On the equivalence of orientation error and positive definiteness of matrices

In this paper we show how a continuous set of orientations can be represented as a positive definiteness test on a given matrix. When this continuous set is restricted by the maximum allowed orientation error in some or all directions it is shown that the requirement for an orientation to satisfy these restrictions is equivalent to positive definiteness for a certain matrix. The problem of finding the optimal orientation that satisfies these restrictions is hence transformed into an optimisation problem on the Riemannian manifold of linearly constrained symmetric positive definite matrices. Thus, the problem of finding the optimal orientation can be solved as a standard optimisation problem with the constraints written in the form of linear matrix inequalities or barrier functions. Linear matrix inequalities have been extensively studied in the optimisation communities and good and efficient algorithms are available.