# Are Isokinetic Torque-Angle Models Derived from Healthy Subjects Applicable to Patients with Upper-Limb Neurological Motor Impairment?

Aurélie Tomezzoli, Boris Cassard, Emilie Leblong, Joel Hayter, Carole Anne, Linda Bodet, Charles Pontonnier

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Aurélie Tomezzoli, Boris Cassard, Emilie Leblong, Joel Hayter, Carole Anne, et al. Are Isokinetic Torque-Angle Models Derived from Healthy Subjects Applicable to Patients with Upper-Limb Neurological Motor Impairment?. Annals of Biomedical Engineering, 2026. hal-05512733v2

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Title page

Are Isokinetic Torque-Angle Models Derived from Healthy Subjects Applicable to Patients
with Upper-Limb Neurological Motor Impairment?

Aurélie Tomezzoli<fnref n="1" />*, Boris Cassard<fnref n="2" />, Emilie Leblong<fnref n="2" />, Joël Hayter<fnref n="2" />, Carole Anne<fnref n="2" />, Linda Bodet<fnref n="2" />, Charles Pontonnier<fnref n="1" />

Brief title

Simplified upper limb torque-angle models for patients with motor impairment.

Abstract

**Purpose.** Upper limb motor impairments are common and affect quality of life. Shared-control robotic assistance systems driven by patients’ residual effort generation capacities are being developed. These systems require the integration, into their control scheme, of knowledge about the maximum voluntary torque achievable by patients at each joint angle. This study aimed to quantify the performance of simplified mathematical models primarily developed to fit healthy individuals’ torque-angle curves, as candidates for integration in control scheme.

**Methods.** 15 patients (6F, 9M, 62.7 ± 14.4 years) hospitalized in a French rehabilitation center for stroke (n=10), multiple sclerosis (n=4) or traumatic tetraplegia (n=1) were included. Passive, then maximum concentric and eccentric torques were measured in the seated position, in shoulder external-internal rotation and in elbow flexion-extension, at an imposed speed of 30°/s, using a Con-Trex® isokinetic ergometer.

**Results.** The normalized RMSE between modeled and experimental curves was 3.1 ± 2.0% of corresponding peak torques. Univariate linear models displayed no difference in nRMSE between mathematical models, but differences across patients (p<0.001, R<fnref n="2" />=0.38). The distance between modeled and experimental curves was continuously lower than 10% of the peak torque over 92 ± 13% of the experimental range of motion.

**Conclusion.** Regardless of the mathematical model used, torque-angle curve modeling was globally less effective than for healthy individuals, while still allowing consideration for future use for robotic assistance, for the majority of patients. Further investigation of patient-related factors affecting model quality will be necessary to assess results’ generalizability.

**Trial registration number.** ID-RCB: 2024-A01007-40, clinicaltrial.gov ID: NCT06608121.

Keywords

muscle strength; effort generation capacities; isokinetic dynamometer; stroke; multiple sclerosis; spinal cord injury

<footnote n="1">¹Univ Rennes, Inria, CNRS, IRISA-UMR 6074, 35000 Rennes, France</footnote>

<footnote n="2">²Pôle Saint-Hélier, 35000 Rennes, France</footnote>

<footnote>*Corresponding author: aurelie.tomezzoli@inria.fr</footnote>
Introduction

Upper limb motor impairments resulting from neurological conditions are common and affect quality of life [1–3]. They are mainly due to neurological conditions such as stroke, multiple sclerosis, and traumatic spinal cord injury [4, 5]. Technological solutions for upper limb assistance have been integrated into patient care since the 1970s, ranging from simple non-motorized arm supports to complex robotic systems [2, 6, 7]. In the context of assistance for daily activities, the primary objective of these technological solutions is to enable the patient to achieve greater functional autonomy [8]. However, adapting assistive robotic devices to the end users’ various anthropometry and diverse clinical pictures [9–11] remains a challenge [12, 13]. To benefit from assistance, patients should not be overly spastic. For assistance in everyday life, heterogeneous tasks must also be performed in diverse environments. In this context, the high demand for versatility renders assistive systems particularly challenging to control and personalize.

