# Peter Stephen DonelanVictoria University of Wellington · School of Mathematics and Statistics

Peter Stephen Donelan

PhD in Mathematics, Southampton, 1984

## About

37

Publications

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250

Citations

Introduction

I am currently working on several projects:
- with Andreas Müller on singularities of manipulators and non-holonomic systems
- Marco Carricato on invariants of multi-screws, serial manipulators and persistent manifolds
- with Hamed Amirinezhad on kinematic constraint models of manipulators.
Also mathematics and poetry remains an interest.

Additional affiliations

Education

October 1974 - July 1977

## Publications

Publications (37)

A syzygy is a relation between invariants. In this paper a syzygy is presented between invariants of sequences of six screws
under the action of the Euclidean group. This relation is useful in simplifying the computation of the determinant of a robot
Jacobian and hence can be used to investigate the singularities of robot manipulators.

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 Whit...

The analysis of singularities is a central aspect in the design of robotic manipulators. Such analyses are usually based on the use of geometric parameters like DH parameters. However, the manipulator kinematics is naturally described using the concept of screws and twists, associated to Lie groups and algebras. These give rise to general and coord...

The analysis of manipulator singularities is a burgeoning area of research. This chapter has touched brieﬂy on some key themes. There are signiﬁcant open problems, including understanding over-constrained mechanisms, genericity theorems for manipulator architectures, higher-order analysis and topology of the singularity loci, singularities of compl...

The signicance 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...

The Euclidean group of proper isometries SE(3) acts on its Lie algebra, the vector space of twists by the adjoint action. This extends to multi-twists and screw systems. Invariants of these actions encode geometric information about the objects and are fundamental in applications to robot kinematics. This paper explores relations between known inva...

Polynomial invariants for robot manipulators and their joints arise from the adjoint action of the Euclidean group on its Lie algebra, the space of infinitesimal twists or screws. The aim of this paper is to determine basic sets of generating polynomials for multiple screws. Techniques from the theory of SAGBI bases are introduced. As a result, a c...

We develop a differential-geometric approach to kinematic modelling for manipulators which provides a framework for analysing singularities for forward and inverse kinematics via input and output mappings defined on the manipulator’s configuration space.

We develop a differential-geometric approach to kinematic modelling for manipulators which provides a framework for analysing singularities for forward and inverse kinematics via input and output mappings defined on the manipulator's configuration space.

We develop a differential-geometric approach to kinematic modelling for manipulators which provides a framework for analysing singularities for forward and inverse kinematics via input and output mappings defined on the manipulator's configuration space.

Transversality is a mathematical concept, widely used in singularity theory, which generalises the idea of regularity of a function. One of its uses is in providing a method for certifying that a property of a given set of functions or mappings is generic, which is to say that it holds for almost all members of that set. Its application in kinemati...

Singularities of the configuration space for planar parallel mechanisms involving prismatic joint are investigated using a kinematic constraint map approach. Eliminating pose parameters from the singularity conditions together with the constraint equations leads to Grashof-type conditions. We illustrate the method by the planar RRRP and 3-RPR paral...

Singularities of the configuration space for planar parallel mechanisms involving prismatic joint are investigated using a kinematic constraint map approach. Eliminating pose parameters from the singu-larity conditions together with the constraint equations leads to Grashof-type conditions. We illustrate the method by the planar RRRP and 3-RPR para...

Transversality is a mathematical concept, widely used in sin-gularity theory, which generalises the idea of regularity of a function. One of its uses is in providing a method for certifying that a property of a given set of functions or mappings is generic, which is to say that it holds for almost all members of that set. Its application in kinemat...

Kinematic singularities are classically defined in terms of the rank of Jacobians of associated maps, such as forward and inverse kinematic mappings. A more inclusive definition should take into account the Lie algebra structure of related tangent spaces. Such a definition is proposed in this paper, initially for serial manipulators and non-holonom...

By incorporating gearing into a planar 3R mechanism, one obtains a family of mechanisms in which the gear ratios play a central kinematic role. Special choices of these parameters result in interesting simplifications of the kinematic mapping. An explicit expression for the mapping can be derived using the ‘matroid method’ of Talpasanu et al. [6]....

The kinematics of a robot manipulator are described in terms of the mapping connecting its joint space and the 6-dimensional Euclidean group of motions $SE(3)$. The associated Jacobian matrices map into its Lie algebra $\mathfrak{se}(3)$, the space of twists describing infinitesimal motion of a rigid body. Control methods generally require knowledg...

The kinematics of parallel mechanisms are defined by means of a kinematic constraint map (KCM) that captures the constraints imposed on its links by the joints. The KCM incorporates both pose parameters describing the configuration of every link and the design parameters inherent in the mechanism architecture. This provides a coherent approach to d...

Mathematics and poetry typically operate in different realms, employing language and symbol in apparently disjoint semantic domains. In this article we explore the creative output of the Romanian poet/mathematician Ion Barbu/Dan Barbilian. Although he actively pursued these disciplines at different times, he wrote extensively on the parallels he pe...

A non-zero element of the Lie algebra $\mathfrak{se}(3)$ of the special
Euclidean spatial isometry group $SE(3)$ is known as a {\em twist} and the
corresponding element of the projective Lie algebra is termed a {\em screw}.
Either can be used to describe a one-degree-of-freedom joint between rigid
components in a mechanical device or robot manipula...

The Denavit-Hartenberg (DH) notation for kinematic chains makes use of a set of parameters that determine the relative positions of and between successive joints. The corresponding matrix representation of a chain's kinematics is a product of two exponentials in the homogeneous representation of the Euclidean group. While the DH notation is based o...

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 th...

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 regul...

A rigid body, three of whose points are constrained to move on the coordinate planes, has three degrees of freedom. Bottema
and Roth [2] showed that there is a point whose trajectory is a solid tetrahedron, the vertices representing corank 3 singularities.
A theorem of Gibson and Hobbs [9] implies that, for general 3-parameter motions, such singula...

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 manipu-lator's joint variables, may impede control algorithms, lead to large joint velocities, forces and torques and...

Screw systems describe the infinitesimal motion of multi–degree-of-freedom rigid
bodies, such as end-effectors of robot manipulators. While there exists an exhaustive
classification of screw systems, it is based largely on geometrical considerations
rather than algebraic ones. Knowledge of the polynomial invariants of the adjoint
action of the Eucl...

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....

The instantaneously singular trajectories of points of a rigid body, able to move with several degrees of freedom, are determined by the associated screw system. We present a classification of screw systems based on a Lie group representation of motions and deduce the corresponding forms of the instantaneous singular sets. The idea of a generic pro...

Abstract 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 unfold- ings of these model singularities give rise to computer generated pictures describing the family of trajectories arising from small deformations of the tracing point, and de...

Plücker's and Klein's equations provide an upper bound on the number of real inflections on the coupler curve of a hinged planar four-bar mechanism. Generally, for any configuration of the four-bar, the coupler points whose trajectories exhibit inflections lie on a circle. The coupler plane is partitioned by the envelope of the inflection circles i...

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...

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...

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 unfold- ings 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 t...

Polynomial invariants for robot manipulators and their joints arise from the adjoint action of the Euclidean group on its Lie algebra, the space of infinitesimal twists or screws. The aim of this paper is to determine basic sets of generating polynomials for multiple screws. Techniques from the theory of SAGBI bases are introduced. As a result, a c...

## Projects

Projects (3)

Introduce a comprehensive methodology for studying mechanism kinematics including singularities. This methodology will be able to describe all topological properties of a given mechanism, locally and globally.