ROBUST ADAPTIVE IMPEDANCE CONTROL OF A PROSTHETIC LEG

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1 Proceedings of ASME 25 DSCC ASME Dynamic Systems and Control Conference October 28-3, 25, 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 v.azimi@csuohio.edu Dan Simon Department of Electrical and Computer Engineering, Cleveland State University, Cleveland, OH, USA d.j.simon@csuohio.edu Hanz Richter Department of Mechanical Engineering, Cleveland State University, Cleveland, OH, USA 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 ±3% on the system parameters results in an increase of the cost function by only %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 []-[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-]. 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 []. However, modeling inaccuracies are unavoidable. Robust controllers can be used to reduce these effects on the performance and stability of the system [2-3]. 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. Nonadaptive 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 [4-6]. Pure adaptive control approaches may become unstable when Copyright 25 by ASME

2 disturbances, unmodeled dynamics, or external forces affect the system. Robust control can alleviate instabilities in these cases [7-9]. Several adaptive control schemes and sliding surface theories have also been proposed for robotics [2-23]. 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 [4-6], our work has the most similarity to the controller presented in [5]. 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 = 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 [5] 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 3% and +3%, the total cost increases by only 7.7% and % respectively, and the tracking performance component of the cost increases by.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.. 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], [], [24]. Fig.. Prosthetic leg model with rigid ankle The three degree-of-freedom model can be written as follows [9]: Mq + Cq + g + R = u T e () where q T = [q q 2 q 3] is the vector of generalized joint displacements (q is the vertical displacement, q 2 is the thigh angle, and q 3 is the knee angle); M(q), C(q, q ), g(q), andr(q, q ) are the inertia matrix, Coriolis matrix, gravity vector, and nonlinear damping vector respectively; T e is the effect of the combined horizontal (F x ) and vertical (F z ) 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 []. The effect T e of the GRF is given as follows [24]: L z = q + l 2 sin(q 2 ) + l 3 sin(q 2 + q 3 ) (2) F z = {, L z < s z k b (L z s z ), L z > s z (3) F x = βf z (4) T e = [ F z (l 2 cos(q 2 ) + l 3 cos(q 2 + q 3 )) F x (l 2 sin(q 2 ) + l 3 sin(q 2 + q 3 ))] F z (l 3 cos(q 2 + q 3 )) F x (l 3 sin(q 2 + q 3 ) (5) F z 2 Copyright 25 by ASME

3 where l 2 and l 3 are the length of the thigh and shank respectively; L z is the vertical position of bottom of the foot in the world frame (x, y, z ); s z is the treadmill standoff (vertical distance between the origin of the world frame and the belt); k b is the belt stiffness; and β is a friction coefficient. See Fig. for details. The states and control inputs are defined as x T = [q q 2 q 3 q q 2 q 3] u T = [f hip τ thigh τ knee ] (6) We convert the left hand side of Eq. () into the following parameterized form [25]-[26]: Mq + Cq + g + R = Yʹ(q, q, q )pʹ (7) where Yʹ(q, q, q ) R n 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 R r is 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]: M r (q r q d) + B r (q r q d) + K r (q r q d ) = T e (8) where the desired mass M r, the damping coefficient B r, and the spring stiffness K r are the positive definite matrices of the target model. For the sake of simplicity, we suppose these matrices are diagonal: M r R n n = diag (M M 22 M nn ) B r R n n = diag (B B 22 B nn ) K r R n n = diag(k K 22 K nn ) where q r R n and q d R n 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, q, 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 = e + λe (9) v = q r λe () e = q q r () λ = diag(λ, λ 2,, λ n ), λ i > (2) In place of the regressor model of Eq. (7), we define an acceleration-free regressor model as follows: Mq + Cq + g + R = Y(q, q, v, v )p (3) Y(q, q, v, v ) is a linear combination of q, q, v, and v. One realization of the regressor matrix Y(q, q, v, v ) and the associated parameter vector p is given as follows: p = [ m + m 2 + m 3 m 3 l 2 + m 2 l 2 + m 2 c 2 m 3 c 3 I 2z + I 3z + m 2 c m 3 c m 2 l m 3 l m 2 c 2 l 2 m 3 c 3 l 2 m 3 c I 3z b f ] (4) v g Y 2 Y 3 sign(q ) Y(q, q, v, v ) = [ Y 22 Y 23 v 2 Y 25 v 3 q 2 ] Y 33 Y 35 v 2+v 3 Y 2 = v 2cos (q 2 ) v 2 q 2sin (q 2 ) Y 3 = (v 2 + v 3)cos (q 3 + q 2 ) (v 2 q 3+v 2 q 2+v 3 q 2+v 3 q 3)sin (q 3 + q 2 ) Y 22 = (v g)cos (q 2 ) Y 23 = Y 33 = (v g) cos(q 3 + q 2 ) Y 25 = (2v 2 + v 3)cos (q 3 ) (v 2 q 3+v 3 q 3+v 3 q 2)sin (q 3 ) Y 35 = v 2 cos(q 3 ) + sin(q 3 ) v 2 q 2 (5) By substituting Eqs. (9), (), () and (2) in Eq. (), we rewrite the model in the following form: Ms + Cs + g + R + Mv + Cv = u T e (6) Since the system of Eq. () is a second-order dynamic system, the error vector of Eq. (9) is derived from the following firstorder sliding surface: s = ( d dt + λ) e (7) where s is n-element vector. Perfect tracking q = q r (e = ) is equivalent to s =. In order to reach the sliding manifold s = in finite time, the following reaching condition must be attained [2]: sgn(s)s γ (8) γ > 3 Copyright 25 by ASME

