J. M. Selig

Clifford algebra of points, lines and planes

The Clifford algebra for the group of rigid body motions is described. Linear elements, that is points, lines and planes are identified as homogeneous elements in the algebra. In each case the action of the group of rigid motions on the linear elements is found. The relationships between these linear elements are found in terms of operations in the algebra. That is, incidence relations, the conditions for a point to lie on a line for example are found. Distance relations, like the distance between a point and a plane are found. Also the meet and join of linear elements, for example, the line determined by two planes and the plane defined by a line and a point, are found. Finally three examples of the use of the algebra are given: a computer graphics problem on the visibility of the apparent crossing of a pair of lines, an assembly problem concerning a double peg-in-hole assembly, and a problem from computer vision on finding epipolar lines in a stereo vision system.

The spatial stiffness matrix from simple stretched springs

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 (1900).

A simple approach to invariant hybrid control

We give a geometrical description of Raibert and Craigs hybrid force/position control method (1981). Our description is coordinate free, hence answering the criticism of the original work that it was not transformation invariant. However, our approach avoids the complications introduced in the work of Lipkin and Duffy (1988). This simplification is achieved by recognising that velocity screws and wrenches are different geometrical objects and then keeping them separate throughout the discussion. So we do not use any metric properties of the screw space of infinitesimal rigid body motions. Rather, we employ the duality between the vector space of screws and the linear functionals on them. We give several examples and show how changes of coordinates should be handled.

A Screw Theory of Timoshenko Beams

In this work, the classic theory of Timoshenko beams is revisited using screw theory. The theory of screws is familiar from robotics and the theory of mechanisms. A key feature of the screw theory is that translations and rotations are treated on an equal footing and here this means that bending, torsion, and extensions can all be considered together in a particularly simple manner. By combining forces and torques into a six-dimensional vector called a wrench, Hookes law for the Timoshenko beam can be written in a very simple form. From here simple expressions can be found for the kinetic and potential energy densities of the beam. Hence equations of motion for small vibrations of the beam can be easily derived. The screw theory also leads to a new understanding of the boundary conditions for beams. It is demonstrated that simple boundary conditions are closely related to mechanical joints. In order to set up the boundary conditions for a beam attached to a joint, a system of wrenches dual to the screws representing the freedoms of the joint must be found. Finally, a screw version of the Rayleigh-Ritz numerical method is introduced. An example is investigated in which the boundary conditions on the beam lead to vibrational modes of the beam involving bending, torsion, and extension at the same time.

Dynamics of Vibratory Bowl Feeders

In this work we construct a simple dynamical model for vibratory bowl feeders. The symmetrical arrangement of the springs supporting the bowl allow us to predict a simple structure for the stiffness matrix of the system. The cylindrical symmetry of the bowl itself then means that the linearized rigid body dynamics of the system can be simplified to a 2-dimensional system. The solutions to this system are elliptical motions of the bowl, vibrating about the symmetry axis and along it at the same time. We are able to find a condition for the system to be at resonance. There is some debate about how the parts move up the helical track inside the bowl. We are able to show that one alternative, a “slip-stick” motion, is unlikely.

Quadratic Constraints on Rigid-Body Displacements

In this work, the solution to certain geometric constraint problems are studied. The possible rigid displacements allowed by the constraints are shown to be intersections of the Study quadric of rigid-body displacements with quadratic hypersurfaces. The geometry of these constraint varieties is also studied and is found to be isomorphic to products of subgroups in many cases. This information is used to find extremely simple derivations for general solutions to some problems in kinematics. In particular, the number of assembly configurations for RRPS and RRRS mechanisms are found in this way. In order to treat planes and spheres on an equal footing, the Clifford algebra for the Mobius group is introduced.

Characterisation of Frenet-Serret and Bishop motions with applications to needle steering

Frenet-Serret and Bishop rigid-body motions have many potential applications in robotics, graphics and computer-aided design. In order to study these motions, new characterisations in terms of their velocity twists are derived. This is extended to general motions based on any moving frame to a space curve. Furthermore, it is shown that any such general moving frame motion is the product of a Frenet-Serret motion with a rotation about the tangent vector. These ideas are applied to a simple model of needle steering. A simple kinematic model of the path of the needle is derived. It is then shown that this leads to Frenet-Serret motions of the needle tip but with constant curvature. Finally, some remarks about curves with constant curvature are made.

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).

Three Problems in Robotics

Abstract 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.

Manipulating robots along helical trajectories

Current industrial robots arc highly non-linear systems. However, the control strategies in most commercially available robots largely ignore the non-linearity. The resulting linear approximations are only valid at low speeds. Any improvement would allow robots to move faster and hence be more productive. There has been much academic research into robot control, but this has almost always separated the control and the trajectory planning. In this work we seek to combine these tasks and hence simplify the computations required. We investigate how to control a general robot in such a way that its gripper follows straight line, circular or helical paths. These simple paths are both one parameter subgroups for the group of proper rigid motions and geodesics for certain metrics on the group. This suggests a non-linear feedback control law which turns the closed loop dynamics of the robot into the equations for geodesics. Although these equations are not completely stable we are able to modify the control law so that the resulting closed loop dynamics are stable. Hence, the end-effector of the robot will follow straight line, helical or circular trajectories.