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Further Results on Predictor-Based Control of Neuromuscular Electrical Stimulation Naji Alibeji, Nicholas Kirsch, Shawn Farrokhi, and Nitin Sharma, Member, IEEE
Abstract—Electromechanical delay (EMD) and uncertain nonlinear muscle dynamics can cause destabilizing effects and performance loss during closed-loop control of neuromuscular electrical stimulation (NMES). Linear control methods for NMES often perform poorly due to these technical challenges. A new predictorbased closed-loop controller called proportional integral derivative controller with delay compensation (PID-DC) is presented in this paper. The PID-DC controller was designed to compensate for EMDs during NMES. Further, the robust controller can be implemented despite uncertainties or in the absence of model knowledge of the nonlinear musculoskeletal dynamics. Lyapunov stability analysis was used to synthesize the new controller. The effectiveness of the new controller was validated and compared with two recently developed nonlinear NMES controllers, through a series of closed-loop control experiments on four able-bodied human subjects. Experimental results depict statistically significant improved performance with PID-DC. The new controller is shown to be robust to variations in an estimated EMD value. Index Terms— Electromechanical delay, functional electrical stimulation, input delay, Lyapunov–Krasovskii functionals, Lyapunov Methods, neuromuscular electrical stimulation (NMES), nonlinear control.
I. INTRODUCTION
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EUROLOGICAL dysfunctions caused by spinal cord injury (SCI), stroke, multiple sclerosis, traumatic brain injury, etc. are the leading causes of mobility disorders among adults across the world. Loss of, or limited, limb functionality in mobility impaired individuals hampers their ability to perform activities of daily living. Neuromuscular electrical stimulation (NMES) is an artificial application of electrical potential across a muscle group to elicit a muscle contraction. NMES is prescribed as an intervention to rehabilitate or restore functionality in mobility-impaired individuals [1]. For example, NMES can be used to produce advanced functional tasks such as walking [2]–[4], upper extremity reaching and grasping [5]–[8], rowing [9], single leg extension [10]–[17], and stationary bicycling [18]–[20]. Closed-loop control is typically employed to achieve accurate and precise limb control during NMES-elicited tasks. Manuscript received July 10, 2014; revised November 01, 2014; accepted March 03, 2015. Date of publication April 02, 2015; date of current version November 04, 2015. (Corresponding author: Nitin Sharma.) N. Alibeji, N. Kirsch, and N. Sharma are with the University of Pittsburgh, Department of Mechanical Engineering and Materials Science, Pittsburgh, PA 15261 USA (e-mail: [email protected]; [email protected]; [email protected]). S. Farrokhi is with University of Pittsburgh, Department of Physical Therapy, Pittsburgh, PA 15260 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNSRE.2015.2418735
However, closed-loop control of NMES is challenging due to the uncertain and nonlinear musculoskeletal dynamics, presence of muscle fatigue and unmodeled disturbances such as muscle spasticity or external changes in muscle loads, electromechanical delay (EMD), etc. Linear control methods such as in [21]–[23] or methods that assume a linear model (e.g., linear quadratic gaussian control, pole placement method, gain scheduling control [24]–[26]) often lack in performance. Moreover, linear control techniques do not guarantee stability during NMES-elicited tasks. Recently, nonlinear control techniques have been implemented for NMES applications [7], [10]–[15], [17], [27], [28] to obtain improved error performance and/or asymptotic tracking. Lyapunov based control synthesis and stability analysis is an exciting tool that has been used to design nonlinear and robust NMES controllers [13], [14], [17], [19], [29]–[31]. For example, Lyapunov stability analysis was used to design NMES controllers that are robust to uncertainties and bounded exogenous disturbances in a single leg extension task [13], [15], [17], [29]–[31] and cycling [19], [20]. Further results in [14] used a backstepping control design to incorporate calcium dynamics and muscle fatigue dynamics with a motivation to develop an NMES controller that compensates for muscle fatigue. In this paper, we design a new robust controller to compensate for EMDs using Lyapunov stability analysis. EMD arises due to a time lag between electrical excitation of the motor-neurons and tension development in a muscle. It is a function of a number of phenomena including: finite propagation time of the chemical ions in the muscle, cross-bridge formation between actin-myosin filaments, stretching of the series elastic components in response to the external electrical input, synaptic transmission delays, and others [32], [33]. In results such as [10], [11], [29], the EMD is modeled as an input delay in the musculoskeletal dynamics. Input delays can cause performance degradation and have also been reported to potentially cause instability during human stance experiments [34]. Despite the fact that EMDs are exhibited in muscle response and can lead to instability, NMES controllers that actively compensate for this phenomena are lacking. Previous results, such as in [23], [34]–[36], examine EMD effects by implementing a standard proportional derivative (PD) controller during stance (or quiet standing) experiments that show robustness to the delays. The controllers in such results are not modified to include a delay compensation (DC) term and have no mathematical proof of stability when the plant has uncertainties, nonlinearities, and/or disturbances. Various methods exist in the general time-delay control literature to compensate for actuator or input delays, but the ex-
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isting approaches such as Smith predictor methods [37], Artstein model reduction [38], finite spectrum assignment [39], and continuous pole placement [40] are applicable to only linear plants. The control problem becomes especially complicated when parametric uncertainties, nonlinearities, and additive disturbances are considered. Recently, two predictor-based control methods were developed for an uncertain input delayed system with additive disturbances [41]. These results suggest that a PD or a proportional integral derivative (PID) controller can be augmented with a delay compensator that contains a finite integral of past control values to transform the delayed system into a delay-free system. Motivated by the modified PD control result in [41], a tracking controller for a nonlinear musculoskeletal system with known constant input delay, the proportional derivative with delay compensation (PD-DC), was developed in [29]. Only the modified PD control result was extended in [29] as the modified PID controller in [41] requires a knowledge of inertia matrix. Thus, the PID-type delay compensating control design proposed in [41] cannot be implemented because of many uncertainties in the musculoskeletal system that form an unknown nonlinear input gain to the voltage input. These uncertainties include the unknown moment of inertia, muscle force-length and force-velocity relationships, moment arm, etc. This paper presents a PID-type delay compensating controller (with a desire to include integral feedback) that overcomes EMDs during NMES. A preliminary work on the control development and stability analysis was presented in [30]. Due to a different control structure employed in this paper, as opposed to the control design approach in [41], the requirement of input gain or inertia matrix knowledge was relaxed in the new result. Therefore, the new controller can be implemented despite parametric uncertainty and additive bounded disturbances. The key contributions of this effort are the design of a delay compensating auxiliary signal to obtain a time delay free open-loop error system, incorporation of a constant control gain that compensates for unknown nonlinear input gain function, and the construction of Lyapunov Krasovskii (LK) functionals which are used in a Lyapunov-based analysis that proves the tracking error is semi-globally uniformly ultimately bounded. The new controller was applied as an amplitude modulated current to external electrodes attached to the distal-medial and proximal-lateral portion of the quadriceps femoris muscle group in 4 non-impaired volunteers. The new controller was compared with its predecessors: PD-DC [29] and the Robust Integral of the Sign of the Error (RISE) [13]. The experimental results indicated that the new controller reduced the steady state root mean squared tracking error (SSRMSE) compared to the two previously developed nonlinear controllers, and was found to be robust to variations in the estimated EMD value.
Fig. 1. Schematic of a subject sitting in the leg extension machine with a knee torque generated via NMES.
where , , are the angular position, velocity, and acceleration of the knee joint, respectively. In (1), is the moment of inertia of the lower shank, denotes the elastic effects due to joint stiffness, denotes the gravitational forces, and denotes the viscous effects due to the damping in the knee joint. Further details of these terms and their definitions are given in [30]. In (1), denotes any unmodeled phenomena or disturbances in the system. The torque produced at the knee due to NMES is denoted as , and denotes the constant input delay caused by electromechanical processes associated with NMES. The torque produced at the knee is generated through NMES of the quadriceps muscles. This torque is defined as (2) denotes the unknown moment arm function, where denotes the unknown force-length and force-velocity relationships, and denotes the artificial voltage/current applied across the quadriceps muscle. The muscle activation dynamics were neglected in order to simplify the control design. The following assumptions and notations are used to facilitate the subsequent control development and stability analysis. Assumption 1: The moment arm is assumed to be a non-zero, positive, bounded function [42], [43] whose first time derivative exists and is continuous. The function is assumed to be a non-zero, positive, and bounded function with a bounded and continuous first time derivative based on the empirical data [44], [45]. Assumption 2: The auxiliary non-zero unknown scalar function , which acts as a nonlinear input gain function to the applied voltage on the muscle, is defined as (3)
II. MUSCULOSKELETAL SYSTEM The musculoskeletal system is a model of a subject sitting in a leg extension machine (LEM) as depicted in Fig. 1. NMES, via surface electrodes, elicits contractions in the quadriceps muscle group that produce a knee-joint torque to extend the swing arm of LEM. The knee-joint dynamics are modeled as (1)
where the first time derivative of is assumed to exist, be bounded, and continuous (see Assumption 1). Assumption 3: The unknown disturbance is bounded. Its first time derivative exists and is bounded and continuous. Based on Assumptions 1 and 2 the ratio / denoted as , is also assumed to be bounded and its first time derivative exists and is bounded and continuous.
ALIBEJI et al.: FURTHER RESULTS ON PREDICTOR-BASED CONTROL OF NEUROMUSCULAR ELECTRICAL STIMULATION
Assumption 4: Based on Assumption 1, the ratio / denoted as , can be upper bounded as
,
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Based on (8) and to facilitate the subsequent analysis, the voltage input is designed as
(4) where are known constants. The force-length and force-velocity values will never allow the lower bound of to be zero (i.e.; ) within the operating region of the knee joint. Assumption 5: The EMD, denoted by , is assumed to be a known constant. Factors, such as fatigue, may cause it to be a time-varying phenomenon; however, the influence of these factors on EMDs are ignored. Assumption 6: The desired trajectory, , and its time derivatives, , are bounded and continuous. Notation: A delayed state in the subsequent control development and analysis is denoted as or as while a non-delayed state is denoted as or as . Remark 1: Assumptions 1–5 are made mainly for stability analysis and control design. These assumptions are based on empirical results as cited above. These assumptions, except Assumption 5, are standard assumptions and have no or little practical significance as shown in the experimental results presented in [13], [17], [29].
(9) where
is a known control gain that can be expanded as (10)
, , and are known constants. where in (9) can be written Lemma 1: The time derivative of as (11) Proof: See Appendix. After differentiating (8) with respect to time and using (11), the closed loop error system can be written as (12)
III. CONTROL DEVELOPMENT The control objective is to track a continuously differentiable desired trajectory, . The tracking error is defined as
In (12), the auxiliary function are defined as
and
(5) To facilitate the control design and stability analysis, the auxiland are defined as iary error signals (6) (7)
where
,
,
where , , and are inertial, viscous, and stiffness terms (defined in (1)) that are exand velocity , pressed in terms of desired limb position and is defined as
are known positive control gains.
