# Uncertainty estimation in marker-based motion capture of the tennis serve

Simon Ozan, Loic Fourel, P Touzard, Richard Kulpa, Caroline Martin

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Simon Ozan, Loic Fourel, P Touzard, Richard Kulpa, Caroline Martin. Uncertainty estimation in marker-based motion capture of the tennis serve. Journal of Biomechanics, 2026, 195, pp.113119. 10.1016/j.jbiomech.2025.113119. hal-05453654

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# Uncertainty estimation in marker-based motion capture of the tennis serve

Simon Ozan<sup>a,*</sup>, Loïc Fourel<sup>a</sup>, Pierre Touzard<sup>a</sup>, Richard Kulpa<sup>a,1</sup>, Caroline Martin<sup>a,1</sup>

<sup>a</sup> Univ Rennes, Inria, M2S, 35000 Rennes, France

## ARTICLE INFO

**Keywords:**

Tennis serve

Kinematics

Uncertainty

Marker-based Motion Capture

Error propagation

## ABSTRACT

Marker-based motion capture (MoCap) systems are widely used to analyse human movement. However, they are affected by measurement uncertainties, particularly marker placement errors (MPE) and soft tissue artefacts (STA). Here, we quantify the individual and combined effects of these two sources of uncertainty on joint angles and angular velocities in the case of the tennis serve. A Monte-Carlo approach was used to simulate 3000 perturbed marker trajectories for each uncertainty source and their combination. We applied a random offset for MPE, while sinusoidal perturbations were used to simulate for STA. The resulting joint kinematics were compared across all degrees of freedom. Confidence intervals (5–95 %), root mean square deviation (RMSD) and Minimal Detectable Changes (MDC) were calculated for key biomechanical variables. Results showed that STA predominantly affected angular velocities, while MPE had a greater impact on joint angles. The combined simulation consistently produced the largest variability, with mean confidence intervals ranging from 5.1° to 30.8° for joint angles and from 70.5°.s⁻<fnref n="1" /> to 248.5°.s⁻<fnref n="1" /> for joint angular velocities, and RMSD values ranging from 1.6° to 8.4° for joint angles and from 16.8°.s⁻<fnref n="1" /> to 68.0°.s⁻<fnref n="1" /> for joint angular velocities. To our knowledge, this is the first quantification of MPE and STA effects on ballistic movement kinematics. These results provide critical reference values, enabling more accurate comparisons across subjects and studies while accounting for measurement uncertainties.

## 1. Introduction

Since the serve is considered as the most crucial stroke in competitive tennis (Ma et al., 2013) and represents approximately 45–60 % of all strokes performed during Grand Slam matches (Johnson et al., 2006), extensive biomechanical analyses have been carried out to understand the mechanisms underlying performance and to prevent athlete injuries (Fett et al., 2021; Martin et al., 2014, 2016; Reid et al., 2007, 2013; Reid & Giblin, 2015; Touzard et al., 2019). By integrating human movements measured in the laboratory with models of the skeletal system, it is possible to estimate biomechanical outputs such as joint kinematics. However, these estimates are often subjected to various sources of uncertainty that must be considered for accurate interpretation and to provide effective feedback to tennis players and coaches for improving their technique.

The analysis of tennis serve usually relies on optoelectronic motion capture (MoCap) systems (Martin et al., 2014, 2016; Reid et al., 2012; Whiteside et al., 2013). These systems measure the marker positions at

each timeframe with an error of less than 1 mm (Topley & Richards, 2020). The accuracy of this method depends on the position and the movements of the markers relative to the anatomical points, which can introduce measurement errors. Marker Placement Error (MPE) are due to factors such as the experimenter's ability to accurately locate anatomical landmarks by palpation and the subject's morphology (de Groot 1997; Della Croce et al., 1999; Fonseca et al., 2023; Moriguchi et al., 2009). Additionally, markers can shift relatively to the underlying segment due to the unavoidable deformation of the soft tissue between bone and markers, known as the soft-tissue artefact (STA) (Blache et al., 2017; Bonci et al., 2014; Cappozzo et al., 1996; Grimpampi et al., 2014; Leardini et al., 2005; Thouzé et al., 2016). STA arises from soft tissue volumetric change due to muscular contraction and gravity, skin sliding and inertial effects (Bonci et al., 2014). STA are task, subject and segment dependent (Blache et al., 2017; Bonci et al., 2014; Thouzé et al., 2016).

During biomechanical analysis, it becomes crucial to quantify how these uncertainties affect all the kinematic variables. Indeed, accurate

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: simon.ozan@univ-rennes2.fr (S. Ozan), loic.fourel@univ-rennes2.fr (L. Fourel), pierre.touzard@univ-rennes2.fr (P. Touzard), richard.kulpa@univ-rennes2.fr (R. Kulpa), caroline.martin@univ-rennes2.fr (C. Martin).</footnote>

<footnote n="1">¹ These authors contributed equally to this work and share last authorship.</footnote>
quantification of measurement uncertainties will provide guidelines to
differentiate true biomechanical change from measurement errors.
Some studies have quantified the error of marker-based MoCap systems
by comparing it to an even more accurate method like bi-planar video-
radiography (Akbarshahi et al., 2010; Fiorentino et al., 2017; Miranda
et al., 2013; Wang et al., 2023) or using intracortical pins (Begon et al.,
2015; Benoit et al., 2006; Blache et al., 2017). However, biplane fluo-
roscopy has a restricted analysis volume and intracortical pins must be
inserted directly in the bones making them unsuitable for evaluating the
marker-based uncertainty during larger and more dynamic movements
like the tennis serve. Probabilistic approaches, such as Monte-Carlo
simulations, are effective to assess the influence of uncertainties on
model outputs (Lamberto et al., 2017; Lavaill et al., 2022; McFadden
et al., 2021; Myers et al., 2015). Monte-Carlo simulations showed an
uncertainty in lower limb joint kinematics ranging from 2.7 to 15°
during walking (Lamberto et al., 2017; Myers et al., 2015). This
approach was also used for shoulder movements with reported un-
certainties between 2.4° and 7.7° (Lavaill et al., 2022). These studies
focused on slow and isolated joint movements, and the approach has
never been applied to ballistic, complex, and coordinated actions
involving both the upper and lower limbs, such as the tennis serve. In the
context of tennis serve analysis, Monte-Carlo simulations can be
employed to create variations in the markers positions by introducing
realistic errors due to MPE and STA that may occur during motion
capture. By incorporating these uncertainties into the model, the simu-
lations generate output distributions that reflect the variability in ki-
nematic data caused by these sources of uncertainty.