Shared-control robotic assistance systems driven by the patients’ residual effort generation capacities are currently being developed [14]. These systems measure the effort performed by the patient through force sensors or electromyography, interpret measurements in terms of the patient’s intention, and provide patients with personalized assistance. The control loop must operate within a timeframe short enough for the assistance to be perceived as responsive and interactive by the user. To this end, a personalized and computationally efficient mathematical model of patients’ residual effort generation capacity can be embedded in the control loop, to infer patients’ intention and compute the level of assistance required [15]. This mathematical model must reflect reasonably well patients’ effort generation capacities across most of the range of motion (ROM) of the joints being assisted. It must also be as simple as possible, to reduce the risk of overfitting.

Various mathematical functions have been proposed to model the relationship between upper limb joint torques and joint angles in healthy individuals. Among the five models most frequently encountered in the literature, namely the normal, quadratic, cubic, cosine, or sinus exponential models, Haering et al. [16] found that the quadratic model best fitted elbow maximum voluntary torque-angle curves. In patients with neurological motor impairments, models have been developed to fit torque-angle curves during passive tests [1]. These models account for spasticity, an undesirable phenomenon due to involuntary contraction of antagonist muscles, for example of elbow flexors during passive elbow extension. Two components are usually modeled: a hypertonia, present over all or part of the ROM, and a stretch reflex, triggered at patient- and speed-specific joint angles [1]. The literature regarding torque-angle relationships at low velocities, during patients’ voluntary (i.e., active) upper limb muscle contraction, remains scarce [17].

The objective of this study was to quantify the performance of simplified mathematical models primarily developed to fit healthy individuals’ maximum voluntary torque-angle curves, as candidates for integration in the control scheme for robotic assistance of patients with neurological motor impairment. The secondary objective is to identify factors explaining model performance.
# Material and methods

## Participants

After ethical approval and registration of the protocol on clinicaltrials.gov (registration number: NCT06608121), 15 patients (6 females [F], 9 males [M]; age 62.7 ± 14.4 years) were included in the study between November 7, 2024, and April 28, 2025. Their mean height was 168.6 ± 10.3 cm (F: 158.3 ± 5.8 cm, M: 175.4 ± 5.8 cm), their mean weight 67.6 ± 10.3 kg (F: 59.8 ± 7.5 kg, M: 72.7 ± 8.8 kg). They were hospitalized at one rehabilitation center for subacute or chronic stroke (n=10), multiple sclerosis (n=4), or traumatic spinal cord injury (n=1). Clinically, patients had a maximum motor deficit grade of 3/5 on the Medical Research Council (MRC) scale [18] for both shoulder internal-external rotation and elbow flexion-extension (Fig. 1) and a maximum spasticity grade of 3/4 on the modified Ashworth scale [19].

<figure><img src="image_2.png" /><figcaption>**Fig. 1** patients' clinical grades at shoulder and elbow. a) Motor deficit grade on the MRC scale. b) Spasticity grade on the modified Ashworth scale. ER: external rotators. Ext: extensors. Fle: flexors. IR: internal rotators</figcaption></figure>

## Isokinetic torque measurements

Ergometric data were collected at the rehabilitation center, by physiotherapists and adapted exercise instructors experienced in isokinetic ergometry. They were trained in the specifics of the protocol. Passive, then maximum voluntary torques were measured in shoulder internal-external rotation, then in elbow flexion-extension, using a Con-Trex MJ® isokinetic ergometer (CMV AG, Dübendorf, Switzerland), at a sampling rate of 256 Hz. Patients were seated in standardized positions. They were stabilized, according to the ergometer's manual guidelines and investigators' expertise of tests achievable by patients with mild to severe motor impairment (Fig. 2): isokinetic shoulder torques were thus measured at 45° of elevation in the scapular plane, with the elbow flexed at 90°. Elbow torques were measured at 10° of elevation in the scapular plane. Additionally, since patients' wrist motor control was often impaired, all of them wore a rigid glove which maintained their hand and forearm in a stable position during measurements. The ergometer axis was visually aligned with the anatomical axis of the joint rotation under investigation.
The ROM which was comfortable for patients was selected for assessment, by setting the ergometer stops. Finally, since the ergometer only measures its own angular position related to the vertical, its position was related to the patient's joint angle using one standardized goniometric measurement (Fig. 2), which reliability is considered acceptable [20] to almost perfect [21] when performed by experienced investigators. The familiarization procedure involved presenting the protocol instructions to patients and included brief submaximal practice on the ergometer, to ensure understanding of the task while preventing fatigue.

