Peter Donelan | Victoria University of Wellington (original) (raw)

Papers by Peter Donelan

Mechanisms and machine science, 2019

We develop a differential-geometric approach to kinematic modelling for manipulators which provid... more 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.

arXiv (Cornell University), Nov 20, 2017

The kinematics of a robot manipulator are described in terms of the mapping connecting its joint ... more 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 se(3), the space of twists describing infinitesimal motion of a rigid body. Control methods generally require knowledge of an inverse for the Jacobian. However for an arm with fewer or greater than six actuated joints or at singularities of the kinematic mapping this breaks down. The Moore-Penrose pseudoinverse has frequently been used as a surrogate but is not invariant under change of coordinates. Since the Euclidean Lie algebra carries a pencil of invariant bilinear forms that are indefinite, a family of alternative hyperbolic pseudoinverses is available. Generalised Gram matrices and the classification of screw systems are used to determine conditions for their existence. The existence or otherwise of these pseudoinverses also relates to a classical problem addressed by Sylvester concerning the conditions for a system of lines to be in involution or, equivalently, the corresponding system of generalised forces to be in equilibrium.

Acta Applicandae Mathematicae, Jul 1, 1988

The motion of a rigid body in a Euclidean space E n is represented by a path in the Euclidean iso... more 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.

Engineers have for some time known that singularities play a significant role in the design and c... more 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.

Springer eBooks, Oct 9, 2006

Checking the regularity of the inverse jacobian matrix of a parallel robot is an essential elemen... more 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.

The kinematics of parallel mechanisms are defined by means of a kinematic constraint map (KCM) th... more 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 determining C-space singularities and generalised Grashof conditions on the design parameters under which these can occur.

Acta Applicandae Mathematicae, Sep 1, 1993

Springer proceedings in advanced robotics, Jul 18, 2020

The Euclidean group of proper isometries SE(3) acts on its Lie algebra, the vector space of twist... more 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 invariants and applies them to serial manipulators.

Journal of Geometry, Jul 14, 2015

A non-zero element of the Lie algebra {\mathfrak{se}(3)}$$se(3) of the special Euclidean spatia... more A non-zero element of the Lie algebra {\mathfrak{se}(3)}$$se(3) of the special Euclidean spatial isometry group SE(3) is known as a twist and the corresponding element of the projective Lie algebra is termed a screw. Either can be used to describe a one-degree-of-freedom joint between rigid components in a mechanical device or robot manipulator. This leads to a practical interest in multiple twists or screws, describing the overall instantaneous motion of such a device. In this paper, invariants of multiple twists under the action induced by the adjoint action of the group are determined. The ring of the polynomial invariants for the adjoint action of SE(3) acting on a single twist is well known to be finitely generated by the Klein and Killing forms, while a theorem of Panyushev (Publ. Res. Inst. Math. Sci. 4:1199–1257, 2007) gives finite generation for the real invariants of the induced action on two twists. However we are not aware of a corresponding theorem for k twists, where {k\geq3}$$k≥3. Following Study, Geometrie der Dynamen, (1903), we use the principle of transference to determine fundamental algebraic invariants and their syzygies. We prove that the ring of invariants for triple twists is rationally finitely generated by 13 of these invariants.

arXiv (Cornell University), Jan 15, 2020

Polynomial invariants for robot manipulators and their joints arise from the adjoint action of th... more 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 complete description is provided of the polynomial invariants for screw pairs and some results for screw triples are obtained. The invariants are shown to be related to Denavit-Hartenberg parameters.

arXiv (Cornell University), Apr 2, 2015

A non-zero element of the Lie algebra se(3) of the special Euclidean spatial isometry group SE(3)... more A non-zero element of the Lie algebra se(3) of the special Euclidean spatial isometry group SE(3) is known as a twist and the corresponding element of the projective Lie algebra is termed a screw. Either can be used to describe a one-degree-of-freedom joint between rigid components in a mechanical device or robot manipulator. This leads to a practical interest in multiple twists or screws, describing the overall instantaneous motion of such a device. In this paper, invariants of multiple twists under the action induced by the adjoint action of the group are determined. The ring of the polynomial invariants for the adjoint action of SE(3) acting on a single twist is well known to be finitely generated by the Klein and Killing forms, while a theorem of Panyushev [9] gives finite generation for the real invariants of the induced action on two twists. However we are not aware of a corresponding theorem for k twists, where k ≥ 3. Following Study [16], we use the principle of transference to determine fundamental algebraic invariants and their syzygies. We prove that the ring of invariants for triple twists is rationally finitely generated by 13 of these invariants. Euclidean isometry group and twist and polynomial invariant and dual number

Birkhäuser Basel eBooks, 2006

A rigid body, three of whose points are constrained to move on the coordinate planes, has three d... more A rigid body, three of whose points are constrained to move on the coordinate planes, has three degrees of freedom. Bottema and Roth [?] showed that there is a point whose trajectory is a solid tetrahedron, the vertices representing corank 3 singularities. A theorem of Gibson and Hobbs [?] implies that, for general 3-parameter motions, such singularities cannot occur generically. However motions subject to this kind of constraint arise as interesting examples of parallel motions in robotics and we show that, within this class, such singularities can occur stably.

InTech eBooks, Apr 1, 2010

Robotica, Nov 1, 2007

The significance of singularities in the design and control of robot manipulators is well known a... more 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.

Local models are given for the singularities which can appear on the trajectories of general moti... more 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

Mechanisms and machine science, Nov 2, 2014

The Denavit-Hartenberg (DH) notation for kinematic chains makes use of a set of parameters that d... more 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 on sound kinematic intuition, it is not obviously natural in mathematical terms. In this paper, we use the principle of transference to determine fundamental algebraic (polynomial) invariants of the Euclidean group SE(3) acting on sets of twists, elements of the group's Lie algebra, se(3), representing joints, and show that the DH parameters are algebraic functions of these invariants. We make use of the fact that for a set of three twists, there is an algebraicgeometric duality with the corresponding set of Lie brackets, so that link lengths of one correspond to offsets of the other.

Quarterly Journal of Mathematics, 1993

THERE are a number of ways of describing the motion of a rigid body in R2. The most general is to... more THERE are a number of ways of describing the motion of a rigid body in R2. The most general is to regard the motion as a subset of the total configuration space of the body. If the body is under no constraints then this space can be represented by the Euclidean group E(2) of ...

Kinematic singularities are classically defined in terms of the rank of Jacobians of associated m... more 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-holonomic platforms. The definition can be interpreted as a change in the number of successive infinitesimal motions required for the system to reach an arbitrary configuration in the vicinity of the given configuration. More precisely, it is based on the filtration of a controllability distribution.

Springer proceedings in advanced robotics, Jul 27, 2017