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# Impedance & Admittance-Based Coordination Control Strategies for Robotic Lower Limb ProsthesesPUBLIC ACCESS

[+] Author Notes
Brian E. Lawson, Michael Goldfarb

Department of Mechanical Engineering, Vanderbilt University

Mechanical Engineering 136(09), S12-S17 (Sep 01, 2014) (6 pages) Paper No: ME-14-SEP6; doi: 10.1115/9.2014-Sep-6

## Abstract

This article presents and compares two different control systems for a powered knee and ankle prosthesis for transfemoral amputees, which were constructed to provide the user a safe, intuitive, and well-coordinated interaction with the prosthesis. The piecewise-passive impedance (PPI) controller utilizes only impedance-like behaviors, while the second – a hybrid impedance-admittance (HIA) controller – utilizes both impedance-like and admittance-like behaviors in a hybrid approach. The HIA approach maintains many of the desirable characteristics of the PPI controller while also reducing the number of selectable control parameters. The HIA approach essentially incorporates the PPI control structure during the early and middle stance phases of gait, and a trajectory tracking control approach in terminal stance and swing. These controllers were implemented on a powered knee and ankle prosthesis and tested in walking trials by a transfemoral amputee. Data from these trials indicate that both controllers achieve comparable performance with respect to healthy subject data, despite some substantial structural differences between the two.

## Article

Robotic prostheses have been emerging in the engineering literature recently. Such prostheses have the ability to better reproduce the variety of behaviors exhibited by the healthy limb during locomotion 15 , relative to passive prostheses. Powered prostheses require a controller to coordinate the movement of the prosthesis with that of the user; accordingly, various control approaches have been recently described 612. Such controllers should provide to the user safe, intuitive, and well-coordinated interaction with the prosthesis. This article presents and compares two different control systems for a powered knee and ankle prosthesis for transfemoral amputees that were constructed to provide such functionality (i.e., to enable power delivery from the prosthesis in a manner that is safe, natural, and coordinated with the motion of the user). The first controller utilizes only impedance-like behaviors, while the second utilizes both impedance-like and admittance-like behaviors in a hybrid approach. The controllers were implemented on a powered knee and ankle prosthesis and tested in walking trials by a transfemoral amputee. Data from these trials indicates that both controllers achieve comparable performance with respect to healthy subject data, despite some substantial structural differences between the two.

## A piecewise-passive impedance control framework

###### A control structure for providing local passivity

Prosthesis motion can be decomposed into two components: movement associated with the internal configuration of the prosthesis, and movement of the prosthesis through space. The former can be described by the prosthesis joint angles, and the latter by joint angles combined with a set of generalized coordinates locating the center of mass and principal axes of the prosthesis relative to an inertial reference frame. For the control problem considered here, the control of movement of the prosthesis through space is assumed to be primarily governed by the user, in the same manner that a user would control movement of a passive prosthesis. As such, the control framework described here considers only the dynamics associated with the internal configuration of the prosthesis, hereafter referred to as the prosthesis dynamics (i.e., the dynamics described by the set of generalized coordinates consisting of prosthesis joint angles).

Two distinct sources of power can impart energy to the prosthesis: the amputee and the prosthesis actuators. The generalized forces associated with the user, fu , are typically imparted at the mechanical interface between the user and prosthesis, and/or at the interface between the foot and ground. The generalized forces associated with the actuators, fa , are the set of torques imparted by each joint actuator on the respective joints of the prosthesis. If the vector f describes the union of user and actuator generalized forces, the state equations describing the prosthesis dynamics can be expressed asDisplay Formula

(1)
$q˙=gq,f$

where q is the state vector corresponding to the internal configuration of the prosthesis. The set of generalized forces associated with the user, fu , cannot be controlled by the prosthesis control system, and so it is regarded as a vector of exogenous inputs. The set of generalized forces associated with the actuators, fa, however, is governed by the prosthesis control system, and can be constructed asDisplay Formula

