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It was probably Study who first considered the possible positions of a rigid body as points in a non-Euclidian space; see Study [118]. His idea was to specify the position of the body by attaching a coordinate frame to it. He called these ‘points’ soma, which is Greek for . He then used dual quaternions as coordinates for the space. As we saw in Section 9.3, using the dual quaternion representation, the elements of the group of rigid body motions can be thought of as the points of a six-dimensional projective quadric (excluding a 3-plane of ‘ideal’ points). If we fix a particular position of the rigid body as the home position, then all other positions of the body can be described by the unique transformation that takes the home configuration to the present one. In this way, we see that Study’s somas are just the points of the six-dimensional projective quadric, the Study quadric, (not forgetting to exclude the points on the special 3-plane).

We begin here by introducing the notion of a . These are elements of *(3), the dual of the Lie Algebra; see Section 7.5. Co-screws are linear functionate on the velocities, that is, functions , where and are constants. The map (s) is usually called the evaluation map of the functional. The space of all such functionate forms a vector space with the same dimension as the original space of velocity vectors. In linear algebra, this vector space of functions is usually called the dual vector space. But to avoid confusion with the dual numbers, six-component velocity vectors will be called screws and the linear functionate co-screws. In older language, the screws would be covariant vectors and the co-screws contravariant vectors.

For a rigid body, the velocity is given by a screw s, and the momentum is given by a dual vector, or co-screw, . The pairing between the velocity and momentum gives the kinetic energy of the body , where , with and the usual three-dimensional angular and linear momenta.

In this chapter we extend the ideas of the previous chapter to more complicated manipulator designs. Specifically we look at the dynamics of tree and star-structured mechanisms. Then we look at robots with constrained end-effectors; we only consider time invariant holonomic constraints here. This type of constraint can occur when the end-effector of the robot interacts with the environment. When these ideas are combined with the dynamics of tree and star-structured robots, it is possible to derive the dynamics of some kinematic loops and parallel robots. Finally, we look at some examples.

This final chapter is slightly different in character from the preceding ones. The aim is to present some less elementary examples. The examples are loosely related by their use of some concepts from differential geometry, hence the title. We will begin by looking at some differential geometry on the manifold of the group (3).