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Let g1, g2,...,gn be a sequence of elements of a Lie group, (knot points). Our problem is to find a smooth, parameterised curve in the group that passes through these elements at parameter values t1, t2,...,tn. There are many variations on this basic problem. For example we could take account of velocities. Perhaps we might only require the curve to be near the knot points. [ABSTRACT FROM AUTHOR]

At the beginning of Chapter 4, we stated that Ball's instantaneous screws are rays through the origin in the Lie algebra se(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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For a rigid body, the velocity is given by a screw s, and the momentum is given by a dual vector, or co-screw, M. The pairing between the velocity and momentum gives the kinetic energy of the body $$ \mathcal{M}^T s = j \cdot \omega + p \cdot v = 2E_K $$, where $$ s = \left( {\begin{array}{*{20}c} \omega \\ v \\ \end{array} } \right) and \mathcal{M} = \left( {\begin{array}{*{20}c} j \\ p \\ \end{array} } \right) $$, with j and p the usual three-dimensional angular and linear momenta. [ABSTRACT FROM AUTHOR]

We begin here by introducing the notion of a co-screw. These are elements of se*(3), the dual of the Lie Algebra; see Section 7.5. Co-screws are linear functionate on the velocities, that is, functions $$ \mathcal{F}:se(3) \to \mathbb{R}satisfying\mathcal{F}(as_1 + bs_2 ) = a\mathcal{F}(s_1 ) + b\mathcal{F}(s_2 ) $$, where a and b are constants. The map $$ \mathcal{F} $$(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. [ABSTRACT FROM AUTHOR]

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 SE(3). [ABSTRACT FROM AUTHOR]

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 body. 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). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 SO(3). So consider a three-joint wrist, as in Figure 5.1, with the joint axes aligned along the unit vectors v1, v2 and v3. 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. [ABSTRACT FROM AUTHOR]

In the previous chapter we saw how the Clifford algebra Cl(0,3,1) contains a representation of SE(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. [ABSTRACT FROM AUTHOR]

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 σ-matrices and Dirac's γ-matrices turned out to be generators of Clifford algebras. [ABSTRACT FROM AUTHOR]

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? [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]