stride), and x, r, and u are given

as follows:

6 Copyright © 2015 by ASME

𝑥𝑇 = [𝑞1 𝑞2 𝑞3 �̇�1 �̇�2 �̇�3] 𝑟𝑇 = [𝑟𝑑1

𝑟𝑑2𝑟𝑑3

𝑟𝑑4𝑟𝑑5

𝑟𝑑6] = [𝑞𝑑1

𝑞𝑑2𝑞𝑑3

�̇�𝑑1�̇�𝑑2

�̇�𝑑3] 𝑢𝑇 = [𝑓ℎ𝑖𝑝 𝜏𝑡ℎ𝑖𝑔ℎ 𝜏𝑘𝑛𝑒𝑒]

(32)

We then define the normalized cost components as follows:

𝐶𝑜𝑠𝑡𝐸𝑖 =𝑅𝑀𝑆𝐸𝑖

maxt∈[0,T]

|𝑥𝑖 − 𝑟𝑑𝑖| 𝐶𝑜𝑠𝑡𝑈𝑗 =

𝑅𝑀𝑆𝑈𝑗

maxt∈[0,T]

|𝑢𝑗|

(33)

The total tracking cost, total control cost, and total combined cost

are finally given as follows.

𝐶𝑜𝑠𝑡𝐸 = ∑𝐶𝑜𝑠𝑡𝐸𝑖 ,

6

𝑖=1

𝐶𝑜𝑠𝑡𝑈 = ∑𝐶𝑜𝑠𝑡𝑈𝑖

3

𝑗=1

𝐶𝑜𝑠𝑡 = 𝐶𝑜𝑠𝑡𝐸 + 𝐶𝑜𝑠𝑡𝑈

(34)

The Cost variable in the previous equation is the objective

function of the PSO algorithm.

SIMULATION RESULTS

The desired trajectory in this paper is walking data obtained

by the Motion Studies Laboratory (MSL) of the Cleveland

Department of Veterans Affairs Medical Center (VAMC). In this

section we show the effectiveness of the proposed controller of

Fig. 3 by performing simulation studies on the prosthesis robot

model.

In the system model considered here, we have 𝑞 ∈ 𝑅3,

so𝑀𝑟 = diag(𝑀11 𝑀22 𝑀33), 𝐵𝑟 = diag(𝐵11 𝐵22 𝐵33),

and 𝐾𝑟 = diag(𝐾11 𝐾22 𝐾33). To have two equal real roots

for each of joint displacements (critically damped responses for

the hip vertical displacement 𝑞1, and the thigh and knee angles

𝑞2and 𝑞3) in the target impedance model of Eq. (8), 𝐵𝑖𝑖must be

equal to 2√𝐾𝑖𝑖𝑀𝑖𝑖 and the two roots can be calculated

as−√𝐾𝑖𝑖/𝑀𝑖𝑖. To have two different real roots, 𝐵𝑖𝑖must be

greater than 2√𝐾𝑖𝑖𝑀𝑖𝑖. We define the target impedance model

with two real roots at−27and −72 for both the knee and thigh,

and two real roots at −52 and −947 for the hip displacement. We

choose these values heuristically so that the target impedance

model is stable, behaves similarly to an able-bodied leg, and

gives near-perfect tracking. This approach results in the

following impedance model matrices:

𝑀𝑟 = diag (10 10 10)

𝐾𝑟 = diag (500000 20000 20000)

𝐵𝑟 = diag (10000 1000 1000)

Particle Swarm Optimization

We use PSO to tune the controller and estimator parameters [27].

We use the following parameters for PSO: optimization problem

dimension=14, number of iterations=20, population size=72,

maximum rates for cognition and social learning=2.05, damping

ratio for inertia rate=0.9, and scale factor=0.1.

We consider the following values for the minimum and

maximum values of the search domain of 𝜇, 𝐾𝑑, and 𝜆:

𝜇𝑖 ∈ [0.001, 0.01] , 𝑖 = 1, … , 8

𝐾𝑑𝑖∈ [50, 150] , 𝑖 = 1, … , 3

𝜆𝑖 ∈ [50, 150] , 𝑖 = 1, … ,3

where we use 𝜇 in Eqs. (9) and (10) to build the signal and error

vectors, 𝐾𝑑 in Eq. (20) to design the controller, and 𝜆in Eq. (27)

to design the update law. After some trial and error, we find good

performance with a boundary layer thicknesses for the

trajectories 𝑠∆(shown in Fig.2) for all joint displacements as𝜑1 =𝜑2 = 𝜑3 = 0.5. The initial state of the system is given as

follows:

𝑥0𝑇 = [0.019 1.13 0.09 0.09 0 1.6].

