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# Profiling biomechanical abilities during sprint front-crawl swimming using IMU and functional clustering of variabilities

Antoine Bouvet, Robin Pla, Erwan Delhaye, Guillaume Nicolas, Nicolas Bideau

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Antoine Bouvet, Robin Pla, Erwan Delhaye, Guillaume Nicolas, Nicolas Bideau. Profiling biomechanical abilities during sprint front-crawl swimming using IMU and functional clustering of variabilities. Sports Biomechanics, 2024, pp.1 - 21. 10.1080/14763141.2024.2368064. hal-04632780

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1 Title of the Article: Profiling biomechanical abilities during sprint front-crawl swimming using

2 IMU and functional clustering of variabilities

3 Submission Type: Research Article

4 Keywords: swimming, inertial measurement unit, technical skills, movement variability,

5 functional data analysis

6 Antoine Bouvet¹,²,³, Robin Pla⁴,⁵, Erwan Delhaye¹,², Guillaume Nicolas¹,², Nicolas Bideau¹,²

7 M2S laboratory (Movement, Sport & Health), University Rennes 2, ENS Rennes, 35170 Bruz,

8 France; ²MIMETIC-Analysis-Synthesis Approach for Virtual Human Simulation, INRIA

9 Rennes Bretagne Atlantique, Campus de Beaulieu, 263 Av. Général Leclerc, 35042 Rennes,

10 France ; ³CREST (Center for Research in Economics and Statistics), ENSAI, Rennes, 35170

11 Bruz, France ; ⁴French Swimming Federation, 92110 Clichy, France ; ⁵IRMES—URP 7329,

12 Institut de Recherche Médicale et d'Epidémiologie du Sport, Université Paris Cité, Paris,

13 France

14 Corresponding author: Antoine Bouvet, M2S laboratory, +33 2 90 09 15 80,

15 antoine.bouvet@ens-rennes.fr

16 Abstract word count: 200

17 Text-Only word count: 5832

18 Number of Figures: 7

19 Number of Tables: 3

20
21 No conflict of interest stated for all authors.

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43 Abstract :

This study aims to profile biomechanical abilities during sprint front-crawl by identifying
technical stroke characteristics, in light of performance level. Ninety-one **recreational** to world-
class swimmers equipped with a sacrum-worn IMU performed 25m all-out. Intra and inter-
cyclic 3D kinematical variabilities were clustered using a functional double partition model.
Clusters were analyzed according to (1) swimming technique using continuous visualization
and discrete **features (standard deviation and jerk cost)** and (2) performance regarding speed
and competition caliber using respectively one-way ANOVA and Chi-squared test as well as
Gamma statistics. Swimmers displayed specific technical profiles of intra-cyclic (*smoothy* and
*jerky*) and inter-cyclic stroke regulation (*low*, *moderate* and *high repeatability*) significantly
discriminated by speed (p<0.001, η²=0.62) and performance caliber (p<0.001, V=0.53). We
showed that combining high levels of both kinds of variability (*jerky+low repeatability*) are
associated with highest speed (1.86±0.12 m/s) and competition caliber (γ=0.75, p<0.001).
It highlights the crucial importance of variabilities combination. Technical skills might be driven
by a specific alignment of stroke pattern and its associated dispersion according to the task
constraints. This data-driven approach can assist eyes-based technical evaluation. Targeting the
development of an explosive swimming style with a high level of body **stability** should be
considered during training of sprinters.

61 Introduction

Performance analysis in swimming requires the identification of individual skills and the way
the best swimmers master their movement (Bartlett, 2001). Therefore, monitoring their
biomechanical abilities through technical assessment is at the heart of the understanding of elite
performances. Swimming is a sequence of repeated but not identical body movements targeting
effective motor patterns to achieve optimal performance (Maglischo, 2003). Thus, variability
is inherent to swimming technique and occurs within a stroke cycle or across multiple stroke
cycles, respectively known as intra and inter-cycle variability (Fernandes et al, 2022a). Such
variabilities are of crucial interest as it can affect the swimming speed (Figueiredo et al., 2013)
and reflect underlying regulation of stroke mechanics. Hence, evaluation of both intra and inter-
cyclic kinematical variability, as well as their combination, may provide valuable insights on
the relationship between movement variability and technical skills during swimming.

Previous research focused on intra-cycle variability and its relations with performance and expertise (Fernandes et al., 2022b). Nevertheless, associations between intra-cyclic aspects of kinematical variability and technical skills remain unclear. Intra-cyclic velocity variations (IVV) were primarily investigated leading to contradictory findings since opposite correlations between IVV and performance were observed (Barbosa et al., 2012 ; Barbosa et al., 2005; Schnitzler et al., 2008). Moreover, divergent outcomes were pointed out regarding the impact of expertise since better skilled swimmers can show higher, lower or similar IVV (Fernandes et al., 2023). Despite the identification of IVV as a relevant way to monitor swimming biomechanical development (Vilas-Boas et al., 2011), inconsistency of literature leads to a crucial lack of information on regulation of stroke variability. In that respect, considering other variables than the forward linear velocity to assess intra-cyclic variability may be meaningful. Indeed, the understanding of the stroke cycle effectiveness refers to the interplay between the propulsive and drag forces that produce body accelerations (Gonjo et al., 2021). Hence, this dynamical interaction reflected by kinematics is considered as an efficiency estimator (Barbosa et al., 2012 ; Figueiredo et al., 2012a). Therefore, analyzing the intra-cyclic acceleration may be of great interest as it provides relevant diagnostic of front-crawl technical skills (Ichikawa, 1999) and propelling actions (Buchner & Reischle, 2002) by a much direct evaluation of the
capacity to propel the body with minimal velocity variations (Gonjo et al., 2021). Acceleration has been shown as a sensitive measure of swimming performance since correlation between velocity and root mean square of acceleration was reported (Tella et al., 2008; Tella et al., 2010). Then, it may be considered as a consistent indicator of stroke mechanics performance and expertise, reflecting a possible advantageous technique (Holmer, 1979) or higher stroke efficiency (Gil et al., 2012). Moreover, time variation of acceleration (i.e. jerk) may also reflect important technical skills because it quantifies the smoothness that leads to a more efficient movement (Hreljac, 1993). In that sense, smoothness is a universal feature of skilled movement which may be used as a window into athletic coordinative proficiency (Kiely et al., 2019). Thus, the jerk cost has been recently used in swimming to evaluate the technique (Ganzevles et al., 2019) and stroke regulations (Bouvet et al., 2023). Finally, the acceleration pattern and its derivative may be valuable for intra-cyclic variability regulation assessment.

