Koopman Model Predictive Control of An Origami-Inspired Soft Exoskeleton for Knee Rehabilitation

Effective rehabilitation methods are essential for recovering lower limb dysfunction caused by stroke, and this work introduces a new control framework for soft rehabilitation robots. The gist: data-driven modeling using a Deep Koopman Network, incorporating EMG signals and PWM duty cycle inputs, allows for well-performed yet computation efficient tracking control on a human-robot coupled soft exoskeleton system.

System Design and Hardware

The research designs a soft lower-limb rehabilitation exoskeleton based on an origami-inspired pneumatic actuator. This actuator module embeds a Quadrangular Expanded origami frame made of PVC sheet within a flexible air bladder, achieving bi-directional deformation for extension force under positive pressure and flexion force under negative pressure. The final lightweight design weighs 0.7 kg and provides an output torque of 17 Nm at 30 kPa/20°, offering a movable range from 10 to 350 degrees. To ensure even force distribution, two 3D-printed connectors are attached to both ends of the actuator, fabricated from high-strength plastic. These connectors are designed to provide sufficient structural stiffness and enable the actuator’s force to be distributed more evenly across the thigh and lower-leg.

Human-Robot Interaction Modeling

The core of the control framework involves modeling the complex human-robot interaction dynamics using a Deep Koopman Network approach. The system is formulated in discrete time with a sample time of 20ms, where the state vector includes:

1. The knee joint angle measured by IMU sensors.

2. The PWM duty cycle controlling the pump and valve (input u(1)).

3. Electromyography (EMG) signals from muscle sensors, specifically using a time-shifted signal u(2) = s k - δ, where δ = 10, to account for the EMG signal being approximately 200ms ahead of actual muscle contraction.

The lifting function Φ maps the original state into a higher-dimensional latent representation z k, which is governed by linear dynamics in this lifted space: z k+1 = A z k + B u k. The pair (A, B) serves as a finite-dimensional approximation of the infinite-dimensional Koopman operator, identified using data.

Control Strategy: KMPC Implementation

Based on the obtained Koopman model, Model Predictive Control (MPC) is employed to control the soft robot. The overall control objective is to let this human-robot coupled system track a given reference signal. The KMPC minimizes a quadratic cost function over a time horizon of N steps:

min over u(1) k for k=0 to N-1 of q · x k+tk − x rk+t squared + r · u(1) k+t-1k squared, subject to the system dynamics in the lifted space and control constraints (u min ≤ u(1) k+tk ≤ u max). The weight q=1 penalizes tracking error, and r=0.25 penalizes control effort. This formulation ensures that the problem is reformulated into a compact form of convex quadratic programming at each time step, allowing for efficient solution within the 20ms sample time.

Data-Driven Training and Personalization

The Koopman model training is performed using collected data D = ∪ i D i, where each dataset includes knee joint angles (IMU), control inputs (PWM duty cycle), and EMG signals. The training involves defining a prediction loss function L(Φ, A, B) based on the difference between the predicted lifted state zˆ k and the actual lifted state Φ(x k). This network is optimized using the Adam optimizer. A key finding is that integrating EMG signals into the Koopman model improve[s] the model accuracy to great extent, leading to better prediction error compared to models without EMG inputs. Furthermore, a personalized Koopman model trained from an individual’s own data performs better than the non-personalized model, demonstrating that the framework adapts to different participants.

Performance Validation

Experiments validate three main aspects: 1) The accuracy of modeling human-robot interaction dynamics, showing that models incorporating EMG signals demonstrate a lower prediction error across all prediction steps. 2) The ability of the Koopman model to capture personalized dynamics, as personalized models consistently outperform non-personalized ones in both passive and active modes. 3) The capability of the KMPC method to provide effective rehabilitation training, where it outperforms conventional control methods (PID) in both passive and active training modes, resulting in lower tracking errors compared to PID controllers. This demonstrates that the proposed framework can adapt to different participants and provides a better human-robot interaction performance.

Here are the specific improvements that can be made to AI systems, derived from this scientific paper, and what those improved systems could achieve:

) Improved System Capability: Real-time Tracking Control for Soft, Compliant Human-Robot Coupled Systems.

The core improvement lies in replacing traditional rigid or overly simplified nonlinear models with a data-driven approach using the Koopman Operator to model complex, soft robot dynamics coupled with human biomechanics (EMG signals).

Specific AI/Modeling Improvements:

1. A Deep Koopman Network architecture that uses neural networks (specifically an MLP encoder and a learned linear system representation, i.e., matrices A and B) to approximate the infinite-dimensional nonlinear dynamics of the soft exoskeleton.

2. Integration of multimodal inputs into the Koopman model: The system explicitly takes both electromyography (EMG) signals (representing muscle intent/torque) and PWM duty cycle signals (representing pneumatic actuator input/force) as inputs to accurately capture human-robot interaction dynamics.

3. Implementation of personalized modeling: Instead of a single, generalized model, the AI framework is designed to train a unique Koopman operator specific to an individual participant's data, leading to personalized control strategies.

4. Application of Model Predictive Control (MPC) based on the Koopman model (KMPC): The learned linear dynamics are used within an MPC framework to generate optimal control inputs in real-time, minimizing a defined cost function that balances tracking error and control effort.

Specific Performance Gains for the Improved AI System:

1. Active and Passive Rehabilitation Training with High Accuracy: The system can reliably guide a user to track complex reference signals (e.g., sinusoidal motions) during rehabilitation training, whether the user is fully assisted (passive mode) or actively contributing (active mode).

2. Superior Tracking Performance over Conventional Control: By using KMPC informed by the Koopman model, the system achieves significantly lower Root Mean Square Error (RMSE) compared to traditional PID controllers across both modes.

3. Adaptive and Personalized Assistance: The personalized nature of the Koopman model allows the control strategy to adapt to individual biomechanics (limb mass, muscle strength, latency), leading to a more comfortable and effective rehabilitation experience tailored precisely to the patient's needs.

4. Robustness Against Noise: The pre-processing steps (band-pass filtering and RMS calculation) applied to EMG signals ensure that the data fed into the Koopman model is clean, preventing noise from corrupting the learned dynamics and improving prediction accuracy.

5. Efficient Real-Time Control: By formulating the control problem as a convex quadratic programming problem at each time step, the system ensures that control commands can be computed and executed efficiently within a tight real-time budget (e.g., 20ms sample time).

In summary, this improved AI system transforms rehabilitation robotics from relying on fixed physical models or simple feedback loops into an intelligent, adaptive control framework capable of modeling complex human-robot interactions to deliver highly personalized and precise physical therapy in real-time.

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.