Koopman-Based Model Predictive Control of Functional Electrical Stimulation for Ankle Dorsiflexion and Plantarflexion Assistance

TL;DR

The paper tackles the challenge of delivering real-time, personalized gait assistance via FES by transforming nonlinear ankle dynamics into a linear framework with Koopman Operator Theory and implementing a phase-aware MPC. It introduces a data-driven pipeline (EDMD) to learn a finite-dimensional Koopman predictor, builds a phase-dependent linear state-space model, and solves a real-time quadratic program to optimally schedule FES for plantarflexion and dorsiflexion across the entire gait cycle. Experimental results from three able-bodied subjects and one MS participant show precise trajectory tracking and improved gait with FES-driven KMPC, though fatigue and phase stability at higher speeds remain as considerations for future work. The approach offers a scalable, real-time method to personalize gait rehabilitation and has potential applicability to diverse neuromuscular impairments such as stroke, SCI, and MS.

Abstract

Functional Electrical Stimulation (FES) can be an effective tool to augment paretic muscle function and restore normal ankle function. Our approach incorporates a real-time, data-driven Model Predictive Control (MPC) scheme, built upon a Koopman operator theory (KOT) framework. This framework adeptly captures the complex nonlinear dynamics of ankle motion in a linearized form, enabling application of linear control approaches for highly nonlinear FES-actuated dynamics. Utilizing inertial measurement units (IMUs), our method accurately predicts the FES-induced ankle movements, while accounting for nonlinear muscle actuation dynamics, including the muscle activation for both plantarflexors, and dorsiflexors (Tibialis Anterior (TA)). The linear prediction model derived through KOT allowed us to formulate the MPC problem with linear state space dynamics, enhancing the real-time feasibility, precision and adaptability of the FES driven control. The effectiveness and applicability of our approach have been demonstrated through comprehensive simulations and experimental trials, including three participants with no disability and a participant with Multiple Sclerosis. Our findings highlight the potential of a KOT-based MPC approach for FES based gait assistance that offers effective and personalized assistance for individuals with gait impairment conditions.

Figures (7)

• Figure 1: Prediction results - Plot shows the ankle motion prediction during a gait cycle under test FES actuation for different observables ($states$, $custom$, $trigonometric$). The dynamics approximated from (\ref{['eq:gen_linear_pred']}) are utilized to predict the approximate dynamics.
• Figure 2: Comparison of actual and predicted ($z_{k}$) ankle motion angles using different embedding lengths ($L$) in the Koopman-based prediction framework. The black solid line represents the actual ankle motion trajectory, while the dashed lines correspond to predictions with embedding lengths $L=1$ (yellow), $L=8$ (orange), and $L=50$ (red). Increasing the embedding length improves prediction accuracy, as evidenced by the closer alignment of the L = 50 prediction with the actual trajectory. The results highlight the importance of appropriate embedding length selection in achieving accurate Koopman-based predictions.
• Figure 3: The experimental setup DDMPC framework for FES-driven gait assistance are illustrated - . The participant walks on a treadmill equipped with ground reaction force (GRF) sensors to detect gait phase transitions (stance and swing phases). FES electrodes are placed on the Gastrocnemius (GAS) and Tibialis Anterior (TA) muscles to induce plantarflexion and dorsiflexion, respectively, with stimulation parameters set at $f=33$ Hz, $i=u_{k|k}$ mA. Kinematic data sensors record ankle motion dynamics, while the treadmill enables constant-speed walking. Phase-based data collection captures state measurements ($x_{k}$) and FES inputs ($u_{k}$) during walking, dividing the gait cycle into stance and swing phases. The raw data is lifted to a higher-dimensional space using Koopman observables, which capture nonlinear dynamics in a linear framework. The Koopman operator predicts the system dynamics via the lifted representation: $z_{k+1+j|k}=(1-\sigma_{k})(A^{P}z_{k}+B^{P}u_{k}^{P})+\sigma_{k}(A^{D}z_{k}+B^{D}u_{k}^{D})$, where $\sigma_{k}$ distinguishes stance ($\sigma_{k}=0$) and swing ($\sigma_{k}=1$) phases. The Koopman MPC optimizes FES inputs to minimize a task-specific performance measure while adhering to state and control constraints, enabling real-time, phase-specific gait assistance. This framework effectively coordinates plantarflexion and dorsiflexion to support natural walking patterns.
• Figure 4: Trajectory tracking of ankle motion using DDMPC FES for subjects A1, A2, A3, and S1 (clockwise). The figure illustrates ankle motion trajectory tracking performance when the muscles are fully rested. The tracking achieved in this condition demonstrates a Root Mean Square Error (RMSE) of $1.625^{\circ}$, highlighting the system’ s effectiveness in accurate control under optimal muscle conditions. Note, only absolute values of FES stimulation, $u$, are provided for both phases.
• Figure 5: Ankle motion trajectory tracking results averaged over the final trial of each session for each participant A1, A2, A3, and S1 (clockwise).. The figure illustrates trajectory tracking performance after $3{\b@ld{bold} \text{\bfseries\textendash}}\textendash4$ walking trials of 60 seconds each, reflecting the effects of muscle fatigue. The trajectory Root Mean Square Error (RMSE) is $3.1^{\circ}$, indicating the onset of fatigue-induced deviations in tracking accuracy.