# Predictive Simulations for Delayed Fatigue Onset in FES Hand Cycling

Kevin Co, Pierre Puchaud, Florent Moissenet, Mickaël Begon

► To cite this version:

Kevin Co, Pierre Puchaud, Florent Moissenet, Mickaël Begon. Predictive Simulations for Delayed Fatigue Onset in FES Hand Cycling. Multidisciplinary Biomechanics Journal, 2025, 50th congress of the., 10.46298/mbj.16197. hal-05191750v2

HAL Id: hal-05191750

https://hal.science/hal-05191750v2

Submitted on 23 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Predictive Simulations for Delayed Fatigue Onset in FES Hand Cycling

Kevin Co<sup>a*</sup>, Pierre Puchaud<sup>b</sup>, Florent Moissenet<sup>c</sup>, Mickaël Begon<sup>a, d</sup>

a Laboratoire de Simulation et Modélisation du Mouvement, Université de Montréal, Montréal (Qc), Canada

b Centre Inria, Université de Bordeaux, 33400, Talence, France

c Laboratoires de Biomécanique et de Cinésiologie, Hôpitaux Universitaires de Genève et Université de Genève, Genève, Suisse

d Centre de recherche Azrieli du CHU Sainte-Justine, Montréal (Qc) Canada

* Corresponding author: kevin.co@umontreal.ca

Received date: 05/04/2025

Accepted date: 27/06/2025

Publication date: 27/10/2025

**Keywords:** receding horizon estimator, musculoskeletal model, functional electrical stimulation, hand cycling, muscle fatigue

© 2025 The Authors

Licence CC-BY 4.0

Published by Société de Biomécanique

## 1. Introduction

Functional electrical stimulation (FES) is a recognized tool in neurological, especially during repetitive tasks like cycling to enhance neuroplasticity. Unfortunately, FES induces rapid muscle fatigue, limiting the exercise duration. This issue has been previously explored using optimal control relying on differential equations modeling both muscle force and fatigue, but mostly limited to isometric contractions on one muscle (S. Gayrard 2023).

To address these limitations, the present study aims to design optimal pulse trains for a multi-muscle FES-driven hand cycling task, hypothesizing that distributing stimulations among muscles and employing a longer time-horizon would prolong cycling sessions. Indeed, the longer the time-horizon, the earlier the re-distribution of forces is anticipated. A receding-horizon optimal control framework was used to compare three cost functions (minimizing stimulations, forces, or fatigue) across two window sizes.

## 2. Methods

### 2.1 System dynamics

The upper-limb musculoskeletal model of Seth et al. (2019) was adapted to function exclusively in the sagittal plane. It was composed of five musculotendon units (biceps long and short head, triceps, anterior and posterior deltoids) and three degrees of freedom (shoulder, elbow, hand-crank joint). The equations of motion were coupled with the Ding model (Ding et al. 2003; Ding et al. 2007) to estimate muscle forces and fatigue.

### 2.2 FES force model

The Ding model is a physiological model designed to estimate muscle force and fatigue in response to electric stimulation. For one muscle, the equations for force production $F^m$ are:

$$ 
\begin{array}{l}
\dot{C}_N = \frac{1}{\tau_c} \left[ \sum_{i=1}^{n} R_i e^{-\frac{t-t_i}{\tau_c}} \right] - \frac{C_N}{\tau_c}, \qquad (1) \\
\text{with } R_i = 1 + (R_0 - 1)\exp\left[\frac{-t_i-t_{i-1}}{\tau_c}\right],
\end{array}
$$

$$ 
\begin{array}{l}
\dot{F}^m = A \frac{C_N}{K_m+C_N} - \frac{F}{\tau_1+\tau_2 \frac{C_N}{2K_m+C_N}}, \qquad (2) \\
\text{with } A = a\left(1 - \exp\left[\frac{-pd-pd_0}{pd_t}\right]\right),
\end{array}
$$

where $C_N$ is the rate-limiting step for force generation, with $\tau_c$ (11 ms) shaping its transient response. Stimulation times $t_i$, combined in $R_i$ account for nonlinear summations when pulses are closely spaced, and $R_0$ ($R_0 = K_m + 1.04$) quantifies the magnitude of $C_N$ enhancement from successive pulses. $K_m$ sets the sensitivity of force to $C_N$, while the scaling factor $A$ depends on the pulse width $pd$, $pd_0$ (131.405 µs) being the minimum effective width and $pd_t$ (194.138 µs) controlling how sharply $A$ increases. Two decay constants, $\tau_1$ and $\tau_2$ (1 ms), model force loss due to viscoelastic properties. $a'$ is the maximal muscle force scale. Force-length-velocity relationships are included (De Groote et al., 2016) to transform isometric forces $F^m$ given by Ding into $F$:
$$F = F^m \cdot f_l^a(\tilde{l}^m) \cdot f_v^a \quad (3)$$

with $l^m$ muscle length, $v^m$ muscle velocity and $f_l$ the force-length, $f_v$ the force-velocity and $F^p$ passive force-length functions.

