Peter Donelan

Singularity-theoretic methods in robot kinematics

The significance of singularities in the design and control of robot manipulators is well known, and there is an extensive literature on the determination and analysis of singularities for a wide variety of serial and parallel manipulators—indeed such an analysis is an essential part of manipulator design. Singularity theory provides methodologies for a deeper analysis with the aim of classifying singularities, providing local models and local and global invariants. This paper surveys applications of singularity-theoretic methods in robot kinematics and presents some new results.

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.

On the regularity of the inverse Jacobian of parallel robots

Checking the regularity of the inverse jacobian matrix of a parallel robot is an essential element for the safe use of this type of mechanism. Ideally such check should be made for all poses of the useful workspace of the robot or for any pose along a given trajectory and should take into account the uncertainties in the robot modeling and control. We propose various methods that facilitate this check. We exhibit especially a sufficient condition for the regularity that is directly related to the extreme poses that can be reached by the robot.

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.

SINGULARITIES OF ROBOT MANIPULATORS

Engineers have for some time known that singularities play a significant role in the design and control of robot manipulators. Singularities of the kinematic mapping, which determines the position of the end–effector in terms of the manipulator’s joint variables, may impede control algorithms, lead to large joint velocities, forces and torques and reduce instantaneous mobility. However they can also enable fine control, and the singularities exhibited by trajectories of the points in the end–effector can be used to mechanical advantage. A number of attempts have been made to understand kinematic singularities and, more specifically, singularities of robot manipulators, using aspects of the singularity theory of smooth maps. In this survey, we describe the mathematical framework for manipulator kinematics and some of the key results concerning singularities. A transversality theorem of Gibson and Hobbs asserts that, generically, kinematic mappings give rise to trajectories that display only singularity types up to a given codimension. However this result does not take into account the specific geometry of manipulator motions or, a fortiori , to a given class of manipulator. An alternative approach, using screw systems, provides more detailed information but also shows that practical manipulators may exhibit high codimension singularities in a stable way. This exemplifies the difficulties of tailoring singularity theory’s emphasis on the generic with the specialized designs that play a key role in engineering.

Generic properties in Euclidean kinematics

The motion of a rigid body in a Euclidean space Enis 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.

Singularities of Regional Manipulators Revisited

The workspace singularities of 3R regional manipulators have been much analyzed. The presence of cusps in the singularity locus is known to admit singularity-avoiding posture change. Cusps arise in singularity theory as second-order phenomena – specifically they are Σ1,1 Thom– Boardman singularities. The occurrence of such singularities requires that the kinematic mapping be generic (in the sense of Pai and Leu [1]). Genericity and the occurrence of higher-order singularities in families of regional manipulators are investigated using Lie-theoretic properties of the Euclidean group.

Kinematic Singularities of Robot Manipulators

Kinematic singularities of robot manipulators are configurations in which there is a change in the expected or typical number of instantaneous degrees of freedom. This idea can be made precise in terms of the rank of a Jacobian matrix relating the rates of change of input (joint) and output (end-effector position) variables. The presence of singularities in a manipulator’s effective joint space or work space can profoundly affect the performance and control of the manipulator, variously resulting in intolerable torques or forces on the links, loss of stiffness or compliance, and breakdown of control algorithms. The analysis of kinematic singularities is therefore an essential step in manipulator design. While, in many cases, this is motivated by a desire to avoid singularities, it is known that for almost all manipulator architectures, the theoretical joint space must contain singularities. In some cases there are potential design advantages in their presence, for example fine control, increased load-bearing and singularityfree posture change. There are several distinct aspects to singularity analysis—in any given problem it may only be necessary to address some of them. Starting with a given manipulator architecture, manipulator kinematics describe the relation between the position and velocity (instantaneous or infinitesimal kinematics) of the joints and of the end-effector or platform. The physical construction and intended use of the manipulator are likely to impose constraints on both the input and output variables; however, it may be preferable to ignore such constraints in an initial analysis in order to deduce subsequently joint and work spaces with desirable characteristics. A common goal is to determine maximal singularity-free regions. Hence, there is a global problem to determine the whole locus of singular configurations. Depending on the architecture, one may be interested in the singular locus in the joint space or in the work space of the end-effector (or both). A more detailed problem is to classify the types of singularity within the critical locus and thereby to stratify the locus. A local problem is to determine the structure of the singular locus in the neighbourhood of a particular point. For example, it may be important to know whether the locus separates the space into distinct subsets, a strong converse to this being that a singular configuration is isolated. Typically, there will be a number of design parameters for a manipulator with given architecture—link lengths, twists and offsets. Bifurcation analysis concerns the changes in both local and global structure of the singular locus that occur as one alters design parameters in a given architecture. The design process is likely to involve optimizing some desired characteristic(s) with respect to the design parameters. 2

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

Genericity Conditions for Serial Manipulators

A generic, or more properly 1-generic, serial manipulator is one whose forward kinematic mapping exhibits singularities of given rank in a regular way. In this paper, the product-of-exponentials formulation of a kinematic mapping together with the Baker-Campbell-Hausdorff formula for Lie groups is used to derive an algebraic condition for the regularity.