4 where the inequality is interpreted element-wise. From Eq. (8) we see that in the worst case,sgn(s)s = γ, so we can calculate the worst-case reaching time to s = of the tracking error trajectories as follows: s() sgn(s)ds = γ T dt s() sgn(s) = γ T T = s() γ (9), s φ s = { 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. where this component has n different reaching time and s() is the initial error. It is seen from Eq. (9) that increasing γ results in a smaller reaching time T. Since the parameters of system are unknown, we use a control law [9] to not only consider parameter uncertainties but also to satisfy the reaching condition of Eq. (8): u = M v + C v + g + R + T e K d sgn(s) (2) where M, C, g, R and T e are estimates of M, C, g, R, and T e respectively; K d is a robust control design coefficient with K d = diag(k d, K d2,, K dn ), andk >. Since the function sgn(s) di 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. (2) as follows: 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]: u = M v + C v + g + R + T e K d sat(s/φ) (2) V(s, p ) = 2 (s T M s ) + 2 (p Tμ p ) (25) where φ is the width of the saturation function. The control law of Eq. (2) comprises two different parts. The first part, M v + C v + g + R, is an adaptive control term that is responsible for handling the uncertain parameters. The second part,t e K d sat(s/φ), is a robust control term that is responsible for dealing with the condition of Eq. (8) and the variations of the external inputs T e. Substituting Eq. (2) into Eq. (6) and defining M = M M, C = C C, g = g g, R = R R, and p = p p, we derive the closed-loop system as follows: Ms + Cs + K d sat(s/φ) + ( T e T e) = (M v + C v + g + R ) (22) We can separate the right side of Eq. (22) into two different parts: the regressor matrix Y(q, q, v, v ) and the parameter estimation error vectorp. Therefore, we can present Eq. (22) in the following regressor (linear parametric) form: Ms + Cs + K d sat(s/φ) + ( T e T e) = Y(q, q, v, v )p (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 [2]-[25]: where μ is a design parameter such that μ = diag(μ, μ 2,, μ r ), with μ i >. We find the derivative of the Lyapunov function as follows: V (s, p ) = 2 (s T M s + s T M s ) + 2 (s T M s ) + 2 (p T μ p + p Tμ p ) = s T M s + 2 (s T M s ) + p T μ p Inside the boundary layer s =, and outside of it s = s, so V (s, p ) = s T ( Cs Kd sat ( s φ ) + (T e T e ) Y(q, q, v, v )p ) + 2 (s T M s ) + p T μ p = 2 s T (M 2C)s s T Y(q, q, v, v )p s T K d sat(s/φ) + s T (T e T e ) + p T μ p Matrix M 2C is skew-symmetric, so s T (M 2C)s =. Also, s T sat(s/φ) is equal to s T, so we can simplify the derivative as follows: V (s, p ) = K d s + s T (T e T e ) + p T μ p s T Y(q, q, v, v )p (26) 4 Copyright 25 by ASME

