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Research Papers

# Adaptive Impedance Control of Parallel Ankle Rehabilitation RobotOPEN ACCESS

[+] Author and Article Information
Prashant K. Jamwal

Department of Electrical and
Electronics Engineering,
Nazarbayev University,
53 Kabanbay Batyr Avenue,
Astana 010000, Kazakhstan
e-mail: prashant.jamwal@nu.edu.kz

Shahid Hussain

School of Mechanical,
Materials, Mechatronic and
Biomedical Engineering,
University of Wollongong,
Northfields Avenue,
Wollongong, NSW 2522, Australia
e-mail: shussain@uow.edu.au

Mergen H. Ghayesh

School of Mechanical Engineering,

Svetlana V. Rogozina

Department of Rehabilitation,
Institute for Scientific Research of
Traumatology and Orthopedics,
Astana 010000, Kazakhstan
e-mail: svetlanarogozina@yahoo.com

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 21, 2016; final manuscript received March 31, 2017; published online July 20, 2017. Assoc. Editor: Evangelos Papadopoulos.

J. Dyn. Sys., Meas., Control 139(11), 111006 (Jul 20, 2017) (7 pages) Paper No: DS-16-1608; doi: 10.1115/1.4036560 History: Received December 21, 2016; Revised March 31, 2017

## Abstract

Robots are being increasingly used by physical therapists to carry out rehabilitation treatments owing to their ability of providing repetitive, controlled, and autonomous training sessions. Enhanced treatment outcomes can be achieved by encouraging patients' active participation besides robotic assistance. Advanced control strategies are required to be designed and implemented for the rehabilitation robots in order to persuade patients to contribute actively during the treatments. In this paper, an adaptive impedance control approach is developed and implemented on a parallel ankle rehabilitation robot. The ankle robot was designed based on a parallel mechanism and actuated using four pneumatic muscle actuators (PMAs) to provide three rotational degrees-of-freedom (DOFs) to the ankle joint. The proposed controller adapts the parallel robot's impedance according to the patients' active participation to provide customized robotic assistance. In order to evaluate performance of the proposed controller, experiments were conducted with stroke patients. It is demonstrated from the experimental results that the robotic assistance decreases as a result of patients' active participation. Similarly, increased robotics assistance is recorded in response to decrease in patient's participation in the rehabilitation process. This work will aid in the further development of customized robot-assisted physical therapy of ankle joint impairment.

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## Introduction

Robot-assisted ankle joint rehabilitation of patients suffering from neurologic impairments such as stroke is gaining popularity among the rehabilitation engineering community [17]. Several robotic devices have been developed during the last two decades for the ankle joint rehabilitation of neurologically impaired subjects [8]. These robotic devices could be categorized into the robotic orthoses [1,9,10] which are used for treadmill or over ground training and parallel robots which are used by the patients while in seated position [8,11,12]. Among these, the parallel robots are more suitable for the ankle rehabilitation application owing to their higher stiffness and accuracy over small range of motions.

Several parallel robots have been proposed in literature for providing physical therapy to the stroke survivors [1117]. These parallel robots provide motions to the ankle joint in three axes of rotation [18]. The initial prototypes of parallel ankle robots are powered by electromagnetic actuators which have high end-point impedance and may not be suitable for human–robot interaction [15]. Later, parallel robots powered by intrinsically compliant actuators such as pneumatic muscle actuators (PMAs) have also been reported in literature [14]. These actuators have high power/weight ratio and can provide safer human–robot interaction as compared to electromagnetic actuators.

The type and nature of physical therapy provided by these parallel robots are also an emerging research area [8]. Different control algorithms have been developed for these parallel robots to provide therapeutic exercises. Most of the initial prototypes of these parallel robots guide the ankle joint on predefined trajectories, without taking into account the disability level and the muscular effort contributed by the patients [14]. The terms of positon control or trajectory tracking control have been used in literature for these kinds of conventional control algorithms. Later, some control algorithms have also been proposed which can modify the parallel robot assistance according to the disability level and effort incorporated by the patients [1921]. The terms of patient-cooperative [22] or assist-as-needed (AAN) training strategies have been used in literature for these control algorithms.