A. Closed Loop Error System The open-loop tracking error system can be developed by multiplying (7) by and utilizing the expressions in (1)–(3), (5), and (6) to obtain
(13)
Using the mean value theorem, the auxiliary functions and can be upper bounded as (8)
where the nonlinear functions are defined as
,
, and
(14) is a known constant, the bounding function is a positive globally invertible non-decreasing is defined as function, and
In (14),
(15)
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Based on the subsequent stability analysis, LK functionals: and are defined as (16)
(17) where ,
are known constants. IV. STABILITY ANALYSIS
Theorem 1: The controller given in (9) ensures semi-globally uniformly ultimately bounded tracking:
Fig. 2. Testbeds used in this study: (A) leg extension machine (LEM) and (B) brace. The LEM was used to compare the three different controllers, and the brace was used to test the robustness of PID-DC delay setting.
(18) where , , denote constants, provided the control gains , , , , and introduced in (6), (7), (9), and (10), respectively, are selected according to the following sufficient conditions:
(19) In (19), the known positive constants , , , and are defined in (4), (16), and (17), respectively, is the input delay and is a subsequently defined constant. Proof: See Appendix. Remark 2: From the gain conditions in (19), a sufficient condition on the time delay can be derived, . Therefore, the stability of the controller is only guaranteed for a certain magnitude of the input delay, which can be increased to a certain limit provided that . V. EXPERIMENTAL PROTOCOL The experiments were conducted on two testbeds as shown in Fig. 2: a modified LEM (A) and a brace (B) to mimic lower leg swing during walking. The LEM was modified to have physical stops to prevent hyperflexion and hyperextension, and an incremental optical encoder [Calt, CN] with 1024 pulses per revolution (ppr) resolution to measure the knee angle. The brace consisted of a motor at the hip joint and an incremental optical encoder [Hengxiang, CN] with 1024 ppr resolution at the knee joint. The QPIDe [Quanser Inc, Ontario Canada] data acquisition board was used to power and measure the output of the optical encoder and run the controllers in real time. An FNS-8 channel stimulator [CWE Inc., PA USA] was used to generate the biphasic pulse train transmitted to the surface electrodes. The pulse trains that were used had a frequency of 35 , which studies have shown is the optimal frequency for NMES [46], and a pulse width of 400 . Current amplitude modulation was used for the experiments. All control algorithms were coded in Simulink [MathWorks Inc, USA] and implemented using the Quarc real-time software [Quanser Inc, Ontario Canada]. Four able bodied male subjects between the ages of 24–30 years were selected for the experiments. Prior to any experi-
mentation, an approval from the Institutional Review Board at the University of Pittsburgh was obtained. The participants were instructed to relax and avoid any voluntary interference during the electrical stimulation. Two sets of experiments were conducted to evaluate the performance of the new controller and its robustness to variations in the estimated EMD value. Each experimental session was run for a duration of 30 seconds with a rest period of 3 minutes in between the sessions to prevent muscle fatigue. In the first set of experiments, we compared the new controller (PID-DC) with two previously developed nonlinear controllers (PD-DC and RISE). The details of the RISE control law and PD-DC control law can be found in [13] and [29], respectively. These controllers were chosen because, just like the controller developed in this paper, they fall under the category of strictly feedback tracking controllers that are designed based on a nonlinear musculoskeletal system and are synthesized using a Lyapunov stability analysis. In addition, these controllers are easily implementable and do not require any model knowledge, unlike controllers such as the sliding mode controller (SMC) in [47]. During the experiments, the 4 subjects were unaware of the controller being tested during the session. For each subject, the order of the controllers was selected at random and were tested on separate days (every other day). The three controllers were used to track a sinusoidal signal with a period of 2-seconds and alternating peaks. The desired trajectory started from the equilibrium and oscillated between a minimum amplitude of 15 and an alternating maximum amplitudes of 35 and 25 , this prevented any subject from anticipating the desired motion and subconsciously interfering with the performance of the controller. Each controller was evaluated in five consecutive trials for each subject. A rest period of 3 minutes was given in between the 30 second trials. Also, the subjects were not allowed to view the desired trajectory or the performance in real time. In addition to these experiments, the three controllers were tested on one subject with a larger range of motion (5–50 ) to see if the controllers could maintain their performance for larger movements. As these experiments were conducted on human subjects, the gain tuning procedure could only be done over a finite interval of time in order to prevent muscle fatigue and subject discomfort. Prior to experimentation on a subject, the three controllers were tuned to find an initial guess of the control gains. Before con-
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Fig. 4. Results obtained from running the three controllers on subject H1 With a larger trajectory ranging from 5–50 .
movement occurred. The EMD value was calculated as the time difference from when the first stimulation pulse was applied and when the knee angle began to change. Five measurements of the EMD value were taken, and the average of the five values was used as the measured EMD value, which is used in the control implementation. In the second set of experiments, the PID-DC controller was tested for its robustness to variations in the EMD value. This was done by assuming an estimated EMD value, used in the PID-DC controller, different from the measured EMD value in the previous experiments. A subject (H1) was held in a gait like configuration using a brace as shown in Fig. 2(b). The thigh of the subject was fixed at a certain angle (using the motor in the brace), while the new controller was used to track a sinusoidal trajectory with a 2 second time period. Robustness of the controller to an estimate EMD value was tested by evaluating its performance for estimated EMD values that ranged between 2 standard deviations (SD) from the mean of the measured EMD value. VI. EXPERIMENTAL RESULTS
Fig. 3. Experimental results obtained from the representative trial for each of the three controllers. These plots show the desired and actual angular position (top plots), error (middle plots), and stimulation current amplitude (bottom plots).
ducting the five trials for each controller, the controller was further tuned, beginning at the initial guess. The control gains were fine tuned till the error over a 10 second trial was minimized. Since the PD-DC and PID-DC require the knowledge of the EMD value, it was determined empirically for each test subject. This was done by applying a step stimulation at a time instant and measuring the time when the resulting knee joint
Figs. 3–6 and Tables I–IV illustrate the results from the two sets of experiments. In the first set of experiments, each controller was tested in five trials on each subject and then again on one subject with a larger range of motion. Trials for the PD-DC and PID-DC used the empirically found EMD values as reported in Table III. A representative trial from each controller is shown in Fig. 3 and the control gains used to produce those results are given in Table I. Table III contains the three criteria used to measure the performance of each controller: the root mean squared of the error (RMSE), steady state RMSE (SSRMSE), and root mean squared of the current (RMSC). The RMSC was normalized by the body mass index of each subject in order to scale the control effort with respect to the size of the subjects. Since the RISE and PID-DC controller have integral control, they require a transient period for the memory component to build up, that is why the SSRMSE is the primary focus of the comparison. The differences in the aforementioned performance criteria were assessed using a one-way analysis of variance (ANOVA)
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Fig. 5. Graphical representation of the results from the LEM experiments. The three criteria: RMSE, SSRMSE, and RMSC were normalized by the maximum of each criterion after the ANOVA analyses were done. indicates statistically significant differences between the controllers at a 95% confidence level and p refers to the p-value.
with repeated measures. Post-hoc testing consisting of paired t-tests with a Bonferroni correction was performed when a significant ANOVA tests was identified. A Bonferroni adjustment was used to avoid potential type I errors associated with performing multiple t-tests in the post-hoc analysis. As a result, the critical threshold for significance was reduced to (0.05 divided by 3). The results for the statistical analyses are given in Fig. 5, where each criterion's amplitude has been normalized to the maximum criterion value. This normalization is performed strictly for plotting the results and is completely independent of the ANOVA analyses. The SSRMSE of the PID-DC controller was found to be significantly lower than that of the other controllers ( between PD-DC and PID-DC and between PID-DC and RISE). The RISE controller's SSRMSE was found to be lower than that of the PD-DC controller ( ). Although the SSRMSE was used to make the main comparison, the RMSE results were found to be concurrent with the SSRMSE results and can be seen in Table III. Lack of integral control, which gives a controller a memory component in order to compensate for steady state errors, in the PD-DC controller seems to have played a role in its greater RMSE and SSRMSE as compared to the other two controllers. While tuning the controllers, it was observed that in order to maintain stability with the RISE controller, the gain associated with integral control in the PID component of the controller, (i.e., ), was required to be kept low. This could be due to the combination of controller's increased responsiveness due to integral action and EMD-induced oscillations. Unlike the RISE controller, the PID-DC did not have this issue because the DC component provided the controller more robustness by removing the ill-timed excess energy that induced these oscillations. One thing to note is that because of the DC component the gains for the PD-DC and PID-DC were required to be much
Fig. 6. Tracking performance of PID-DC with mismatched EMDs that were set to 2 SDs away from the mean measured EMD. With mismatched EMD settings, the performance degraded but the PID-DC maintained stability. Top plot: the control performance, before tuning . Bottom plot: the tracking performance improved after tuning .
