Computer Aided Synthesis of Piecewise Rational Motions for Planar 2R and 3R Robot Arms

This paper deals with the problem of synthesizing piecewise rational motions of an object that satisfies kinematic constraints imposed by a planar robot arm with revolute joints. This paper brings together the kinematics of planar robot arms and the recently developed freeform rational motions to study the problem of synthesizing constrained rational motions for Cartesian motion planning. Through the use of planar quaternions, it is shown that for the case of a planar 2R arm, the problem of rational motion synthesis can be reduced to that of circular interpolations in two separate planes and that for the case of a planar 3R arm, the problem can be reduced to a combination of circular interpolation in one plane and a constrained spline interpolation in a circular ring on another plane. Due to the limitation of circular interpolation, only $C1$ continuous rational motions are generated that satisfy the kinematic constraints exactly. For applications that require $C2$ continuous motions, this paper presents a method for generating $C2$ continuous motions that approximate the kinematic constraints for planar 2R and 3R robot arms.