Deterministic Reconstruction of Tennis Serve Mechanics: From Aerodynamic Constraints to Internal Torques via Rigid-Body Dynamics

TL;DR

This work tackles the problem of explaining tennis serve mechanics from first principles by constructing a 12-DOF rigid-body model of the upper body and solving a trajectory optimization problem to meet aerodynamic targets for flat, slice, and kick serves. It then performs inverse dynamics via the Principle of Virtual Power to reveal the joint torques required to realize the optimized motions. A key finding is that kinematically similar serves can demand drastically different torque profiles, with distal joints like the wrist showing large, time-varying torques driven by inertia, gravity, and inter-joint coupling. The approach provides a transparent, mechanistic framework that connects aerodynamic outcomes to musculoskeletal demands and offers practical value for coaching, injury prevention, and rehabilitation.

Abstract

Most conventional studies on tennis serve biomechanics rely on phenomenological observations comparing professional and amateur players or, more recently, on AI-driven statistical analyses of motion data. While effective at describing \textit{what} elite players do, these approaches often fail to explain \textit{why} such motions are physically necessary from a mechanistic perspective. This paper proposes a deterministic, physics-based approach to the tennis serve using a 12-degree-of-freedom multi-segment model of the human upper body. Rather than fitting the model to motion capture data, we solve the inverse kinematics problem via trajectory optimization to rigorously satisfy the aerodynamic boundary conditions required for Flat, Slice, and Kick serves. We subsequently perform an inverse dynamics analysis based on the Principle of Virtual Power to compute the net joint torques. The simulation results reveal that while the kinematic trajectories for different serves may share visual similarities, the underlying kinetic profiles differ drastically. A critical finding is that joints exhibiting minimal angular displacement (kinematically ``quiet'' phases), particularly at the wrist, require substantial and highly time-varying torques to counteract gravitational loading and dynamic coupling effects. By elucidating the dissociation between visible kinematics and internal kinetics, this study provides a first-principles framework for understanding the mechanics of the tennis serve, moving beyond simple imitation of elite techniques.

Figures (9)

• Figure 1: Schematic representation of the multi-segment kinematic model showing the twelve joint angles. The global coordinate system is defined with $x$ pointing laterally toward the dominant side, $y$ pointing toward the net, and $z$ pointing vertically upward. The angular variables illustrated in the figure are $(\theta_1, \theta_2, \ldots, \theta_{12}) = (-25^\circ, 5^\circ, 25^\circ, 20^\circ, -5^\circ, 80^\circ, 170^\circ, -20^\circ, -40^\circ, 35^\circ, 50^\circ, 30^\circ)$.
• Figure 2: Sequential illustration of the tennis flat serve motion from the preparation phase to the moment of impact. The vectors $\vec{n}$ and $\vec{v}$ denote the normal vector and velocity vector of the racket head, respectively. Here, $s$ represents the normalized time elapsed from the preparation phase.
• Figure 3: Sequential illustration of the tennis slice serve motion from the preparation phase to the moment of impact. The vectors $\vec{n}$ and $\vec{v}$ denote the normal vector and velocity vector of the racket head, respectively. Here, $s$ represents the normalized time elapsed from the preparation phase.
• Figure 4: Sequential illustration of the tennis kick serve motion from the preparation phase to the moment of impact. The vectors $\vec{n}$ and $\vec{v}$ denote the normal vector and velocity vector of the racket head, respectively. Here, $s$ represents the normalized time elapsed from the preparation phase.
• Figure 5: Time dependence of arm and wrist joint angles ($\theta_7$--$\theta_{10}$) for flat (red), slice (green), and kick (blue) serves. Line styles indicate specific angles: $\theta_7$ (thin solid), $\theta_8$ (thin dotted), $\theta_9$ (thick solid), and $\theta_{10}$ (thick dashed). The time axis is normalized to the impact moment ($T=1$).