TABLE I GAINS USED TO PRODUCE THE EXPERIMENTAL RESULTS SHOWN IN FIG. 3
TABLE II TABULATED RESULTS FOR THE SINUSOIDAL TRAJECTORY WITH 2 SECOND TIME PERIOD RANGING FROM 5–50 IMPLEMENTED ON SUBJECT H1 WITH ALL THREE CONTROLLERS
larger than the gains for the RISE controller. This does not necessarily mean the PD-DC and PID-DC controllers were using more control effort. This is simply due to the mechanics of the DC part of the controllers. The DC part of the controller integrates the control signal over an interval the duration of the input
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TABLE III TABULATED RESULTS FOR THE SINUSOIDAL TRAJECTORY WITH 2 SECOND TIME PERIOD EXPERIMENTS INCLUDE THE AVERAGE (AVG) AND STANDARD DEVIATION (SD) OF THE EMD VALUES MEASURED FOR EACH VOLUNTEER SUBJECT (S) ACROSS FIVE SAMPLES, THE AVERAGE ROOT MEAN SQUARED ERROR (RMSE), THE AVERAGE STEADY STATE RMSE (SSRMSE), AND THE AVERAGE ROOT MEAN SQUARED CURRENT (RMSC) NORMALIZED BY THE BODY MASS INDEX OF EACH SUBJECT
TABLE IV TABULATED RESULTS FOR THE ROBUSTNESS OF THE ESTIMATED EMD VALUE IN THE PID-DC CONTROLLER AS OPPOSED TO THE MEASURED EMD . NOTE THAT THE CONTROLLER WAS ABLE TO MAINTAIN VALUE OF 85 CONSISTENT PERFORMANCE EVEN WITH MISMATCHED EMDS AFTER TUNING (ATB). THE RESULTS FOR THE ESTIMATED EMD VALUES BEFORE TUNING (BTB) WERE INCLUDED TO DEMONSTRATE THE EFFECT OF TUNING
the DC portion of the controller over compensated due to the larger integral interval; therefore, decreasing helped in offsetting the effect of mismatched EMD values. Similarly, increasing helped in offsetting the effect of mismatched EMD values, when an estimated EMD value was smaller than the measured EMD value. VII. DISCUSSION
delay and subtracts it from the PID component of the controller. Therefore, most of the control effort generated from the PID component was dissipated due to the DC component. Although results of the PD-DC controller reported the lowest RMSC as compared to the other two controllers, the statistical analysis determined that the differences in the RMSCs were not statistically significant ( between PD-DC and PID-DC, between PD-DC and RISE, between PID-DC and RISE). From Fig. 4 and Table II, it can be seen that even with larger ranges of motion, the PID-DC continued to outperform the other two controllers. In the second set of experiments, the measured EMD value was empirically found to be 85 for Subject H1. The experimental results for tracking the sinusoidal trajectory with a 2 second time period, where the PID-DC controller used estimate EMD values different from the measured EMD value, are given in Fig. 6 and Table IV. As the estimated EMD value strayed from the measured EMD value, before tuning (BTB), the control performance deteriorated but the controller maintained stability. However, after tuning (ATB), the PID-DC controller not only maintained stability but also provided the same level of performance as in the first set of experiments. This was observed even when the estimated EMD value was varied by 2 SDs from the measured EMD value. Results when the estimated EMD values were assumed to be 2 SDs away from the measured EMD values are plotted in Fig. 6. The decrease in control performance, resulting from mismatched delay estimates, was compensated for by tuning the gain . For example, for the estimate EMD values greater than the measured EMD value,
Advanced NMES controllers that account for the nonlinearities and delays in the musculoskeletal system can play a significant role in the rehabilitation of limb function in mobility-impaired persons. Potentially, these controllers can be applied for various NMES-elicited tasks such as gait training or restoration, upper limb reaching, functional electrical stimulation (FES) cycling, rowing, etc. Therefore, it is important to investigate and choose the most suitable NMES controller among different available options. The new nonlinear controller, PID-DC, showed promising results compared to its predecessors: the RISE controller [13] and the PD-DC controller [41]. These three controllers were compared because they are all strictly feedback NMES controllers that were synthesized using Lyapunov stability and control design methods. The PID-DC controller showed statistically significant improvements in the tracking performance compared to the other controllers. Moreover, the new controller was shown to be robust to variations in the EMD values. Although the RISE and PID-DC controllers both have a PID based structure, the RISE controller was not designed to compensate for EMDs. Similarly, the PD-DC controller was designed to compensate for EMDs but lacks integral control. Due to these reasons, the RISE and the PD-DC controllers showed poor performance compared to the new controller. We also want to emphasize on the significance of Lyapunovbased control design approach that was used to develop the new controller. Lyapunov-based control synthesis is an exciting control design tool for nonlinear systems, such as a musculoskeletal system. More importantly, the nonlinear system is not required to be linearized, which results in a better match between the system model and the physical system. With Lyapunov stability analysis, a guarantee on the controller stability can be provided. This is especially crucial in applications such as NMES where safety risks could occur if the controller is unstable. Moreover, as shown in the control development, the controller can be implemented despite no knowledge of
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the musculoskeletal model. This is important as the musculoskeletal model is typically uncertain and its parameters are subject to day-to-day variations. Clinically, it also suggests that the NMES controller can be implemented without having to estimate the model parameters of a subject, which can be time consuming. Although this strictly feedback controller showed promising results, it does not prove that model-free controllers are the best option. The addition of a feedforward component to a feedback controller such as in [17], [49] can further improve closed-loop performance of the system. A model-based controller, or an artificial neural network controller, that works in conjunction with PID-DC based feedback may be a more promising control strategy. Limitations of the Study: Although the controller was determined to be robust to variations in the estimated EMDs, the controller was designed with the assumption that the EMD is constant. This is a limitation because the EMD could be time varying; e.g., variance of the EMD over time was linked to muscle fatigue [48]. Our study evaluated the controller on able bodied subjects and was designed for a single degree of freedom system. The effectiveness of the new controller remains to be proven on persons with stroke or SCI and its performance on FES-based rehabilitation systems with multiple degrees of freedom remains to be seen. Our control design also neglected the effects of activation dynamics, as accounted for by other studies in [10], [14], [47]. The latter two issues are the focus of our future research. VIII. CONCLUSION In this paper, a PID-based delay compensation (PID-DC) controller was developed for an NMES-driven musculoskeletal system with EMDs. Lyapunov-based stability analysis yielded semi-globally uniformly ultimately bounded tracking despite model uncertainties and EMDs. The main motivation for developing the new controller was to add an integral action in our previous control design for EMD compensation. The addition of the integral action resulted in improved performance that was validated on four able bodied subjects and empirically compared to two nonlinear controllers: PD-DC (previous control design for EMD compensation) and RISE controller. The results showed that the PID-DC has a superior tracking performance (statistically significant) vis-á-vis the other two controllers. Further, the new controller was shown to be robust to variations in the measured EMDs. Future work will focus on testing this controller on persons with stroke or SCI, investigating the benefits of adding activation dynamics, considering spasticity and fatigue [49], and extending the controller to multiple degrees of freedom and more general systems.
The two integrals in (20) can be rearranged to
(21) , perform the following change of variables, For take , , where the new bounds of integration can be computed as: , . Thus, . By further separating the integral, it yields . For , another change of variables such that , where the bounds can be computed as , , yields . Assuming that in the interval , the second integral is equal to zero. Because we are only concerned with positive values of time, this assumption is not intrusive to the result. Thus, the integrals in (21) can be expressed as
Take , , and taking the Laplace transform of both sides we get
, after
Thus,
Taking the inverse Laplace transform results in
Theorem 1 Proof: Let
be defined as (22)
A positive definite Lyapunov functional candidate is defined as (23) and satisfies the following inequalities
APPENDIX
(24)
Lemma 1 Proof: Using the fact that and (7) the time derivative of (9) can be expressed as
where
,
are known constants defined as
(20) where
and
are defined in (4).
ALIBEJI et al.: FURTHER RESULTS ON PREDICTOR-BASED CONTROL OF NEUROMUSCULAR ELECTRICAL STIMULATION
After using (6), (7), and (12), and canceling the common terms, the time derivative of (23) can be expressed as
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Rearranging the terms and canceling the common terms results in
(25)
After applying the Young's Inequality and utilizing the definition of in (13), the following terms in (25) can be bounded as
(31) After utilizing (10), completing the squares, and provided that , (31) can be upper bounded as
(26) (27) (32)
Further, by using the Cauchy Schwarz inequality, the following term in (27) can be upper bounded as Since (28) Using (14), (26), and (28), (25) can be bounded as
and after utilizing (15), (16), and (17), the inequality in (32) can be expressed as (33) (29) Note that by choosing control gains and such that , the following Young's inequality can be used
where
is defined as
Using the definition of be upper bounded as
in (22), the expression in (33) can
(30) After adding and subtracting using (28), provided that be upper bounded as
to (29) and , (29) can
(34) where
is defined as
In order to further bound (34), it is required that which is true if Consider a set defined as
.
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can be lower bounded by a constant
as
and the condition, , is satisfied. By further utilizing (24), the inequality in (34) can be expressed as (35) The linear differential equation in (35) can be solved as (36) provided the control gains , , , and are selected according to the sufficient conditions in (19) (i.e. a semi-global result). The result in (18) can now be obtained from (36). Based on the definition of , the result in (36) indicates that , , in . Since , in , then (13) indicates that in . Given that , , , in , (5) and (6) indicate that and in . Since , , , , , in , then (7) indicates that in . Given that , , , , in , (8) and Assumptions 3 and 4 indicate that in . REFERENCES [1] P. H. Peckham and J. S. Knutson, “Functional electrical stimulation for neuromuscular applications,” Annu. Rev. Biomed. Eng., vol. 7, pp. 327–360, 2005. [2] V. Nekoukar and A. Erfanian, “A decentralized modular control framework for robust control of FES-activated walker-assisted paraplegic walking using terminal sliding mode and fuzzy logic control,” IEEE Trans. Biomed. Eng., vol. 59, no. 10, pp. 2818–2827, Oct. 2012. [3] N. Sharma, V. Mushahwar, and R. Stein, “Dynamic optimization of FES and orthosis-based walking using simple models,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 22, no. 1, pp. 114–126, Jan. 2014. [4] M. M. Skelly and H. J. Chizeck, “Real-time gait event detection for paraplegic FES walking,” IEEE Trans. Neural Sys. Rehab. Eng., vol. 9, no. 1, pp. 59–68, 2001. [5] T. A. Thrasher, V. Zivanovic, W. McIlroy, and M. R. Popovic, “Rehabilitation of reaching and grasping function in severe hemiplegic patients using functional electrical stimulation therapy,” Neurorehab. Neural. Re., vol. 22, no. 6, pp. 706–714, 2008. [6] E. K. Chadwick, D. Blana, J. D. Simeral, J. Lambrecht, S. P. Kim, A. S. Cornwell, D. M. Taylor, L. R. Hochberg, J. P. Donoghue, and R. F. Kirsch, “Continuous neuronal ensemble control of simulated arm reaching by a human with tetraplegia,” J. Neural Eng., vol. 8, no. 3, p. 034003, 2011. [7] P. Cooman and R. F. Kirsch, “Control of a time-delayed 5 degrees of freedom arm model for use in upper extremity functional electrical stimulation,” in Proc. Int. IEEE EMBC, 2012, pp. 322–324, IEEE. [8] E. M. Schearer, Y.-W. Liao, E. J. Perreault, M. C. Tresch, W. D. Memberg, R. F. Kirsch, and K. M. Lynch, “Multi-muscle FES force control of the human arm for arbitrary goals,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 22, no. 3, pp. 654–663, May 2014. [9] R. Davoodi, B. J. Andrews, G. D. Wheeler, and R. Lederer, “Development of an indoor rowing machine with manual FES controller for total body exercise in paraplegia,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 10, no. 3, pp. 197–203, 2002. [10] S. Jezernik, R. G. V. Wassink, and T. Keller, “Sliding mode closedloop control of FES: Controlling the shank movement,” IEEE Trans. Biomed. Eng., vol. 51, pp. 263–272, 2004. [11] T. Schauer, N. O. Negard, F. Previdi, K. J. Hunt, M. H. Fraser, E. Ferchland, and J. Raisch, “Online identification and nonlinear control of the electrically stimulated quadriceps muscle,” Control Eng. Pract., vol. 13, pp. 1207–1219, 2005.
