Numerical Orientation Workspace Analysis with Different Parameterization Methods

For numerical orientation workspace analysis, a finite partition of the orientation workspace in its parametric domain is necessary. Among various parameterization methods for rigid-body rotations, it has been realized that the Euler angles, the tilting-and-torsion (T&T) angles, and the exponential coordinates are appropriate for finite partition. With these three parameterization methods, the rigid body rotation group, i.e., the special orthogonal group (SO(3)), can be mapped to a rectangular parallel-piped (for Euler angles), a solid cylinder (for T&T angles), and a solid sphere (for exponential coordinates). To simplify the computation, isotropic/equi-volumetric partition schemes are proposed for the three geometric entities so that each of them can be geometrically divided into finite elements with equal volumes. As a result of parameterizations, the volume of orientation workspace, i.e., the volume of SO(3), can be numerically computed as a weighted volume sum of its constituent equi-volumetric elements in which the weightages are the element associated integration measures. Using such partition schemes, various global performance measures can be readily implemented so as to evaluate the quality of the orientation workspace. A comparison study for the three parameterization methods has shown that the exponential coordinates method is more effective for numerical orientation workspace analysis because it has no formulation singularity and exhibits higher computation accuracy