# Electronic Supplementary Materials 2 (ESM2) – Additional results and methodological details of the forehand motion capture protocol

This document is an electronic supplementary material (ESM2) for the article entitled “Electromyography-informed musculoskeletal modelling provides new insight into hand tendon forces during tennis forehand”

## Article information

Electromyography-informed musculoskeletal modelling provides new insight into hand tendon forces during tennis forehand

Goislard de Monsabert Bª*, Herbaut Aᵇ, Cartier Tª, Vigouroux Lª.

ªAix-Marseille University, CNRS, ISM, Marseille, France; ᵇMovement Sciences Department, Decathlon SportsLab Research and Development, Villeneuve d'Ascq, France

*Corresponding author: Benjamin Goislard de Monsabert, Ph.D.
[benjamin.goislard-de-monsabert@univ-amu.fr](mailto:benjamin.goislard-de-monsabert@univ-amu.fr).

## Table of contents

<table><tr><td>ARTICLE INFORMATION</td><td>1</td></tr><tr><td>TABLE OF CONTENTS</td><td>1</td></tr><tr><td>FOREHAND ADDITIONAL RESULTS</td><td>2</td></tr><tr><td>RACKET CENTRE TRAJECTORY</td><td>2</td></tr><tr><td>RACKET VELOCITY AND ACCELERATION</td><td>3</td></tr><tr><td>JOINT ANGLES</td><td>4</td></tr><tr><td>FOREHAND MOTION CAPTURE METHODOLOGICAL DETAILS</td><td>5</td></tr><tr><td>MARKER PLACEMENT</td><td>5</td></tr><tr><td>JOINT ANGLE COMPUTATION</td><td>5</td></tr><tr><td>REFERENCES</td><td>7</td></tr></table>
# Electromyography-informed musculoskeletal modeling provides new insight into hand tendon forces during tennis forehand

Benjamin Goislard de Monsabert, Alexis Herbaut, Théo Cartier, Laurent Vigouroux

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

## To cite this version:

Benjamin Goislard de Monsabert, Alexis Herbaut, Théo Cartier, Laurent Vigouroux. Electromyography-informed musculoskeletal modeling provides new insight into hand tendon forces during tennis forehand. Scandinavian Journal of Medicine and Science in Sports, 2023, 10.1111/sms.14434. hal-04136109

HAL Id: hal-04136109

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Submitted on 26 Jun 2023

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<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Electronic Supplementary Materials 1 (ESM1) – Results and methodological details of the grip calibration protocol

This document is an electronic supplementary material (ESM1) for the article entitled “Electromyography-informed musculoskeletal modelling provides new insight into hand tendon forces during tennis forehand”

## Article information

Electromyography-informed musculoskeletal modelling provides new insight into hand tendon forces during tennis forehand

Goislard de Monsabert Bª*, Herbaut Aᵇ, Cartier Tª, Vigouroux Lª.

ªAix-Marseille University, CNRS, ISM, Marseille, France; ᵇMovement Sciences Department, Decathlon SportsLab Research and Development, Villeneuve d'Ascq, France

*Corresponding author: Benjamin Goislard de Monsabert, Ph.D.
[benjamin.goislard-de-monsabert@univ-amu.fr](mailto:benjamin.goislard-de-monsabert@univ-amu.fr).

## Table of contents

<table><tr><td>ARTICLE INFORMATION</td><td>1</td></tr><tr><td>TABLE OF CONTENTS</td><td>1</td></tr><tr><td>GRIP CALIBRATION RESULTS</td><td>2</td></tr><tr><td>FDS ACTIVATION-GRIP FORCE RELATIONSHIP</td><td>2</td></tr><tr><td>GRIP FORCE DISTRIBUTION</td><td>2</td></tr><tr><td>FINGER POSTURE ON THE HANDLE</td><td>3</td></tr><tr><td>GRIP CALIBRATION METHODOLOGICAL DETAILS</td><td>3</td></tr><tr><td>MARKER PLACEMENT</td><td>3</td></tr><tr><td>JOINT ANGLE COMPUTATION</td><td>4</td></tr><tr><td>REFERENCES</td><td>5</td></tr></table>
# Electronic Supplementary Materials 3 (ESM3) – Additional results and methodological details of the musculoskeletal model

This document is an electronic supplementary material (ESM3) for the article entitled “Electromyography-informed musculoskeletal modelling provides new insight into hand tendon forces during tennis forehand”

## Article information

Electromyography-informed musculoskeletal modelling provides new insight into hand tendon forces during tennis forehand

Goislard de Monsabert Bª*, Herbaut Aᵇ, Cartier Tª, Vigouroux Lª.

ªAix-Marseille University, CNRS, ISM, Marseille, France; ᵇMovement Sciences Department, Decathlon SportsLab Research and Development, Villeneuve d'Ascq, France

*Corresponding author: Benjamin Goislard de Monsabert, Ph.D.
[benjamin.goislard-de-monsabert@univ-amu.fr](mailto:benjamin.goislard-de-monsabert@univ-amu.fr).

## Table of contents

<table><tr><td>ARTICLE INFORMATION</td><td>1</td></tr><tr><td>TABLE OF CONTENTS</td><td>1</td></tr><tr><td>MUSCULOSKELETAL MODEL ADDITIONAL RESULTS</td><td>2</td></tr><tr><td>WRIST NET MOMENT COMPONENTS</td><td>2</td></tr><tr><td>MUSCLE FORCES AND LENGTHS ESTIMATED VIA FORCE-LENGTH-ACTIVATION MODELS</td><td>3</td></tr><tr><td>MUSCLE FORCES ESTIMATED BY THE EMG-INFORMED MUSCULOSKELETAL MODEL</td><td>5</td></tr><tr><td>SENSITIVITY ANALYSIS</td><td>8</td></tr><tr><td>EMG-INFORMED MUSCULOSKELETAL MODEL METHODOLOGICAL DETAILS</td><td>10</td></tr><tr><td>FORCE-LENGTH-ACTIVATION MODEL</td><td>10</td></tr><tr><td>Step 1 - Muscle-tendon unit kinematics</td><td>10</td></tr><tr><td>Step 2 - muscle belly excursion</td><td>11</td></tr><tr><td>Step 3 - Force-length-activation relationship</td><td>13</td></tr><tr><td>Associated data</td><td>14</td></tr><tr><td>REFERENCES</td><td>15</td></tr></table>
# Forehand additional results

## Racket centre trajectory

<figure><img src="image_1.png" /><figcaption>B. View from right side of player</figcaption></figure>

<figure><img src="image_2.png" /></figure>

Figure s2-1. Racket centre trajectory from above (A) and from the right side of the player (B) for the Advanced and Intermediate player. Lateral axis is pointing toward the right side of the player (negative values correspond to the left). Frontal axis is pointing toward the target, anterior to the player (negative values correspond to posterior). Upward axis is pointing up (negative values corresponding to value below the ground).
# Electromyography-informed musculoskeletal modelling provides new insight into hand tendon forces during tennis forehand

Goislard de Monsabert, Benjamin<sup>a*</sup> ; Herbaut, Alexis<sup>b</sup> ; Cartier, Théo<sup>a</sup> ; Vigouroux, Laurent<sup>a</sup>.

<sup>a</sup>Aix-Marseille University, CNRS, ISM, Marseille, France; <sup>b</sup>Human Factors & Ergonomics Department, Decathlon SportsLab Research and Development, Lille, France

*Corresponding author:

Benjamin Goislard de Monsabert, Ph.D.

UMR 7287 CNRS & Aix-Marseille University
Faculty of Sport Science, CP 910
163 av. de Luminy
13288 Marseille cedex 09 FRANCE

E-mail: benjamin.goislard-de-monsabert@univ-amu.fr.

## Section

Biomechanics and Motor Control
Grip calibration results

FDS activation-grip force relationship

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

Figure s1-1. Experimental data fitting of the EMG-grip force relationship (first column) and comparison between experimental and estimated grip force for three representative examples (second-fourth columns) of the Advanced (first row) and Intermediate (Second row) player. Fgrip: Grip force; α<sup>FDS</sup>: flexor digitorum superficialis (FDS) muscle activation; MGF: maximal grip force. See article for detailed explanation of the methodology.

Grip force distribution

<figure><img src="image_2.png" /></figure>

Figure s1-2. Normalised data from pressure map sensor (left panels) and grip force distribution across 25 anatomical areas (right panels) for the Advanced (upper panels) and Intermediate (lower panels) player. See article for detailed explanation of the methodology.
# Musculoskeletal model additional results

## Wrist net moment components

<figure><img src="image_1.png" /><figcaption>Figure s3-1. Mean time-patterns (N=5 trials) of wrist net moment components during the forehand drive of both players (Advanced and Intermediate) for the two shot speeds (Slow, s; Fast, s) performed with the racket R2. Shaded areas represent ± one standard deviations around the mean. Lighter nuances represent the Slow (s) speed shots and darker nuances the Fast (f) ones. The vertical solid black bar represents the impact frame, the dashed vertical bars represent the beginning and end of forward acceleration phase, and the different points correspond to the frames on which musculoskeletal model was ran (see corpus of the article).
Ang: angular acceleration moment component; Lin: Linear acceleration moment component; Grav: gravitation moment component; Net: sum of three components</figcaption></figure>
Racket velocity and acceleration

<figure><img src="image_3.png" /></figure>

Figure s2-2. Mean time-patterns (N=5 trials) of racket velocities and accelerations during the forehand drive of both players (Advanced and Intermediate) for the two shot speeds (Slow, s; Fast, s) performed with the three rackets (R1, R2, R2w). Shaded areas represent ± one standard deviations around the mean. Lighter nuances represent the Slow (s) speed shots and darker nuances the Fast (f) ones. The vertical solid black bar represents the impact frame, the dashed vertical bars represent the beginning and end of forward acceleration phase, and the different points correspond to the frames on which musculoskeletal model was ran (see corpus of the article). $V^{\text{rack-front}}$: racket centre velocity along the frontal axis (pointing toward the target). $A^{\text{rack-front}}$: racket centre acceleration along the frontal axis. $V^{\text{rack-vert}}$: racket centre velocity along the vertical axis (pointing up). $A^{\text{rack-vert}}$: racket centre acceleration along the vertical axis. Racket centre is computed as the midpoint between lateral markers RK9D and RK9U (see details below).
# Electromyography-informed musculoskeletal modelling provides new insight into hand tendon forces during tennis forehand drive

## Abstract

Lateral epicondylitis, also known as tennis elbow, is a major health issue among tennis players. This musculoskeletal disorder affects hand extensor tendons, results in substantial pain and impairments for sporting and everyday activities and requires several weeks of recovery. Unfortunately, prevention remains limited by the lack of data regarding biomechanical risk factors, especially because in vivo evaluation of hand tendon forces currently remains challenging. Electromyography-informed musculoskeletal modelling is a non-invasive approach to provide physiological estimation of tendon forces based on motion capture and electromyography but was never applied to study hand tendon loading during tennis playing. The objective of this study was to develop such electromyography-informed musculoskeletal model to provide new insight into hand tendon loading in tennis players. The model was tested with three-dimensional kinematics and electromyography data of two players performing forehand drives at two shot speeds and with three rackets. Muscle forces increased with shot speed but were moderately affected by racket properties. Wrist prime extensors withstood among the highest forces, but their relative implication compared to flexors depended on the player-specific grip force and racket motion strategy. When normalising wrist extensor forces by shot speed and grip strength, up to three-fold differences were observed between players, suggesting that gesture technique, e.g., grip position or joint motion coordination, could play a role in the overloading of wrist extensor tendons. This study provided a new methodology for in situ analysis of hand biomechanical loadings during tennis gesture and shed a new light on lateral epicondylitis risk factors.

**Keywords:** Tennis; lateral epicondylitis; muscle force; musculoskeletal model; motion capture; electromyography
## Finger posture on the handle

<figure><img src="image_3.png" /><figcaption>Figure s1-3. Marker 3D coordinates (left panels) and finger joint angles (right panels) for the Advanced (upper panels) and Intermediate (lower panels) player. See article and section below for detailed explanation of the methodology.</figcaption></figure>

## Grip calibration methodological details

### Marker placement

Grip calibration methodological details

Grip calibration methodological details

Finger segment kinematics were tracked using a motion capture system consisting of seven cameras (MX T40, Vicon, Oxford, UK, 100Hz) tracking 29 reflective hemispheric markers with 6-mm radius placed on the dorsal aspects of the forearm and the hand (Figure s1-4 and Table s1-1). The radius was tracked by three markers placed on the dorsal aspect of the distal forearm. The first metacarpal and the thumb proximal phalanx were tracked by three markers on the thumb dorsal aspect including one placed on a plastic frame. Each long finger metacarpal was tracked by two markers placed on the dorsal aspects of hand. For each long finger, three markers were placed to track the phalanges motion.
Muscle forces and lengths estimated via Force-Length-Activation models

The following two graphs present the data estimated via the muscle-specific Force-Length-Activation models from joint angle and muscle activation used to provide EMG-informed guidance to the inverse-dynamics musculoskeletal model (see corpus). The model details can be found at the end of this document.

<figure><img src="image_2.png" /></figure>

Figure s3-2. Mean time-patterns (N=5 trials) of muscle forces estimated using the muscle-specific Force-Length-Activation models during the forehand drive of both players (Advanced and Intermediate) for the two shot speeds (Slow, s; Fast, f) performed with the three rackets (R1, R2, R2w). Shaded areas represent ± one standard deviations around the mean. Lighter nuances represent the Slow (s) speed shots and darker nuances the Fast (f) ones. The vertical solid black bar represents the impact frame, the dashed vertical bars represent the beginning and end of forward acceleration phase, and the different points correspond to the frames on which musculoskeletal model was ran (see corpus of the article). ECR: extensor carpi radialis, EDC: extensor digitorum communis, FCR: flexor carpi radialis.
Joint angles

<figure><img src="image_4.png" /></figure>

Figure s2-3. Mean time-patterns (N=5 trials) of racket velocities and accelerations during the forehand drive of both players (Advanced and Intermediate) for the two shot speeds (Slow, s; Fast, s) performed with the three rackets (R1, R2, R2w). Shaded areas represent ± one standard deviations around the mean. Lighter nuances represent the Slow (s) speed shots and darker nuances the Fast (f) ones. The vertical solid black bar represents the impact frame, the dashed vertical bars represent the beginning and end of forward acceleration phase, and the different points correspond to the frames on which musculoskeletal model was ran (see corpus of the article). θ<sup>WR-fe</sup>: wrist flexion-extension, positive in flexion. θ<sup>WR-rud</sup>: wrist radial-ulnar deviation, positive in ulnar deviation. θ<sup>FA-ps</sup>: forearm pronation-supination, positive in supination. θ<sup>ELB-fe</sup>: elbow flexion-extension, positive in flexion. See below for details on joint angle computation. Details on joint angle computation are provided below
Introduction

Tennis involves both highly dynamic and accurate gestures to intercept and hit the ball to reach a specific target on the opponent’s field. This sport places high loads on the joints and muscles of the players that are repeated hundreds of times during each match, hence exposing players to injuries. While the lower limbs of tennis players are affected by acute injuries, e.g., ankle sprain, their upper limb are more affected by chronic overuse syndromes, with lateral epicondylitis (LE), also known as tennis elbow, being the most prevalent affections¹. LE is affecting the tendon of wrist radial extensors, i.e., the extensor carpi radialis brevis (ECRB) and longus (ERCL), as well as in some cases those of finger extensors, i.e., extensor digitorum communis (EDC)². The pathology incidence is around 30-50% among tennis players¹ and results in substantial pain and impairments for everyday life³,⁴ as well as modified muscle coordination during gripping⁵. Understanding the factors inducing LE thus appears crucial to reduce injuries.

