Materials and Methods
Seventeen male professional volleyball players from local volleyball facilities voluntarily participated in this study (15 right-handed and two left-handed players). Their age, height, weight, and volleyball experience were expressed as mean ± SD [range], respectively: 20.70 ± 4.88 years (17–30 years), 1.94 ± 0.06 m (1.83–2.08 m), 85.17 ± 10.45 kg (64–104 kg), and 8.29 ± 3.09 years (5–12 years). Within their respective teams, nine of them serve as outside hitters, two as opposite spikers, and six as opposites. All players had prior experience with the jump serve, having practised it during training sessions and in national or regional competitions. Consequently, they exhibited proficiency in performing the jump serve. The participants had standard Chinese language skills, allowing them to complete the experiment by following the instructions and providing written informed consent. None of the participants had suffered neuro-muscular or skeletal injuries in 3 months before data collection. The study was conducted following the Declaration of Helsinki and approved by the Shanghai Research Institute of Sports Science Ethics Committee (protocol code LLSC20230015; September 21, 2023). Before the experiment, all participants provided written informed consent to confirm that they understood the purpose and potential risks of the study.

The experimental setup is shown in Figure 1 . The experiment was conducted in a regulation-sized volleyball hall (length: 18 m; width: 9 m), and the height of the net was set at 2.43 m. All participants wore close-fitting clothing and their own sports shoes. We utilised a traditional marker-based 3D motion capture system because of its ability to capture intricate movement details and achieve higher accuracy in spatial coordinate positions ( Scataglini et al., 2024 ). A set of 47 markers (diameter: 14 mm) as shown in Figure 2 were placed on the head vertex, left/right tragus, anterior/posterior shoulders, left/right acromions, suprasternal notch, xiphoid process, 5th cervical vertebrae, 10th thoracic vertebrae, left/right lateral costal borders (10th rib), 3rd metacarpal hands, medial/lateral wrist joints, medial/lateral elbow joints, left/right toes, 1st and 3rd metatarsal, calcaneal tuberosities, medial/lateral malleolus, medial/lateral knee joints, left/right greater trochanters, left/right anterior iliac spines, and left/right superior iliac spines. Three-dimensional coordinate data of the reflective markers during the jump serve were captured at 200 Hz using a 12-camera motion capture system (Qualisys Track Manager, Qualisys, Gothenburg, Sweden). Before each data collection session, the cameras were adjusted and calibrated according to the specific environment of the site. The cameras were arranged as depicted in Figure 1 , with the angle of each camera aligned to ensure coverage of the global coordinate system (L frame). A global coordinate system was set as a right-handed orthogonal reference frame, as shown in Figure 1 . The positive Z-axis was vertically upward, the Y-axis was in the opposite direction of the direction of motion of the spiked ball, and the X-axis was perpendicular to both the Y- and Z-axes.

Before the actual jump serve measurement, the participants were allowed to warm up following their usual routines ( e.g ., stretching, running, and spikes). Subsequently, the participants were allowed to practice jump serves (approximately 3–5 times), and they were allowed to decide whether they could participate in the formal experiments. A ‘T-pose’ static trial was captured for each participant, which included placing 47 markers on their body. Each jump serve was executed along a straight line behind the end line of the court. The ready position at which the jump began was decided independently by each individual. At the final step before take-off, both feet had to be planted in an area (length: 1.2 m; width: 1 m) composed of four force plates (FP 600 × 500 × 50, Kistler Group, Winterthur, Switzerland) to confirm that both feet left the ground. The force plates were zeroed repeatedly at the beginning and end of each trial. In addition, a runway made of hard plastic, measuring 6 m in length, 1 m in width, and 0.05 m in height, was installed behind the force plates. The participants were required to stand on this runway and then execute a ball toss and run up along the runway.

The participants were instructed to serve a volleyball into the target zone (length: 6 m, width: 3 m) with maximal effort. A successful trial was defined as one in which the ball cleared the net and landed in the target zone. Two experimental assistants were present: the first stood behind the target area, observing whether the ball landed inside or outside the target area, and the other subjectively evaluated whether the ball impacted the net. When a participant completed a successful trial, he was asked to evaluate his athletic performance subjectively using a five-point Likert scale (five points indicating an excellent trial and four points indicating a good trial). The participants were asked to repeat the jump serve until at least three successful trials with scores of four or five points were obtained. The trial with the highest ball speed, treated as the best performance trail, was selected for the subsequent motion analysis. The ball speed was measured using a volleyball pocket radar gun (Smart Coach, Pocket Radar Inc., Santa Rosa, CA, USA), which is the measurement method previously used in volleyball biomechanical studies ( Bujang & Samsudin, 2022 ; Lima et al., 2021 ).