PSO decreases the 𝐶𝑜𝑠𝑡 of Eq. (34) is from 3.33 at the initial

generation to 2.6 at the 20th generation, which means that the

total cost improves by 22%. Fig. 4 shows the best cost values

over the 20 generations. The best solution found by PSO is given

as follows:

The robust term coefficients: 𝐾𝑑 = diag(89, 117, 103)

The adaptation rates: 𝜇 = diag(0.013 0.010 0.011 0.015 0.015 0.014 0.019 0.01)

The sliding term coefficients: 𝜆 = diag(105, 121, 98)

Fig. 4. PSO on the proposed system to find designing

parameters of the proposed controller

Robustness to System Model Parameter Variations

We assume that the treadmill parameters are constant so that

𝑠𝑧 = 0.905 (meters), 𝑘𝑏 = 37000 (N/m), and 𝛽 = 0.2.

However, we assume that the other system parameters are

unknown to the controller and can vary ±30% from their nominal

0 5 10 15 202.55

2.7

2.85

3

3.15

3.3

3.45

3.6

Iteration

Best C

ost

7 Copyright © 2015 by ASME

values. We list the nominal values of the system parameters in

Table I.

Fig. 5 compares the states of the closed-loop system with the

desired trajectories (VAMC data) when values of the system

parameters are varied 30% and+30% relative to their nominal

values. Fig. 5 shows that the controller tracks the desired

trajectories not only with nominal parameter values, but also

when the parameter values vary 30% and +30% from their

nominal values. This demonstrates good robustness for the

control system.

Fig. 5. Tracking performance: desired trajectory (magenta

dotted line), nominal response (blue solid line), response with

+30% parameter deviations (black dash-dot line), and

response with 30% parameter deviations (red dashed line)

Fig. 6 shows the control signals of the system (the active control

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

knee) for the nominal parameter values, and also for the

maximum and minimum deviations of the parameters. We see

that the control magnitudes for the nominal case and the off-

nominal cases are similar. This shows that the controller

structure is suitable for dealing with parameter variations

without large changes in control effort.

Table I. 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 from 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

Fig. 6. Control signals: nominal system (blue solid line),

+30% parameter deviations (black dash-dot line) and −𝟑𝟎%

parameter deviations (red dashed line)

Fig. 7 depicts the horizontal and vertical forces for different

parameter values. We see that the generated forces appear similar

to able-bodied GRFs. Fig. 7 shows that when the system

parameters deviate from their nominal values by −30%, the mean

of the squares of both 𝐹𝑋 and 𝐹𝑧 increase by 7%. When the

system parameters deviate from their nominal values by +30%,

the mean of the squares of 𝐹𝑋 and 𝐹𝑧 decrease by 14%.

0 1 2 3 4-0.04

-0.02

0

0.02

0.04

time(s)

hip

dis

pla

cem

ent(

m)

0 1 2 3 4-0.4

-0.2

0

0.2

0.4

time(s)

hip

velo

city(m

/s)

0 1 2 3 40.5

1

1.5

2

time(s)

thig

h a

ngle

(ra

d)

0 1 2 3 4-4

-2

0

2

4

time(s)

thig

h a

ngula

r velo

city(r

ad/s

)

0 1 2 3 4-0.5

0

0.5

1

1.5

time(s)

knee a

ngle

(rad)

0 1 2 3 4-10

-5

0

5

10

time(s)

knee a

ngula

r ve

locity(r

ad

/s)

0 1 2 3 4

-500

0

500

time(s)

hip

fo

rce

(N)

0 1 2 3 4

-400

-300

-200

-100

0

100

time(s)

thig

h t

orq

ue(N

.m)

0 1 2 3 4

-200

-150

-100

-50

0

time(s)

knee t

orq

ue(N

.m)

8 Copyright © 2015 by ASME

Fig. 7. The horizontal and vertical forces (GRFs) for nominal

parameter values (blue solid line), +30% parameter

deviations (black dash-dot line) and 30% parameter

deviations (red dashed line). The heel strikes are shown in the

right subplot with circles on the x-axis

Fig.8 illustrates the tracking errors of the system state for

different values of the system parameters, and it is seen from the

figure that the tracking errors are well within desired bounds,

even for +30% and 30% parameter deviations.

