Autonomously Adapting Robotic Assistance for ...

Autonomously Adapting Robotic Assistance for Rehabilitation Therapy Duygun Erol, Vishnu Mallapragada, Nilanjan Sarkar

Gitendra Uswatte, Edward Taub

Robotics and Autonomous Systems Laboratory Vanderbilt University VU Station B 351592, 2301 Vanderbilt Place, Nashville, TN, 37235, USA {duygun.erol, vishnu.g.mallapragada, nilanjan.sarkar} @vanderbilt.edu

Psychology Department University of Alabama at Birmingham 1530 3rd Ave S, CPM712, Birmingham, AL, 35294-0018, USA {etaub,guswatte}@uab.edu

Abstract - The goal of our research is to develop a novel control framework to provide robotic assistance for rehabilitation of the hemiparetic upper extremity after stroke. The control framework is designed to provide an optimal time-varying assistive force to stroke patients in varying physical and environmental conditions. An artificial neural network (ANN) based Proportional-Integral (PI) gain scheduling direct force controller is designed to provide optimal force assistance in a precise and smooth manner. The human arm model is integrated within the control framework where ANN uses estimated human arm parameters to select the appropriate PI gains. Experimental results are presented to demonstrate the effectiveness and feasibility of the proposed control framework. Index Terms-robot-assisted therapy, direct force controller, human arm parameter estimation, PI-gain scheduling.

I. INTRODUCTION Stroke is a highly prevalent condition [1], especially among the elderly, that results in high costs to the individual and society [2]. It is a leading cause of disability, commonly involving deficits of motor function. In recent years, new techniques adopting a task-oriented approach have been developed to encourage active training of the affected limb, which assume that control of movement is organized around goal-directed functional tasks [3]-[4]. “Shaping” is one of the task oriented behavioural training techniques employed in Constraint-Induced Movement (CI) therapy [3],[5] pioneered by the co-authors Taub and Uswatte, which has the effect of placing optimal adaptive task practice procedures into a systematic, standardized, and quantified format. It may be viewed as an elaboration of conventional physical rehabilitation practice. The availability of such training techniques, however, is limited by the amount of costly therapist’s time they involve and the ability of the therapist to provide controlled, quantifiable and repeatable assistance to complex arm movement. Consequently, robot assisted rehabilitation that can quantitatively monitor and adapt to patient progress, and ensure consistency during rehabilitation may provide a solution to these problems. Robotic devices are general-purpose aids used to assist, enhance and quantify rehabilitation therapy, which has been an active research area for the last few years [6][8]. The clinical results from use of these devices have been promising, but it is still unclear how robots should assist in therapies. Patients exhibit a wide range of arm impairment levels. For example, some subjects are able to move through a large range of motion at normal or nearly

normal velocity, while others have severe range and velocity limitations. Providing optimal assistance (assistance as needed) to each patient, which can be defined as the least amount or “just enough” assistance, is desirable from a rehabilitation perspective since it enables the patient to make a specific movement with the least amount of assistance [9]. It would be useful if a robot could provide controllable, quantifiable assistance specific to a particular patient by adapting the level of assistance provided. However, there is a paucity of research in designing such controllers that can offer optimal assistance in a person-specific and task-specific manner. We are working on designing such a controller, which has both high-level and low-level components. The highlevel component is a supervisory controller, which determines the desired optimal assistive force for a given situation; it is not discussed here. The low-level component, which is the primary engine for actually generating the force for providing the optimal assistance, is the focus of this paper. The low-level component is an artificial neural network (ANN) based PI gain scheduling direct force controller that has the ability to provide optimal assistance for a wide range of people and tasks under varying conditions. Initial simulation results of the proposed control framework had been presented in [10]. In this paper, we further develop the initial concept for realtime implementation and present the results obtained from application of the proposed controller on unimpaired participants. II. CONTROL ARCHITECTURE In the literature, position based Proportional-IntegralDerivative (PID) control or impedance control [6][8]approaches have been traditionally used for rehabilitation robotics. A PID controller requires continuous adjustment of control gains to accommodate a wide range of patients for a variety of tasks and conditions. In addition, a position-based PID controller is not suitable for direct force control. On the other hand, an impedance controller, which is shown to be stable for a wide range of patients, achieves force control in an indirect manner. We did not pursue an impedance control approach because we are interested in a direct force control approach where the desired time varying assistive force can be directly specified for the robot. We propose an ANN based PI gain scheduling direct force controller in this paper (Fig.1). The proposed controller has the ability to automatically adjust

control gains to accommodate a large number of patients with varying physical conditions, and to “fade out” assistance in a progressive manner to complement improvement produced by therapy.