5 In order to ensure semi-negative definiteness forv (s, p ), we constrain the term p T μ p s T Y(q, q, v, v )p to zero in Eq. (26), which allows us to derive a stable update law as follows: p = μ Y T (q, q, v, v )s (27) By using the Lyapunov function of Eq. (25), and the adaptation law of Eq. (27), and by considering K d > F, the i 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. By defining K d = F + γ for some n-element vectors F and γ, i which comes from the inequality T e T e F (that is, the difference between T e and T e is bounded element-wise) and from the inequality sgn(s)s γ, we obtain the final form of V (s, p ) as follows: V (s, p ) γ s (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. V (t, x) is negative semi-definite. III. V (t, x) is bounded (V (t, x)is uniformly continuous) then V (t, x) as t, which means that the closed-loop system is asymptotically stable. Theorem: The tracking errors defined in Eq. () asymptotically converge to zero, which in turn results in asymptotically perfect tracking (q q r ). 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, p is bounded, and sinces is bounded, s is bounded. V (s, p ) γs and in the worst case we have V (s, p ) = γs = γm ( Cs K d sat(s/φ) + (T e T e ) Y(q, q, v, v )p ) (29) Since the model and controller parameters (M, C, Y, γ, and K d ) are bounded, T e T e F and p and s are bounded, we conclude that V is bounded. Consequently, since we confirm premises I, II, and III in Barbalat s lemma, we conclude that V (s, p ) as t. This implies that γ s in Eq. (28) converges to, and we easily conclude that the inequality of Eq. (28) can be written as V (s, p ) = γ s. V (s, p ) γ s s e and e and eventually q converges to q r to attain perfect tracking. 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 proved that the errors of Eq. () between the states of the target impedance model (q ri and q ri ) and the states of the system (q i and q i ) converge to zero. PSO optimizes the controller design parameters (λ, μ, and K d ) so that the errors between the states of the system and the desired trajectories (q di and q di ) 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: RMSE i = T T (x i r d i )2 dt RMSU j = T T u j 2 dt, i =,, 6 (3), j =,,3 (3) where T is the time period (one stride), and x, r, and u are given as follows: 5 Copyright 25 by ASME

6 x T = [q q 2 q 3 q q 2 q 3] r T = [r d r d 2 r d 3 r d 4 r d 5 r d 6] = [q d q d 2 q d 3 q d q d2 q d3 ] u T = [f hip τ thigh τ knee ] (32) We then define the normalized cost components as follows: Cost Ei = RMSE i RMSU j max x i r d t [,T] i Cost Uj = max u j t [,T] (33) The total tracking cost, total control cost, and total combined cost are finally given as follows. 6 Cost E = Cost Ei, i= 3 Cost U = Cost Ui j= Cost = Cost E + Cost U (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 R 3, som r = diag(m M 22 M 33 ), B r = diag(b B 22 B 33 ), and K r = diag(k K 22 K 33 ). To have two equal real roots for each of joint displacements (critically damped responses for the hip vertical displacement q, and the thigh and knee angles q 2 and q 3 ) in the target impedance model of Eq. (8), B ii must be equal to 2 K ii M ii and the two roots can be calculated as K ii /M ii. To have two different real roots, B ii must be greater than 2 K ii M ii. 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: M r = diag ( ) K r = diag (5 2 2) B r = diag ( ) Particle Swarm Optimization We use PSO to tune the controller and estimator parameters [27]. We use the following parameters for PSO: optimization problem dimension=4, number of iterations=2, population size=72, maximum rates for cognition and social learning=2.5, damping ratio for inertia rate=.9, and scale factor=.. We consider the following values for the minimum and maximum values of the search domain of μ, K d, and λ: μ i [.,.], i =,, 8 K di [5, 5], i =,, 3 λ i [5, 5], i =,,3 where we use μ in Eqs. (9) and () to build the signal and error vectors, K d in Eq. (2) 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 asφ = φ 2 = φ 3 =.5. The initial state of the system is given as follows: x T = [ ]. PSO decreases the Cost of Eq. (34) is from 3.33 at the initial generation to 2.6 at the 2th generation, which means that the total cost improves by 22%. Fig. 4 shows the best cost values over the 2 generations. The best solution found by PSO is given as follows: The robust term coefficients: K d = diag(89, 7, 3) The adaptation rates: μ = diag( ) The sliding term coefficients: λ = diag(5, 2, 98) Best Cost Iteration 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 s z =.95 (meters), k b = 37 (N/m), and β =.2. However, we assume that the other system parameters are unknown to the controller and can vary ±3% from their nominal 6 Copyright 25 by ASME

