Numerical geometry of the coupler motion of a planar 4R closed kinematic chain

This paper studies the problem of analyzing the motion of the coupler link of a planar 4R closed kinematic chain from the viewpoint of constrained motion interpolation. The kinematics of the planar 4R chain is formulated using planar quaternions. The four-point interpolatory subdivision scheme for curves in the field of Computer Aided Geometric Design (CAGD) is extended to planar quaternions for the generation of an inbetween candidate position of the coupler link from a given set of four key positions. The candidate position is then checked and modified according to the kinematic constraints of the planar 4R chain. The resulting new inbetween position that satisfies the kinematic constraints is then inserted into the given set of key positions. In the early stage of this refinement process, each new inbetween position must be made to satisfy the 4R kinematic constraints exactly to ensure the correct motion. When there is sufficient number of coupler positions, one can use the unconstrained four-point interpolatory scheme to generate the inbetween positions to allow for fast animation of the coupler motion.