The aim of this study is to evaluate the propagation of measurement
uncertainties caused by MPE, STA and their combined effect through the
biomechanical analysis of the tennis serve. This evaluation seeks to
provide a clearer understanding of how such uncertainties impact key
kinematic performance indicators. More precisely, the study aims to
estimate the confidence intervals associated with each variable and to
determine their Minimal Detectable Change (MDC). The MDC helps
researchers assess whether observed changes in performance indicators
reflect true biomechanical differences or fall within the range of mea-
surement error (Kovacs et al., 2008).

## 2. Method

Prior to experimentation, a voluntary participant (male, age: 24
years old, height: 1.74 m, weight: 77.0 kg, weekly tennis training: 3 h/
week, 11 years of tennis experience) was informed of the benefits and

risks of the investigation before signing an institutionally approved
informed consent document to participate in the study. This study was
conducted following the principles of the Declaration of Helsinki and
approved by a national ethics committee (Comité de Protection des
Personnes Ile de France I) (approval code: 2021-A02133-38; approval
date: 15/12/2021). Following a self-directed warm-up session similar to
that typically performed before a competition, the participant was
equipped with 34 *retro*-reflective markers (Fig. 1), placed on anatomical
landmarks as detailed in Table 1. To facilitate the labelling process,
additional markers were positioned as follows: two on the proximal and
distal heads of the third metacarpal bones, four on the head, two on the
heads of the fifth metatarsals, four asymmetric markers on both thighs
and both upper arms, and one on the right scapula. Five additional
markers were attached to the racket (Fourel et al., 2024; Martin et al.,
2016). Marker trajectories were recorded at 300 Hz using a 10-camera
motion capture system (Arqus A12, Qualisys AB., Göteborg, Sweden)
with a calibration accuracy of 0.44 mm. The participant performed
seven successful flat serves in a target area (1 m x 2 m bordering the “T”
zone) in the opponent’s service box on the deuce side. The player was
asked to serve at his best speed as in an official tournament. The fastest
successful trial was selected for subsequent analysis.

### 2.1. *Marker perturbation*

Following the approach of Myers et al. (2015) and Lavaill et al.
(2022), two types of experimental perturbations were added to raw
marker trajectories: (1) marker placement errors (MPE) and (2) soft
tissue artefacts (STA). A third condition was modelled by combining
MPE and STA, reflecting situations typically encountered in a motion
capture laboratory. MPE and STA perturbations ($P^{MPE}$ and $P^{STA}$) were
applied to each marker in the segment coordinate system defined by the
International Society of Biomechanics (ISB) recommendations (Wu
et al., 2002, 2005), and then transformed into the global coordinate
system to obtain the perturbed trajectories.

#### 2.1.1. *Marker placement error perturbation ($P^{MPE}$)*

MPE comes from variability in the experimenter's ability to accu-
rately place markers on body landmarks though palpation. These errors
are supposed to be constant through the entire motion in the segment
reference coordinate system. The variability of marker placement be-
tween operators was modelled following de Groot (1997) regarding
upper limbs and thorax misplacement, and Fonseca et al. (2023) for the
pelvis and lower limbs. The offsets used to perturb marker trajectories in

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Marker placement and motion reconstruction. a) Anterior view of the participant with reflective markers. b) Posterior view of the participant with reflective markers. c) 3D reconstruction of the tennis serve from motion capture data.</figcaption></figure>
<caption>**Table 1**</caption>

Inter-operator variability (1 standard deviation) in locating body landmarks on the lower and upper body.

<table><thead><tr><th>Segment</th><th>Skeletal Landmarks</th><th>Medio-lateral axis (mm)</th><th>Antero-posterior axis (mm)</th><th>Superior axis (mm)</th><th>References</th></tr></thead><tbody><tr><td rowspan="4">Pelvis</td><td>Left anterior superior iliac spine (LASIS)</td><td>4.5</td><td>4.2</td><td>5.6</td><td rowspan="4">Fonseca et al., 2023</td></tr><tr><td>Right anterior superior iliac spine (RASIS)</td><td>4.9</td><td>4.6</td><td>4.7</td></tr><tr><td>Left posterior superior iliac spine (LPSIS)</td><td>5.7</td><td>1.8</td><td>5.6</td></tr><tr><td>Right posterior superior iliac spine (RPSIS)</td><td>4.1</td><td>1.8</td><td>6.1</td></tr><tr><td rowspan="4">Thigh</td><td>Left femur medial epicondyle (LME)</td><td>4.7</td><td>8.9</td><td>7.0</td><td rowspan="4">Fonseca et al., 2023</td></tr><tr><td>Right femur medial epicondyle (RME)</td><td>5.4</td><td>6.6</td><td>8.9</td></tr><tr><td>Left femur lateral epicondyle (LLE)</td><td>2.6</td><td>7.9</td><td>8.2</td></tr><tr><td>Right femur lateral epicondyle (RLE)</td><td>4</td><td>9.2</td><td>9.4</td></tr><tr><td rowspan="4">Shank</td><td>Left medial malleolus (LMM)</td><td>2</td><td>3.9</td><td>3.2</td><td rowspan="4">Fonseca et al., 2023</td></tr><tr><td>Right medial malleolus (RMM)</td><td>3</td><td>5.9</td><td>3.9</td></tr><tr><td>Left lateral malleolus (LLM)</td><td>4.6</td><td>3.0</td><td>5.1</td></tr><tr><td>Right lateral malleolus (RLM)</td><td>1.9</td><td>3.0</td><td>4.2</td></tr><tr><td rowspan="4">Foot</td><td>Left calcaneus (LHEE)</td><td>9.9</td><td>2.5</td><td>9</td><td rowspan="4">Fonseca et al., 2023</td></tr><tr><td>Right calcaneus (RHEE)</td><td>4.3</td><td>2.3</td><td>8</td></tr><tr><td>Left hallux (LTOE)</td><td>6</td><td>5.5</td><td>2.7</td></tr><tr><td>Right hallux (RTOE)</td><td>3.3</td><td>5.3</td><td>2.1</td></tr><tr><td rowspan="6">Thorax</td><td>Incisura Jugularis (IJ)</td><td>1.4</td><td>1.9</td><td>1.6</td><td rowspan="6">de Groot, 1997</td></tr><tr><td>7th cervical vertebrae (C7)</td><td>2.3</td><td>1.9</td><td>3.3</td></tr><tr><td>Processus Xiphoideus (STRN)</td><td>2.1</td><td>2.2</td><td>1.6</td></tr><tr><td>10th thoracic vertebrae (T10)</td><td>1.3</td><td>3.1</td><td>1.2</td></tr><tr><td>Acromioclavicular joint (right and left) (SHO)</td><td>2.3</td><td>2.7</td><td>1.0</td></tr><tr><td>Humerus medial epicondyle (right and left) (HUM)</td><td>1.8</td><td>1.8</td><td>2.3</td></tr><tr><td rowspan="2">Arm</td><td>Humerus lateral epicondyle (right and left) (RAD)</td><td>1.8</td><td>1.8</td><td>2.3</td><td rowspan="2">de Groot, 1997</td></tr><tr><td>Radial Styloid (right and left) (WRA)</td><td>1.8</td><td>1.8</td><td>2.3</td></tr><tr><td rowspan="2">Forearm</td><td>Ulnar Styloid (right and left) (WRB)</td><td>1.8</td><td>1.8</td><td>2.3</td><td rowspan="2">Hypothesis</td></tr><tr><td>Head of the 2nd metacarpal (right and left) (MC2h)</td><td>1.8</td><td>1.8</td><td>2.3</td></tr><tr><td rowspan="2">Hand</td><td>Head of the 5th metacarpal (right and left) (MC5h)</td><td>1.8</td><td>1.8</td><td>2.3</td><td rowspan="2">Hypothesis</td></tr><tr><td></td><td></td><td></td><td></td></tr></tbody></table>