estimate human arm parameters in real-time within the control loop, which is new in the rehabilitation robotics literature. A bilinear transformation is discretized by replacing ∆ x& with ⎛ 2 ⎞⎛⎜ 1 − z − 1 ⎞⎟ to obtain: ⎜ T ⎟⎜ −1 ⎟ ⎝ ⎠⎝ 1 + z

⎠

⎡ ⎛ 2 ⎞ ⎛ 1 − z− 1 ⎞ ⎤ ⎟ + k ⎥ ∆x ∆F = ⎢b ⎜ ⎟ ⎜ T ⎜ −1⎟ ⎣⎢ ⎝ ⎠ ⎝ 1 + z ⎠ ⎦⎥

Note that ∆F = F − F

Fig.1. Block Diagram of Control Architecture

In this controller, an artificial neural network is trained with human arm parameters as inputs, and the controller gains as outputs. Since human arm characteristics vary for different patients as well as for the same patient during the execution of different tasks, the controller requires estimation of the arm parameters to be used for selecting variable control gains. The idea here is to automatically tune the gains of the force controller based on the condition of each patient as characterized by his/her arm parameters so that it can apply the desired assistive force in an efficient and precise manner. This ability to apply precise assistive force in varying conditions is directly related to optimal assistance in the following sense. In our overall framework for generating optimal assistive force, a supervisory controller determines a reference optimal assistive force trajectory for the robot for a given task. This reference trajectory is determined by an algorithm that takes into account both the total desired force required for the task and how much of it the patient can generate. It is clear that this assistive force computation will not be useful unless a low-level controller can apply this force in a precise and efficient manner in a variety of arm conditions. Herein lies the need for the low-level controller for optimal assistance presented here. In what follows, human arm parameter estimation, design of the ANN-based PI-gain scheduling direct force controller techniques is presented. A. Estimation of Human Arm Characteristics A human arm can be characterized by its impedance parameters [11]. Since the muscle of a person is mechanically analogous to a linear spring-damper system, the following second-order equation can be used to model the arm dynamics [12]: (1) ∆F = b∆x& + k∆x where b is the damping coefficient and k is the stiffness of the human arm. ∆x , ∆ x& and ∆F represent the difference between the position, velocity and force at the tip of the human operator’s arm and the equilibrium point, respectively. Various system identification techniques can be used to estimate the damping and stiffness parameters in (1). In this work we choose an ARX (Auto Regressive eXogenous) model to estimate the damping and stiffness parameters of the human arm. ARX model structure [13] is one of the simplest parametric structures one can use with very little numerical difficulty and is chosen as the first estimation technique for our problem. An ARX model has been previously used to estimate human arm parameters [11]. We use this system identification technique to

(2)

eq

and ∆x = x − x

eq

. F and x are

the force and position at the robot end-effector. F and x are the force and position at the equilibrium eq

eq

point. T is the sampling period and z −1 represents a shift of one step in the time domain. Let k be the time-step index. We obtain the following difference equation from (2):

(

)

⎡ ⎛2⎞ −1 −1 ⎤ ⎟ 1 − z + k (1 + z ) ⎥ ∆x ⎣ ⎝T ⎠ ⎦ ⎛ ⎛2⎞ ⎞ ⎛ ⎛ 2 ⎞⎞ ∆F [ k ] + ∆F [ k − 1] = ⎜ b⎜ ⎟ + k ⎟ ∆x[ k ] + ⎜ k − b⎜ ⎟ ⎟ ∆x[ k − 1] T ⎝ ⎠ ⎝ T ⎠⎠ ⎝ ⎠ ⎝ ∆F (1 + z

−1

) = ⎢b ⎜

(3)