Swimming stroke analyses generally consider only intra-cyclic aspects of motor pattern in isolation of adjacent cycles that reduces the understanding of variability. However, considering also the movement variability between cycles, may provide complementary information about performance and technical skills characterization (Fernandes et al., 2022b). Given the traditional approach of movement variability as noise (Preatoni et al., 2013), swimmers with high inter-cyclic variability may exhibit a greater dispersion in their technique and a lower repeatability in their kinematics leading to reduced performance. Fernandes et al., (2022a) suggested that slowest swimmers displayed higher mean velocity oscillations between two consecutive cycles with higher numbers of cycles presenting extreme variations than the fastest ones. However, Fernandes et al., (2022b) showed that elite swimmers' velocity in backstroke were more unstable and complex despite similar intra-cycle variations. Thus, inter-cyclic variability may contain important information about motor organization and not only refer to deleterious limitations to performance which should therefore be eliminated.

Variability analysis has been achieved using a variety of devices, including video, speedometer, motion capture and inertial measurement units (IMU) (Fernandes et al., 2023). Among them, IMU is a relevant embedded tool to assess 3D swimming kinematics in a real environment (Mooney et al., 2016). Moreover, IMU directly measures the acceleration which is convenient for movement variability analysis. Indeed, it may avoid errors on acceleration signals that can arise using numerical derivatives from position and velocities. Therefore, IMU is suitable to monitor intra-cyclic acceleration pattern allowing to track technical skills and changes in efficiency (e.g. Engel et al., 2022; Ganzevles et al., 2019; Tella et al., 2008). Furthermore, as swimming is composed of translational and rotational motions, IMU may enable more insights on movement variability because it also measures angular velocities. Since IMU data are multivariate time series, it seems appropriate to consider their functional form by continuous instead of discrete measures of variabilities (Ryan et al., 2006). To this end, motor control methodologies present a useful framework for variability analysis based on nonlinear dynamics in time series (Bravi et al., 2011). Among them, features of complexity, such as the approximate and the sample entropy, allowed to quantify the temporal structure in the signals through the unpredictability of its variations. This methodology was applied in swimming kinematics to provide new insights into variability (Barbosa et al., 2017; Fernandes et al., 2022b). However, even if entropy features proceeds over the entire signal to capture temporal dynamics, and thus could be considered as an easy-to-use continuous measure, they finally result in a restricted scalar value (i.e., discretization of the information in a numeric feature), whereas variability analysis could be improved by considering it as a function of the time. Indeed, function regression provided more informative and intuitive interpretations of function-valued data, as time series, than conventional single scalar approaches (Yen et al., 2015). Moreover, entropy features can't handle the multivariate nature of IMU data, referring to interactions between
139 dimensions of swimmers' motion variability. As a consequence, this kind of discrete parameters neglect a part of the temporal and dynamical information in functional data leading to restricted details on motricity (Richter et al., 2014). Then, numerous studies in swimming were limited to these non-function-valued methodologies in process-driven approaches, mainly due to the absence in literature of a suitable statistical model capturing the functional nature of the IMU swimming data. Thus, it leads to limited data-driven insights regarding kinematical variabilities profiling and performance explanations. Therefore, taking into account the need to, on the one hand, consider the entire sequence of data points, and on the other, directly model swimming variabilities through functional data analysis (FDA) applied to IMU data is crucial.

148 In order to better investigate the kinematical variability recorded by IMU and its relation with technical skills and performance, FDA depicts a useful way. Actually, FDA can take into account the shape of kinematical curves, by relying on statistical modeling of data, giving indications on the motor pattern and helping to classify behavior and relevant characteristics of performance (Preatoni et al., 2013). Indeed, FDA can outpace traditional approaches relying on process-driven analysis and computation of discrete features that leads to limited kinematical information (Mallor et al., 2010). For example, it provides evidence on technical differences in national and international race-walkers (Donà et al., 2009). Moreover, literature demonstrated the usefulness of FDA for motion pattern analysis and description of different performance groups on cyclic biomechanical data such as gait, running and cycling (Andrade et al., 2014; Forrester & Townend, 2015; Helwig et al., 2016). In this way, Barbosa et al., 2021 highlighted differences between international and national-level sprinters using a general FDA approach, without automatic classification of swimmers. Furthermore, these methods can be extended, regarding the scope of application and the data specificities, to allow clustering of functional data (Jacques & Preda, 2014) and then group similar time series of biomechanical or performance parameters (Leroy et al., 2018). Then, functional data clustering has been growingly applied in applications of various fields of science and gathering statistical methods coming from FDA and clustering community (Zhang & Parnell, 2023). Hence, it may provide identification of trends and features leading to new insights about technique and performance profiles in sport applications. More generally, cluster analysis is a powerful data-driven tool for detect performance and technique patterns in various sport applications such as athletes' visual search strategies (Murray & Hunfalvay, 2017), anthropometric attributes (Zhang et al., 2018) and coordination pattern in swimming (Seifert et al., 2016; Bideault et al., 2013). Then, in its functional extension, clustering may be a powerful approach to deal with IMU data in order to profile variabilities in swimming. However, such methodology of functional data clustering generally required the use of a customized statistical model able to deal with both the specificity of the input data and the aim of the analysis (Bouveyron et al., 2019). To this end, recent developments in statistical literature provided an approach explicitly devoted to functional clustering of swimmers through the modeling of kinematical variabilities recorded by IMU (Bouvet et al., 2024). It embodies a promising way to perform profiling of biomechanical abilities during swimming based on both intra and inter-cyclic kinematical variability.

