Constrained Convex Generators: A Tool Suitable for Set-Based Estimation With Range and Bearing Measurements

Abstract

Autonomous vehicles in GPS-denied areas or when cooperating in missions might have access to bearing and range measurements corrupted by noise, rendering the reachable set to be nonconvex since the measurement set is a segment of an annulus in 2D or a spherical shell in 3D. There are various alternatives that could be used in the literature to over-approximate the set by a convex one. However, given the circular part caused by the range measurement, adopting an exact polytopic description would require an infinite number of hyperplanes. In a similar fashion, using ellipsoids suffers from the same problem due to the hyperplane constraints arising from the bearing part. Moreover, if only bearing measurements are available, the measurement set should be unbounded. Motivated by these observations, we propose a generalization of the definition for constrained zonotopes recently introduced in the literature to also consider the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula> norm and cones (or any other convex set for that matter) as to represent these sets with less conservatism. Given the exact nature of the propagation, these can serve as a worst-case bounds for the true state which is relevant in some applications such as collision avoidance. In simulations, we also illustrate the performance of the computations to be suitable for relatively small sampling times.