# EMG-to-torque models for exoskeleton assistance: a framework for the evaluation of in situ calibration

Lucas Quesada, Dorian Verdel, Olivier Bruneau, Bastien Berret, Michel Ange Amorim, Nicolas Vignais

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## ► To cite this version:

Lucas Quesada, Dorian Verdel, Olivier Bruneau, Bastien Berret, Michel Ange Amorim, et al., EMG-to-torque models for exoskeleton assistance: a framework for the evaluation of in situ calibration. The International Journal of Robotics Research, 2026, 10.1177/02783649251414884. hal-05384159

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# EMG-to-torque models for exoskeleton
assistance: a framework for the
evaluation of *in situ* calibration

Journal Title
XX(X):1–18
©The Author(s) 2016
Reprints and permission:

DOI: 10.1177/02783649251414884

Lucas Quesada<fnref n="1" />,<fnref n="2" />,<fnref n="3" /> Ⓓ, Dorian Verdel<fnref n="4" /> Ⓓ, Olivier Bruneau<fnref n="3" /> Ⓓ, Bastien Berret<fnref n="1" />,<fnref n="2" /> Ⓓ,
Michel-Ange Amorim<fnref n="1" />,<fnref n="2" /> Ⓓ, Nicolas Vignais<fnref n="5" /> Ⓓ

## Abstract

In the field of robotic exoskeleton control, it is critical to accurately predict the intention of the user. While surface electromyography (EMG) holds the potential for such precision, current limitations arise from the absence of robust EMG-to-torque model calibration procedures and a universally accepted model. This paper introduces a practical framework for calibrating and evaluating upper-limb EMG-to-torque models, accompanied by a novel nonlinear model. The framework includes an *in situ* procedure that involves generating calibration trajectories and subsequently evaluating them using standardized criteria. A comprehensive assessment on a dataset with 17 participants, encompassing single-joint and multi-joint conditions, suggests that the novel model outperforms the others in terms of accuracy while conserving computational efficiency. This contribution introduces an efficient model and establishes a versatile framework for EMG-to-torque model calibration and evaluation, complemented by a dataset made available. This further lays the groundwork for future advancements in EMG-based exoskeleton control and human intent detection.

## Keywords

Exoskeleton, Electromyography, EMG-to-torque models, Intention detection, Human-machine interaction

## Introduction

Active exoskeletons are a major potential tool to support human movements in a broad variety of applications. In particular, they may become important tools to support the work of physiotherapists during neurorehabilitation protocols (Mehrholz et al. 2020). For instance, they have been used to improve gait and upper limb control during daily activities for those who have suffered a stroke or a spinal cord injury, by alleviating the muscle effort required to perform them (Federici et al. 2015; Shi et al. 2019; Colucci et al. 2022; Proietti et al. 2016). Similarly, exoskeletons are powerful tools that could contribute to reducing the prevalence of musculoskeletal disorders (MSD) at work (Moeller et al. 2022), in particular when the workspace cannot be adapted to improve ergonomics (Moore et al. 2011). Despite these promising advances, there is currently only limited evidence of the beneficial impact of active exoskeletons at work (Theurel and Desbrosses 2019; McFarland and Fischer 2019). A major venue to overcome the current limitations of exoskeletons is to better estimate, and adapt to, human intent (Li et al. 2022), which is critical to ensure safe, well proportioned, and intuitive assistance (Bi et al. 2019). In the present paper, we focus on providing a framework to better estimate human intent by relying on electromyographic (EMG) inputs and exoskeleton's measurements, thereby allowing to perform accurate, *in situ* calibration of assistive controllers.

In the context of exoskeletons, estimating human intent corresponds to continuously inferring the human's joints motion or torque, for which several approaches have been developed in the literature. For instance, kinematics can

be used to anticipate human movements based on prior knowledge of their typical shape and/or distribution, often by learning movement primitives (Saveriano et al. 2023) or the weighting of a predefined cost function (Berret et al. 2011; Mainprice et al. 2015). Such techniques have been shown to be usable for human-robot interaction in general (Mainprice and Berenson 2013; Mainprice et al. 2015), as well as to assist users with an active exoskeleton (Jamsek et al. 2021). Alternatively, it is possible to perform this inference by measuring human bio-electric signals, that are processed and fed to a model, whether analytical or black-box (e.g. artificial neural networks), to estimate the movement. Among all the possible biological signals that can be used to perform this estimation (Losey et al. 2018), muscle activity measurements using surface EMG activity are particularly promising. Specifically, they allow to leverage delays in the human sensorimotor system—in this case the latency between EMG signal and muscle force production known as the *electromechanical delay* (Cavanagh and Komi 1979)—while remaining non-invasive

## Corresponding author:

Lucas Quesada, CIAMS, Université Paris-Saclay, 91405 Orsay, France.
Email: lucas.quesada@ens-paris-saclay.fr

<footnote n="1">¹ CIAMS, Université Paris-Saclay, 91405 Orsay, France</footnote>

<footnote n="2">² CIAMS, Université d'Orléans, Orléans, France</footnote>

<footnote n="3">³ LURPA, École Normale Supérieure Paris-Saclay, Gif-sur-Yvette, France</footnote>

<footnote n="4">⁴ Human Robotics Group, Department of Bioengineering, Imperial College of Science, Technology and Medicine, W12 0BZ London, United-Kingdom.</footnote>

<footnote n="5">⁵ M2S, Université Rennes 2, Rennes, France</footnote>
and practicable for human-robot interaction applications. Therefore, they allow to obtain predictive information before the onset of the movement itself. This potential predictive capability can be advantageously used in real-time assistive systems, making EMG signals an appealing choice for human/machine interactions (Kiguchi and Hayashi 2012; Li et al. 2020; Huang et al. 2022).

Several approaches can be employed to predict intent based on EMG signals, but two general categories can be defined: i) categorical classification and ii) continuous regression. Movement classification tends to use convolutional neural networks (Jiang et al. 2020; Treussart et al. 2019), support vector machine (Tsai et al. 2014; Phinyomark et al. 2018), gaussian mixture models (Trigili et al. 2019) and other pattern recognition techniques (Jaramillo-Yánez et al. 2020). Such techniques can efficiently predict movement direction or the involved joints (e.g., elbow flexion/extension), but rely on prior knowledge regarding the task at hand, preventing their generalization. Alternatively, continuous movement regression aims to predict a continuous feature of the movement, such as joints angle or torques. This is generally achieved through a mathematical model linking EMG signals to the predicted feature. For example, neuromusculoskeletal (NMS) models (Lotti et al. 2020; Durandau et al. 2018; Grimmelsmann et al. 2023; Buongiorno et al. 2018; Pan et al. 2019), simple linear regressions (Camardella et al. 2021; Kiguchi and Hayashi 2012; Pan et al. 2019; Ziai and Menon 2011; Li et al. 2023), synergy-based approaches (Camardella et al. 2021; Li et al. 2023; Buongiorno et al. 2019; Liu and Gutierrez-Farewik 2021; Chen and Qiao 2021) and other custom models (Li et al. 2014; Lu et al. 2019; Wu and Chen 2023; Dai et al. 2017; Ameri et al. 2014) currently coexist. Among these models, NMS approaches are the most advanced, as their physiology-based formulation theoretically allows to capture the nonlinearities and multiple dependencies (e.g. to kinematics) of the relationship between muscle activity and joints torques.

In this study, we will focus on EMG-to-torque continuous regression approaches, as they theoretically allow to extract rich information that would be critical to provide an adaptive assistance tailored to each user. Unfortunately, comparing the actual usability and accuracy of the above-mentioned EMG-to-torque models based on their reported performance in the literature is impossible, due to important differences in the methodology used for their calibration and evaluation. Specifically, continuous regression generally requires the model to be calibrated first, in order to adjust its internal parameters to an individual (Buongiorno et al. 2018). This step is critical as it conditions the subsequent ability of the model to predict human movements in a wide range of situations. Therefore, calibration movements should not be limited to isometric tasks (Huang et al. 2018; Dai et al. 2017; Ziai and Menon 2011; Buongiorno et al. 2018; Camardella et al. 2021), and specific or periodic dynamic tasks (Pan et al. 2019; Peternel et al. 2016; Li et al. 2023; Wu and Chen 2023; Grimmelsmann et al. 2023), but cover the widest possible range of movements and levels of effort. In addition, usual calibration procedures can induce computationally intensive and long parametric optimizations (in particular for

NMS approaches) or specialized equipment, reducing their applicability for future commercial or clinical applications.

In these contexts, it will be critical to develop efficient procedures (i) limited to the equipment directly available (i.e. the EMG and the exoskeleton with its sensors), (ii) allowing to calibrate the model for all the possible tasks performed with the assisted joints, and (iii) with reduced computational burden to calibrate the EMG-to-torque model, but without impacting the accuracy of the torque estimation.

In the present paper, we address these requirements through the following contributions:

* We introduce an *in situ* calibration framework enabling a generic calibration of EMG-to-torque models through a trajectory tracking task performed in an exoskeleton with viscous resistive control. The trajectories are designed to smoothly cover the whole workspace while exciting as much as possible the model parameters through varied effort requirements, thereby allowing to obtain an accurate model, adapted for a wide range of tasks.

