  To enhance accuracy of robot state estimation, active sensing (or
perception-aware) methods seek trajectories that maximize the information
gathered by the sensors. To this aim, one possibility is to seek trajectories
that minimize the (estimation error) covariance matrix output by an extended
Kalman filter (EKF), w.r.t. its control inputs over a given horizon. However,
this is computationally demanding. In this article, we derive novel
backpropagation analytical formulas for the derivatives of the covariance
matrices of an EKF w.r.t. all its inputs. We then leverage the obtained
analytical gradients as an enabling technology to derive perception-aware
optimal motion plans. Simulations validate the approach, showcasing
improvements in execution time, notably over PyTorch's automatic
differentiation. Experimental results on a real vehicle also support the
method.