∆F [ k ] + ∆F [ k − 1] = A ∆x[ k ] + A ∆x[ k − 1] 1 2

where A = ⎛⎜ b⎛⎜ 2 ⎞⎟ + k ⎞⎟ and A = ⎛⎜ k − b⎛⎜ 2 ⎞⎟ ⎞⎟ (4) 1 ⎝ ⎝T ⎠ 2 ⎝ ⎝ T ⎠⎠ ⎠ Equation (3) can be cast into the regressor form as follows (5) y[ k ] = ϕ T [ k ]θ [ k ] with θ [ k ] = [ A [ k ] A [ k ]]T 1

representing

2

the

parameter

matrix, ϕ[ k ] = [x[k ] x[k − 1]]T the regression vector, and y[k ] = ∆F [ k ] + ∆F [ k − 1] the output vector. The RLS solution to determine parameters A and A is given by 1

2

T

θ [ k ] = θ [ k − 1] + G[ k ]( y[ k ] − ϕ [ k ]θ [ k − 1])

(6) where the gain factor G[k ] determines how the current

prediction error y[ k ] − ϕ T [k ]θ [k − 1] affects the update of parameter estimation. G[k ] is determined using: G[ k ] =

P[ k − 1]ϕ [ k ] T λ + ϕ [ k ]P[ k − 1]ϕ [ k ]

(7)

where λ is the forgetting factor that influences the weight given to earlier data relative to the newly acquired data. P[k ] is the covariance matrix of the estimated parameters, which is calculated using Equation (8). 1⎡ P[ k − 1]ϕ [ k ]ϕ T [ k ]P[ k − 1] ⎤ (8) P[ k ] = ⎢ P[ k − 1] − ⎥ λ⎢ λ + ϕ T [ k ]P[ k − 1]ϕ [ k ] ⎦⎥ ⎣ The initial guess for the covariance matrix P and the forgetting factor λ are specified by the user. Once the parameters A1 and A2 are estimated, the estimates of environment parameters b and k are obtained by solving the set of linear equations (4). ⎛ A − A2 ⎞ and ˆ ⎛ A + A2 ⎞ (9) ⎟⎟ ⎟⎟ bˆ ( k ) = T ⎜⎜ 1 k ( k ) = ⎜⎜ 1 4 ⎝ 2 ⎠ ⎝ ⎠

B. Artificial Neural Network (ANN) based PI gain scheduling An ANN based PI gain scheduling direct force controller is expected to provide better performance in robot-assisted rehabilitation than a traditional fixed-gain PI force controller. This is due to the fact that a variable gain controller will have the ability to adjust the control gains to accommodate a large number of patients in varying physical and environmental conditions. It has been found that a multi-layer ANN with back-propagation method works well in most control applications. In this work, we ∧

use human arm stiffness estimate k as the sole input vector and Proportional (P) and Integral (I) gains as the corresponding target (output) vectors to train the neural network. Note that a derivative gain in force control generally destabilizes the system and thus this gain is not used for this controller. It was experimentally found out during pilot work that for unimpaired participants, the ∧

values of arm damping b were almost the same for all participants and thus were not needed to train the ANN. However, if it was found that in the next phase of work with real patients, the damping may vary among different

In (12), y represents a new input. The new control input y is designed so as to allow tracking of the desired force F . The control law is selected as follows: d

y = J (q)

where

Md

−1

( − K d x& + K p ( x f − x ) − M d J& ( q , q& ) q& )

(13)

is a suitable reference to be related to force error. f

J (q ) is the Jacobian matrix. M (mass), K (damping), d d

and K p (stiffness) matrices specify the target impedance of the robot. x and x& are the position and velocity of the end-effector in the Cartesian coordinates, respectively. The relationship between the joint space and the Cartesian space acceleration is used to determine position control equation. &x& = J ( q ) q&& + J& ( q , q& ) q& and &x& = J ( q ) y + J& ( q , q& ) q& (14) By substituting (13) into (14), we obtain &x& = J ( q)( J (q) − 1 M − 1 ( − K x& + K ( x − x) − M J& ( q, q& ) q& )) + J& (q, q& )q& d d p f d &x& = − M − 1 K x& + M − 1 K ( x − x ) d d d p f

(15)

