We discuss a unified approach to a class of geometric combinatorics incidence problems in two dimensions, of the Erdös distance type. The goal is obtaining the second moment estimate. That is, given a finite point set $S$ in two dimensions, and a function $f$ on $S\times S$, find the upper bound for the number of solutions of the equation $f(p,p') = f(q,q')\neq 0, (p,p',q,q')\in S\times S\times S\times S$; e.g., $f$ is the Euclidean distance in the plane, sphere, or a sheet of the two-sheeted hyperboloid. Our ultimate tool is the Guth--Katz incidence theorem for lines in $\mathbb{RP}^3$, but we focus on how the original problem in two dimensions gets reduced to its application. The corresponding procedure was initiated by Elekes and Sharir, based on symmetry considerations. The point we make here is that symmetry considerations, not necessarily straightforward and potentially requiring group representation machinery can be bypassed or made implicit. The classical Plücker--Klein formalism for line geometry enables one to directly interpret a solution to the above relation as an intersection of two lines in $\mathbb{RP}^3$. This, e.g., allows for a very brief argument as to how the Euclidean plane distance argument extends to the spherical and hyperbolic distances. The space of lines in the projective three-space, the Klein quadric $\mathcal K$, is four dimensional. Thus, we start out with an injective map $\mathfrak F:\,S\times S\to\mathcal K$, that is from a pair of points $(p,q)$ to a line $l_{pq}$ and seek a corresponding combinatorial problem in the above form in two dimensions, which can be solved by applying the Guth--Katz theorem to the set of lines $\{l_{pq}\}$ in $\mathbb{RP}^3$. We identify a few arguably new such problems, and hence applications of the Guth--Katz theorem and make generalizations of the existing ones. It is the dimension of the direct approach in question that is the main purpose of this paper.

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).

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.

This work looks at several problems concerned with interpolating rigid-body motions and their application in robotics. Two recently proposed interpolation methods are shown to produce the same results. We also discuss how it might be possible to control a robot in such a way as to follow one of these interpolated motions

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.

We derive the deflection equation of a simple beam using the screw theory. The effects of tension, torsion and bending of the beam can be unified into a single equation. We begin by looking at the compliance matrix for small elements of the beam. This is loosely based on the work by von Mises (1924). We reproduce von Mises' results for the compliance matrix of the entire beam by integrating along the beam. This allows us to make some general comments concerning the use of beams as compliant design element. Then we derive the equation for the deflection of the beam considering the deflection as a rigid body motion.

Develops a simple linear model for the motion of a Stewart platform in a stationary position. That is, the situation where the platform is at rest and is then subject to an impulsive disturbance. The hydraulic actuators are modelled as simple spring-dashpot systems and the stiffness and damping matrices of the system are derived. It is found that the damping and stiffness matrices are simply proportional to each other and this simplifies the dynamics greatly. The general solution to the equations of motion is a linear combination of eigensolutions. The eigensolutions are damped oscillations about the harmonic screws of the undamped system. The characteristic values associated with each eigensolution can also be found in terms of the undamped frequencies. Finally some remarks are made concerning the possibility of the system being fully damped.

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).

If a rigid body is subject to a set of holonomic constraints then the set of allowed positions and orientations comprise a subspace in the group of rigid body motions. Consideration of the geometry of these constraint subspaces leads us to consider their second fundamental form. This allows us to predict the constraint forces and torques generated by a constrained motion. The result has implications for the control of robots that have constrained end-effectors. A slight modification is proposed to the standard methods of shared control and hybrid control. Effectively this modification accounts for the curvature of the constraint space. Finally, the situation where the constraints are in the form of a physical surface in space is considered. In particular some comments are made regarding the problem of measuring the curvature of the surface using a robot.

We give a geometrical description of Raibert and Craig's 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.

Recent experimental and analytical evidence indicates that direct drive robots become very practical and economical at miniature and microscopic scales, so it is interesting to understand quantitatively the properties of direct drive robots under scaling transformations. This leads to a study of how screws and their dual co-screws behave under the group of similarity transforms. This group is the group of isometries together with dilations. Several different representations are found on the space of screws and complementary representations are found on the dual space of co-screws. From the electromagnetic theory of the force and torque on a magnet in a magnetic field, we derive the scaling properties of the electromagnetic wrench. Hence, these results can be directly applied to the scaling of direct drive motors (A. Haruhiko and K. Youcef-Toumi, 1986). We conclude by proposing a scale invariant measure for direct drive actuator performance.