<figure><img src="image_3.png" /><figcaption>**Fig. 2** setup for isokinetic shoulder torque measurements. Photographs: ©Inria / Photo B. Fourrier</figcaption></figure>

Patients performed voluntary maximum muscle contractions in shoulder internal-external rotation, then in elbow flexion-extension. These contractions were performed at an imposed angular velocity of 30°/second, as preliminary tests showed that measurements at higher velocity were not achievable for many patients, and that lower velocities induced fatigue due to increased periods of exertion. Patients performed 5 repetitions of maximum concentric, then eccentric muscle contractions. Physiotherapists reminded the patients of the instructions and encouraged them throughout each repetition. Hereafter, the combination of the muscle assessed and the contraction type will be referred to as “muscle test types”.

To limit the effect of fatigue, the muscle test types considered least tiring, based on physiotherapists' experience, were performed first: at the shoulder, concentric internal rotation was assessed first, followed by concentric external rotation, eccentric internal rotation, and eccentric external rotation. At the elbow, concentric flexion was assessed first, followed by concentric extension, etc. In addition, each measurement was followed by a break of at least 60 seconds. The order of measurements was not randomized, to maximize the possibility of comparing identical muscle test types from one patient to another.

The ergometer measures an overall torque at each angular position. This overall torque is the sum of several components: the patient's voluntary torque if present, the torque due to the effect of gravity on both the ergometer arm and patients' upper limb, the passive joint torque, and the torque due to antagonist muscles' co-contraction [22, 23], the last component being expected to increase with spasticity. To enable later extraction of net voluntary maximum
torques from overall torques recorded during active tests, the active tests were preceded at each joint by a passive test, also performed at 30°/second, and which contains all the other components of the overall torques. During the passive tests, patients were instructed to remain relaxed during the 5 repetitions of the ergometer's back-and-forth motion. All torques and angles recorded by the ergometer were then exported as raw data.

## Torques pre-processing

Data were processed using MATLAB software v. R2023b. Only the portions of records displaying a steady velocity i.e., exceeding 95% of nominal velocity were analyzed, to prevent artifacts caused by dynamic effects [22]. Torques recorded during active and passive tests were filtered in the angle domain using two successive 4th-order zero-phase Butterworth filters, with visual inspection of the filter effect: to remove a known ergometer's geometric artifact, a band-stop filter was first applied (cutoff frequencies 0.153-0.173/° and 0.316-0.336/°), followed by a low-pass filter (cutoff frequency 0.5/°, 15 Hz equivalent). Torques due to friction were then subtracted from records. Finally, torques recorded during passive tests were averaged across the 5 repetitions performed in the same direction, and subtracted, at each joint angle, from torques recorded during active tests, to extract maximum voluntary torques.

Repetitions of maximum voluntary torques were pooled. Repetitions with average torque below 20% of the mean of the five repetitions were first considered failed and removed. For future use in the control scheme of assisting devices, models of the reproducible part of the patients' maximum voluntary torques at each joint angle were sought. For this reason, averaging the repetitions was preferred over selecting the maximum value of all repetitions at each angle [22]. Only the portion of the resulting curve where torques were consistently exercised by the patient in the expected direction was retained for modeling related muscles' effort generation capacities.