M d &x& + K d x& + K p x = K p x f

Equation (15) shows the position control tracking of

∧

patients, estimation of b will need to be included in ANN training. We use supervised training with an error backpropagation algorithm to train the ANN. The input training data set for the backpropagation algorithm is generated by obtaining the human arm stiffness of different subjects. The output training data are the PI gains that can provide an acceptable force response in these subjects. The training data set is chosen in such a way that it spans a large range of the human arm characteristics. The number of the hidden layers and neurons in each layer are determined based on actual data. Once the ANN is trained offline, it can work online in conjunction with the system identification module to generate suitable control gains, which are used in the direct force controller described below, based on estimated human arm characteristics. C. Direct Force Controller The proposed control framework allows us to directly specify the desired force as shown in Fig. 1. This control framework uses an inner position loop for force control similar to what is described in [14]. Using inverse dynamics control, manipulator dynamics are linearized and decoupled via feedback. The dynamic equation of the robotic manipulator can be written in the form (10) u = M ( q ) q&& + V ( q , q& ) + G ( q ) where u is the input to the manipulator, M (q ) is the mass matrix of the manipulator, V ( q , q& ) is a vector of centrifugal and Coriolis terms, and G (q ) is an vector of gravity terms. Control input u to the manipulator is designed as follows: (11) u = M ( q ) y + V ( q , q& ) + G ( q ) which leads to the system of double integrators (12) q&& = y

x

−1

x with dynamics specified by the choices of K d , K p and M d matrices. Impedance is attributed to a mechanical

system characterized by these matrices that allows specifying the dynamic behavior. Let Fd be the desired force reference. The relationship between x

f

and the force

error is expressed in (16) as: (16) ∫ ( F d − F h ) dt where P and I are the proportional and integral gains, respectively, and F is the actual applied force. Equations x

f

= P (F

d

− F ) + I h

h

(15) and (16) are combined to obtain the following equation: M &x& + K x& + K p x = K p ( P ( F − F ) + I ∫ ( F − F ) dt ) (17) d

d

d

h

d

h

We can observe from (17) that the desired force, which can be directly specified, is achieved by controlling the position of the manipulator. The PI gains are obtained from the ANN gain scheduler described earlier. D. Gain Switching Control gains are switched to the values predicted by the ANN when the performance of the initial PI gains is found to be unsatisfactory. Instantaneous switching of control gains, however, may sometimes destabilize the system or cause a large overshoot. Hence, we designed a position-reference-compensation based gain switching strategy to achieve a smooth transfer of control gains. The key to a smooth switching is the position variable ( x ) f

that transforms the force error into a position reference as given in (16). The bump in the velocity response occurs because changing the gains causes a change in ( x ) . f

III. EXPERIMENTAL RESULTS A. Experimental Setup Fig. 2 illustrates an experimental setup where a human subject is interacting with a PUMA 560 robotic manipulator. PUMA 560 has been used for rehabilitation of the shoulder joint and was shown to be safe to use with patients [8]. We have designed several additional safety features such as a wireless emergency stop, a pneumatic stop, removal of multiple inverse kinematic solutions, and others that will be added to the arm before clinical trials in the future. We have replaced the microcontroller board of the PUMA to develop an open architecture system and interfaced the robot with Matlab and Realtime Workshop to allow fast and easy system development. ATI Gamma, NC force/torque is used for force feedback. B.ANN based PI Gain Scheduling Direct Force Controller Two experiments simulating tasks that could be relevant for upper extremity rehabilitation were conducted with unimpaired participants to evaluate the controller. B. 1. Experiment 1 Arm guiding is one of the therapeutic intervention techniques used in rehabilitation for muscle strengthening and increasing the active range of motion (AROM). The therapist applies a constant force to each patient and asks them to follow his/her arm movement without losing contact with him/her. In this experiment, we simulated a similar situation using the PUMA 560 robotic manipulator.

Fig. 2. Experimental Setup.

The robot applied a constant force on the participants’ extended arm and they were asked to keep their arms in the starting position in contact with the robot. Two female and three male participants within the age range of 25-32 years took part in this experiment. Four of the participants were right-handed and one of them was left-handed. Let us first describe why this task demonstrates a need for the control framework we are proposing. Consider Fig. 3 that shows the force response of 3 of the participants who were asked to maintain 15N contact force with the robot. The robot used a traditional fixed-gain PI based direct force controller with the same set of PI gains for each participant to generate a constant 15N contact force. It was observed that different participants had different abilities to keep their arms in the starting position, which required applying 15N force on the robot. As can be seen from Fig. 3, Participant 1 was able to track the desired force well. On the other hand, this set of PI gains resulted in an overshoot in force response for Participant 2, which is not desirable for rehabilitation therapies. Furthermore, Participant 3 could not even track the desired force. Thus, it was clear that the same control gains are not appropriate for different participants. This was because of the fact that there existed variation of arm characteristics among different participants. This variation must be taken into account by the controller to obtain good force response. In order to design the ANN based gain scheduled controller that can accommodate people with different arm characteristics, we first determined the arm stiffness of the 5 participants using the ARX model described earlier (Table I). 30 Desired Force Participant 1 Participant 2 Participant 3

