Navigating a set of discrete regions in body position space

This paper defines a mathematical object that expresses physiological aspects of motor control, besides physical aspects. Based on observations in human development and rehabilitation, this paper lays a foundation for unifying discrete with continuous aspects of motor control. Discreteness in motor control arises from the fact that movements are planned by the organism, based on movement limits set by the organism, the task, and the environment, rather than following simply from physical laws. This paper defines two relations, an ordering and a contiguity relation, between discrete regions of joint space crossed with physical space, to make a d-space. Within the d-space, a particular body and physical position is represented by a completely ordered set, a coincidence. A trajectory through a continuous space is represented by a coincidence sequence. Examples are drawn mostly from human locomotion, with particular interest in balance. Only in certain regions of body and physical position space can a body maintain balance, for example, a standing human. Slow steps proceed from one balanced region to another. A poset (partially ordered set) of sets of steps and its relationship to a person's balance regions can be used in the study of development or rehabilitation.