179 The present study aimed at clustering intra and inter-cyclic variability considering the functional nature of 3D IMU data during sprint front-crawl swimming in order to identify technical strokes characteristics according to the performance level leading to biomechanical abilities profiling.

183 **Materials and methods:**

184 *Participants*
91 subjects (31 women, age: 18.7±3.20 y.o.; height: 170±6.37 cm; mass: 60.1±6.67 kg; FINA points on 50m freestyle short course: 457±175) and 60 men, age: 21.4±6.65 y.o.; height: 183±8.01 cm; mass: 71.8±12.3 kg; FINA points on 50m freestyle short course: 527±195) participated in this study including recreationally active to world-class swimmers (McKay et al., 2022). All participants signed an informed consent form in agreement with the French Ethical Committee (approval obtained under reference 2021-A00250-41).

## Data collection
Each swimmer performed a 25m wall-start in front-crawl at maximal intensity with the instruction to not perform underwater kicks and breathing. Participants were instrumented with one waterproof 3D IMU (Xsens DOT, Movella, Enschede, The Netherlands) sampled at 60 Hz using a full scale set at 16 g for the accelerometer and 1000 deg/s for the gyroscope. The sensor was placed on the sacrum, then fixed with double-sided tape and secured with waterproof medical adhesive (Tegaderm, 3M, Cergy-Pontoise, France). The IMU described a coordinate system defined with x-axis pointing cranially, y-axis pointing laterally, and z-axis pointing posteriorly.

## IMU data preprocessing and functional data clustering model

Raw data were filtered using a second order Butterworth low-pass filter with a 10 Hz cut-off frequency. The algorithm of Delhaye et al., 2022 was used, on the one hand, to extract only the front-crawl frames from the signal based on the human activity recognition provided by the classifier, and to the other hand, to compute average swimming speed as the pool length (i.e., 25m) divided by the lap time automatically provided by the model.
In order to avoid kinematical disturbances in data during the first stroke after swimmer's breakout and finish, one-second time windows before and after the first and the last front-crawl frame were removed. In order to prevent temporal drift in swimmer's time series, curve alignment is performed to enable proper functional modeling. It is achieved by adjusting the signal starting point at the frame corresponding to a common reference for all swimmers (Bouvet et al., 2024). To do so, we used a specific and generalizable feature in the IMU signals, defined as an operational definition of a stroke cycle beginning in swimming literature. The latter was identified as the first zero-crossing on the medio-lateral acceleration filtered with a second-order Butterworth band-pass filter between 0.1 and 1 Hz (Ganzevles et al., 2019). For each cycle, data were time normalized using the previously described zero-crossing detection. This feature allows, through a highly filtered signal processing approach, to only extract the waveform characteristics about that IMU axis. Consequently, the repeating elemental pattern obtained is representative of the body roll during front-crawl (Mooney et al., 2016) and allowed to both determine the periodicity of the signal as well as extracting the single-cycle data (Zhuang & Xue, 2019). This method allows to identify the start and end of a stroke cycle while dealing with high levels of intra and inter-subject kinematical variability as well as noise in sensor's measurement. Finally, its generalization ability defines consistent time within a stroke cycle, which is a required step in the functional clustering of kinematics detailed below. Figure 1 shows the preprocessed data for two participants of different levels.

**[FIGURE 1 NEAR HERE]**
These data were then clustered using the double partition model of Bouvet et al., (2024). This FDA framework enables a fully integrated curve clustering based on both kinds of variabilities and also considers dependency between both partitions. Modeling this dependency during the clustering process is mandatory since the difficulty of reproducing a swimming pattern (i.e., inter-cyclic variability) depends on its shape (i.e., intra-cyclic variability). Then, this approach

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is able to model and analyze the complex multivariate dynamics of swimming patterns by aggregating the dependent kinematical variabilities linked to swimming technical skills into interrelated partitions. As a consequence, this extended FDA approach belongs to the model-based functional clustering (Jacques & Preda, 2014; Bouveyron et al., 2019), and has been developed for taking into account all of the specificities of the kinematical swimming data recorded by IMU through a powerful statistical model devoted to technical skills profiling (Bouvet et al., 2024). Based on a multivariate functional decomposition of IMU signals as well as squared residuals into Fourier basis, the model considers both the kinematical information related to stroke pattern as well as the ability to reproduce it. Finally, it led to two partitions respectively related to intra and inter-cyclic variability clusters. The degrees of each Fourier basis were selected between four and twenty using a leave-one-out cross-validation. Model selection was performed using Bayesian Information Criterion considering clusters values between one and five. Visualization was performed by computing the mean cluster curves (i.e. stroke pattern representing intra-cyclic variability) as well as credibility region (i.e. ability to reproduce the stroke pattern representing inter-cyclic variability).

