Development of Mirror Image Motion System with semg for Shoulder Rehabilitation of Post-stroke Hemiplegic Patients
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1 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 8, pp AUGUST 2012 / 1473 DOI: /s Development of Mirror Image Motion System with semg for Shoulder Rehabilitation of Post-stroke Hemiplegic Patients Kihan Park 1, Dong Ju Lee 1, Pilwon Heo 1, and Jung Kim 1,# 1 Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, Republic of Korea, # Corresponding Author / jungkim@kaist.ac.kr, TEL: , FAX: KEYWORDS: Surface electromyography, Bimanual shoulder flexion, Robot-assisted rehabilitation, Mirror image motion A robot-assisted bimanual shoulder flexion rehabilitation system using surface electromyography (semg) for post-stroke hemiplegic patients is presented. Impedance compensation of the actuators using a disturbance observer (DOB) was applied for back-drivable operation. The semg signal processing was utilized to obtain a desired assistive torque. Four modes of motion (passive, mirror image, shared, and voluntary) were suggested as an appropriate training platform for the various patient statuses and levels of individual recovery. Then the performance of the impedance compensation and assistive operation of the system was verified by experiments with healthy participants. The DOB decreased resistive torque by 99% compared to the open loop performance. The shoulder torque was estimated using the semg and linear regression (CORR = ± 0.011, NRMSE = 7.31 ± 1.32%) and an artificial neural network (ANN) (CORR = ± 0.005, NRMSE = 6.96 ± 1.08%) methods for generating system input based on the user s motion intention. Every mode had less than a 6% NRMSE motion error in experiments without discomforts or resistance during shoulder flexion motion of mirror the arm. Manuscript received: October 27, 2011 / Accepted: February 27, Introduction Hemiplegic disorder occurs in over 80% of stroke patients. Approximately 750,000 individuals experience a stroke every year, which is a rate of about 140 patients per 10,000 people in the United States. 1 It is necessary to perform sufficient amounts of voluntary, intensive, repeated, continuous, and goal-oriented exercise for rehabilitation. 2-6 However, in the early stage after stroke, passive joint movement by physical therapists or continuous passive movement (CPM) machines are the only ways to stimulate the paretic parts of the body in conventional physical therapy. Moving patients limb manually is labor-, time-, and cost-intensive work. Furthermore, passive motion of the patient is less effective for rehabilitation than active motion. 6 The need for a more efficient exercise therapy is amplified by the growing evidence that more intensive exercise is an important factor in motor recovery after stroke. 2-4,7-9 A surface electromyography based (semg) robotic rehabilitation system that induces active motion from the patient can help the recovering neural/musculoskeletal system by providing intensive exercise while evaluating the states of the patients quantitatively. Furthermore, the motivation resulting from intentional, voluntary motion enables faster recovery. However, robotic systems have difficulties in anticipating the patient s motion intention due to the limited joint motion caused by atrophy and contracture. Several research groups have studied rehabilitation systems for voluntary exercise using the motional intention extracted from the semg or bimanual movement to overcome the difficulties in anticipating the motions of a paralyzed body part. The MIT-Manus 10 assists shoulder and elbow exercises in the horizontal plane. Its assistance is triggered by the patient s semg activity. Lum et al. 2,3 developed a mirror image movement enabler (MIME) robot for motor recovery. This robot enables bimanual exercise by assisting the paretic limb s movement to mirror the healthy limb. Hesse et al. 4 also tested the effect of a bimanual rehabilitation robot on the distal motor impairment and functionality of wrist flexion/extension and forearm pronation/supination. However, there are limitations of these assistive robots. The MIT-Manus requires the paretic limb s intermittent muscle contraction only during the initial trigger. In addition, active movement of the paretic limb is excluded in the two KSPE and Springer 2012