The mechanism of LE has been associated to repetitive microtrauma of the tendons progressively initiating an angio-fibroblastic degeneration⁶. Although LE is multifactorial, biomechanical risk factors, such as awkward joint angulations and high tendon tensile forces, are believed to induce tendon damage outside of its optimal physiological re-modelling capacities and accelerate degenerative processes⁴,⁷,⁸. One of the central mechanism of LE is that hand extensors tendons are indeed highly involved during gripping activities because the wrist joint balance requires high co-contraction levels⁹⁻¹¹. Nevertheless, the data regarding hand biomechanical loading during tennis playing remain scarce. The main reason is that the *in vivo* evaluation of tendon forces currently remains challenging for biomechanician since direct measures are highly invasive. Furthermore, indirect measurements, such as forces at the hand/handle interface and joint kinetics, are challenging to obtain in complex motions as tennis
<figure><img src="image_4.png" /><figcaption>Figure s1-4. Illustration of the marker placement. See Table s1.1 for exact placement.</figcaption></figure>

<table><caption>Table s1-1. Name and locations of the markers.</caption><thead><tr><th>Name</th><th>Location</th></tr></thead><tbody><tr><td><strong>Forearm</strong></td><td></td></tr><tr><td>US</td><td>Most distal-dorsal point on ulnar styloid</td></tr><tr><td>RS</td><td>Most distal-dorsal point on radial styloid</td></tr><tr><td>FA</td><td>Three fingerbreaths in the proximal direction from RS along the radius</td></tr><tr><td><strong>Thumb</strong></td><td></td></tr><tr><td>MC1b</td><td>Most proximal point of the 1<sup>st</sup> metacarpal dorsal midline while flexing trapeziometacarpal joint</td></tr><tr><td>MC1h</td><td>Most distal point on the 1<sup>st</sup> metacarpal dorsal midline while flexing metarcapophalangeal joint</td></tr><tr><td>MC1s</td><td>1.5cm in the ulnar direction from the axis defined by MC1h and MC1b</td></tr><tr><td>PP1h</td><td>Most distal point of the proximal phalanx head while flexing interphalangeal joint</td></tr><tr><td>PP1s</td><td>1.5cm in the ulnar direction from the axis defined by PP1h and MC1h</td></tr><tr><td>DP1h</td><td>Most distal point on the dorsal aspect of the nail.</td></tr><tr><td><strong>Fingers</strong></td><td></td></tr><tr><td>MCib</td><td>Three fingerbreaths in the proximal direction from MC1h along the i<sup>th</sup> metacarpal</td></tr><tr><td>MCih</td><td>Most distal point on the i<sup>th</sup> metacarpal head dorsal aspect while flexing metacarpophalangeal joint</td></tr><tr><td>PPih</td><td>Most distal point on the i<sup>th</sup> finger proximal phalanx head dorsal midline while flexing proximal interphalangeal joint</td></tr><tr><td>MPih</td><td>Most distal point on the i<sup>th</sup> finger middle phalanx head dorsal midline while flexing distal interphalangeal joint</td></tr><tr><td>DPih</td><td>Most distal point on the dorsal aspect of the i<sup>th</sup> finger distal phalanx</td></tr></tbody></table>

## Joint angle computation

The posture of the fingers on the handle during forehand drives was considered fixed during the motion and finger joint angles were calculated from the position of reflective markers during grip calibration protocol using a previously developed method (Goislard de Monsabert et al., 2012). The marker coordinates were averaged on 100 frames during a grip calibration trial to determine joint angles. The 100-frame window was chosen during a grip force exertion period such that all markers were visible. The trapeziometacarpal (TMC) and metacarpophalangeal (MCP) joints were described by two DoF in flexion/extension and radial/ulnar deviation and the proximal (PIP) and distal (DIP) joints by one DoF in flexion/extension. For each joint, a distal and a
<figure><img src="image_3.png" /></figure>

Figure s3-3. Mean time-patterns (N=5 trials) of muscle lengths estimated using the muscle-specific Force-Length-Activation models during the forehand drive of both players (Advanced and Intermediate) for the two shot speeds (Slow, s; Fast, f) performed with the three rackets (R1, R2, R2w). Shaded areas represent ± one standard deviations around the mean. Lighter nuances represent the Slow (s) speed shots and darker nuances the Fast (f) ones. The vertical solid black bar represents the impact frame, the dashed vertical bars represent the beginning and end of forward acceleration phase, and the different points correspond to the frames on which musculoskeletal model was ran (see corpus of the article). ECR: extensor carpi radialis, EDC: extensor digitorum communis.
# Forehand motion capture methodological details

## Marker placement

Upper limb segment kinematics were assessed using a seven-camera motion capture system (MX T40, Vicon, Oxford, UK) tracking 20 reflective markers with a 14-mm diameter that were placed on the right upper limb. The arm was tracked by three markers placed on the acromion and both epicondyles as well as a supplementary one placed on the lateral aspect to control the internal rotation of the shoulder. The forearm was tracked with three markers placed on styloid and on the distal part of the radius. The hand was tracked by four markers placed on the dorsal aspect of metacarpals. The racket was tracked by 6 markers placed all around the frame.

<figure><img src="image_5.png" /><figcaption>Figure s2-4. Illustration of the marker placement. See Table s1.1 for exact placement.</figcaption></figure>

## Joint angle computation

The upper limb joint angles were estimated from the filtered marker coordinates on the impact-centred window. For each trial, a 1.5-s analysis window centred around the ball impact time frame was first determined and kept for further analysis. The
gestures and without disturbing the player. Consequently, hand muscle/tendon loading has mainly been inferred from either grip force, wrist net moment, or electromyography (EMG) recordings.

Grip force was assessed either by using pressure sensors equipped on the handle<sup>12</sup> or the hand<sup>13</sup> or by replacing the handle with a dynamometer<sup>14</sup>. Those studies observed a single-peak curve with the highest values, around 200-300N, occurring just before the impact. The instrumentation however required to equip the handle with sensors altering the hand/racket interface thus potentially affecting the player's natural gesture. Removing force sensors is possible by using EMG to estimate grip force<sup>15-17</sup> and was used to study the role of grip force in modulating shock vibration transmission<sup>18</sup>. Despite representing crucial information on the forces at the hand/handle interface, the grip force is the result of the action of over 30 muscles which implication also depends on the mechanical demands at the wrist<sup>19</sup>.

Net joint moments, reflecting mechanical load balanced by muscles, can be quantified via inverse dynamics approaches relying on accurate measurement of segment kinematics and inertial parameters of body segment. Such approaches were used to study the upper limb during tennis playing and a few studies provided values at the wrist during serve<sup>20-22</sup> and during the forehand<sup>12,23</sup>. Those studies showed that both flexion-extension (FE) and radial-ulnar deviation (RUD) are facing high torques with values around 5-15 Nm and reaching sometimes 30 Nm, thus similar to maximal isometric contraction performances<sup>24</sup>. Authors also showed that the distribution of the wrist torque between the two degrees of freedom can vary importantly according to the player's individual technique, with for instance larger RUD torques in professional players and larger FE torques in intermediate ones<sup>23</sup>. Although the net moment traduces
proximal segment coordinate system were calculated from the marker coordinates
(Table s1.2).

<table><caption>Table s1-2 – Equations used to compute axes of segment coordinate systems</caption><thead><tr><th></th><th>1<sup>st</sup> axis</th><th>2<sup>nd</sup> axis</th><th>3<sup>rd</sup> axis</th></tr></thead><tbody><tr><td>Radius</td><td>x&#773; = u&#773;<sub>(RS,FaR)</sub></td><td>y&#773; = u&#773;<sub>(US,RS)</sub> &times; x&#773;</td><td>z&#773; = x&#773; &times; y&#773;</td></tr><tr><td>1<sup>st</sup> metacarpal</td><td>x&#773; = u&#773;<sub>(MC1h,MC1b)</sub></td><td>y&#773; = u&#773;<sub>(MC1s,MC1h)</sub> &times; x&#773;</td><td>z&#773; = x&#773; &times; y&#773;</td></tr><tr><td>2<sup>nd</sup> Metacarpal</td><td>x&#773; = u&#773;<sub>(MC2h,MC2b)</sub></td><td>y&#773; = u&#773;<sub>(MC3h,MC2h)</sub> &times; x&#773;</td><td>z&#773; = x&#773; &times; y&#773;</td></tr><tr><td>3<sup>rd</sup> Metacarpal</td><td>x&#773; = u&#773;<sub>(MC3h,MC3b)</sub></td><td>y&#773; = u&#773;<sub>(MC3h,MC2h)</sub> &times; x&#773;</td><td>z&#773; = x&#773; &times; y&#773;</td></tr><tr><td>4<sup>th</sup> Metacarpal</td><td>x&#773; = u&#773;<sub>(MC4h,MC4b)</sub></td><td>y&#773; = u&#773;<sub>(MC4h,MC3h)</sub> &times; x&#773;</td><td>z&#773; = x&#773; &times; y&#773;</td></tr><tr><td>5<sup>th</sup> Metacarpal</td><td>x&#773; = u&#773;<sub>(MC3h,MC3b)</sub></td><td>y&#773; = u&#773;<sub>(MC5h,MC4h)</sub> &times; x&#773;</td><td>z&#773; = x&#773; &times; y&#773;</td></tr><tr><td>Trapezium</td><td>x&#773; = <sup>TRA</sup>R<sub>MC3</sub>x&#773;<sub>MC3</sub></td><td>y&#773; = <sup>TRA</sup>R<sub>MC3</sub>y&#773;<sub>MC3</sub></td><td>z&#773; = x&#773; &times; y&#773;</td></tr><tr><td>Thumb proximal phalanx</td><td>x&#773; = u&#773;<sub>(PP1h,MC1h)</sub></td><td>y&#773; = u&#773;<sub>(PP1s,PP1h)</sub> &times; x&#773;</td><td>z&#773; = x&#773; &times; y&#773;</td></tr><tr><td>Phalanges</td><td>x&#773; = u&#773;<sub>(Pp,Pd)</sub></td><td>y&#773; = v&#773;<sub>prox</sub> &times; x&#773;</td><td>z&#773; = x&#773; &times; y&#773;</td></tr></tbody></table>

u&#773;<sub>(P1,P2)</sub> represents the unit vector associated to the vector going from point P1 to P2. For the phalanges, Pp and Pd refer to the points representing the proximal and distal ends of the segment, respectively, and v&#773;<sub>prox</sub> refers to sagittal axis of the proximal segment. <sup>TRA</sup>R<sub>MC3</sub> refers to the rotation matrix from the third metacarpal to the trapezium defined from the work of Cooney et al. (1981).

For all segments, three axes were computed with first the longitudinal (x&#773;) axis as the unit vector from the distal to the proximal marker, thus pointing in the proximal direction. The second axis was the coronal (y&#773;), pointing in the dorsal direction, and was orthogonal to the plane defined by x&#773; and a vector pointing radially, either built with a third marker or from the sagittal axis of the proximal vector. The third axis was the sagittal (z&#773;) and was the cross product of the two others. For the trapezium, the segment coordinate system was computed from the 3rd metacarpal segment coordinate system using a fixed rotation matrix defined from the work of Cooney et al. (1981) :

$$TRA R_{MC3} = R(\vec{z}, 46^{\circ}) R(\vec{y}, 35^{\circ}) R(\vec{x}, 82^{\circ})$$

Then, joint angles were extracted from the relative orientation (distal to proximal) matrix segment rotation matrix using a Z-Y-X (flexion/abduction/pronation) sequence of Cardan angles. Flexion, and radial deviation angle were considered positive.

## References

Cooney, W., Lucca, M., Chao, E., & Linscheid, R. (1981). The kinesiology of the thumb trapeziometacarpal joint. *Journal of Bone and Joint Surgery, American Volume, 63*(9), 1371-1381. https://doi.org/PMID:7320028
Muscle forces estimated by the EMG-informed musculoskeletal model

The two figures on the next pages present all the 42 individual muscle forces estimated by the EMG-informed musculoskeletal model. The six rows of figures represent the muscle forces at the thumb, at the index, middle, ring and little fingers and at the wrist, respectively. The muscle abbreviations are detailed in the table below.

