Mathematical methods for personal positioning and navigation

Computing the position of a personal mobile device based on a mix of various types of measurements requires a wide array of mathematical concepts ranging from optimisation to robust estimation and nonlinear filtering theory. Algorithms for positioning and navigation have surfaced concurrently with the development of new measurement equipment and navigation infrastructure. However, most solutions and algorithms pertain only to certain equipment, involving just a single or few measurement sources. This work synthesises existing techniques into a general framework covering static positioning, filtering, batch positioning and dead reckoning. Measurements are not restricted to any specific technology, equation form or distribution assumption. The static positioning problem, deducing position from a set of simultaneous measurements, is considered first. Parallels between geometric, least squares and statistical approaches are given. The more complex problem of time series estimation can be solved by navigation filters that also make use of all past measurements and information about the system dynamics. Different filter implementations can be derived from the ideal Bayesian filter by choosing different probability density function (pdf) approximation schemes. The standard methods are briefly introduced in this context along with a novel generalisation of a piecewise defined grid filter. Finally, given the wide variety of existing and potential filter implementations, fair and expressive methods for comparing the quality and performance of nonlinear filters are discussed.