Statistical analysis

Statistical analyses were conducted using RStudio (Version 4.2.2, RStudio, Inc., Boston, MA, USA). The level of significance was set at p<0.05.

Technical analysis of clusters

To identify cluster technical profiles, classes were analyzed regarding clustering visualization and biomechanical parameters computed on each IMU axis. Indeed, considering clusters mean curves, standard deviation was calculated as a reflection of cycle variation (Dadashi et al., 2016) and jerk cost (for accelerometric dimension) as a practical evidence of stroke smoothness (Ganzevles et al., 2019). Regarding cluster credibility region curves, standard deviation was calculated as the amount of kinematical dispersion.

Performance analysis of clusters

To identify cluster performance profiles defining relevant biomechanical abilities, two analyses were conducted considering different partitions: both single and the cross-partition (i.e. combination of inter and intra-cyclic variability clusters for each subject, reflecting technical skills profiling). Firstly, a one-way analysis of variance (ANOVA) or t-test was performed to investigate the link between swimming speed and the partitions. If appropriate, post-hoc tests were conducted with Tukey test and Bonferroni correction. The magnitude of differences was interpreted using generalized eta-squared ($\eta_g^2$) for ANOVA or Cohen's d (d) for t-test with threshold values of respectively 0.01, 0.06, 0.14 and 0.20, 0.50 and 0.80 defining trivial, small, medium and large effects (Cohen, 1988). Secondly, a Pearson's Chi-squared test was performed to investigate the dependence between a performance caliber ordinal variable (defined as Tier 1, Tier2-3 and Tier4-5 respectively for recreationally active (tier 1) swimmers, trained/developmental (tier 2) or highly trained/national level (tier 3) swimmers and elite/international level (tier 4) or world class (tier 5) swimmers according to McKay et al., 2022) and the clusters ranked as ordinal variables based on speed. These modalities of the performance caliber ordinal variables displayed FINA points on 50m freestyle short course of 308±128, 455±78 and 742±73 for respectively Tier1, Tier2-3 and Tier4-5. The magnitude of differences was interpreted using V Cramer (V) and threshold values of 0.10, 0.20, 0.40, 0.60, 0.80 and 1 defining negligible, weak, moderate, relatively strong, strong and very strong effects respectively (Cramer, 1946). The strength of the association between performance caliber and clusters was measured using the Goodman Kruskal's Gamma statistic ($\gamma$) (Goodman & Kruskal, 1954) with 95% confidence intervals and threshold value of 0.01, 0.10, 0.30, 0.60, 0.75 and 1
defining no association, negligible, low, moderate, strong, and very strong association respectively (Rea & Parker, 2014).

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## Results

Technical profiles according to cluster

Mean cluster curves and credibility regions for the accelerometric and gyroscopic dimensions are presented on Figure 2 and Figure 3, respectively. The best model displayed two clusters of stroke patterns defining intra-cyclic variability and three clusters of ability to reproduce the stroke pattern defining inter-cyclic variability. Intra-cyclic variability clusters displayed a specific stroke pattern, especially discriminated by signal amplitude and number of peaks in longitudinal and anteroposterior acceleration as well as signal amplitude in mediolateral acceleration. Then, clusters of intra-cyclic variability were named as the *jerky* and the *smoothy* class. The biomechanical features for the accelerometric and gyroscopic dimensions are displayed on Table 1 and Table 2, respectively. The *jerky* cluster displayed stroke pattern with higher variations of 74% and 178% on longitudinal and anteroposterior accelerations and of 181% and 8% on mediolateral and anteroposterior angular velocities, as well as lower variations of 15% on mediolateral acceleration and 11% on longitudinal angular velocity. The stroke smoothness of the *jerky* cluster was lower of 830% and 1675% on longitudinal and anteroposterior accelerations but lower of only 14% on mediolateral acceleration in comparison with the *smoothy* cluster (see Table 1). Inter-cyclic variability clusters were ordered according to the magnitude of stroke pattern dispersion. Thus, the resulting three kinds of inter-cyclic variability were referred as *low* (*LR*), *moderate* (*MR*), and *high* (*HR*) *repeatability* clusters. Indeed both the credibility region around the mean cluster curve (see Figure 2 and Figure 3) and the corresponding kinematical variation (see Table 2) were higher as clusters moved closer to *LR*. For example, the dispersion in longitudinal acceleration was 633% higher in *MR* than *HR* and 117% in *LR* than *MR* clusters.

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[FIGURE 2 NEAR HERE]

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[FIGURE 3 NEAR HERE]

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[TABLE 1 NEAR HERE]

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[TABLE 2 NEAR HERE]

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Performance profiles according to cluster

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Swimming speeds according to the different classes coming from the intra and inter-cyclic variability clustering are presented on Figure 4, as well as the classes coming from the cross-partition on Figure 5. Regarding clustering of mean stroke pattern (i.e. intra-cyclic variability), the *jerky* class showed a significantly higher swimming speed of 29.6% compared to the *smoothy* class with a large effect size (1.84 [1.79 ; 1.89] ±0.12 vs. 1.42 [1.38 ; 1.47]±0.18, t(68.0)=12.8, p<0.001, d=2.72). Regarding clustering the ability to reproduce the stroke pattern (i.e. inter-cyclic variability), there was a large significant swimming speed difference between the three clusters (F(2.88)=21.7, p<0.01, η<sup>2</sup>=0.33). Regarding the cross-partition, there was a large significant swimming speed difference between the five clusters (F(4.86)=34.6, p<0.01, η<sup>2</sup>=0.62). Table 3 presented the significant pairwise comparisons for both the inter-cyclic variability partition and the crossing partition, with all p<0.01.

