Direct and indirect effects of joint torque inputs during an induced speed analysis of a swinging motion

This study proposed a method to quantify direct and indirect effects of the joint torque inputs in the speed-generating mechanism of a swinging motion. Linear and angular accelerations of all segments within a multi-linked system can be expressed as the sum of contributions from a joint torque term, gravitational force term and motion-dependent term (MDT), where the MDT is a nonlinear term consisting of centrifugal force, Coriolis force and gyroscopic effect moment components. Direct effects result from angular accelerations induced by a joint torque at a given instant, whereas indirect effects arise through the MDT induced by joint torques exerted in the past. These two effects were quantified for the kicking-side leg during a rugby place kick. The MDT was the largest contributor to the foot centre of gravity (CG)'s speed at ball contact. Of the factors responsible for generating the MDT, the direct and indirect effects of the hip flexion-extension torque during both the flight phase (from the final kicking foot take-off to support foot contact) and the subsequent support phase (from support foot contact to ball contact) were important contributors to the foot CG's speed at ball contact. The indirect effect of the ankle plantar-dorsal flexion torque and the direct effect of the knee flexion-extension torque during the support phase showed the largest positive and negative contributions to the foot CG's speed at ball contact, respectively. The proposed method allows the identification of which individual joint torque axes are crucial and the timings of joint torque exertion that are used to generate a high speed of the distal point of a multi-linked system.