Kinematic Convexity of Rigid Body Displacements

Abstract

In this paper, we explore the notion of kinematic convexity for rigid body displacements. Previously, we have shown that when spatial rigid body displacements are represented by dual quaternions, an oriented projective space is better suited for the image space of displacements. Geometric algorithms for rigid body motions become more general and elegant when developed from the perspective of oriented projective geometry. By extending the concept of convexity in affine geometry to oriented projective geometry of the image space of rigid body displacements, we define the concept of kinematic convexity. This concept, apart from being theoretically significant, facilitates localization of a displacements and provides a measure of the kinematic separation useful in collision prediction, interference checking, and geometric analysis of swept volumes.

Bézier motions with end-constraints on speed

“…where (in the more general case) the N j (t) are the B-spline basis functions for the appropriate degree d [1]. (It is this form that is used to generate motions using dual quaternions in [13]. )…”

“…The quaternions can handle rotations but the dual quaternions are needed to handle translations as well [7,13,14]. Use γ to denote the dual number (with γ 2 = 0).…”

“…Variable 4 × 4 matrices [9,10] can have problems with robustness and clarity of geometric meaning of smoothing metrics. Quaternions [9,11,12] handle rotations, and their extension to dual quaternions [7,13,14] allows translations as well. The slerp (spherical linear interpolation) construction [11] generates free-form motions based on combining control poses multiplicatively, although this requires the use of exponentiation and logarithms.…”

Kinematic Interpretation of the Study Quadric’s Ambient Space

“…In this context it should also be mentioned that the PSH-map is the analogue of the map in P 5 sending a screw $ to its axis (cf. [17] and [9,Remark 10] [13] and [14].…”

“…It is known (cf. [14,Section 3.2], [17,Section 4], [16,Section 3]) that straight lines in P 7 are in general send by the PSH-map to vertical Darboux motions (see e.g. [2, pp.…”