Approximate Motion Synthesis of Spherical Kinematic Chains

In this paper we present a novel dimensional synthesis technique for approximate motion synthesis of spherical kinematic chains. The methodology uses an analytic representation of the spherical RR dyad’s workspace that is parameterized by its dimensional synthesis variables. A two loop nonlinear optimization technique is then employed to minimize the distance from the dyad’s workspace to a finite number of desired orientations of the workpiece. The result is an approximate motion dimensional synthesis technique that is applicable to spherical open and closed kinematic chains. Here, we specifically address the spherical RR open and 4R closed chains however the methodology is applicable to all spherical kinematic chains. Finally, we present two examples that demonstrate the utility of the synthesis technique. INTRODUCTION The novel dimensional synthesis technique presented utilizes an analytic representation of the spherical RR dyad’s workspace that is parameterized by its dimensional synthesis variables. The parameterized workspace represents the geometric constraint imposed on the motion of the moving body or workpiece. This constraint is a result of the geometric and kinematic structure of the dyad; e.g. its length and the location of its fixed and moving axes (i.e. lines). The workspace is an analytical representation of the workspace of the dyad that is parameterized ∗Address all correspondence to this author. 1 by the dyad’s dimensional synthesis variables. Here we derive the parameterized workspace of spherical RR dyads using 3×3 elements of SO(3) (also known as rotation matrices) and utilize this representation to perform dyadic dimensional synthesis for approximate rigid body guidance. The derivation of the parameterized workspace involves writing the kinematic constraint equations of the dyad using homogeneous coordinate transformations. The result is an analytical representation of the workspace of the dyad that is parameterized by its joint variables. The synthesis goal is to vary the design variables such that all of the prescribed locations are either: (1) in the workspace, or, (2) the workspace comes as close as possible to all of the desired locations. Recall that in general five is the largest number of locations for which an exact solution is possible for the spherical RR dyad, see [1, 2]. In related works, kinematic mappings have been used to derive the constraint manifold representation of the kinematic constraint equations. The derivation of the constraint manifold involves writing the kinematic constraint equations using the image space representation of displacements, see [3], [4], and [5]. In [6] the constraint manifold of the spherical RR dyad is used to solve the 5 orientation Burmester problem. Previous works discussing constraint manifold fitting for an arbitrary number of locations include [7], [8], [9], and [4]. All of these works employ implicit representations of the dyad constraint manifolds. The constraint manifolds, that are known to be highly nonlinear [10], are approximated by tangent hyperplanes by using a standard Taylor series linearization strategy. The distance from Copyright c © 2007 by ASME