[12] M. M. Ebrahimpour and A. Erfanian, “Comments on “sliding mode closed-loop control of FES: controlling the shank movement”,” IEEE Trans. Biomed. Eng., vol. 55, no. 12, p. 2842, 2008. [13] N. Sharma, K. Stegath, C. M. Gregory, and W. E. Dixon, “Nonlinear neuromuscular electrical stimulation tracking control of a human limb,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 17, no. 6, pp. 576–584, 2009. [14] N. Sharma, P. Patre, C. Gregory, and W. E. Dixon, “Nonlinear control of NMES: Incorporating fatigue and calcium dynamics,” in Proc. ASME Dyn. Syst. Control Conf., Hollywood, CA, Oct. 2009, pp. 705–712. [15] R. J. Downey, T.-H. H. Cheng, and W. E. Dixon, “Tracking control of a human limb during asynchronous neuromuscular electrical stimulation,” in Proc. IEEE 52nd CDC, Dec. 2013, pp. 139–144. [16] W. K. Durfee and J. M. Hausdorff, “Regulating knee joint position by combining electrical stimulation with a controllable friction brake,” Ann. Biomed. Eng., vol. 18, no. 6, pp. 575–596, 1990. [17] N. Sharma, C. Gregory, M. Johnson, and W. E. Dixon, “Closed-loop neural network-based NMES control for human limb tracking,” IEEE Trans. Control Syst. Technol., vol. 20, pp. 712–725, 2012. [18] K. J. Hunt, B. Stone, N. O. Negard, T. Schauer, M. H. Fraser, A. J. Cathcart, C. Ferrario, S. A. Ward, and S. Grant, “Control strategies for integration of electric motor assist and functional electrical stimulation in paraplegic cycling: utility for exercise testing and mobile cycling,” IEEE Trans Neural Syst. Rehab. Eng., vol. 12, no. 1, pp. 89–101, March 2004. [19] M. J. Bellman, T.-H. H. Cheng, R. J. Downey, and W. E. Dixon, “Stationary cycling induced by switched functional electrical stimulation control.,” in Proc. ACC, Portland, OR, 2014. [20] H. Kawai, M. Bellman, R. J. Downey, and W. E. Dixon, “Tracking control for FES-cycling based on force direction efficiency with antagonistic bi-articular muscles,” in Proc. ACC, 2014. [21] J. J. Abbas and H. J. Chizeck, “Feedback control of coronal plane hip angle in paraplegic subjects using functional neuromuscular stimulation,” IEEE Trans. Biomed. Eng., vol. 38, no. 7, pp. 687–698, 1991. [22] N. Lan, P. E. Crago, and H. J. Chizeck, “Feedback control methods for task regulation by electrical stimulation of muscles,” IEEE Trans. Biomed. Eng., vol. 38, no. 12, pp. 1213–1223, 1991. [23] A. H. Vette, K. Masani, and M. R. Popovic, “Implementation of a physiologically identified PD feedback controller for regulating the active ankle torque during quiet stance,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 15, no. 2, pp. 235–243, Jun. 2007. [24] F. Previdi, M. Ferrarin, S. M. Savaresi, and S. Bittanti, “Gain scheduling control of functional electrical stimulation for assisted standing up and sitting down in paraplegia: a simulation study,” Int. J. Adapt. Control Signal Process., vol. 19, pp. 327–338, 2005. [25] H. Gollee, K. J. Hunt, and D. E. Wood, “New results in feedback control of unsupported standing in paraplegia,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 12, no. 1, pp. 73–91, 2004. [26] K. J. Hunt and M. Munih., “Feedback control of unsupported standing in paraplegia-part 1: Optimal control approach,” IEEE Trans. Rehab. Eng., vol. 5, pp. 331–340, 1997. [27] V. Nekoukar and A. Erfanian, “Adaptive terminal sliding mode control of ankle movement using functional electrical stimulation of agonist-antagonist muscles,” in Proc. Int. IEEE EMBC, Aug. 2010, pp. 5448–5451. [28] M. Benoussaad, K. Mombaur, and C. Azevedo-Coste, “Nonlinear model predictive control of joint ankle by electrical stimulation for drop foot correction,” in Proc. IEEE/RSJ Int. Conf. IROS, 2013, pp. 983–989, IEEE. [29] N. Sharma, C. Gregory, and W. E. Dixon, “Predictor-based compensation for electromechanical delay during neuromuscular electrical stimulation,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 19, no. 6, pp. 601–611, 2011. [30] N. Sharma, “A predictor-based compensation for electromechanical delay during neuromuscular electrical stimulation-II,” in Proc. ACC, 2012, pp. 5604–5609. [31] Q. Wang, N. Sharma, M. Johnson, C. M. Gregory, and W. E. Dixon, “Adaptive inverse optimal neuromuscular electrical stimulation.,” IEEE Trans. Cybern., vol. 43, no. 6, p. 1710, 2013. [32] S. Zhou, D. L. Lawson, W. E. Morrison, and I. Fairweather, “Electromechanical delay in isometric muscle contractions evoked by voluntary, reflex and electrical stimulation,” Eur. J. Appl. Physiol. Occup. Physiol., vol. 70, no. 2, pp. 138–145, 1995. [33] P. R. Cavanagh and P. V. Komi, “Electromechanical delay in human skeletal muscle under concentric and eccentric contractions,” Eur. J. Appl. Physiol. Occup. Physiol., vol. 42, no. 3, pp. 159–163, 1979.
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[34] K. Masani, A. H. Vette, N. Kawashima, and M. R. Popovic, “Neuromusculoskeletal torque-generation process has a large destabilizing effect on the control mechanism of quiet standing,” J. Neurophysiol., vol. 100, no. 3, p. 1465, 2008. [35] A. H. Vette, K. Masani, and M. R. Popovic, “Neural-mechanical feedback control scheme can generate physiological ankle torque fluctuation during quiet standing: A comparative analysis of contributing torque components,” in Proc. IEEE Int. CCA, 2008, pp. 660–665, IEEE. [36] K. Masani, A. H. Vette, and M. R. Popovic, “Controlling balance during quiet standing: proportional and derivative controller generates preceding motor command to body sway position observed in experiments.,” Gait & Posture, vol. 23, no. 2, pp. 164–172, Feb. 2006. [37] O. J. M. Smith, “A controller to overcome deadtime,” J. ISA, vol. 6, pp. 28–33, 1959. [38] Z. Artstein, “Linear systems with delayed controls: A reduction,” IEEE Trans. Autom. Control, vol. 27, no. 4, pp. 869–879, 1982. [39] A. Manitius and A. Olbrot, “Finite spectrum assignment problem for systems with delays,” IEEE Trans. Autom. Control, vol. 24, no. 4, pp. 541–552, 1979. [40] W. Michiels, K. Engelborghs, P. Vansevenant, and D. Roose, “Continuous pole placement for delay equations,” Automatica, vol. 38, no. 5, pp. 747–761, 2002. [41] N. Sharma, S. Bhasin, Q. Wang, and W. E. Dixon, “Predictor-based control for an uncertain Euler-Lagrange system with input delay,” Automatica, vol. 47, no. 11, pp. 2332–2342, 2011. [42] W. L. Buford, Jr., F. M. Ivey, Jr., J. D. Malone, R. M. Patterson, G. L. Peare, D. K. Nguyen, and A. A. Stewart, “Muscle balance at the knee—moment arms for the normal knee and the ACL—minus knee,” IEEE Trans. Rehab. Eng., vol. 5, no. 4, pp. 367–379, 1997. [43] J. L. Krevolin, M. G. Pandy, and J. C. Pearce, “Moment arm of the patellar tendon in the human knee,” J. Biomech., vol. 37, pp. 785–788, 2004. [44] R. Nathan and M. Tavi, “The influence of stimulation pulse frequency on the generation of joint moments in the upper limb,” IEEE Trans. Biomed. Eng., vol. 37, pp. 317–322, 1990. [45] T. Watanabe, R. Futami, N. Hoshimiya, and Y. Handa, “An approach to a muscle model with a stimulus frequency-force relationship for FES applications,” IEEE Trans. Rehab. Eng., vol. 7, no. 1, pp. 12–17, 1999. [46] S. A. Binder-Macleod, E. E. Halden, and K. A. Jungles, “Effects of stimulation intensity on the physiological responses of human motor units,” Med. Sci. Sport Exer., vol. 27, no. 4, pp. 556–565, 1995. [47] C. L. Lynch and M. R. Popovic, “A comparison of closed-loop control algorithms for regulating electrically stimulated knee movements in individuals with spinal cord injury,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 20, no. 4, pp. 539–548, July 2012. [48] F. Bouillon, “Measure, modeling and compensation of fatigue-induced delay during neuromuscular electrical stimulation,” Ph.D. dissertation, University of Florida, , 2013. [49] P. Cooman, “Nonlinear feedforward-feedback control of an uncertain, time-delayed musculoskeletal arm model for use in functional electrical stimulation,” Ph.D. dissertation, , Case Western Univ., Cleveland, 2014.
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Naji Alibeji received the B.S. degree (Summa Cum Laude) from the University of Pittsburgh, Pittsburgh, PA, USA, in mechanical engineering in 2012. He is currently pursuing a Ph.D. in mechanical engineering for his research in nonlinear controls under the advisement of Dr. Sharma. As an undergraduate, he spent two summers working at Acutronic USA as a mechanical assembler/designer.
Nicholas Kirsch received the B.S. degree in mathematics with a physics minor from St. Vincent College, Latrobe, PA, USA, in 2008. At the University of Pittsburgh, Pittsburgh, PA, USA, he received the B.S. and M.S. degrees in mechanical engineering, and is currently pursuing a Ph.D. degree in mechanical engineering for his research in restorative gait control.
Shawn Farrokhi received the Ph.D. degree in biokinesiology from the University of Southern California, Los Angeles, CA, USA. He is an Associate Professor in the Departments of Physical Therapy and Bioengineering and the Co-director of the Human Movement Research Laboratory at the University of Pittsburgh, Pittsburgh, PA, USA.
Nitin Sharma (M’14) received the Ph.D. degree in 2010 from the Department of Mechanical and Aerospace Engineering at the University of Florida, Gainesville, FL, USA. He was an Alberta Innovates Health Solutions postdoctoral fellow in the Department of Physiology at the University of Alberta, Edmonton, Canada. Since 2012, he has been an Assistant Professor in the Department of Mechanical Engineering and Material Science at the University of Pittsburgh, Pittsburgh, PA, USA. His research interests are robust control of functional electrical stimulation (FES) and modeling and control of FES-elicited walking.