## Torque-angle curves modeling

The five mathematical models compared by Haering et al. [24] in healthy individuals were first fitted to the patients' experimental torque-angle curves (Fig. 3). The models' coefficients were scaled along the x-axis to reach $Γ_{mod}$ ($±\frac{π}{2}$) = 0 while keeping $Γ_{maxMod}$ = 1 (Table 1). Three parameters were then optimized: the peak torque ($Γ_{maxMod}$), an offset ($δ$) in joint angles ($α_{joint}$), and a parameter ($k$) for scaling the curve along the x-axis. The initial value of $Γ_{maxMod}$ was set equal to the experimental peak torque $Γ_{maxExp}$. The initial value of $δ$ was defined to superimpose the modeled and experimental optimal angles i.e., the angles at which peak torques occurred. The initial value of $k$ was 1. Optimized model parameters for each experimental curve were determined by minimizing the sum of the squared errors ($ε$) between each model and the experimental curve at each joint angle i.e., by minimizing $\sqrt{\sum ε_i^2}$, using a Nelder-Mead algorithm (MATLAB fminsearch function).
<figure><img src="image_4.png" /><figcaption>**Fig. 3** mathematical models used to fit the experimental torque-angle curves: shape on the [$\pi/2$, $\pi/2$] radian interval. Red text: parameters to be optimized. Red arrows: effect of their optimization. k: parameter for scaling the curve along the x-axis. Quadr: quadratic model. SinExp: sinus exponential model.</figcaption></figure>

<table><caption>**Table 1** mathematical models used to fit experimental torque-angle curves: formula</caption><thead><tr><th>Model name</th><th>Formula</th></tr></thead><tbody><tr><td>Normal</td><td>Γ<sub>mod</sub>(α<sub>joint</sub>) = Γ<sub>maxMod</sub> * e<sup>-(4/π * kα̃)<sup>2</sup></sup></td></tr><tr><td>Cosine</td><td>Γ<sub>mod</sub>(α<sub>joint</sub>) = Γ<sub>maxMod</sub> * cos(kα̃)</td></tr><tr><td>Sinus exponential</td><td>Γ<sub>mod</sub>(α<sub>joint</sub>) = Γ<sub>maxMod</sub> * (0.5 + 0.5 * sin(1 + 1.919 π e<sup>-kα̃/π</sup>))</td></tr><tr><td>Quadratic</td><td>Γ<sub>mod</sub>(α<sub>joint</sub>) = Γ<sub>maxMod</sub> * (1 - (2/π)kα̃<sup>2</sup>)</td></tr><tr><td>Cubic</td><td>Γ<sub>mod</sub>(α<sub>joint</sub>) = Γ<sub>maxMod</sub> * (27/4 * (1/8 - (1/4 * (kα̃/π) - (1/2 * (kα̃/π)<sup>2</sup>) + (kα̃/π)<sup>3</sup>))</td></tr></tbody></table>

α<sub>joint</sub>: joint angle. α̃: corrected angle defined as α̃ = α<sub>joint</sub> + δ. δ: offset in joint angles. Γ<sub>mod</sub>: modeled torques. Γ<sub>maxMod</sub>: modeled peak torque. k: scaling parameter.

## Outcomes and statistics

Two outcomes were computed to account for models' performance. The first, hereafter referred to as nRMSE, quantified their overall performance across the ROM. After optimization, nRMSE was computed as the square root of the sum of residual errors (ε) values, normalized to the experimental peak torque (Equation 1):

$$nRMSE = \frac{\sqrt{\frac{1}{n} \sum \epsilon_i^2}}{\Gamma_{maxExp}} \quad (1)$$

The second outcome, hereafter referred to as ROMinf10%, quantified the proportion of the experimental ROM where the distance to the model was continuously lower than 10% of the experimental peak torque, the 10% error threshold being set arbitrarily to define the points of the experimental curve considered incorrectly fitted.
Univariate linear models were fitted to assess the influence of the following explanatory factors on nRMSE: nature of the mathematical model, joint (shoulder versus elbow), contraction type (eccentric versus concentric), torque expected direction (torques that bring the forearm toward the thorax versus away from the thorax), experimental peak torque value, experimental ROM, patient. After fitting each statistical model, the normality of residuals was visually checked to ensure model validity. Then, the curves with ROMinf10% over at least 50% of the experimental ROM were considered incorrectly fitted. They were described in terms of frequency and their aspect was qualitatively described.