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[TABLE 3 NEAR HERE]

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[FIGURE 4 NEAR HERE]

[FIGURE 5 NEAR HERE]

Performance caliber distribution according to the different classes coming from the intra and inter-cyclic variability clustering are presented on Figure 6, as well as the classes coming from the cross-partition on Figure 7. Clusters of mean stroke pattern are strongly dependent on performance caliber ($\chi^2(2)=49.9$, p<0.001, V=0.74) with a very strong significant association between the more the stroke pattern is *jerky*, the higher is the performance caliber ($\gamma=0.97$ [0.93; 1.00], p<0.001). Clusters of ability to reproduce the stroke pattern are moderately dependent of performance caliber ($\chi^2(4)=28.0$, p<0.001, V=0.39) with a strong significant association between the more there is *low repeatability* stroke pattern, the higher is the performance caliber ($\gamma=0.63$ [0.42; 0.84], p<0.001). Clusters of the cross-partition are relatively strongly dependent of performance caliber ($\chi^2(10)=51.5$, p<0.001, V=0.53) with a very strong significant association between the more the stroke pattern is *jerky+low repeatability* the higher is the performance caliber ($\gamma=0.75$ [0.60; 0.89], p<0.001).

[FIGURE 6 NEAR HERE]

[FIGURE 7 NEAR HERE]

## Discussion and Implication

This study aimed at clustering intra and inter-cyclic 3D kinematics during sprint front-crawl to identify technical strokes characteristic according to performance level based on an embedded IMU on the sacrum. To this end, a data-driven process based on a large database and including an heterogeneous level, ranging from recreationally active to world-class swimmers, was applied for multivariate functional clustering considering the specificities of IMU swimming data. Specific profiles of technique variability were identified. Indeed, two clusters of intra-cyclic stroke regulation (*smoothy* and *jerky*) and three clusters of inter-cyclic stroke regulation (*LR*, *MR*, and *HR*) were identified. All of them were analyzed according to swimming speed and performance caliber supporting the identification of relevant biomechanical abilities for front-crawl sprint.

Regarding the intra-cyclic clusters defining magnitude of variability, the *smoothy* class presents a more fluent acceleration pattern than the *jerky* one with continuity of propulsive actions. It is characterized by a lower acceleration variation along the longitudinal and anteroposterior axes and a higher variation of longitudinal angular velocity in addition to mediolateral acceleration. The *jerky* class includes a more explosive and rough stroke pattern leading to a higher jerk cost along longitudinal and anteroposterior reflecting degradation of stroke smoothness. Indeed, smooth swimming movements referring to a low jerk cost, are those without abrupt, intermittent and discontinuous changes in accelerations, whereas non-smooth stroke patterns reflect abruptness and erratic discordance (Kiely et al., 2019). A higher jerk cost might lead to a faster swimming (Ganzevles et al., 2019) as it implies greater intra-cycle acceleration and deceleration that can be converted into higher speed with appropriate technical skills. Indeed, our results are in line with such behavior because the *jerky* participants swim significantly faster than the swimmers of the *smoothy* cluster while being the ones demonstrating the higher performance caliber. Thus, the *jerky* cluster gathers almost all of the elite and world-class swimmers, whereas the *smoothy* cluster includes all the recreationally active swimmers and the majority of the trained and highly trained swimmers. It may be explained by a larger speed curve amplitude in faster sprinters (Barbosa et al., 2021) as well as higher peaks in speed with similar minimal
speeds (Leblanc et al., 2007). However, the lower stroke smoothness induced by higher intra-cyclic variability in the *jerky* stroke pattern may also be associated with higher energy expenditure leading to a lower stroke efficiency (Fernandes et al., 2023). Nevertheless, the jerky cluster displays reduced variations in both mediolateral acceleration and longitudinal angular velocity than the smoothy one, reflecting more core stability and a better overall alignment of the body *by possibly a decrease of body roll*. However, *body roll is not directly assessed in this work*. To the author's knowledge, only the biomechanical model of Bächlin & Tröster (2012) tracked body roll angle from IMU data under strong mechanical assumptions. Indeed, body angles assessment from IMU data requires a sensor-to-segment calibration, combined with a reliable computing of the sensor orientation, which involves very complex processings in the water environment (Mangia et al., 2017). Hence, in this work, we consider the association between medio-lateral acceleration recorded from a sacrum IMU during front-crawl and body roll previously used in literature (Mooney et al., 2016; Callaway, 2015). Since the medio-lateral acceleration with respect to the swimmer mainly reflects the roll movements during front-crawl (Ganzevles et al., 2019), we use this indirect relationship as a proxy for interpretation of our functional clustering. Then, the higher stability that could be induced by this reduced body roll seems to especially occur during the critical propelling phases of the cycle (i.e., extrema of longitudinal acceleration). This finding is in line with previous literature pointing out a reduction of whole-body roll, shoulder roll and hip roll amplitude in front crawl as pace increases (Gonjo et al., 2021). It allows to improve stroke rate by reducing the stroke cycle duration (Psycharakis & Sanders, 2008) while avoiding disturbance of propulsion caused by a large amount of torque around longitudinal axis (Yanai, 2003). Then, the *jerky* stroke pattern matches the requirements of the maximal exercise performed since reducing amplitude of body roll is expected from sprinters as they use higher stroke frequency and a possible more continuous inter-limb coordination (Seifert et al., 2010).