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Further Results on Predictor-Based Control of Neuromuscular Electrical Stimulation Naji Alibeji, Nicholas Kirsch, Shawn Farrokhi, and Nitin Sharma, Member, IEEE
Abstract—Electromechanical delay (EMD) and uncertain nonlinear muscle dynamics can cause destabilizing effects and performance loss during closed-loop control of neuromuscular electrical stimulation (NMES). Linear control methods for NMES often perform poorly due to these technical challenges. A new predictorbased closed-loop controller called proportional integral derivative controller with delay compensation (PID-DC) is presented in this paper. The PID-DC controller was designed to compensate for EMDs during NMES. Further, the robust controller can be implemented despite uncertainties or in the absence of model knowledge of the nonlinear musculoskeletal dynamics. Lyapunov stability analysis was used to synthesize the new controller. The effectiveness of the new controller was validated and compared with two recently developed nonlinear NMES controllers, through a series of closed-loop control experiments on four able-bodied human subjects. Experimental results depict statistically significant improved performance with PID-DC. The new controller is shown to be robust to variations in an estimated EMD value. Index Terms— Electromechanical delay, functional electrical stimulation, input delay, Lyapunov–Krasovskii functionals, Lyapunov Methods, neuromuscular electrical stimulation (NMES), nonlinear control.
I. INTRODUCTION
N
EUROLOGICAL dysfunctions caused by spinal cord injury (SCI), stroke, multiple sclerosis, traumatic brain injury, etc. are the leading causes of mobility disorders among adults across the world. Loss of, or limited, limb functionality in mobility impaired individuals hampers their ability to perform activities of daily living. Neuromuscular electrical stimulation (NMES) is an artificial application of electrical potential across a muscle group to elicit a muscle contraction. NMES is prescribed as an intervention to rehabilitate or restore functionality in mobility-impaired individuals [1]. For example, NMES can be used to produce advanced functional tasks such as walking [2]–[4], upper extremity reaching and grasping [5]–[8], rowing [9], single leg extension [10]–[17], and stationary bicycling [18]–[20]. Closed-loop control is typically employed to achieve accurate and precise limb control during NMES-elicited tasks. Manuscript received July 10, 2014; revised November 01, 2014; accepted March 03, 2015. Date of publication April 02, 2015; date of current version November 04, 2015. (Corresponding author: Nitin Sharma.) N. Alibeji, N. Kirsch, and N. Sharma are with the University of Pittsburgh, Department of Mechanical Engineering and Materials Science, Pittsburgh, PA 15261 USA (e-mail: [email protected]; [email protected]; [email protected]). S. Farrokhi is with University of Pittsburgh, Department of Physical Therapy, Pittsburgh, PA 15260 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNSRE.2015.2418735
However, closed-loop control of NMES is challenging due to the uncertain and nonlinear musculoskeletal dynamics, presence of muscle fatigue and unmodeled disturbances such as muscle spasticity or external changes in muscle loads, electromechanical delay (EMD), etc. Linear control methods such as in [21]–[23] or methods that assume a linear model (e.g., linear quadratic gaussian control, pole placement method, gain scheduling control [24]–[26]) often lack in performance. Moreover, linear control techniques do not guarantee stability during NMES-elicited tasks. Recently, nonlinear control techniques have been implemented for NMES applications [7], [10]–[15], [17], [27], [28] to obtain improved error performance and/or asymptotic tracking. Lyapunov based control synthesis and stability analysis is an exciting tool that has been used to design nonlinear and robust NMES controllers [13], [14], [17], [19], [29]–[31]. For example, Lyapunov stability analysis was used to design NMES controllers that are robust to uncertainties and bounded exogenous disturbances in a single leg extension task [13], [15], [17], [29]–[31] and cycling [19], [20]. Further results in [14] used a backstepping control design to incorporate calcium dynamics and muscle fatigue dynamics with a motivation to develop an NMES controller that compensates for muscle fatigue. In this paper, we design a new robust controller to compensate for EMDs using Lyapunov stability analysis. EMD arises due to a time lag between electrical excitation of the motor-neurons and tension development in a muscle. It is a function of a number of phenomena including: finite propagation time of the chemical ions in the muscle, cross-bridge formation between actin-myosin filaments, stretching of the series elastic components in response to the external electrical input, synaptic transmission delays, and others [32], [33]. In results such as [10], [11], [29], the EMD is modeled as an input delay in the musculoskeletal dynamics. Input delays can cause performance degradation and have also been reported to potentially cause instability during human stance experiments [34]. Despite the fact that EMDs are exhibited in muscle response and can lead to instability, NMES controllers that actively compensate for this phenomena are lacking. Previous results, such as in [23], [34]–[36], examine EMD effects by implementing a standard proportional derivative (PD) controller during stance (or quiet standing) experiments that show robustness to the delays. The controllers in such results are not modified to include a delay compensation (DC) term and have no mathematical proof of stability when the plant has uncertainties, nonlinearities, and/or disturbances. Various methods exist in the general time-delay control literature to compensate for actuator or input delays, but the ex-
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isting approaches such as Smith predictor methods [37], Artstein model reduction [38], finite spectrum assignment [39], and continuous pole placement [40] are applicable to only linear plants. The control problem becomes especially complicated when parametric uncertainties, nonlinearities, and additive disturbances are considered. Recently, two predictor-based control methods were developed for an uncertain input delayed system with additive disturbances [41]. These results suggest that a PD or a proportional integral derivative (PID) controller can be augmented with a delay compensator that contains a finite integral of past control values to transform the delayed system into a delay-free system. Motivated by the modified PD control result in [41], a tracking controller for a nonlinear musculoskeletal system with known constant input delay, the proportional derivative with delay compensation (PD-DC), was developed in [29]. Only the modified PD control result was extended in [29] as the modified PID controller in [41] requires a knowledge of inertia matrix. Thus, the PID-type delay compensating control design proposed in [41] cannot be implemented because of many uncertainties in the musculoskeletal system that form an unknown nonlinear input gain to the voltage input. These uncertainties include the unknown moment of inertia, muscle force-length and force-velocity relationships, moment arm, etc. This paper presents a PID-type delay compensating controller (with a desire to include integral feedback) that overcomes EMDs during NMES. A preliminary work on the control development and stability analysis was presented in [30]. Due to a different control structure employed in this paper, as opposed to the control design approach in [41], the requirement of input gain or inertia matrix knowledge was relaxed in the new result. Therefore, the new controller can be implemented despite parametric uncertainty and additive bounded disturbances. The key contributions of this effort are the design of a delay compensating auxiliary signal to obtain a time delay free open-loop error system, incorporation of a constant control gain that compensates for unknown nonlinear input gain function, and the construction of Lyapunov Krasovskii (LK) functionals which are used in a Lyapunov-based analysis that proves the tracking error is semi-globally uniformly ultimately bounded. The new controller was applied as an amplitude modulated current to external electrodes attached to the distal-medial and proximal-lateral portion of the quadriceps femoris muscle group in 4 non-impaired volunteers. The new controller was compared with its predecessors: PD-DC [29] and the Robust Integral of the Sign of the Error (RISE) [13]. The experimental results indicated that the new controller reduced the steady state root mean squared tracking error (SSRMSE) compared to the two previously developed nonlinear controllers, and was found to be robust to variations in the estimated EMD value.
Fig. 1. Schematic of a subject sitting in the leg extension machine with a knee torque generated via NMES.
where , , are the angular position, velocity, and acceleration of the knee joint, respectively. In (1), is the moment of inertia of the lower shank, denotes the elastic effects due to joint stiffness, denotes the gravitational forces, and denotes the viscous effects due to the damping in the knee joint. Further details of these terms and their definitions are given in [30]. In (1), denotes any unmodeled phenomena or disturbances in the system. The torque produced at the knee due to NMES is denoted as , and denotes the constant input delay caused by electromechanical processes associated with NMES. The torque produced at the knee is generated through NMES of the quadriceps muscles. This torque is defined as (2) denotes the unknown moment arm function, where denotes the unknown force-length and force-velocity relationships, and denotes the artificial voltage/current applied across the quadriceps muscle. The muscle activation dynamics were neglected in order to simplify the control design. The following assumptions and notations are used to facilitate the subsequent control development and stability analysis. Assumption 1: The moment arm is assumed to be a non-zero, positive, bounded function [42], [43] whose first time derivative exists and is continuous. The function is assumed to be a non-zero, positive, and bounded function with a bounded and continuous first time derivative based on the empirical data [44], [45]. Assumption 2: The auxiliary non-zero unknown scalar function , which acts as a nonlinear input gain function to the applied voltage on the muscle, is defined as (3)
II. MUSCULOSKELETAL SYSTEM The musculoskeletal system is a model of a subject sitting in a leg extension machine (LEM) as depicted in Fig. 1. NMES, via surface electrodes, elicits contractions in the quadriceps muscle group that produce a knee-joint torque to extend the swing arm of LEM. The knee-joint dynamics are modeled as (1)
where the first time derivative of is assumed to exist, be bounded, and continuous (see Assumption 1). Assumption 3: The unknown disturbance is bounded. Its first time derivative exists and is bounded and continuous. Based on Assumptions 1 and 2 the ratio / denoted as , is also assumed to be bounded and its first time derivative exists and is bounded and continuous.
ALIBEJI et al.: FURTHER RESULTS ON PREDICTOR-BASED CONTROL OF NEUROMUSCULAR ELECTRICAL STIMULATION
Assumption 4: Based on Assumption 1, the ratio / denoted as , can be upper bounded as
,
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Based on (8) and to facilitate the subsequent analysis, the voltage input is designed as
(4) where are known constants. The force-length and force-velocity values will never allow the lower bound of to be zero (i.e.; ) within the operating region of the knee joint. Assumption 5: The EMD, denoted by , is assumed to be a known constant. Factors, such as fatigue, may cause it to be a time-varying phenomenon; however, the influence of these factors on EMDs are ignored. Assumption 6: The desired trajectory, , and its time derivatives, , are bounded and continuous. Notation: A delayed state in the subsequent control development and analysis is denoted as or as while a non-delayed state is denoted as or as . Remark 1: Assumptions 1–5 are made mainly for stability analysis and control design. These assumptions are based on empirical results as cited above. These assumptions, except Assumption 5, are standard assumptions and have no or little practical significance as shown in the experimental results presented in [13], [17], [29].
