# Modelling torque-angle curves in patients with upper limb motor impairment: a preliminary study

Aurélie Tomezzoli, Boris Cassard, Emilie Leblong, Charles Pontonnier

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Aurélie Tomezzoli, Boris Cassard, Emilie Leblong, Charles Pontonnier. Modelling torque-angle curves in patients with upper limb motor impairment: a preliminary study. Multidisciplinary Biomechanics Journal, 2025, 50th congress of the.., 10.46298/mbj.16195. hal-05191748v2

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# Modelling torque-angle curves in patients with upper limb motor impairment: a preliminary study

Aurélie Tomezzoli<sup>a*</sup>, Boris Cassard<sup>b</sup>, Emilie Leblong<sup>b</sup>,
Charles Pontonnier<sup>a</sup>

a Univ Rennes, Inria, CNRS, IRISA, Rennes, France

b Fondation Saint-Hélier, Rennes, France

* Corresponding author: aurelie.tomezzoli@inria.fr

Received date: 04/04/2025

Accepted date: 27/06/2025

Publication date: 27/10/2025

**Keywords:** muscle strength, upper limb, motor impairment, modeling, torque-angle curves

© 2025 The Authors
Licence CC-BY 4.0
Published by Société de Biomécanique

## 1. Introduction

Upper limb motor impairments resulting from neurological conditions are common and affect quality of life. To assist patients, exoskeletons controlled by their residual effort generation capacities are currently being developed (Gallois et al. 2025). One of today's challenges is to render their use as seamless as possible in everyday life. To this end, providing a concise, personalized representation of patients' effort generation capacities can be an asset.

Various mathematical functions have been proposed to model the relationship between joint torques and joint angles in healthy persons (Haering et al. 2017). In patients with motor impairment, such models have been developed to account for spasticity, an undesirable phenomenon due to involuntary muscle hyperactivity, during passive joint mobilization (Cha et Arami 2020). However, the literature concerning torque-angle relationship during voluntary (i.e., active) upper limb muscle contraction in these patients remains scarce. The goal of this study is to assess the performance of mathematical models primarily developed to fit healthy persons' torque-angle curves, when applied to torque-angle curves of patients with motor impairments, for future use in the control of assistive exoskeletons.

## 2. Methods

### 2.1 Data acquisition

After approval from the ethics committee (2024-A01007-40), the first 7 of the 60 planned patients

(5 females, 2 males, 55.7±16.6 years, 163.6±9.8 cm, 62.9±9.2 kg), hospitalized at one rehabilitation center for subacute or chronic stroke (n=5) or multiple sclerosis (n=2), with spasticity ≤3/4 on the Ashworth scale (6, 3 and 5 tests/14 for Ash 0, 1, 2) and muscle strength ≤3/5 on the MRC scale (2, 13, 13 test/28 for MRC 1, 2, 3) were included. Maximal torques were measured using a Con-Trex® ergometer at an imposed angular speed of 30°/s, during 5 repetitions of maximal concentric and eccentric muscle contractions, in shoulder internal-external rotation, then in elbow flexion-extension. Patients' positioning, stabilization, and resting procedures were standardized according to the ergometer's manual guidelines and the experimenter's expertise.

### 2.2 Data processing

Data were processed using MATLAB software v. R2023b. The portions of trials exceeding 95% of the nominal speed were extracted. Torques were filtered using a 4th-order Butterworth filter in the angle domain, first with a band-stop filter (cutoff frequencies 0.153-0.173/° and 0.316-0.336/°, to remove a known ergometer's geometric artifact) and then with a low-pass filter (cutoff frequency 0.5/°, 15 Hz equivalent), with visual check of the filter effect. The ergometers' default gravity correction was applied. The 5 repetitions of the same measurement type were averaged.
## 2.3 Torque-angles curves modelling

Five mathematical models were fitted to the experimental curves (Fig. 1.) by optimizing 3 parameters: their peak torque $Γ_{maxMod}$, an $α_0$ shift of the joint angle $α$ (used to compute $\bar{α} = α - α_0$), and an angle-scaling parameter controlling curves' range of motion:

Normal: $Γ_{mod}(\bar{α}) = Γ_{maxMod} * e^{-(\frac{4}{π} * \bar{α})^2}$

Cosinus: $Γ_{mod}(\bar{α}) = Γ_{maxMod} * \cos\bar{α}$

Sinus exponential: $Γ_{mod}(\bar{α}) =$

$$Γ_{maxMod} * (\frac{1}{2} + \frac{1}{2} * \sin(1 + 1.919πe^{-\frac{\bar{α}}{π}}))$$

Quadratic: $Γ_{mod}(\bar{α}) = Γ_{maxMod} * (1 - (\frac{2}{π} * \bar{α})^2)$

Cubic: $Γ_{mod}(\bar{α}) =$

$$\frac{Γ_{maxMod} * 27}{4} * (\frac{1}{8} - \frac{1}{4} * \bar{α} - \frac{1}{2} * (\frac{\bar{α}}{π})^2 + (\frac{\bar{α}}{π})^3)$$

The models' coefficients were deduced from Haering's et al. (2017) models, scaled along the a-axis to reach $Γ_{model}(\pm\frac{π}{2}) = 0$ and keep $Γ_{model}(\bar{α}_{max}) = 1$. Model parameters' initialization set $Γ_{maxMod} = $ experimental peak torque, $α_0 = $ angle shift aligning the experimental and modelled peak torques, and angle-scaling parameter = 1. The least-squared distances between models and experimental curves were minimized using the fminsearch MATLAB function (Needler-Mead algorithm). Each optimization residual was then normalized by the related experimental peak torque.