each plane were randomly selected from a normal distribution of the inter-rater variability reported in these studies (Table 1). Due to the lack of data on the inter-operator variability at wrist and hand levels, it was assumed that the amounts of variability were equivalent to the medial and lateral epicondyles of the humerus from de Groot (1997).

### 2.1.2. Soft tissue artefact perturbation ($P^{STA}$)

STA were modelled as sinusoidal functions of time expressed as $Asin(\omega t + \varphi)$, where A is the amplitude, $\omega$ the frequency and $\varphi$ the phase, each defined along the three axes of the segment coordinate system (Chèze et al., 1995; Dumas & Chèze, 2009; Lamberto et al., 2017; Lu & O'Connor, 1999; Soodmand et al., 2025). The parameters A, $\omega$ and $\varphi$ were randomly selected within the ranges detailed in Table 2. A ranges were defined according to literature data (Dumas & Chèze, 2009; Leardini et al., 2005; Ryu et al., 2006; Scheiterer et al., 2024). Frequencies $\omega$ were drawn between 0 and $25 \text{ rad.s}^{-1}$, corresponding to up to three times the movement frequency (Chèze et al., 1995; Lu & O'Connor, 1999), and $\varphi$ ranged from 0 to $2\pi$ (Chèze et al., 1995; Lu & O'Connor, 1999). As the markers placed on the foot were attached to the shoes, which were considered sufficiently rigid, foot markers were not disturbed by STA.

### 2.2. Monte-Carlo simulation

The Monte-Carlo approach (Pataky, 2022) consists in perturbing measured marker trajectories a large number of times to simulate $P^{MPE}$ and $P^{STA}$ with random processes as previously described.

The trajectories obtained are similar to that illustrated in Fig. 2. The two types of disturbances (i.e., MPE, STA) and their combination were analysed through three separate Monte-Carlo simulations. For the placement-error-only analysis, $P^{STA} = 0$, while for the STA-only analysis, $P^{MPE} = 0$. Each Monte-Carlo simulation consisted of 3 000 iterations.

### 2.3. Analysis

Once Monte-Carlo simulations are done, the analysis follows the traditional procedures used in biomechanical investigation of the tennis serve (Martin et al., 2016; Reid et al., 2013; Reid & Giblin, 2015). First, the perturbed 3D markers trajectories were filtered with a zero-lag second-order low-pass Butterworth filter with a cutoff frequency of 20 Hz, determined by residual analysis (Winter, 2009). Joint angles were calculated from filtered marker trajectories using Euler angles relative to the proximal segment, following the sequence recommended by the ISB guidelines (Wu et al., 2002, 2005) and by Bonnefoy-Mazure et al. (2010) for the shoulder motion. An exception was made for the trunk, whose orientation was expressed relative to the global coordinate system using the rotation sequence ZYX (Z: longitudinal axis, Y: antero-posterior axis, and X: medio-lateral axis) (Martin et al., 2016; Reid & Giblin, 2015; Whiteside et al., 2013). Key variables commonly studied in serve biomechanics, such as peak flexion angles of the front and back ankles and knees, maximum shoulder external rotation, and maximum