(9) where
is a known control gain that can be expanded as (10)
, , and are known constants. where in (9) can be written Lemma 1: The time derivative of as (11) Proof: See Appendix. After differentiating (8) with respect to time and using (11), the closed loop error system can be written as (12)
III. CONTROL DEVELOPMENT The control objective is to track a continuously differentiable desired trajectory, . The tracking error is defined as
In (12), the auxiliary function are defined as
and
(5) To facilitate the control design and stability analysis, the auxiland are defined as iary error signals (6) (7)
where
,
,
where , , and are inertial, viscous, and stiffness terms (defined in (1)) that are exand velocity , pressed in terms of desired limb position and is defined as
are known positive control gains.
A. Closed Loop Error System The open-loop tracking error system can be developed by multiplying (7) by and utilizing the expressions in (1)–(3), (5), and (6) to obtain
(13)
Using the mean value theorem, the auxiliary functions and can be upper bounded as (8)
where the nonlinear functions are defined as
,
, and
(14) is a known constant, the bounding function is a positive globally invertible non-decreasing is defined as function, and
In (14),
(15)
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Based on the subsequent stability analysis, LK functionals: and are defined as (16)
(17) where ,
are known constants. IV. STABILITY ANALYSIS
Theorem 1: The controller given in (9) ensures semi-globally uniformly ultimately bounded tracking:
Fig. 2. Testbeds used in this study: (A) leg extension machine (LEM) and (B) brace. The LEM was used to compare the three different controllers, and the brace was used to test the robustness of PID-DC delay setting.
(18) where , , denote constants, provided the control gains , , , , and introduced in (6), (7), (9), and (10), respectively, are selected according to the following sufficient conditions:
(19) In (19), the known positive constants , , , and are defined in (4), (16), and (17), respectively, is the input delay and is a subsequently defined constant. Proof: See Appendix. Remark 2: From the gain conditions in (19), a sufficient condition on the time delay can be derived, . Therefore, the stability of the controller is only guaranteed for a certain magnitude of the input delay, which can be increased to a certain limit provided that . V. EXPERIMENTAL PROTOCOL The experiments were conducted on two testbeds as shown in Fig. 2: a modified LEM (A) and a brace (B) to mimic lower leg swing during walking. The LEM was modified to have physical stops to prevent hyperflexion and hyperextension, and an incremental optical encoder [Calt, CN] with 1024 pulses per revolution (ppr) resolution to measure the knee angle. The brace consisted of a motor at the hip joint and an incremental optical encoder [Hengxiang, CN] with 1024 ppr resolution at the knee joint. The QPIDe [Quanser Inc, Ontario Canada] data acquisition board was used to power and measure the output of the optical encoder and run the controllers in real time. An FNS-8 channel stimulator [CWE Inc., PA USA] was used to generate the biphasic pulse train transmitted to the surface electrodes. The pulse trains that were used had a frequency of 35 , which studies have shown is the optimal frequency for NMES [46], and a pulse width of 400 . Current amplitude modulation was used for the experiments. All control algorithms were coded in Simulink [MathWorks Inc, USA] and implemented using the Quarc real-time software [Quanser Inc, Ontario Canada]. Four able bodied male subjects between the ages of 24–30 years were selected for the experiments. Prior to any experi-
mentation, an approval from the Institutional Review Board at the University of Pittsburgh was obtained. The participants were instructed to relax and avoid any voluntary interference during the electrical stimulation. Two sets of experiments were conducted to evaluate the performance of the new controller and its robustness to variations in the estimated EMD value. Each experimental session was run for a duration of 30 seconds with a rest period of 3 minutes in between the sessions to prevent muscle fatigue. In the first set of experiments, we compared the new controller (PID-DC) with two previously developed nonlinear controllers (PD-DC and RISE). The details of the RISE control law and PD-DC control law can be found in [13] and [29], respectively. These controllers were chosen because, just like the controller developed in this paper, they fall under the category of strictly feedback tracking controllers that are designed based on a nonlinear musculoskeletal system and are synthesized using a Lyapunov stability analysis. In addition, these controllers are easily implementable and do not require any model knowledge, unlike controllers such as the sliding mode controller (SMC) in [47]. During the experiments, the 4 subjects were unaware of the controller being tested during the session. For each subject, the order of the controllers was selected at random and were tested on separate days (every other day). The three controllers were used to track a sinusoidal signal with a period of 2-seconds and alternating peaks. The desired trajectory started from the equilibrium and oscillated between a minimum amplitude of 15 and an alternating maximum amplitudes of 35 and 25 , this prevented any subject from anticipating the desired motion and subconsciously interfering with the performance of the controller. Each controller was evaluated in five consecutive trials for each subject. A rest period of 3 minutes was given in between the 30 second trials. Also, the subjects were not allowed to view the desired trajectory or the performance in real time. In addition to these experiments, the three controllers were tested on one subject with a larger range of motion (5–50 ) to see if the controllers could maintain their performance for larger movements. As these experiments were conducted on human subjects, the gain tuning procedure could only be done over a finite interval of time in order to prevent muscle fatigue and subject discomfort. Prior to experimentation on a subject, the three controllers were tuned to find an initial guess of the control gains. Before con-
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Fig. 4. Results obtained from running the three controllers on subject H1 With a larger trajectory ranging from 5–50 .
movement occurred. The EMD value was calculated as the time difference from when the first stimulation pulse was applied and when the knee angle began to change. Five measurements of the EMD value were taken, and the average of the five values was used as the measured EMD value, which is used in the control implementation. In the second set of experiments, the PID-DC controller was tested for its robustness to variations in the EMD value. This was done by assuming an estimated EMD value, used in the PID-DC controller, different from the measured EMD value in the previous experiments. A subject (H1) was held in a gait like configuration using a brace as shown in Fig. 2(b). The thigh of the subject was fixed at a certain angle (using the motor in the brace), while the new controller was used to track a sinusoidal trajectory with a 2 second time period. Robustness of the controller to an estimate EMD value was tested by evaluating its performance for estimated EMD values that ranged between 2 standard deviations (SD) from the mean of the measured EMD value. VI. EXPERIMENTAL RESULTS
Fig. 3. Experimental results obtained from the representative trial for each of the three controllers. These plots show the desired and actual angular position (top plots), error (middle plots), and stimulation current amplitude (bottom plots).
ducting the five trials for each controller, the controller was further tuned, beginning at the initial guess. The control gains were fine tuned till the error over a 10 second trial was minimized. Since the PD-DC and PID-DC require the knowledge of the EMD value, it was determined empirically for each test subject. This was done by applying a step stimulation at a time instant and measuring the time when the resulting knee joint
Figs. 3–6 and Tables I–IV illustrate the results from the two sets of experiments. In the first set of experiments, each controller was tested in five trials on each subject and then again on one subject with a larger range of motion. Trials for the PD-DC and PID-DC used the empirically found EMD values as reported in Table III. A representative trial from each controller is shown in Fig. 3 and the control gains used to produce those results are given in Table I. Table III contains the three criteria used to measure the performance of each controller: the root mean squared of the error (RMSE), steady state RMSE (SSRMSE), and root mean squared of the current (RMSC). The RMSC was normalized by the body mass index of each subject in order to scale the control effort with respect to the size of the subjects. Since the RISE and PID-DC controller have integral control, they require a transient period for the memory component to build up, that is why the SSRMSE is the primary focus of the comparison. The differences in the aforementioned performance criteria were assessed using a one-way analysis of variance (ANOVA)
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Fig. 5. Graphical representation of the results from the LEM experiments. The three criteria: RMSE, SSRMSE, and RMSC were normalized by the maximum of each criterion after the ANOVA analyses were done. indicates statistically significant differences between the controllers at a 95% confidence level and p refers to the p-value.
with repeated measures. Post-hoc testing consisting of paired t-tests with a Bonferroni correction was performed when a significant ANOVA tests was identified. A Bonferroni adjustment was used to avoid potential type I errors associated with performing multiple t-tests in the post-hoc analysis. As a result, the critical threshold for significance was reduced to (0.05 divided by 3). The results for the statistical analyses are given in Fig. 5, where each criterion's amplitude has been normalized to the maximum criterion value. This normalization is performed strictly for plotting the results and is completely independent of the ANOVA analyses. The SSRMSE of the PID-DC controller was found to be significantly lower than that of the other controllers ( between PD-DC and PID-DC and between PID-DC and RISE). The RISE controller's SSRMSE was found to be lower than that of the PD-DC controller ( ). Although the SSRMSE was used to make the main comparison, the RMSE results were found to be concurrent with the SSRMSE results and can be seen in Table III. Lack of integral control, which gives a controller a memory component in order to compensate for steady state errors, in the PD-DC controller seems to have played a role in its greater RMSE and SSRMSE as compared to the other two controllers. While tuning the controllers, it was observed that in order to maintain stability with the RISE controller, the gain associated with integral control in the PID component of the controller, (i.e., ), was required to be kept low. This could be due to the combination of controller's increased responsiveness due to integral action and EMD-induced oscillations. Unlike the RISE controller, the PID-DC did not have this issue because the DC component provided the controller more robustness by removing the ill-timed excess energy that induced these oscillations. One thing to note is that because of the DC component the gains for the PD-DC and PID-DC were required to be much
Fig. 6. Tracking performance of PID-DC with mismatched EMDs that were set to 2 SDs away from the mean measured EMD. With mismatched EMD settings, the performance degraded but the PID-DC maintained stability. Top plot: the control performance, before tuning . Bottom plot: the tracking performance improved after tuning .