<table><caption>Table s3-1 – List of muscles included in the EMG-informed musculoskeletal model, their abbreviation, and their corresponding muscle group</caption><thead><tr><th>Abb.</th><th>Definition</th><th>Type</th><th>Group</th></tr></thead><tbody><tr><td colspan="4"><strong>Thumb</strong></td></tr><tr><td>FPL</td><td>Flexor pollicis longus</td><td>Extrinsic</td><td>-</td></tr><tr><td>FPB</td><td>Flexor pollicis brevis</td><td>Intrinsic</td><td>-</td></tr><tr><td>OPP</td><td>Opponens pollicis</td><td>Intrinsic</td><td>-</td></tr><tr><td>APB</td><td>Abductor pollicis brevis</td><td>Intrinsic</td><td>-</td></tr><tr><td>ADPt</td><td>Aductor pollicis transverse head</td><td>Intrinsic</td><td>-</td></tr><tr><td>ADPo</td><td>Aductor pollicis oblique head</td><td>Intrinsic</td><td>-</td></tr><tr><td>APL</td><td>Abductor pollicis longus</td><td>Extrinsic</td><td>-</td></tr><tr><td>EPL</td><td>Extensor pollicis longus</td><td>Extrinsic</td><td>-</td></tr><tr><td>EPB</td><td>Extensor pollicis brevis</td><td>Extrinsic</td><td>-</td></tr><tr><td colspan="4"><strong>Index</strong></td></tr><tr><td>FDP2</td><td>Flexor digitorum profundus index finger compartment</td><td>Extrinsic</td><td>F-flex</td></tr><tr><td>FDS2</td><td>Flexor digitorum superficialis index finger compartment</td><td>Extrinsic</td><td>F-flex</td></tr><tr><td>LU2</td><td>Lumbrical of index finger (1<sup>st</sup> lumbrical)</td><td>Intrinsic</td><td>-</td></tr><tr><td>RI2</td><td>Radial interosseus of index finger (1<sup>st</sup> dorsal interosseus)</td><td>Intrinsic</td><td>-</td></tr><tr><td>UI2</td><td>Ulnar interosseous of index finger (1<sup>st</sup> palmar interosseus)</td><td>Intrinsic</td><td>-</td></tr><tr><td>EDC2</td><td>Extensor digitorum communis index finger compartment</td><td>Extrinsic</td><td>F-ext</td></tr><tr><td>EDI</td><td>Extensor indicis</td><td>Extrinsic</td><td>-</td></tr><tr><td colspan="4"><strong>Middle</strong></td></tr><tr><td>FDP3</td><td>Flexor digitorum profundus middle finger compartment</td><td>Extrinsic</td><td>F-flex</td></tr><tr><td>FDS3</td><td>Flexor digitorum superficialis middle finger compartment</td><td>Extrinsic</td><td>F-flex</td></tr><tr><td>LU3</td><td>Lumbrical of middle finger (2<sup>nd</sup> lumbrical)</td><td>Intrinsic</td><td>-</td></tr><tr><td>RI3</td><td>Radial interosseus of middle finger (2<sup>nd</sup> dorsal interosseus)</td><td>Intrinsic</td><td>-</td></tr><tr><td>UI3</td><td>Ulnar interosseous of middle finger (3<sup>rd</sup> dorsal interosseus)</td><td>Intrinsic</td><td>-</td></tr><tr><td>EDC3</td><td>Extensor digitorum communis middle finger compartment</td><td>Extrinsic</td><td>F-ext</td></tr><tr><td colspan="4"><strong>Ring</strong></td></tr><tr><td>FDP4</td><td>Flexor digitorum profundus ring finger compartment</td><td>Extrinsic</td><td>F-flex</td></tr><tr><td>FDS4</td><td>Flexor digitorum superficialis ring finger compartment</td><td>Extrinsic</td><td>F-flex</td></tr><tr><td>LU4</td><td>Lumbrical of ring finger (3<sup>rd</sup> lumbrical)</td><td>Intrinsic</td><td>-</td></tr><tr><td>RI4</td><td>Radial interosseus of ring finger (2<sup>nd</sup> palmar interosseus)</td><td>Intrinsic</td><td>-</td></tr><tr><td>UI4</td><td>Ulnar interosseous of ring finger (4<sup>th</sup> dorsal interosseus)</td><td>Intrinsic</td><td>-</td></tr><tr><td>EDC4</td><td>Extensor digitorum communis ring finger compartment</td><td>Extrinsic</td><td>F-ext</td></tr><tr><td colspan="4"><strong>Little</strong></td></tr><tr><td>FDP5</td><td>Flexor digitorum profundus little finger compartment</td><td>Extrinsic</td><td>F-flex</td></tr><tr><td>FDS5</td><td>Flexor digitorum superficialis little finger compartment</td><td>Extrinsic</td><td>F-flex</td></tr><tr><td>FDQ</td><td>Flexor digit quinti</td><td>Intrinsic</td><td>-</td></tr><tr><td>LU5</td><td>Lumbrical of little finger (4<sup>th</sup> lumbrical)</td><td>Intrinsic</td><td>-</td></tr><tr><td>RI5</td><td>Radial interosseus of little finger (3<sup>rd</sup> palmar interosseus)</td><td>Intrinsic</td><td>-</td></tr><tr><td>UI5</td><td>Ulnar interosseous of little finger (abductor digiti quinti)</td><td>Intrinsic</td><td>-</td></tr><tr><td>EDC5</td><td>Extensor digitorum communis little finger compartment</td><td>Extrinsic</td><td>F-ext</td></tr><tr><td>EDQ</td><td>Extensor digit quinti</td><td>Extrinsic</td><td>-</td></tr><tr><td colspan="4"><strong>Wrist</strong></td></tr><tr><td>FCR</td><td>Flexor carpi radialis</td><td>Wrist mover</td><td>W-flex</td></tr><tr><td>FCU</td><td>Flexor carpi ulnaris</td><td>Wrist mover</td><td>W-flex</td></tr><tr><td>PL</td><td>Palmaris longus</td><td>Wrist mover</td><td>W-flex</td></tr><tr><td>ECRB</td><td>Extensor carpi radialis brevis</td><td>Wrist mover</td><td>W-ext</td></tr><tr><td>ERCL</td><td>Extensor carpi radialis longus</td><td>Wrist mover</td><td>W-ext</td></tr><tr><td>ECU</td><td>Extensor carpi ulnaris</td><td>Wrist mover</td><td>W-ext</td></tr></tbody></table>
impact frame was considered as the minimal 3D relative distance between the ball and the racket centre, calculated as the mean of the two side markers, using raw coordinates. The marker coordinates were low-pass filtered (Butterworth, 5Hz, order 2, zero-phase). At each time frame, segment coordinate systems were determined from filtered marker coordinates (Table s1.2).

<table><caption>Table s2-1. Name and locations of the markers used in the kinematic process.</caption><thead><tr><th>Name</th><th>Location</th></tr></thead><tbody><tr><td><strong>Arm</strong></td><td></td></tr><tr><td>AC</td><td>Most dorsal point on the acromion prominence</td></tr><tr><td>LE</td><td>Most distal point on the lateral epicondyle</td></tr><tr><td>ME</td><td>Most distal point on the medial epicondyle</td></tr><tr><td><strong>Forearm</strong></td><td></td></tr><tr><td>US</td><td>Most distal point on the ulnar styloid dorsal aspect</td></tr><tr><td>RS</td><td>Most distal point on the radial styloid dorsal aspect</td></tr><tr><td>FA</td><td>Three fingerbreaths in the proximal direction from RS along the radius</td></tr><tr><td><strong>Hand</strong></td><td></td></tr><tr><td>MC3h</td><td>Most distal point on the 3<sup>rd</sup> metacarpal head dorsal aspect while flexing metacarpophalangeal joint</td></tr><tr><td>MC3b</td><td>Three fingerbreaths in the proximal direction from MCih along the i<sup>th</sup> metacarpal</td></tr><tr><td>MC2h</td><td>Most distal point on the 2<sup>nd</sup> metacarpal head dorsal aspect while flexing metacarpophalangeal joint</td></tr><tr><td>MC5h</td><td>Most distal point on the 5<sup>th</sup> metacarpal head dorsal aspect while flexing metacarpophalangeal joint</td></tr><tr><td><strong>Racket</strong></td><td></td></tr><tr><td>RKH</td><td>Between the 8<sup>th</sup> and 9<sup>th</sup> main (longitudinal) string on the racket top</td></tr><tr><td>RK9U</td><td>At the level of the 9<sup>th</sup> cross string on the side pointing down (floor) when the racket is at the impact while the Yoke marker pointing posterior</td></tr><tr><td>RK9D</td><td>At the level of the 9<sup>th</sup> cross string on the side pointing up (ceiling) when the racket is at the impact while the Yoke marker pointing posterior</td></tr></tbody></table>

<table><caption>Table s2-2 – Equations used to compute axes of segment coordinate systems</caption><thead><tr><th></th><th>1<sup>st</sup> axis</th><th>2<sup>nd</sup> axis</th><th>3<sup>rd</sup> axis</th><th>Origin</th></tr></thead><tbody><tr><td><strong>Elbow</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Humerus</td><td>ŷ = ū<sub>(MidE,AC)</sub></td><td>x̂ = ŷ × ū<sub>(ME,LE)</sub></td><td>ẑ = x̂ × ŷ</td><td>MidE=0.5(LE+ME)</td></tr><tr><td>Forearm</td><td>ŷ = ū<sub>(US,ME)</sub></td><td>x̂ = ŷ × ū<sub>(US,RS)</sub></td><td>ẑ = x̂ × ŷ</td><td>US</td></tr><tr><td><strong>Forearm</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Ulna</td><td>ŷ = ū<sub>(US,MidE)</sub></td><td>x̂ = ŷ × ū<sub>(ME,LE)</sub></td><td>ẑ = x̂ × ŷ</td><td>US</td></tr><tr><td>RadiusFa</td><td>ŷ = ū<sub>(RS,LE)</sub></td><td>x̂ = ŷ × ū<sub>(US,RS)</sub></td><td>ẑ = x̂ × ŷ</td><td>RS</td></tr><tr><td><strong>Wrist</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Radius</td><td>ŷ = ū<sub>(RS,FA)</sub></td><td>x̂ = ŷ × ū<sub>(US,RS)</sub></td><td>ẑ = x̂ × ŷ</td><td>RS</td></tr><tr><td>Hand</td><td>ŷ = ū<sub>(MC3h,MC3b)</sub></td><td>x̂ = ŷ × ū<sub>(MC2h,MC3h)</sub></td><td>ẑ = x̂ × ŷ</td><td>MC3h</td></tr><tr><td><strong>Syst. Dynamics</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Hand-Racket</td><td>ŷ = ū<sub>(MC2h,MC3b)</sub></td><td>x̂ = ŷ × ū<sub>(MC2h,MC5h)</sub></td><td>ẑ = x̂ × ŷ</td><td>MC3h</td></tr></tbody></table>

ū<sub>(P1,P2)</sub> represents the unit vector associated to the vector going from point P1 to P2. Radius1 and Radius2 are two different coordinate systems used for computation of pronation-supination and wrist joint angles, respectively. Hand-Racket corresponds to the coordinate system used to compute the acceleration of the hand-racket system in the global coordinate system.

For all segments, three axes were computed with first the longitudinal (ŷ) axis as the unit vector from the distal to the proximal marker, thus pointing in the proximal direction. The second axis was the transverse (x̂) axis, pointing in the dorsal direction,
the overall mechanical demand at the joint, it does not allow to characterize how forces
are shared across individual muscles.

EMG analyses during tennis playing demonstrated wrist extensors are among
the most implicated forearm muscles during both backhand²⁵⁻²⁹ and forehand³⁰⁻³³
drives. This high activity was observed around the impact and corresponded to the need
for muscle actions to balance the increase in wrist mechanical loading associated to both
variations of racket dynamics and player’s grip force, necessary to control racket
orientation and absorb ball/racket impact vibrations. Nevertheless, EMG is not linearly
related to the tendon forces and hence cannot fully explain injury mechanisms.

Although previous works informed on grip forces, net joint moment and muscle
electrical activity during tennis playing, understanding the biomechanical risks of
tendinopathies such as LE\_requires an approach that allows characterizing forces
withstood by hand and wrist muscle-tendon units. Musculoskeletal models represent the
only non-invasive method to quantify internal mechanics in complex musculoskeletal
systems, such as the hand. Some models rely on a forward-dynamics approach and
estimate muscle force from EMG using muscle contraction models. Only one study
used such technique to estimate ECRB mechanics during tennis gesture to investigate
the influence of skill level³⁴ but the model was derived from a thumb muscle, whereas
hand/forearm muscles present specific mechanical behaviours³⁵,³⁶. Other models rely on
an inverse-dynamics approach and estimate muscle loading from kinematics and
external force data using multibody rigid mechanics and optimization technique³⁷. Our
group used such an approach to estimate hand extensor loading during tennis forehand
and showed that racket handle size might represent a way to modulate this loading to
potentially reduce the risk of LE¹². Nevertheless, the muscle force estimation was only
based on kinematics or kinetics data while it has been shown that guiding the muscle
Goislard de Monsabert, B., Rossi, J., Berton, E., & Vigouroux, L. (2012).

Quantification of hand and forearm muscle forces during a maximal power grip task. *Medicine and science in sports and exercise, 44*(10), 1906-1916.

https://doi.org/10.1249/MSS.0b013e31825d9612
<figure><img src="image_4.png" /><figcaption>Figure s3-4. Mean time-patterns (N=5 trials) of muscle forces estimated by the EMG-informed musculoskeletal during the forehand drive for the Advanced player for the two shot speeds (Slow, s; Fast, f) performed with the three rackets (R1, R2, R2w). Shaded areas represent ± one standard deviations around the mean. Lighter nuances represent the Slow (s) speed shots and darker nuances the Fast (f) ones. The vertical solid black bar represents the impact frame, the dashed vertical bars represent the beginning and end of forward acceleration phase, and the different points correspond to the frames on which musculoskeletal model was ran (see corpus of the article). Muscle abbreviations are defined in the table above</figcaption></figure>
and was orthogonal to the plane defined by $\vec{x}$ and a vector pointing radially. The third
axis was the sagittal ($\vec{z}$) and was the cross product of the two others. The definition for
the Humerus, Forearm, Ulna and Radius1 to compute elbow and prono-supination were
taken from Wu *et al*. (2005)

The joint angles were extracted from relative orientation matrices using Cardan
sequences. The elbow rotations were extracted from the orientation of the Forearm
segment relatively to the Humerus with a z-x-y (flexion-rotation-abduction) sequence
but only the flexion/extension joint angle was kept. The forearm pronation-supination
rotation was extracted from the orientation of the RadiusFa segment relatively to the
Ulna using a x-z-y (abduction-flexion-pronation) sequence. The wrist rotations were
extracted from the orientation of the Radius segment relative to the Hand using a z-y-x
(flexion-rotation-deviation) and both flexion-extension and radial-deviation were kept.

References

Wu, G., van der Helm, F. C. T., Veeger, H. E. J. D., Makhsous, M., Van Roy, P.,
Anglin, C., Nagels, J., Karduna, A. R., McQuade, K., Wang, X., Werner, F. W.,
& Buchholz, B. (2005). ISB recommendation on definitions of joint coordinate
systems of various joints for the reporting of human joint motion--Part II :
Shoulder, elbow, wrist and hand. *Journal of Biomechanics*, *38*(5), 981-992.
load sharing optimisation using EMG data provides a better estimation of muscle co-
contraction³⁸,³⁹. The development of such EMG-informed models appears crucial to
quantify internal loading during tennis gestures and provide new insight on the potential
risk of musculoskeletal disorders such as LE.

The objective of this study was thus to propose a method to characterise hand
biomechanical loading and estimate hand tendon forces during forehand drives using an
EMG-informed musculoskeletal model. A previously developed multibody rigid model
of the hand¹⁰ was adapted to study dynamical tasks and to include an EMG guidance at
two levels: grip force estimation and muscle load sharing optimisation. The grip force
during tennis gestures was estimated from the EMG signal of the *flexor digitorum superficialis* (FDS) based on a player-specific calibration avoiding instrumentation of
the racket and minimising perturbation of player gesture. The EMG guidance in the
muscle load sharing optimisation consisted in a force-tracking constraint for three
muscles (ECRB, EDC and *flexor carpi radialis*: FCR) using previously developed
force-length-activation relationships³⁵,³⁶. The method was evaluated using motion
capture and EMG data measured of two players with different levels and grip position
performing forehand drives at two shot speeds and with three rackets.

**Methods**

Participants

Two right-handed men with similar anthropometrics volunteered for the study:
an Advanced player (ITN 3; Age: 19y; Height 185cm: Weight: 64kg Hand length:
19.7cm) and an Intermediate player (ITN 7; Age: 19y; Height: 183cm; Weight: 73kg
Hand length: 20.4cm). The two players used different grip position during forehand,
meaning the palmar side of their second metacarpal head was in contact with a different
<figure><img src="image_5.png" /></figure>

Figure s3-5. Mean time-patterns (N=5 trials) of muscle forces estimated by the EMG-informed musculoskeletal during the forehand drive for the Intermediate player for the two shot speeds (Slow, s; Fast, f) performed with the three rackets (R1, R2, R2w). Shaded areas represent ± one standard deviations around the mean. Lighter nuances represent the Slow (s) speed shots and darker nuances the Fast (f) ones. The vertical solid black bar represents the impact frame, the dashed vertical bars represent the beginning and end of forward acceleration phase, and the different points correspond to the frames on which musculoskeletal model was ran (see corpus of the article). Muscle abbreviations are defined in the table above
bevel of the racket handle, thus resulting in specific wrist and forearm at impact. The Advanced player used an eastern grip position, i.e., using the bevel parallel to the racket head plane, whereas the Intermediate used a semi-western grip, i.e., using the adjacent bevel in the little finger direction, oriented at 45° of the racket head plane. Both players did not suffer from any injuries within the past 24 months and signed an informed consent before beginning the experiment. The protocol was approved by the local ethics committee of Aix-Marseille University.

