Hyperbolic pseudoinverses for kinematics in the Euclidean group

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

The complex of lines from successive points and the horopter

The Horopter is the set of points in space which project to the same image points in the two cameras of a stereo vision system. Modern proofs are given for many of the classical results about the Horopter. Some of these rely on another classical construction, the quadratic complex of lines joining successive points. A modern derivation of this is also given. In particular the circumstances in which the image of the Horopter can degenerate is discussed in some detail. Finally, these ideas are extended to the case of optical flow where there is a single camera observing the velocities of points in space. ©2008 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works

Curves of stationary acceleration in SE(3)

The concept of curves of minimal acceleration seems to have been introduced by Žefran and Kumar and independently by Noakes, Heinzinger and Paden. In part, the motivation was to extend the notion of spline curves to curves in groups, specifically the groups associated with robotics. A curve in the rigid-body motion group SE(3), e.g. can be thought of as a trajectory of a rigid body. Hence, these ideas have applications to motion planning and interpolation. In this work, the analysis is repeated but using bi-invariant metrics on the group. Since these metrics are not positive definite, the curves specified by the equations derived are only stationary, not minimal. It is possible to solve these non-linear coupled differential equations in some simple cases. However, these simple cases turn out to be highly relevant to robotics and mechanism theory. © 2007 Oxford University Press. This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Mathematical Control and Information following peer review. The version of record is available online at: http://dx/doi/org/10.1093/imamci/dnl01

Three problems in robotics

Three rather different problems in robotics are studied using the same technique from screw theory. The first problem concerns systems of springs. The potential function is differentiated in the direction of an arbitrary screw to find the equilibrium position. The second problem is almost identical in terms of the computations; the least-squares solution to the problem of finding the rigid motion undergone by a body given only data about points on the body is sought. In the third problem the Jacobian of a Stewart platform is found. Again, this is achieved by differentiating with respect to a screw. Furthermore, second-order properties of the first two problems are studied. The Hessian of second derivatives is computed, and hence the stability properties of the equilibrium positions of the spring system are found

Some rigid-body constraint varieties generated by linkages

© Springer Science+Business Media Dordrecht 2012. The set of rigid-body displacements allowed by three simple open-chain linkages are studied. These linkages consist of a cylindrical and spherical joint: The CS dyad, a revolute, a prismatic and a spherical joint: The RPS linkage, two revolutes and a spherical joint: The RRS linkage. Using the Study quadric to represent the group of all rigid-body displacements the constraint varieties for these examples are found. In the case of the CS and RPS linkages these are found to be quartic hypersurfaces while the constraint variety for the RRS linkage is a hypersurface of degree 8. Finally it is shown that all three constraint varieties are linear projections of a Segre variety in P15

On the instantaneous acceleration of points in a rigid body

This work re-examines some classical results in the kinematics of points in space using modern vector-matrix methods. In particular, some very simple Lie theory allows the velocities and accelerations of points to be found in terms of the instantaneous twist of the motion and its derivative. From these results many of the classical results follow rather simply. Although most of the results are well known, some new material is presented. In particular, the discriminant curve that separates cases with one or three real acceleration axes is found and plotted. Another new result concerns the chords to the cubic of inflexion points. It is shown that for points on such a chord the osculating planes of the point's trajectories are parallel. Also a new result is found which distinguishes between cases where the Bresse hyperboloid of points whose velocities and accelerations are perpendicular, has one or two sheets. © 2011 Elsevier Ltd. All rights reserved

A Class of Explicitly Solvable Vehicle Motion Problems

A small but interesting result of Brockett is extended to the Euclidean group SE(3) and is illustrated by several examples. The result concerns the explicit solution of an optimal control problem on Lie groups, where the control belongs to a Lie triple system in the Lie algebra. The extension allows for an objective function based on an indefinite quadratic form. Applying the result requires explicit knowledge of the Lie triple systems of the Lie algebra se(3). Hence, a complete classification of the Lie triple systems of this Lie algebra is derived. Examples are considered for optimal trajectories in three cases. The first case concerns cars moving in the plane. The second looks at motions that rigidly follow the Bishop frame to a space curve. The final example does not have a particular name as it does not seem to have been studied before. The appendix gives a brief introduction to Screw theory. This is essentially the study of the Lie algebra se(3)

Spatial stiffness matrix from simple stretched springs

This work looks at the stiffness matrix of some simple but very general systems of springs supporting a rigid body. The stiffness matrix is found by symbolically differentiating the potential function. After a short example attention turns to the general structure of the stiffness matrix and in particular the principal screws introduced by Ball

On the geometry of the homogeneous representation for the group of proper rigid-body displacements

This work investigates the geometry of the homogeneous representation of the group of proper rigid-body displacements. In particular it is shown that there is a birational transformation from the Study quadric to the variety determined by the homogeneous representation. This variety is shown to be the join of a Veronese variety with a 2-plane. The rest of the paper looks at sub-varieties, first those which are sub-groups of the displacement group and then some examples defined by geometric constraints. In many cases the varieties are familiar as sub-varieties of the Study quadric, here their transforms to the homogeneous representation is considered. A final section deals with the map which sends each displacements to its inverse. This is shown to be a quadratic birational transformation