* We propose a novel *non-linear mapping* EMG-to-torque model aimed at combining the advantages of existing approaches, i.e. the ability to capture nonlinearities between muscle activities and torques, as allowed by NMS modeling, and the low computational cost of linear approaches.

Importantly, although the present study could be extended to the lower limb, we validate our developments on upper limb movements, as these display more diverse trajectories during daily tasks, thus enabling a more general evaluation of the methodology. Specifically, we analyze elbow and shoulder flexion and extension movements, as these are very common in everyday activities and still allow some comparison with existing literature. Finally, we validate the performance of our novel model using systematic cross-validation techniques to limit the risk of overfitting and misinterpretation, which techniques are rarely exploited in the literature (see (Ziai and Menon 2011; Pan et al. 2019) for examples).

The structure of this document is as follows: first, we provide an overview of the experimental protocol that produced our dataset, including the exoskeleton control law, arbitrary task generation, and EMG placement; next, we explain the data processing and model evaluation procedures; then, we describe the four EMG-to-torque models we evaluated; and finally, we present our results, along with their limitations and potential future directions.

# Methods

## Participants

17 healthy subjects (11 males, age 28.2 ± 7 years, height 175.4 ± 7 cm, weight 70 ± 11 kg) took part in the experiment. The experimental protocol was approved by the ethics committee of Université Paris-Saclay (CER-PS-2021-048/A1). Participants signed a written informed consent form before starting the experiment.

## Material

**Exoskeleton** The ABLE upper limb active exoskeleton (Garrec et al. 2008) was used to impose a viscous resistive
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Participant with EMG sensors and reflective markers in the exoskeleton. Visible reflective markers and EMG sensors are respectively highlighted with pink and blue lines. The setup corresponds to the multi-joint condition of Fig. 2.</figcaption></figure>

force on the participant's wrist to induce torques in their elbow and shoulder. This viscous resistance allowed to increase the range of necessary muscle activities to perform the task, thereby allowing to optimize the EMG-to-torque models' parameters on a large range of efforts. As previously mentioned, movements were limited to a parasagittal plane, only allowing flexion and extension of the elbow and the shoulder. Participants were attached to the exoskeleton through a previously developed mechanical interface (Verdel et al. 2022b) composed of an orthosis fixed on a force/torque (FT) sensor (1010 Digital FT, ATI, Apex, USA) itself attached to a slider on the exoskeleton's forearm. This mechanical interface includes a passive rotation between the exoskeleton and the human limb, which reduces unwanted hyperstatic constraints and increases comfort while permitting the measurement of interaction forces (Verdel et al. 2023).

**Electromyography** Electromyographic signals were recorded using wireless MiniWave sensors with a WavePlus receiver (Cometa, Bareggio, Italy). The receiver was connected via USB to a computer, and the acquisition was performed through the Qualisys Track Manager (QTM) software (Qualisys, Göteborg, Sweden). Muscle head recording locations were identified following SENIAM recommendations (Hermens et al. 2000). The skin was locally shaved, cleaned with alcohol pads, and dried before placing the Ag/AgCl EMG electrodes (F3010, Fiab, Firenze, Italy). The muscle set included the long and short head of the biceps (BICShort, BICLong), long, lateral, and median heads of the triceps (TRILong, TRILat, TRIMed), brachioradialis (BRD), brachialis (BRA), posterior, medial, and anterior heads of the deltoids (DELTPost, DELTMed, DELTAnt), clavicular head of pectoralis major (PECT), and latissimus dorsi (LATI). The detailed placement of EMG sensors on the muscles is provided in supplementary material S2.

**Motion capture** Kinematic data of the participant and the exoskeleton were captured using a total of 10 Oqus 500+ motion capture cameras (Qualisys, Göteborg, Sweden).

Eight 10 mm markers were placed on anatomical landmarks, including the acromion, seventh cervical vertebra, sternal end of the clavicle, styloid processes of the ulna and radius, lateral and medial epicondyles and distal end of the second metacarpal bone (see supplementary S1.a). Additionally, two markers were placed on the anterior and posterior sides of the arm, specifically at the midpoint of the upper arm. Additional markers were placed on the exoskeleton and orthosis (see Fig. 1 and supplementary S1.b) to track their movement for later data synchronization and frame transformation purposes. Motion capture data was recorded in synchronization with EMG using the QTM software.

## Procedure

The experimental session was designed to allow a systematic comparison of the performance of different EMG-to-torque models. Specifically, we investigated (i) multi-variate linear regressions (MVLR), (ii) muscle synergies (SYN), (iii) neuromusculoskeletal modeling (NMS), and (iv) the novel non-linear mapping model (NLMap), combining the low computational cost of MVLR and the NMS's ability to capture nonlinearities (see *EMG-to-torque Models* Section).

**MVC and anthropometric measurements** Participants first completed maximum voluntary contraction (MVC) tasks for the flexion and extension of the elbow and shoulder. During the elbow tests, the participants' upper-arm was placed vertically along their trunk while their forearm was placed horizontally, with their wrist pushing downwards (extension) or upwards (flexion) on a horizontal bar with foam for comfort. The same process was applied to the shoulder joint, with the bar placed at the height of the shoulder and the elbow fully extended. Each task was repeated twice and lasted for 3 s.

After these MVC trials, the reflective markers were placed on specific anatomical landmarks as described above, and participants were instructed to stand still within the motion capture area to extract their individual anthropometric information.

**Motor tasks.** The experiment was divided in two trajectory-tracking tasks: (i) a single-joint task where participants used elbow flexion/extension movements, and (ii) a multi-joint task where participants used shoulder and elbow flexion/extension movements. In both tasks, the exoskeleton was controlled to impose a viscous resistance at the elbow and shoulder joints, increasing the required muscle effort. This allowed to increase the range of measured muscle activities, thereby increasing the range of validity of the resulting EMG-to-torque model calibration. The corresponding kinematics for both tasks are described in Fig. 2a, and the visual cues provided to the participant, including the desired trajectory and the current position of the participant, are detailed in Fig. 2b. Importantly, participants earned a point whenever their current position coincided with the desired trajectory, which points were displayed as an interactive score to enhance engagement. A video of the experiments in single-joint and multi-joint conditions can be viewed in supplementary material S3 and S4.

Before the main trials, an initial calibration was performed to individualize the viscous resistance applied
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Illustration of the different experimental conditions. a) Kinematic configurations of the single- and multi-joints tasks. Movements were respectively limited to elbow flexion/extension and shoulder and elbow flexion/extension during the single- and multi-joint task. For the multi-joints task, participants were only attached to the exoskeleton at the wrist level, and the slider was mechanically blocked to allow the transmission of efforts between the human and the exoskeleton. To compute accurate human joint torques through inverse dynamics, we accounted for the misalignment between $Z_s$ and $Z_o$, respectively the frames of the FT sensor and of the orthosis. b) Participants were positioned in front of a screen during the tasks. For the multi-joint condition, current and future desired trajectory tracked by participants were respectively provided using red dots (over a 500 ms horizon) and white dots (over a 2 s horizon). A yellow dot represented the participant's current elbow joint angle (single-joint condition) or projected wrist position (multi-joint condition). When participants successfully hit one of the white dots with their cursor, it turned green and their score (displayed on the screen) increased by one. c) B-splines were used to produce the trajectories with control points (gray dots) generated into the light-gray area that delimited the reachable workspace. Example trajectories are represented using black lines, the dotted sections representing the 2 s horizon provided to the participant.</figcaption></figure>

by the exoskeleton, thereby ensuring the feasibility for all participants. This calibration was as follows: a baseline viscous coefficient was first selected, and participants were requested to rate their perceived effort on a Borg scale. Based on this rating, the viscous coefficient was then iteratively adjusted by dichotomy until participants reported a moderate effort.

Then, ten 30 s trials with a different desired trajectory (see below for details on the trajectory generation) were performed for each condition. Before each trial, participants were instructed to stand ready, and a 3 s countdown accompanied with an auditive cue was displayed on the screen. After each trial, a one-minute rest period was observed to prevent muscle fatigue.

**Exoskeleton Control:** As previously mentioned, the exoskeleton applied a viscous resistance to the participant's movements. Furthermore, to mitigate the asymmetry between upward and downward movements due to the exoskeleton weight, we identified and compensated the gravity torques at the exoskeleton shoulder and elbow based on previous works (Verdel et al. 2021, 2024). The resulting exoskeleton control torque $\tau_e$ was computed as follows:

$$ \tau_e = g_e(\mathbf{q}_e) - \mu_e \mathbf{J}_e(\mathbf{q}_e) \cdot \dot{\mathbf{p}}_h \quad (1) $$

where $\mathbf{J}_e(\mathbf{q}_e)$ and $\mathbf{q}_e$ respectively represent the Jacobian matrix and the angular positions of the exoskeleton and $\dot{\mathbf{p}}_h$

the velocity vector of the hand expressed in the task space. In both equations, $g_e$ or $g_e$ denotes the gravity model of the exoskeleton. As mentioned above, the viscous coefficient $\mu_e$ was identified separately for each task and for each participant, according to a subjective evaluation of effort using a Borg scale.