Fig. 8. Tracking errors with nominal parameter values (blue

solid line), +30% parameter deviations (black dash-dot line),

and 30% parameter deviations (red dashed line)

Table II summarizes the root mean square tracking error of

each state, and the mean square value of each control signal

(𝑅𝑀𝑆𝐸𝑖 and 𝑅𝑀𝑆𝑈𝑗 respectively, as shown in Eqs. (30) and

(31)), the normalized sum of the two sets of cost values (𝐶𝑜𝑠𝑡𝐸

and 𝐶𝑜𝑠𝑡𝑈 from Eq. (33)), and the total cost (𝐶𝑜𝑠𝑡 from

Eq. (34)). Table II shows that there is not a large difference

in𝑅𝑀𝑆𝐸𝑖 (𝑖 = 1,… ,6) and 𝐶𝑜𝑠𝑡𝐸 whether the system

parameters are at their nominal values, or at ±30% deviations,

which shows that the system demonstrates good robustness to

parameter variations. Table II also shows that the total control

cost and the total cost does not change by large amounts whether

the system parameters are at their nominal values, or at ±30%

deviations.

If the system parameters deviate by −30% from their

nominal values, the total cost (𝐶𝑜𝑠𝑡) increases by 7.7% and the

tracking performance cost (𝐶𝑜𝑠𝑡𝐸) increases by 10.7%. If the

system parameters deviate by +30% from their nominal values,

𝐶𝑜𝑠𝑡 and 𝐶𝑜𝑠𝑡𝐸 increase by 10% and 39% respectively.

Table II. The root mean squares of the tracking errors

(𝑹𝑴𝑺𝑬𝒊) and control signals (𝑹𝑴𝑺𝑼𝒊), the total tracking

error (𝑪𝒐𝒔𝒕𝑬), the total control cost (𝑪𝒐𝒔𝒕𝑼), and the total

cost (𝑪𝒐𝒔𝒕) for different values of the system parameters Nominal value Min parameter

value (30%)

Max parameter

value (+30%)

𝑅𝑀𝑆𝐸1(𝑚) 0.0014 0.0015 0.0050

𝑅𝑀𝑆𝐸2(𝑟𝑎𝑑) 0.0078 0.0090 0.0070

𝑅𝑀𝑆𝐸3(𝑟𝑎𝑑) 0.0049 0.0053 0.0049

𝑅𝑀𝑆𝐸4(𝑚/𝑠) 0.0089 0.0101 0.0148

𝑅𝑀𝑆𝐸5(𝑟𝑎𝑑/𝑠) 0.0683 0.0760 0.0695

𝑅𝑀𝑆𝐸6(𝑟𝑎𝑑/𝑠) 0.0317 0.0335 0.0410

𝑅𝑀𝑆𝑈1(𝑁) 514 544 464

𝑅𝑀𝑆𝑈2(𝑁.𝑚) 170 180 146

𝑅𝑀𝑆𝑈3(𝑁.𝑚) 102 109 87

𝐶𝑜𝑠𝑡𝐸 1.12 1.24 1.56

𝐶𝑜𝑠𝑡𝑈 1.48 1.57 1.30

𝐶𝑜𝑠𝑡 2.60 2.80 2.86

Fig. 9 shows the trajectories of the estimated parameter

vector 𝑝of Eq. (14) for the system for the nominal value of the

system parameters, and also for the cases when the system

parameters vary by ±30%. As anticipated, the parameter

estimates do not perfectly match their true values, but exact

parameter matching is not our goal; our goal is that the

trajectories of 𝑠 remain inside the boundary layer after the

adaptation period as shown in Fig. 10.

Fig. 9 also shows the trajectories of 𝑠∆ as shown in Eq. (24).

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

parameter adaptation is only active when the trajectories of 𝑠are

outside the boundary layer, which results in nonzero values for

𝑠∆.

0 1 2 3 40

100

200

300

time(s)

horizonta

l fo

rce(N

)

0 1 2 3 40

500

1000

1500

time(s)ve

rtic

al fo

rce

(N)

0 1 2 3 4-10

-5

0

5x 10

-3

time(s)

hip

dis

pla

ce

me

nt(

m)

0 1 2 3 4-0.01

0

0.01

0.02

0.03

time(s)

thig

h a

ngle

(rad)

0 1 2 3 4-5

0

5

10

15x 10

-3

time(s)

knee a

ngle

(rad)

0 1 2 3 4-0.06

-0.04

-0.02

0

0.02

0.04

time(s)

hip

ve

locity(m

/s)

0 1 2 3 4-0.8

-0.6

-0.4

-0.2

0

0.2

time(s)

thig

h a

ngu

lar

ve

locity(r

ad

/s)

0 1 2 3 4-0.2

-0.1

0

0.1

0.2

time(s)

knee a

ngula

r velo

city(r

ad/s

)

9 Copyright © 2015 by ASME

For instance, among the 𝑠trajectories with minimum

deviations of the system parameters in Fig. 10 (dashed lines), the

trajectory 𝑠 for the hip is the only trajectory that exceeds the

boundary layer (the region between the lines −𝜑1 and +𝜑1),

which occurs in 𝑡 ∈ [0.136, 0.326]. The other trajectories 𝑠 remain in the boundary layer. So the

𝑠∆trajectory for the hip is equal to 𝑠ℎ𝑖𝑝−𝜑1as shown in Eq. (24),

while the other s∆ trajectories are zero. In this case the adaptation

law of Eq. (27) is a function of only 𝜇𝑖 and the 𝑠∆ trajectory for

the hip.