Models' parameters were described. Mean ± SD modeled peak torques and optimal angles were computed. The models' ROM endpoints were defined as the first joint angle at which modeled torques reached zero on either side of the optimal angle. Finally, the relationships between experimental and modeled curves' peak torques, optimal angles, and ROM were assessed using Pearson's tests for linear correlation, after graphically verifying the shape of the relationship. All statistical analyses were performed using the R software v. 3.6.1 [25].

## Results

The experimental ROM was 47 ± 15° at the shoulder and 54 ± 16° at the elbow. The experimental ROM was almost entirely retained for modeling (98.8 ± 5.2%), with a minimum value of 67.7%.

## Models' performances

The average nRMSE across all modeled curves (n=515) was 3.1 ± 2.0% of the corresponding peak torques (details in Table 2). No statistically significant differences were found among the mathematical models (p = 1.00, Fig. 4), the joints (p = 0.47), and the contraction types (p = 0.11). In contrast, nRMSE varied depending on the experimental ROM (p <0.001, adjusted coefficient of determination R²=0.13), with values increasing by an average of 0.05 per degree of ROM. The nRMSE also differed depending on torque expected direction (p <0.001, adjusted R²=0.01), with higher values for torques that bring the forearm away from the thorax (+0.5%). Finally, the nRMSE differed among patients (p <0.001, R²=0.38), ranging from 1.3 ± 0.5% to 6.2 ± 1.8% of the peak torque.

The average ROMinf10% across all modeled curves was 92 ± 13% of the experimental ROM. The distribution of ROMinf10% values across modeled curves was however markedly asymmetric (Fig. 5a): in 90% of modeled curves, less than 10% of the ROM was incorrectly fitted. Conversely, at least 50% of the experimental ROM was incorrectly fitted in only 3% of modeled curves, representing 6% of experimental curves but 27% of patients. Visually, these incorrectly fitted curves exhibited torque instability, either localized or persistent throughout the ROM (Fig. 5b). The instability that was less pronounced in the other curves (Fig. 5c).

## Models' parameters description

Modeled peak torques and optimal angles, displayed in Table 2, were highly correlated to experimental values (Pearson's correlation coefficient r=1.00, p <0.001). Modeled ROMs were higher than experimental ROMs, with moderate [26] correlation between these parameters (r=0.18, p <0.001). Modeled ROM endpoints i.e., joint angles with torque value equal to zero, ranged from -29° to +27° of elbow extension across all muscle test types at the elbow, from
152° to 185° of elbow flexion, from 58° to 87° of maximum shoulder internal rotation, and from -117° to -68° of maximum shoulder external rotation.