**Desired trajectory generation:** Before each trial, 100 candidate trajectories were generated using b-splines with sets of random control points in time and space (see Fig. 2c). We used 90 and 60 points for single- and multi-joints conditions respectively. This approach based on random desired trajectory generation can produce a broad spectrum of trajectories, ranging from flat to highly curved and intricate. To guarantee that the selected trajectory, among the 100 generated, required a broad range of muscle activities and covered as much as possible of the workspace, two optimization criteria were established. These criteria were the amount of mechanical work and the evenness of its distribution in the workspace. For each component $i$ of the trajectory (e.g., $i \in [Y, Z]$ for multi-joint condition), according to equation 1, the absolute value of the work due to the viscous field can be computed as

$$ W_i = \mu_e \int_0^T \dot{s}_i^2(t) dt \quad (2) $$
where $T$ was the duration of the trajectory, and $s$ corresponds to $q_e$ and $p_h$ for single and multi-joint conditions, respectively. The distribution of mechanical work within the movement can be assessed by analyzing partial work, which quantifies the amount of work dedicated to movement in either the positive or negative direction:

$$w_i^+ = \mu_c \int_0^T [\dot{s}_i(t)]_+^2 dt \quad w_i^- = \mu_c \int_0^T [\dot{s}_i(t)]_-^2 dt \quad (3)$$

where $w_i^+$ and $w_i^-$ respectively denote the work of the movement in the upward (i.e., positive and denoted $[\dot{s}_i(t)]_+$) and downward (i.e., negative and denoted $[\dot{s}_i(t)]_-$) directions. With the total work as the sum of individual components work $W = \sum_{i=1}^D W_i$, the normalized work balance criterion is thus expressed as the mean of relative partial work contributions:

$$\tilde{C} = \frac{1}{2D} \sum_{i=1}^{D} \left[ 2 - \frac{|w_i^+ - W/(2D)|}{W/(2D)} - \frac{|w_i^- - W/(2D)|}{W/(2D)} \right] \quad (4)$$

here, $D$ represents the dimension of the task space (e.g., $D = 1, 2$ for single and multi-joint conditions, respectively). For example, in multi-joint condition, forward, backward, upward, and downward movements should all account for a quarter of the total work. With equation 4, trajectories with an evenly distributed work throughout space maximize $\tilde{C}$. Before executing a trial, scores for each of the 100 generated trajectories were computed and the highest scoring one was retained based on the following optimization metric:

$$C = W \times \tilde{C} \quad (5)$$

## Data processing

**Data synchronization** EMG and motion capture data were synchronized using the QTM software. However, the exoskeleton kinematic data obtained from internal encoders and force/torque measurements were recorded separately, necessitating temporal synchronization with the EMG and motion capture data. To address this, the exoskeleton's positions were measured using its internal encoders and the motion capture system. The alignment process consisted of determining the optimal temporal shift that minimized the root mean square error between the encoder-informed and motion-capture-informed positions of the exoskeleton arm. Once this optimal shift was determined, it was applied to the force/torque data to ensure proper alignment across all data sources.

**EMG feature extraction** EMG signals underwent a series of filtering stages. Initially, to eliminate noise from movement artifacts and external electromagnetic disturbances, a fourth-order Butterworth bandpass filter with a frequency range of 20-450Hz was utilized. Following this, the signal was rectified. Subsequently, to obtain useful information from the signal, the EMG envelope was extracted using a 3Hz fourth-order Butterworth low-pass filter (Potvin and Brown 2004). These filtering operations were conducted using a forward-backward method to prevent any phase shift. The same filtering technique was also applied to the EMG data from MVC tests, allowing to extract the maximum muscle activity for each participant and muscle involved, which was then used for normalization.

**Anthropometrics** Data collected during the static motion capture phase was used to scale an upper limb model (Holzbaur et al. 2005) using OpenSim's scaling tool (Seth et al. 2018). This process involved adjusting the dimensions of the model's bones to match the marker spacing on anatomical landmarks. By adapting the bone dimensions, the model accurately represents the proportions of the individual's limb. Furthermore, the limb masses were computed by considering the ratio of the new bone dimensions to the original dimensions of Holzbaur's model. This ensured that the model incorporates the appropriate mass distribution (mass matrix) to simulate the upper limb dynamics accurately.

**Kinematics** Although human kinematics can be deduced from the exoskeleton's internal encoders for practical scenarios, it was instead computed from motion capture data to improve accuracy and compensate for a known misalignment between the human and exoskeleton joint axes (Verdel et al. 2022a). Positions of the reflective markers were extracted and labeled using the automatic identification models (AIM) feature provided by the QTM software. The methods used for inverse kinematics varied depending on the condition. In single-joint trials, the angle of the elbow was directly computed using the marker coordinates. Conversely, in multi-joint trials, although the main focus was on shoulder and elbow flexions and extensions, it was essential to consider the potential movements of the entire upper body to take into account its full dynamics. Consequently, a comprehensive inverse kinematic analysis was conducted using OpenSim. Finally, kinematic data were filtered (4th order 3Hz low-pass Butterworth) to align the harmonic content of kinematic data to the EMG envelope and facilitate the extraction of time derivatives using numerical differentiation. Joint angles, velocities and accelerations are referred to as $q_h, \dot{q}_h$ and $\ddot{q}_h$.

**Human dynamics** Human dynamics were automatically computed from kinematics and anthropometric data previously introduced using Opensim. Classically, the motion of the human upper limb is described by the following dynamics, for $J$ joints:

$$\tau_h = M(q_h) \cdot \ddot{q}_h + c(q_h, \dot{q}_h) + g_h(q_h) + \tau_i \quad (6)$$

where, $\tau_h \in \mathbb{R}^J$ represents the torque exerted at the human's joints, $M \in \mathbb{R}^{J \times J}$ denotes the mass matrix, $c \in \mathbb{R}^J$ corresponds to the vector of Coriolis and centrifugal forces, $g_h \in \mathbb{R}^J$ is the vector of gravitational torques, and $\tau_i$ represents the vector of human-exoskeleton interaction torques. By considering these factors, the equation provides a comprehensive representation of the complex dynamics involved in the motion of the human upper limb. Interaction torques are computed by first transforming the FT data from its original frame to the forearm's frame and using the upper limb's jacobian matrix $J_h(q_h)$ so that

$$\tau_i = J_h^T(q_h) \cdot f_s + \tau_s \quad (7)$$

where $f_s$ and $\tau_s$ are the forces and torques measured from the sensor underneath the orthosis.
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Cross-validation method. The data is split in two (or more) folds, one is used to perform the calibration and the other the evaluation. Roles are then reversed and the scores are averaged.</figcaption></figure>

**Resampling** All signals were resampled at a rate of 30Hz to reduce the number of samples and speed up the calibration of the models. All signals being 3Hz low-pass filtered, no information was lost.

## Analysis

As the present study aims to evaluate and compare EMG-to-torque models, it is necessary to properly define the criteria for selecting one model over another. In particular, for future implementations, the model accuracy and its cost of use are essential performance criteria that need to be systematically tested as detailed below.

**Cross-validation** It is more conservative to provide different calibration and evaluation datasets to mitigate the risk of evaluating overfitted models. To that end, a K-fold cross-validation approach was employed. First, the dataset was divided into k folds. k iteration of calibration and evaluation were then performed. For each iteration, k − 1 folds were used for the calibration and the remaining fold was used for evaluation (Refaeilzadeh et al. 2009). Due to the limited amount of data (10 trials per condition) and to the computational burden associated with large values of k, we applied a 2-fold division to all the participants and conditions, and tested the effects of a 5-fold on the single-joint condition to assess the robustness of our findings. For the 2-fold case, each participant and condition (single and multi-joint), the ten 30s trials were randomly divided into two folds, each containing five trials. Therefore, for each model, condition, and participant combination, we obtained two values of the evaluation metrics, which were then averaged for further analyses (see Fig. 3). For the 5-fold investigation, each fold contained eight trials for calibration and two for evaluation, yielding five score values.

## Performance metrics

**Joint torque estimation:** The accuracy of the models was evaluated using two metrics, (i) the normalized root mean square error (NRMSE), reflecting the model's ability to predict joint torques with the proper magnitude, and (ii) Pearson's coefficient of correlation (R), reflecting the model's ability to estimate the changes in human joint torques. The NRMSE is defined as follows

$$ NRMSE_j = \frac{\sqrt{\frac{1}{N} \sum_{i=0}^{i=N} (\tau_{h,j}(t_i) - \hat{\tau}_{h,j}(t_i))^2}}{\max(\tau_{h,j}) - \min(\tau_{h,j})} \quad (8) $$

where N is the number of samples in the fold and tᵢ is the i'th time sample. The reference torque τᵢⱼ for the j'th joint is calculated from equation 6, while τᵢⱼ is an estimated torque obtained from the tested EMG-to-torque model. The NRMSE is normalized with respect to the amplitude of the reference torque, obtained using max(τᵢⱼ) and min(τᵢⱼ). The Pearson's coefficient of correlation (Rⱼ) is computed as

$$ R_j = \frac{\operatorname{cov}(\tau_{h,j}, \hat{\tau}_{h,j})}{\sigma_{\tau_{h,j}} \sigma_{\hat{\tau}_{h,j}}} \quad (9) $$

where cov(·, ·) denotes the covariance, and σᵢⱼ represents the standard deviation. The NRMSE and R values were compared during the cross-validation process to evaluate and compare the performance of the models.

