Switched Robust Tracking/Impedance Control for an Active Transfemoral Prosthesis*

Transcription

1 Switched Robust Tracking/Impedance Control for an Active Transfemoral Prosthesis* Holly Warner 1, Dan Simon, Hadis Mohammadi, and Hanz Richter Abstract A novel control method for an active transfemoral prosthesis is developed based on the combination of two previously published controllers, robust tracking/impedance control and switched impedance control. This controller is simulated and optimized to provide desired gait features and energy regeneration. Specifically, an evolutionary optimization algorithm termed biogeography-based optimization is used. Across all optimization trials a tracking error on the order of 1 2 rad rms or less results for both the knee and ankle joints. This is accompanied by ground reaction forces approximating able-bodied gait data and an energy regeneration capacity of about 9.6 J per stride. I. INTRODUCTION Above-knee amputees are frequently equipped with passive prostheses. Though microprocessor controlled knees like the C-leg offer improved gait compared to mechanical solutions like the Mauch SNS, stance phase knee flexion and push-off ankle plantarflexion remain elusive, often leading to health issues [1]. For example, amputees have a significantly increased probability of suffering from osteoarthritis, osteoporosis, and back problems compared to able-bodied individuals [2], [3], [4]. Furthermore, above-knee amputees use up to 5% more energy for ambulation than able-bodied persons [5]. This is partly because damping and spring elements compose passive prostheses; unlike the natural leg no energy can be added to the system. One solution to these shortcomings is to design a prosthesis that provides motor actuation at both the knee and ankle joints. Commercialized active prostheses, the Power Knee and iwalk BiOM ankle, address a single joint in the suggested manner but lose the benefit of a unified control approach [6]. Alternatively, an active prosthesis prototype at Vanderbilt University addresses both joints [7]. However, all are limited by battery life [6], [7]. The natural leg possesses some excess energy at the knee that could be harvested to power the energy consuming ankle, a fact unaccounted for in current powered devices [8]. Consequently, a controller combining accurate kinematics and kinetics for both the knee and ankle with energy regeneration is sought in this research. The remainder of the paper is organized as follows. The dynamic model used for this work is conveyed in Section II. In Sections III-A and III-B precedent research will be covered, namely robust tracking/impedance control and switched impedance control. A method of combining the two will then *This work was supported by National Science Foundation grant The authors are with Cleveland State University, Cleveland, OH, United States. 1 Corresponding author: h.warner@vikes.csuohio.edu be discussed in Section III-C. Next, Section IV will present the details of the optimization. This will be followed by results in Section V. A discussion, Section VI, will conclude. II. HIP ROBOT AND PROSTHESIS DYNAMIC MODEL A planar robotic hip for prosthesis testing has been developed at Cleveland State University. The robot includes a vertical joint and rotary joint. A prosthesis can then be attached via a standard pyramid connector. A dynamic model of this robotic hip and a prosthesis was obtained for studying its control in simulation. This model is developed in detail up to the ankle joint [9]. Several modifications and the addition of a foot model have been made since then [1]. The version used in this work is described fully in [11]. The complete system was derived in the robotics framework. M(q) q + C(q, q) q + G(q) + R( q) + T e = u (1) q is a four-element vector. Its variables correspond to the vertical hip displacement, hip flexion, knee flexion, and ankle plantarflexion degrees of freedom. The positive sense for the hip displacement is defined as downward relative to a horizontal threshold at the hip. T e is the external force effects equal to the sum of the forces or torques applied by the heel and toe ground reaction forces reflected to the various joints. The linearity in parameters property holds true for (1), and it can therefore be written as follows: Y (q, q, q)θ = u T e, (2) where Y is the regressor and Θ is a vector of the parameters. The inertia matrix M, Coriolis matrix C, gravity vector G, and friction and damping vector R are shown in terms of Θ in [11]. No actuator dynamics are included in this model. III. CONTROLLER DEVELOPMENT A. Robust Tracking/Impedance Control Overview Robust tracking/impedance control is a combination of two control methods, robust passivity-based control and impedance control. Robust passivity-based control is a motion control strategy that takes advantage of the passivity property of the robotic equations and is capable of handling parameter uncertainties, which is particularly important in real-world implementation [12]. Impedance control does not independently control force or motion, but rather produces stable motion in response to arbitrary external forces [13], [14], [15]. A controller combining these two approaches is desirable for the system of interest because the upper two joints, hip vertical displacement and hip rotation, are required