2 1474 / AUGUST 2012 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 8 bimanual systems. This paper focuses on the development of a semg-based robotic assistive bimanual shoulder flexion system utilizing mirror image motion. Additionally, a training platform for various patient statuses from acute to chronic phase is presented. Quantitative evaluation is presented for semg based rehabilitation robot inducing continuous active muscle contraction. Assistive operation of bimanual shoulder flexion system with healthy participants using semg is tested to verify its performance. 2. Materials and Methods 2.1 Bimanual Shoulder Flexion Device Design The assistive system consists of two devices with a symmetrical design. Each device is designed to rotate the arm uni-axially along the sagittal plane for the shoulder flexion/extension in a seated position. The assistive torque is generated by a geared direct current (DC) motor (RE40 with a gearhead of 91:1, Maxon Precision Motors, Switzerland) equipped with a magnetic brake. The motor shaft is connected to a torque sensor (SDN-2kgf.m, CTA Plus, Korea) via timing pulleys with a reduction ratio of 1.5:1 as shown in Fig. 1. The maximum available torque of the unit is Nm, which is about 2 times larger than the maximum required torque calculated based on the American s anthropometry statistics. A slider rail is attached at the bottom of the actuator unit for passive alignment of the output shaft with the center of rotation of the arm during shoulder movement. The linkage part is composed of two plastic splints for the upper/lower arm, support links, and an elbow angle adjustable plate. The elbow angle can be changed manually to 45, 90, 135, 180 (straight) degrees. However, the elbow angle is fixed to 90 degrees, which is functional position frequently used in daily activity and is felt most comfortable during experiment. The supporting links for the upper/lower arm are designed to have an adjustable length of mm and; mm, with a 10 mm incremental step, respectively, fit an arm length of the 5 th through 95 th percentiles of Koreans. 11 The system is controlled using Matlab/Simulink (The MathWorks Inc., USA) and the real-time control software QuaRC using a Q8-USB DAQ board (both from Quanser, Canada). 2.2 Controller Design with Disturbance Observer (DOB) The gear train attached to the DC motor, which magnifies the generated torque, decreases the back-drivability. The approach similar to that of Kong et al. 12,13 and Pan et al. 14 using a disturbance observer (DOB) was adopted to compensate for the mechanical impedance of the actuator. DOB is a well known technique used to cancel out disturbance to the system and was proposed by Ohnishi. 15 Because the inverse model of plant is generally unrealizable and very susceptible to high frequency noise, a DOB should include a Q-filter, which is a low-pass filter. The cutoff frequency was selected to be 20 Hz, which is a sufficiently large bandwidth compared to the maximum frequency of human body movement, 8 Hz. 16 Specifically, the usual frequency of shoulder motion is less than 2 Hz. Therefore the cutoff frequency of Q-filter is high enough to preserve the shoulder motion. The block diagram of the overall joint control system with a DOB is presented in Figs. 2 and 3. A feed-forward filter is applied to improve the tracking performance. The desired angular position of the motor consists of the weighted summation of healthy arm motion (see θ mirror in Fig. 3) and the translated angular position from the desired torque obtained from the semg signal in case of a paretic arm. A nominal model of the actuator is needed to apply the DOB. The governing equation of motion for a geared DC motor is I && θ + C & θ = τ (1) m m m where & θ, && θ, and τ m are the angular velocity, angular acceleration and torque at output shaft, respectively. The angular moment of inertia of the rotor I m and the damping coefficients of the geared motor C m were identified empirically. The nominal model of the motor obtained is presented as a thick continuous line in Fig. 4. The frequency responses show variation among each model in the low frequency range due to the velocity limitation of the motor. Because the model s variation at low frequencies interrupts the human-robot interaction, a proportional-derivative (PD) controller is used to attenuate the open loop model variation before applying the disturbance observer. The linear quadratic (LQ) method provides the optimal PD controller gains quantitatively if the state variables Fig. 2 Block diagram of the mirrored arm Fig. 1 The actuator unit of the device composed of a motor and a torque sensor connected serially with an output shaft Fig. 3 Block diagram of the paretic arm
