Investigation on the analytical form of the transition matrix in inertial geodesy

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The error behavior of inertial survey systems can best be described by a system of differential equations. Its solution in analytical form, by way of a transition matrix, is discussed in this report. After a review of the methods available to solve systems of differential equations, the dynamics matrix of the local-level system operating in three dimensions is treated in detail. Two methods are used to derive the analytical form of the transition matrix: the inverse Laplace transform technique and the series of expansion of the matrix exponential. Analytical and numerical comparisons show that the two derived solutions are not completely equivalent but agree very well for time intervals up to 1000 seconds. For large time spans the inverse Laplace transform solution is more accurate. The report concludes with a brief discussion of the effects which the variation of certain parameters has on the error behaviour.