AAN robotic training by the parallel ankle robot (ARBOT) has been developed based on the impedance control law [15,19]. Initially, an extended compliance control algorithm has been proposed [15]. The compliance controller provides motion assistance or resistance in the direction of patients' effort. However, ARBOT and its AAN control scheme have only been evaluated with five healthy subjects [19]. An impedance control scheme has been developed for the four degrees-of-freedom (DOF) parallel ankle robot [23]. The impedance control scheme is implemented in the form of a cascaded position and force controller. Different range of exercises has been designed and the impedance values are adjusted accordingly by the physical therapist. Details of the experimental evaluations of the impedance control scheme have not been provided in literature.

Impedance control based AAN robotic ankle rehabilitation strategy has been developed for the intrinsically compliant parallel robot powered by PMA [20,24]. The impedance control scheme modifies the robotic assistance according to the active muscular contributions of the human subject. Inverse dynamics algorithm has been developed to extract the active muscle contributions of the human ankle complex from the remaining passive and inertial effects of the combined human–robot system. The impedance control algorithm has four modes, namely: zero impedance control mode, position control mode, impedance control mode with low compliance, and impedance control mode with high compliance [20]. However, the intervention of a physical therapist is required to switch among the four control modes of the impedance control scheme depending on the patient's progress [20]. Impedance control-based AAN robotic ankle rehabilitation strategy has also been evaluated with only healthy subjects [20].

In order to overcome the limitations of the impedance control-based AAN strategy of PMA parallel ankle robot [20], the authors have developed an adaptive impedance control law. The adaptive impedance control law can modify the robotic assistance online without the intervention of the physical therapists. The adaptive impedance control law for the parallel ankle robot is evaluated with three subjects suffering from stroke. According to authors' best knowledge, no adaptive impedance control scheme for the intrinsically compliant parallel ankle rehabilitation robot has been evaluated with neurologically impaired subjects prior to this work.

## Methodology

In order to provide an effective robot-assisted rehabilitation treatment for the ankle joint impairments, an adaptive impedance control is implemented on a parallel ankle rehabilitation robot. Modeling of the ankle robot and details of the implementation of proposed adaptive impedance controller are being discussed in this section.

###### Parallel Ankle Rehabilitation Robot.

The ankle robot so developed [13,14,17] has an end effector (close to the ground) or the moving platform which is used to fix the foot–ankle body and it has actuators running parallel to the shinbone of the subject. The ankle robot can provide motions to the ankle joint in 3DOFs and is redundantly actuated by four PMAs (Fig. 1). Readers are referred to Refs. [13], [14], and [17] for further details regarding the parallel ankle rehabilitation robot mechanism.

###### Dynamic Modeling.

The dynamic model of the complete system comprises of the models of the parallel ankle robot and the human subject. The details of the overall dynamic model have been provided in the authors' previous work [20]; however, a brief description is provided here to facilitate the readers. The overall dynamics of human subject and parallel robot can be modeled as below Display Formula

(1)$M(θ)θ¨+C(θ,θ˙)θ˙+G(θ)=Trob+Tha$

Also, $Trob=J−TF$.

Here, $θ$, $θ˙$, and $θ¨$ are the generalized position, velocity, and acceleration vectors, respectively. M(θ) is the system inertia tensor. C is a vector of moments resulting from centrifugal and Coriolis forces, and G is the vector of moments applied by the gravitational forces. Trob is the vector of torques resulting from the actuator forces (F) of the parallel robot. The Jacobian matrix (J) is derived from the geometrical modeling of the robot and is used to provide mapping between the joint forces (F) and the task space torques. Trob is the robot applied torque which guides subject's ankle joint motions along the reference trajectories. On the other hand, Tha is used to depict active torque applied by the subject. All these terms have been elaborated and discussed in the authors' previous work [20].