3 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 8 AUGUST 2012 / 1475 are angular position and angular velocity. The state space model of the motor is expressed as x& = Ax + Bu = x u 0 Cm/ I + m 1/ I m y = Cx+ Du = x [ 1 0] where x = [ θ & θ ] T is the state variable and; u and θ are the system input (i.e. τ m ) and output, respectively. The objective function of the LQ method is T T J = xqx + uru dt 0 (2) (3) T where Q = C C and R = 0.05 are the weighting factor that relate the importance between performance (i.e. θ ) and cost (i.e. τ m ). The control law to minimize the objective equation (3) is given by τ m = Kx, where K 1 T = R B P (4) and P, 2 2 matrix in (4) is the positive definite solution of the Riccati equation 17 & = + + = 0 (5) T 1 T T P A P PA PBR B P C C where A, B, and C matrices are as defined in the state space model of (2). The closed loop model in which the model variation is attenuated by the PD controller is considered as the nominal model in the disturbance observer. The model variation is moved to a high frequency range over 30 Hz which rarely affect human motion as shown in Fig. 4. The magnitude of the gain for the closed loop nominal model is close to one (i.e. 0 db in Fig. 4) below the 10 Hz range. 2.3 Surface Electromyography The shoulder joint is a complex joint consisting of 3 bones, 5 joints, 12 ligaments, and over 15 muscles. 18 Through preliminary experiments, 5 (flexion: the anterior deltoid, long head of biceps, coracobrachialis, extension: the posterior deltoid, long head of triceps) out of the 10 muscles used in shoulder flexion/extension were selected for their muscle activity and small signal distortion. (Fig. 5) The activities of the muscles were measured using bipolar noninvasive surface electrodes (DE-2.1, Delsys, USA) with a builtin amplifier system (Bagnoli TM 8-channel system, Delsys, USA) Mean Absolute Value (MAV) with Linear Regression The mean absolute value (MAV) takes the moving average of the absolute value of the signal. Despite its simplicity, it is able to represent the features of the high frequency bio-signals such as EMG. The linear regression equation for the MAV of the torque/angle estimation is Y a n L i i= 1 L k= 1 x = (6) where Y is the estimation result, n is the number of muscles involved, L is the window length, a i is the linear constant, and x ik, is the k th raw signal acquired from the i th channel. Joint torque and angle are estimated by linearly combining the acquired signal for each individual muscle. The variables that determine the performance of the model are the window length (L) and the linear constant ( a i ). The mean absolute value attained using various window lengths is linearly regressed to find an estimate and is then compared to the measured data to find the optimal condition that yield a high correlation coefficient and a low normalized root mean square error (NRMSE). The linear weighting of each muscle can be determined using the linear coefficient Artificial Neural Network (ANN) An artificial neural network (ANN) was utilized to estimate the relationship between the EMG signal and the shoulder flexion torque/angle without a known musculoskeletal model. The number of hidden neurons and layers determine the complexity of an ANN. Complex neural networks are able to match the non-linear input and output characteristics better at the cost of a longer network training time. Moreover, the ANN model may become more complex than the actual system, which causes over-fitting. On the other hand, as the number of layers and hidden neurons decreases, the time required for training may decrease. However, the network might become too simple to model the actual system, which causes under-fitting. It was determined through simulation that the estimation performance converged at about 15 neurons per layer, and increasing the number of layers did not show any clear effect while ik, Fig. 4 The frequency response of the actuator; open loop nominal model (thick continuous line, red), closed loop nominal model (thick dotted line, red), and model variation (thin continuous line, gray) (a) Anterior view (b) Posterior view Fig. 5 Electrodes placement for the semg data acquisition