In summary, faster front-crawl sprinters display specific regulations of stroke pattern variability, reflected as *jerky*, targeting two goals. Firstly, producing high intra-cyclic acceleration fluctuations along both forward and upward axes leading to a degradation in overall stroke smoothness and an increase in energy expenditure. Secondly, presenting a higher body stability *during the critical instants of forward propulsion, that is the stroke cycle timings with the higher forward speed variations (i.e., peaks and valleys on longitudinal acceleration), might lead to faster swimming*. More precisely, it may be enabled by reducing body roll in order to increase the stroke rate.

Regarding the inter-cyclic clusters defining the structure of variability, three profiles arise from two characteristics. On the one hand, the non-constant credibility regions (as depicted on Figure 2 and Figure 3) refer to unequal variance in inter-cyclic kinematics. It indicates various stroke pattern dispersions and uneven variability according to the instant within the stroke cycle. More precisely, the highest dispersion occurs during the peaks on accelerations that lead to swimming velocity generation. Thus, such findings are in line with previous outcomes showing that the higher velocity oscillations in extreme cycles occurred mainly during propulsive phases of the cycle (Fernandes et al., 2022a). On the other hand, faster swimmers display significantly lower repeatability of stroke pattern indicating that higher kinematical inter-cyclic variation is associated with better sprint performance. Indeed, 62% of the *LR* cluster is composed of elite to world class swimmers. Fernandes et al., (2022b) showed that elite swimmer's kinematics (*i.e.* velocity variability) in sprint backstroke were more unstable than non-elite's ones which is in line with our result as the *LR* cluster is the fastest. However, these authors assess the inter-cyclic variability in a process-driven approach, using features of nonlinear dynamics (*i.e.*, sample entropy and Largest Lyapunov Exponent) that captures the notion of stability and complexity in the raw kinematical signal. Nevertheless, in the present approach, relying on
435 multivariate functional clustering, inter-cyclic variability is assessed through a modeling of the
436 ability to reproduce the swimming pattern. This methodology presents relevant advantages in
437 comparison to discrete features of nonlinear dynamics (Bouvet et al., 2024). Indeed it allows,
438 on the one hand, to deal with multivariate time series in order to consider at the same time all
439 the IMU dimensions as well as their interactions, and on the other hand, to quantify both the
440 magnitude of inter-cyclic variability as well as the timings in stroke cycle at which it occurs.
441 It opens the door to data-driven approach by applying curve clustering in sport settings (Chu et
442 al., 2020 ; Leroy et al., 2018). By gathering of FDA tools and clustering tools, respectively
443 towards suitable functional basis decomposition of the swimming IMU signals and double
444 partition of multivariate data allowing for dependency, the methodology used in this work
445 enable the identification of heterogeneous morphological patterns in swimming kinematics
446 (Zhang & Parnell, 2023). Despite the inherent methodological differences with features of
447 nonlinear dynamics, that could lead to further works comparing or combining them with
448 specific FDA approach to help in the understanding of the inter-cyclic variability (Barbosa et
449 al., 2017), congruent results are highlighted regarding the association between inter-cyclic
450 variability of kinematics and sprint swimming performance. Nonetheless, it seems
451 counterintuitive as traditional approaches of literature generally perceive movement variability
452 as noise indicating poor technique or inefficiencies in movement patterns detrimental to
453 performance (Abernethy et al., 2008). This apparent paradoxical effect of inter-cyclic
454 variability on performance could be explained by the specificity of our population and sprint
455 task. Indeed, we consider a large database with multiple skill levels. Actually, the impact of
456 expertise on kinematical variability management remains unclear and would imply an
457 adaptation to the task constraints rather than reproducing a predetermined motor pattern
458 (Fernandes et al., 2022a). Thus, the inverse relationship between movement variability and
459 expertise, skills or performance is not fully acknowledged in the literature, leading to a
460 contentious role in sport technique (Preatoni et al., 2013 ; Glazier & Davids, 2009). Indeed,
461 recent studies about swimming provided evidence that highly skilled athletes display larger
462 levels of movement variability to maximize their swimming speed during 50m freestyle
463 (Ganzevles et al., 2019 ; Morais et al., 2023). Moreover, Fernandes et al., 2022a demonstrated
464 that variability characterizes the sprint swimming. In that respect, sprint performance would be
465 associated with kinematical variability since elite swimmers adopted strategies to manage it,
466 referring the movement variability as an emergent behavior under highly constrained aquatic
467 environment and sprint task (Fernandes et al., 2022b). Then, movement variability is not always
468 detrimental to performance (Seifert et al., 2013). It can be adaptive as world class swimmers
469 must explore the environment to provide functional regulations of motor patterns leading to
470 effective outcomes (Seifert et al., 2016 ; Seifert et al., 2011). As a consequence, movement
471 variability is a way to distinguish elite athletes as previously shown in race walking (Preatoni
472 et al, 2010), and may also contribute to human movement optimization to ensure success in
473 technique as highlighted in gymnastics (Hiley & Yeadon, 2013). Furthermore, swimming at
474 high intensity in front-crawl implies biomechanical constraints leading to technical adaptations
475 of movement variability and coordination pattern (Seifert et al., 2016). The latter may be related
476 both to swimmer's speciality (i.e. sprint or distance swimming) and swimmer's characteristics
477 (i.e. expertise level or gender) (Figueiredo et al., 2012b ; Bideault et al., 2013). However,
478 variability may also reflect fatigue which can negatively affect the performance.
479 Finally, it is crucial to monitor kinematical variability in a data-driven framework to identify
480 when it reflects functional adaptations supporting performance versus when it indicates a need
481 for technical corrections. In that respect, the curve clustering framework used in the present
482 study can help on-field end-users for this purpose of technical profiling by using a functional
483 statistical model suited with the specificities of swimming kinematical data.
484
Regarding the cross-partition clusters defining technical skills profiling, both the jerky+LR and the jerky+MR are the fastest clusters. Indeed, they are both composed of more than 70% by elite to world-class swimmers with no one recreationally active. Such finding is consistent with aforementioned results highlighting distinct association of intra and inter-cyclic variability clusters with performance. More precisely, specific congruence of variabilities by the combination of both high levels of intra and inter-cyclic variability turns out to be a highly discriminative aspect in the context of a sprint task. The present analysis did not reveal any swimmers belonging to the jerky+HR cluster that represents a non-congruence of the two kinds of variability (i.e. high level of intra but low level of inter-cyclic variability). Actually, a jerky but repeatable swimming may refer to a lack of technical skills to convert kinematical variations to effective propulsion, while a jerky and unsteady swimming could refer to proficient functional adaptation to sustain the highest possible speed during sprint swimming. By this way, it can be hypothesized that the biomechanical ability of swimmers to establish congruence between two kinds of stroke variabilities (i.e. mean stroke pattern and associated dispersion) according to the constraints of the task (i.e. sprint swimming vs. long distance swimming) define technical skills and swimming expertise. Indeed, the opposite cluster of congruent variability (i.e. smoothy+HR) displays low performance level with only 2% of elite to world-class swimmers. Such results could be explained by the specificity of the sprint exercise involving large constraints on swimmer's motricity through the requirement of maximal propulsive forces by large kinematical variations at the expense of the swimming economy. Indeed, swimmers are not concerned about saving energy during maximal trials and adopt a strategy based on generating greater propulsive forces and less efficient technique (Lopes et al., 2023). Nevertheless, we may expect the smoothy+HR cluster to be a relevant technique for other tasks including longer duration and lower intensity. Indeed, the technical requirements are likely to be different between long, middle and short-distance events as resistive and propulsive forces differ between low and high speed in front-crawl (Takagi et al., 2021). For example, longer distance swimming as open water races requires conservative strategy in order to improve stroke efficiency by lowering energy cost from a given pace (Bouvet et al., 2023) whereas overcoming a large drag by maximizing thrust force through mechanical power output is essential in sprint swimming technique to achieve high velocities (Gatta et al., 2018). These specific task constraints can then widely affect the kinematics of swimmers by modifying the management of movement variabilities. By the way, swimmers adapt their technique in order to maximize performance.

