Kinematic Analysis and Derivation of Equations of Motion for Mechanisms with Loops of Different Motion Spaces

Kinematic analysis and derivation of equations of motion for a mechanism with loops of different motion spaces are developed. First, it is shown that screw coordinates of a joint which expresses relative motion between two links must be expressed by the basis vectors of motion spaces of the loops to which it belongs. Then, it becomes possible to derive the same number of linear equations for joint velocities and accelerations of a loop as the dimension of its motion space, and joint velocities and accelerations can be determined by solving these linear equations. Next, it is described that the force analysis can be performed by expressing the joint wrench as a linear combination of basis vectors of a reciprocal space of a motion space and other linearly independent screw coordinates. Using the developed theorem, the computational scheme for kinematic and force analysis is derived. The method to derive the equations of motion for a mechanism is also clarified.