Generic properties in Euclidean kinematics

Abstract

The motion of a rigid body in a Euclidean space E n is represented by a path in the Euclidean isometry group E(n). A normal form for elements of the Lie algebra of this group leads to a stratification of the algebra which is shown to be Whitney regular. Translating this along invariant vector fields give rise to a stratification of the jet bundles Jk(R, E(n)) for k = 1, 2 and, hence, via the transversality theorem, to generic properties of rigid body motions. The relation of these to the classical centrodes and axodes of motions is described, together with applications to planar 4-bar mechanisms and the dynamics of a rigid body.

Real inflections of hinged planar four-bar coupler curves

“…Whenθ(t 0 ) = 0 butθ(t 0 ) = 0 points x satisfying (8), whose trajectories necessarily possess inflections at that instant, form a line asymptotic to the moving centrode. For generic four-bars the above condition on the derivatives of θ necessarily holds [3]. This ensures that such instants t 0 are isolated.…”

“…The motion of a generic four-bar may be regarded as a regular parametrised motion by mapping its configuration space, defined as a real algebraic variety by equations (1) with w = 1, onto the configuration space for the coupler bar alone, as was done in Donelan [3] and Marsh [8]. That it has a smooth local parametrisation is a simple consequence of the Implicit Function Theorem applied to the equations defining the variety; the parameter may be chosen from among the variables x i , y i , although in practice it is more usual to employ the angular displacement of one of the moving bars.…”

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.…”

Singular Phenomena in Kinematics

“…For a 1-manifold N , there are I-invariant stratifications of the jet bundles J k (N, SE(p)) with k = 1, 2 to which transversality of the jet extension of a motion implies the following [12]. For p even, there is a unique instantaneous centre, except at a discrete set of points in the configuration space N where the instantaneous rotational component vanishes; moreover the locus of instantaneous centres will be an immersion in R p .…”

“…The first such families are those corresponding to the K-type of the germ t, 0 → t 3 , 0 ; the A-types are labelled E 6k , E 6k+2 in [9]. It was shown in [12] that these types can only occur when the centrode curve (the locus of ISSs) itself has a singularity. This presages further relations between singularity type and the ISSs for 2-dof motions.…”