TABLE I GAINS USED TO PRODUCE THE EXPERIMENTAL RESULTS SHOWN IN FIG. 3
TABLE II TABULATED RESULTS FOR THE SINUSOIDAL TRAJECTORY WITH 2 SECOND TIME PERIOD RANGING FROM 5–50 IMPLEMENTED ON SUBJECT H1 WITH ALL THREE CONTROLLERS
larger than the gains for the RISE controller. This does not necessarily mean the PD-DC and PID-DC controllers were using more control effort. This is simply due to the mechanics of the DC part of the controllers. The DC part of the controller integrates the control signal over an interval the duration of the input
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TABLE III TABULATED RESULTS FOR THE SINUSOIDAL TRAJECTORY WITH 2 SECOND TIME PERIOD EXPERIMENTS INCLUDE THE AVERAGE (AVG) AND STANDARD DEVIATION (SD) OF THE EMD VALUES MEASURED FOR EACH VOLUNTEER SUBJECT (S) ACROSS FIVE SAMPLES, THE AVERAGE ROOT MEAN SQUARED ERROR (RMSE), THE AVERAGE STEADY STATE RMSE (SSRMSE), AND THE AVERAGE ROOT MEAN SQUARED CURRENT (RMSC) NORMALIZED BY THE BODY MASS INDEX OF EACH SUBJECT
TABLE IV TABULATED RESULTS FOR THE ROBUSTNESS OF THE ESTIMATED EMD VALUE IN THE PID-DC CONTROLLER AS OPPOSED TO THE MEASURED EMD . NOTE THAT THE CONTROLLER WAS ABLE TO MAINTAIN VALUE OF 85 CONSISTENT PERFORMANCE EVEN WITH MISMATCHED EMDS AFTER TUNING (ATB). THE RESULTS FOR THE ESTIMATED EMD VALUES BEFORE TUNING (BTB) WERE INCLUDED TO DEMONSTRATE THE EFFECT OF TUNING
the DC portion of the controller over compensated due to the larger integral interval; therefore, decreasing helped in offsetting the effect of mismatched EMD values. Similarly, increasing helped in offsetting the effect of mismatched EMD values, when an estimated EMD value was smaller than the measured EMD value. VII. DISCUSSION
delay and subtracts it from the PID component of the controller. Therefore, most of the control effort generated from the PID component was dissipated due to the DC component. Although results of the PD-DC controller reported the lowest RMSC as compared to the other two controllers, the statistical analysis determined that the differences in the RMSCs were not statistically significant ( between PD-DC and PID-DC, between PD-DC and RISE, between PID-DC and RISE). From Fig. 4 and Table II, it can be seen that even with larger ranges of motion, the PID-DC continued to outperform the other two controllers. In the second set of experiments, the measured EMD value was empirically found to be 85 for Subject H1. The experimental results for tracking the sinusoidal trajectory with a 2 second time period, where the PID-DC controller used estimate EMD values different from the measured EMD value, are given in Fig. 6 and Table IV. As the estimated EMD value strayed from the measured EMD value, before tuning (BTB), the control performance deteriorated but the controller maintained stability. However, after tuning (ATB), the PID-DC controller not only maintained stability but also provided the same level of performance as in the first set of experiments. This was observed even when the estimated EMD value was varied by 2 SDs from the measured EMD value. Results when the estimated EMD values were assumed to be 2 SDs away from the measured EMD values are plotted in Fig. 6. The decrease in control performance, resulting from mismatched delay estimates, was compensated for by tuning the gain . For example, for the estimate EMD values greater than the measured EMD value,
Advanced NMES controllers that account for the nonlinearities and delays in the musculoskeletal system can play a significant role in the rehabilitation of limb function in mobility-impaired persons. Potentially, these controllers can be applied for various NMES-elicited tasks such as gait training or restoration, upper limb reaching, functional electrical stimulation (FES) cycling, rowing, etc. Therefore, it is important to investigate and choose the most suitable NMES controller among different available options. The new nonlinear controller, PID-DC, showed promising results compared to its predecessors: the RISE controller [13] and the PD-DC controller [41]. These three controllers were compared because they are all strictly feedback NMES controllers that were synthesized using Lyapunov stability and control design methods. The PID-DC controller showed statistically significant improvements in the tracking performance compared to the other controllers. Moreover, the new controller was shown to be robust to variations in the EMD values. Although the RISE and PID-DC controllers both have a PID based structure, the RISE controller was not designed to compensate for EMDs. Similarly, the PD-DC controller was designed to compensate for EMDs but lacks integral control. Due to these reasons, the RISE and the PD-DC controllers showed poor performance compared to the new controller. We also want to emphasize on the significance of Lyapunovbased control design approach that was used to develop the new controller. Lyapunov-based control synthesis is an exciting control design tool for nonlinear systems, such as a musculoskeletal system. More importantly, the nonlinear system is not required to be linearized, which results in a better match between the system model and the physical system. With Lyapunov stability analysis, a guarantee on the controller stability can be provided. This is especially crucial in applications such as NMES where safety risks could occur if the controller is unstable. Moreover, as shown in the control development, the controller can be implemented despite no knowledge of
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the musculoskeletal model. This is important as the musculoskeletal model is typically uncertain and its parameters are subject to day-to-day variations. Clinically, it also suggests that the NMES controller can be implemented without having to estimate the model parameters of a subject, which can be time consuming. Although this strictly feedback controller showed promising results, it does not prove that model-free controllers are the best option. The addition of a feedforward component to a feedback controller such as in [17], [49] can further improve closed-loop performance of the system. A model-based controller, or an artificial neural network controller, that works in conjunction with PID-DC based feedback may be a more promising control strategy. Limitations of the Study: Although the controller was determined to be robust to variations in the estimated EMDs, the controller was designed with the assumption that the EMD is constant. This is a limitation because the EMD could be time varying; e.g., variance of the EMD over time was linked to muscle fatigue [48]. Our study evaluated the controller on able bodied subjects and was designed for a single degree of freedom system. The effectiveness of the new controller remains to be proven on persons with stroke or SCI and its performance on FES-based rehabilitation systems with multiple degrees of freedom remains to be seen. Our control design also neglected the effects of activation dynamics, as accounted for by other studies in [10], [14], [47]. The latter two issues are the focus of our future research. VIII. CONCLUSION In this paper, a PID-based delay compensation (PID-DC) controller was developed for an NMES-driven musculoskeletal system with EMDs. Lyapunov-based stability analysis yielded semi-globally uniformly ultimately bounded tracking despite model uncertainties and EMDs. The main motivation for developing the new controller was to add an integral action in our previous control design for EMD compensation. The addition of the integral action resulted in improved performance that was validated on four able bodied subjects and empirically compared to two nonlinear controllers: PD-DC (previous control design for EMD compensation) and RISE controller. The results showed that the PID-DC has a superior tracking performance (statistically significant) vis-á-vis the other two controllers. Further, the new controller was shown to be robust to variations in the measured EMDs. Future work will focus on testing this controller on persons with stroke or SCI, investigating the benefits of adding activation dynamics, considering spasticity and fatigue [49], and extending the controller to multiple degrees of freedom and more general systems.
The two integrals in (20) can be rearranged to
(21) , perform the following change of variables, For take , , where the new bounds of integration can be computed as: , . Thus, . By further separating the integral, it yields . For , another change of variables such that , where the bounds can be computed as , , yields . Assuming that in the interval , the second integral is equal to zero. Because we are only concerned with positive values of time, this assumption is not intrusive to the result. Thus, the integrals in (21) can be expressed as
Take , , and taking the Laplace transform of both sides we get
, after
Thus,
Taking the inverse Laplace transform results in
Theorem 1 Proof: Let
be defined as (22)
A positive definite Lyapunov functional candidate is defined as (23) and satisfies the following inequalities
APPENDIX
(24)
Lemma 1 Proof: Using the fact that and (7) the time derivative of (9) can be expressed as
where
,
are known constants defined as
(20) where
and
are defined in (4).
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After using (6), (7), and (12), and canceling the common terms, the time derivative of (23) can be expressed as
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Rearranging the terms and canceling the common terms results in
(25)
After applying the Young's Inequality and utilizing the definition of in (13), the following terms in (25) can be bounded as
(31) After utilizing (10), completing the squares, and provided that , (31) can be upper bounded as
(26) (27) (32)
Further, by using the Cauchy Schwarz inequality, the following term in (27) can be upper bounded as Since (28) Using (14), (26), and (28), (25) can be bounded as
and after utilizing (15), (16), and (17), the inequality in (32) can be expressed as (33) (29) Note that by choosing control gains and such that , the following Young's inequality can be used
where
is defined as
Using the definition of be upper bounded as
in (22), the expression in (33) can
(30) After adding and subtracting using (28), provided that be upper bounded as
to (29) and , (29) can
(34) where
is defined as
In order to further bound (34), it is required that which is true if Consider a set defined as
.
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can be lower bounded by a constant
as
and the condition, , is satisfied. By further utilizing (24), the inequality in (34) can be expressed as (35) The linear differential equation in (35) can be solved as (36) provided the control gains , , , and are selected according to the sufficient conditions in (19) (i.e. a semi-global result). The result in (18) can now be obtained from (36). Based on the definition of , the result in (36) indicates that , , in . Since , in , then (13) indicates that in . Given that , , , in , (5) and (6) indicate that and in . Since , , , , , in , then (7) indicates that in . Given that , , , , in , (8) and Assumptions 3 and 4 indicate that in . REFERENCES [1] P. H. Peckham and J. S. Knutson, “Functional electrical stimulation for neuromuscular applications,” Annu. Rev. Biomed. Eng., vol. 7, pp. 327–360, 2005. [2] V. Nekoukar and A. Erfanian, “A decentralized modular control framework for robust control of FES-activated walker-assisted paraplegic walking using terminal sliding mode and fuzzy logic control,” IEEE Trans. Biomed. Eng., vol. 59, no. 10, pp. 2818–2827, Oct. 2012. [3] N. Sharma, V. Mushahwar, and R. Stein, “Dynamic optimization of FES and orthosis-based walking using simple models,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 22, no. 1, pp. 114–126, Jan. 2014. [4] M. M. Skelly and H. J. Chizeck, “Real-time gait event detection for paraplegic FES walking,” IEEE Trans. Neural Sys. Rehab. Eng., vol. 9, no. 1, pp. 59–68, 2001. [5] T. A. Thrasher, V. Zivanovic, W. McIlroy, and M. R. Popovic, “Rehabilitation of reaching and grasping function in severe hemiplegic patients using functional electrical stimulation therapy,” Neurorehab. Neural. Re., vol. 22, no. 6, pp. 706–714, 2008. [6] E. K. Chadwick, D. Blana, J. D. Simeral, J. Lambrecht, S. P. Kim, A. S. Cornwell, D. M. Taylor, L. R. Hochberg, J. P. Donoghue, and R. F. Kirsch, “Continuous neuronal ensemble control of simulated arm reaching by a human with tetraplegia,” J. Neural Eng., vol. 8, no. 3, p. 034003, 2011. [7] P. Cooman and R. F. Kirsch, “Control of a time-delayed 5 degrees of freedom arm model for use in upper extremity functional electrical stimulation,” in Proc. Int. IEEE EMBC, 2012, pp. 322–324, IEEE. [8] E. M. Schearer, Y.-W. Liao, E. J. Perreault, M. C. Tresch, W. D. Memberg, R. F. Kirsch, and K. M. Lynch, “Multi-muscle FES force control of the human arm for arbitrary goals,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 22, no. 3, pp. 654–663, May 2014. [9] R. Davoodi, B. J. Andrews, G. D. Wheeler, and R. Lederer, “Development of an indoor rowing machine with manual FES controller for total body exercise in paraplegia,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 10, no. 3, pp. 197–203, 2002. [10] S. Jezernik, R. G. V. Wassink, and T. Keller, “Sliding mode closedloop control of FES: Controlling the shank movement,” IEEE Trans. Biomed. Eng., vol. 51, pp. 263–272, 2004. [11] T. Schauer, N. O. Negard, F. Previdi, K. J. Hunt, M. H. Fraser, E. Ferchland, and J. Raisch, “Online identification and nonlinear control of the electrically stimulated quadriceps muscle,” Control Eng. Pract., vol. 13, pp. 1207–1219, 2005.
