| This dissertation deals with the problem of synthesizing rational motions of a rigid body that satisfy kinematic constraints imposed by planar, spherical, and spatial kinematic chains. The dissertation brings together the well-known kinematics of various kinematic chains and the recently developed freeform rational motions to synthesize the constrained rational motions for Cartesian motion planning. The kinematic constraints under consideration are workspace related constraints that limit the position of the end link of open chains and the coupler link of closed chains.;Planar quaternions, quaternions, and dual quaternions are used to represent planar, spherical, and spatial displacements, respectively. In this way, displacements of a rigid body in Cartesian space are mapped into points in quaternion space, and the kinematic constraints are transformed into geometric constraints, such as circle, circular ring, spherical and hyperboloidal shells in quaternion space. Thus, the problem of rational motion interpolation is transformed into that of rational curve interpolation, where the standard scheme for curve interpolation in Computer Aided Geometric Design (CAGD) can be applied. For the constrained curve interpolation this dissertation develops several efficient numerical algorithms that include smooth piecewise rational Bezier interpolation on a circle, smooth rational B-spline interpolation inside an n-spherical shell and within intersection of two hyperboloidal shells.;The last portion of the dissertation adopts a different approach for rational motion interpolation of planar chains in a parametric space defined by the elements of planar displacement matrix. This approach has the advantage of being direct and yields lower degree motions.;The results of this dissertation have applications in Cartesian motion planning in robotics and task specification for task driven design of robots and mechanisms. |