The physical rehabilitation treatment needs to be objective and at the same time requires appropriate interventions apart from continuous monitoring. To emulate the entire treatment methodology and implement using robots is a very complicated process. Various phases of therapeutic procedures are designed considering subjects' levels of disabilities apart from their physical differences. A successful robot-assisted rehabilitation treatment requires cooperative efforts from the subject and the robot. It is desired that the robot's role be limited to scaffold or support the subject and provide compensation for motions and torques during the rehabilitation program. In other words, the robotic assistance is required to be adapted according to the subjects' capabilities both in terms of the range of motions and strengths.

Underlying concept behind the development of an adaptive impedance control is attune the robotic assistance based on subjects' participation. This further means that the robot applied moments should be increased when subjects' are not participating actively or their contribution is small. Similarly, the robot-applied moments should be reduced in the wake of active participation from the subjects. Robot-applied moments are required to be increased to guide the subjects' ankle joint motions on reference trajectories. Similarly, the reduced moments from the robot are expected to allow subjects to deviate from the reference physiological trajectories. The above two approaches are required at different stages of the rehabilitation program. During the initial phase of the rehabilitation, it is required that the subjects' ankle joint be moved by the robot on physiological reference trajectories in order to increase the neuroplasticity and improve their motion capabilities. Further, during later phases of the treatment, it is desired that the subjects' are allowed to deviate from the reference physiological trajectories and take over the robot. In order to implement this treatment protocol, the impedance of the parallel robot should be changed to a low level which will allow the deviation (within some threshold range) from the reference trajectories. However, the extent of such deviation should be decided on the basis of the subjects' disability level and efforts. Further, if the deviation from reference trajectories goes beyond the set threshold range, it is desired that the robot-applied moments be adjusted so that the subjects' ankle joint can be restrained to remain within the threshold range of the reference trajectories.

In the present research, the intended adaptive impedance control architecture is implemented in the task space of the parallel ankle robot (Fig. 2). Output signals from the linear potentiometers, placed in parallel to the actuators, are used to measure actuator displacements and subsequently, the ankle joint trajectories $θi$ (about three axes). It is assumed that, while taking readings from the sensors, there was no relative motion between the ankle robot and the subject's ankle joint. From available joint trajectories, angular velocities $θ˙i$ about three axes were computed by numerical differentiation. The joint torques T that guides ankle joint motions on reference trajectories were derived from the trajectory tracking errors from joint angles $θ⌣i(θi=θi*−θi)$ and joint velocities $θ⌣˙i$. To begin with a BASMC law was used as the basic position controller in the overall adaptive impedance control scheme. The aim of this controller was to guide the subject's limbs on the reference trajectories overcoming the structured uncertainties present in the model of PMA [20] (Fig. 2).

The interaction torques $T⌣t$ between the parallel ankle robot and the human subject is determined using the load cells which are placed in series with each PMA [20]. It should be noted here that the force measurements provided by the load cells comprised of gravitational and inertial components as well as the torques produced by subject's joints Th. Here, the human torque component Th can be further divided into the active and passive torque components. While the active torque component was offered by the human skeletal muscles, the passive component of the torque comes from the viscoelastic components like ligaments and other tissues surrounding the skeletal joints Display Formula

(2)$Th=Tha+Thp$

Here, the active and passive human torque components are represented by Tha and Thp, respectively [20]. A double exponential equation by Riener and Edrich [25] was used to model and implement the passive joint torque component in this study. Further, in order to extract the active human torque component Tha from the remaining passive and kinetic effects, an inverse dynamics model of the parallel robot and human subject was used Display Formula

(3)$T⌣ha=M⌣(θ)θ¨+C⌣(θ,θ˙)+G⌣(θ)−T⌣hp−T⌣t$

Here, $M⌣(θ)θ¨$ stands for the inertia of the parallel robot and human subject combine, whereas $C⌣(θ,θ˙)$ and $G⌣(θ)$ are used for the torques resulting from Coriolis and gravitational forces, respectively. For further details on the implementation of the inverse dynamics model for the parallel ankle robot, readers are referred to Ref. [20].

