Observability optimization for the nonholonomic integrator

A growing number of control problems are concerned with vehicles that have insufficient sensors to directly measure all system states and require active control actuation to observe all necessary states (e.g. underwater navigation, simultaneous localization and mapping, parameter identification). In this paper, we consider first-order nonholonomic systems in canonical form as a model system for exploring the development of active control that optimizes system observability characteristics. Observability analysis of this system shows that all nonholonomic states must be measured in the output, and actuation in a minimum of two control channels is required for observability. Analytical trajectories are derived that allow almost arbitrary placement of the observability gramian eigenvalues, which are shown to be inversely related to state estimation covariances. Simulation results show that the optimal trajectories provide four to five times faster estimator convergence and three times lower state estimate covariances than suboptimal trajectories.