A minimal delay-programmed model unifying kinetic-chain timing and stretch--shortening-cycle enhancement in tennis racket acceleration

Abstract The stretch--shortening cycle (SSC) is among the best-documented mechanisms of enhanced power output in striking skills, yet it emerges from a complex interplay of tendon recoil, pre-activation, and reflex action that is difficult to embed in the simple mechanical models commonly used in racket-sport engineering. We present a minimal planar body--arm--racket model in which (i)~each joint stores elastic energy in a latched posture and is released after an explicitly programmed delay, and (ii)~the shoulder and wrist may additionally engage a reduced SSC whose active character is captured by only two parameters: a rebound stiffness gain $\alpha$ and a velocity-dependent firing gain $\beta$. Because both mechanisms enter the equations of motion through the same joint-torque algorithm, kinetic-chain timing and SSC enhancement can be optimized within a single computational framework. With optimally tuned delays, racket-tip speed grows superadditively from \SI{16.6}{m.s^{-1}} (one active joint) to \SI{30.2}{m.s^{-1}} (two joints) and \SI{43.1}{m.s^{-1}} (three joints); engaging the SSC within the same firing schedule raises the peak to \SI{46.7}{m.s^{-1}} ($\approx 8\%$), at essentially the same impact frame. The model thus provides a drop-in, analytically transparent surrogate for muscle-level simulation of racket strokes.