**Model cost of use:** While EMG-to-torque model accuracy is undoubtedly the most critical feature for exoskeleton control, it is also crucial to be as low cost as possible. This implies that the model should require minimum (i) calibration data for a given level of performance, referred to as *data cost*, (ii) duration for the optimization of the model's parameters, referred to as *calibration cost*, and (iii) duration for the evaluation of the model in real-time, referred to as *evaluation cost*. For future applications, these three points are critical as they directly influence the usability of the control on a daily basis, with different users, and the controller's responsiveness. As such, during cross-validation, the calibration and evaluation costs were respectively measured as the time required to optimize each model's parameters and the time to compute joint torques from muscle activities. These two costs were measured on the same platform, that is with the same computational resources, and with optimized implementations (see implementation details below) allowing their comparison between models. Note that evaluating the optimal size of the training dataset for each model, i.e. the minimum amount of data above which its calibration performance does not improve, requires massive computations and is out of the scope of the present study. Therefore, the *data cost* of the different models will be discussed based on their relative performance with the 2-fold division, and comparison trends consistency will be tested with 5-fold division.

## Statistical analysis

The 2-fold cross-validation process resulted in two values for each criterion, which were then averaged, leaving one value per subject and model as data. R scores were normalized using the Fisher transform. Mauchly's test were first conducted to verify the sphericity of data in repeated measurements. Subsequently, a repeated measures analysis of variance (rm-ANOVA) was conducted with the models as the within-subjects factor, and a Greenhouse-Geisser correction was applied to correct for sphericity violations, which was when Mauchly's test was significant. The null hypothesis was that models produced identical scores. Model-to-model post-hoc comparisons were performed with
a paired sample t-test and a Holm-Bonferroni correction. In the results, ± corresponds to the standard deviation (SD). Datapoints with a Z-score over 3 were considered to be outliers and not included in the statistical analysis.

## EMG-to-torque Models

The following section describes the different EMG-to-torque models investigated in the present study. The mathematical descriptions are provided in a general fashion, using K as the number of muscles, J the number of joints, and N the number of recorded EMG samples. All models were calibrated on a computer embedding a 16-core Intel Xeon processor with 32Go of RAM.

### Linear mapping model

The multivariate linear regression (MVLR) model is a straightforward formulation used to compute joint torques by linearly combining muscle excitations (Camardella et al. 2021). This is achieved with a mapping matrix, denoted as $H_{\ell} \in \mathbb{R}^{J \times K}$, which establishes the link between the estimated torques $\hat{\tau}_{h}(t) \in \mathbb{R}^{J}$ and the muscle excitations $m(t) \in \mathbb{R}_{+}^{K}$. This relationship can be expressed as

$$ \hat{\tau}_{h}(t) = H_{\ell} \cdot m(t) \quad (10) $$

By construction, this model is unable to capture the non-linear relationships between muscle activities and force production.

**Calibration implementation:** Calibrating the MVLR model required to establish the linear relationship between filtered EMG signals and joints torques obtained using upper-limb dynamics (see Eq. 6). However, the real-time estimation of joint torques only requires EMG signals.

The model is calibrated by first defining the data matrices $T \in \mathbb{R}^{J \times N}$ and $M \in \mathbb{R}_{+}^{K \times N}$, which contain torque and muscle excitation samples from the training dataset, respectively. The calibration process involves finding the mapping matrix $H_{\ell}$ that satisfies the equation:

$$ T = H_{\ell} \cdot M \quad (11) $$

To achieve this, the *mvregress* function from Matlab (MathWorks, Natick, MA, US) is employed on T and M to determine the optimal values for $H_{\ell}$ in the least squares sense.

**Expected cost of use:** In terms of computational load, the number of parameters to optimize is $J \times K$ with a least squares approach, which is a negligible *calibration cost* for our applications if performed by a modern computer.

Similarly, the real-time evaluation of the model requires to multiply a matrix of dimension $J \times K$ with a vector of length K, which represents a negligible *evaluation cost*.

### Neuromusculoskeletal model

Neuromusculoskeletal (NMS) models are a combination of a neuromuscular Hill-type model (Hill 1938) and a musculoskeletal representation of the body. Such models were designed to capture as well as possible the complexity of human neuromuscular dynamics and musculoskeletal biomechanics, including their nonlinearities and the complex

geometry of tendon insertions to bones. In such models, muscle activations are first derived from EMG signals via a nonlinear activation function (Buchanan et al. 2004), which reflects the dynamics of muscle activation from neural excitation as follows

$$ a_k(t) = \frac{e^{A_k m_k(t)} - 1}{e^{A_k} - 1} \quad (12) $$

where $A_k$, $a_k(t)$, and $m_k(t)$ are, respectively, the shape factor, activation and excitation of the $k^{th}$ muscle. For a given muscle, the force can then be expressed as follows (see also (Millard et al. 2013)),

$$ F^M = F_{iso}^M (a(t)f^L(\tilde{l}^M)f^V(\tilde{v}^M) + f^{PE}(\tilde{l}^M)) \cos \alpha \quad (13) $$

where $F_{iso}^M$ denotes the maximal isometric force, while $\tilde{l}^M$ and $\tilde{v}^M$ represent the normalized fiber length and velocity, respectively. The function $f^L(\tilde{l}^M)$ characterizes the force-length relationship, describing how a muscle fiber's maximum active force varies with its length. The normalized fiber length is given by $\tilde{l}^M = l^M/l_0^M$, where $l_0^M$ corresponds to the optimal fiber length at which $f^L(\tilde{l}^M)$ reaches its maximum value (i.e., $\tilde{l}^M = 1$). Similarly, $f^V(\tilde{v}^M)$ describes the force-velocity relationship, reflecting the changes in active force generation of the muscle with fiber velocity (i.e., $f^V(0) = 1$ for isometric contractions), and $f^{PE}(\tilde{l}^M)$ accounts for the passive elastic behavior of the fiber. Lastly, $\alpha$ denotes the pennation angle, which quantifies the angle between the axis of muscle force and the orientation of its fibers. This model incorporates a serial elastic element representing the tendon to ensure equilibrium between the muscle fiber and tendon forces and compute their lengths. This is captured by ensuring the following constraint during forward dynamics:

$$ F^M - F_{iso}^M f^T(\tilde{l}^T) = 0 \quad (14) $$

where $\tilde{l}^T$ represents the normalized tendon length, and $f^T(\tilde{l}^T)$ describes the force-length relationship of the tendon's passive elastic behavior. The normalized tendon length is given by $\tilde{l}^T = l^T/l_0^T$, where $l_0^T$ corresponds to the tendon slack length, such that $f^T(\tilde{l}^T) = 0$ when $\tilde{l}^T < 1$. Equation 14 is solved at each time step during forward estimation of muscle force. Once the muscle force is obtained, the resulting estimated joint torques can then be derived as

$$ \hat{\tau}_{h,j}(t) = \sum_{k=1}^{k=K} F_k^M(t) \times r_{k \to j}(t) \quad (15) $$

where $r_{k \to j}(t)$ is the moment arm of $k^{th}$ muscle on the $j^{th}$ joint.

**Calibration implementation:** A previously developed (Holzbaur et al. 2005) and benchmarked (Saul et al. 2014) NMS model, implemented in the OpenSim simulation software (Seth et al. 2018), was used in the present study. For the calibration and evaluation in real-time of the model, both the muscle force and the moment arm, using multibody simulation (Sherman et al. 2013), were computed via OpenSim. For the calibration, When a muscle is not acting on a joint (e.g., brachioradialis on shoulder flexion),
$r_{k→j} \triangleq 0$ was set. As the NMS model is dependent on the velocity of muscle shortening and on joints angular positions, its calibration required filtered EMG signals, kinematic information and joints torques estimated from upper-limb dynamics. Its evaluation in real-time required both EMG signals and kinematic data to estimate joint torques.

