Coordinate Frames and Gradient Calculation

Computational Sensor Networks1 depend on phenomenological models which describe spatio-temporal relations between physical quantities. This generally requires a common coördinate frame of reference. Almost all calculation depends on functions defined with respect to x, y, z, and t (e.g., the heat equation relates the partial derivative of temperature with respect to time to the second derivative of temperature with respect to space). Other quantities of interest, such as velocity, acceleration, momentum, etc., all depend on a frame of reference. Generally, such a frame is assumed known or given; however, this is not usually the case for SELs in an S-Net. Such nodes are typically restricted in terms of hardware due to energy concerns and are used in places where GPS is not available (in buildings, cities, forests, etc.). In many cases it is sufficient to construct a local coördinate frame, that is, one given in terms of the SELs which define it; this is also called a relative frame. Of course, such a relative frame may include a large number of spatially distributed SELs so long as the necessary conditions hold (see below). It is also possible to anchor SELs to a global, or absolute, frame, like that provided by GPS, if necessary, and given enough global information. This chapter provides the necessary tools to construct coördinate frames, and to relate them to mobile agents who desire to exploit them for localization and navigation