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In May 2000 there was a meeting at the National Science Foundation in Arlington Virginia on “The Interplay between Mathematics and Robotics”. Many leading experts in the U.S. discussed the importance of mathematics in robotics and also the role that robotic problems could play in the development of mathematics. The experts gave a broad overview of the problems they saw as important and worth studying. Their list was long and touched on many branches of mathematics and many areas in robotics.

The concept of a group was introduced into mathematics by Cayley in the 1860s, generalising the older notion of “substitutions”. The theory of substitutions studied the symmetries of algebraic equations generated by permutations of their roots. The theory was already highly developed; in particular Galois had developed a method to determine whether an algebraic equation can be solved by radicals. Although the work was done before 1832, it was not until 1843 that it gained a wide audience when it was popularised by Liouville.

For any group, a is a subset of elements of the original group that is closed under the group operation. That is, the product of any two elements of the subgroup is again an element of the subgroup. For Lie groups, we have the concept of a Lie subgroup. In addition to the closure requirement, the subgroup must also be a submanifold of the group manifold of the original group. It is quite possible to have subgroups of Lie groups that are not Lie subgroups. However, when we talk about the subgroups of a Lie group we will always mean a Lie subgroup. So, for consistency, the group manifold of a discrete group will be thought of as a zero-dimensional manifold. For example, the trivial group has just a single element, the identity element. We will write this group as 0 = {e}; notice that 0 is a subgroup of every group.

In 1900 R.S. Ball first published his treatise on “the Theory of Screws” [6]. The finite screws he describes are clearly rigid body motions. Ball also describes instantaneous rigid body motions as ‘twists’ these clearly correspond to elements of the Lie algebra of the group (3). Ball’s instantaneous screws are elements of the projective Lie algebra of the group (3), that is rays through the origin in the Lie algebra. Although roughly contemporary with the work of Lie and Killing, Ball’s work had a rather different focus from the emerging theory of Lie groups and algebras. We hope to show the connections here. We begin by looking at Lie algebras in general.

In Section 3.7, it was stated that the general problem of inverse kinematics for robots is rather difficult. However, for the 3-R wrist it is possible to give a general solution. This is because we are only dealing with three joints, and since the joints are all revolute, intersecting at a single point, we can reduce everything to the rotation group (3). So consider a three-joint wrist, as in Figure 5.1, with the joint axes aligned along the unit vectors v, v and v. These are the home positions of the axes, and we will assume that these vectors are linearly independent. If the home position of the wrist has linearly dependent joint axes, then we can always move it a little and use a non-singular home position.

Line geometry is not as popular these days as it was even fifty years ago. This is perhaps because many of the original problems of the subject have been solved. Algebraic geometers think of ruled surfaces as line bundles over a curve or even more abstract descriptions. Differential geometers usually worry about the extrinsic geometry of ruled surfaces—that is, how such surfaces can sit in three dimensions—their curvature, and so forth. Symplectic geometers have all but forgotten that their subject began with the study of the symmetries of line complexes.

In the latter half of the twentieth century a large part of group theory was concerned with the theory of group representations. This followed from the nineteenth century’s concentration on invariants and covariants. Crudely speaking invariants are trivial representations and covariants are just elements of some non-trivial representation. The significance of these ideas is that if we want to write down equations and relations in terms of coordinates then we expect that if we change coordinates then the geometry or mechanics expressed in our equations should not be altered. The simplest way to do this is to make sure that the expressions are equalities between invariants, but we can also use relations between covariants—so long as the covariants we compare correspond to the same representation. The upshot of this that whenever we want to study some new kind of object, a line, an ellipsoid or an inertia matrix perhaps, then we should always ask how the new object transforms under a change of coordinates, that is, which representation does the object belong to?

At the beginning of Chapter 4, we stated that Ball’s instantaneous screws are rays through the origin in the Lie algebra (3). Clifford referred to the elements of the Lie algebra as motors. In the following, the practice of referring to Lie algebra elements as screws will continue, with the hope that no confusion will arise.

At the turn on the nineteenth century there was a vituperative dispute about which was the ‘correct’ notation to use in modern geometry. The matrix-vector methods promoted by Gibbs won and the quaternion-Clifford algebra methods lost. This is why modern students in science and engineering no longer learn about quaternions. However, news of this revolution was slow to spread in some areas, particularly in kinematics. So Study and latter Blaschke [12] and Dimentberg [27] continued to develop ‘dual quaternions’ and applied them to the theory of mechanisms. Mathematicians never really forgot about these things, although the real impetus to look at these structures afresh came when physicists rediscovered them. Pauli’s -matrices and Dirac’s -matrices turned out to be generators of Clifford algebras.

In the previous chapter we saw how the Clifford algebra (0,3,1) contains a representation of (3), the group of rigid body motions. Here we will see that this algebra also contains representations of the points, lines and planes of Euclidean space. Moreover, the usual constructions of Euclidean geometry can be modelled by standard algebraic operations in the algebra. This provides us with a very neat setting for performing geometric computations.