<table><caption>**Table 2**
Per-marker parameter ranges for STA simulation.</caption><thead><tr><th>Segment</th><th>Marker acronym</th><th>Parameters (A, ω, φ)</th><th>References</th></tr></thead><tbody><tr><td>Pelvis</td><td>ASIS, PSIS</td><td>A ≤ 15mm, ω ≤ 25rad.s<sup>-1</sup>, φ ≤ 2π</td><td>(Chèze et al., 1995; Scheiterer et al., 2024)</td></tr><tr><td>Thigh</td><td>ME, LE</td><td>A ≤ 15mm, ω ≤ 25rad.s<sup>-1</sup>, φ ≤ 2π</td><td>(Chèze et al., 1995; Dumas & Chèze, 2009)</td></tr><tr><td>Shank</td><td>MM, LM</td><td>A ≤ 10mm, ω ≤ 25rad.s<sup>-1</sup>, φ ≤ 2π</td><td>(Chèze et al., 1995; Leardini et al., 2005)</td></tr><tr><td>Foot</td><td>HEE, TOE</td><td>—</td><td>Hypothesis</td></tr><tr><td>Thorax</td><td>IJ, C7, STRN, T10, SHO</td><td>A ≤ 10mm, ω ≤ 25rad.s<sup>-1</sup>, φ ≤ 2π</td><td>Hypothesis; (Chèze et al., 1995)</td></tr><tr><td>Arm</td><td>HUM, RAD</td><td>A ≤ 10mm, ω ≤ 25rad.s<sup>-1</sup>, φ ≤ 2π</td><td>Hypothesis; (Chèze et al., 1995)</td></tr><tr><td>Forearm</td><td>WRA, WRB</td><td>A ≤ 10mm, ω ≤ 25rad.s<sup>-1</sup>, φ ≤ 2π</td><td>Hypothesis; (Chèze et al., 1995)</td></tr><tr><td>Hand</td><td>MC2h, MC5h</td><td>A ≤ 5mm, ω ≤ 25rad.s<sup>-1</sup>, φ ≤ 2π</td><td>(Chèze et al., 1995; Ryu et al., 2006)</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Right anterior superior iliac spine (RASIS) marker perturbed trajectory (dashed line) compared to measured trajectory (grey line). Perturbations were applied using simulated MPE (a), STA (b), and their combination (c). The illustrated MPE were: 4.7 mm along the pelvis medio-lateral axis, -8.5 mm along the pelvis antero-posterior axis, and 6.4 mm along the pelvis superior axis. For the STA simulation, the amplitude Ax, Ay and Az were 7.0, 14.7 and 14.4, respectively.</figcaption></figure>

velocities of both ankle and knee extension, trunk flexion, shoulder-over-shoulder rotation, trunk longitudinal rotation, elbow extension, forearm pronation and wrist flexion were analysed (Martin et al., 2016; Reid et al., 2013; Reid & Giblin, 2015). The angular velocity vector $\omega$ was computed in the global frame for each segment as $\omega = \dot{R}R^T$ where R is the segment rotation matrix. The relative angular velocity between child and parent segments was then expressed in the child segment's local frame, except for elbow flexion, which was defined around the upper arm's medio-lateral axis (parent segment).

## 2.4. Statistical analysis

The root mean square deviation (RMSD) and peak to peak amplitude ($\Delta p_{max}$) of the STA simulations were calculated and compared with published STA measurements to assess their representativeness (Cereatti et al., 2017).

Confidence limits of 5 to 95 % were calculated for each Degree of Freedom (DoF) between the ball toss and the ball impact, representing the range within which 90 % of the results fall. Additionally, the variability of each perturbed simulation (i.e., MPE, STA and Combined) was assessed by calculating the difference between the 5 % and 95 % confidence limits. The accuracy of the perturbed joint angles and angular velocities was evaluated by computing the average RMSD between the unperturbed and perturbed simulations (Fonseca et al., 2020; Lavaill et al., 2022).

Convergence of the results was considered achieved when the convergence criterion $\epsilon$, given by Equation (1), reached $5 \times 10^{-4}$ (Lavaill et al., 2022). $i$ corresponds to the number of iterations. $\epsilon$ was computed for all joint angles across the combined simulation. The convergence criterion ($\epsilon < 5 \times 10^{-4}$) was met between 705 and 2243, indicating that 3000 simulations are sufficient.

$$ \epsilon(i) = \frac{\sum_{j=1}^{i+9} \left| \frac{\Delta_{5-95\%}^{\max j+1} - \Delta_{5-95\%}^{\max j}}{\Delta_{5-95\%}^{\max j}} \right|}{10} \quad (1) $$

In the context of evaluating the serve kinematics, it is critical to distinguish between true experimental effects and measurement variability caused by MPE and STA. To address this, the MDC was calculated

for key biomechanical variables. The MDC was calculated based on the intra-simulation variance using the formula: $1.96*\sqrt{2*SEM}$. The standard error of measurement (SEM) was estimated as the standard deviation of the repeated measurements (Kovacs et al., 2008). MDC values derived from STA only were also considered separately to provide an estimate of MDC values independent of marker placement errors (MDC<sub>STA</sub>), thus allowing interpretation of data from the literature that used the same marker placement between two conditions, as in the study by Martin et al. (2016). The relative MDC (rMDC) was then computed by dividing the MDC by the key variable mean value.

## 3. Results

Table 3 presents the RMSD and $\Delta p_{max}$ of the STA simulation. The highest RMSD and $\Delta p_{max}$ were obtained in the pelvis and thigh segment whereas the hand showed the lowest RMSD and $\Delta p_{max}$.

Table 4 shows the differences between the lower and upper limits of the 90 % confidence intervals and RMSD for each method of perturbation across joints and DoF. As expected, for all joints, the confidence intervals are higher for the combined method.

The variability in joint angular velocities was evaluated across all Monte-Carlo simulations. Table 5 summarizes the differences between the lower and upper limits of the 90 % confidence intervals and RMSD. Unlike angle measurements, STA has a significantly greater effect on angular velocities than MPE. In total, STA accounts for approximately 86 % of the uncertainty in angular velocities, highlighting the dominant influence of STA on angular velocity outputs.

<table><caption>**Table 3**
Mean (± standard deviation) of RMSD and $\Delta p_{max}$ of STA values from 3000 iterations.</caption><thead><tr><th>Segment</th><th>RMSD (mm)</th><th>Δp<sub>max</sub> (mm)</th></tr></thead><tbody><tr><td>Pelvis</td><td>10.2 ± 03.0</td><td>27.1 ± 08.0</td></tr><tr><td>Thigh</td><td>10.2 ± 03.0</td><td>26.9 ± 08.0</td></tr><tr><td>Shank</td><td>6.8 ± 02.0</td><td>18.1 ± 05.4</td></tr><tr><td>Thorax</td><td>6.8 ± 02.0</td><td>18.0 ± 05.4</td></tr><tr><td>Arm</td><td>6.8 ± 02.0</td><td>18.2 ± 05.3</td></tr><tr><td>Forearm</td><td>6.8 ± 02.0</td><td>18.1 ± 05.3</td></tr><tr><td>Hand</td><td>3.4 ± 01.0</td><td>9.0 ± 02.7</td></tr></tbody></table>
<caption>**Table 4**</caption>

Mean 5–95% confidence limits and RMSD for each degree of freedom of each joint and for each source of uncertainty for the angles.

