NEW ZEALAND JOURNAL OF MATHEMATICS Volume 25 (1996), 141-163 SINGULARITY AND BIFURCATION FOR GENERAL TWO-DIMENSIONAL PLANAR MOTIONS C . G . G i b s o n a n d C.A. H o b b s* (Received March 1995) Abstract. Local models are given for the singularities which can appear on the trajectories of general two-dimensional motions of the plane. 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. 1. Introduction This paper is the second in a series devoted to a very basic question in theoret­ ical kinematics, of considerable potential importance in engineering robotics. The question is that of describing the surprisingly complex geometry associated to sin­ gularities of trajectories forgeneral motions of the plane and space. This research programme has its origins in two publications: a detailed study of the singularities of the planar four bar mechanism [17] and the seminal work of Donelan [9] on gen­ eral properties in Euclidean kinematics. The general framework is as follows. Let SE(p) denote the Lie group of proper rigid motions of R p. By an n-dimensional motion of R p we mean a smooth mapping A N: —> SE(p), where AT is a smooth manifold of dimension n. That yields a smooth mappingM\ : N x R p — > R p defined by (t, w) h-► X(t)(w): and for any fixed choice oftracing point w G R p it yields a smooth mappingM \)W : N —> R p defined by 1 t-> A(t)(w), which we refer to as the trajectory of w G R p under A. We think of M\ as a ^-parameter family of trajectories. The first paper in the series [18] established a basic transversality lemma, roughly speaking that a general n-dimensional motion of p-space gives rise to general sin­ gularities on the family of resulting trajectories. (An exact statement is given in Section 2 below.) That reduces the problem to one of classifying singularities from n-space to p-space of ‘codimension’ < p, in a certain sense. In the case of one­ dimensional motions of the plane we used standard techniques of singularity theory to establish an explicit finite list of models for the local behaviour of trajectories: these models are correct up to diffeomorphism, bar one exception correct only up to homeomorphism. Versal unfoldings of these models then yield complete descrip­ tions of the behaviour resulting from small changes in the position of the tracing point. Of course one-parameter motions of the plane are relatively well understood geometric objects, representing the core of classical kinematics, and the resulting pictures do little more than confirm the kind of instincts which a working kinemati- cian is likely to have: rather less is understood about one-parameter motions of 1991 AMS Mathematics Subject Classification: 70B10, 57R45. ’ Supported by a grant from the Science and Engineering Research Council. 142 C.G. GIBSON AND C.A. HOBBS space, and already the pictures throw up objects which do not appear to have been studied. An account of this work for an engineering audience appeared in [15]. However the question changes quite dramatically when one moves forward to the multi-parameter motions of the plane and space, closer to the daily concern of working kinematicians, but already on the edge of classical kinematics, if not beyond. In this paper we present a complete description of the local models asso­ ciated to general two-dimensional motions of the plane: we believe these results to be completely new to kinematics. (A description of the results for an engineering audience will appear in [20]). The trajectories are (in general) two-dimensional subsets of the plane: roughly speaking, singularities are exhibited on the “bound­ aries” of these sets, and can exhibit surprisingly complex behaviour. An easily visualized example is provided by the following construction, which we like to call the “parallel motion” [3, page 438]. Suppose we are given a parametrized curve C in a (fixed) planeE i, and a pointp in a (moving) planeE 2, and constrain the motion of E 2 by requiring p to lie on C. This generates a motion ofE 2 with two degrees of freedom - one for movement along C, and (for a fixed position ofp on C ) one given by rotation ofE 2 about p. The trajectory of a point inE 2 distant d from p is simply the union of the parallels ofC at distance <d: and the singular set is the union of the two “boundary” parallels at distanced. As in the one-dimensional case, for ageneral motion of the plane the changes in the models obtained by small perturbations of the tracing point are described by versal unfoldings of the local models. It takes an exceptional geometric ‘feel’ to guess even the simplest kinds of picture which can arise: thus there is little question that in the two-dimensional case the mathematical analysis really has something to offer the kinematician, especially since the interface is a graphical one. The pictures associated to singularities of corank 1 are well-known. However it proved to be a surprisingly difficult problem in computer graphics to obtain the pictures associated to corank 2 singularities, and we believe them to be new. For any family of smooth mappings a key role is provided by thebifurcation set i.e. the set of parameters for which the corresponding mapping exhibits non-stable singularities. Thus for the familyM\ : N x R p —► R p it comprises the set of tracing points w G R p for which the trajectory M \jW exhibits a non-stable singularity. The crucial importance of the bifurcation set in this situation is that it represents the boundary between different types of kinematic behaviour, and hence is of consid­ erable potential interest in engineering robotics. The basic transversality lemma alluded to above ensures that the local structure of the bifurcation set is determined by versal unfoldings of the generic singularity types. In the case of planar motions the bifurcation set is the union of a number of curves (one for each non-stable generic singularity type) and it appears to be a serious problem in computer alge­ bra to nail down their local structure. For one-dimensional motions there are three classical bifurcation curves, namely the cusp curve (or moving centrode), the tacn- ode (or transition) curve and the triple point curve. For two-dimensional motions there are six bifurcation curves, which (to the best of our knowledge) have never been observed in the kinematics literature. Even for mechanically simple examples it appears to be a substantial problem in computer graphics to display these curves on a computer screen. SINGULARITY AND BIFURCATION 143 We wish to acknowledge the considerable help we have received from others, in particular to Professor J.W. Bruce whose idea it was to reduce the multigerm clas­ sification to a monogerm classification (under a geometric subgroup of the Mather group *4) using recent results on determinacy. In order to make this idea work we use a key theoretical result of Gaffney and Wilson [13] on C°°-normalisations of real germs. On the computer graphics front, we received invaluable advice from Dr R.J. Morris and DrN . Kirk on using the A l g c u r v e program to obtain renderings for some of the versal unfoldings. In this connexion we would like to thank Dr W. Hawes for helping us to clarify questions about the corank 2 singularities, and for kindly agreeing to let us use her excellent sketches of some versal unfoldings. We enjoyed numerous helpful conversations with Dr F. Tari around some of the technical questions of the paper. And finally, we would like to thank the referee for picking up a missing case on the lists of multigerms. We adopt standard notations [24] used in singularity theory. Thus 7£,C and A are the standard Mather groups. And for any subgroupH C A we write H i for the subgroup of equivalences whose 1-jets are the identity. It will be convenient to reserve the term planar germ for a smooth germ R 2 —► R 2. 2. Generalities The starting point for listing local models of trajectories lies in a fundamen­ tal transversality result established in [18]. In order to state the result we re­ quire some preliminary definitions. Given positive integers r and k the mapping M \:W : N — > R p induces a multijet extension rjkMKw-.N^ ^ rJk(N,B?) and since M \jW depends smoothly on w that yields a mapping rj kM\ : iV(r) x R p — * rJk{N, R p). Theorem 2.1. Let T be a finite stratification of rJk( N ,R p). The set of n-dimensional motions A : N — * E{p) with rj kM\ transverse to T is residual in C °° (AT, E (p)), endowed with the Whitney C °° topology. (Basic Transversality Lemma.) We need now to be more explicit about the stratifications of the jet-space which concern us. Consider an .4-invariant stratificationS of the fibre rJk(n,p) of the multi-jet bundle with the property that for any stratumX all the .4fc-orbits con­ tained in X have the same dimension: then we may refer to themodality m of X , i.e. the common codimension of all the ^ -o r b its in the ambient spaceX. Such a stratificationS gives rise to an ^-invariant stratificationT of the multi-jet bundle space rJk(N ,R p). Let Y be a stratum in T.