The model was calibrated by first scaling the musculoskeletal component using anthropometric data from motion capture, adjusting the length and width of each limb. Then, the muscle parameters were optimized using a single objective genetic algorithm (Mohammadi et al. 2017; Buongiorno et al. 2018) with a stalling criterion of 20 generations, 16 chromosomes per generation, and a maximum of 1000 generations. In the present study, chromosomes were composed of 5 parameters repeated for each of the K muscles, resulting in 5 × K parameters to optimize. To make the calibration possible, the evaluation of the chromosomes' performance was rendered parallel by implementing a custom C++ multi-threaded program taking advantage of OpenSim's application programming interface. This approach was designed to provide faster implementation comparatively to other methods relying on simulated annealing optimization that is not well-suited for parallelization and entails high calibration cost (Pizzolato et al. 2015; Bennett et al. 2022). Crossover fraction and mutation rate were determined with a parametric study showing that a value of 0.7 for each provided the best calibration. The cost function was defined as the quadratic mean of the estimation mean square error of each joint:

$$C(\tau_h, \hat{\tau}_h) = \sqrt{\sum_{j=1}^{J} \frac{1}{N} \sum_{n=0}^{n=N} (\tau_{h,j}(t_n) - \hat{\tau}_{h,j}(t_n))^2} \quad (16)$$

The five parameters stored K times in each chromosome were the maximal isometric force $F_{iso}^M$, optimal fiber length $l_0^M$, tendon slack length $l_0^T$, pennation angle at optimal fiber length $\tilde{\alpha}$ and shape factor A.

**Expected cost of use:** When compared to MVLR, NMS models require to optimize more parameters for $J \le 5$ if one assumes that all the $J \times K$ elements of the MVLR matrix are non-zero. Beyond the number of parameters, the computational complexity of the NMS model is incomparable to the complexity of the MVLR approach. This is because of (i) its nonlinearity, implying more costly parameters' optimization, and (ii) the computation of the muscle equilibrium (see Eq. 14) at each iteration, requiring an iterative solver. Therefore, its *calibration cost* is expected to be very high, and the NMS may require more data than simpler models to sufficiently excite all its parameters and reach a similar performance, probably resulting in a high *data cost* or in the necessity of deeper anatomical measurements such as muscles cross-section areas and tendon insertions.

Similarly, the *evaluation cost* of the model is expected to be high due to its nonlinearity and to the computation of the moment arms requiring real-time multibody simulations.

## Synergy-based model

Synergies are described as a *coherent activation, in space or time, of a group of muscles* (d'Avella et al. 2003). From a human motor control perspective, these synergies

would result from the central nervous system using common descending motor commands to simultaneously control groups of muscles. Similarly to the MVLR, synergy based models are linear, thus they cannot capture the nonlinearities of the relationship between neural excitation and muscle activation. A variety of synergy decompositions, including spatial, temporal, spatiotemporal, or space-by-time, can be extracted from a set of muscle excitations depending on the specific analysis or application (Delis et al. 2014). However, since our dataset consists of randomly generated trajectories without specific gait or periodic movements, we do not anticipate the emergence of temporal synergies. Therefore, we have decided to focus on a spatial synergy model for our analysis. Considering the number of spatial synergies as S, the muscles EMG activity matrix $M \in \mathbb{R}_+^{K \times N}$ can be factorized into two nonnegative matrices as follows:

$$M = W \cdot A + \varepsilon \quad (17)$$

with $W \in \mathbb{R}_+^{K \times S}$ the synergy weight matrix, mapping a synergy activation to a group of muscles, and $A \in \mathbb{R}_+^{S \times N}$ the synergies activation matrix, containing the corresponding synergy activation for each sample of the data. $\varepsilon \in \mathbb{R}^{K \times N}$ is the muscle activity residual matrix that contains the residual error resulting from the factorization. After factorization, the synergy-based EMG-to-torque models are formulated similarly to the MVLR,

$$\hat{\tau}_h(t) = H_s \cdot \alpha(t) \quad (18)$$

where $H_s \in \mathbb{R}^{J \times S}$ is the synergy mapping matrix, which maps a synergy to a joint, and $\alpha(t) \in \mathbb{R}_+^S$ is the synergy activation vector (i.e., one of the columns of A). In this framework, $\alpha(t)$ is not measured directly and must be calculated from the previously factorized synergy weight matrix:

$$\alpha(t) = W^+ \cdot m(t) \quad (19)$$

with $\bullet^+$ the pseudo-inverse operator. Therefore, the final formulation of the synergy-based model is

$$\hat{\tau}_h(t) = H_s \cdot W^+ \cdot m(t) \quad (20)$$

**Calibration implementation:** Similarly to the MVLR, calibrating the SYN models required filtered EMG signals and joints torques, and the real-time estimation of joint torques only requires the filtered EMG signals.

The model is calibrated in several steps. First, the muscle excitation matrix M from the training data (with a corresponding torque matrix T) is factorized via Matlab's non-negative matrix factorization (NNMF) function to obtain the synergy weights and activation matrices W and A. Then, the calibration problem corresponds to finding $H_s$ such that

$$T = H_s \cdot A \quad (21)$$

Similarly to the MVLR model, this is done by using Matlab's *mvregress* function on T and A. In this paper, synergy-based models are abbreviated SYN; if the number of synergies is relevant in the context, the model is abbreviated SYNX, with X replaced by the number of synergies. To ensure at least one synergy per degree of freedom and direction, a minimum of 2 and 4 synergies were used for single and multi-joint conditions, respectively.
**Expected cost of use:** The *calibration cost* of the SYN models is comparable to that of the MVLR, although it is higher due to the NNMF algorithm. Specifically, the step for identifying the **H**<sub>s</sub> matrix requires to identify *J* × *S* parameters, where *S* < *K*, implying a lower cost than the MVLR. However, the NNMF algorithm is iterative and may not converge fast depending on the **M** matrix.

The *evaluation cost* of SYN models is negligible as the computation of the joints torques in real-time only requires two low-dimension matrices multiplications.

## Nonlinear mapping model

The models that we introduced in the previous sections range from very simple and straightforward with the MVLR, to complex and nonlinear with the NMS. While the MVLR model is easy to formulate and calibrate, its main limitation stems from its inability to capture the nonlinearities of the relationship between neural excitation and muscle activation, and hence between EMG signal and force production. On the other hand, while the NMS model, built through physiological observations, theoretically allows to capture these nonlinearities, its cost of use is expected to be high, limiting its usability. To address these limitations, we propose to combine the shape factor, allowing the NMS to capture some of the nonlinearities, with the linear mapping of the MVLR, allowing to easily scale the contribution of each muscle's activity to the joint torque, to obtain a novel *nonlinear mapping* (NLMap). By construction, this NLMap is thus be able to capture relevant nonlinearities while remaining sufficiently simple to limit its cost of use. The model formulation is as follows:

$$ 
\begin{aligned}
a_k(t) &= \frac{e^{A_k m_k(t)} - 1}{e^{A_k} - 1} \\
\hat{\tau}_h(t) &= \mathbf{H}_{nl} \cdot \mathbf{a}(t)
\end{aligned}
\qquad (22) $$

where $A_k$, $m_k$ and $a_k$ are the corresponding shape factor, excitation, and nonlinear activation of the $k^{th}$ muscle, and $\mathbf{H}_{nl} \in \mathbb{R}^{J \times K}$ the nonlinear mapping matrix. $\mathbf{a}$ is the activation vector. Note that, by construction, the MVLR model is an ablated version of the NLMap, removing its nonlinear activation component to only consider the linear mapping.

**Calibration implementation:** As moment-arm estimation and muscle-shortening velocity did not need to be considered for this model, the NLMap only required filtered EMG signals and joint torques for the calibration. Similarly to simpler linear mapping alternatives, only the filtered EMG signal was required for joint torque estimation during the real-time evaluation.

The model was calibrated iteratively using a single-objective genetic algorithm with multiple steps in each iteration. Initially, the genetic algorithm generated a set of shape factors {$A_k$} for each muscle, which served as a chromosome. These parameters were then used to compute the nonlinear muscle activations of the training dataset, which were compiled into an activation matrix **A**. Next, the mapping matrix $\mathbf{H}_{nl}$ was computed using the torque data matrix **T** through the *mvregress* function as for the MVLR model. Since the mapping problem has a unique solution, a specific set of shape factors corresponds to a unique

mapping, and hence, the matrix coefficients are not part of the chromosome. The score of the chromosome was then computed with the cost function described in equation 16, as for the NMS model. The best chromosome was determined through iterative crossover and mutation operations as is typically done when using a genetic algorithm. We used a 0.8 crossover rate, 200 chromosomes per generation (allowed by the small evaluation cost), and the other parameters were the same as for the NMS calibration.

**Expected cost of use:** The expected *calibration cost* of the NLMap model was much lower than the one of the NMS, although higher than the MVLR and SYN linear alternatives. Specifically, the number of parameters to optimize with the genetic algorithm was five times lower than for the NMS ($K$ elements in the chromosome), as only the shape factor for each muscle was included. This implied that less generations were needed to converge to the optimum on average. However, as the *mvregress* function was called for each chromosome in each generation (i.e. 200 × in a single generation), the *calibration cost* was expected to be much higher than for the linear models.

On the other hand, the real-time *evaluation cost* of the NLMap model was negligible for our applications, as it did not include the costly estimation of the moment arms required by the NMS.

## Summary of model properties

In Table 1, we summarized the properties of the different models presented above to facilitate their comparison.

