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

This paper deals with the problem of synthesizing smooth piecewise rational spherical motions of an object that satisfies the kinematic constraints imposed by a spherical robot arm with revolute joints. This paper brings together the kinematics of spherical robot arms and recently developed freeform rational motions to study the problem of synthesizing constrained rational motions for Cartesian motion planning. The kinematic constraints under consideration are workspace related constraints that limit the orientation of the end link of robot arms. This paper extends our previous work on synthesis of rational motions under the kinematic constraints of planar robot arms. 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. We present algorithms for the generation of $C1$ and $C2$ continuous rational motion of spherical 2R and 3R robot arms.