The adaptation law (Fig. 2) designed for the adjustment of impedance/compliance of the parallel ankle robot is given by Display Formula

(4)$δi=1γi=w1i+w2i sat[w3i(Tha)+w4i]$

Here, $δi$ and $γi$ are the parallel robot stiffness and compliance, respectively. Also, $w1i…w4i$ stands for the indices of the adaptation matrix along three ankle joint trajectories. Further, the joint compliance of the ankle robot actuated by PMA [26,27] is given by Display Formula

(5)$γi=1δi=θi2ri2K0iθi+K1i(ri2πP0fi−P0ixeiri)+K1iri2Δpi$

In relation (5), $Δpi$ are the controllable pressures inside PMA and represented as arbitrary functions of time. The moment arms or the radial distance of the actuator connection points on the foot platform from the center of rotation which is the ankle joint are shown as ri. Further, K0i and K1i are the parameters (i.e., spring element) of the numerical model of PMA [28]. xei is the length of PMA expressed in terms of $θi$. $Pofi$ is used for the nominal pressure of PMAs and $Poi$ stands for the difference in nominal pressures of antagonistic PMAs.

In the overall adaptive impedance control scheme, the basic position controller (based on the BASMC law [20]), provides the parallel robot torques (T) based on the trajectory tracking errors (Fig. 2). It is important to mention here that the position controller does not consider human subjects' active torque contribution (3) while tacking the trajectories. In fact, the adjustment of parallel ankle robot impedance based on the extent of human subjects' active joint torque contribution (Fig. 2) is achieved by the adaptation law given in Eq. (4). In other words, the joint impedance modification of the parallel robot will in turn modify the amount of robotic assistance (Trob) (Fig. 2). Therefore, the adaptation law given in Eq. (4) actually maps the joint active torque (Tha) to a saturation function “sat,” which was characterized in a linear manner between maximum and minimum saturation levels. Additional feedback loop was used to partially compensate the gravitational effects (identified in the inverse dynamic model) in order to improve the performance of the adaptive impedance controller (Fig. 2) [20].

## Experiments

The parallel ankle robot being controlled by the proposed adaptive impedance scheme is used during all the experiments. Three subjects (male, age 28–52 years) volunteered to participate in the experiments who were all mildly impaired stroke survivors (4–6 months post stroke) with ankle joint dysfunctions. Appropriate ethics approval was obtained to conduct experiments with the subjects. Initially, the subjects were asked to fix the ankle robot to their foot and shin bone and remain relaxed for 5 mins. Later, the subjects were asked to move their ankle joint around without actuating the ankle robot. This initial session was intended to make the subjects acquainted with the ankle robot and be prepared for subsequent experiments.

###### Experimental Protocol.

In order to evaluate performance of the ankle robot during zero impedance and maximum impedance modes, two experimental protocols were used, namely, inactive mode or trajectory tracking control mode and active mode or zero impedance control mode. During the first mode, i.e., inactive mode, the ankle robot was actuated while the patients were asked to remain completely inactive. In order to move ankle robot during trajectory tracking control mode, the BASMC scheme was implemented. The trajectory tracking controller is not an adaptive impedance controller. Aim of the trajectory tracking controller is to move subjects' ankle joint on selected reference trajectories in such a manner that the trajectory tracking errors are kept at minimum. The target trajectories along three axes were derived from the available literature [14]. To begin with the experiments, the ankle robot (appropriately fixed on subjects' limb) was moved along the three predominant ankle joint trajectories, namely, flexion, inversion–aversion, and adduction–abduction. A trajectory tracking controller was implemented and used during the first set of experiments. Each of the subjects was given 15 mins to complete one experiment mode. The real‐time ankle motion is sensed through four linear potentiometers connected in parallel to four PMA in the ankle robot. Actuator lengths are transformed into the pose of the ankle robot using forward kinematics [29]. The ankle robot motions during trajectory tracking mode are recorded and plotted for further analysis.