<table><caption>**Table 1. Summary of the different models properties and requirements.**</caption><thead><tr><th></th><th>Requirement/ Property</th><th>MVLR</th><th>NMS</th><th>SYN</th><th>NLMap</th></tr></thead><tbody><tr><td rowspan="4">Calibration</td><td>Req. EMG</td><td></td><td>YES</td><td></td><td></td></tr><tr><td>Req. FT sensor</td><td></td><td></td><td>YES</td><td></td></tr><tr><td>Req. Kinematics</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td></tr><tr><td>Cost</td><td>LOW</td><td>HIGH</td><td>LOW</td><td>MEDIUM</td></tr><tr><td rowspan="5">Evaluation</td><td>Req. EMG</td><td></td><td>YES</td><td></td><td></td></tr><tr><td>Req. FT sensor</td><td></td><td></td><td>NO</td><td></td></tr><tr><td>Req. Kinematics</td><td>NO</td><td>YES</td><td>NO</td><td>NO</td></tr><tr><td>Cost</td><td>LOW</td><td>HIGH</td><td>LOW</td><td>LOW</td></tr><tr><td>Nonlinearities</td><td>NO</td><td>YES</td><td>NO</td><td>YES</td></tr></tbody></table>

# Results

## *Trajectory generation*

Our trajectory generation process resulted in Gaussian-like distributions for elbow positions during single-joint and multi-joint conditions. The center of the elbow movements' distribution was around the 90° position, with the majority of positions falling within [45°, 135°] (see Fig. 4). Shoulder positions were concentrated within the [0°, 90°] range. Importantly, the distributions generated by our framework are consistent with the angular distributions observed during daily living tasks by Haverstock et al. (Haverstock et al. 2020) and Chapman et al. (Chapman et al. 2019). This shows that our trajectory generation and selection process resulted in calibration trajectories that allow to cover the
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Generated trajectory kinematics and reference torque distributions. For the single joint condition, each line represents a participant. For the multi-joint condition, lines and fills represent the mean and standard deviation of the data distribution across all participants. Blue solid and red dashed lines represent the elbow and shoulder joints, respectively. A 0° angle corresponds to a fully extended elbow and no shoulder elevation. Areas around the lines correspond to the standard deviation between participants. Black lines represent reference positions observed during daily living tasks for the elbow (Haverstock et al. 2020) and the shoulder (Chapman et al. 2019).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Single-joint model accuracy. a) Pearson's correlation coefficient b) Normalized root mean square error</figcaption></figure>

<caption>**Figure 6.** Effect of mapping nonlinearization on Elbow torques. Comparison between MVLR and NLMap of reference ($\tau_h$) to estimated ($\hat{\tau}_h$) torques correlations. Each line corresponds to one participant and represents the smoothed data from the concatenated ten 30s trials.</caption>

necessary movements for a wide variety of use cases. In terms of effort, during single-joint tasks, desired elbow torques were uniformly distributed within [-10, 20] Nm. A Gaussian-like distribution emerged during multi-joint tasks, featuring mainly positive elbow torques within [-5, 20] Nm and shoulder torques [-10, 30] Nm.

## Torque estimation performance

**Single-joint** Qualitatively, Fig. 5 shows that, for the single-joint condition, there is a consistent trend in the performance in terms of torque magnitude (NRMSE) and variations (Pearson-R) estimations. That is, models that better predict the torque magnitude, also better predict the torque variations. Remarkably, this first scoring suggests that the NLMap outperforms the NMS on these two criteria. Furthermore, as could be expected, the NMS seems to outperform the linear alternatives, with SYN models being the less accurate.

Quantitatively, the repeated measures ANOVA exhibited a main effect of the chosen model on the NRMS error and the correlation coefficient ($p < 0.001$; $\eta^2 > 0.38$). As detailed in Fig. 5, the NLMap and NMS models attained NRMS errors of $4.66 \pm 0.54\%$ and $5.94 \pm 0.96\%$, respectively. In contrast, the MVLR had a score of $7.11 \pm 1.17\%$, and SYN models ranged from $7.31 \pm 1.28\%$ (SYN5) to $8.46 \pm 2.37\%$ (SYN2). Similarly, the correlation coefficient for the NLMap and NMS models were $0.971 \pm 0.011$ and $0.958 \pm 0.013$, respectively, while the MVLR was $0.942 \pm 0.019$ and SYN models exhibited a range spanning from $0.943 \pm 0.016$ (SYN6) to $0.918 \pm 0.0418$ (SYN2). Overall, the NLMap model proved to be significantly better in terms of NRMSE and R with a large effect size (2-fold: $p < 0.001$, Cohen's $d > 1.65$; 5-fold: $p < 0.001$, Cohen's $d > 1.65$). The NMS model was also significantly better than MVLR and SYN models for both criteria, with a large effect size (2-fold: $p < 0.05$, Cohen's $d > 1$; 5-fold: $p < 0.05$, Cohen's $d > 1.03$). No trend difference were observed between the 2-fold and the 5-fold cross-validation schemes (supplementary Figure S5).

Importantly, Fig. 6 shows the role of the shape factor included in the NLMap model in capturing nonlinearities
and reducing errors by comparing the correlations between estimated and measured joints torques. The S-shaped correlation curve of the MVLR model indicates the presence of nonlinear residual errors. However, incorporating the shape factor term in the NLMap model contributes to their mitigation, resulting in a straighter curve that aligns more closely with the ideal 45° line representing fully correlated torques. Combined with the results provided in Fig. 5, this shows that ablating the shape factor from the NLMap model, thereby falling back to the linear MVLR, results in significantly decreased torque estimation performance.

**Multi-joints** Fig. 7a shows representative traces of EMG signals used to estimate the joints torque, illustrating their smoothness after processing and the contributions of the different muscles. Fig. 7b illustrates the large variety of positions and velocities that can be obtained during a 30 s trial, and Fig. 7c shows the large variety of corresponding joints torques, as well as a representative estimation using the NLMap model (see supplementary Fig. S6 for similar graphs with the other models). Qualitatively, Fig. 8 shows a consistent trend in the performance in terms of torque magnitude (NRMSE) and variations (Pearson-R) estimations, as was the case for the single-joint condition. Remarkably, the NLMap seems to once again outperform the NMS and all the investigated models on these two criteria. More surprisingly, the MVLR also seems to outperform the NMS despite its simplicity and inability to capture nonlinearities.

Quantitatively, a repeated measures ANOVA confirmed a main effect of the EMG-to-torque model on the NRMS error and correlation coefficient for the shoulder and elbow joints ($p < 0.01$; $\eta^2 > 0.08$ and $p < 0.001$; $\eta^2 > 0.16$, respectively). Prior to analyzing the performance of the difference models for each joint, an ANOVA showed a main effect of the joint observed on the correlation coefficient after Fisher transform ($p < 0.001$, $\eta^2 = 0.72$). Further comparisons showed that the correlation between the measured and predicted torques was higher for the elbow than for the shoulder with all models (in all cases: $p < 0.001$, Cohen's $d > 1.87$). Below, we detail the results of post-hoc comparisons for the elbow and shoulder separately.

**Elbow:** The NLMap and NMS models respectively obtained NRMS errors of 6.58 ± 0.73% and 8.97 ± 1.51%. The MVLR model had an error of 8.54 ± 1.75%, which is slightly lower than the NMS, while SYN models ranged from 9.07 ± 2.12% (SYN9) to 10.24 ± 2.72% (SYN4). Likewise, NLMap and NMS models, respectively, obtained a correlation score of 0.913 ± 0.019 and 0.844 ± 0.043, while MVLR was 0.863 ± 0.041, which is once again slightly better than the NMS, and SYN ranged from 0.857 ± 0.042 (SYN9) to 0.798 ± 0.082 (SYN4). Overall, the NLMap model proved to be significantly better in terms of NRMSE and R than any other model with a large effect size (in all cases: $p < 0.01$, Cohen's $d > 1.45$). Despite the differences reported above, the NMS model's performance was not significantly lower than the performance of the MVLR model.

**Shoulder:** The NRMS error for the NLMap and NMS models were 6.87 ± 1.4% and 8.0 ± 2.1%, respectively. The MVLR model had an error of 7.79 ± 1.4%,

which is slightly lower than the NMS as for the elbow joint, while SYN models exhibited a range from 8.08 ± 1.5% (SYN7) to 8.47 ± 1.9% (SYN10). Similarly, the correlation scores for the NLMap and NMS models were 0.885 ± 0.030 and 0.840 ± 0.052, while MVLR was once again slightly better than the NMS with a R of 0.855 ± 0.033. SYN models ranged from 0.849 ± 0.034 (SYN9) to 0.828 ± 0.075 (SYN10). In terms of NRMSE, the NLMap model demonstrated statistically significant superiority with a moderate to strong effect (in all cases: $p < 0.05$, Cohen's $d > 0.62$). Regarding the correlation coefficient, the NLMap model was also significantly better than all the other models (in all cases: $p < 0.01$, Cohen's $d > 0.96$), while, as for the elbow joint, the NMS and MVLR models did not lead to significantly different performance.