4 1476 / AUGUST 2012 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 8 maintaining a high estimation performance. Therefore, an artificial neural network with a single layer of 15 hidden neurons was used. 2.4 Bimanual Assistive Shoulder Flexion The appropriate exercise for the status of the patient is important for better recovery of the musculoskeletal system. Because patients lose consciousness right after a stroke, passive joint movement by external forces such as physical therapy or continuous passive movement (CPM) machines is the only way to stimulate the paretic part of the body. After recovering consciousness, patients can move their paretic arm by moving the healthy arm. As the neural system of the patient recovers, the paretic arm is assisted by the combination of the mirror arm motion and the estimated motion from the semg from the paretic arm. This results in assistive voluntary motion of the paretic arm. Fig. 6 shows the 4 modes explained above; 1) continuous passive movement, 2) mirror image motion, 3) shared motion, 4) voluntary motion. In addition to these modes, the resistive mode can be used instead of assistance as previously shown. 19 The joint contracture of post-stroke hemiplegic patients should also be considered. With the mirror image motion (mode 2), patients become aware of range of motion of paretic side by oneself using mirror image motion of healthy part. Then repeated external (passive) and internal (active) muscle stimulation mitigates the joint contracture. As patients get recover, the modes 3 and 4 of the system can be used. Three healthy male participants (26.67 ± 1.53 yrs) were asked to exercise with this system using mode 2 through mode 4 in a functional position (90 degrees elbow joint angle) and to move both their arms simultaneously to induce voluntary muscle contraction of the assumed paretic arm in mode 3 and 4. The weight of the semg ( w shown in Fig. 3) that represents the dependency of voluntary contraction of the paretic arm can be adjusted between 0 and 1. Assistive operation experiments with a healthy participant were performed for 5 cases of bimanual mode with 5 trials in each mode; mirror imaging motion only (i.e. semg weight w = 0), shared motion (selected semg weight w = 0.25, 0.5, 0.75) and semg driven motion (w = 1). In mode 2, the participant was asked to not induce any voluntary contraction of the arm with the semg electrodes in order to represent the paretic arm while they moved the other arm slowly from rest to approximately 90 degrees. 3. Experiments and Results 3.1 Actuator Impedance Compensation The effectiveness of a DOB in reducing actuator impedance was verified by measuring the resistive torque with zero control input. Fig. 7(a) shows that the nonlinearity such as the discontinuity at ω = 0 and the undesired resistive torque due to the linear damping torque were reduced using the DOB. Then a torque linearity test was performed to measure the desired and output torque with zero angular velocity. Fig. 7(b) shows the linear relation between the input and output torques. The nonlinear dead zone of the actuator within ± 0.7 Nm disappeared after impedance compensation. 3.2 Shoulder Joint Torque Estimation from semg Fig. 8 shows the experimental setup for the shoulder joint torque estimation using the semg in a functional position. The isometric shoulder flexion torque and the displacement angle of the shoulder in flexion was measured separately in 4 healthy males (25.75 ± 2.22 yrs), and the semg signals measured from the 5 muscles were used to estimate the torque/angle. The device s weight and driving impedances were compensated using a disturbance observer (DOB). The participants were asked to freely perform 3 to 5 shoulder flexions per set while keeping the contact between their back and seat back in order to prevent compensatory motion of the trunk. Both the torque and angle estimation experiment were performed 10 times. To find the contribution of each muscle and the optimum Fig. 6 Four modes of the bimanual assistive shoulder flexion system (a) (b) Fig. 7 Actuator impedance (a) and torque linearity (b) Fig. 8 Experimental setup for shoulder joint torque estimation and bimanual assistive shoulder flexion (left: healthy arm, right: simulated paretic arm)
5 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 8 AUGUST 2012 / 1477 combination of muscles to estimate the flexion torque, estimations using 1 flexion muscle, 3 flexion muscles, 1 flexion with 1 extension Fig. 9 The torque estimation results (solid, dashed and dotteddashed line indicate measured torque from a participant on isometric condition, estimated torque using MAV with linear regression and ANN, respectively) (a) w = 0 (b) w = 0.25 muscle, and 3 flexion and 2 extension muscles were analyzed using the ANN and MAV using the linear regression method. Fig. 9 shows the result of the torque estimation. Table 1 shows the estimation performance of each method using the normalized root mean square error (NRMSE) and the correlation coefficient (CORR). Using all muscles (3F 2E) demonstrated better estimation performance than using only 1 flexion muscle (1F) with both the ANN and MAV. The ANN s estimation performance increased with the increase in the input data (1F < 1F 1E < 3F < 3F 2E). On the other hand, for the MAV, the case using 1 flexion with 1 extension muscle (1F 1E) performed better than the case using 3 flexion muscles (3F) (1F < 3F < 1F 1E < 3F 2E). These results suggest that the extension muscle s contribution is quite large when using the MAV method. In terms of performance in estimating the torque excluding the 1F 1E case, the ANN performed slightly better than the MAV with a 1% change in the NRMSE. 