<table><caption>**Table 5**</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th rowspan="2">Joint</th><th colspan="3">5–95% confidence limits (°)</th><th colspan="3">RMSD (°)</th></tr><tr><th>MPE</th><th>STA</th><th>Combined</th><th>MPE [maximum]</th><th>STA [maximum]</th><th>Combined [maximum]</th></tr></thead><tbody><tr><td rowspan="10">Joint Angle</td><td rowspan="10">Flexion-Extension</td><td>Hip</td><td>7.9 ± 0.2</td><td>8.5 ± 0.2</td><td>11.3 ± 0.3</td><td>1.9 [8.1]</td><td>2.5 [5.3]</td><td>3.3 [10]</td></tr><tr><td>Knee</td><td>5.0 ± 0.6</td><td>4.3 ± 0.4</td><td>6.6 ± 0.7</td><td>1.3 [5.0]</td><td>1.3 [2.6]</td><td>1.9 [6.0]</td></tr><tr><td>Ankle</td><td>9.4 ± 1.3</td><td>4.1 ± 0.9</td><td>10.3 ± 1.5</td><td>2.4 [14.3]</td><td>1.2 [3.0]</td><td>2.8 [16.3]</td></tr><tr><td>Shoulder</td><td>3.5 ± 0.1</td><td>7.3 ± 0.4</td><td>8.2 ± 0.4</td><td>0.9 [3.7]</td><td>2.1 [4.9]</td><td>2.4 [6.1]</td></tr><tr><td>Elbow</td><td>2.3 ± 0.2</td><td>4.8 ± 0.3</td><td>5.3 ± 0.2</td><td>0.6 [2.3]</td><td>1.4 [3.1]</td><td>1.6 [3.9]</td></tr><tr><td>Wrist</td><td>5.2 ± 0.5</td><td>9.9 ± 0.8</td><td>11.2 ± 1.0</td><td>1.3 [5.4]</td><td>2.9 [6.2]</td><td>3.3 [9.7]</td></tr><tr><td>Hip</td><td>6.4 ± 0.3</td><td>7.8 ± 0.3</td><td>10.1 ± 0.5</td><td>1.6 [6.8]</td><td>2.3 [5.1]</td><td>2.9 [7.8]</td></tr><tr><td>Knee</td><td>14.4 ± 3.9</td><td>11.2 ± 2.5</td><td>18.5 ± 4.7</td><td>3.7 [15.5]</td><td>3.3 [7.9]</td><td>5.3 [19]</td></tr><tr><td>Ankle</td><td>12.4 ± 0.7</td><td>9.9 ± 1.0</td><td>15.8 ± 1.2</td><td>6.5 [29.2]</td><td>4.6 [10.8]</td><td>8.4 [33]</td></tr><tr><td>Shoulder</td><td>2.6 ± 0.3</td><td>6.2 ± 0.7</td><td>6.8 ± 0.8</td><td>0.7 [3.0]</td><td>1.8 [4.3]</td><td>2.0 [4.3]</td></tr><tr><td>Wrist</td><td>5.6 ± 0.7</td><td>11.0 ± 1.4</td><td>12.4 ± 1.6</td><td>1.4 [5.9]</td><td>3.2 [7.2]</td><td>3.6 [8.8]</td></tr><tr><td rowspan="6">Abduction-Adduction</td><td rowspan="6">Internal-External</td><td>Hip</td><td>18.9 ± 1.4</td><td>15.3 ± 1.0</td><td>24.6 ± 1.6</td><td>4.7 [19.8]</td><td>4.4 [10.8]</td><td>6.9 [24.0]</td></tr><tr><td>Knee</td><td>15.2 ± 3.8</td><td>14.0 ± 2.4</td><td>20.7 ± 4.6</td><td>4.1 [15.5]</td><td>4.1 [10]</td><td>6.1 [18.1]</td></tr><tr><td>Ankle (Inv/Eversion)</td><td>26.2 ± 2.0</td><td>15.8 ± 1.6</td><td>30.8 ± 2.6</td><td>3.0 [14.6]</td><td>2.8 [7.6]</td><td>4.4 [16.2]</td></tr><tr><td>Shoulder</td><td>4.5 ± 0.8</td><td>9.3 ± 1.7</td><td>10.5 ± 1.9</td><td>1.2 [5.2]</td><td>2.8 [6.0]</td><td>3.1 [7.8]</td></tr><tr><td>Elbow (Pro/Supination)</td><td>7.4 ± 0.4</td><td>16.1 ± 0.8</td><td>17.6 ± 0.8</td><td>1.8 [8.3]</td><td>4.6 [10.5]</td><td>5.1 [12.4]</td></tr><tr><td>Trunk flexion</td><td>2.6 ± 0.3</td><td>4.9 ± 0.8</td><td>5.6 ± 0.8</td><td>0.7 [2.4]</td><td>1.5 [3.0]</td><td>1.6 [3.5]</td></tr><tr><td rowspan="2">Segmental Angle</td><td>Shoulder-over-shoulder</td><td></td><td>1.9 ± 0.2</td><td>4.8 ± 0.6</td><td>5.1 ± 0.6</td><td>0.5 [2.2]</td><td>1.4 [3.5]</td><td>1.5 [3.6]</td></tr><tr><td>Trunk longitudinal rotation</td><td></td><td>3.6 ± 0.3</td><td>7.5 ± 0.8</td><td>8.4 ± 0.8</td><td>0.9 [3.7]</td><td>2.1 [5.2]</td><td>2.4 [7.9]</td></tr></tbody></table>

Mean 5–95% confidence limits and RMSD for each degree of freedom of each joint and for each source of uncertainty for the angular velocities.