<figure><img src="image_8.png" /><figcaption>**Figure 7.** Multi-joint torque estimation with the NLMap model for a representative participant. a) Processed EMG signals. b) Angular position and velocity of the Shoulder (solid) and Elbow (dotted). c) Shoulder ($\tau_{sho}$) and Elbow ($\tau_{elb}$) reference and estimated torques. Reference and estimated torques are respectively represented using a solid and a dashed line.</figcaption></figure>

## Cost of use of EMG-to-torque models

Table 2 summarizes the cost of use of the different models as defined in the Methods. As previously mentioned, establishing the *data cost* was outside the scope of the present study so we only mention it based on accuracy and comparisons between the different *k*-fold validations. Therefore, we provide here estimations of the common costs related to the preparation of the participant and MVC, which,
<figure><img src="image_9.png" /><figcaption>**Figure 8.** Multi-joint model accuracy. In both panels, data for the elbow joint is represented in gray, and data for the shoulder joint in white. a) Pearson's correlation coefficient b) Normalized root mean square error</figcaption></figure>

<table><caption>**Table 2.** Cost of use of the different models. The *calibration cost* is the time necessary to perform the complete calibration of a model for a given participant. The *evaluation cost* is the time necessary to compute one estimation time step. Results with a value of < $10^{-3}$ indicate that the obtained time was lower than the resolution of the measurement.</caption><thead><tr><th></th><th>NLMap</th><th>NMS</th><th>MVLR</th><th>SYN6</th><th>SYN2</th></tr></thead><tbody><tr><td><strong>Single-joint</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Prepa. [min]</td><td></td><td colspan="4">Common: ~ 15 ± 5</td></tr><tr><td>One trial [min]</td><td></td><td colspan="4">Common: ~ 1.5 ± 0.5</td></tr><tr><td>Calibration [s]</td><td>22.1</td><td>1216</td><td>1.096</td><td>2.02</td><td>1.37</td></tr><tr><td>SD [s]</td><td>0.27</td><td>538</td><td>0.33</td><td>0.32</td><td>0.5</td></tr><tr><td>Evaluation [ms]</td><td>0.08</td><td>10.8</td><td>&lt; 10<sup>-3</sup></td><td>&lt; 10<sup>-3</sup></td><td>&lt; 10<sup>-3</sup></td></tr><tr><td>SD [ms]</td><td>0.003</td><td>0.84</td><td>&lt; 10<sup>-3</sup></td><td>&lt; 10<sup>-3</sup></td><td>&lt; 10<sup>-3</sup></td></tr><tr><td><strong>Multi joints</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Prepa. [min]</td><td></td><td colspan="4">Common: ~ 20 ± 5</td></tr><tr><td>One trial [min]</td><td></td><td colspan="4">Common: ~ 1.5 ± 0.5</td></tr><tr><td>Calibration [s]</td><td>44.7</td><td>3662</td><td>1.49</td><td>2.45</td><td>1.89</td></tr><tr><td>SD [s]</td><td>2.6</td><td>1200</td><td>0.13</td><td>0.22</td><td>0.22</td></tr><tr><td>Evaluation [ms]</td><td>0.1</td><td>15.6</td><td>&lt; 10<sup>-3</sup></td><td>&lt; 10<sup>-3</sup></td><td>&lt; 10<sup>-3</sup></td></tr><tr><td>SD [ms]</td><td>0.005</td><td>5.7</td><td>&lt; 10<sup>-3</sup></td><td>&lt; 10<sup>-3</sup></td><td>&lt; 10<sup>-3</sup></td></tr></tbody></table>

added to the duration of a trial multiplied by the number of necessary trials to reach a plateau in performance, would allow to obtain the total *data cost*. Then, we provide the *calibration cost* and the *evaluation cost* obtained with the 2-fold division (2.5 min of calibration data).

**Data cost:** With the same amount of calibration data, the analyses provided on the performance of the torque estimation show that the NLMap has the lowest cost of all the tested models. That is, for the same amount of calibration data, it provides the best performance. Interestingly, the NMS model has a lower cost than the MVLR model for the single-joint but not for the multi-joint condition. Finally, the SYN models are the most expensive when considering only the calibration data collection. Importantly, one can note that there was no difference between the accuracy of the calibrated models with the 2-fold and 5-fold divisions, that is between 2.5 min and 4 min of training data. This suggests that our framework is efficient to generate relevant

calibration data, thereby minimizing the *data cost* and making it comparable for all models.

**Calibration cost:** As predicted based on the complexity of the models and associated parameters optimization processes, the NMS model has a very high calibration cost with approximately 20 ± 10 min and 60 ± 20 min for the single- and multi-joint conditions respectively. These results also suggest that increasing the size of the calibration dataset to improve its performance, and possibly equal the NLMap, could only lead to even more prohibitory *calibration cost* when it comes to applications. Conversely, the cost of the NLMap remains below one minute in both cases, which would not hamper the model's usability, and the cost of the linear alternatives is negligible, as was expected.

**Evaluation cost:** Here again the NMS is the only model that results in a cost susceptible to hamper the applications, as even for single-joint movements each evaluation of the torque systematically takes more than 10 ms, which is a lot less reactive than most active exoskeletons (e.g. our exoskeleton's control loop lasts 1 ms). All the other models are easily usable with the control frequency of such robots, with the linear ones being so low-cost that their evaluation times are not really measurable with standard methods.

**Total cost of use:** Overall, these results show that the total cost of use of linear EMG-to-torque models such as the MVLR or SYN approaches is only related to the *data cost*, and their limitation lies in the lack of accuracy. They also show that the cost of use of the NLMap is primarily accounted for by the *data cost*, the *calibration cost* and *evaluation cost* being slightly higher than with linear alternatives without hampering its usability. Finally, the cost of use of the NMS models is very high, hampering their usability both from a calibration and exoskeleton control perspective, with a performance in terms of accuracy that is significantly lower than the NLMap.

## Discussion

The present paper introduced a general framework for the *in situ* calibration and evaluation of EMG-to-torque models, with the aim of enabling predictive exoskeleton control to assist arbitrary human movements. Through this paper, we outlined and evaluated a set of common models, their ontology, and the associated calibration technique. We further showed the respective limits of these existing approaches, and proposed a novel NLMap model allowing to mitigate them. This model is able to capture nonlinearities in the relationship between EMG signals and muscle force, but at a low cost of use. Our framework is supported by the generation of random, non-periodic calibration trajectories, enabling us to make these models fit for each individual user and varied kinematic and dynamic situations. Importantly, we investigated the usability of our framework and model for single- and multi-joint conditions, directly confronting our developments to the difficulty of predicting torques with different degrees of freedom. Although our evaluation remains limited to a parasagittal plane, they represent a significant step towards the assistance of arbitrary human movements (Bi et al. 2019). Finally, we systematically evaluated the models using a cross-validation approach to prevent biases related to overfitting.
In order to obtain robust EMG-to-torque models suited to a wide variety of applications, creating a task to generate data with a wide range of kinematic and dynamic situations is essential. While recent studies have used predefined or periodic movements (Grimmelsmann et al. 2023; Lotti et al. 2022), the proposed framework was able to generate arbitrary random trajectories. Although the experimental setup, specifically the joints limits of our exoskeleton, prevented the full coverage of the workspace of the human elbow, the distributions of angular positions were representative of daily tasks (Haverstock et al. 2020; Chapman et al. 2019). This suggests that the presented framework is able to produce a model calibration that is valid in a workspace covering most daily activities and use cases.

Evaluating the quality of the EMG-to-torque model calibration requires well-defined metrics, that need to be systematically compared to behavioral measurements. This study utilizes Pearson's correlation coefficient and the normalized root mean square error. While both metrics lead to similar conclusions about the models, their implications differ. Pearson's correlation coefficient measures the linear relationship between the predicted and actual torques, inherently missing any nonlinear relationships. As depicted in Fig. 6, the MVLR model exhibited a nonlinear correlation with the reference torques, prompting the development of the NLMap model with a shape factor adjustment to enhance linear correlation. Conversely, the NRMSE measures the normalized difference between the estimated and actual torques, facilitating comparisons across different subjects and studies. However, it is highly sensitive to peaks in the actual torques, which can compress the normalized error and obscure discrepancies in error magnitude between low and high torque conditions. To address this issue, evaluations were conducted on trajectories that avoided high torque peaks, as confirmed in Fig. 6, showing no error variation between high and low torque for the NLMap model.