In summary, it seems that the intra-cyclic variability is the most impacting kind of variability on performance for sprint swimming. Indeed, the present results do not highlight any significant differences in terms of speed for the different inter-cyclic pattern given the intra-cyclic pattern, albeit all of the three smoothy clusters are significantly different from the two jerky clusters. Therefore, it results in distinctive profiles combining intra and inter-cyclic variability and defining relevant biomechanical abilities.

Further assessment of variabilities regarding tasks constraints is necessary to fully understand stroke regulations leading to swimming performance. Extending the current approach to longer swimming duration can be a powerful way to interpret variabilities profiles according to race distance and define effective swimming technical skills. A limitation of this study is the lack of anthropometric and gender data that may impact the preferential kinematics. Moreover, combining present functional clusters of variabilities with conventional stroke parameters as stroke length, stroke rate and stroke index would contribute to the improvement of the identification of relations between technical regulations and their spatio-temporal outcomes.
In conclusion, the present study provided the first data-driven approach to profile technical stroke profiles during sprint front-crawl based on kinematical *curve profiling of both* intra and inter-cyclic variability. In this way, it provides a novel method, *based on an adapted functional clustering model*, for variabilities determination in swimming using embedded sensors, as it is a hot topic in literature (Fernandes et al., 2023). This study demonstrates the importance of variabilities combination through their congruence to adapt the swimming technique. Indeed, technical skills might be driven by a specific management of both variabilities, in order to align stroke pattern and its associated dispersion, according to the performance features required by the task constraints. Therefore, sprint swimming requires both a high level of intra and inter-cyclic variability through a jerky and unsteady stroke pattern. In contrast, polarization of variabilities seems dysfunctional. The contributions of this study can assist eyes-based technical evaluation using an automatic and non-invasive continuous method, *explicitly devoted to swimming kinematical clustering*, to provide objective data on variabilities regulations. Consequently, it allows coaches to profile technical skills while providing benchmarks to target the development of an explosive swimming style with a high level of body *stability* for their sprinters.

## Acknowledgments

We are grateful to the athletes and the staff for their kind collaboration.

## Fundings

This research was funded by the French Agence Nationale de la Recherche within the NePTUNE project (ANR-19-STPH-004) and conducted within the framework of the PIA EUR DIGISPORT project (ANR-18-EURE-0022). Antoine Bouvet was supported by a Ph.D. scholarship from the Ecole Normale Supérieure de Rennes.

## Declaration of interest statement

The authors have no conflict of interests.