3.3 Bimanual Assistive Shoulder Flexion The experimental results shows that bimanual exercises with little motion error (Fig. 10) and low resistive torque during mirror arm motion (Fig. 11) could be performed. Quantitative errors are presented in Table 2. The motion error became larger as the semg weight increased due to the uncertainty in the conversion from the semg to the joint torque. Participant trying to match both arms angular position showed a tendency to be trained for this task and had the tendency to reduce the angular position difference between both arms by controlling muscle contraction. The errors associated with the angular position error between both arms got smaller as the trials proceeded (Fig. 12). (c) w = 0.5 (d) w = 0.75 (e) w = 1 Fig. 10 The result of joint angle during bimanual assistive shoulder flexion Fig. 11 The result of resistive torque in mirror arm (representative case for w = 0) Table 1 Estimation performance of the shoulder flexion torque using the MAV with a linear regression and ANN Muscles MAV ANN CORR NRMSE (%) CORR NRMSE (%) A.Del ± ± ± ± F* Bic ± ± ± ± 2.62 Cor ± ± ± ± F A.Del. + Bic. + Cor ± ± ± ± F 1E A.Del. + P.Del ± ± ± ± 1.33 A.Del. + Tri ± ± ± ± F 2E A.Del. + Bic. + Cor. + P.Del. + Tri ± ± ± ± 1.08 *F: flexion muscle, E: extension muscle
6 1478 / AUGUST 2012 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No Conclusion A bimanual shoulder rehabilitation system using semg is presented as a preliminary study before clinical tests. For backdrivability, the actuator impedance was compensated using a DOB. The estimation of the torque and angle of the shoulder flexion from the semg signals using the MAV with a linear regression method and the ANN was performed with healthy participants to verify the bimanual shoulder flexion system. The MAV performed better at torque estimation than at angle estimation, and the best estimation performance was achieved using 1 flexion with 1 extension muscle. In the case of the ANN, estimation performance for both torque and angle increased with an increase in the number of muscles involved. Although the ANN performed better, it cannot be generalized, and the high frequency component of the estimation makes it difficult to apply without signal processing. Furthermore, both methods are based on purely numerical models that do not consider any kind of physiological model. Therefore, the model itself does not hold any physical meaning. A physiologically-based model, such as Hill s model, will be required to estimate isokinetic, isotonic and even general motion intent. In addition, the system s performance can be improved by applying impedance control to minimize the system s resistance to the user s movements. Currently, the DOB receives torque values and converts these values into the desired angular position. The control scheme would be simplified if a time delay control (TDC) were adopted. For the specific case using a TDC, the system will be further simplified as the number of system parameters decrease. When muscle fatigue occurs, the semg signal s amplitude will Table 2 Errors for bimanual shoulder flexion assist Joint angle e nrms [%] Resistive torque e rms [Nm] semg weight w = ± ± 0.01 semg weight w = ± ± 0.02 semg weight w = ± ± 0.02 semg weight w = ± ± 0.02 semg weight w = ± ± 0.02 increase for the same amount of force exerted and the frequency range of the signal will compress towards the lower frequencies. 