A framework for active estimation: Application to Structure from Motion

Nonlinear ego-motion estimation from optical flow for online control of a quadrotor UAV

“…Compared to the previously discussed EKF, this estimation scheme does not require any linearization/approximation step of the system dynamics. This results in an overall cleaner design which, following the analysis reported in [17], also allows for a full characterization of the estimation error transient response in case of a camera traveling with a constant linear acceleration norm ( C κ(t) = const). In particular, we will discuss how one can impose to the estimation error a transient response equivalent to that of reference linear second-order system with desired poles by suitably acting on the estimation gains and on the UAV acceleration.…”

“…Nevertheless, exponential convergence of the estimation error e(t) to 0 can still be proven by resorting to Lyapunov theory and by noting that the spurious term g(e, t) is a vanishing perturbation of an otherwise globally exponentially stable nominal system, i.e., g(0, t) = 0, ∀t. We refer the reader to [17], [25], [26] for additional discussion and proofs of these facts. We note that the design of observer (42) did not require any linearization step as for the previous EKF thanks to the more general class of (nonlinear) systems spanned by formulation (35).…”

“…In this Section, we summarize the derivations of the alternative nonlinear observer for retrieving the plane distance d based on a framework originally proposed in [25], [26] for dealing with Structure from Motion (SfM) problems, and recently revisited in [17], [18] in the context of nonlinear active estimation. Compared to the previously discussed EKF, this estimation scheme does not require any linearization/approximation step of the system dynamics.…”

“…1) Shaping the Estimation Transient Response: We now apply the theoretical analysis developed in [17] to the case at hand, and aimed at characterizing the transient response of the error system (43) in the unperturbed case, i.e., with g(e, t) = 0.…”

“…However, despite being widespread, the use of a EKF-based scheme presents two (often overlooked) drawbacks: (i) it necessarily involves a linearization of the (nonlinear) system dynamics (such as when dealing with visual-inertial estimation problems), and (ii) as a consequence, it does not allow for an explicit characterization of the estimation error behavior (e.g., its convergence rate). Therefore, as an additional contribution, in this paper we propose a novel visual-inertial estimation scheme exploiting optical flow and IMU measurements based on a recently-developed nonlinear observation framework for active structure from motion [17], [18]. Compared to a classical EKF, the use of this nonlinear filter yields an estimation error dynamics with a fully characterized convergence behavior when the camera is moving with a constant acceleration norm w.r.t.…”