Geometric methods for designing optimal filters on Lie groups

In control theory, the problem of having available good measurements is of primary importance in order to perform good tracking and control. Unfortunately, in real-life applications, sensing systems do not provide direct measurements about the pose (and its rate) of mechanical systems, while, in other situations, measurements are so noisy that require pre-processing to filter out disturbances and biases. These problems could be faced by using filters and observers. In this thesis, we apply a second-order optimal minimum-energy filter constructed on Lie groups to several planar bodies. We start by studying the application of the filter to the matrix Lie group TSE(2), i.e. the tangent bundle of the Special Euclidean group SE(2); moreover, a comparison with the extended Kalman filter is presented. After that, we consider the Chaplygin sleigh case, that is a mechanical system with a nonholonomic constraint. Then, we move our attention to the case of an articulated convoy with hooking constraints. Finally, we apply the filter to a real case scenario consisting of a scaled model representing a parking truck semi-trailer system. Particular attention is posed to the description of the geometric structure that underlies the dynamics and to the choice of the measurement equation, the affine connection, and the other parameters that define the filters. Simulations show the effectiveness of the proposed filters. The use of Lie groups theory for designing the filters is challenging, but the accuracy of the results, obtained considering the geometric structure and the symmetries of the system justifies the effort.