[12] M. M. Ebrahimpour and A. Erfanian, “Comments on “sliding mode closed-loop control of FES: controlling the shank movement”,” IEEE Trans. Biomed. Eng., vol. 55, no. 12, p. 2842, 2008. [13] N. Sharma, K. Stegath, C. M. Gregory, and W. E. Dixon, “Nonlinear neuromuscular electrical stimulation tracking control of a human limb,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 17, no. 6, pp. 576–584, 2009. [14] N. Sharma, P. Patre, C. Gregory, and W. E. Dixon, “Nonlinear control of NMES: Incorporating fatigue and calcium dynamics,” in Proc. ASME Dyn. Syst. Control Conf., Hollywood, CA, Oct. 2009, pp. 705–712. [15] R. J. Downey, T.-H. H. Cheng, and W. E. Dixon, “Tracking control of a human limb during asynchronous neuromuscular electrical stimulation,” in Proc. IEEE 52nd CDC, Dec. 2013, pp. 139–144. [16] W. K. Durfee and J. M. Hausdorff, “Regulating knee joint position by combining electrical stimulation with a controllable friction brake,” Ann. Biomed. Eng., vol. 18, no. 6, pp. 575–596, 1990. [17] N. Sharma, C. Gregory, M. Johnson, and W. E. Dixon, “Closed-loop neural network-based NMES control for human limb tracking,” IEEE Trans. Control Syst. Technol., vol. 20, pp. 712–725, 2012. [18] K. J. Hunt, B. Stone, N. O. Negard, T. Schauer, M. H. Fraser, A. J. Cathcart, C. Ferrario, S. A. Ward, and S. Grant, “Control strategies for integration of electric motor assist and functional electrical stimulation in paraplegic cycling: utility for exercise testing and mobile cycling,” IEEE Trans Neural Syst. Rehab. Eng., vol. 12, no. 1, pp. 89–101, March 2004. [19] M. J. Bellman, T.-H. H. Cheng, R. J. Downey, and W. E. Dixon, “Stationary cycling induced by switched functional electrical stimulation control.,” in Proc. ACC, Portland, OR, 2014. [20] H. Kawai, M. Bellman, R. J. Downey, and W. E. Dixon, “Tracking control for FES-cycling based on force direction efficiency with antagonistic bi-articular muscles,” in Proc. ACC, 2014. [21] J. J. Abbas and H. J. Chizeck, “Feedback control of coronal plane hip angle in paraplegic subjects using functional neuromuscular stimulation,” IEEE Trans. Biomed. Eng., vol. 38, no. 7, pp. 687–698, 1991. [22] N. Lan, P. E. Crago, and H. J. Chizeck, “Feedback control methods for task regulation by electrical stimulation of muscles,” IEEE Trans. Biomed. Eng., vol. 38, no. 12, pp. 1213–1223, 1991. [23] A. H. Vette, K. Masani, and M. R. Popovic, “Implementation of a physiologically identified PD feedback controller for regulating the active ankle torque during quiet stance,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 15, no. 2, pp. 235–243, Jun. 2007. [24] F. Previdi, M. Ferrarin, S. M. Savaresi, and S. Bittanti, “Gain scheduling control of functional electrical stimulation for assisted standing up and sitting down in paraplegia: a simulation study,” Int. J. Adapt. Control Signal Process., vol. 19, pp. 327–338, 2005. [25] H. Gollee, K. J. Hunt, and D. E. Wood, “New results in feedback control of unsupported standing in paraplegia,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 12, no. 1, pp. 73–91, 2004. [26] K. J. Hunt and M. Munih., “Feedback control of unsupported standing in paraplegia-part 1: Optimal control approach,” IEEE Trans. Rehab. Eng., vol. 5, pp. 331–340, 1997. [27] V. Nekoukar and A. Erfanian, “Adaptive terminal sliding mode control of ankle movement using functional electrical stimulation of agonist-antagonist muscles,” in Proc. Int. IEEE EMBC, Aug. 2010, pp. 5448–5451. [28] M. Benoussaad, K. Mombaur, and C. Azevedo-Coste, “Nonlinear model predictive control of joint ankle by electrical stimulation for drop foot correction,” in Proc. IEEE/RSJ Int. Conf. IROS, 2013, pp. 983–989, IEEE. [29] N. Sharma, C. Gregory, and W. E. Dixon, “Predictor-based compensation for electromechanical delay during neuromuscular electrical stimulation,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 19, no. 6, pp. 601–611, 2011. [30] N. Sharma, “A predictor-based compensation for electromechanical delay during neuromuscular electrical stimulation-II,” in Proc. ACC, 2012, pp. 5604–5609. [31] Q. Wang, N. Sharma, M. Johnson, C. M. Gregory, and W. E. Dixon, “Adaptive inverse optimal neuromuscular electrical stimulation.,” IEEE Trans. Cybern., vol. 43, no. 6, p. 1710, 2013. [32] S. Zhou, D. L. Lawson, W. E. Morrison, and I. Fairweather, “Electromechanical delay in isometric muscle contractions evoked by voluntary, reflex and electrical stimulation,” Eur. J. Appl. Physiol. Occup. Physiol., vol. 70, no. 2, pp. 138–145, 1995. [33] P. R. Cavanagh and P. V. Komi, “Electromechanical delay in human skeletal muscle under concentric and eccentric contractions,” Eur. J. Appl. Physiol. Occup. Physiol., vol. 42, no. 3, pp. 159–163, 1979.
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[34] K. Masani, A. H. Vette, N. Kawashima, and M. R. Popovic, “Neuromusculoskeletal torque-generation process has a large destabilizing effect on the control mechanism of quiet standing,” J. Neurophysiol., vol. 100, no. 3, p. 1465, 2008. [35] A. H. Vette, K. Masani, and M. R. Popovic, “Neural-mechanical feedback control scheme can generate physiological ankle torque fluctuation during quiet standing: A comparative analysis of contributing torque components,” in Proc. IEEE Int. CCA, 2008, pp. 660–665, IEEE. [36] K. Masani, A. H. Vette, and M. R. Popovic, “Controlling balance during quiet standing: proportional and derivative controller generates preceding motor command to body sway position observed in experiments.,” Gait & Posture, vol. 23, no. 2, pp. 164–172, Feb. 2006. [37] O. J. M. Smith, “A controller to overcome deadtime,” J. ISA, vol. 6, pp. 28–33, 1959. [38] Z. Artstein, “Linear systems with delayed controls: A reduction,” IEEE Trans. Autom. Control, vol. 27, no. 4, pp. 869–879, 1982. [39] A. Manitius and A. Olbrot, “Finite spectrum assignment problem for systems with delays,” IEEE Trans. Autom. Control, vol. 24, no. 4, pp. 541–552, 1979. [40] W. Michiels, K. Engelborghs, P. Vansevenant, and D. Roose, “Continuous pole placement for delay equations,” Automatica, vol. 38, no. 5, pp. 747–761, 2002. [41] N. Sharma, S. Bhasin, Q. Wang, and W. E. Dixon, “Predictor-based control for an uncertain Euler-Lagrange system with input delay,” Automatica, vol. 47, no. 11, pp. 2332–2342, 2011. [42] W. L. Buford, Jr., F. M. Ivey, Jr., J. D. Malone, R. M. Patterson, G. L. Peare, D. K. Nguyen, and A. A. Stewart, “Muscle balance at the knee—moment arms for the normal knee and the ACL—minus knee,” IEEE Trans. Rehab. Eng., vol. 5, no. 4, pp. 367–379, 1997. [43] J. L. Krevolin, M. G. Pandy, and J. C. Pearce, “Moment arm of the patellar tendon in the human knee,” J. Biomech., vol. 37, pp. 785–788, 2004. [44] R. Nathan and M. Tavi, “The influence of stimulation pulse frequency on the generation of joint moments in the upper limb,” IEEE Trans. Biomed. Eng., vol. 37, pp. 317–322, 1990. [45] T. Watanabe, R. Futami, N. Hoshimiya, and Y. Handa, “An approach to a muscle model with a stimulus frequency-force relationship for FES applications,” IEEE Trans. Rehab. Eng., vol. 7, no. 1, pp. 12–17, 1999. [46] S. A. Binder-Macleod, E. E. Halden, and K. A. Jungles, “Effects of stimulation intensity on the physiological responses of human motor units,” Med. Sci. Sport Exer., vol. 27, no. 4, pp. 556–565, 1995. [47] C. L. Lynch and M. R. Popovic, “A comparison of closed-loop control algorithms for regulating electrically stimulated knee movements in individuals with spinal cord injury,” IEEE Trans. Neural Syst. Rehab. Eng., vol. 20, no. 4, pp. 539–548, July 2012. [48] F. Bouillon, “Measure, modeling and compensation of fatigue-induced delay during neuromuscular electrical stimulation,” Ph.D. dissertation, University of Florida, , 2013. [49] P. Cooman, “Nonlinear feedforward-feedback control of an uncertain, time-delayed musculoskeletal arm model for use in functional electrical stimulation,” Ph.D. dissertation, , Case Western Univ., Cleveland, 2014.
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Naji Alibeji received the B.S. degree (Summa Cum Laude) from the University of Pittsburgh, Pittsburgh, PA, USA, in mechanical engineering in 2012. He is currently pursuing a Ph.D. in mechanical engineering for his research in nonlinear controls under the advisement of Dr. Sharma. As an undergraduate, he spent two summers working at Acutronic USA as a mechanical assembler/designer.
Nicholas Kirsch received the B.S. degree in mathematics with a physics minor from St. Vincent College, Latrobe, PA, USA, in 2008. At the University of Pittsburgh, Pittsburgh, PA, USA, he received the B.S. and M.S. degrees in mechanical engineering, and is currently pursuing a Ph.D. degree in mechanical engineering for his research in restorative gait control.
Shawn Farrokhi received the Ph.D. degree in biokinesiology from the University of Southern California, Los Angeles, CA, USA. He is an Associate Professor in the Departments of Physical Therapy and Bioengineering and the Co-director of the Human Movement Research Laboratory at the University of Pittsburgh, Pittsburgh, PA, USA.
Nitin Sharma (M’14) received the Ph.D. degree in 2010 from the Department of Mechanical and Aerospace Engineering at the University of Florida, Gainesville, FL, USA. He was an Alberta Innovates Health Solutions postdoctoral fellow in the Department of Physiology at the University of Alberta, Edmonton, Canada. Since 2012, he has been an Assistant Professor in the Department of Mechanical Engineering and Material Science at the University of Pittsburgh, Pittsburgh, PA, USA. His research interests are robust control of functional electrical stimulation (FES) and modeling and control of FES-elicited walking.