(2)
$fa=hq$

such that the function h(·) ensures strict passivity between the state inputs and the torque outputs. In particular, considering an actuated joint on the prosthesis, let the actuator torque be given by τ and the associated joint angle and angular velocity by θ and Display Formula$θ˙$, respectively. The prosthesis will exhibit passive behavior if the actuator torque output is controlled according toDisplay Formula

(3)
$τ=zθ,θ˙$

where z(·) is a diagonal, odd function of its arguments. Although a number of functions will satisfy this requirement, one simple form is the polynomialDisplay Formula

(4)
$τ=∑n=1Nk2n−1θ−θ02n−1−∑m=1Mb2n−1θ˙2n−1$

where N and M define the highest order of each odd polynomial, and kn and bm are coefficients associated with each of the polynomial terms. In the case of a first-order polynomial, the actuator torque simplifies toDisplay Formula

(5)
$τ=kθ−θ0+bθ˙$

Defining the prosthesis controller in this form (i.e., a strictly passive impedance function), the system dynamics can be reformulated as shown in Figure 1. Since h(·)is energetically passive by selection, and since the dynamics described by g(·) are also passive, the system is comprised of two passive interconnected systems, and thus the closed loop is also passive (i.e., as described by the passivity theorem). As a result, the feedback-controlled prosthesis can be reduced to a single passive system that interacts with the user in a manner similar to a passive prosthesis. Among other properties, this characteristic of passivity ensures that the powered prosthesis is stable, and in the absence of excitation from the user, the prosthesis will come to rest in a known state. The control structure therefore possesses inherent characteristics that foster safe human-robot interaction.

Figure 1 Amputee/prosthesis dynamic system with passive impedance control law.

Despite these desirable properties, a control structure that maintains strict passivity defeats the point of a powered prosthesis, which (like the healthy neuromuscular system) should be capable of power generation (in addition to storage and dissipation). In order to provide net power generation, the prosthesis control structure is modified such that (exogenous) input generated by the user is used to switch h(·) between successive passive behaviors. Such an approach can be implemented as a finite state machine (FSM), where state transitions are selected based upon biomechanical events. With mindful selection of state transitions, the user must be actively engaged in a given activity in order to maintain the succession of switching between passive behaviors. In this manner, the powered prosthesis is globally active (i.e., is able to generate net power over time), but locally passive (in the absence of excitation from the user, switching will not occur, and the prosthesis will come to rest).

###### Emulating biomechanical functionality of the healthy limb

The previously presented control structure was described without regard to its ability to emulate the biomechanical functionality of the healthy limb. Consider the example of level ground walking. Figure 2 shows the (averaged) knee joint angle, angular velocity, and torque for a group of healthy subjects during normal level walking. Within the context of the passive control structure, the biomechanical behavior of the joints is characterized by the relationship between the joint motion (i.e., angle and angular velocity) input and the joint torque output. Therefore, the proposed control structure should emulate the biomechanical behavior of the healthy joint if a passive function of the general form given in Reference 3 can be constructed such that, given the (healthy) angle and angular velocity profiles shown in Figure 2 as input, the function will result in the (healthy) torque profile also shown in Figure 2. Since the biomechanical behavior of the healthy joint is in general not passive, such construction will in general require the construction of h(·) with a series of piecewise passive functions. Using the data shown in Figure 2 as an example, and incorporating six linear passive functions of the form 5 with the parameters listed in Table 1 Table1 to construct h(·) over one period of the gait cycle, the control method will generate an approximation of healthy knee joint torque as indicated by the red circles in the torque plot. As indicated in Figure 2, the set of simple linear passive functions provides a reasonably faithful representation of healthy joint behavior. In the implementation described subsequently, the Piecewise Passive Impedance (PPI) controller is implemented as an FSM with the states as labeled in Figure 2.

Figure 2 Healthy knee kinematics reprinted from Reference 2. The knee torque has been scaled for a 79 kg subject (the mass of the subject used in the subsequent experiments). The red circles overlaid on the knee torque plot indicate the torque reference that would be generated from the piecewise linear fit using parameters specified in Table I. It is clear from these data that the knee behavior is well modeled in middle and early stance, and in swing, by the impedance model, although the behavior during late stance (ankle push off) is not as well represented by the model.