<table><caption>Table 2 models' performances and models' parameters: mean ± SD for each muscle test type</caption><thead><tr><th>Muscle</th><th>Contr<br>type</th><th>nRMSE (%)</th><th>Peak<br>Torque<br>(Nm)</th><th>Optimal<br>angle (°)</th><th>ROM of<br>models<br>(°)</th><th>Min angle<br>(°)</th><th>Max<br>angle (°)</th><th>n</th></tr></thead><tbody><tr><td rowspan="2">Sh IR</td><td>Concentric</td><td>2.6 ± 1.8</td><td>9.0 ± 4.4</td><td>-26 ± 16</td><td>181 ± 100</td><td>-117 ± 56</td><td>65 ± 61</td><td>70</td></tr><tr><td>Eccentric</td><td>2.7 ± 1.5</td><td>13.8 ± 6.4</td><td>-24 ± 15</td><td>148 ± 72</td><td>-88 ± 28</td><td>58 ± 61</td><td>70</td></tr><tr><td rowspan="2">Sh ER</td><td>Concentric</td><td>3.5 ± 2.2</td><td>6.1 ± 4.3</td><td>-18 ± 15</td><td>152 ± 76</td><td>-85 ± 35</td><td>87 ± 69</td><td>60</td></tr><tr><td>Eccentric</td><td>3.3 ± 1.7</td><td>7.4 ± 5.3</td><td>-16 ± 18</td><td>140 ± 70</td><td>-68 ± 34</td><td>70 ± 73</td><td>65</td></tr><tr><td rowspan="2">Elb fle</td><td>Concentric</td><td>2.7 ± 2.1</td><td>8.5 ± 6.0</td><td>74 ± 20</td><td>185 ± 95</td><td>-29 ± 84</td><td>166 ± 55</td><td>65</td></tr><tr><td>Eccentric</td><td>3.5 ± 2.1</td><td>15.9 ± 8.3</td><td>82 ± 18</td><td>143 ± 67</td><td>22 ± 33</td><td>164 ± 51</td><td>65</td></tr><tr><td rowspan="2">Elb ext</td><td>Concentric</td><td>3.2 ± 2.4</td><td>10.5 ± 9.5</td><td>75 ± 22</td><td>187 ± 102</td><td>-4 ± 56</td><td>185 ± 62</td><td>60</td></tr><tr><td>Eccentric</td><td>3.1 ± 1.9</td><td>17.7 ± 10.1</td><td>82 ± 16</td><td>125 ± 53</td><td>27 ± 30</td><td>152 ± 39</td><td>60</td></tr></tbody></table>

By convention, a zero joint angle corresponds to full elbow extension or neutral shoulder rotation, and positive values correspond to elbow flexion and shoulder internal rotation.

Contr: contraction. Elb ext: elbow extensors. Elb fle: elbow flexors. n: number of modeled curves. Sh IR: shoulder internal rotators. Sh ER: shoulder external rotators.

<figure><img src="image_5.png" /><figcaption>Fig. 4 mean nRMSE depending on mathematical models and muscle test types. Error bars = 1 SD. Quadr: quadratic model. SinExp: sinus exponential model.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Fig. 5** proportion of the experimental ROM with distance to the experimental curves continuously <10% of the experimental peak torque (ROMinf10%). a) Number of occurrences across modeled curves (n=515). b) Example of curve incorrectly fitted (ROMinf10% = 41.2 to 42.8% of the experimental ROM, depending on mathematical models). c) Curve with ROMinf10% = 97.6% of the experimental ROM, for all mathematical models</figcaption></figure>

# Discussion

## Models' performances: acceptable for the majority of patients in this study

The objective of this study was to quantify the performance of mathematical models primarily developed to fit healthy individuals' maximum voluntary torque-angle curves, as candidates for integration in the control scheme for robotic assistance of patients with neurological motor impairment. In our study, the overall nRMSE was 2.5 to 22.5 times higher than those reported in healthy individuals, whose nRMSE ranged from 0.14% to 1.3% at the elbow, for identical muscle test types [24]. Nevertheless, only a few curves were considered incorrectly fitted. For the other curves, the quality of the mathematical representations of patients' effort generation capacities therefore allows consideration of future applications for robotic assistance. It should also be noticed that the choice of the quality threshold -a distance between the experimental curve and the model continuously below 10% of the experimental peak torque - is not a definitive criterion. Indeed, its ability to reasonably represent patients' effort generation capacities within the frame of a shared control scheme depends on the robustness of the control strategy i.e., its capacity to compensate for deviations between the model and reality. Furthermore, a failure to simultaneously achieve satisfactory modeling of all muscles, observed in 27% of patients in our study, may increase the risk of inadequate assistance, leading to discomfort.

In contrast to the results observed in healthy individuals, the nRMSE was not significantly influenced by the choice of the mathematical models [16]. One of the important findings of our study is, however, the predominant weight, among the explanatory variables considered,
of the patient effect, as it explained 38% of nRMSE variance. High inter-individual variability has already been reported for post-stroke and multiple sclerosis patients' kinematics [11, 27]. This finding therefore suggests the need to select homogeneous groups of patients, in terms of torque-angle curve shapes, for developing effective assistive devices. Ideally, routinely assessable clinical factors enabling this selection prior to ergometric assessment should be investigated.