2 to follow set trajectories, pure motion control, while the prosthesis, knee and ankle joints, should be more flexible while still following reference data. In this way the natural motion of a human hip is enforced regardless of the force and torque requirements on the hip robot. Further, the prosthesis can be tuned to act and react like a human leg even though it is not part of the natural system. This divides the system s joints into motion controlled (MC) joints q 1 and q 2 and impedance controlled (IC) joints q 3 and q 4 [1]. The general control method, particularly the way in which these two control strategies are interrelated, will be presented next. A joint space impedance controller was selected over a task space impedance controller partially because the latter introduces unnecessary complexity and further because the results of this work using the joint space form showed that it could be tuned to effectively meet the aforementioned objectives. The desired impedance is described as follows: I q IC + b q IC + k q IC = T IC. (3) q is the tracking error and T IC is the external force and moment effects. Diagonal matrices I, b, and k are the desired impedance values. The control is then defined in the form utilized in robust passivity-based control. u = ˆM(q)a + Ĉ(q, q)v + ĝ Kr + T IC (4) K is a diagonal matrix of four gains. The first two values of K are for MC joints, and the last two values are for IC joints. This can also be expressed as in the form indicating linearity in the parameters. ˆΘ will be discussed later. u = Y (q, q, v, a) ˆΘ Kr + T IC (5) According to robust passivity-based control, v, a, and r are defined as follows: v MC = q d MC Λ MC q MC (6) a MC = v = q d MC Λ MC q MC (7) r MC = q v = q MC + Λ MC q MC. (8) For the impedance controlled joints v, a, and r include an extra term. v IC = q d IC Λ IC q IC F r z (9) a IC = v = q d IC Λ IC q IC F r ż (1) r IC = q v = q IC + Λ IC q IC + F r z (11) Λ is a diagonal matrix composed of four elements, as seen in the motion control set of equations for v, a, and r and the impedance control set of equations for v, a, and r. It is divided into two diagonal matrices, Λ MC and Λ IC, composed of two elements relative to each set of v, a, and r. The term added to v, a, and r is the product of a gain matrix F r and a dynamic compensator. The dynamic compensator is a state equation with the state z. ż = Az + K p q IC + K d q IC + K f T IC (12) F r, K p, and K d can be calculated from the selected impedance gains. Justification for these values is provided in [1]. F r = I 1 (13) K p = k + AIΛ IC (14) K d = b IΛ IC + AI (15) At this point ˆΘ can be determined according to the robust passivity-based control framework. ˆΘ = Θ + δθ (16) δθ is a switching term that is determined by Lyapunov techniques, detailed in [1]. This controller is prone to chattering, and therefore, a deadzone solution was implemented rather than the original version [1]. ρ Y T r δθ = Y T r, Y T r > ɛ ρ ɛ Y T r, Y T r (17) ɛ In applying this controller to the hip robot and prosthesis model there are a total of 14 gains to be tuned. There are two values of K and two values of Λ for the motion control joints, two values of K and two values of Λ for the impedance control joints, and lastly, the three impedance values, I, b, and k, for each of the two lower joints. Ten of these gains are associated with the prosthetic joints. All were originally tuned by trial and error [1]. B. Switched Impedance Control Overview An alternative control method for prostheses, also based on impedance control, was developed by Sup and co-workers [7], [16]. In this case the control equation is simpler, but the gains are switched according to a finite state machine based on measurable features of the gait cycle. The controller was implemented for both the knee and ankle joints and tested with an active prosthesis prototype. Consisting of a spring term, a damper term, and an equilibrium angle, the impedance controller is defined as τ i = k i (θ θ ki ) + b i θ. (18) i is the index of the current state as determined by a finite state machine. As opposed to the model presented in Section III-A, this model does not include an inertia term. The finite state machine includes a total of five modes. These modes are early stance, middle stance, late stance, swing flexion, and swing extension, phases through 4. Contact of the ball of the foot, indicated by a contact load greater than a defined threshold, signals the transition from early to middle stance. The switch to late stance is associated with the body being centered over the ankle, physically, the ankle angle being greater than a given threshold. To progress to swing flexion, the foot must leave the ground; this is equivalent to the load at the ball of the foot being less than a selected threshold. Advancing to the next phase, swing extension, the knee must begin to swing forward, indicated by a knee flexion velocity less than zero. Completing

3 TABLE I: Threshold values selected for switching of the finite state machine Threshold Value Units Ball of Foot Load 25 N Ankle Angle 1 Degrees Heel Load 25 N the cycle, the heel strikes the ground, characterized by a heel contact load greater than a chosen threshold, and the transition to early stance occurs. Each of the measurements required to transition between states are dependent only upon data that could be obtained from the prosthesis, making them hardware-feasible selections [16]. Considering a total of five states, three parameters per control law, and two joints, there are 3 control gains requiring tuning for this method. Initial tuning was completed via a least squares fit against reference torque, angle, and angular velocity profiles. Further tuning was completed by hand during prototype testing [16]. C. Switched Robust Tracking/Impedance Control The controllers discussed in Sections III-A and III-B both have merit with respect to the human system. In the robust tracking/impedance controller it is assumed that a human will use his or her residual limb the same as before his or her amputation. Additionally, in both cases the application of impedance control implements the capacity to