###### A Hybrid Impedance-Admittance Control Framework

Although the PPI controller provides desirable control behavior, it also requires a potentially large number of selectable parameters. In an effort to maintain many of the desirable characteristics of the PPI controller while also reducing the number of selectable control parameters, the authors modified the control approach with a hybrid impedance-admittance (HIA) approach. The HIA approach essentially incorporates the PPI control structure during the early and middle stance phases of gait, and a trajectory-tracking control approach in terminal stance and swing.

With this approach, both the passivity and naturalness aspects of the PPI controller are compromised, but neither in a substantial manner. With regard to passivity, the swing phase is a time-based trajectory, and therefore is transient and bounded by nature, so it does not substantially compromise the previously described inherent passivity of the PPI control structure. With regard to naturalness, although the trajectory control is characterized by a high joint impedance, the impedance is high during a period of interaction in which the user is relatively insensitive to the joint impedance. Specifically, the mechanical impedance interacting with the user is comprised of the internal impedance of the leg (i.e., the impedance imposed by the controller), in series with the impedance of the environment. In the stance phase, when the leg is on the ground, the impedance felt by the user is approximately the impedance of the joints of the leg (since the ground impedance is high). In the swing phase, when the leg is in the air, the impedance felt by the user is approximately the inertial impedance of the leg (since the impedance of air is essentially non-existent). Since the user is relatively insensitive to the nature of joint impedances in swing phase, implementing a high-impedance (i.e., admittance-type) trajectory-tracking controller in swing does not substantially compromise the naturalness properties of the PPI controller.

In the implementation described subsequently, the HIA controller consists of two superstates: an impedance-based state in the majority of the stance phase (early and middle stance), and an admittance-based state during terminal stance (powered push off) and swing phase. Note that the ability to change the impedance between early and middle stance is primarily used for slope walking (i.e., the two states share a single impedance in level walking).

## Implementation, demonstration, and comparison of controllers

###### Powered Prosthesis Prototype

The PPI and HIA controllers were each implemented on a self-contained powered prosthesis prototype previously developed by the authors. A photograph of the powered prosthesis is shown on a transfemoral amputee in Figure 3. Both the knee and ankle units are actuated by the combination of a brushless DC motor and a three-stage belt/chain speed reduction transmission. The knee is capable of generating a maximum torque of approximately 85 Nm, and the ankle approximately 110 Nm. The actuator output at the ankle joint is supplemented by a parallel carbon-fiber leaf spring (stiffness of 6 Nm/deg, engagement angle of 0 deg). The mass of the current prosthesis prototype, configured for a 50th percentile male, is approximately 5 kg.

Figure 3 A subject walking with the powered prosthesis.

###### Walking Trials

Each controller was implemented in the powered prosthesis prototype and tested in walking trials by a transfemoral amputee subject. Prior to the experiments, informed consent was obtained in accordance with the requirements set forth by the Vanderbilt University Institutional Review Board. For each controller, the controller parameters were tuned during treadmill walking at a self-selected treadmill speed of 0.89 m/s (2.0 mph). The control parameters corresponding to the PPI controller are given in Table 2. The impedance parameters for the first two phases of the HIA controller are the same as those used in the stance phases of the PPI controller. Following controller parameter selection, knee and ankle joint angle and motor current data from 20 consecutive strides were logged internally by the embedded system on the prosthesis for assessment and comparison.

###### Results and Discussion

The kinematics of the prosthetic joints in the sagittal plane are plotted in Figure 4, along with averaged data from 19 healthy subjects. The nature of control activity is indicated by the corresponding motor current references for each joint, shown in Figure 5. Both controllers achieve knee and ankle kinematics that contain the salient features of healthy level ground walking. The current references in Figure 5 best illustrate the (minor) differences in the behavior of these two controllers. Because the first two phases have identical impedance parameters in both controllers, any differences in the current references in these two states are due to reactions and interactions with the user. Such differences are small, however, and they appear to be due to slightly varied timings relative to the rest of the stride. At push off, however, it is clear that the HIA controller is more active at the knee, generating a flexive torque that better resists the knee's tendency to hyperextend during the initial portion of push off.