During the zero impedance control mode, i.e., active mode, the ankle robot was not actuated and the subjects were asked to move their ankle joint actively along the specified trajectories. Subjects were shown visual feedbacks of the target ankle trajectories and their own tracking performances in real time in order to encourage them to track specified trajectories to their best capabilities. A wash out time of 15 mins was provided to all the subjects between the trajectory tracking and zero impedance control modes.

Subsequent to the above, two sets of experiments were carried out in order to evaluate whether the adaptive impedance controller is able to modify robot-applied moments (Fig. 2). During the first experimental mode (inactive to active mode), the subjects were asked to remain completely passive for 15 mins. Continuing the experiment, the subjects were further instructed to actively track the reference joint angle trajectories for next 15 mins. During the second experimental mode, the subjects were asked to actively track the ankle trajectories for the first 15 mins. However, for the next 15 mins, the subjects were instructed to remain completely passive and allow the robot to guide their ankle joint on reference trajectories. Visual feedback was provided to the subjects during adaptive impedance control modes. The objective of conducting these two experimental modes was to assess whether the proposed adaptive impedance control scheme is able to adjust (increase or decrease) the moment contribution from the ankle robot in accordance to the active contribution extended by the subjects. In other words, the robot commanded moment should decrease when the subject is participating actively, similarly, the robot-applied moment should increase if the subject remains inactive.

###### Investigations.

The data sampling rate from sensors during above experiments was kept as 50 Hz. Appropriate information was extracted from the data such as expected values and standard deviations of the angular deviations and moments obtained during all the experiments. Owing to the small sample size (three subjects), a nonparametric statistical tests, namely, Wilcoxon signed-rank test [30] was performed in order to evaluate the significance of proposed controller for the ankle rehabilitation treatment. matlab-r2009b (Math Works, Inc., Natick, MA) software was used for the data analysis including the statistical treatment.

###### Results.

Experiments were conducted on the parallel ankle robot whereby the robot was controlled using the proposed adaptive impedance control scheme. Initially, in order to evaluate the overall performance of the ankle robot, it was operated between two extreme impedance conditions, i.e., zero impedance and maximum impedance condition or trajectory tracking control mode. During zero impedance mode, the robot remains completely passive and the target trajectories are followed purely based on the subjects' efforts. On the other hand during maximum impedance control mode, the subject remained completely passive and the robot tracked the desired trajectories actively. Results from these two modes are displayed in Figs. 3 and 4. Maximum absolute values of the Euler angle deviations for both control modes and averaged over subjects are also provided in Table 1. In order to provide further insight in the experimental observations, expected values and standard deviations are also provided. Close trajectory tracking performance was observed during the experiments involving trajectory tracking control mode (Fig. 3). The maximum absolute deviations observed during trajectory tracking mode were 9.3 deg, 6.7 deg, and 6.6 deg along three axes of the ankle joint (Table 1). Predominantly, these deviations arise out of the structured uncertainty existing in the model of PMA used as actuators in the ankle robot. During the zero impedance control mode, however, higher trajectory tracking errors were recorded (Fig. 4). Nevertheless, since the subjects were all stroke survivors and had not fully recovered from impairment, higher values of deviations can be justified (Table 1).

Later, experiments were performed to evaluate the efficacy of the proposed adaptive impedance controller (Figs. 5 and 6). It was expected that the controller should adapt the impedance resulting due to the interaction between the robot and the subjects. Two sets of experiments were performed and, in the first experiment, the subjects were asked to remain inactive initially and then get actively involved during the later half of the experiment (Fig. 5). Conversely, in the second experiment, the subjects remained active during the first half cycle and then became inactive for the remaining time during the experiment (Fig. 6). In order to evaluate the controller intervention, sample data (for 60 s) were captured at the interface time when the subjects' role changed from inactive to active and vice versa.