25 20 Force (N)

Hence to achieve a smooth velocity response, we perform the following steps: 1) When the gain switching mechanism is activated, a modified position reference (MPR) is given to the controller. The MPR is linearly extrapolated from the actual position reference (APR) at the time of activation. Giving a linearly extrapolated position reference causes the robot to continue moving at a constant velocity. A MPR is given to the controller so that the control gains can be switched without affecting the velocity response. 2) At this point, the control gains are switched to the values predicted by the ANN. Note that switching the control gains causes a change in the APR, but this does not affect the velocity response. 3) The desired force reference (DFR) is modified to make the force error zero. The error integrator is also reset to make the APR zero. 4) After the APR becomes zero, the desired force reference is shifted to the original reference value so that the force error increases the APR using the ANN predicted gains. During the above steps, velocity response is not affected, since a MPR is given to the controller. 5) At the instant when the APR equals the MPR, input to the controller is switched back to the APR Switching from compensated position reference to the APR occurs when both are equal, hence, bump in the velocity response is reduced and the desired velocity is tracked using the ANN predicted gains.

15 10 5 0

0

5

10 Time (s)

15

20

Fig. 3. Controller Response for Three Participants with Same PI gains.

Force and position were recorded, and velocity was computed from the position data. ∆F and ∆x which are the inputs to ARX model, are used to estimate the human arm characteristics of each participant. We then used a traditional PI based direct force controller to determine the suitable control gains (PI) for these 5 participants. The ANN was trained using the Levenberg-Marquardt (LM) backpropagation method where human arm stiffness k

Participant # Stiffness k (N/m)

TABLE I 2 3 4000 1000

1 2200

4 2750

5 2300

A 6th Participant whose data was not used for neural network training was selected to demonstrate the validity of the proposed ANN-based PI gain scheduling controller. A constant force reference of 15N was applied to the participant. The controller used an initial set of PI gains ( P = 4 x10

−4

, I = 4 x10

−4

) that did not result in good performance (e.g., large overshoot and no convergence) (Fig 4.). As a result, based on the online estimate of the arm stiffness of this new participant, which was 3500 N/m, the ANN predicted new gains

( P = 2.5 x10

−4

, I = 3 x10

−4

that were applied by the controller at B resulting in a much better performance. All of the above processes were performed autonomously and in real-time. This experiment demonstrates that a rehabilitation robot that can accommodate the variation of arm characteristics of different patients can achieve a better performance. )

35

B

Force (N)

25 20 15 10

A 0

5

10 Time (s)

15

x 10

-3

5 4 3 Trial 1 Max* Jerk = 5.1 m/s

2

Trial 2 Max Jerk = 1.9 m/s Trial 3 Max Jerk = 0.6 m/s Desired Velocity

1 0

0

5

10

15 Time (s)

20

3

3 3

25

30

Fig. 5. Velocity Profiles for Training Participant (Max* Maximum)

Performanc e Index

5 0

6

Hence, a velocity error – jerk based performance index was calculated for each trial. This performance index was used to evaluate the suitability of a particular PI gain for each participant. The performance index was calculated using the below equation: (18) J max

Desired Force Participant

30

performance for each participant because of variation in the arm characteristics. Smoothness of motion (measured by jerk) and velocity error (difference between desired and actual velocity) were the two criteria that characterized the performance of the PI gains. Fig. 5 shows the velocity profiles for one participant during three trials with different PI gains. We can observe from Fig. 5 that minimizing jerk during motion results in poor velocity tracking (Trial 3). We can also observe that good velocity tracking results in higher jerk (Trial 1). Hence, to achieve an acceptable performance, jerk and velocity error had to be optimized.

Velocity (m/s)

was the input vector, and P and I gains were the output vectors. The network with 15 neurons for P gains and 15 neurons for I gains were sufficient for this application.

20

Fig. 4. Controller Response for New Participant.