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<table><caption>**Table 1.** Biomechanical features computed by IMU axis from the mean curves of the intra-cyclic variability cluster. LON: longitudinal. MED: mediolateral. ANT: anteroposterior</caption><thead><tr><th rowspan="2">Cluster</th><th colspan="3">Acceleration standard deviation (x10 g)</th><th colspan="3">Angular velocity standard deviation (deg/s)</th><th colspan="3">Jerk cost (x10 g²/s²)</th></tr><tr><th>LON</th><th>MED</th><th>ANT</th><th>LON</th><th>MED</th><th>ANT</th><th>LON</th><th>MED</th><th>ANT</th></tr></thead><tbody><tr><td><i>Jerky</i></td><td>15.4</td><td>47.7</td><td>9.12</td><td>101</td><td>11.9</td><td>18.1</td><td>17.3</td><td>22.8</td><td>7.10</td></tr><tr><td><i>Smoothy</i></td><td>8.84</td><td>56.3</td><td>3.28</td><td>113</td><td>4.23</td><td>16.7</td><td>1.86</td><td>20.0</td><td>0.40</td></tr></tbody></table>

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<table><caption>**Table 2.** Biomechanical features computed by IMU axis from the credibility regions of the inter-cyclic variability cluster. LON: longitudinal. MED: mediolateral. ANT: anteroposterior. LR: low repeatability. MR: moderate repeatability. HR: high repeatability</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Cluster</th><th colspan="3">Acceleration (x10<sup>2</sup> g)</th><th colspan="3">Angular velocity (deg/s)</th></tr><tr><th>LON</th><th>MED</th><th>ANT</th><th>LON</th><th>MED</th><th>ANT</th></tr></thead><tbody><tr><td rowspan="3">Standard deviation</td><td>LR</td><td>2.39</td><td>1.32</td><td>0.39</td><td>426</td><td>334</td><td>87.0</td></tr><tr><td>MR</td><td>1.10</td><td>1.14</td><td>0.36</td><td>284</td><td>96.8</td><td>81.0</td></tr><tr><td>HR</td><td>0.15</td><td>0.38</td><td>0.07</td><td>117</td><td>21.3</td><td>28.1</td></tr></tbody></table>
<table><caption>**Table 3.** Significant pairwise comparisons of clusters for both the inter-cyclic variability partition and the crossing partition. LR: low repeatability. MR: moderate repeatability. HR: high repeatability</caption><thead><tr><th>Partition</th><th>Cluster pairwise comparisons</th><th>Mean±SD and 95% confidence intervals of swimming speed (m/s)</th><th>Difference (%)</th><th>Effect size (Cohen's d)</th></tr></thead><tbody><tr><td rowspan="2">Inter-cyclic variability</td><td>LR vs. HR</td><td>1.75 [1.63;1.87]±0.27 vs. 1.40 [1.35;1.46]±0.17</td><td>+25.0</td><td>1.67</td></tr><tr><td>MR vs. HR</td><td>1.62 [1.53;1.70]±0.21 vs. 1.40 [1.35;1.46]±0.17</td><td>+15.7</td><td>1.15</td></tr><tr><td rowspan="6">Crossing intra-cyclic and inter-cyclic variability</td><td>Jerky+LR vs. Smoothy+HR</td><td>1.86 [1.80;1.92]±0.12 vs. 1.40 [1.35;1.46]±0.17</td><td>+32.9</td><td>2.86</td></tr><tr><td>Jerky+LR vs. Smoothy+LR</td><td>1.86 [1.80;1.92]±0.12 vs. 1.30 [0.88;1.71]±0.26</td><td>+43.1</td><td>3.68</td></tr><tr><td>Jerky+LR vs. Smoothy+MR</td><td>1.86 [1.80;1.92]±0.12 vs. 1.50 [1.42;1.58]±0.15</td><td>+24.0</td><td>2.60</td></tr><tr><td>Jerky+MR vs. Smoothy+HR</td><td>1.81 [1.72;1.90]±0.13 vs. 1.40 [1.35;1.46]±0.17</td><td>+29.3</td><td>2.49</td></tr><tr><td>Jerky+MR vs. Smoothy+LR</td><td>1.81 [1.72;1.90]±0.13 vs. 1.30 [0.88;1.71]±0.26</td><td>+39.2</td><td>2.97</td></tr><tr><td>Jerky+MR vs. Smoothy+MR</td><td>1.81 [1.72;1.90]±0.13 vs. 1.50 [1.42;1.58]±0.15</td><td>+20.7</td><td>2.15</td></tr></tbody></table>
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1000 **Figure 1.** Preprocessed data of two participants with different levels (blue: world-class swimmer (tier 5), red: recreationally active swimmer (tier 2)). Performance caliber defined according to the classification of McKay et al., 2022

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1004 **Figure 2.** Intra-cyclic clustering (*jerky* and *smoothy* stroke patterns) and inter-cyclic clustering (*low*, *moderate* and *high repeatability* credibility regions) of 3D accelerations

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1007 **Figure 3.** Intra-cyclic clustering (*jerky* and *smoothy* stroke patterns) and inter-cyclic clustering (*low*, *moderate* and *high repeatability* credibility regions) of 3D angular velocities

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1010 **Figure 4.** Swimming speed according to the classes coming from the partitions of (A) intra-cyclic variability clustering, (B) inter-cyclic variability clustering

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1013 **Figure 5.** Swimming speed according to the classes coming from the cross-partition

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1015 **Figure 6.** Performance caliber distribution according to the classes coming from the partitions of (A) intra-cyclic variability clustering, (B) inter-cyclic variability clustering

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1018 **Figure 7.** Performance caliber distribution according to the classes coming from the cross-partition

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