Figure 4 Knee and ankle controllers as compared kinematics for both to healthy subject data from Reference 2. The gray areas represent ±1, 2, and 3 standard deviations from the mean of healthy subjects.

Figure 5 Knee and ankle current references for the finite state-based impedance controller and the impedance/admittance controller as functions of percentage of stride. Positive currents indicate positive motor torques, where a positive torque, should it be the net torque on a joint,

In swing, the HIA controller generally matches healthy norms better than the PPI controller. The knee joint in the HIA controller is leading the trajectory, and so the controller behaves essentially as a damper in swing, while the PPI controller provides active torque in the form of virtual springs to generate sufficient knee flexion. Although this behavior could perhaps be altered with a different set of controller parameters for the PPI controller, it was in this case achieved inherently with the HIA controller. Finally, as one might expect, the high-gain nature of the HIA controller requires increased electrical power relative to the PPI, although for the walking trials shown, the increase was small (∼5%).

###### Conclusion

Powered lower limb prostheses are emerging, and in theory they have the capacity to better emulate the functionality of the healthy limb. In order to be useful, such prostheses must provide biomechanical levels of torque and power. Consequently, these prostheses are powerful robots that are firmly attached to a human. It is imperative to provide a control structure that coordinates the movement of the prosthesis with the movement of the user in a safe and natural manner. The authors describe here a control structure that provides these characteristics, and a modified version of it that maintains favorable characteristics, while greatly reducing the number of required control parameters. The authors additionally showed that both controllers provide similar behavior and provide biomechanics representative of healthy walking.

Competing InterestsB.E.L. and M.G. hold patent applications through Vanderbilt University that have been licensed to Freedom Innovations, a United States-based prosthetics manufacturer.

## References

B. J. McFadyen, and D. A. Winter, “An integrated biomechanical analysis of normal stair ascent and descent,” J Biomech, vol. 21, no. 9, pp. 733– 744 1988. [PubMed]
D. A. Winter, The biomechanics and motor control of human gait: Normal, elderly and pathological, Waterloo, Ontario, Canada: University of Waterloo Press, 1991.
R. Riener, M. Rabuffetti, and C. Frigo, “Stair ascent and descent at different inclinations,” Gait Posture, vol. 15, no. 1, pp. 32– 44, Feb. 2002. [PubMed]
A. Forner Cordero, H. F. Koopman, and F. C. van der Helm, “Multiple-step strategies to recover from stumbling perturbations,” Gait Posture, vol. 18, no. 1, pp. 47– 59, Aug. 2003. [PubMed]
A. S. McIntosh, K. T. Beatty, L. N. Dwan, and D. R. Vickers, “Gait dynamics on an inclined walkway,” J Biomech, vol. 39, no. 13, pp. 2491– 2502 2006. [PubMed]
S. Au, M. Berniker, and H. Herr, “Powered ankle-foot prosthesis to assist level-ground and stair-descent gaits,” Neural Netw, vol. 21, pp. 654– 666, 2008. [PubMed]
S. K. Au, J. Weber, and H. Herr, “Powered Ankle-Foot Prosthesis Improves Walking Metabolic Economy,” IEEE Transactions on Robotics, , vol. 25, no. 1, pp. 51– 66, 2009.
E. C. Martinez-Villalpando, and H. Herr, “Agonist-antagonist active knee prosthesis: a preliminary study in level-ground walking,” J Rehabil Res Dev, vol. 46, no. 3, pp. 361– 373 2009. [PubMed]
J. K. Hitt, T. G. Sugar, M. Holgate, and R. Bellman, “An Active Foot-Ankle Prosthesis With Biomechanical Energy Regeneration,” Journal of Medical Devices, vol. 4, no. 1, Mar. 2010.
M. A. Holgate, A. Bohler, and T. G. Sugar, “Control algorithms for ankle robots: A reflection on the state-of-the-art and presentation of two novel algorithms,” Biomedical Robotics and Biomechatronics, pp. 97– 102, 2008.
C. D. Hoover, G. D. Fulk, and K. B. Fite, “Stair Ascent With a Powered Transfemoral Prosthesis Under Direct Myoelectric Control,” IEEE/ASME Trans on Mechatronics, vol. 18, no. 3, 2013.
M. F. Eilenberg, H. Geyer, and H. Herr, “Control of a powered ankle-foot prosthesis based on a neuromuscular model,” IEEE Trans Neural Syst Rehabil Eng, vol. 18, no. 2, pp. 164– 173 Apr. 2010. [PubMed]
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## References