A key issue when calibrating and using EMG-to-torque models is their accuracy. We found that the NLMap model was the most accurate for both single-joint and multi-joint conditions. Given its simplicity, the MVLR model also demonstrated remarkable accuracy and should not be ignored for potential applications. Moreover, the NLMap and MVLR models do not require kinematic data as input features, making their implementation easier and necessitating fewer sensor types. The NMS model, which is implemented in widely used tools such as OpenSim or CEINMS (Pizzolato et al. 2015; Bennett et al. 2022), was surprisingly under-performing: while it had a satisfactory accuracy for single-joint estimation, it did not perform better than the MVLR in the multi-joint condition. This is likely due to the wide variety and the number of internal parameters that must be fitted, making it difficult to find a global minimum during calibration. Thus, while the NMS model should theoretically have the best accuracy due to its knowledge-based nature, this same nature mitigates the practicability of its calibration for in-field applications. The SYN models did not offer any notable advantage compared to their counterparts. Although the decrease in dimensionality may provide insight into how the central nervous system utilizes muscle redundancy during different movements, this does not appear to enhance the accuracy

of movement prediction. It is difficult to compare the accuracy of other models in the literature because the evaluation method differs. Generally, models are evaluated on a different limb, with a different metric, feature, and task, and more often than not, without any cross-validation procedure. However, Table 3 provides an overview of some model scores reported in the literature using the NMRSE metric. The NRMSE scores for single-joint and multi-joint conditions indicated that the accuracy of the estimation for a single joint (i.e. the elbow) may be reduced when a more complex movement (e.g., shoulder rotations) is included. This effect could be caused by various factors, such as bi-articular muscles and movement artifacts due to the EMG sensor cables.

While we tested the models on a powerful computer, exoskeleton controllers are typically embedded systems with limited computing power. Thus, it was important to assess the computational complexity of the models, as it played a major role in their *cost of use*. However, to our knowledge, this kind of analysis has not been done in the literature. We found that simpler models, such as MVLR and SYN, were quicker to calibrate and provided faster real-time torque estimation, resulting in negligible *calibration cost* and *evaluation cost*. Similarly, the NLMap model *calibration cost* was low, less than one minute, which is an acceptable waiting time for most applications. The *evaluation cost* was less than one millisecond, allowing for a 1 kHz exoskeleton control frequency. On the contrary, the NMS model did not demonstrate the same level of performance. The *calibration cost* ranged from 20 min to an hour, depending on the condition, and the *evaluation cost* was more than ten milliseconds. This is due to the number of parameters that need to be adjusted during calibration (five parameters per muscle, sixty in total for the multi-joint condition). Moreover, two steps are necessary for the computation of this model: muscle equilibrium and moment arm computation. The muscle equilibrium step is an iterative process that must be repeated at each time step, and for each muscle to balance the lengths of the fibers and tendons, it therefore influences computation duration both during calibration and real-time estimation. Because muscle paths and insertions are not adjusted, the time taken to calculate the moment arm does not significantly impact the calibration duration since it is only necessary to do it once. However, during real-time estimation, each muscle's moment arm must be computed for each actuated degree of freedom, impacting the *evaluation cost*. Consequently, numerical methods have been developed to accelerate this process (Sartori et al. 2012).

This study offers new insights into the estimation of joint torques from EMG signals. However, certain limitations restrict the scope of these findings. To begin with, this research did not include all existing models as it would not have been feasible: we compared models commonly encountered in the literature and developed our model with practicality in mind. However, we proposed an evaluation protocol that enables anyone to test their model against others. To that end, we provided a standardized dataset that can be used to test new models and compare them with our results (Quesada et al. 2024). Currently, replicating the visual feedback of the trajectory and the motion-capture-based
<table><caption>**Table 3. Overview of NMRSE model evaluations.** Results are reported taking the best score showcased in each paper along with their standard deviation (SD) when communicated. These results must be compared cautiously as they are reported using various evaluation methods, limbs, movements, and features. KF, LSTM, CNN, and NNMF, respectively, stand for Kalman Filter, Long-Short Term Memory, Convolutional Neural Network, and Non-negative Matrix Factorization.</caption><thead><tr><th>Ref</th><th>Type</th><th>Feature</th><th>Joint</th><th>NRMSE [%]</th><th>SD [%]</th></tr></thead><tbody><tr><td>-</td><td>NLMap</td><td>Torque</td><td>Elbow</td><td>6.50</td><td>0.73</td></tr><tr><td>-</td><td>NLMap</td><td>Torque</td><td>Shoulder (Flex/Ext)</td><td>6.87</td><td>1.4</td></tr><tr><td>Ziai and Menon (2011)</td><td>NMS</td><td>Torque</td><td>Wrist (Flex/Ext)</td><td>2.02</td><td>0.92</td></tr><tr><td>Zhang et al. (2022)</td><td>LSTM</td><td>Torque</td><td>Hip (Flex/Ext)</td><td>3.6</td><td>n.c.</td></tr><tr><td>Lu et al. (2019)</td><td>KF</td><td>Torque</td><td>Elbow</td><td>5.81</td><td>n.c.</td></tr><tr><td>Li et al. (2021)</td><td>CNN-LSTM</td><td>Torque</td><td>Elbow</td><td>12.76</td><td>5.1</td></tr><tr><td>Li et al. (2021)</td><td>CNN-LSTM</td><td>Torque</td><td>Shoulder (Flex/Ext)</td><td>12.44</td><td>2.21</td></tr><tr><td>Pan et al. (2019)</td><td>NMS</td><td>Angle</td><td>Wrist &amp; Elbow</td><td>13</td><td>n.c.</td></tr><tr><td>Li et al. (2023)</td><td>NNMF-NMS</td><td>Angle</td><td>Wrist (Pro/Sup)</td><td>17</td><td>n.c.</td></tr></tbody></table>

kinematic measurements for in-field use is challenging. Nevertheless, this visual feedback can be modified to display on wearable devices like smartphones or virtual reality headsets, and the human's angular positions can be matched with those of the exoskeleton. Note that, although it improves the range of torques requested from the user for the model calibration, our resistive viscous task could still be completed with static directional force fields and quick perturbations. This would allow to better model situations where unusual muscle pattern appear, such as short bursts of activity following an unexpected perturbation.

Importantly, even though sagittal movements encompass many potential applications of exoskeleton assistance, these results should be extended to include all the degrees of freedom of the upper limbs. Such an extension of our findings would be straightforward from the exoskeleton control and trajectory generation standpoints. It however necessitates a 3D visual feedback, which may be difficult to process by the human CNS without the implementation of detailed virtual or augmented environments. Additionally, some of the supplementary EMG signals may be harder to measure due to muscle placement. Specifically, predicting the shoulder's internal/external rotation torque implicates deep muscles not suited for surface EMG, and a similar issue can be encountered in the lower limb when generalizing beyond knee and hip flexion/extension. Intramuscular electrodes are a potential venue to mitigate this problem, as they would allow to obtain accurate measurements from deep muscles (Jung et al. 2022), although such invasive methods still pose significant ethical and practical challenges.

Furthermore, increasing the number of assisted joints would necessarily increase the *data cost* allowing to obtain a robust calibration. In the present study this *data cost* was between 25 min and 30 min for all the models, in the single- and multi-joints conditions respectively, which was because we have not addressed the question of the amount of data necessary to reach a plateau in calibration quality. Instead, we discussed the *data cost* relatively to performance for a given dataset, thereby showing that the NMS is more expensive if one wants to obtain the level of performance of the NLMap in our study. In addition, research has shown that EMG signals can be degraded through time due to changes in the electrode/skin impedance over time, affecting the accuracy of the model estimation (Ziai and Menon 2011). Consequently, sweat and skin degradation

over time could impede the validity of the calibration and the associated exoskeleton assistance in a workplace environment, potentially making it unsafe. Although current EMG sensors allow the measurement of the electrode/skin impedance, no model has yet been developed to consider it in the torque estimation. Finally, muscle fatigue may emerge as a factor impacting torque estimation accuracy over time. Fatigued muscles typically exhibit a lower mean EMG frequency (Chandra et al. 2020) that current models fail to consider. Finally, other non-ideal factors such as electrode shift and inter-day differences are known to induce accuracy variations (Zhu et al. 2022), but could not be considered in this paper.

We anticipate conducting further research based on the results of this study. First, we will explore EMG feature extraction and reducing the number of measured muscles to enhance the robustness and widespread use of EMG-to-torque models. Comparing various feature extraction methods and muscle sets will be essential for improving the information embedded in the signal. Several studies have examined feature selection for EMG-based movement classification (Tsai et al. 2014; Phinyomark et al. 2012, 2018), but to our knowledge, no comprehensive comparison of EMG features has been conducted for continuous movement regression. Also, minimizing the number of measured muscles can simplify the placement of EMG sensors, which is a time-consuming and specialized task. Second, using our dataset and evaluation protocol, we plan to investigate artificial neural network-based (ANN) models, including Long Short-Term Memory (LSTM) networks. ANN models have the potential to offer more accurate estimations while maintaining a short calibration (Li et al. 2023; Treussart et al. 2019). They can be adapted to incorporate multiple EMG features and other relevant factors, such as kinematics. Finally, we aim to assess the impact of the estimation error on exoskeleton assistance and user feedback, considering potential safety concerns related to unexpected feedback loops and system instabilities resulting from variable human behavior and signal degradation over time. Despite these challenges, real-time estimation of human torques holds significant promises for the development of versatile assistive control strategies.
## Data availability

The dataset used in this paper has been made publicly available on Zenodo: https://doi.org/10.5281/zenodo.11209324.

## Funding statement

This work was funded by the french National Research Agency (ANR-19-CE33-0009) and by the École Normale Supérieure Paris-Saclay.

## Declaration of conflicting interests

The authors declare no conflict of interest.

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