<table><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th rowspan="2">Joint</th><th colspan="3">5–95% confidence limits (°.s<sup>-1</sup>)</th><th colspan="3">RMSD (°.s<sup>-1</sup>)</th></tr><tr><th>MPE</th><th>STA</th><th>Combined</th><th>MPE [maximum]</th><th>STA [maximum]</th><th>Combined [maximum]</th></tr></thead><tbody><tr><td rowspan="10">Joint Angular velocities</td><td rowspan="10">Flexion-Extension</td><td>Hip</td><td>16.2 ± 18.9</td><td>123.1 ± 4.1</td><td>124.6 ± 8.3</td><td>6.3 [29.2]</td><td>34.1 [106.2]</td><td>35.0 [107.2]</td></tr><tr><td>Knee</td><td>13.3 ± 11.5</td><td>69.0 ± 7.2</td><td>72.3 ± 9.4</td><td>4.8 [20.0]</td><td>19.9 [49.6]</td><td>21.0 [60.9]</td></tr><tr><td>Ankle</td><td>23.5 ± 25.2</td><td>43.5 ± 8.7</td><td>56.9 ± 21.5</td><td>8.9 [56.0]</td><td>12.8 [33.2]</td><td>17.2 [78.1]</td></tr><tr><td>Shoulder</td><td>4.8 ± 4.4</td><td>92.8 ± 16.2</td><td>93.7 ± 16.2</td><td>1.8 [6.5]</td><td>26.7 [81.7]</td><td>27.0 [78.8]</td></tr><tr><td>Elbow</td><td>6.0 ± 5.4</td><td>70.2 ± 3.8</td><td>70.5 ± 3.8</td><td>2.3 [7.2]</td><td>20.3 [59.2]</td><td>20.4 [53.7]</td></tr><tr><td>Wrist</td><td>8.9 ± 10.9</td><td>134.3 ± 10.4</td><td>134.1 ± 12.3</td><td>3.9 [14.7]</td><td>38.0 [115.5]</td><td>38.0 [110.7]</td></tr><tr><td>Hip</td><td>41.2 ± 47.0</td><td>121.0 ± 17.3</td><td>133.0 ± 37.6</td><td>15.4 [66.1]</td><td>34.8 [99.7]</td><td>39.6 [120.1]</td></tr><tr><td>Knee</td><td>33.5 ± 29.5</td><td>167.5 ± 34.0</td><td>173.5 ± 35.2</td><td>11.2 [51.2]</td><td>46.7 [167.8]</td><td>49.0 [142.6]</td></tr><tr><td>Ankle</td><td>51.1 ± 67.1</td><td>145.7 ± 27.6</td><td>160.9 ± 58.5</td><td>10.8 [43.6]</td><td>65.1 [202.3]</td><td>68.0 [269.0]</td></tr><tr><td>Shoulder</td><td>6.1 ± 5.1</td><td>95.2 ± 11.7</td><td>96.6 ± 12.0</td><td>2.2 [7.6]</td><td>27.3 [78.6]</td><td>27.8 [78.7]</td></tr><tr><td>Wrist</td><td>10.2 ± 13.4</td><td>145.3 ± 18.9</td><td>147.1 ± 20.5</td><td>4.5 [18.2]</td><td>41.5 [123.1]</td><td>42.1 [123.0]</td></tr><tr><td rowspan="6">Abduction-Adduction</td><td rowspan="6">Internal-External</td><td>Hip</td><td>11.4 ± 9.6</td><td>220.6 ± 15.6</td><td>221.1 ± 16.0</td><td>4.0 [13.8]</td><td>60.6 [219.7]</td><td>61.0 [182.4]</td></tr><tr><td>Knee</td><td>9.7 ± 8.4</td><td>201.2 ± 35.1</td><td>203.5 ± 35.7</td><td>3.6 [11.1]</td><td>57.1 [163.0]</td><td>57.8 [179.8]</td></tr><tr><td>Ankle (Inv/Eversion)</td><td>28.8 ± 26.5</td><td>236.6 ± 20.9</td><td>248.5 ± 23.4</td><td>20.8 [93.6]</td><td>40.2 [144.0]</td><td>47.6 [139.1]</td></tr><tr><td>Shoulder</td><td>8.5 ± 10.9</td><td>121.9 ± 22.0</td><td>123.2 ± 23.6</td><td>3.8 [14.3]</td><td>35.8 [93.8]</td><td>36.4 [91.2]</td></tr><tr><td>Elbow (Pro/Supination)</td><td>8.7 ± 11.4</td><td>235.3 ± 15.0</td><td>237.3 ± 16.1</td><td>4.0 [15.3]</td><td>64.4 [220.2]</td><td>64.7 [243.0]</td></tr><tr><td>Trunk flexion</td><td>4.4 ± 5.9</td><td>58.5 ± 4.6</td><td>110.7 ± 4.0</td><td>2.5 [7.6]</td><td>16.5 [45.8]</td><td>16.8 [45.6]</td></tr><tr><td rowspan="2">Segmental Angle</td><td>Shoulder-over-shoulder</td><td></td><td>5.9 ± 7.3</td><td>59.1 ± 6.1</td><td>111.6 ± 4.5</td><td>3.0 [8.6]</td><td>16.7 [53.6]</td><td>17.0 [46.7]</td></tr><tr><td>Trunk longitudinal rotation</td><td></td><td>2.7 ± 2.9</td><td>108.4 ± 2.6</td><td>144.2 ± 3.1</td><td>2.3 [4.6]</td><td>28.2 [121.8]</td><td>28.8 [10.7]</td></tr></tbody></table>
To facilitate the interpretation of kinematic changes, we computed MDC values for key joint angles and key joint angular velocities representative of the serve kinematic chain (Table 6).

## 4. Discussion

First, concerning MPE, the average RMSD for knee flexion-extension is lower than that reported by Fonseca et al. (2020), who applied MPE along each segmental axis; a 10 mm misplacement of the lateral epi- condyle marker along the antero-posterior axis resulted in an RMSD of 4.2°. In our study, RMSD is 1.3° but reaches a maximum of 5.0° in simulations with large placement errors (Table 4), indicating that inaccurate marker placement can substantially affect knee kinematics.

Regarding ankle angles, MPE contributed more to variability than STA (Table 4). This is probably because only the shank was perturbed by STA, as foot markers were placed on shoes, which were assumed to be sufficiently rigid. In addition, the amount of MPE on foot markers is higher than for other segments (Table 1), especially on medio-lateral axis, increasing the uncertainty for abduction-adduction and inversion-eversion angles.

