Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Figure 11:

A relative motion system.

12 Serial-Link Manipulators

It will now be argued that there is a profound relation between serial-link manipulators in robotics, and modulation in music. Both are decompositional means of reaching a point by hierarchical transfer. Therefore their mathematical structure is identical. This section considers serial-link manipulators, and the next deals with musical modulation. Both illustrate our theory of inheritance, and in particular our theory of relative motion.

Standardly in a serial-link manipulator (such as the human arm), one says that the frames of two successive links are related by a special Euclidean transformation $Ai$ , and thus the overall relationship between the hand coordinate frame and the base coordinate frame is given by the product of matrices

A1A2 ...An

(14)

corresponding to the succession of links.

Now, in setting up the object-oriented structure of such manipulators, one usually stipulates that a distal link is a child of the next proximal link, and so on, successively along the manipulator. Our argument is that this arises from the transfer structure: The distal link has a space of actions that is transferred through the environment by the next proximal link. This exemplifies our claim that the basis of inheritance is the deeper notion of transfer. It is this that allows us to formulate inheritance algebraically in terms of wreath products. Thus, we argue that the group of a serial-link manipulator has the following wreath-product structure:

SE(3)1 wO SE(3)2 wO ...wO SE(3)n

(15)

where each level $SE(3) i$ is isomorphic to the special Euclidean group $SE(3)$ , and the succession from left to right corresponds to the succession from hand to base (distal to proximal).

The entire group we have given in (15) for the serial-link manipulator, is very different from the group that is normally given in robotics for serial-link manipulators. Standardly, it is assumed that, because one is multiplying the matrices in (14) together, and therefore producing an overall Euclidean motion $T$ between hand and base, the group of such motions $T$ is simply $SE(3)$ . However, we argue that this is not the case. The group is the much more complicated group given in expression (15). The conventional group $SE(3)$ necessarily models the arm as a rigid structure, whereas the wreath product (15) models the arm as a structure we call semi-rigid: a group where rigidity breaks down at a discrete set of points. Most crucially the wreath product models the object-oriented structure, which is basic to all computation concerning the kinematics.