Computer aided synthesis of piecewise rational motions for spherical 2R and 3R robot arms

This paper deals with the problem of synthesizing piecewise rational spherical motions of an object that satisfies the kinematic constraints imposed by a spherical robot arm with revolute joints. The paper brings together the kinematics of spherical robot arms and the recently developed freeform rational motions to study the problem of synthesizing constrained rational motions for Cartesian motion planning. Using quaternion kinematics of spherical arms, it is shown that the problem of synthesizing the Cartesian rational motion of a 2R arm can be reduced to that of circular interpolation in two separate planes. Furthermore, the problem of synthesizing the Cartesian rational motion of a spherical 3R arm can be reduced to that of constrained spline interpolation in two separate planes. Due to the limitation of circular interpolation, for spherical 2R robot arm, only C¹ continuous rational motions are generated. In this case, for applications that require C² continuous motions, the paper presents a method for generating a C² continuous joint motion that approximates a given C¹ rational motion of the end-effector. For spherical 3R arm, C² continuous rational motions are generated exactly.