Koopman Model Predictive Control of An Origami-Inspired Soft Exoskeleton for Knee Rehabilitation
Authors: Junxiang Wang, Han Zhang, Zehao Wang, Huaiyuan Chen, Pu Wang, Weidong Chen
Organizations: School of Automation and Intelligent Sensing, Institute of Medical Robotics, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China · Seventh Affiliated Hospital, SunYat-sen University, Shenzhen 510275, China
Knee rehabilitation plays a critical role in restoring patients' mobility and functional independence. Traditional rigid rehabilitation exoskeletons are often bulky and cumbersome to wear, whereas soft pneumatic exoskeletons offer lightweight, wearable, and intrinsically compliant solutions that are better suited for human--robot interaction. However, achieving precise motion control for soft exoskeletons remains challenging due to the difficulty of accurately modeling pneumatic actuators and the patient-specific human--robot coupled dynamics during rehabilitation training. To address these challenges, this paper proposes a Koopman-based Model Predictive Control (KMPC) framework for soft knee rehabilitation exoskeletons. The nonlinear human--robot coupled system is represented through a lifted linear Koopman model, enabling predictive control with explicit handling of constraints. In addition to actuation commands used to control valves and pumps, electromyography (EMG) signals are incorporated as system inputs, allowing the Koopman model to capture voluntary neuromuscular contribution and individual neuromuscular characteristics. Experimental results on both healthy participants and patients demonstrate that the proposed framework improves model prediction accuracy and effectively captures subject-specific behaviors, thereby supporting EMG-informed subject-specific assistance within the tested seated knee-rehabilitation setting. Compared with conventional Proportional--Integral--Derivative (PID) control, the proposed KMPC approach achieves lower tracking errors and reduced actuation effort in both passive and active rehabilitation modes. Additional comparisons with Iterative Learning Control (ILC) further validate the tracking performance of the proposed controller.
Figures & tables
Fig. 1 : System overview of the proposed EMG-informed Koopman-MPC framework for soft knee rehabilitation. The framework integrates pneumatic assistance, IMU/EMG sensing, subject-specific Koopman modeling, and KMPC-based online control.
Fig. 2 : The origami-inspired pneumatic actuator illustrating bi-directional torque output, large Range of Motion (RoM), and kinematic alignment with the knee joint.
Fig. 3 : (a) Parameterization of the origami-inspired actuator [ 29 ] . (b) Torque-to-pressure ratio as a function of actuator angle α with 16 air chambers, computed using l1=101 mm, l2=70 mm, and l3=20 mm.
Fig. 4 : The overall exoskeleton design and its main components. (a) Assembled exoskeleton mounted on a dummy leg. (b) Pneumatic actuator. (c) Primary fixing strap. (d) 3D-printed connector. (e) Patella-opening strap.
Fig. 5 : Placement of EMG and IMU sensors on the lower limb. EMG sensors are placed on the thigh muscles, and IMUs are mounted on the thigh and lower leg to measure joint angles.
Fig. 6 : Hardware architecture of the pneumatic drive and control unit. The diagram shows the integration of the host computer, microcontroller, power module, pump and pump driver, valve and valve driver, together with the corresponding control signals and airflow paths.
Fig. 7 : Deep Koopman operator network framework. The encoder maps the physical state xk into learned features, which are concatenated with the original state to form the lifted state zk . The lifted state evolves linearly under the influence of the delayed EMG input sk−δ and the control input uk(1) , and the physical state is recovered through a linear output map.
Fig. 8 : Overview of the experimental framework, including data acquisition, model training, and rehabilitation control.
ID
Gender
Age (y)
Weight (kg)
Height (cm)
Brunnström Stage
P1
male
24
88
188
/
P2
male
25
72
185
/
P3
male
26
85
181
/
P4
male
23
83
183
/
P5
male
25
66
172
/
S1
female
31
70
163
4
TABLE I : Participant Demographics
Fig. 9 : (a) GUI for knee angle tracking. (b) Experimental setup for healthy participants. (c) Experimental setup for patients.
Fig. 10 : The prediction evaluation of the Koopman models for the healthy participants and patients.
Ref. Traj. Freq. (Hz)
Ref. Traj. Peak Knee Motion Range ( ∘ )
Model Used
P1
P2
P3
P4
P5
0.027
90–120
Pers.
2.2933
1.8421
2.0262
0.7189
1.0640
Non-Pers.
2.8923
2.0143
2.4298
1.3553
2.1641
0.032
90–120
Pers.
3.2650
1.5989
2.2230
0.9069
1.2427
Non-Pers.
3.9000
2.4224
2.7117
1.1039
2.0770
0.04
90–120
Pers.
4.4369
3.1267
2.7982
1.2030
3.0558
Non-Pers.
9.4003
4.1941
3.1707
1.9197
3.1998
TABLE II : The tracking RMSE errors in degrees ( ∘ ), compared between personalized and non-personalized models under different frequencies and angle ranges in passive mode for healthy participants P1 to P5.
Mode & Model
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
Active
Pers
10.78
3.59
7.11
2.55
11.30
7.35
7.60
6.56
4.84
4.96
6.72
Non-Pers
10.13
4.73
8.71
5.21
13.81
7.03
8.47
7.92
5.76
5.12
9.01
Passive
Pers
12.95
11.84
10.30
11.05
13.57
14.53
11.82
7.17
5.19
5.05
9.42
Non-Pers
13.84
13.69
13.23
10.68
16.09
17.59
13.01
9.83
5.90
5.27
9.77
TABLE III : Comparison of tracking RMSE in degrees ( ∘ ) between personalized and non-personalized models for stroke patients.
Fig. 11 : The average tracking errors of personalized and non-personalized human-robot coupled dynamics in both active and passive modes among the healthy participants.
Mode & Model
P1
P2
P3
P4
P5
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
Active
KMPC
1.147
1.203
1.211
1.103
1.785
10.781
3.585
7.114
2.545
11.296
7.349
7.604
6.559
4.837
4.958
6.715
PID
1.410
1.360
1.588
1.705
1.639
10.830
5.945
7.389
6.608
13.863
7.976
10.776
7.428
6.103
7.953
9.567
ILC
–
–
–
–
–
–
–
–
–
–
–
–
6.861
4.444
6.144
7.288
Passive
KMPC
3.265
1.599
2.223
0.907
1.243
12.950
11.839
10.298
11.054
13.568
14.529
11.818
7.169
5.188
5.053
9.421
PID
5.640
2.380
3.082
2.254
2.131
15.794
13.002
11.359
15.731
14.820
14.041
12.977
8.591
6.918
6.642
12.907
ILC
–
–
–
–
–
–
–
–
–
–
–
–
7.890
5.381
5.055
9.929
TABLE IV : Comparison of tracking RMSE in degrees ( ∘ ) between KMPC and controller baselines for healthy participants (P1–P5) and stroke patients (S1–S11). For participants S8–S11, the comparison with ILC is included.
Fig. 12 : The tracking error histograms of the KMPC and PID controllers for both healthy participants and stroke patients. (a) Tracking error distribution in passive mode. (b) Tracking error distribution in active mode.
Fig. 13 : Comparison of the average absolute PWM duty-cycle values in both active and passive modes.
Fig. 14 : The histogram of the computation time of the proposed KMPC framework.
Fig. 15 : Comparison of RMS amplitudes across the following three conditions: (1) no-exoskeleton assistance, (2) KMPC-assisted active mode, and (3) KMPC-assisted passive mode.