## Experimental protocol

Each player participated in a three-step protocol over a single session (Figure 1). The first step consisted in the electromyography setting-up, including the placement of electrodes and normalisation contraction. The second step concerned grip calibration tasks to determine a relationship between FDS muscle activation and grip force as well as participant-specific maximal grip force, finger joint postures and grip force distribution. Finally, the third step consisted in forehand tasks where upper limb and racket kinematics and EMG activities were recorded to estimate biomechanical loadings during forehand drives.
## Sensitivity analysis

A sensitivity analysis was run to quantify how the estimated muscle forces are affected by a modification of ±15% of each of the experimental data inputted in the EMG-informed musculoskeletal model. The data presented below correspond to the variations of muscle forces at the Pre-Impact frame of a single trial of the Advanced player performing a Fast shot with the R2 racket. The figure below presents the results of modifying eight input data, listed below.

* $F^{grip}$: grip force estimated using FDS activation

* $\theta^{WR-fe}$: wrist flexion-extension joint angle

* $\theta^{WR-rud}$: wrist radial ulnar joint angle

* $M^{WR-fe}$: wrist flexion-extension net moment

* $M^{WR-rud}$: wrist radial-ulnar net moment

* $F^{FCR}$: FCR muscle force estimated using the Force-Length-Activation model

* $F^{ECR}$: FCR muscle force estimated using the Force-Length-Activation model

* $F^{EDC}$: EDC muscle force estimated using the Force-Length-Activation model
<figure><img src="image_3.png" /><figcaption>Figure 1. Framework to estimate muscle forces during forehand drives from forehand and grip calibration data using the EMG-informed musculoskeletal model.</figcaption></figure>

Electromyography setting-up

The activities of four muscles (ECRB; EDC; Flexor Carpi Radialis: FCR; Flexor Digitorum Superficialis: FDS) were recorded during both grip calibration and forehand tasks using a wireless EMG system (Trigno, Delsys, Natick, MA, 2000 Hz) Before placing electrodes, the skin was shaved, sanded and rinsed with an alcoholic solution. The placement of the electrodes followed those established to use previously developed EMG-informed force-length-activation models³⁵,³⁶. Functional contractions were used to ensure crosstalk was minimised and included finger flexion and extension, wrist flexion and extension, grip force exertion and radial deviation. After equipment, each
<figure><img src="image_6.png" /><figcaption>Figure s3-6. Normalized muscle force variations induced by ±15% modification of each of the eight experimental input data (see list above). The analysis was run for a single trial of the Advanced player performing a Fast shot with the R2 racket and only the variations at Pre-Impact frame are shown. Variations are expressed as difference between perturbed (±15% input modification) and nominal trial (no modification), expressed as percentage of the highest force estimated in the nominal trial for that muscle. For each modified input variable, only the 15 muscles with the highest variations are shown.</figcaption></figure>
participant performed an isometric maximal voluntary contraction consisting in producing a maximal wrist extension moment while firmly gripping a racket handle for normalisation purposes of ECRB and EDC EMG signals. Participants were encouraged verbally. Participants performed two trials separated by a 2-minute rest. No additional maximal contraction was performed for flexors normalisation purpose as grip calibration tasks already included maximal force exertions.

## Grip calibration

*Task.* Participants performed isometric grip force tracking tasks. During each trial, both the target time-force curve and their current force-time curve were plotted on a screen. Their instruction was to follow the target profile as closely as possible. The participants were standing and were instructed to position their fingers on the handle the same way they grip the racket during forehand drives. The upper limb posture was let free to perform the task in a comfortable way while ensuring the optical markers (see below) were seen by cameras.

*Protocol.* First, each participant performed three maximum grip force trials with the instruction of reaching the maximal value as fast as possible and maintaining that level for 5 seconds. No target force-time curve was plotted, and participants were encouraged verbally. The maximal RMS value (50-ms window) of all three trials was taken as a maximal force reference to scale all the other tasks. Participants then performed the grip force tracking tasks following three types of target curve: constant, single-sine, double-sine. During constant trials, participants aimed at maintaining for 5s a grip force level indicated by a horizontal line positioned. The three force levels tested were 75%, 50% and 25% of the subject maximal force. During single-sine curve, the participants had to follow a force-time curve describing first a 3s constant force and then a sine cycle
# EMG-informed musculoskeletal model methodological details

## Force-length-activation model

The EMG-informed musculoskeletal model was guided by estimations of muscle forces from electromyography and joint angles, i.e., a forward dynamics model. This model was based on Force-Length-Activation relationships and was used to estimate the force ($F^m$) of extensor digitorum communis (EDC), flexor carpi radialis (FCR) and extensor carpi radialis (ECR). The model was divided in three steps (Figure s3-7): muscle-tendon unit kinematics, muscle belly excursion, and force-length-activation relationships. The equations and data corresponding to muscle belly excursion and force-length-activation were taken from previous studies (Goislard de Monsabert et al., 2020; Hauraix et al., 2018) that combined motion capture, ergometer, ultrasound imaging and electromyography data during maximal isometric contractions.

### Step 1 - Muscle-tendon unit kinematics

The current muscle-tendon unit (MTU) length ($L^{mtu}$) was determined from the measured reference MTU length ($L_r^{mtu}$) and joint angles ($\theta^j$) using geometrical models representing the path of the tendon around each joint as a function of time. The current MTU length was calculated by adding to $L_r^{mtu}$ the total MTU excursion corresponding to the sum of excursion at each individual joint ($\Delta L_j^{mtu}$).

$$L^{mtu} = L_r^{mtu} + \sum_j \Delta L_j^{mtu} \qquad \text{Equation 1}$$

With $j$ corresponding to one of the four joints, i.e., wrist (WR) and index finger distal (DIPi) or proximal (PIPi) interphalangeal or metacarpophalangeal (MCPi) joint. The excursion was null when tendon was not crossing the joint such that $\Delta L_{DIPi}^{MTU} = \Delta L_{PIPi}^{MTU} = \Delta L_{MCPi}^{MTU} = 0$ for wrist prime movers (ECR, FCR).
(increase and decrease) over a period of 10s starting and ending at the level of the initial constant level. The amplitude of the sine was 25% of the maximal force reference with a beginning either at 5, 15 or 30% of the subject maximal force, hence corresponding to three grip force ranges, i.e., 5-30%, 15-40%, 30-55%. During double-sine, the participants performed the same task as the single-sine, except that, after the initial level, the target curve described two sine cycles over the 10-s period. The same amplitude and initial levels were used as in single-sine trials. Each task was repeated twice for a total of 18 trials (3 tasks × 3 force levels × 2 trials) separated by a 30-sec rest. Task and repetitions were randomised for each participant.

*Material.* The systems to record grip force and hand kinematics were the same as in a previous study¹⁰. Grip force was recorded at 2000 Hz with a cylindrical dynamometer (3.3-cm diameter, Sixaxes, Argenteuil, FR). The dynamometer is divided into 6 beams, each equipped with strain gauge sensors and was mounted on a mechanical arm which helped participants maintaining the upper limb posture, but which could move slightly to prevent applying any unrequired mechanical actions, such as pulling or pronation moment. The grip force distribution across hand segments was assessed using a pressure sensor map (Hoof 3200F-scan mobile; TekScan, Boston, MA, 50Hz). The pressure map consisting in 1089 transducers (33 rows by 33 columns, 0.5-cm spacing) each having a 0-255 range was wrapped around the handle, and they were both squeezed during all trials. Finger segment kinematics were tracked using a motion capture system consisting of seven cameras (MX T40, Vicon, Oxford, UK, 100Hz) tracking 29 reflective markers placed on the dorsal aspects of the forearm and the hand (Figure s1-4 and Table s1-1 in Electronic Supplementary Material ESM1). Muscle activity of FDS was recorded as mentioned in above section. An external trigger was used to synchronise the acquisition of all four recording systems.
The MTU excursion model at the index finger joints (DIPi, PIPi, MCPi) relied on the normative tendon path data and the geometrical models provided by Chao *et al.* (1989). For finger extensors (EDC), the excursion was determined by considering a constant moment arm. For finger flexors (FDS), the excursion was determined using a bow-string model. The MTU excursion at the wrist joint was determined using regression equation taken from Lemay & Crago (1996).

<figure><img src="image_7.png" /><figcaption>Figure s3-7. Illustration describing the Force-Length-Activation model principle. All symbols and variables are described in the text.</figcaption></figure>

### Step 2 - muscle belly excursion

The current muscle belly length ($L^m$) was determined removing the active muscle belly shortening ($\Delta L^m$) from the passive belly length ($L^p$), i.e., before contraction.

$$L^m = L^p - \Delta L^m \qquad \text{Equation 2}$$

The passive belly length varies with the tendon lengthening/shortening when joint posture changes and was described here as a function of the current MTU length:
## Forehand task

*Task.* The forehand task was inspired from previous works of our group¹². Participants were instructed to hit a target with a ball using a flat forehand drive, i.e., with as less top spin as possible. The target was a 50-cm square located at 1.5m from the ground fixed and fixed on a large net used to cushion the ball shock. The participant was facing the target, standing 6m away from it and was instructed to move as less as possible in both frontal and lateral directions. The balls were sent by a ball machine at 2.5 m/s (Tennis twist, Sports Tutor, Burbank, CA) positioned at the net level and 3m on the left side of the target. The ball machine was adjusted to ensure participants could hit the ball after one rebound with minimal displacements.

*Protocol.* Participants performed the forehand task described above with three different rackets and at two different shot speeds. The properties of the three rackets are presented in Table 1. The R2w racket is the same model as R2 but with a 15-g added weight on the top. The R1 racket was chosen because of its light weight and lower moment of inertia values. Participants were instructed to perform the forehand task using either the minimal or maximal shot speed, further called “Slow” and “Fast”, allowing them to repeatably reach the target. Those speeds were adjusted by the participants for the first tested racket and the participants were instructed to reproduce them as closely as possible for the following rackets. To allow those adjustments, the participants hit approximately 40 balls when a new racket condition was presented. Racket order was randomised, and the speed conditions were randomised within each racket condition. For each shot speed with a racket, five trials were recorded.

*Material.* Upper limb and racket kinematics were assessed using again a same seven-camera motion capture system (MX T40, Vicon, Oxford, UK) tracking 20 reflective
$$L^p(L^{mtu}) = p_2(L^{mtu})^2 + p_1L^{mtu} + p_0 \qquad \text{Equation 3}$$

Where $d = \{d_2; d_1; d_0\}$ are polynomial coefficients that were determined in previous studies(Goislard de Monsabert et al., 2020; Hauraix et al., 2018) from ultrasound imaging and motion capture experimental data. The values of those coefficients are provided in Table s3-1.

The active muscle belly shortening ($\Delta L^m$), corresponding to muscle length decrease with active contraction of the fibres was determined from activation ($a^m$) with two relationships. The first relationship estimates a virtual muscle force level ($f^*$) using a sigmoid function (Equation 4).

$$F^*(a^m) = \alpha_1 \left[ \frac{1}{1 + \exp(-\alpha_2 (a^m - \alpha_3))} - 0.5 \right] + \alpha_4 \qquad \text{Equation 4}$$

Where $\alpha = \{\alpha_3; \alpha_2; \alpha_1\}$ are coefficients describing the sigmoid function that were determined in previous studies from electromyography and dynamometry experimental data (Goislard de Monsabert et al., 2020; Hauraix et al., 2018). The values of those coefficients are provided in Table s3-2. The second relationship estimates the muscle active shortening $\Delta L^m$ during contraction from the virtual force level and was described by an exponential function (Equation 5).

$$\Delta L^m(F^*) = \gamma_1 [1 - \exp(-\gamma_2 F^*)] \qquad \text{Equation 5}$$

Where $\gamma = \{\gamma_2; \gamma_1\}$ are coefficients describing the exponential function and were determined in previous studies from dynamometry, ultrasound and motion capture experimental data (Goislard de Monsabert et al., 2020; Hauraix et al., 2018). The values of those coefficients are provided in Table s3-3.
markers placed on upper trunk, arm, forearm, and hand of right upper limb (Figure s2-4 and Table s2-1 in Electronic Supplementary Material ESM2). For each participant, five balls covered with reflective tape were used to track the ball position during recorded trials. Muscle activities of FCR, ECR, FDS, and EDC were recorded at 2000Hz (see “Electromyography setting-up” section for details). EMG signals were synchronously recorded with kinematic data using an external trigger.

<table><caption>Table 1. Properties of the three rackets used during the study.</caption><thead><tr><th></th><th>R1</th><th>R2</th><th>R2w<sup>*</sup></th></tr></thead><tbody><tr><td>Model</td><td>Décathlon TR560 2017</td><td>Décathlon TR990 2017</td><td>Décathlon TR990 2017</td></tr><tr><td>String Tension (kg)</td><td>25/23</td><td>25/23</td><td>25/23</td></tr><tr><td>Handle Size</td><td>3</td><td>3</td><td>3</td></tr><tr><td>Length (mm)</td><td>680</td><td>685</td><td>685</td></tr><tr><td>Balance<sup>a</sup> (mm)</td><td>344</td><td>323</td><td>340</td></tr><tr><td>Mass (g)</td><td>270</td><td>305</td><td>320</td></tr><tr><td>Polar MOI<sup>b</sup> (Mg/mm<sup>2</sup>)</td><td>1.34</td><td>1.39</td><td>1.40</td></tr><tr><td>Transverse MOI<sup>b</sup> (Mg/mm<sup>2</sup>)</td><td>12.5</td><td>14.3</td><td>16.2</td></tr><tr><td>Lateral MOI<sup>b</sup> (Mg/mm<sup>2</sup>)</td><td>13.9</td><td>15.7</td><td>17.6</td></tr></tbody></table>

## Forehand kinematics

### Kinematic data processing

The kinematics of the racket, ball and upper limb were determined from marker coordinates during forehand trials. For each trial, a 1.5-s analysis window centred around the ball impact time frame was first determined and kept for further analysis. The impact frame was considered as the minimal 3D relative distance between the ball and the racket centre, calculated as the mean of the two side markers, using raw coordinates. The marker coordinates were low-pass filtered (Butterworth, 5Hz, order 2, zero-phase). Because of its discontinuous trajectory, the ball coordinates before and after impact frame were filtered separately and the raw coordinates at impact were kept as the position at impact.