respond to the environment in a dynamic fashion. Within the switched impedance controller the fact that humans do not present a constant impedance during dynamic action is applied [17]. Taking each of these features and combining them, therefore, should produce a prosthesis controller in better agreement with the human system. To perform this combination, the exact form of the robust tracking/impedance controller from Section III-A is utilized. It is then augmented with a gain switching algorithm, where the motion control and impedance control gains for the knee and ankle are each allowed to vary over the five discrete intervals defined in Section III-B. Within this control framework robustness holds for each gain set in accordance with the guarantees shown in [1]. The robustness of the overall controller is not considered here. There are a total of 5 gains for each joint, forming a combined 1 gains per state. Overall, there are 5 gains to be determined in tuning this controller. In addition, there are several switching thresholds. The threshold values for phase switching were selected as given in Table I. To avoid premature switching due to any stubbing of the heel or toe, the heel and ball of foot load thresholds were set above the contact level. By considering two criteria, the ankle angle threshold was selected. First, because the reference data was prepared by inverse kinematics and the inverse kinematics solution could select from an infinite number of revolutions about the ankle joint, though an equilibrium at approximately 9 o or 27 o was most likely, the ankle angle threshold needed to be set informed of the trajectory s equilibrium. Secondly, the ankle angle threshold value was selected based on observation of the TABLE II: Biogeography-based optimization parameters used for optimization of the switched robust tracking/impedance controller Parameter Value Population Size 5 Number of Generations 2 Number of Elite Individuals 2 Probability of Mutation.2 TABLE III: Optimization parameter ranges for the switched robust tracking/impedance controller. The gains for the knee joint are in the upper portion while the ankle joint gains compose the lower half Parameter Minimum Value Maximum Value Λ 3 or Λ IC,1 1 4 K I b k Λ 4 or Λ IC,2 1 4 K I b k dataset; the center of mass was judged to be over the ankle when the ankle is slightly dorsiflexed. IV. GAIN OPTIMIZATION A global evolutionary optimization algorithm called biogeography-based optimization (BBO) was selected for use in determining the gains of the controller developed in Section III-C [18], [19]. BBO is based on the immigration and emigration of species between isolated habitats according to a measure termed the habitat suitability index (HSI), which is a function of multiple suitability index variables (SIVs). More suitable habitats (high HSI) have a greater variety of species and generally share species with other habitats. Conversely, less suitable habitats have fewer species and are open to accepting more species. To form the parallel to a problem solution, habitats represent candidate solutions, and habitat features correspond to features of those solutions. Immigration and emigration are the means by which solution features are shared probabilistically. Additionally, elitism, which is a method for saving the best solutions from the previous generation and injecting them into the next generation, and mutation, which is a means of introducing new information, were implemented within the algorithm. The BBO algorithm parameters were set as given in Table II for each trial. The population size was selected based on the number of individual parameters to be varied. After several iterations the number of generations was determined according to the observed rate of convergence. Each population member included a set of 5 gains, those previously described. For each of the 1 gains per state ranges within which the population features could be varied and from which the initial population would be selected were set; refer to Table III. The population was initialized randomly with

4 TABLE IV: Initial optimization candidate solutions for the switched robust tracking/impedance controller. The table is divided by grouping the knee joint gains in the upper portion and the ankle joint gains below Parameter Initial Candidate 1 Initial Candidate 2 Λ 3 or Λ IC, K I b k Λ 4 or Λ IC,2 3 3 K I b k TABLE V: Gains used for the hip joint Hip Joint Gain Value Λ 1 or Λ MC,1 155 Λ 2 or Λ MC,2 155 K K 2 15 exception of two candidate solutions, one high impedance case and one low impedance case from [1]. To define these two predetermined initial gain sets for use in the multiple state switching format, the 1 published gains were repeated for each of the 5 states, forming 5 gain candidate solutions. These initial candidates are presented in Table IV. Lastly, the gains of the hip motion controller, none of which were optimized, were selected to provide accurate tracking and based on [1]. It was found that the gains related to the hip vertical displacement q 1 could be reduced relative to those selected in [1]. The values associated with the hip rotation q 2 were replicated from [1]. The gains are given in Table V. A composite cost function including tracking, ground reaction force, and energy regeneration was defined. To bring each component into a general range of magnitude such that each would have an equal contribution to the cost, the following weights were applied cost = q 3,cost + q 4,cost E 3,cost GRF vert,cost 1 q 3,cost = (q 3,i,sim q 3,i,ref ) n 2 i 1 q 4,cost = (q 4,i,sim q 4,i,ref ) n 2 i t 2 E 3,cost = (E 3,t 2,sim E 3,t1,sim) P 3,ref dt t 1 1 GRF vert,cost = (GRF vert,i,sim GRF vert,i,ref ) n 2, i (19) where i is the time index of each point