Fig. 9. Trajectories of the estimated parameter vector with

nominal parameter values (blue solid line), +30% parameter

deviations (black dash-dot line), and 30% parameter

deviations (red dashed line). The true values are shown with

magenta dotted lines

Fig. 10. Trajectories of 𝒔∆ and 𝒔 with nominal parameter

values (blue solid line), +30% parameter deviations (black

dash-dot line), and 30% parameter deviations (red dashed

line)

CONCLUSIONS AND FUTURE WORK

We designed a regressor-based nonlinear robust model

reference adaptive impedance controller for a prosthesis robot

model. We first defined a target impedance model with two real

poles for each degree of freedom. We then designed a robust

model reference adaptive impedance controller, not only for

estimating the uncertain parameters of the system, but also for

driving the system trajectories to a sliding manifold while

compensating for the variations of GRF. We then proved the

stability of the closed-loop system with Lyapunov theory and

Barbalat’s lemma. We used PSO to find the optimal control and

estimator design parameters to minimize tracking error and

control signal magnitude. PSO decreased the cost function by

22%.

We performed simulations with ±30% parameter deviations,

and we saw that tracking performance was accurate. Tracking

errors for the nominal values of the parameter vector were 1.4

mm for the hip vertical displacement, 0.45 deg for the thigh

angle, and 0.28 deg for the knee angle). We achieved fast

transient responses with nominal parameter values and also with

parameter deviations as large as ±30%.

0 1 2 3 430

40

50

60

70

p1(k

g)

time(s)0 1 2 3 4

2

4

6

8

10

p2

(kg

-m)

time(s)

0 1 2 3 4-2

0

2

4

6

p3(k

g-m

)

time(s)0 1 2 3 4

0

5

10

15

p4(k

g-m

2)

time(s)

0 1 2 3 4-1

-0.5

0

0.5

1

p5(k

g-m

2)

time(s) 0 1 2 3 40

1

2

3

p6(k

g-m

2)

time(s)

0 1 2 3 46

8

10

12

14

p7

(N-m

-s)

time(s)0 1 2 3 4

40

60

80

100

120

p8(N

)

time(s)

0 1 2 3 4-1

-0.5

0

0.5

1

time(s)

s f

or

hip

0 1 2 3 4-1

-0.5

0

0.5

1

time(s)

s f

or

thig

h

0 1 2 3 4-1

-0.5

0

0.5

1

time(s)

s f

or

kn

ee

0 1 2 3 4-0.05

0

0.05

0.1

0.15

time(s)

s f

or

hip

0 1 2 3 4-0.06

-0.04

-0.02

0

0.02

time(s)

s f

or

thig

h

0 1 2 3 40

0.01

0.02

0.03

0.04

time(s)

s f

or

kn

ee

10 Copyright © 2015 by ASME

With ±30 % parameter variations, the total cost increased by

10% which demonstrates good robustness. Although the

parameter estimates did not converge to their true values, the

trajectories of 𝑠 remained inside their boundary layers after the

adaptation period, which resulted in good tracking performance.

For future work, we will consider other important aspects of

the proposed controller, including the following: the effect of the

boundary layer thickness on system performance; the robustness

of the system for variations of the effect of GRF; alternative 𝑠∆

trajectoriesand adaptation laws to improve system performance

and parameter estimation accuracy; reduction of the control

signal magnitudes; the effect of control signal saturation; and

reduction of GRF and its tradeoff with tracking accuracy. We

will also add the rotary and linear actuator models to the system

to obtain the required voltages for driving DC motors. We will

also implement the proposed method on a prosthetic leg

prototype in the Control, Robotics and Mechatronics Lab at

Cleveland State University. We will extend the controller to a 4-

DOF model that includes an active ankle joint. Finally, we will

use muli-objective optimization to achieve better tradeoffs of the

tracking error costs and the control signal magnitudes.

ACKNOWLEDGMENTS

This research was supported by NSF Grant 1344954. The

authors express their sincere gratitude to Jean-Jacques Slotine,

Antonie van den Bogert and Elizabeth C. Hardin for their

guidance that has improved the quality of this paper.

REFERENCES

[1] Ziegler-Graham, K., 2008, “Estimating the prevalence of limb loss in the United States: 2005 to 2050,” Archives of Physical Medicine and

Rehabilitation, vol. 89, no. 3, pp. 422–429.

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