Dynamics of Open and Closed Chain Spherical Mechanisms Using Quaternion Coordinates

Abstract

Abstract This paper presents a general technique for deriving the equations of motion for any open or closed chain spherical mechanism. The technique uses quaternion coordinates to represent the position of each rigid body in the mechanism. Thus, if there are n moving bodies in the chain, there are 4n generalized coordinates in the equations of motion. The use of quaternion coordinates results in standardized quadratic constraint relations representing the hinged connections between bodies in the mechanism. These constraint equations augment the equations of motion. This technique has two important features. First, it is specifically adapted to spherical mechanisms and presents all positions as rotations. Second, the quadratic form of the constraint equations simplifies the computation of velocities and accelerations compatible with the constraints. As an example the equations of motion for a closed six bar spherical chain are derived.