Kinematics of a Novel Serial-Parallel, Compliant, Three-Legged Robot

In this article, the solution to the inverse and forward kinematics of a novel three-legged, serial-parallel, compliant robot are presented. The design of the robot aims to combine properties from serial and parallel architectures, thereby targeting an agile, yet precisely controllable robot that possibly allows both for dynamic locomotion and accurate manipulation tasks. The robot embodies the series connection of two parallel architectures, a planar and a spherical mechanism, realized in a highly dense mechanical assembly, allowing for lightweight, functionally redundant and compliant 4-DOF legs. A hybrid compliance behaviour is achieved, serving as a threshold between a stiff and compliant systemic state of the robot. Based on the mechanism design, the study involves the derivation of an alternative, yet complete solution for the inverse and forward kinematics of the spherical parallel manipulator (SPM). The approach utilizes spherical trigonometry and spatial vector geometry and yields a unique solution, while being both easy to implement and numerically efficient, thereby being applicable to real time implementations. Conceptually, a reduction of the mechanism assembly and working modes was directly integrated into the solution terms, drastically simplifying the expressions as they only represents the mechanically meaningful configuration relevant for actual physical systems. In addition to the active joint coordinates, the solution yields a unique set of all passively driven joints. The derived robot kinematics are furthermore verified through a 3D simulation model, showing the robot performing several motions. Thereby, the simulations portray the characteristics of the motor units by means of their corresponding phase profiles, revealing a balanced utilisation.