<footnote>*R2w is the same racket as R2 with an added 15-g weight at the racket top. <sup>a</sup>The balance corresponds to the distance from the handle extremity to the centre of mass along the longitudinal axis. <sup>b</sup>MOI: moment of inertia.</footnote>
Step 3 - Force-length-activation relationship

The muscle force was estimated from muscle belly length (L^m) and muscle activation (a^m) using a force-length-activation relationship. This relationship relied on the equations of Kaufman et al. (1989) (Equations 7a-d) describing force-length behaviour at maximal activation (a^m = 1) and was adapted to include activation-dependency of the three parameters, i.e., index of architecture (i_a), optimal belly length (L_0) and maximal isometric force (F_0) (Equations 6a-c). The first step was thus to compute these parameters using the following polynomial relationship

$$i_a = b_5(a^m)^5 + b_4(a^m)^4 + b_3(a^m)^3 + b_2(a^m)^2 + b_1a^m + b_0 \qquad \text{Equation 6a}$$

$$L_0 = c_5(a^m)^5 + c_4(a^m)^4 + c_3(a^m)^3 + c_2(a^m)^2 + c_1a^m + b_0 \qquad \text{Equation 6b}$$

$$F_0 = d_3(a^m)^3 + d_2(a^m)^2 + d_1a^m \qquad \text{Equation 6c}$$

Where **b** = {b_5; b_4; b_3; b_2; d_1; d_0}, **c** = {c_3; c_2; c_1; c_0} and **d** = {d_3; d_2; d_1; d_0} are polynomial coefficients that were determined in previous studies (Goislard de Monsabert et al., 2020; Hauraix et al., 2018). The values of those coefficients are provided in Table s3-4, s3-5 and s3-6. Muscle force was then estimated from muscle belly length using the calculated parameters in the equation of Kaufman et al.

$$F^m(L^m) = F_0 \cdot \exp \left[ - \left( \frac{(\varepsilon^m + 1)^\beta - 1}{\omega} \right)^\rho \right] \qquad \text{Equation 7a}$$

with

$$\varepsilon^m = \frac{L^m - L_0}{L_0} \qquad \text{Equation 7b}$$
## Performance indexes

The performance during forehand drives was analysed qualitatively using racket trajectory and quantitatively using racket and ball velocities. The racket trajectory was plotted in the transverse and sagittal planes using the racket centre position at impact as origin. The racket centre and ball velocities and accelerations were obtained through central differentiation from filtered marker coordinates using one frame before and one frame after the current frame and low-pass filtering (Butterworth, 5Hz, order 2, zero-phase) after each step of differentiation. Inbound ($V_{in}^{ball}$ and $V_{in}^{rack}$) and outbound ($V_{out}^{ball}$ and $V_{out}^{rack}$) ball and racket velocities were determined by averaging their norm on the window going from 25 to 5 frames before and 5 frames to 25 frames after the impact frame. The after/before ratio of the ball and racket velocities were also determined.

## Upper limb joint angles

The upper limb joint angles were estimated from the filtered marker coordinates on the impact-centred window. The wrist was described by two degrees of freedom (DoF) in flexion/extension and radial/ulnar deviation, the forearm by one DoF in pronation/supination and the elbow by one DoF in flexion/extension. At each time frame, segment coordinate systems were determined, and the joint angles were extracted from relative orientation matrices using Cardan sequences (see Table s2-2 and corpus of Electronic Supplementary Material 2, ESM2). Flexion, pronation and ulnar deviation angle correspond to positive value.

## Finger posture on the handle

The posture of the fingers on the handle during forehand drives was considered constant during the forehand motion and was extracted from the position of reflective markers during the grip calibration protocol. The marker coordinates were averaged on
$$ \omega = 0.35327(1 - i_a) \qquad \text{Equation 7c} $$

$$ \beta = 0.96343 \left( 1 - \frac{1}{i_a} \right) \qquad \text{Equation 7d} $$

Where $\varepsilon^m$ was the muscle strain and $\omega$, $\beta$ and $\rho$ were shape parameters of the force-length relationship corresponding to width, skewness, and roundness, respectively. Roundness $\rho$ was equal to 2 (Goislard de Monsabert et al., 2020; Hauraix et al., 2018).

## Associated data

<table><caption>Table s3-2 – Coefficients of the polynomial regression between normalized passive belly length ($l^p$) and normalized muscle-tendon unit length ($l^{mtu}$). To use as follows $l_1^m = p_2(l^{mtu})^2 + p_1l^{mtu} + p_0$. Muscle-tendon unit length was normalized by dividing by the reference muscle-tendon unit length ($L_r^{MTU}$). Passive muscle belly length was normalized by dividing it by its value at reference posture, i.e., using $L^{MTU} = L_r^{MTU}$ in equation 2.</caption><thead><tr><th></th><th>p2</th><th>p1</th><th>p0</th></tr></thead><tbody><tr><td>FCR</td><td>-0.66</td><td>2.70</td><td>-1.05</td></tr><tr><td>ECR</td><td>25.97</td><td>-48.21</td><td>23.25</td></tr><tr><td>EDC</td><td>15.71</td><td>-29.60</td><td>14.90</td></tr></tbody></table>

<table><caption>Table s3-3 – Coefficients of the relationship between normalized virtual force ($f^*$) and muscle activation ($a^m$). To use as follows: $f^*(a^m) = \alpha_1 \left[ \frac{1}{1+\exp(-\alpha_2(a^m - \alpha_3))} - 0.5 \right] + \alpha_4$. Virtual force was normalized by dividing by $F^{max}$, i.e., the maximal isometric force $F^0$ at maximal activation ($a^m = 1$). Muscle activation $a^m$ was unitless and equal to 1 during a maximal voluntary contraction.</caption><thead><tr><th></th><th>α<sub>1</sub></th><th>α<sub>2</sub></th><th>α<sub>3</sub></th><th>α<sub>4</sub></th></tr></thead><tbody><tr><td>FCR</td><td>0</td><td>0</td><td>2.07</td><td>2.95</td></tr><tr><td>ECR</td><td>0.096</td><td>0.13</td><td>2.01</td><td>2.72</td></tr><tr><td>EDC</td><td>0.33</td><td>0.40</td><td>1.27</td><td>4.47</td></tr></tbody></table>

<table><caption>Table s3-4 – Coefficients of the relationship between normalized muscle active shortening ($\Delta l^m$) and normalized virtual force $f^*$. To use as follows: $\Delta l^m(f^*) = \gamma_1 [1 - \exp(-\gamma_2 f^*)]$. Virtual force was normalized by dividing by $F^{max}$, i.e., the maximal isometric force $F^0$ at maximal activation ($a^m = 1$). Muscle active shortening was normalized by dividing by passive muscle belly length at reference posture, i.e., using $L^{MTU} = L_r^{MTU}$ in equation 2. A positive $\Delta l^m$ corresponds to muscle shortening.</caption><thead><tr><th></th><th>γ<sub>1</sub></th><th>γ<sub>2</sub></th></tr></thead><tbody><tr><td>FCR</td><td>0.043</td><td>4.37</td></tr><tr><td>ECR</td><td>0.069</td><td>3.61</td></tr><tr><td>EDC</td><td>0.028</td><td>3.72</td></tr></tbody></table>
100 frames when the participant was exerting a grip force on the handle. For the Advanced player, the data was taken from a maximum grip force trial. For the Intermediate player, not all markers could be seen on maximum grip trials, so the marker coordinates were taken during a 15-40% single sine trial. The finger posture was characterised by 23 joint angles calculated using a previously developed method¹⁰. The trapeziometacarpal (TMC) and metacarpophalangeal (MCP) joints were described by two DoF in flexion/extension and radial/ulnar deviation and the proximal (PIP) and distal (DIP) joints by one DoF in flexion/extension. For each joint, a distal and a proximal segment coordinate system were calculated from marker positions (see Table s1-2 and corpus of Electronic Supplementary Material ESM1) and joint angles were extracted from the relative orientation matrix using a Z-Y-X (flexion/pronation/abduction) sequence of Cardan angles. Flexion, and ulnar deviation angle correspond to positive value.

## Hand biomechanical loadings during forehand

### Muscle activation

The muscle activation was calculated from EMG signals the same way for both protocols. For each muscle, the signal was processed to be used with our force-length-activation relationships³⁵ with first the application of a bandpass filter (Butterworth, 10-400Hz, order 2, zero-phase) and then obtaining EMG envelope by rectifying and applying a lowpass filter (Butterworth, 5Hz, order 2, zero-phase). The muscle activation was then calculated by normalizing each envelope value in a trial by the maximal envelope value observed among all trials for that muscle, i.e., both MVC, grip calibration and forehand drives.
<table><caption>Table s3-5 – Coefficients of the polynomial regression between the index of architecture ($i_a$) and the activation level ($a^m$). The coefficients are used as follows: $i_a = b_5(a^m)^5 + b_4(a^m)^4 + b_3(a^m)^3 + b_2(a^m)^2 + b_1a^m + b_0$. Muscle activation $a^m$ was unitless and equal to 1 during a maximal voluntary contraction. Index of architecture $i_a$ was unitless.</caption><thead><tr><th></th><th>b<sub>5</sub></th><th>b<sub>4</sub></th><th>b<sub>3</sub></th><th>b<sub>2</sub></th><th>b<sub>1</sub></th><th>b<sub>0</sub></th></tr></thead><tbody><tr><th>FCR</th><td>0.714</td><td>-2.399</td><td>2.463</td><td>-0.701</td><td>0.069</td><td>0.169</td></tr><tr><th>ECR</th><td>-0.664</td><td>2.575</td><td>-3.404</td><td>1.712</td><td>-0.157</td><td>0.231</td></tr><tr><th>EDC</th><td>-0.105</td><td>0.308</td><td>-0.697</td><td>0.792</td><td>-0.167</td><td>0.173</td></tr></tbody></table>

<table><caption>Table s3-6 – Coefficients of the polynomial regression between normalized optimal length ($l_0$) and muscle activation ($a^m$). To use as follows: $l_0 = c_5(a^m)^5 + c_4(a^m)^4 + c_3(a^m)^3 + c_2(a^m)^2 + c_1a^m + c_0$. Optimal length $L_0$ was normalized by dividing by the passive muscle belly length at reference posture, i.e., using $L^{MTU} = L_r^{MTU}$ in equation 2. Muscle activation $a^m$ was unitless and equal to 1 during a maximal voluntary contraction. Index of architecture $i_a$ was unitless.</caption><thead><tr><th></th><th>c<sub>5</sub></th><th>c<sub>4</sub></th><th>c<sub>3</sub></th><th>c<sub>2</sub></th><th>c<sub>1</sub></th><th>c<sub>0</sub></th></tr></thead><tbody><tr><th>FCR</th><td>0.096</td><td>-0.335</td><td>0.330</td><td>-0.042</td><td>-0.099</td><td>1.002</td></tr><tr><th>ECR</th><td>0</td><td>0</td><td>-0.048</td><td>0.196</td><td>-0.211</td><td>0.971</td></tr><tr><th>EDC</th><td>0</td><td>0</td><td>-0.050</td><td>0.107</td><td>-0.083</td><td>1.016</td></tr></tbody></table>

<table><caption>Table s3-7 – Coefficients of the polynomial regression between the normalised maximal force ($f_0$) and the activation level ($a^m$). The coefficients are used as follows: $f_0 = d_3(a^m)^3 + d_2(a^m)^2 + d_1a^m$. Maximal force was normalized by dividing by $F^{max}$, i.e., the maximal isometric force $F^0$ at maximal activation ($a^m = 1$).</caption><thead><tr><th></th><th>d<sub>3</sub></th><th>d<sub>2</sub></th><th>d<sub>1</sub></th></tr></thead><tbody><tr><th>FCR</th><td>0.297</td><td>-1.430</td><td>2.120</td></tr><tr><th>ECR</th><td>-0.029</td><td>-0.336</td><td>1.361</td></tr><tr><th>EDC</th><td>-0.528</td><td>0.025</td><td>1.484</td></tr></tbody></table>

## References

Chao, E. Y., An, K. N., Cooney III, W. P., & Linscheid, R. L. (1989). *Biomechanics of the hand: A basic research study*. World Scientific.

Goislard de Monsabert, B., Hauraix, H., Caumes, M., Herbaut, A., Berton, E., & Vigouroux, L. (2020). Modelling force-length-activation relationships of wrist and finger extensor muscles. *Medical & Biological Engineering & Computing*, 58(10), 2531-2549. https://doi.org/10.1007/s11517-020-02239-0

Hauraix, H., Goislard de Monsabert, B., Herbaut, A., Berton, E., & Vigouroux, L. (2018). Force-Length Relationship Modeling of Wrist and Finger Flexor Muscles. *Medicine & Science in Sports & Exercise*, 50(11), 2311-2321. https://doi.org/10.1249/MSS.0000000000001690

Kaufman, K. R., An, K. N., & Chao, E. Y. (1989). Incorporation of muscle architecture into the muscle length-tension relationship. *Journal of Biomechanics*, 22(8-9), 943-948.

Lemay, M. A., & Crago, P. E. (1996). A dynamic model for simulating movements of the elbow, forearm, and wrist. *Journal of Biomechanics*, 29(10), 1319-1330. https://doi.org/10.1016/0021-9290(96)00026-7
Grip force

The grip force during forehand drives was estimated from FDS muscle activation using a player-specific relationship derived from grip calibration experimental data, i.e., dynamometer and EMG. First, the six signals from the dynamometer were low-pass filtered (Butterworth, 5Hz, order 2, zero-phase) and converted to Newtons via a calibration matrix. The grip force was then computed as the sum from the six forces recorded by the beams of the dynamometer and normalised by the maximal grip force (MGF), determined as the maximal RMS value (500-ms window) among the three maximum grip trials performed at the beginning of the grip calibration session. The FDS activation was calculated the same way as explained above. For each trial, only the data corresponding to the points where normalised grip force was higher than 5% were kept and FDS muscle activation and grip force were then linearly interpolated to obtain 200 data points for a total of 3600 points of each signal (18 trials).

The EMG to grip force relationship was a sigmoid function (equation 1) which parameters were estimated by a constrained least square optimisation (fmincon, MATLAB R2021b) aiming to minimise the squared difference between measured ($f^{grip}$) and estimated ($\hat{f}^{grip}$) grip force (equation 2). The criterion and sigmoid function were as follows:

$$ \hat{f}^{grip}(a^{FDS}) = K \cdot \left( \frac{1}{1+e^{-\beta \cdot (a^{FDS}-a_0)}} - 0.5 \right) + f_0 \quad (1) $$

$$ G(f_0, a_0, K, \beta) = \sum_t (f^{grip}(t) - \hat{f}^{grip}(a^{FDS}))^2 \quad (2) $$

Where $t$ represents a time sample, $a^{FDS}$ is the muscle activation of FDS, and $p = \{f_0, a_0, K, \beta\}$ are the parameters of the sigmoid relationship. The optimisation included boundaries on parameters, i.e., $0 \le a_0 \le 1$, $0 \le f_0 \le 1$, $1.01 \le K \le 10$, $0.1 \le \beta \le 1$
10. Two supplementary constraints regarding the value of estimated grip force for activation at 0 and 1, i.e., $\hat{f}^{grip}(0) = 0$ and $\hat{f}^{grip}(1) = 1$.

## Grip force distribution in the hand

The grip force distribution across hand segments during forehand drives was estimated for each player using the pressure sensor data measured during the grip calibration session using a previously developed method¹⁰. The pressure data was taken from the 75% constant force task, using only the trial with the highest maximal value. Briefly, a single pressure map was obtained by averaging each of the 1089 transducer signal on a 500-ms window corresponding to the maximal 500-ms RMS value of the grip force. The pressure map was then normalised by dividing each transducer value by the sum of all transducer values. From this normalised pressure map, 25 normalised forces were obtained by manually segmenting the pressure map to identify 25 anatomical areas, i.e., five by fingers. Each normalised force was computed as the sum of the normalised transducer values within the corresponding segmented area. The force applied on an anatomical area during forehand drive is obtained by multiplying the normalised force by the estimated grip force.