B. J. McFadyen, and D. A. Winter, “An integrated biomechanical analysis of normal stair ascent and descent,” J Biomech, vol. 21, no. 9, pp. 733– 744 1988. [PubMed]
D. A. Winter, The biomechanics and motor control of human gait: Normal, elderly and pathological, Waterloo, Ontario, Canada: University of Waterloo Press, 1991.
R. Riener, M. Rabuffetti, and C. Frigo, “Stair ascent and descent at different inclinations,” Gait Posture, vol. 15, no. 1, pp. 32– 44, Feb. 2002. [PubMed]
A. Forner Cordero, H. F. Koopman, and F. C. van der Helm, “Multiple-step strategies to recover from stumbling perturbations,” Gait Posture, vol. 18, no. 1, pp. 47– 59, Aug. 2003. [PubMed]
A. S. McIntosh, K. T. Beatty, L. N. Dwan, and D. R. Vickers, “Gait dynamics on an inclined walkway,” J Biomech, vol. 39, no. 13, pp. 2491– 2502 2006. [PubMed]
S. Au, M. Berniker, and H. Herr, “Powered ankle-foot prosthesis to assist level-ground and stair-descent gaits,” Neural Netw, vol. 21, pp. 654– 666, 2008. [PubMed]
S. K. Au, J. Weber, and H. Herr, “Powered Ankle-Foot Prosthesis Improves Walking Metabolic Economy,” IEEE Transactions on Robotics, , vol. 25, no. 1, pp. 51– 66, 2009.
E. C. Martinez-Villalpando, and H. Herr, “Agonist-antagonist active knee prosthesis: a preliminary study in level-ground walking,” J Rehabil Res Dev, vol. 46, no. 3, pp. 361– 373 2009. [PubMed]
J. K. Hitt, T. G. Sugar, M. Holgate, and R. Bellman, “An Active Foot-Ankle Prosthesis With Biomechanical Energy Regeneration,” Journal of Medical Devices, vol. 4, no. 1, Mar. 2010.
M. A. Holgate, A. Bohler, and T. G. Sugar, “Control algorithms for ankle robots: A reflection on the state-of-the-art and presentation of two novel algorithms,” Biomedical Robotics and Biomechatronics, pp. 97– 102, 2008.
C. D. Hoover, G. D. Fulk, and K. B. Fite, “Stair Ascent With a Powered Transfemoral Prosthesis Under Direct Myoelectric Control,” IEEE/ASME Trans on Mechatronics, vol. 18, no. 3, 2013.
M. F. Eilenberg, H. Geyer, and H. Herr, “Control of a powered ankle-foot prosthesis based on a neuromuscular model,” IEEE Trans Neural Syst Rehabil Eng, vol. 18, no. 2, pp. 164– 173 Apr. 2010. [PubMed]

## Tables

Table 1 Knee Level Walking Parameters from Healthy Subject Data
*Because the knee joint is not well modeled by a single, linear impedance in this state, a sub-state has been integrated allowing a change in stiffness and equilibrium parameters once the knee passes a threshold angle.
Table 2 Impedance Parameters for the PPI controller
*Because the knee joint is not well modeled by a single, linear impedance in this state, a sub-state has been integrated allowing a change in stiffness parameters once the knee passes a threshold angle.

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