Representative data for the maximum absolute values of the controller output (i.e., robot commanded) moments for both sets of experiments along with their expected values and standard deviations have been provided in Table 2 for further analysis. It is apparent from Fig. 5 that the adaptive impedance controller has decreased robot applied moments as soon as the subject's role changed from in inactive to active. Similarly, it is evident from Fig. 6 that when subject turned inactive from active, the robot-applied moments also increased. Although, the estimated changes in the robot-applied moments (Table 2) are quite considerable, statistical tests were required to be conducted in order to establish the significance of the controller. Owing to the small data sample size, a nonparametric approach, namely, Wilcoxon signed rank test, was considered to perform the statistical test. Experimental data between two modes (i.e., active and inactive modes) were tested with the null hypothesis that there is no statistically significant difference between the data across the two modes. Significance threshold of 0.001, which is a commonly accepted value, was considered, while evaluating the statistical significance. A p-value which is less than 0.001 should indicate the rejection of the null hypothesis and vice versa. Results from the Wilcoxon signed-rank test are displayed in Table 2. Based on the small p-values obtained after the signed-rank test, it can be concluded that the null hypothesis can be rejected for all the observations. In other words, the change observed in the robot-applied moments is statistically significant and do not come from the chance causes. This further strengthens the presumption that the adaptive impedance controller is able to increase or decrease the robot-applied moments when the subjects' role is changed from active to inactive and vice versa.

## Discussion and Conclusion

In the present work, an adaptive impedance control scheme was developed for an intrinsically compliant parallel ankle rehabilitation robot. The adaptive impedance control scheme was designed to provide robotic assistance according to the disability level and stage of rehabilitation of neurologically impaired subjects. The overall adaptive impedance control scheme utilizes a BASMC scheme as the basic position controller. It was necessary to use BASMC in order to cope with the structured uncertainties in the model of PMA [20]. The proposed control scheme was evaluated with three stroke survivors. The authors believe that employing adaptive impedance control scheme during the robotic rehabilitation, it is possible to enhance the therapeutic outcomes of the neurologically impaired subjects.

The impedance controller owing to its inherent adaptive mechanism adjusts to the subjects' disability levels by taking inputs in the form of interaction moments. By adaptation, it is meant that the robotic assistance is adjusted in real time in order to compensate the subjects' participation. As soon as some active participation from subjects is detected through force sensors, the robotic assistance is reduced and vice versa.

Ahead of the experiments involving impedance adaptation, the parallel ankle robot was operated between two extreme modes, such as zero impedance control and maximum impedance control or trajectory tracking control modes. During the trajectory tracking control mode, the maximum absolute angular deviations in all the three trajectories were observed to be 9.3 deg. Angular deviations were anticipated due to the structured uncertainties in the actuators employed with the ankle robot. Further, the maximum fractional angular deviation was observed in the adduction–abduction trajectory, whereas the minimum deviation was found during the flexion trajectory. Eventually, the BASMC controller was found to be tracking the desired trajectories with reasonable accuracy in the presence of uncertainties arising from the use of PMA and external disturbance from the human user. The trajectory tracking control mode is vital and can be used in the early stage of the rehabilitation program to train severely impaired subjects. During zero impedance control mode, the robot was not actuated at all and the subjects were expected to move their ankle joint voluntarily along the targeted trajectories. Subjects were provided visual feedback of the target as well as their actual trajectories in real time and were encouraged to follow the intended motions. Perceptibly, the angular deviations, during zero impedance mode, were higher compared to the trajectory tracking mode. Maximum absolute angular deviation recorded during zero impedance mode was 32 deg which may be present due to the lack of capabilities of the subjects.

In order to assess the effectiveness of the adaptive impedance controller, two sets of experiments were carried out. It was expected that the inherent adaptive mechanism in the controller should be able to increase the robotic assistance when the subjects become inactive from being active initially and vice versa. It is evident from the results displayed in Figs. 5 and 6 that the robotic assistance is well adapted based on the subjects' contribution in moving the ankle robot. The adaptation or the adjustment in the robot applied moments provided by the controller was observed to be varying from 11% to 18%. The adaptation scheme works in both the directions, i.e., robotic assistance is decreased when the subject is participating actively (Fig. 5). Similarly, the robotic assistance can also be observed to have been increased in the response of subject being inactive. It can be concluded here that the adaptive impedance control scheme allows subjects to participate actively in the rehabilitation treatments by lowering the robot-applied moments.

