C. G. Gibson

Geometry of screw systems—1: Screws: Genesis and geometry

Abstract In this two-part paper (Part 2: Mech. Mach. Theory 25 , 11–27; 1990) we claim to present, definitively and exhaustively, all “screw systems”, namely all possible linear combinations of given instantaneous screws. In kinematics, screw systems are central to multi-freedom devices, in particular those prevalent in robotics. We justify our work mainly because, first, there appears to be uncertainty in the minds of some as to the completeness of earlier efforts to define all screw systems, and, second, our approach, via projective five-space, throws valuable light on screw theory, and so has promise of other applications. In Part 1 we introduce lines and screws, and present the relevant geometry of projective five-space for their representation in it.

Geometry of screw systems—2: classification of screw systems

Abstract Here we apply and extend the material of Part 1 (Mech. Mach. Theory25, 1–10; 1990), systematically studying all the screw systems in the screw space, corresponding to motions of one, two and three freedoms. We establish a “normal form” for every screw system, namely the most concise representation in terms of base screws. Since, in Part 1, the principle of reciprocity of screws is shown to correspond to a polarity in the screw space, there is no need for a detailed survey of screw systems of a higher order than three.

On the geometry of the planar 4-bar mechanism

The paper seeks to elucidate the geometry of a simple engineering mechanism, comprising four bars smoothly jointed together to form a movable quadrilateral with one fixed side. The configurations of this mechanism correspond to the points of an elliptic curve, to which is associated interesting geometry and Morse theory. By appropriate projection, this curve yields the 2-parameter family of plane curves described by points rigidly attached to the side of the quadrilateral opposite the fixed side: the geometry of the general projection is related to the configuration of lines on a Segre quartic surface.

Simple singularities of space curves

The genesis of this paper lies in theoretical questions in kinematics where a central role is played by naturally occurring families of rigid motions of 3-space. The resulting trajectories are parametrized families of space curves, and it is important to understand the generic singularities they can exhibit. For practical purposes one seeks to classify germs of space curves of fairly small A e -codimension. It is however little harder to list the A-simple germs, which includes all germs of A e -codimension ≤11: that then is the principal objective of this paper.

A dexterity measure for the kinematic control of a multifinger, multifreedom robot hand

This work examines the properties of the manifold gener ated as the configuration space of the linkage used for each finger of the Salisbury hand. We begin with an exhaustive catalog of design types for the finger based on an analysis of its branch loci. We then study the condi tions under which the forward kinematic map becomes singular. These singularities define a submanifold that partitions the linkages configuration space into a number of open sheets, each of which maps diffeomorphically onto a corresponding open region in the fingers reach able work space. Next we consider the determinant func tion of the fingers Jacobian matrix. The stationary points of this function reveal those configurations where the Jacobian determinant is a maximum. The Jacobian deter minant can be thought of as an oriented volume in the tangent space to the fingers work space, and the orienta tion of this volume reveals the most favorable direction(s) for effecting tip motion or, reciprocally, for applying tip forces. From this we establish a simple criterion that can be used to find the optimal grasp configuration(s) for a given finite displacement of the workpiece.

On the Hierarchy of screw systems

LetE(3) be the Lie group of proper rigid motions of Euclidean 3-space. The adjoint action ofE(3) on its Lie algebrae(3) induces an action on the Grassmannian of subspaces of given dimensiond. Projectively, these subspaces are the screw systems of classical kinematics. The authors show that existing classifications of screw systems give rise to Whitney regular stratifications of the Grassmannians, and establish diagrams of specialisations for the strata. A list is given of the screw systems which can appear generically for motions of 3-space with at most three degrees of freedom.

First-order invariants of Euclidean motions

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.

Trajectory singularities of general planar motions

Local models are given for the singularities which can appear on the trajectories of general motions of the plane with more than two degrees of freedom. Versal unfoldings of these model singularities give rise to computer generated pictures describing the family of trajectories arising from small deformations of the tracing point, and determine the local structure of the bifurcation curves. 2 Supported by a grant from the Science and Engineering Research Council. AMS 1991 Classification: 70B10, 57R45

On Versal Unfoldings of Singularities for General Two-Dimensional Spatial Motions

Local models are given for singularities which can appear on the trajectories of general two-dimensional spatial motions. Versal unfoldings of these model singularities give rise to computer generated pictures describing the family of trajectories arising from small deformations of the tracing point.

Movable hinged spherical quadrilaterals—I

Abstract The underlying idea of this paper is that the possible motions of spherical quadrilaterals are represented by a family of algebraic curves in a four-dimensional configuration space, the parameters of the family being the angles A, B, C, D associated to the quadrilateral. In general these curves are elliptic octics with four ordinary double points, but acquire extra singularities precisely when A ± B ± C ± D ≡0 mod 2 π, relating to the classical Grashof conditions.