## Wrist net moment

The wrist moment was determined from the racket and upper limb marker coordinates recorded during the forehand drives using a Newton-Euler inverse dynamics method. The hand and racket were considered as a single rigid body. The hand-racket system inertial properties were deduced from those of the racket (Table 1) and the hand. The hand mass was estimated from participant's hand length, width as well as hand and wrist circumferences using regression equations⁴⁰. Because those equations assume a “flat” hand posture, with straight fingers, the hand centre of mass
(CoM) and inertia were determined differently assuming the hand as a cylinder wrapping around the racket handle. The hand CoM was the projection of the 3rd metacarpal head marker on the racket longitudinal axis. The hand inertia was determined by assuming a cylinder whose internal diameter was the racket diameter (Table 1). The cylinder inter-radii distance was equal to the distance between the 3rd metacarpal head marker and the hand CoM. The cylinder length was equal to the participant's hand width. The hand-racket system CoM (G) was then determined as the weighted average of the two points, i.e., considering their mass. The inertia matrices were rotated to the hand-racket coordinate system and translated to the hand-racket CoM. The intersegmental force and moment of the hand-racket system on the radius were then determined:

$$F^{WR} = ma - mg \qquad (3)$$

$$M^{WR} = [I^G]\dot{\omega} + \omega \times [I^G]\omega - d \times F^{WR} \qquad (4)$$

Where $F_{WR}$ and $M_{WR}$ are the intersegmental force and moment, $a$ and $\omega$ are the linear and angular velocities of the system, $m$ is the mass, $g$ the gravity vector, $[I^G]$ is the inertia matrix expressed at the CoM and $d$ is the vector going from the CoM to the wrist joint centre calculated as the unweighted barycentre of the ulnar and radial styloid markers. The wrist intersegmental moment was then rotated to the radius coordinate system. The ball-racket impact force was not considered in these calculations.

## Estimation of muscles forces

The muscle forces during forehand drives were determined via an EMG-informed musculoskeletal model adapted from a previous study¹⁰. Briefly, this model considers the musculoskeletal chains of the wrist and the five fingers by included 21
rigid segments (three phalanges and one metacarpal per long finger, two phalanges, one metacarpal and the trapezium for the thumb and the radius). The segments were linked by 23 DoF, with one rotation for the nine interphalangeal joints, two rotations for the five metacarpophalangeal joints, the trapeziometacarpal joint and the wrist joints. The DoFs were actuated by 42 muscle-tendon units, including 18 hand intrinsic muscles, 18 extrinsic and 6 wrist prime movers¹⁰. The list of all muscles is provided in Table s3-1 of Electronic Supplementary 3 (ESM3).

The 42 muscle forces were estimated using a static optimization procedure aiming to minimise muscle stress (equation 5) while ensuring mechanical equilibrium (equation 6) and respecting EMG-informed muscle force constraints and extensor mechanism force transmission constraints. The muscle stress criterion and mechanical equilibrium were described as follows:

$$J(F^M) = \sum_m \left( \frac{F^m}{PCSA^m} \right)^4 \qquad (5)$$

$$[R^M]F^M + M^E + M^P = 0 \qquad (6)$$

Where $F^M$ is the 42×1 muscle force vector, $PCSA^m$ is the physiological cross sectional area of a muscle, $[R^M]$ is a 23×42 matrix describing moment arm of each muscle at all DoF calculated from finger posture and wrist joint angle, $M^E$ is the 23×1 external moment vector including the wrist moment calculated above (equation 3) and those calculated from the forces acting on phalanges deduced from grip force level and force distribution data (see above), $M^P$ is a 23×1 vector including moments of passive structures at the TMC and MCP joints. Phalanx segment lengths, points describing tendon path of the finger muscle-tendon units and their PCSA were scaled using hand length and hand breadth. The action of extrinsic finger muscles was included in the
wrist balance mechanical equilibrium. The force distribution in the extensor mechanism was also solved in the optimisation procedure using constraints equation. More details on the data and calculations described above can be found in our previous study¹⁰.

In addition to constraints regarding mechanical equilibrium (equation 6) and extensor mechanism, the optimization procedure was guided by constraining the forces of FCR, ECRB and the four EDC comportments to be equal to an EMG-informed estimation of muscle force. This EMG-informed estimation was based on muscle-specific force-length-activation relationships experimentally derived from in vivo motion capture data and already presented in previous studies⁹,³⁵,³⁶. Electronic Supplementary Material 3 (ESM3) provide additional results (Figure s3-2 and s3-3), parameters (Table s3-2 to s3-7) and methodological details (Figure s3-7) of these EMG-informed estimations. Briefly, the current muscle-tendon length is first estimated from wrist and finger joint angles using geometric models. The muscle length due to active contraction is deduced from muscle-tendon length and muscle activation. Finally, muscle force is estimated from muscle length and activation using a force-length relationship considering a non-linear activation-dependency of the maximal isometric force, optimal length, and shape of the curve.

To reduce computational time, the muscle forces were estimated on five specific time frames related to racket motion, illustrated on Figure s2-2 of Electronic Supplementary Material ESM2 and corresponding to :

* Beginning of forward acceleration: first frame where antero-posterior component of racket velocity in the laboratory frame is positive, i.e., toward the net, on the 1.5-s window kinematic analysis.
* Peak racket forward acceleration: frame corresponding to maximal value of antero-posterior component of racket acceleration in the laboratory frame before impact frame

* Pre-Impact: three frames before the ball-racket impact

* Peak racket vertical acceleration: frame corresponding to maximal value of vertical component of racket acceleration in the laboratory frame after impact frame

* End of forward acceleration: last frame where antero-posterior component of racket velocity in the laboratory frame is positive, i.e., toward the net, on the 1.5-s window kinematic analysis.

The static optimisation procedure was run for the five frames of interest mentioned above using as input the time-varying wrist joint angles and moments (equation 3) and grip force level and fixed finger posture and grip force distribution in the hand (see above for details).

To provide a global understanding of tendon loadings, the 42 muscle forces were reduced to 4 muscle group forces, namely wrist prime extensors (W-ext; including Ecu, ECRB and ECRL), wrist prime flexors (W-flex; including PL, FCR and FCU), finger extrinsic extensors (F-ext; including the four compartments of EDC) and finger extrinsic flexors (F-flex; including the four compartments of FDS and FDP). Each group muscle force is the sum of the individual muscle forces in the group. The results of the 42 individual forces are provided in Figure s3-4 and s3-5 Electronic Supplementary Material 3 (ESM3).
## Data analysis

The analysis aimed at studying the influence of shot speed (Slow vs Fast) and racket properties (R1 vs R2 vs R2w) on hand biomechanical loading data and identify differences between players, e.g., due to skill level or general gesture technique. For comparison between players, kinetic variables were normalised and expressed as percentage of MGF for grip force and muscle forces or of the product of MGF and hand length (HL) for wrist net moment. To express the kinetic and muscle efficiency of each player gesture, the percentage-normalised values were divided by the ball velocity after impact ($V_{out}^{ball}$) in m/s². A low efficiency index value corresponds to a greater efficiency. The comparisons will not include statistical tests as only two players were analysed and only five repetitions per condition were acquired.

## Results

### Grip force calibration results

All results associated with the grip calibration protocol (relationship between grip force and electromyography, grip force distribution and finger posture) are presented in Figures s1-1, s1-2 and s1-3 of Electronic Supplementary Material ESM1. A brief description of the key elements will be provided here. The measured MGF was 653.2 N and 1195.2 N for the Advanced and the Intermediate player, respectively. The estimated grip force obtained from FDS activation, with the player-specific least-square fitted relationship (equation 1), correlated well with experimental data (R²=0.87 for Advanced; R²=0.82 for Intermediate). The average root mean square error (RMSE) calculated from all estimated and experimental values was moderate (RMSE = 9% of MGF for Advanced and RMSE = 12% of MGF for Intermediate player).
## Forehand performance

The forehand performance indexes related to racket and ball velocities are presented in Table 2. The inbound ball velocity was approximately 2.3 m/s and similar across shot speeds, racket properties, and players. The outbound ball velocity $V_{ball}^{out}$ was influenced by speed shots and was approximately 10 m/s higher during Fast speed shots compared to Slow ones and were slightly lower for the Advanced player than for the Intermediate. The results suggest an influence of racket properties on $V_{ball}^{out}$. For the Advanced player, the use of R2 resulted in slightly higher velocities for both shot speeds and the use of R1 resulted in lower velocities during Fast shots. For the Intermediate player, the use of R2w resulted in slightly lower velocities during Slow shots.

<table><caption>Table 2. Mean ± one standard deviation values (N=5 trials) of racket and ball velocities before (in) and after (out) the ball/racket impact of both players (Advanced and Intermediate) for each of the two shot speeds (Slow, Fast) performed with one of the three rackets (R1, R2 R2w)</caption><thead><tr><th rowspan="2"></th><th colspan="3">Ball velocity</th><th colspan="3">Racket velocity</th></tr><tr><th>V<sub>ball</sub><sup>in</sup> (m/s)</th><th>V<sub>ball</sub><sup>out</sup> (m/s)</th><th>V<sub>ball</sub><sup>out</sup>/V<sub>ball</sub><sup>in</sup></th><th>V<sub>rack</sub><sup>in</sup> (m/s)</th><th>V<sub>rack</sub><sup>out</sup> (m/s)</th><th>V<sub>rack</sub><sup>out</sup>/V<sub>rack</sub><sup>in</sup></th></tr></thead><tbody><tr><td colspan="7"><strong>Advanced player</strong></td></tr><tr><td colspan="7"><strong>Slow</strong></td></tr><tr><td>R1</td><td>2.4 ± 0.2</td><td>12.3 ± 0.5</td><td>5.2 ± 0.4</td><td>5.6 ± 0.4</td><td>2.9 ± 0.4</td><td>0.5 ± 0.1</td></tr><tr><td>R2</td><td>2.4 ± 0.2</td><td><strong>15.0 ± 0.4</strong> *</td><td>6.2 ± 0.4</td><td>6.4 ± 0.2</td><td>3.0 ± 0.3</td><td>0.5 ± 0.0</td></tr><tr><td>R2w</td><td>2.4 ± 0.2</td><td>13.3 ± 0.7</td><td>5.5 ± 0.3</td><td>6.0 ± 0.3</td><td>3.0 ± 0.3</td><td>0.5 ± 0.0</td></tr><tr><td colspan="7"><strong>Fast</strong></td></tr><tr><td>R1</td><td>2.3 ± 0.2</td><td><strong>23.7 ± 2.0</strong> #</td><td>10.2 ± 1.5</td><td>8.2 ± 0.6</td><td>4.1 ± 0.1</td><td>0.5 ± 0.0</td></tr><tr><td>R2</td><td>2.4 ± 0.1</td><td><strong>26.4 ± 0.9</strong> *</td><td>11.0 ± 0.4</td><td>8.8 ± 0.4</td><td>4.6 ± 0.3</td><td>0.5 ± 0.0</td></tr><tr><td>R2w</td><td>2.2 ± 0.2</td><td>25.7 ± 1.2</td><td>11.6 ± 1.0</td><td>8.5 ± 0.2</td><td>4.1 ± 0.4</td><td>0.5 ± 0.1</td></tr><tr><td colspan="7"><strong>Intermediate player</strong></td></tr><tr><td colspan="7"><strong>Slow</strong></td></tr><tr><td>R1</td><td>2.3 ± 0.2</td><td>20.4 ± 2.4</td><td>8.8 ± 1.6</td><td>7.6 ± 0.6</td><td>3.0 ± 0.6</td><td>0.4 ± 0.1</td></tr><tr><td>R2</td><td>2.3 ± 0.1</td><td>20.2 ± 2.3</td><td>8.7 ± 1.1</td><td>7.4 ± 0.7</td><td>3.0 ± 0.7</td><td>0.4 ± 0.1</td></tr><tr><td>R2w</td><td>2.3 ± 0.1</td><td><strong>18.5 ± 1.7</strong> #</td><td>8.0 ± 0.7</td><td>6.9 ± 0.5</td><td>2.9 ± 0.4</td><td>0.4 ± 0.1</td></tr><tr><td colspan="7"><strong>Fast</strong></td></tr><tr><td>R1</td><td>2.4 ± 0.1</td><td><strong>28.5 ± 0.9</strong> #</td><td>12.1 ± 0.5</td><td>8.1 ± 0.3</td><td>5.1 ± 0.5</td><td>0.6 ± 0.1</td></tr><tr><td>R2</td><td>2.3 ± 0.1</td><td>31.1 ± 1.4</td><td>13.3 ± 0.7</td><td>9.0 ± 0.3</td><td>5.6 ± 0.4</td><td>0.6 ± 0.1</td></tr><tr><td>R2w</td><td>2.2 ± 0.2</td><td>31.0 ± 2.4</td><td>14.1 ± 0.4</td><td>8.8 ± 0.1</td><td>5.2 ± 0.4</td><td>0.6 ± 0.1</td></tr></tbody></table>

Bold font and * symbol highlight a high value across the three rackets. Bold italic font and # symbol highlight a low value across the three rackets
The inbound racket velocity ($V_{rack}^{in}$) was influenced by shot speeds, with an increase of approximately 2.5 and 1.5 m/s during Fast shots for Advanced and Intermediate player, respectively. Inbound and outbound racket velocities were not influenced by racket properties.

Electronic Supplementary Material [ESM2] provides additional graphics presenting racket trajectories (Figure s2-1) and time-patterns of racket (Figure 2-2) and joint kinematics (Figure s3-3).

## Hand biomechanical loadings during forehand

### Forearm muscle activation

The muscle activation time-patterns during forehand drives measured using EMG are presented in Figure 2. The shot speed influenced activation levels with globally higher peak values during Fast shots, except for EDC. For the Advanced player, the greatest increases were around 20% for the wrist extensor (ECR) and the finger flexors (FDS). For the Intermediate player the greatest increase in maximal activation level was for the wrist flexor (FCR) with around 40% higher maximal values during Fast shots. The ECR activation time-pattern of the Advanced player also changed from a single peak during Slow shots to a double peak during Fast shots. The racket properties had no influence on muscle activation pattern of the Advanced player. For the Intermediate player, the use of the R2w led to a decrease of FCR activation during Slow shots and an increase of FDS activation during Fast shots compared to other rackets. The comparison of players suggested different timing of muscle activations relative to ball/racket impact and different muscle coordination. Compared to the Intermediate player, the Advanced player peak activation values were reached earlier, and the muscle coordination suggested lower wrist (FCR) and finger (FDS)
flexors activation levels and higher activation levels of wrist extensor (ECR) during
Fast shots.