in the time interval of interest and E 3,sim is the integral of the joint power TABLE VI: Cost function and energy results for five optimization trials Measure Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 q 3,cost (rad) q 4,cost (rad) E 3,cost (J) GRF cost (N) cost E 3 (J) calculated from simulation outputs. Each of the tracking costs and the GRF cost are rms values. The energy cost is calculated such that the change in energy over the simulated gait cycle will approach the total excess of energy available at the knee, determined by integration of the knee power reference data. The energy cost was defined to approach the reference energy because the capacity to gain or lose energy is directly related to the joint torque, assuming the velocity is fixed by accurate tracking performance. Therefore, if realistic torque values are met for the actuation system, it is technically possible for the energy to increase by an amount greater than would be indicated by human reference data. The cost to this, however, is the potential for high reaction forces or torques at other joints, making a limit desirable. While the tracking/impedance controller includes guarantees of robust stability and convergence to the desired impedance for fixed gains, the introduced variable gain structure does not. Unstable solutions were, therefore, anticipated during the optimization process. Any gain combinations that resulted in divergence were penalized by forcing them to have a cost value of infinity. In practice a number of solutions would be penalized early in the generational progression. The number of penalized solutions would then decrease as the population improved throughout ongoing generations. V. RESULTS Across five optimization trials in simulation, each using the same set of reference data, the final cost measures and amount of energy gained were remarkably consistent. These results are summarized in Table VI. The accuracy of the tracking error is sufficient for gait across all trials. According to integration of the precalculated power data from the reference dataset, the available energy at the knee is J, which each of the solutions approaches as desired. The gains determined by the optimization process for all trials are detailed in [11]. Variation in the selected gains from trial to trial is fairly extensive. This might suggest that there are many combinations possible to form the same level of performance upon which the cost functions appeared to be converging. However, one trend amid the variation is particularly notable. There is a high stiffness predicted for the knee upon heel strike (Phase ), and a clear reduction during Phase 1. This might replicate the reduction in torque resisting knee flexion after the initial impact, as seen in [8]. To further evaluate the results, examples will be provided from Trial 1. The convergence behavior of the minimum cost is shown in Fig. 1. The tracking performance is provided in

5 Minimum Cost Generation Fig. 1: Example overall cost convergence Trial 1 Current State Fig. 4: Example state switching results Trial (a) Hip Displacement q 1 (Meters) 1 5 (c) Knee Angle q 3 (Degrees) (b) Hip Angle q 2 (Degrees) (d) Ankle Angle q 4 (Degrees) Fig. 2: Example tracking results Trial (a) u 1 (N) (b) u 2 (Nm) (c) u 3 (Nm) (d) u 4 (Nm) Fig. 5: Example control signal results Trial 1 Fig. 2. Excellent tracking is seen for both hip trajectories and the knee trajectory. The ankle trajectory is also quite good. It is of particular interest because it portrays the expected behavior of an impedance controller. It allows divergence from the path but quickly regains tracking accuracy. The knee shows the same behavior though on a more limited scale. The vertical ground reaction force compares favorably with the reference data. In Fig. 3 one can note that the magnitude and timing are fairly accurate and a double peak has been formed, though it is not smooth. Some of the sharp portions of the GRF curve can be associated with the diversions of the ankle joint from its intended trajectory as well as the mechanics of the contact model. Based upon the previously presented figures, one can Vertical GRF (N) Fig. 3: Example vertical GRF results Trial 1 consider the accuracy of the state switching method. The timing of the states is presented in Fig. 4. Fig. 5 illustrates the control signal profiles. The preprocessed three-dimensional estimates of the required joint efforts from the reference data can be used to provide a baseline comparison between these control torques and those used by the human subject. This data is available for the hip torque, knee torque, and ankle torque. A peak reference magnitude of 27 Nm is given for the hip torque, significantly less than the simulated control torque. Comparing the peak knee torque to the reference peak, the simulated control torque is more than ten times the magnitude of the reference value, 44 Nm. The peak simulated ankle control torque, ignoring the instantaneous spike, is consistently smaller than the reference peak, 16 Nm. It is possible that this is related to the fact that the ankle was more compliant. Lastly, the energy profile is shown in Fig. 6. The general shape is consistent with the result of integrating the reference power data, though the magnitude is greater because it is dependent on u 3. Furthermore, the timing of the largest positive change in energy corresponds with the latter part of swing phase. There is also an small increase associated with the early portion of stance phase. These are two of the periods when excess energy is typically dissipated. The other significant rise is early swing phase as the gait cycle approaches toe off. The remainder of the shape is associated with various periods of energy usage [8].