B. 2. Experiment 2 In the second experiment, the task was designed to move the robot end-effector between two points with a desired velocity trajectory. This task models the point-topoint motion of a goal directed functional task during rehabilitation. During the task, participants grasped the robot end-effector and followed the motion of the robot. When the force applied by the participant is not sufficient to follow the desired trajectory, robot provides assistance to complete the task. Assistance provided by the robot is decreased when the subject applies enough force to follow the desired trajectory. During the motion, velocity error is transformed into a force reference using an outer PD (Proportional-Derivative) velocity loop. Desired assistance is computed as a force reference and given as input to the direct force controller. To obtain data for training the ANN, two different female and three different male participants, 25-40 years old, four right-handed, and one left-handed were selected. Participants were asked to perform the task using a traditional PI based force controller. This task was repeated four times for each participant with different PI gains. Predictably, all the PI gains did not show good

=

∫ Verror dt

∫ F dt

where J max is the maximum value of jerk. ∫ V error dt and ∫ F dt are the integrals of absolute velocity error and force, respectively. Performance index is chosen in the above form since maximum jerk captures the smoothness of motion and the ratio of integrals captures the deviation from the desired velocity profile. Note that as the force (applied by the participant) increases, velocity error also increases. Hence, ratio of force to velocity error is used to normalize the performance index. For all the participants, the performance index was calculated offline from the position and force data for each trial. The range of the performance index was observed to be from 2000 to 500000. Performance index in the lower range signifies a smooth motion which deviates significantly from the desired trajectory. Performance index in the higher range indicates a jerky motion. It was observed from the data that a performance index ranging from 30000 to 100000 shows reasonably smooth motion with good trajectory tracking. Hence, for each participant PI gains were chosen so that the performance index was within this optimum range. Performance index within this range assures the suitability of the PI gains for the subject. During the performance of the task, force and position data is also used to estimate the human arm characteristics of each participant. According to the equilibrium point

hypothesis, during a point-to-point motion, equilibrium point of the human arm shifts considerably [15]. Hence the equilibrium force and position ( F and x ) are updated

-4

5

TABLE II Performance Index T = 20 s T = 40 s 42636 48642 5800 27900

As can be seen from Table II, the performance index for Participant 1 (P1) was within the optimum range, hence the PI gains were not switched (Fig. 6 - solid line). On the other hand, performance index for Participant 2 (P2) was out of the optimum range; hence, PI gains were switched (Fig. 6 – dashed line). Switching to the gains predicted by the gain scheduler based on participant’s arm characteristics resulted in a performance index very close to the optimum range (Table II). IV. CONCLUSION In this work a new control framework for robot assisted rehabilitation has been presented. In this framework, online human arm parameter estimation is integrated in with an ANN based PI gain-scheduling direct force controller for applying optimal assistive force to a patient in a precise and smooth manner. The results with unimpaired participants demonstrate the efficacy and adaptability of the proposed controller. No existing control framework in rehabilitation robotics, to our knowledge, can accommodate variation of human arm characteristics in providing assistance to the patients. We believe that the proposed system can enhance treatment gains by fading out assistance more effectively than when carried out by a therapist based on a patient’s condition. Further work in this area would involve experimentation with real patients to determine the efficacy of this controller in a therapy.

x 10

-4

4.5 4 1

Integral Gain

every second. The output vector for the training data was created by selecting the best PI gains for each participant based on the performance index. The input vector was created with the corresponding arm characteristics for each participant. A two-layer feedforward LM backpropagation method with 15 neurons for P gain and 15 neurons for I gain for each gain was sufficient for this application. To test the efficacy of the ANN based gain scheduling direct force controller, two new participants were instructed to perform the same task. The duration of the experiment was 60 seconds. Initial PI gains for the direct force controller were chosen arbitrarily. During the experiment, the performance index was evaluated online. The control algorithm checked the performance index every 20 seconds. If the performance index was within the optimum range, PI gains were not switched. If the performance index was beyond the optimum range, the controller switched the gains to the ANN predicted values. Table II shows the performance indices for the two new participants at 20 and 40 seconds. Participant 1 2

x 10

eq

Proportional Gain

eq

1.5

0.5

0

20 40 Time (s)

3 2.5 2 1.5

P1 P2 0

3.5

P1 P2

1 60

0.5

0

20 40 Time (s)

60

Fig. 6. PI gain Changes for Participant 1 and Participant 2.

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