<figure><img src="image_4.png" /><figcaption>Figure 2. Mean time-patterns (N=5 trials) of muscle activations during the forehand drive of both players (Advanced and Intermediate) for the two shot speeds (Slow; Fast) performed with the three rackets (R1, R2, R2w). Shaded areas represent ± one standard deviation around the mean. Lighter nuances represent the Slow (s) speed shots, and darker nuances the Fast (f) ones. The vertical solid black bar represents the impact frame, the dashed vertical bars represent the beginning and end of forward acceleration phase, and the different points correspond to the frames on which musculoskeletal model was run (see corpus). ECR: Extensor Carpi Radialis, EDC: Extensor Digitorum Communis, FCR: Flexor Carpi Radialis, FDS: Flexor Digitorum Superficialis.</figcaption></figure>
Grip force

<figure><img src="image_5.png" /></figure>

Figure 3. Mean time-patterns (N=5 trials) of estimated grip force ($\hat{F}^{grip}$) and wrist flexion/extension ($M^{WR-fe}$) and radial-ulnar deviation ($M^{WR-rud}$) moments during the forehand drives of both players (Advanced and Intermediate) for the two shot speeds (Slow and Fast) performed with the three rackets (R1, R2, R2w). Shaded areas represent ± one standard deviation around the mean. Lighter nuances represent the Slow (s) speed shots and darker nuances the Fast (f) ones. The vertical solid black bar represents the impact frame, the dashed vertical bars represent the beginning and end of forward acceleration phase, and the different points correspond to the frames on which musculoskeletal model was run (see corpus). Normalised $\hat{F}^{grip}$ values expressed in percentage of maximal grip force (MGF) are represented on the right axis of upper panels. Wrist moment components are expressed from an internal point of view, such that positive values represent a flexion and ulnar moment exerted by muscles.

The estimated grip force during forehand drives estimated from FDS activation are presented in Figure 3 and Table 3. The shot speed influenced grip force with higher forces during Fast shots. At Pre-Impact, this increase was up to 40% and 20% for the
Advanced and Intermediate players, respectively. The racket properties influenced the grip force with a decrease of 10% for the Advanced player using R2 during slow shots and an increase of 10% for the Intermediate player using the R2w.

<table><caption>Table 3. Mean ± one standard deviation values of estimated grip force ($\hat{F}^{grip}$) and wrist flexion/extension ($M^{WR-fe}$) and radial-ulnar deviation ($M^{WR-rud}$) moments at the Pre-Impact time (three frames before impact) for both players (Advanced and Intermediate) for each of the two shot speeds (Slow and Fast) performed with one of the three rackets (R1, R2, R2w)</caption><thead><tr><th rowspan="2">Kinetics<br>at Pre-Impact</th><th colspan="2">Normalised value<br>(%MGF or %MGF · HL)</th><th colspan="2">Efficiency index<br>(%/V<sub>ball</sub><sup>out</sup>)</th></tr><tr><th>Slow</th><th>Fast</th><th>Slow</th><th>Fast</th></tr></thead><tbody><tr><td rowspan="12"><i>F</i><sup>grip</sup></td><td colspan="2">Advanced</td><td colspan="2"></td></tr><tr><td>R1</td><td>42.4 ± 12.1</td><td>74.4 ± 11.1</td><td>3.47 ± 1.06</td><td>3.13 ± 0.32</td></tr><tr><td>R2</td><td><b>30.6 ± 3.9 #</b></td><td>76.1 ± 8.1</td><td><b>2.04 ± 0.26 #</b></td><td>2.88 ± 0.29</td></tr><tr><td>R2w</td><td>42.6 ± 8.2</td><td>74.6 ± 7.1</td><td>3.21 ± 0.70</td><td>2.90 ± 0.26</td></tr><tr><td colspan="2">Intermediate</td><td colspan="2"></td></tr><tr><td>R1</td><td>53.1 ± 14.1</td><td>75.2 ± 5.7</td><td>2.58 ± 0.43</td><td>2.63 ± 0.15</td></tr><tr><td>R2</td><td>58.3 ± 7.1</td><td>73.7 ± 11.3</td><td>2.89 ± 0.15</td><td>2.37 ± 0.34</td></tr><tr><td>R2w</td><td><b>66.5 ± 6.4 #</b></td><td><b>87.2 ± 5.4 #</b></td><td><b>3.63 ± 0.54 #</b></td><td><b>2.84 ± 0.41 #</b></td></tr><tr><td rowspan="12"><i>M</i><sup>WR-fe</sup></td><td colspan="2">Advanced</td><td colspan="2"></td></tr><tr><td>R1</td><td><b>-2.4 ± 0.4 #</b></td><td><b>-6.8 ± 0.7 #</b></td><td><b>-0.19 ± 0.03 #</b></td><td><b>-0.29 ± 0.01 #</b></td></tr><tr><td>R2</td><td>-3.9 ± 0.3</td><td><b>-9.1 ± 1.0 #</b></td><td>-0.26 ± 0.02</td><td><b>-0.34 ± 0.03 #</b></td></tr><tr><td>R2w</td><td>-3.5 ± 0.6</td><td>-8.2 ± 0.7</td><td>-0.26 ± 0.04</td><td>-0.32 ± 0.02</td></tr><tr><td colspan="2">Intermediate</td><td colspan="2"></td></tr><tr><td>R1</td><td>-1.7 ± 0.5</td><td>-2.2 ± 0.2</td><td>-0.08 ± 0.02</td><td>-0.08 ± 0.01</td></tr><tr><td>R2</td><td>-1.5 ± 0.2</td><td>-2.6 ± 0.1</td><td>-0.08 ± 0.02</td><td>-0.08 ± 0.00</td></tr><tr><td>R2w</td><td>-1.4 ± 0.5</td><td>-3.0 ± 0.5</td><td>-0.08 ± 0.03</td><td>-0.10 ± 0.01</td></tr><tr><td rowspan="12"><i>M</i><sup>WR-rud</sup></td><td colspan="2">Advanced</td><td colspan="2"></td></tr><tr><td>R1</td><td>1.0 ± 0.6</td><td>2.7 ± 1.0</td><td>0.08 ± 0.05</td><td>0.11 ± 0.04</td></tr><tr><td>R2</td><td><b>2.1 ± 0.3 #</b></td><td><b>2.1 ± 1.3 #</b></td><td><b>0.14 ± 0.02 #</b></td><td><b>0.08 ± 0.05 #</b></td></tr><tr><td>R2w</td><td>0.8 ± 1.5</td><td>2.7 ± 1.2</td><td>0.06 ± 0.11</td><td>0.11 ± 0.04</td></tr><tr><td colspan="2">Intermediate</td><td colspan="2"></td></tr><tr><td>R1</td><td>-0.9 ± 0.3</td><td>-0.9 ± 0.2</td><td>-0.04 ± 0.01</td><td>-0.03 ± 0.01</td></tr><tr><td>R2</td><td>-0.7 ± 0.2</td><td>-1.3 ± 0.2</td><td>-0.03 ± 0.01</td><td>-0.04 ± 0.01</td></tr><tr><td>R2w</td><td>-0.8 ± 0.4</td><td>-1.0 ± 0.1</td><td>-0.04 ± 0.03</td><td>-0.03 ± 0.01</td></tr></tbody></table>

Normalised forces are expressed as percentage of maximal grip force (MGF) and moments as percentage of the product of MGF and hand length (HL). Efficiency is the percentage-normalised value divided by ball velocity after impact ($V_{ball}^{out}$) in m/s. Bold font and # symbol highlight a high value across the three rackets. Bold italic font and # symbol highlight a low value across the three rackets

The grip force efficiency index at Pre-Impact was not influenced by shot speed or player technique but showed some differences between rackets (Table 3). The efficiency index was lower for the Advanced player using R2 during Slow shots and higher for the Intermediate player using R2w.
## Wrist moment

The components of wrist moment during forehand drives calculated from marker coordinates are presented in Figure 3 and Table 3. The shot speed influenced the maximal intensities of both wrist moment components with higher peak values during Fast shots. The racket properties had no influence on flexion-extension and radial-ulnar moments of the Intermediate player. For the Advanced player, the use of racket R2 modified intensities of wrist moments at Pre-Impact with a lower radial-ulnar moment and a higher flexion-extension moment during Fast shots and higher radial-ulnar moment Slow shots (Table 3). The use of R1 for that same player resulted in a slightly lower extension moment intensities at Pre-Impact (Table 3). The comparison of players showed differences in temporal evolution and intensities of both wrist moment components as well as differences in the direction of the radial-ulnar moment. The peak flexion moment value close to the beginning of forward acceleration was lower and reached earlier for the Advanced player. The peak extension moment value during Fast shots around the impact was higher and reached earlier for the Advanced player. The peak radial-ulnar deviation moment values were higher for the Advanced player. At the beginning of the forward acceleration, the advanced player produced a peak radial moment whereas the Intermediate player produced a constant ulnar moment.

The wrist moment efficiency indexes at Pre-Impact were not influenced by shot speed or racket properties for the Intermediate player (Table 3). For the Advanced player, only the extension moment efficiency index changed between speed shots, presenting higher, suggesting less efficient kinetic strategy, during Fast shots. The ulnar moment efficiency index of the Advanced player was higher, suggesting less efficient kinetic strategy, when using R2 during Fast shots. The comparison of players demonstrated that the Advanced player was using a less efficient kinetic strategy with values two to four times higher than for the Intermediate player.
<figure><img src="image_6.png" /><figcaption>Hand muscle forces during forehand drive</figcaption></figure>

Figure 4. Mean time-patterns (N=5 trials) of estimated muscle forces ($\sum F^m$) during the forehand drives of both players (Advanced and Intermediate) for the two shot speeds (Slow and Fast) performed with the three rackets (R1, R2, R2w). Shaded areas represent $\pm$ one standard deviation around the mean. Lighter nuances represent the Slow (s) speed shots and darker nuances the Fast (f) ones. The vertical solid black bar represents the impact frame, the dashed vertical bars represent the beginning and end of forward acceleration phase (see corpus). W-ext: wrist extensors (ECRB, ECRL, ECU); F-ext: finger extensors (four compartments of EDC); W-flex: wrist flexors (FCR, FCU, PL); F-flex: finger flexors (four compartments of FDS and four compartments of FDP)

The hand muscle group forces during forehand drives estimated using the musculoskeletal model are presented in Figure 4 and Table 4. The shot speed influenced muscle forces with higher values at Pre-Impact during Fast shots, except for the finger
extensors F-ext. For the Advanced player, these increases were more pronounced for the wrist extensors (W-ext) and finger flexors (F-flex) with up to twice higher values than during Slow shots. For the Intermediate player, the muscle force increase reached less important values and was more evenly distributed across muscle groups, from about 35% for finger flexors to 75% for wrist flexors. The racket properties had no influence on F-ext and W-flex forces and influenced forces of other muscle groups differently for each player. For the Advanced player, the use of R2 increased W-ext force during Fast shots and reduced F-flex force during Slow shots. For the Intermediate player, the use of R2w increased W-ext muscle force during Fast shots and F-flex muscle forces for both shot speeds. The comparison of players suggested different muscle coordination. Compared to the Intermediate player, the Advanced player presented higher muscle forces of W-ext during Fast shots and lower W-flex and F-flex muscle forces for both speeds.

The muscle force efficiency index at Pre-Impact was not influenced by shot speed for the Intermediate player (Table 3). For the Advanced player, only the muscle force efficiency index of W-ext changed between shot speeds, presenting higher values, suggesting a less efficient strategy, during Fast shots. The influence of racket properties was different for each player. When the advanced player used R2, the muscle force efficiency of W-ext was higher, suggesting a less efficient strategy, during Fast shots and that of F-flex was lower, suggesting a more efficient strategy, during Slow shots. When the Intermediate player used R2w, the muscle force efficiency of W-ext and F-flex were higher, suggesting a less efficient strategy, during Fast and Slow shots, respectively. The comparison of players showed that the Advanced player presented three times higher W-ext efficiency index, suggesting a less efficient strategy.
Additional graphics presenting time-patterns of individual muscle forces are presented in Figures s3-4 and s3-5 of Electronic Supplementary Material ESM3.

<table><caption>Table 4. Mean ± one standard deviation values of estimated muscle group forces (ΣF<sup>m</sup>) at Pre-Impact (three frames before impact) for both players (Advanced and Intermediate) for each of the two shot speeds (Slow and Fast) performed with the three rackets (R1, R2, R2w)</caption><thead><tr><th rowspan="2">Muscle forces<br>at Pre-impact</th><th colspan="2">Normalised value<br>(%MGF)</th><th colspan="2">Efficiency index<br>(%V<sub>ball</sub><sup>out</sup>)</th></tr><tr><th>Slow</th><th>Fast</th><th>Slow</th><th>Fast</th></tr></thead><tbody><tr><td colspan="5"><strong>Σ F<sup>W-ext</sup></strong></td></tr><tr><td>Advanced</td><td></td><td></td><td></td><td></td></tr><tr><td>R1</td><td>52.6 ± 6.4</td><td>125.0 ± 15.5</td><td>4.3 ± 0.5</td><td>5.3 ± 0.4</td></tr><tr><td>R2</td><td>70.8 ± 6.2</td><td><strong>147.6 ± 15.2 *</strong></td><td>4.7 ± 0.4</td><td><strong>5.6 ± 0.6 *</strong></td></tr><tr><td>R2w</td><td>63.9 ± 10.6</td><td>132.3 ± 12.6</td><td>4.8 ± 0.7</td><td>5.1 ± 0.3</td></tr><tr><td>Intermediate</td><td></td><td></td><td></td><td></td></tr><tr><td>R1</td><td>35.3 ± 8.0</td><td>46.1 ± 4.3</td><td>1.7 ± 0.2</td><td>1.6 ± 0.1</td></tr><tr><td>R2</td><td>34.1 ± 4.0</td><td>51.3 ± 4.9</td><td>1.7 ± 0.2</td><td>1.6 ± 0.2</td></tr><tr><td>R2w</td><td>31.0 ± 2.4</td><td><strong>55.4 ± 4.9 *</strong></td><td>1.7 ± 0.3</td><td><strong>1.8 ± 0.2 *</strong></td></tr><tr><td colspan="5"><strong>Σ F<sup>F-ext</sup></strong></td></tr><tr><td>Advanced</td><td></td><td></td><td></td><td></td></tr><tr><td>R1</td><td>3.8 ± 1.2</td><td>6.9 ± 2.7</td><td>0.3 ± 0.1</td><td>0.3 ± 0.1</td></tr><tr><td>R2</td><td>3.6 ± 1.4</td><td>4.9 ± 1.5</td><td>0.2 ± 0.1</td><td>0.2 ± 0.1</td></tr><tr><td>R2w</td><td>3.1 ± 0.4</td><td>5.8 ± 2.8</td><td>0.2 ± 0.0</td><td>0.2 ± 0.1</td></tr><tr><td>Intermediate</td><td></td><td></td><td></td><td></td></tr><tr><td>R1</td><td>4.7 ± 1.3</td><td>6.0 ± 0.6</td><td>0.2 ± 0.0</td><td>0.2 ± 0.0</td></tr><tr><td>R2</td><td>4.5 ± 1.3</td><td>6.3 ± 0.5</td><td>0.2 ± 0.0</td><td>0.2 ± 0.0</td></tr><tr><td>R2w</td><td>4.1 ± 0.9</td><td>5.8 ± 0.9</td><td>0.2 ± 0.0</td><td>0.2 ± 0.0</td></tr><tr><td colspan="5"><strong>Σ F<sup>W-flex</sup></strong></td></tr><tr><td>Advanced</td><td></td><td></td><td></td><td></td></tr><tr><td>R1</td><td>11.2 ± 1.5</td><td>21.8 ± 8.4</td><td>0.9 ± 0.1</td><td>0.9 ± 0.3</td></tr><tr><td>R2</td><td>12.8 ± 2.6</td><td>18.9 ± 6.1</td><td>0.9 ± 0.2</td><td>0.7 ± 0.2</td></tr><tr><td>R2w</td><td>11.3 ± 1.7</td><td>14.5 ± 4.9</td><td>0.8 ± 0.1</td><td>0.6 ± 0.2</td></tr><tr><td>Intermediate</td><td></td><td></td><td></td><td></td></tr><tr><td>R1</td><td>11.3 ± 2.4</td><td>14.8 ± 4.1</td><td>0.6 ± 0.1</td><td>0.5 ± 0.1</td></tr><tr><td>R2</td><td>9.8 ± 3.1</td><td>17.3 ± 2.7</td><td>0.5 ± 0.2</td><td>0.5 ± 0.1</td></tr><tr><td>R2w</td><td>8.1 ± 3.9</td><td>17.3 ± 3.5</td><td>0.4 ± 0.2</td><td>0.6 ± 0.1</td></tr><tr><td colspan="5"><strong>Σ F<sup>F-flex</sup></strong></td></tr><tr><td>Advanced</td><td></td><td></td><td></td><td></td></tr><tr><td>R1</td><td>25.1 ± 7.2</td><td>43.8 ± 6.5</td><td>2.1 ± 0.6</td><td>1.8 ± 0.2</td></tr><tr><td>R2</td><td><strong>18.0 ± 2.2 *</strong></td><td>45.0 ± 4.9</td><td><strong>1.2 ± 0.1 *</strong></td><td>1.7 ± 0.2</td></tr><tr><td>R2w</td><td>25.2 ± 4.9</td><td>44.0 ± 4.4</td><td>1.9 ± 0.4</td><td>1.7 ± 0.2</td></tr><tr><td>Intermediate</td><td></td><td></td><td></td><td></td></tr><tr><td>R1</td><td>33.3 ± 9.3</td><td>47.5 ± 3.1</td><td>1.6 ± 0.3</td><td>1.7 ± 0.1</td></tr><tr><td>R2</td><td>35.9 ± 4.9</td><td>46.4 ± 7.4</td><td>1.8 ± 0.1</td><td>1.5 ± 0.2</td></tr><tr><td>R2w</td><td><strong>41.4 ± 4.3 *</strong></td><td><strong>55.3 ± 4.7 *</strong></td><td><strong>2.3 ± 0.3 *</strong></td><td><strong>1.8 ± 0.3 *</strong></td></tr></tbody></table>

