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.
Safe rehabilitation is an interaction-dynamics problem: the controller must regulate a prescribed motion while absorbing involuntary spasm, voluntary effort, actuator compliance, and model mismatch as disturbances. This paper instantiates the predictive interaction-dynamics framework of the base pHRI formulation on a SEA knee joint. SEA feedforward reduces the gravity-compensated knee to the same scalar double integrator as the base framework, while a dynamic-residual measurement from spring deflection supplies an interaction-disturbance observation. A steady-state target converts the estimated disturbance into a cancelling input, and a finite-horizon quadratic program regulates deviations from that target under range-of-motion, torque, and velocity constraints. The evaluation matches stiffness and damping across controllers so gains cannot be attributed to higher impedance. Under a motion-opposing 15\unitNm step, classical impedance and MPC without estimation produce about 500\unitmrad steady-state error, whereas Kalman-augmented interaction MPC reduces this to 1.17\unitmrad at 100Hz and 0.70\unitmrad at 500Hz; the 500Hz peak is 7.27\unitmrad. In 30 randomized trials, the 95th-percentile peak is 21.57\unitmrad. Bounded Assist-as-Needed scheduling, a corrective-channel energy tank, constrained OSQP stress cases, direct MuJoCo execution, and a posture-clamped MyoSuite knee slice are implemented. The framework holds on a single-mass, closed-inner-loop SEA approximation; an explicit two-mass plant with a finite-bandwidth, pole-placed inner torque loop (SectionVIII) confirms this for nominal tracking but shows delivered torque can overshoot the commanded bound by 21.7% near saturation. Scope excludes clinical intent recognition, full-system passivity, safety certification, hardware trials, and multi-joint validation.
Yongyan Cao, Jinshan Tang
Voryx Robotics · George Mason University, Dept. of Health Administration and Policy
Multi-segment soft robotic arms can continuously reconfigure their body shapes for safe interaction, but tip control alone is insufficient for constrained-space tasks. Therefore, shape control is a more important task for multi-segment soft arms than tip control, but remains challenging due to the high dimensionality and nonlinear dynamics of continuum deformation. In existing work, shape control accuracy is defined by the error in the global frame (global shape error). For multi-segment soft arms, using only global shape error as the control objective is insufficient, as segment coupling, gravity-induced loading, and inertial effects become more significant. This difficulty increases with the number of segments. In this paper, we present a Koopman-based model predictive control framework that combines global and local observables, enabling real-time shape control on multi-segment soft robotic arms. The framework is evaluated through numerical and physical experiments. Numerical experiments demonstrate the scalability of the proposed controller by achieving shape control on robots with up to 10 independently actuated segments. The physical experiments demonstrate that the controller is capable of (1) real-time shape control of 3- and 5-segment robotic arms with tip speeds up to 0.6 m/s, (2) robust tracking without retraining, including distal payloads up to 400g and recovery from a 7N lateral disturbance, and (3) the potential for future inspection applications through a confined-space demonstration. These results demonstrate that the proposed framework enables dynamic, scalable, and accurate real-time shape control on multi-segment soft robotic arms.
Personalization of impedance controllers for powered prosthetic legs is critical to accommodating individual gait biomechanics but remains challenging. Existing methods rely on time-intensive human-in-the-loop exploration and/or constrain optimization to low-dimensional, single-joint parameter subspaces. Sim-to-real transfer has enabled high-dimensional locomotion control for legged robots, but in assistive device control the human partner remains un-modelable. We present a replay-constrained simulation framework: a MuJoCo-based simulator reproduces prosthetic knee-ankle dynamics while replaying recorded hip kinematics and feedback-based ground reaction forces from individual walking data, bypassing the need to model complex human neuromuscular control mechanisms. We demonstrate the framework with a deep reinforcement learning policy that personalizes phase-dependent stiffness, damping, and equilibrium angle at both joints simultaneously, maximizing a biomimicry-based reward computed solely from onboard prosthesis measurements. Experiments with three participants with transfemoral amputation during level-ground walking at 0.8~m/s demonstrate strong simulation-to-hardware predictive validity (Pearson r=0.96--0.997). The best-performing policy on hardware was consistently predicted within the top five simulation policies for all participants. The learned controllers improved overall biomimicry rewards by 42--59% relative to the unpersonalized baseline. The framework supports scalable high-dimensional personalization of powered prosthetic legs and is amenable to extension to higher-dimensional controller parameterizations such as neural-network controllers.
Duong Le, Ryan Posh, Shihao Cheng +2
College of Engineering, University of Michigan, Ann Arbor, MI 48109, USA