In order to further evaluate the statistical significance of the proposed controller, Wilcoxon signed-rank test was carried out for the instances when the subjects changed their roles from being active to inactive and vice versa. The nonparametric approach was adopted owing to smaller sample size of data recorded. Here, the null hypothesis considered was that there is no statistically significant difference between the observations taken across the two modes of experiments. A usual significance threshold of 0.05 was considered to evaluate the statistical significance and as such p-values which are less than 0.05 shall indicate rejection of the null hypothesis. Apparently, the null hypothesis was rejected for all the experimental modes owing to smaller p-values obtained after the signed-rank test. This further means that there exists a statistically significant difference in the robot-applied moments when subjects became active after being inactive for some time and vice versa. During all the experiments, identical reference trajectories were used for all the subjects. However, it may not be advisable to use identical trajectories for all the subjects having different levels of neurological impairments. It is proposed to develop algorithms to automatically generate and adapt subject-specific reference trajectories for different levels of disabilities.

Summarizing above, this research presents the adaptive impedance control scheme implemented on the intrinsically compliant parallel ankle rehabilitation robot. It is believed that the outcomes from the robotic rehabilitation treatment can be enhanced by inviting active participation from the subjects. Trajectory tracking control approach does not allow subjects to participate actively during the treatment and therefore, it is essential that the control scheme be modified so that the robotic assistance is reduced gradually depending on the amount of subjects' active participation. The proposed adaptive impedance control scheme was evaluated with three stroke survivors. It is evident from the results that the intended objective of impedance adaptation was successfully achieved and the robotic assistance was adjusted based on subjects' participation during the experiments. According to authors' best knowledge, no prior work has been reported in the literature regarding the development of an adaptive impedance control scheme for an intrinsically compliant parallel ankle rehabilitation robot. Further, there is no evidence of experimental evaluations of similar parallel robotic ankle rehabilitation system and control scheme with stroke survivors as discussed in the present paper.

