First-order invariants of Euclidean motions

Abstract

Let E(n) be the Lie group of proper rigid motions of Euclidean n-space. The paper is concerned with the adjoint action of E(n) on its Lie algebra e(n), and the induced action on the Grassmannian of subspaces of e(n) of a given dimension. For the adjoint action, the authors list explicit generators for the ring of invariant polynomials. In the case n = 3, of greatest physical interest, explicit finite invariant stratifications are given for the Grassmannians, providing a formal listing of the screw-systems familiar in theoretical kinematics.

Singular Phenomena in Kinematics

“…However, the equivalence relation underlying the listing was not explicit, raising doubts about its completeness. This situation was rectified in [27], and a classification was described intrinsically by the present authors in [14] following the original philosophy of Klein. In [15] we showed that the listing gave rise to natural Whitney stratifications, and gave complete descriptions of the specialisations.…”

“…The group SE(p) behaves like a geometrically reductive group, in that the ring of invariant polynomials for the adjoint action is finitely generated. In [14] we gave explicit generators, the form depending on the parity of p. When p = 3 the ring is generated by the Klein and Killing forms, a result implicit in the subject but for which we are not aware of any prior reference.…”

Singularities of Robot Manipulators

“…It inherits from the group structure described in Section 1 the structure of a semi-direct product of the Lie algebras so(3) of the rotation group and t(3) of the translation group. Thus, elements may be represented by a pair (B, v) ∈ so (3) Following ideas in [21,22], the following local equivalence was defined in [23], where it is assumed that coordinates are chosen so that at the configuration x ∈ M , λ(x) = 1 (the group identity). Definition 6.1.…”

“…In particular its Killing form is degenerate and the polynomial invariant theory for reductive algebras does not apply. Donelan and Gibson [23] determined generators for the ring of invariant polynomials for the adjoint action of SE(n) and in particular:…”

“…A mathematical foundation for the classification was provided by Gibson and Hunt [32]. The principles underlying the classification, which is I-invariant, are the following [23].…”