Torque instability was observed on experimental curves incorrectly fitted (Fig. 5b). In our study, torques recorded during passive tests were subtracted from torques recorded during active tests to compute voluntary torques. The underlying assumption is that, at an equal velocity, involuntary muscle contractions are identical at every joint angle in both tests. Failure to meet this assumption may result in instability in computed voluntary torque curves, especially in the event of a sudden spastic stretch reflex occurring at a different joint angle during active and passive joint mobilization [28]. Passive test curves of our dataset did not display any drop-off suggestive of spastic stretch reflexes. This concern does not therefore explain the instability observed in voluntary torque curves. Our patients had however relatively low Ashworth grades, with only one patient reaching a score of 3/4 (Fig. 1). Consequently, this concern could be relevant for patients with higher levels of spasticity, or for tests performed at higher velocities.

Torque instability is graphically characterized by the presence of multiple inflection points (Fig. 5b). The models used by Haering et al. [24] for healthy individuals were inspired by muscle active contraction models at the muscle fiber level [29]. They all display a classic bell shape. In patients with motor impairments and unstable torques, more complex mathematical models, allowing additional inflection points will be needed. It should also be noted that, in our study, modeled curves were obtained by averaging torque values of all repetitions at each angle. This approach results in modeling the reproducible part of patients' maximum voluntary torques and avoids overestimating their residual capacities in case of unsynchronized fluctuations. Nevertheless, this procedure may smooth torque fluctuations present in each repetition, as is the case with any aggregated data. Modeling, or model quality assessment in these patients, should take this risk into account. Finally, as temporal discontinuity is one of the characteristics of motion after stroke [30], instability of voluntary torques may be common in patients with motor impairment.

## Models' parameter values: which implications for assistance?

The average voluntary peak torques in our study were approximately half of those observed in women, and a quarter of those observed in men of a similar age (60-69 years), when performing elbow concentric flexion-extension and shoulder internal-external rotation, at the same angular velocity (30°/s) [31]. The patients included in our study had a maximum MRC grade of 3; the voluntary peak torques observed confirmed that their effort generation capacities were markedly reduced. Developing assistance systems for these patients is therefore warranted. Moreover, peak torques are known to be velocity-dependent in patients [17]; extending models with data obtained at different velocities could therefore be valuable.

At the shoulder, the optimal angles were located between 16 and 26° of external rotation, depending on the muscle test type. In the literature, results vary considerably depending on authors, muscle test types and settings, with the optimal angle ranging from 20° of internal
rotation to 50° of external rotation [32, 33]. The optimal angle values observed in our study can therefore be considered consistent with values observed in healthy individuals. At the elbow, the optimal angles, located between 74 to 82° of flexion for all muscle test types, are also consistent with those observed in healthy individuals, ranging from 60° to 95° of flexion [16, 34, 35]. Due to the possible variety of curves' shapes across patients, this result should be confirmed in a larger population.

Finally, to enable discussion of modeled ROM and its implications for the use of models in assistive devices, it is necessary to clarify the distinction between the patient's clinical ROM, the ROM covered by the isokinetic assessment, and the theoretical ROM obtained by modeling. The clinical ROM corresponds to the maximum ROM patients can achieve. The ROM measured by the isokinetic ergometer is necessarily reduced, as it corresponds only to the ROM that is comfortable for the patient, which is further limited by the presence of the large ergometer seat. The modeled ROM was defined as the ROM associated with non-zero torques in modeled curves, which can be either wider or narrower than the clinical ROM, since no limits were imposed in the models.

