A Formulation of Kinematic Constraints Imposed by Kinematic Pairs Using Relative Pose in Vector Form

This paper presents a formulation that expresses kinematic pairs in form of holonomic constraints as functions of a measure of the relative position and orientation of the connected bodies expressed in vector form. While formulating the relative position measure is straightforward, expressing a suitable measure of the relative orientation requires some care. The problem is addressed by computing the Euler vector of the product of the actual and prescribed relative rotation matrices. By arbitrarily combining the error measures in up to six independent equations, a general family of holonomic rheonomic constraints can be formulated. The relative motion between the bodies can be constrained or specified component-wise, respectively, resulting in scleronomic or rheonomic constraints. The proposed formulation is implemented in a free, general-purpose multibody solver; numerical applications to generic mechanical and aerospace problems are presented.