The kinematic image of RR, PR, and RP dyads

Abstract

We provide necessary and sufficient conditions for admissible transformations in the projectivised dual quaternion model of rigid body displacements and we characterise constraint varieties of dyads with revolute and prismatic joints in this model. Projective transformations induced by coordinate changes in moving and/or fixed frame fix the quadrics of a pencil and preserve the two families of rulings of an exceptional three-dimensional quadric. The constraint variety of a dyad with two revolute joints is a regular ruled quadric in a three-space that contains a "null quadrilateral". If a revolute joint is replaced by a prismatic joint, this quadrilateral collapses into a pair of conjugate complex null lines and a real line but these properties are not sufficient to characterise such dyads. We provide a complete characterisation by introducing a new invariant, the "fiber projectivity", and we present examples that demonstrate its potential to explain hitherto not sufficiently well understood phenomena.

Rational motions with generic trajectories of low degree

“…A motion polynomial P + εD ∈ DH[t] of degree n describes a rational motion. We assume that P + εD is reduced whence also the rational motion is of quaternion degree n. A glance at Equation (8) confirms that its trajectories are generically of degree 2n. But, as already mentioned in the introduction, it is possible that this degree drops.…”

“…The trajectory of an arbitrary point [x 0 + x] ∈ P 3 with x = x 1 i + x 2 j + x 3 k is given by (8). We may re-write this as…”

“…This article will provide a comprehensive answer to the questions above. A first step in this direction was done in [8] where the authors provided a complete geometric characterization of all transformations of the seven-dimensional projective space P 7 over the vector space of dual quaternions that are induced by coordinate changes in fixed and moving frame. These projective transformations not only fix the Study quadric S, the exceptional generator E, and the quadric E in E but also the two families of (complex) rulings on E. Since conjugation interchanges these families, different trajectories of motion C and inverse motion C * may be explained by the position of intersection points of C with E with respect to one family of rulings.…”

The Study Variety of Conformal Kinematics

“…We summarize our findings in Theorem 1. The Study variety S ⊂ P 15 of conformal kinematics is given by the ideal (12) which is generated by ten bilinear equations. It is a projective variety of dimension ten and degree twelve.…”

“…[14], we observe that some typical approaches to problems of robotics or mechanism science via DH have not yet been generalized to conformal kinematics. Examples include the geometry of the Study quadric and its relation to space kinematics in the sense of [11,12,15], the study of constraint varieties [2,13,16], and the factorization theory of motion polynomials [6,10]. We feel that it is worth extending or generalizing these concepts to CGA + .…”