20 This change in semg signal properties with the onset of fatigue affects the accuracy of the torque estimation since the torque estimation parameters are calculated prior to fatigue and fixed afterwards. Therefore, the rehabilitation robot may provide more or less than the desired torque to the patient, possibly decreasing the effectiveness of the rehabilitation exercises. Since fatigue occurs more readily in stroke patients this may become a problem. Thus for our future work, an improved torque estimation strategy that incorporates fatigue detection and compensation is being investigated. From a signal standpoint, the semg signals are noisy and have high frequency components. Many previous studies concerning EMG-based assistive systems utilized artificial intelligence methods such as an ANN or fuzzy networks with low-pass filters to generate a stable control input. If the command input is unstable, then the system performance will be limited regardless of the control method. An optimized semg signal processing algorithm that can extract accurate and stable joint torque profiles will increase the system s performance. Moreover, because the semg signal from the patients is generally very weak relative to that of a healthy person, more sensitive, yet precise, signal processing methods are required to extract the desired assistive torque. In the future, an optimization method to obtain the desired torque from the semg signal and the application of a more robust impedance control must be implemented to reach a successful clinical verification. ACKNOWLEDGEMENT This research was supported by the Happy tech. program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No ). REFERENCES 1. Mackay, J. and Mensah, G., The atlas of heart disease and stroke, World Health Organization, Lum, P. S., Burgar, C. G., and Shor, P. C., Evidence for Improved Muscle Activation Patterns After Retraining of Reaching Movements with the MIME Robotic System in Subjects with Post-Stroke Hemiparesis, IEEE Trans. Neural Syst. Rehabil. Eng., Vol. 12, No. 2, pp , Lum, P. S., Burgar, C. G., Loos, M. V., Shor, P. C., Majmundar, M., and Yap, R., MIME robotic device for upper-limb neurorehabilitation in subacute stroke subjects: a follow-up study, J. Rehabil. Res. Dev., Vol. 43, No. 5, pp , Fig. 12 Normalized root mean square error (NRMSE) of the bimanual shoulder flexion angle 4. Hesse, S., Schulte-Tigges, G., Konrad, M., Bardeleben, A., and Werner, C., Robot-assisted arm trainer for the passive
7 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 8 AUGUST 2012 / 1479 and active practice of bilateral forearm and wrist movements in hemiparetic subjects, Arch. Phys. Med. Rehabil., Vol. 84, No. 6, pp , Stein, J., Harvey, R. L., Macko, R. F., Winstein, C. J., and Zorowitz, R. D., Stroke recovery & rehabilitation, Demos Medical, training machine and its application to biceps exercises, Int. J. Precis. Eng. Manuf., Vol. 12, No. 1, pp , Viitasalo, J. and Komi, P., Signal characteristics of EMG during fatigue, Eur. J. Appl. Physiol. Occup. Physiol., Vol. 37, No. 2, pp , Donatelli, R. A., Physical therapy of the shoulder, 4th ed., Churchill Livingstone, Krebs, H. I., Hogan, N., Aisen, M. L., and Volpe, B. T., Robotaided neuro-rehabilitation, IEEE Trans. Rehabil. Eng., Vol. 6, No. 1, pp , Lum, P. S., Burgar, C. G., Shor, P. C., Majmundar, M., and Van der Loos, M., Robot-Assisted Movement Training Compared With Conventional Therapy Techniques for the Rehabilitation of Upper-Limb Motor Function After Stroke, Arch. Phys. Med. Rehabil., Vol. 83, pp , Fasoli, S. D., Krebs, H. I., Stein, J., Frontera, W. R., and Hogan, N., Effects of robotic therapy on motor impairment and recovery in chronic stroke, Arch. Phys. Med. Rehabil., Vol. 84, No. 4, pp , Dipietro, L., Ferraro, M., Palazzolo, J. J., Krebs, H. I., Volpe, B. T., and Hogan, N., Customized interactive robotic treatment for stroke: EMG-triggered therapy, IEEE Trans. Neural Syst. Rehabil. Eng., Vol. 13, No. 3, pp , Korean Agency for Technology and Standard (KATS), 5th Anthropometric Dimensional Data Report for Korean, Kong, K., Bae, J., and Tomizuka, M., Control of rotary series elastic actuator for ideal force mode actuation in human robot interaction applications, IEEE/ASME Trans. Mechatronics, Vol. 14, No. 1, pp , Kong, K., Moon, H., Hwang, B., Jeon, D., and Tomizuka, M., Impedance Compensation of SUBAR for Back-Drivable Force Mode Actuation, IEEE Trans. Robot., Vol. 25, No. 3, pp , Pan, Y.-R., Shih, Y.-T., Horng, R.-H., and Lee, A.-C., Advanced Parameter Identification for a Linear-Motor-Driven Motion System Using Disturbance Observer, Int. J. Precis. Eng. Manuf., Vol. 10, No. 4, pp , Ohnishi, K., Robust motion control by disturbance observer, J. Robot. Mechatronics, Vol. 8, No. 3, pp , Winter, D., Biomechanics and motor control of human movement, Wiley-Interscience, Nian, X.-H., Yang, Y., and Huang, L., Matrix Approximation with Constraints of Matrix Inequalities and Applications in Robust Control, Acta Automatica Sinica, Vol. 31, No. 3, pp , Cutter, N. C. and Kevorkian, C. G., Handbook of manual muscle testing, McGraw-Hill, Park, J., Kim, K., and Hong, D., Haptic-based resistance
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