The ROMs covered by isokinetic assessment, 54 ± 16° in elbow flexion-extension, were narrower than those observed in healthy individuals (100°) [34] and in post-stroke patients (80°) displaying higher residual effort generation capacities (average peak torque around 40 Nm) [36]. However, a larger experimental ROM was associated with lower model fit quality in our study. As a consequence, trying to extend experimental ROM does not seem advisable for these patients. Due to the less voluminous seat and to the combined movements of the trunk, shoulder, and elbow, joint ROMs may however be greater in patients' everyday life. Moreover, the ROMs used during activities of daily living by healthy individuals are more than double the ergometric ROMs explored in our study [37, 38]. To satisfy patients' need for mobility and promote their motor skill recovery [2, 39–41] over a wider ROM, it is therefore advisable to provide assistance, and thus models, over a ROM exceeding the experimental ROM.

For almost all muscle tests, the ROM endpoints were beyond clinical ROM limits of healthy individuals at similar age [42]. Elbow extension angles during eccentric tests are an exception, as they ranged from 22° to 27° instead of 0° (F) to 20° (M) in healthy individuals [42]. It should be noted that, at clinical ROM limits, for example in complete flexion-extension of the elbow, joint torques are non-zero in healthy individuals [34]. A necessary-but not sufficient- condition for a model to correctly reflect individuals' effort generation capacities is therefore that their modeled ROM exceed their clinical ROM. The patients' models displayed in our study might therefore be appropriate. However, this criterion is not sufficient to guarantee the validity of models, as ROMs might be excessively broad in the event of flat-shaped experimental curves in some patients. Additional constraints, for example provided by incorporating isometric torques measured at clinical ROM limits, could contribute to model validity, by preventing excessively broad or narrow model ROM.

In healthy individuals, torque-angle models resulting in zero joint torques within the clinical ROM have been reported [15]. This observation raises the question of why flatter curves were observed in our study. Two hypotheses might be proposed. First, these flatter shapes may reflect increased passive forces due to muscle retractions or spasticity at clinical ROM limits in
some patients [43, 44]. Second, they may reflect heterogeneity in motor command impairment across muscle groups [11], with altered muscle recruitment at peak torque or involuntary contraction of antagonist muscles [1]. In healthy individuals, models distributing joint torques across individual muscles have been proposed for generalizing the models' use in joint configurations that have not been investigated experimentally [15]. In patients with motor impairment, such models should thus be used with caution due to motor command impairment or altered muscle properties.

Overall, our study shows that torque-angle curve modeling is less effective in patients with motor impairment than in healthy individuals. The modeling was still of sufficient quality, in most patients, to consider future use in robotic assistance systems that integrate their residual capacities in the control scheme, at least for the ROMs covered by the isokinetic assessment. For some patients, more complex mathematical models should be developed to satisfactorily reflect unstable effort generation capacities across the ROM. Further investigation of patient-related factors affecting model quality will be necessary to assess results' generalizability. The choice of mathematical model among those assessed by Haering et al. [24] for healthy individuals does not seem important. Finally, the threshold of tolerable intra- and inter-subject torque variability and the validity of models beyond the experimental ROM remains to be assessed.

### Declarations

### Ethics approval and consent to participate

The study was approved by the French CPP Sud-Méditerranée on July 29, 2024 (registration number 2024-A01007-40). The protocol was registered on clinicaltrials.gov (registration number: NCT06608121). Patients gave their informed written consent before being included in the study.

### Availability of data and material

The datasets generated during the current study are not publicly available because they contain sensitive medical information.

### Consent for publication

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

### Funding

This work is supported by the INRIA exploratory action MusMapS and the Fondation Saint-Hélier.

### Authors' contribution

Conceptualization, Methodology: all authors; Investigation: Boris Cassard, Carole Anne, Emilie Leblong, Aurélie Tomezzoli. Formal analysis and Writing - original draft preparation: Aurélie
Tomezzoli; Writing - review and editing: all authors; Funding acquisition, Resources: Charles Pontonnier, Emilie Leblong; Project administration: Joël Haytier and Linda Bodet; Supervision: Charles Pontonnier.

## Acknowledgements

The authors wish to thank the patients who participated in this study. They also acknowledge Anne Laure Roy, Claire Lemeur, David Hervieux, Vincent Lofficial and Marion Poyau for their involvement in data collection.

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