5–95 % confidence limits for the MPE effect on joint angles during the tennis serve are higher than those reported in the literature for gait (Myers et al., 2015) and for isolated shoulder movement (Lavaill et al., 2022). Our values for hip flexion-extension, abduction-adduction and internal-external rotation are 7.9 ± 0.2°, 6.4 ± 0.3° and 18.9 ± 1.4° (Table 4), compared with 4.2°, 1.5° and 2.6°, reported by Myers et al. (2015). Lavaill et al. (2022) reported 5–95 % confidence limits of 0.8°, 3.1°, and 1.1° for shoulder abduction-adduction, flexion-extension, and internal-external rotation, whereas our study found 2.6 ± 0.3°, 3.5 ± 0.1°, and 4.5 ± 0.8°. Both studies used MKO to process perturbed trajectories and compute kinematics. In contrast, we chose not to use MKO in order to align our methodology with previous tennis serve studies (Martin et al., 2016; Reid et al., 2013; Reid & Giblin, 2015). This comparison suggests that applying MKO could reduce the influence of MPE

<caption>**Table 6**</caption>

Minimum Detectable Change (MDC) and relative MDC (rMDC) of key kinematic variables, as well as the MDC<sub>STA</sub> calculated specifically for STA.

<table><thead><tr><th>Variable</th><th>MDC<sub>STA</sub></th><th>rMDC<sub>STA</sub> (%)</th><th>MDC</th><th>rMDC (%)</th></tr></thead><tbody><tr><td>Maximal front ankle dorsi-flexion (°)</td><td>2.4</td><td>4.6</td><td>7.6</td><td>14.7</td></tr><tr><td>Maximal back ankle dorsi-flexion (°)</td><td>2.5</td><td>6.7</td><td>7.2</td><td>19.0</td></tr><tr><td>Maximal front knee flexion (°)</td><td>3.1</td><td>2.9</td><td>4.5</td><td>4.2</td></tr><tr><td>Maximal back knee flexion (°)</td><td>3.1</td><td>2.9</td><td>4.8</td><td>4.5</td></tr><tr><td>Maximal shoulder external rotation (°)</td><td>6.6</td><td>4.1</td><td>7.3</td><td>4.6</td></tr><tr><td>Maximal front ankle plantar-flexion velocity (°.s<sup>-1</sup>)</td><td>86.0</td><td>10.6</td><td>124.1</td><td>15.3</td></tr><tr><td>Maximal back ankle plantar-flexion velocity (°.s<sup>-1</sup>)</td><td>79.0</td><td>8.9</td><td>105.4</td><td>11.9</td></tr><tr><td>Maximal front knee extension velocity (°.s<sup>-1</sup>)</td><td>62.4</td><td>9.4</td><td>67.9</td><td>10.3</td></tr><tr><td>Maximal back knee extension velocity (°.s<sup>-1</sup>)</td><td>66.5</td><td>11.1</td><td>73.4</td><td>12.2</td></tr><tr><td>Maximal trunk flexion velocity (°.s<sup>-1</sup>)</td><td>101.0</td><td>15.8</td><td>106.0</td><td>16.5</td></tr><tr><td>Maximal shoulder-over-shoulder velocity (°.s<sup>-1</sup>)</td><td>75.9</td><td>14.1</td><td>78.0</td><td>14.4</td></tr><tr><td>Maximal trunk longitudinal rotation velocity (°.s<sup>-1</sup>)</td><td>127.2</td><td>23.5</td><td>133.1</td><td>24.5</td></tr><tr><td>Maximal shoulder internal rotation velocity (°.s<sup>-1</sup>)</td><td>213.9</td><td>11.7</td><td>231.0</td><td>12.7</td></tr><tr><td>Maximal elbow extension velocity (°.s<sup>-1</sup>)</td><td>72.1</td><td>6.2</td><td>73.1</td><td>6.3</td></tr><tr><td>Maximal elbow pronation velocity (°.s<sup>-1</sup>)</td><td>222.4</td><td>27.2</td><td>237.9</td><td>28.9</td></tr><tr><td>Maximal wrist flexion velocity (°.s<sup>-1</sup>)</td><td>199.1</td><td>17.2</td><td>218.2</td><td>18.9</td></tr></tbody></table>

in kinematic analysis.

Regarding the STA, our results are in line with the literature (Table 3). Δp<sub>max</sub> are consistent with those reported by Hara et al. (2014) and Rozumalski et al. (2007). For the thigh, shank, and arm, RMSD and Δp<sub>max</sub> are close to the values reported by Cereatti et al. (2017) during various activities. Regarding the thorax, the STA are within the range of literature data (Heneghan & Balanos, 2010; Zemp et al., 2014). For the hand, the Δp<sub>max</sub> are similar to those described by Ryu et al. (2006).

The effect of STA on joint angles during tennis serve is higher to those reported in the literature for gait (Myers et al., 2015). 5–95 % confidence limits for hip, knee and ankle flexion-extension were 8.5 ± 0.2°, 4.3 ± 0.4° and 4.1 ± 0.9° respectively (Table 4), compared with 5.9°, 2.4° and 5° reported by Myers et al. (2015). In contrast, Lamberto et al. (2017) reported larger uncertainty ranges of 15°, 8° and 6° for hip, knee, and ankle flexion. These discrepancies could be attributed to differences in the methods used to simulate STA and to compute joint angles. Myers et al. (2015) simulated STA by applying a different offset between each stage of the gait cycle (i.e. the offset changes at toe-off and heel strike) and restricted to 15 mm. Nevertheless, STA arises from skin sliding, muscular contraction, and inertial effects on soft tissue masses (wobbling), consequently, the resulting movement is continuous and not limited to discrete gait events. Therefore, modelling STA solely based on gait events may oversimplify the complexity of STA. Differences with Lamberto et al. (2017) may stem from simulation parameters. For example, we applied an amplitude A ≤ 15 mm for the thigh markers, whereas they used A ≤ 30 mm. Additionally, Lamberto et al. (2017) and Myers et al. (2015) have used MKO known in the literature as a solution to reduce the influence of STA (Begon et al., 2015, 2018; Camomilla et al., 2017; Leardini et al., 2005), whereas our method solely followed the ISB recommendations (Martin et al., 2016; Reid et al., 2013; Reid & Giblin, 2015). Moreover, the choice of MKO model has a direct effect on the uncertainty, which can be up to 2.5 times higher depending on the model used (Lamberto et al., 2017), making it difficult to assess the real effect of MKO on reducing STA-induced uncertainty.

