Active Sensing Control for Differentially Flat Systems

This letter proposes an optimal active perception strategy using the Constructibility Gramian (CG) as a metric to quantify the richness of the information acquired along the planned trajectory. A critical issue is the dependence of the CG on the transition matrix, whose closed-form expression is not available for most robotic systems while its numerical computation is usually costly. We leverage differential flatness to transform the nonlinear system in the Brunovsky form, for which the transition matrix reduces to the exponential of a Jordan block. The resulting CG is a measure of the acquired information through the flat outputs about the flat outputs themselves and their derivatives. The inverse flatness change of coordinates is then used to come back to the original state variables, needed for computing the feedback control law. The flat outputs are parameterized through B-Splines with control points determined by actively maximizing CG. We simulate our approach on a unicycle vehicle and a planar UAV that need to estimate their configuration while measuring their distance w.r.t. two fixed markers. Simulations show the effectiveness of our methodology in reducing both the computational time and the estimation uncertainty.