All data processing was performed using MATLAB 2016a (MathWorks, Inc., Natick, MA, USA). The raw coordinates of the marker trajectories were smoothed using a low-parm swing speed Butterworth filter, and the cut-off frequency was set to 13 Hz ( Serrien et al., 2016 ). The entire body was modelled as a 14-link segment consisting of the trunk, head, upper arms, forearms, hands, thighs, shanks, and feet. The location of the centre of mass (CoM) and moment of inertia of the segment were estimated based on the study by Ae, Tang & Yokoi (1992) .

The airborne phase was defined as the interval between take-off and landing. Take-off (TO) was defined as the instant at which both feet were off the force plates, confirming a vertical ground reaction force (GRF) of less than 5 N. Landing-on (LO) was defined as the time of maximum downward vertical velocity of the CoM. MSER was defined as the timing of the maximum external rotation angle of the shoulder of the attack arm. The instant of ball impact (BI) was adopted from the previous studies ( Wagner et al., 2014 ; Zhang, Sado & Fujii, 2022 ), defined as the time at frame 1 (0.005 s) before the instant of detecting the anterior positive horizontal (Y-axis; Figure 1 ) acceleration (+) of the CoM of the attack arm hand. A more detailed analysis of the jump serve in the airborne phase was reported by Reeser et al. (2010) , who divided the jump serve into the arm-cocking (TO to MSER), arm-acceleration (MSER to BI), and follow-up (BI to LO) phases.

The arm swing speed, as adopted from a previous study ( Tilp, Wagner & Muller, 2008 ), was calculated using the square root of the CoM velocity data of the hand of the attack arm, with the value at BI extracted for statistical analysis.

The angular momentum was calculated using the methods reported by Bahamonde (2000) and Robertson et al. (2014) . The whole-body angular momentum was calculated as the sum of 14 body segments. To gain a deeper understanding of the contribution played by each segment in the angular momentum of the whole body, the body was partitioned into segmental systems for a right-handed player: attack arm (right upper arm, right forearm, and right hand), non-attack arm (left upper arm, left forearm, and left hand), trunk-head (trunk, head), attack leg (right thigh, right shank, and right foot), and non-attack leg (left thigh, left shank, and left foot). The units of angular momentum are kg·m 2 /s. To minimise the effects of differences in height and weight for each participant, all angular momentum variables were adopted from Hinrichs (1987) , which were normalised by dividing by the body mass (kg) and square of the height (m) of each participant. Consequently, in this study, the angular momentum variable was expressed in units of s –1 . Each calculated variable was normalised to 100% of the normalised time. The study focused on the sagittal plane (rotated around the X-axis). The counter-clockwise direction (+) corresponded to the direction of the spike ball, and the clockwise direction (−) was the opposite of the spike ball direction. For left-handed players, the data were treated as those of right-handed players during analysis and processing.

To investigate the importance of changes in angular momentum on the arm swing velocity during critical temporal phases (arm-cocking and arm acceleration), we employed the analysis method adopted by Martin et al. (2013) . The angular momentum variables at TO, MSER, and BI (as shown in Figs. 3 and 4 ) and the peak angular momentums were extracted for subsequent analysis of their correlation with the arm swing velocity at BI.

The descriptive statistics are presented as mean ± standard deviation (SD). The data normality was checked using the Shapiro–Wilk test. The Pearson product–moment correlation coefficients were used to identify the relationships between the peak angular momentum parameters and the arm swing velocity at BI. Spearman’s rank correlation test was conducted when the data were not normally distributed. In interpreting the correlations, the magnitude of r was classified as small, 0.1; moderate, 0.3; large, 0.5; very large, 0.7; and extremely large, 0.9 ( Hopkins et al., 2009 ). Stepwise multiple linear regression analyses were performed, with arm swing speed as the dependent variable and peak angular momentum (which showed significant correlations only) as the independent variables. Statistical significance for correlation and regression analyses was set at p < 0.05. All tests and statistical analyses were performed using the free statistical software JASP (JASP Team, University of Amsterdam, Amsterdam, The Netherlands, 2020).