Concerning the combined condition, the largest RMSD was observed for shoulder internal-external rotation (Table 4), with an average RMSD of 3.1° and a maximal of 7.8° which is consistent with Ludewig et al. (2002). As shoulder external rotation is a key component of the tennis serve (Fleisig et al., 2003) and exhibits a high RMSD, this degree of freedom should be analysed with caution.

When computing segmental orientation using markers placed close to one another, small variations in 3D marker locations introduced greater errors into angles and velocities (Chèze et al., 1995), particularly for hip, knee, ankle, and elbow internal-external rotation. However, shoulder internal-external rotation shows lower confidence intervals (Table 4, Table 5). This is because the arm coordinate system is determined by the cross product of the longitudinal axes of the arm and the forearm (Wu et al., 2005), compared to the longitudinal axis and medial and lateral distal markers as for the forearm, thigh, and shank (Wu et al., 2002, 2005). This difference in defining the arm coordinate system also contributes to the lower confidence intervals and MDC observed for elbow flexion-extension angle and angular velocity (Table 4, Table 5, Table 6).

The combined effect of MPE and STA on joint angular velocities remains largely unexplored in the current literature where attention was mostly focused on joint angles and centres uncertainties (de Groot, 1997; Dumas & Chèze, 2009; Fonseca et al., 2020, 2023; Lavaill et al., 2022; Leardini et al., 2005; Myers et al., 2015; Thouzé et al., 2016). Our results suggest that most of the variability in angular velocity is driven by the STA, rather than MPE (Table 5). Given this clear influence, it is essential to consider the effects of STA when interpreting joint angular velocities in tennis serve biomechanical analyses.

MDC values provide essential reference thresholds when interpreting results in past and future studies. Several studies have reported changes in kinematics associated with fatigue or technical modifications in the tennis serve, and these changes can be evaluated with respect to the
MDC established in the present study. For instance, a 5° reduction in maximal knee flexion and a 70°·s⁻¹ reduction in maximal knee extension velocity were observed during a 3-hour match in trained tennis players equipped with markers rigorously positioned in the same location between the start and end of the match (Martin et al., 2016). When players serve in kneeling position, a decrease in maximal elbow extension velocity of 531.5°·s⁻¹ has been reported by Reid et al. (2013). All these reported changes exceed the MDC or MDC<sub>STA</sub>, supporting the idea that such variations likely reflect true biomechanical differences rather than measurement uncertainty. However, Martin et al. (2016) also reported decreases of 6.0 %, 7.5 % and 13.8 % of maximal elbow extension, shoulder internal and wrist flexion velocities respectively due to fatigue, which are lower than the rMDC of 6.2 %, 11.7 % and 17.2 % (Table 6). Exaggerated arabesque landing position have also led to kinematic changes, such as a 54.9°·s⁻¹ increase in maximal back knee extension velocity and with a 1.6° increase of front knee maximal flexion, which does not exceeds the MDC (Reid & Giblin, 2015). From our perspective, in these cases, the observed effect of fatigue and technical change may not solely be attributed to the effect of the task but may be part of measurement uncertainty.

Despite the sources of uncertainties, marker-based MoCap remains the reference system for analysing human motion and is used to assess the validity of new MoCap systems such as inertial measurement unit (Robert-Lachaine et al., 2017; Tham et al., 2021) or markerless MoCap (Chaumeil et al., 2024; Kanko et al., 2021; Pagnon et al., 2022; Ripic et al., 2023; Uhlrich et al., 2023). Recently, a study investigating the validity of markerless MoCap highlighted that the observed inconsistencies were comparable in magnitude to the effects of STA (Bousiques et al., 2025), which our results tend to support. For instance, Kanko et al. (2021) reported a RMSD of 3.3° for the knee flexion-extension angle during walking between marker-based and markerless MoCap systems. This value is higher than the average RMSD observed for this degree of freedom in the present study (1.9°) but lower than the maximum RMSD of 6.0° (Table 4). These results highlight the importance of considering the uncertainty inherent in marker-based systems when validating new motion capture methods, and uncertainty estimates can serve as reference values for this purpose.

A limitation of our study lies in the commonly used STA simulation approach adopted here, which does not account for phase-dependent variations in STA throughout the motion (Ancillao et al., 2021) and represents STA as a single periodic displacement. However, experimental evidence has shown that STA results from a combination of multiple components (Blache et al., 2017). Another limitation is the absence of MKO, which is known to reduce the effects of STA but remains dependent on modelling and scaling assumptions. Since the choice of model can influence the outcomes, reprocessing the marker trajectories with MKO may provide alternative kinematic estimates. Future work should therefore examine the impact of MKO using different models. Another limitation is that the study was conducted with a single participant, and results may vary with athletes of different tennis skill levels.

## 5. Conclusion

This study highlights the significant impact of STA and MPE on joint angles and angular velocities during the tennis serve. By simulating both sources of uncertainty independently and in combination, we provide a methodological framework that enables a better understanding of how measurement errors propagate through biomechanical analyses of sport motion across all joints. These insights are valuable for researchers aiming to assess the effects of technique changes or experimental interventions in tennis serve. They may be also relevant for studies of other similar overhead tasks. Importantly, the proposed approach is not limited to the tennis serve and can be directly applied to other tennis strokes in future work. Accounting for such uncertainties is essential to distinguish between true biomechanical adaptations and variations due

to measurement uncertainty.

### CRediT authorship contribution statement

**Simon Ozan:** Writing – original draft, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Loïc Fourel: Writing – review & editing, Software, Methodology, Investigation, Formal analysis, Conceptualization. Pierre Touzard: Writing – review & editing, Methodology, Investigation, Formal analysis, Conceptualization. Richard Kulpa: Writing – review & editing, Validation, Supervision, Project administration, Methodology, Funding acquisition, Conceptualization. Caroline Martin: Writing – review & editing, Validation, Supervision, Project administration, Methodology, Funding acquisition, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

The authors would like to thank the participant and the Betton Tennis Association (AT Betton) for hosting the experiment. This work was supported by French government funding managed by the National Research Agency under the Investments for the Future program in France 2030 with the grants ANR-18-EURE-0022 (DIGISPORT) and ANR-21-ESRE-0030 (CONTINUUM).

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