## 2.3 FES fatigue model

Ding's model (2007) was combined with the fatigue model of Ding et al. (2003):

$$\dot{a}' = \frac{-a-a'_{rest}}{\tau_{fat}} + \alpha_a F, \quad (4)$$

$$\dot{K}_m = \frac{-K_m-K_{mrest}}{\tau_{fat}} + \alpha_{K_m} F, \quad (5)$$

$$\dot{\tau}_1 = \frac{-\tau_1-\tau_{1rest}}{\tau_{fat}} + \alpha_{\tau_1} F, \quad (6)$$

where $a'_{rest}$ (4920 N/s), $K_{m,rest}$ (0.137 unitless), and $\tau_{1,rest}$ (60.601 ms) are the rested muscle states. $\tau_{fat}$ (127 s) is the time constant controlling the force-model parameter recovery, and $\alpha_a$ (-0.04 s), $\alpha_{K_m}$ (1.9×10$^{-6}$ s$^{-1}$N$^{-1}$), and $\alpha_{\tau_1}$ (2.1×10$^{-6}$ N$^{-1}$), represent coefficients for muscle fatigue onset.

## 2.4 Receding-horizon

A receding-horizon predictive simulation iteratively solves a finite-horizon optimal control problem using the updated initial state at each step, then advances the horizon for the next prediction. Six simulations were run: three cost functions × two window sizes, namely 1 or 3, all performed at 60 rpm. The cost functions minimized (1) the sum of stimulation pulse widths, (2) the muscle forces, or (3) the muscle fatigue (i.e., maximizing the scaling factors $a'$). OCPs were solved with Bioptim (Michaud et al. 2022) using a direct multiple shooting scheme (Runge-Kutta 4, 1 ms step). The decision variables were the states $x = [C_N, F, A, K_m, \tau_1, q, \dot{q}]$ with the pulse widths as controls. The simulation was run until exhaustion when the pedalling motion could not be satisfied three times in a row. Optimization problems were solved using an AMD® Ryzen 9 5550x 16-core processor x 32, 128 GB RAM configuration.

## 3. Results and discussion

As hypothesized, larger number of repetitions was obtained with longer optimization windows. The simulation was able to adapt and apprehend the muscle fatigue leading to longer cycling capacities (Table 1).

<table><caption>**Table 1.** Number of cycles before exhaustion based on cost functions and window size.</caption><thead><tr><th>Window size (cycles)</th><th>Min pulse width</th><th>Min force</th><th>Min fatigue</th></tr></thead><tbody><tr><td>1</td><td>4</td><td>129</td><td>29</td></tr><tr><td>3</td><td>52</td><td>192</td><td>169</td></tr></tbody></table>

Credit: Co.

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Window optimization time stacked based on cost function and window size.</figcaption></figure>

Credit: Co.

To our knowledge, we are the first to combine Ding et al. models (2003 and 2007) with a musculoskeletal model to simulate a multi-muscle motor task using receding-horizon OCP and to explicitly minimize muscle fatigue to increase task duration. The reason why fatigue cost function was less efficient to increase task duration may be because of similar fatigue rate among muscles, despite we know some can resist longer. Additional cost and regulation functions combination should be tested to improve convergence time and task duration.

Limitation such as identifying the FES model parameters for each muscle must be addressed to verify relevance before in vivo applications.

## 4. Conclusions

Using a FES model that can reduce muscle fatigue on-set allows to design pulse trains to extend hand cycling tasks. Further improvements should lead to experimental applications to validate efficiency against traditional stimulation.
**Conflict of Interest Statement**

None.

**Contributor Roles**

KC: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft; PP: Conceptualization, Methodology, Software, Writing - review & editing; FM: Supervision, Writing - review & editing; MB: Supervision, Resources, Funding acquisition, Writing - review & editing.

**Funding**

This work was supported by the Fonds de recherche du Québec - Nature et technologies (FRONT), grant number: 2024-PR-341023.

**Data, software, code availability**

The optimal control problem was set and solved using Bioptim: https://github.com/pyomeca/bioptim, and the simulations were run using CocoFest: https://github.com/pyomeca/cocofest.

**References**

De Groote, F., Kinney, A. L., Rao, A. V., & Fregly, B. J. (2016). Evaluation of direct collocation optimal control problem formulations for solving the muscle redundancy problem. *Annals of Biomedical Engineering*, 44(10), 2922-2936. https://doi.org/10.1007/s10439-016-1591-9

Ding, J., Wexler, A. S., & Binder-Macleod, S. A. (2003). Mathematical models for fatigue minimization during functional electrical stimulation. *Journal of Electromyography and Kinesiology*, 13(6), 575-588. https://doi.org/10.1016/S1050-6411(03)00102-0

Ding, J., Chou, L., Kesar, T. M., Lee, S. C., Johnston, T. E., Wexler, A. S., & Binder-Macleod, S. A. (2007). Mathematical model that predicts the force-intensity and force-frequency relationships after spinal cord injuries. *Muscle & Nerve*, 36(2), 214-222. https://doi.org/10.1002/mus.20806

Gayrard, S. (2023). Fast optimization scheme for the muscular response to FES stimulation to design a smart electrostimulator. In *2023 European Control Conference (ECC)*. Bucharest, Romania, 1-6. https://doi.org/10.23919/ECC57647.2023.10178228

Michaud, B., Bailly, F., Charbonneau, E., Ceglia, A., Sanchez, L., & Begon, M. (2022). Bioptim, a Python

framework for musculoskeletal optimal control in biomechanics. *IEEE Transactions on Systems Man and Cybernetics Systems*, 53(1), 321-332. https://doi.org/10.1109/TSMC.2022.3183831

Seth, A., Dong, M., Matias, R., & Delp, S. (2019). Muscle contributions to Upper-Extremity movement and work from a musculoskeletal model of the human shoulder. *Frontiers in Neurorobotics*, 13. https://doi.org/10.3389/fnbot.2019.00090