Set inversion

Abstract: The problem considered is state estimation in the presence of unknown state and measurement noise, each noise component being assumed to belong to some known interval. In such a bounded-error context, most available results are for linear models, and the purpose of the present paper is to deal with the nonlinear case. Based on interval analysis and the notion of set inversion, a new state estimator is presented, which evaluates a set estimate guaranteed to contain all values of the state that are consistent with the available observations, given the noise bounds and a set containing the initial value of the state. To the best of our knowledge, it is the first time that such a guaranteed estimator is made available. The precision of the set estimate can be improved, at the cost of more computation. The theoretical properties of the estimator are studied, and computer implementation has received special attention. A simple illustrative example is treated.

Abstract: In this paper, initial localization problems are solved by using set-membership estimation. The method can be used with any robot and any kind of sensor(s), provided that a computable model of the environment/sensor interaction is available. With a pedagogical aim in mind, it is detailed in the case of the localization of a vehicle from range measurements in a polygonal environment. Salient properties of the method are as follows. First, it does not need any explicit management of matching hypotheses. Second, it is able to deal with ambiguous situations where several radically different vehicle configurations are consistent with the measurements. Third, it can be made robust to outliers. Fourth, it can deal with nonlinear observation models without any approximation. Fifth, the result is guaranteed in the sense that no configuration consistent with the data and the hypotheses can be missed.