<figure><img src="image_3.png" /><figcaption>Figure 1. Average elbow extensors' concentric torque from one person: experimental curve (ExpCurve), and models fitted.</figcaption></figure>

## 3. Results and discussion

On these first data, the cubic and sinus exponential models had the best performance overall, as in healthy persons, reflecting good performance for almost all muscle groups' eccentric and concentric contractions (Table 1), and residuals were on average 6.3 times higher to that of healthy persons (Haering et al. 2017). Residuals were generally lower for elbow flexors and shoulder internal rotators than for elbow extensors and shoulder external rotators, and lower for eccentric contractions than for concentric contractions. Ongoing patient inclusion will enable statistical assessment of the factors explaining models' performance, to improve models' quality and/or model selection.

## 4. Conclusions

These first data suggest that torque-angle curve modelling, similar to that performed for healthy persons, is possible in some or all patients with motor impairment,

for future use of model parameters to control assistive exoskeletons based on patients' residual capacities.

## Acknowledgements

The authors wish to thank the participants.

## Conflict of Interest Statement

None.

## Contributor Roles

AT: Conceptualization, Methodology, Formal analysis, Writing original draft. BC & EL: Data acquisition, Writing-review & editing. CP: Conceptualization, Methodology, Funding acquisition, Supervision, Validation, Writing-review & editing, Data curation.

## Funding

This work is supported by the INRIA exploratory action MusMaps and the Fondation Saint-Hélier.
<table><caption>**Table 1. Normalized RMS torque residuals (%)**.</caption><thead><tr><th rowspan="3">Mathematical model</th><th colspan="4">Shoulder</th><th colspan="4">Elbow</th><th rowspan="3">Total</th></tr><tr><th colspan="2">External rotators</th><th colspan="2">Internal rotators</th><th colspan="2">Extensors</th><th colspan="2">Flexors</th></tr><tr><th>Ecc</th><th>Con</th><th>Ecc</th><th>Con</th><th>Ecc</th><th>Con</th><th>Ecc</th><th>Con</th></tr></thead><tbody><tr><td>Normal</td><td>3.2<br/>±0.9</td><td>4.1<br/>±3.4</td><td>2.0<br/>±1.0</td><td>3.1<br/>±1.8</td><td>2.9<br/>±1.2</td><td>2.5<br/>±1.5</td><td>2.2<br/>±1.3</td><td>3.1<br/>±2.2</td><td>2.9<br/>±1.8</td></tr><tr><td>Cosinus</td><td>3.1<br/>±1.1</td><td>3.1<br/>±1.7</td><td>2.2<br/>±1.1</td><td>3.0<br/>±1.5</td><td>3.3<br/>±2.5</td><td>5.6<br/>±7.7</td><td>2.1<br/>±0.9</td><td>3.0<br/>±2.3</td><td>3.2<br/>±3.1</td></tr><tr><td>Quadratic</td><td>3.0<br/>±1.1</td><td>3.3<br/>±1.8</td><td>2.3<br/>±1.2</td><td>2.9<br/>±1.4</td><td>3.4<br/>±2.8</td><td>5.7<br/>±7.7</td><td>2.5<br/>±1.1</td><td>2.9<br/>±2.3</td><td>3.3<br/>±3.1</td></tr><tr><td>Cubic</td><td>3.2<br/>±1.4</td><td>3.7<br/>±2.0</td><td>2.2<br/>±1.2</td><td>2.6<br/>±1.5</td><td>2.6<br/>±1.5</td><td>3.1<br/>±2.0</td><td>2.5<br/>±1.3</td><td>2.5<br/>±1.9</td><td>2.8<br/>±1.6</td></tr><tr><td>Sinus exponential</td><td>3.2<br/>±1.2</td><td>4.1<br/>±3.0</td><td>2.1<br/>±1.0</td><td>2.8<br/>±1.6</td><td>2.5<br/>±1.2</td><td>2.8<br/>±1.8</td><td>2.4<br/>±1.4</td><td>2.7<br/>±2.0</td><td>2.8<br/>±1.7</td></tr></tbody></table>

Con: concentric. Ecc: eccentric.

Yellow: highest normalized RMS.

## References

Cha, Yesung, & Arami, A. (2020). Quantitative modeling of spasticity for clinical assessment, treatment and rehabilitation. *Sensors*, 20(18), 5046. https://doi.org/10.3390/s20185046

Gallois, M., Manzano, M., Guégan, S., Vignais, N., Babel, M., & Pontonnier, C. (2025). Effort generation capabilites mapping for personalized robotic assistance of the elbow. In *2025 International Conference On Rehabilitation Robotics (ICORR)* (pp. 1215–1220). Chicago, IL, USA, https://doi.org/10.1109/ICORR66766.2025.11063068

Haering, D., Pontonnier, C., & Dumont, G. (2017). Which mathematical model best fit the maximal isometric torque-angle relationship of the elbow? *Computer Methods in Biomechanics and Biomedical Engineering, 20(sup1)*, 101–102. https://doi.org/10.1080/10255842.2017.1382881