6 E 3 (J) Fig. 6: Example E 3 results Trial 1 VI. DISCUSSION Two ideas previously applied to prosthesis control, namely robust tracking/impedance control and switched impedance control, have been combined. The resulting controller was optimized. The results of this preliminary study indicate that there is potential for such a controller to meet the outlined objectives, natural motion and energy regeneration. Directly, the next step is to formally consider the controller s stability. While the optimization constrained solutions to be stable, there is no guarantee of this controller s stability in general. The study of a switched impedance controller s stability can be completed in a passivity framework. It should also be noted that there are multiple limitations to the modeling process. While the model was expanded to use the limb lengths of the reference subject rather than all robotic parameters, the masses, locations of the centers of mass, and moments of inertia remained unchanged from the original robot model. Perhaps these inconsistencies caused some results to not match the reference data better. In addition, the control efforts, particularly those associated with the hip joint, have the potential of being high because of the tuning of the tracking portion of the controller. The goal of such a controller is to follow a trajectory regardless of external influences. Therefore, excessive force or torque can be used. In [2] optimization of the controller gains for a similar problem, obtaining realistic GRF, illustrates the potential for significant improvement. An alternative to reducing the control efforts by tuning is implementing impedance control, in which case the trajectories are no longer enforced [21]. Other control methods developed for a related system could be investigated as well [22], [23], [24]. Improvements of the optimization portion of this work can also be addressed. Development of the optimization within a formal multi-objective framework could be beneficial; the balance of objectives could be more easily defined and further objectives added, such as reducing the control signals. Finally, extending the optimization across multiple gait cycles would encourage robustness of the controller. APPENDIX The code used to produce the results can be found at prosthetics/research/switched-robustimpedance-control.html. REFERENCES [1] B. Hafner, L. Willingham, N. Buell, K. Allyn, and D. Smith, Evaluation of function, performance, and preference as transfemoral amputees transition from mechanical to microprocessor control of the prosthetic knee, Arch. Phys. Med. Rehab., vol. 88, pp , April 27. [2] P. Struyf, C. van Heugten, M. Hitters, and R. Smeets, The prevalence of osteoarthritis of the intact hip and knee among traumatic leg amputees, Arch. Phys. Med. Rehab., vol. 9, pp , April 29. [3] R. Gailey, K. Allen, J. Castle, J. Kucharik, and M. Roeder, Review of secondary physical conditions associated with lower-limb amputation and long-term prosthesis use, J. Rehabil. Res. Dev., vol. 45, no. 1, pp , 28. [4] D. Ehde, D. Smith, J. Czerniecki, K. Campbell, D. Malchow, and L. Robinson, Back pain as a secondary disability of persons with lower limb amputations, Arch Phys Med Rehabil, vol. 82, pp , June 21. [5] A. Gitter, J. Czerniecki, and K. Weaver, A reassessment of centerof-mass dynamics as a determinate of the metabolic inefficiency of above-knee amputee ambulation, Am. J. Phys. Med. Rehab., vol. 74, no. 5, pp , [6] Z. Harvey, B. Potter, J. Vandersea, and E. Wolf, Prosthetic advances, J. Surg. Orthop. Adv., vol. 21, no. 1, pp , 212. [7] F. Sup, H. Varol, J. Mitchell, T. Withrow, and M. 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