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Jamwal, P. K. , and Hussain, S. , 2016, “ Design Optimization of a Cable Actuated Parallel Ankle Rehabilitation Robot: A Fuzzy Based Multi-Objective Evolutionary Approach,” J. Intell. Fuzzy Syst., 31(3), pp. 1897–1908.
Riener, R. , and Edrich, T. , 1999, “ Identification of Passive Elastic Joint Moments in the Lower Extremities,” J. Biomech., 32(5), pp. 539–544. [PubMed]
Choi, T.-Y. , and Lee, J.-J. , 2010, “ Control of Manipulator Using Pneumatic Muscles for Enhanced Safety,” IEEE Trans. Ind. Electron., 57(8), pp. 2815–2825.
Choi, T.-Y. , Choi, B.-S. , and Seo, K.-H. , 2011, “ Position and Compliance Control of a Pneumatic Muscle Actuated Manipulator for Enhanced Safety,” IEEE Trans. Control Syst. Technol., 19(4), pp. 832–842.
Reynolds, D. B. , Repperger, D. W. , Phillips, C. A. , and Bandry, G. , 2003, “ Modeling the Dynamic Characteristics of Pneumatic Muscle,” Ann. Biomed. Eng., 31(3), pp. 310–317. [PubMed]
Jamwal, P. K. , Xie, S. Q. , Tsoi, Y. H. , and Aw, K. C. , 2010, “ Forward Kinematics Modelling of a Parallel Ankle Rehabilitation Robot Using Modified Fuzzy Inference,” Mech. Mach. Theory, 45(11), pp. 1537–1554.
Wilcoxon, F. , 1946, “ Individual Comparisons of Grouped Data by Ranking Methods,” J. Econ. Entomol., 39(2), pp. 269–270.
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Riener, R. , Lunenburger, L. , Jezernik, S. , Anderschitz, M. , Colombo, G. , and Dietz, V. , 2005, “ Patient-Cooperative Strategies for Robot-Aided Treadmill Training: First Experimental Results,” IEEE Trans. Neural Syst. Rehabil. Eng., 13(3), pp. 380–394. [PubMed]
Yoon, J. , Ryu, J. , and Lim, K.-B. , 2006, “ Reconfigurable Ankle Rehabilitation Robot for Various Exercises,” J. Rob. Syst., 22(S1), pp. S15–S33.
Jamwal, P. K. , and Hussain, S. , 2016, “ Design Optimization of a Cable Actuated Parallel Ankle Rehabilitation Robot: A Fuzzy Based Multi-Objective Evolutionary Approach,” J. Intell. Fuzzy Syst., 31(3), pp. 1897–1908.
Riener, R. , and Edrich, T. , 1999, “ Identification of Passive Elastic Joint Moments in the Lower Extremities,” J. Biomech., 32(5), pp. 539–544. [PubMed]
Choi, T.-Y. , and Lee, J.-J. , 2010, “ Control of Manipulator Using Pneumatic Muscles for Enhanced Safety,” IEEE Trans. Ind. Electron., 57(8), pp. 2815–2825.
Choi, T.-Y. , Choi, B.-S. , and Seo, K.-H. , 2011, “ Position and Compliance Control of a Pneumatic Muscle Actuated Manipulator for Enhanced Safety,” IEEE Trans. Control Syst. Technol., 19(4), pp. 832–842.
Reynolds, D. B. , Repperger, D. W. , Phillips, C. A. , and Bandry, G. , 2003, “ Modeling the Dynamic Characteristics of Pneumatic Muscle,” Ann. Biomed. Eng., 31(3), pp. 310–317. [PubMed]
Jamwal, P. K. , Xie, S. Q. , Tsoi, Y. H. , and Aw, K. C. , 2010, “ Forward Kinematics Modelling of a Parallel Ankle Rehabilitation Robot Using Modified Fuzzy Inference,” Mech. Mach. Theory, 45(11), pp. 1537–1554.
Wilcoxon, F. , 1946, “ Individual Comparisons of Grouped Data by Ranking Methods,” J. Econ. Entomol., 39(2), pp. 269–270.

## Figures

Fig. 1

Parallel ankle rehabilitation robot

Fig. 2

Adaptive impedance control architecture implemented in task space. Position controller works on the basis of boundary layer augmented sliding mode control (BASMC) law. The adaptation law (4) modifies the robotic assistance according to the extent of subjects' active participation.

Fig. 3

Average ankle joint displacement trajectories obtained during inactive mode (i.e., trajectory tracking control mode), averaged over three patients

Fig. 4

Average ankle joint displacement trajectories obtained during active mode (i.e., zero impedance control mode), averaged over three patients

Fig. 5

Robot-applied torque at the ankle joint of patients during adaptive impedance control experiments for inactive to active condition, averaged over three patients for two cycles of 30 s each (cycle 1 begins at 0 s and ends at 30 s whereas cycle 2 begins at 30 s and ends at 60 s). The subjects remained inactive (i.e., passive) during cycle 1. At the end of cycle 1, the subjects participated actively in the training process during cycle 2.

Fig. 6

Robot applied torque at the ankle joint of patients during adaptive impedance control experiments for active to inactive condition, averaged over three patients for two cycles of 30 s each (cycle 1 begins at 0 s and ends at 30 s whereas cycle 2 begins at 30 s and ends at 60 s). The subjects were active during cycle 1. At the end of cycle 1, the subjects remained inactive (i.e., passive) in the training process during cycle 2.

## Tables

Table 1 Maximum absolute values of the Euler angle deviations for trajectory tracking (Fig. 3) and zero impedance (Fig. 4) control modes averaged over all subjects
Table 2 Maximum absolute values of the controller output (i.e., robot commanded) moments for both control modes of adaptive impedance control experiment (Figs. 5 and 6) and averaged over all subjects
Robot commanded torque is the rough indicator of the robotic assistance provided to the subjects. Standard deviations are presented for intersubject variability and p-values from Wilcoxon signed-rank test for both modes are also provided.

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