7 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 3% and+3% 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 3% and +3% from their nominal values. This demonstrates good robustness for the control system. hip displacement(m) thigh angle(rad) knee angle(rad) Fig. 5. Tracking performance: desired trajectory (magenta dotted line), nominal response (blue solid line), response with +3% parameter deviations (black dash-dot line), and response with 3% 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 offnominal cases are similar. This shows that the controller structure is suitable for dealing with parameter variations without large changes in control effort. hip velocity(m/s) thigh angular velocity(rad/s) knee angular velocity(rad/s) Table I. Nominal values of model parameters Parameter Description Nominal Units Value m Mass of link kg m 2 Mass of link kg m 3 Mass of link kg l 2 thigh length.425 m l 3 Length from knee joint.527 m to bottom of shoe c 2 Center of mass on thigh.9 m c 3 Center of mass on shank.32 m f sliding friction in link N b Rotary actuator damping 9.75 N-m-s I 2z Rotary inertia of link 2.38 kg-m^2 I 3z Rotary inertia of link 3.68 kg-m^2 g acceleration of gravity 9.8 m/s^2 hip force(n) thigh torque(n.m) knee torque(n.m) Fig. 6. Control signals: nominal system (blue solid line), +3% parameter deviations (black dash-dot line) and 3% 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 3%, the mean of the squares of both F X and F z increase by 7%. When the system parameters deviate from their nominal values by +3%, the mean of the squares of F X and F z decrease by 4%. 7 Copyright 25 by ASME

8 horizontal force(n) Fig. 7. The horizontal and vertical forces (GRFs) for nominal parameter values (blue solid line), +3% parameter deviations (black dash-dot line) and 3% 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 +3% and 3% parameter deviations. hip displacement(m) knee angle(rad) thigh angular velocity(rad/s) 5 x x Fig. 8. Tracking errors with nominal parameter values (blue solid line), +3% parameter deviations (black dash-dot line), and 3% parameter deviations (red dashed line) vertical force(n) thigh angle(rad) hip velocity(m/s) knee angular velocity(rad/s) Table II summarizes the root mean square tracking error of each state, and the mean square value of each control signal (RMSE i and RMSU j respectively, as shown in Eqs. (3) and (3)), the normalized sum of the two sets of cost values (Cost E and Cost U from Eq. (33)), and the total cost (Cost from Eq. (34)). Table II shows that there is not a large difference inrmse i (i =,,6) and Cost E whether the system parameters are at their nominal values, or at ±3% 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 ±3% deviations. If the system parameters deviate by 3% from their nominal values, the total cost (Cost) increases by 7.7% and the tracking performance cost (Cost E ) increases by.7%. If the system parameters deviate by +3% from their nominal values, Cost and Cost E increase by % and 39% respectively. Table II. The root mean squares of the tracking errors (RMSE i ) and control signals (RMSU i ), the total tracking error (Cost E ), the total control cost (Cost U ), and the total cost (Cost) for different values of the system parameters Nominal value Min parameter value ( 3%) Max parameter value (+3%) RMSE (m) RMSE 2 (rad) RMSE 3 (rad) RMSE 4 (m/s) RMSE 5 (rad/s) RMSE 6 (rad/s) RMSU (N) RMSU 2 (N. m) RMSU 3 (N. m) Cost E Cost U Cost Fig. 9 shows the trajectories of the estimated parameter vector pof Eq. (4) for the system for the nominal value of the system parameters, and also for the cases when the system parameters vary by ±3%. 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.. Fig. 9 also shows the trajectories of s as shown in Eq. (24). Based on the values of φ, φ 2,φ 3, 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. 8 Copyright 25 by ASME

9 For instance, among the strajectories with minimum deviations of the system parameters in Fig. (dashed lines), the trajectory s for the hip is the only trajectory that exceeds the boundary layer (the region between the lines φ and +φ ), which occurs in t [.36,.326]. The other trajectories s remain in the boundary layer. So the s trajectory for the hip is equal to s hip φ as 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 μ i and the s trajectory for the hip. s for hip s for thigh s for knee -.5 s for hip.5 p(kg) p2(kg-m) s for thigh s for knee p3(kg-m) p5(kg-m 2 ) p7(n-m-s) Fig. 9. Trajectories of the estimated parameter vector with nominal parameter values (blue solid line), +3% parameter deviations (black dash-dot line), and 3% parameter deviations (red dashed line). The true values are shown with magenta dotted lines p4(kg-m 2 ) p8(n) p6(kg-m 2 ) Fig.. Trajectories of s and s with nominal parameter values (blue solid line), +3% parameter deviations (black dash-dot line), and 3% 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 ±3% parameter deviations, and we saw that tracking performance was accurate. Tracking errors for the nominal values of the parameter vector were.4 mm for the hip vertical displacement,.45 deg for the thigh angle, and.28 deg for the knee angle). We achieved fast transient responses with nominal parameter values and also with parameter deviations as large as ±3%. 9 Copyright 25 by ASME

10 With ±3 % parameter variations, the total cost increased by % 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 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 [] Ziegler-Graham, K., 28, Estimating the prevalence of limb loss in the United States: 25 to 25, Archives of Physical Medicine and Rehabilitation, vol. 89, no. 3, pp [2] Robbins, J. 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