Normalised forces are expressed as percentage of maximal grip force (MGF). Efficiency is the percentage-normalised value divided by ball velocity after impact (V<sub>ball</sub><sup>out</sup>) in m/s. Bold font and * symbol highlight a high value across the three rackets. Bold italic font and # symbol highlight a low value across the three rackets. W-ext : wrist extensors (ECRB, ECRL, ECU); F-ext : finger extensors (four compartments of EDC); W-flex : wrist flexors (FCR, FCU, PL); F-flex : finger flexors (four compartments of FDS and four compartments of FDP)

Discussion

This study aimed at quantifying hand extensor tendon forces, often affected by lateral epicondylalgia, during tennis gestures based on a methodology combining
motion capture, electromyography, and musculoskeletal modelling. The use of motion capture and electromyography provided an in vivo evaluation of grip force and wrist moment, as well as tendon forces, while avoiding invasive sensors that perturb the player gesture or modify racket properties. The methodology was tested with two players performing forehand drives with three rackets and two different speed shots. The results suggest the methodology provides physiologically realistic estimation of hand biomechanical loadings. The study also shows that muscle forces during forehand drives seemed more influenced by shot speed or player-specific technique than racket properties. The loadings withstood by wrist extensors were the highest among the main hand muscle groups and were modulated by both the grip force and wrist moment that varies according to player's technique, e.g., racket motion dynamics and grip position.

Despite a lack of data and a variability of assessment methods in the literature, the hand biomechanical loading values obtained using the present methodology were consistent with other works. As already observed before in EMG analysis of the forehand, ECR and FCR activation reached similar peak levels³³ in the acceleration phase or early follow-through³². Contrary to a previous study³², EDC activation remained stable and low. However, little information was provided in this early work on the instruction regarding how players performed forehand drives, e.g., ball speed or accuracy constraints, such that it is difficult to understand this difference. FDS activation during forehand has never been evaluated but the time-pattern of its envelope is consistent with previous recordings of grip force, with a peak around the impact time¹²⁻¹⁴. The maximal grip force estimated from FDS activation demonstrated a similar time-pattern than in previous works but reached higher absolute levels than previous works, up to 1000N against 200-300 N¹²,¹⁴. The higher values can be explained by different measurement tools. The MGF was measured using a six-beam handle in the
present study whereas previous evaluations relied on pressure map and unidirectional dynamometer which underestimate grip forces¹⁰. Finally, wrist moment intensities were also comparable to a previous evaluation during forehand drives, i.e., values between 5 and 15 N.m at Pre-impact²³. Those relatively good agreements with the literature confirm the hand biomechanical loading estimated with the present methodology are physiologically realistic.

## Influence of shot speed on hand biomechanical loadings

The results showed that, for the two analysed players, a higher shot speed increased the grip force exerted by the player as well as wrist moment due to racket motion dynamics (Figure 3 and Table 3) and resulted in higher hand tendon forces (Figure 4 and Table 4). These increases could be expected as the players increased by about 30% the inbound racket velocity during Fast shots compared to Slow ones (Table 2). As a result, the wrist moment increased because of the higher linear and angular inertial moments resulting from greater accelerations of the hand-racket system before impact. The player also exerted around 30% higher grip force during Fast shots, probably in an effort to maintain racket motion stability, compensating for increased accelerations and anticipating greater ball/racket impact forces. With higher wrist moments and grip forces, the muscle forces estimated by the musculoskeletal model also increased with shot speed, but this increase was spread differently across muscle groups for each player. Fast shots led to a two-fold increase of wrist extensor loading for the Advanced player whereas it led to lesser and more evenly distributed variations across all muscle group forces for the Intermediate player (+35 to +75%). This difference between players can be explained by a modification of the gesture efficiency. The efficiency indexes indeed showed that, for a same outbound ball velocity, the Advanced player withstood a greater extension moment during Fast shots (Table 3) that
resulted in higher wrist extensor forces (Table 4), suggesting a less efficient strategy than during Slow shots. On the contrary, the efficiency indexes of the Intermediate player remained relatively stable across shot speeds (Table 3 and 4), suggesting a similar gesture strategy between shot speeds resulting in a scaling of mechanical loadings with performance. The degradation of the Advanced player gesture efficiency could be explained by the fact that the difference between the racket velocities of his Slow and Fast shots was higher (40%) than for the Intermediate player (20%). Because of this larger difference in speed, the Advanced player significantly changed his joint motion coordination whereas the Intermediate player moderately changed it (see time-pattern of joint angles in Figure s2-3 of Electronic Supplementary Materials ESM2). Those results suggest the adjustments of hand biomechanical loading to shot speed are not necessarily linear and that considering this gesture parameter is necessary to fully understand the wrist and finger muscle forces associated with tennis gesture.

## Influence of racket properties on hand biomechanical loadings

The results showed that racket properties had a relatively low influence on hand biomechanical loadings for the two analysed players with different trends between players. For the Intermediate player, the efficiency indexes suggested that, for a given shot speed, the use of R2w increased the finger flexor muscle force (Figure 4 and Table 4). This result could be explained by the fact that R2w racket is the heaviest (Table 1) and might thus require a firmer grip to ensure a proper control of its trajectory and orientation at impact which is confirmed with higher grip force indexes (Table 3). As the wrist moment intensities were equivalent, the data suggest the Intermediate player maintained similar racket and joint motion dynamics but had to increase its grip force to face the higher weight of the racket. The R2w racket appears inappropriate for the Intermediate player compared to other rackets as, for a given speed, it resulted in a
supplementary load on finger flexors that could potentially lead to fatigue or musculoskeletal disorders. For the Advanced player, the use of racket R2 appears to have modified its hand biomechanical loading but the trend is unclear. With this racket, the finger flexor force was lower during Slow shots and the wrist extensor force was higher during Fast shots. The reason for these adjustments can be explained by the fact that the use of R2 resulted in a lower grip force during Slow shots and a higher wrist flexion-extension moment intensities during Fast shots. Nevertheless, the fact that the trend was not constant across shot speeds and influenced different muscle groups does not allow to conclude on whether the R2 racket is suited for the Advanced player. Furthermore, the R2 racket is neither the lightest nor the heaviest racket nor possesses extreme moments of inertia values compared to other rackets (Table 1) such that no rationale can be found on the influence on muscle force. The use of R1 also resulted in unclear trends for the Advanced player as it modified the wrist moment intensities but not muscle forces. Overall, the changes in hand biomechanical loadings resulting from different racket use during forehand appears marginal despite large variations in racket weights (from 270 for R1 to 320 g for R2w). A previous study identified that the polar moment could significantly influence wrist moment components²² but these differences were observed during the serve motion where the racket motion dynamics are much higher than in the current study, i.e., Pre-Impact racket speed of 22m/s against 8m/s here. Beyond these differences with literature, only two players were considered so that no trend can be directly expressed from our results and further studies are required to understand the effect of racket properties on wrist and finger muscle forces and potentially confirm whether this effect is player specific.
Influence of player technique on hand biomechanical loadings

The results indicated that the Advanced player muscle force coordination was less efficient than the one of the Intermediate Player with a much higher contribution of wrist extensor muscles (Table 4). The efficiency indexes indicated that, for a given speed shot and considering maximal grip force capacity, the wrist extensor muscle forces of the Advanced player were 2.5 to 3.5 five times higher than those of the Intermediate player. This inefficient coordination of the Advanced player does not result from grip force efficiency, which was comparable with the Intermediate player, but from the upper limb and racket motion dynamics which affected the wrist flexion-extension moment, which was 3.5 times higher once normalised by muscle capacities and shot speed. This higher wrist moment intensities observed for the Advanced player resulted from the larger amplitudes of elbow flexion-extension and forearm pronosupination (Figure s2-3 in Electronic Supplementary Materials ESM2) increasing the inertial effects of the angular acceleration of the hand racket system (see time-pattern of wrist moment components in Figure s3-1 of Electronic Supplementary Materials ESM3). These increased joint angle amplitudes of the Advanced player probably resulted from his use of an eastern grip while the Intermediate player used a semi-western grip. As already observed by Elliott et al.⁴¹, the use of an eastern grip by the Advanced player involved a greater involvement of elbow extension, increasing the relative distance of the racket, to produce racket speed. On the contrary, by using a semi-western grip, the Intermediate player maintained a more flexed elbow and supinated forearm, keeping the hand closer to the trunk and relying on shoulder internal rotation to develop racket speed. This different joint coordination results in different angular and linear accelerations of the hand-racket system and impact differently the wrist moment components. The results from the present study suggest that an eastern grip position might be less efficient from the hand biomechanical loading point of view,
resulting in higher moment intensities at the wrist. More importantly, this inefficiency
might negatively impact wrist extensors by increasing their tensile loads and potentially
lead to a higher risk of LE<sup>7</sup>, confirming the pathology could be related to player's
technique<sup>1</sup>. Again, as only two participants were investigated, a larger population will
be necessary to elucidate the influence of player-specific grip position and joint
coordination on the forces acting on the common extensor tendon. Nevertheless, the use
of the EMG-informed musculoskeletal model highlighted that hand tendon loading
results from an interplay between the level of grip force wrist moment intensities, both
directly implied in the control of racket motion, but that might vary between different
players. Those results corroborate the idea that hand muscle coordination is related to
both wrist moment and grip force<sup>19</sup>.

## Limitations

Some limitations should be considered when interpreting the results from this
study. First, only two players participated in the study to test this methodology so that
the trends observed need to be investigated with larger population samples to be
confirmed, especially regarding the risk of tendinopathies like LE. Second, the accuracy
of the grip force estimation was on average 10% of MGF which is not negligible.
Nevertheless, this error is comparable to previous attempts to estimate grip forces from
electromyography<sup>15-17</sup> and is inherent to the non-linear relationship between EMG and
force. This error remains acceptable considering the use of EMG represents the only
alternative way to avoid the use of sensors modifying the hand/racket interface such as
dynamometer<sup>14</sup> or force sensitive systems<sup>12,13</sup>. A sensitivity analysis (Figure s3-6
Electronic Supplementary ESM3) showed that the grip force estimation uncertainty
mainly modulates the implication of finger extrinsic flexors (FDP and FDS
compartments) as well as few intrinsic muscles but not extensor tendon forces. The
sensitivity analysis also showed that the muscle load sharing is sensitive to the radial-
ulnar deviation moment, so that care should be taken regarding its calculation. Finally,
the musculoskeletal model relied on generic muscle capacities, e.g., PCSA, whereas
studies showed that tennis playing modifies the hand musculature non-uniformly across
hand flexors and extensors²⁴. The obtained muscle load sharing could have thus been
modified if the musculoskeletal model included player-specific capacities. Nevertheless,
obtaining this player-specific muscle strength profile would have required another
measurement session, whereas the protocol already lasted two hours and a half on
average. The muscle forces estimated through this methodology nevertheless provide a
good order of magnitude of the loading withstood by hand muscle tendons during the
tennis forehand drive as well as how shot speed, racket properties and player's
technique can influence them.

Perspective

This study proposed a methodology to evaluate hand biomechanical loading
during tennis gesture using motion capture, electromyography, and musculoskeletal
modelling. The methodology provided estimations of grip force, wrist moment and
muscle forces that were consistent with previous works. The wrist extensors withstood
among the highest muscle forces and the total force exerted on the common extensor
tendon could reach levels close to 1000N for relatively moderate shot speeds, i.e.,
outbound ball velocity below 30m/s. Those values confirm wrist extensors are exposed
to large amount of force during tennis playing, corroborating their exposure to
tendinopathies like LE. Nevertheless, the use of the EMG-informed musculoskeletal
model suggests that this loading is modulated by the interplay between the level of grip
force and the intensities of the two wrist moment components that depends on player-
specific technique, e.g., grip position and joint coordination. Further studies are thus
required to evaluate the role of those player-specific strategies on hand biomechanical
loading. The methodology should also be applied to study hand muscle forces during
backhand drives which seems associated to abnormal wrist extensor loading¹. Future
works should also attempt to reduce the equipment for both the racket and player to
facilitate its use outside laboratory environment.

Acknowledgements

The authors would like to thank the participants for their patience and Dr Hugo Hauraix
for his assistance during the data acquisition.

Data availability statement

The datasets generated and/or analysed during the current study are available from the
corresponding author on reasonable request.

Conflict of interest disclosure

Part of the study was funded through a research collaboration between the Decathlon
company and the Aix-Marseille university. The authors will not receive any financial
benefits from the results of this study and declare no competing interest.

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**Electronic supplementary materials**

Electronic Supplementary Material 1 (ESM1) – PDF File - Results and methodological details of the grip calibration protocol

Electronic Supplementary Material 2 (ESM2) – PDF File - Additional results and methodological details of the forehand motion capture protocol

Electronic Supplementary Material 3 (ESM3) – PDF File - Additional results and methodological details of the musculoskeletal model