Exact and overapproximated guarantees for corner cutting avoidance in a multiobstacle environment

Abstract

The corner cutting avoidance problem is an important but often overlooked part of motion planning strategies. Obstacle and collision avoidance constraints are usually imposed at the sampling time without regards to the intrasample behavior of the agent. Hence, it is possible for an agent to "cut the corner" of an obstacle while apparently respecting the constraints. This paper improves upon state of the art by providing exact and overapproximated descriptions of the undershadow (and of its complement, the visible) region generated by an agent against obstacles. We employ a hyperplane arrangement construction to handle multiple obstacles simultaneously and provide piecewise descriptions of the regions of interest and parametrizations of the corner cutting conditions (useful, eg, in finite horizon optimization problems). Mixed-integer representations are used to describe the regions of interest, leading in the overapproximated case to binary-only constraints. Illustrative proofs of concept, comparisons with the state of the art, and simulations over a standard multiobstacle avoidance problem showcase the benefits of the proposed approach. KEYWORDS corner cutting problem, hyperplane arrangement, mixed-integer programming (MIP), model predictive control (MPC), multiobstacle avoidanceRecent advances in computational resources and the proliferation of (semi)autonomous vehicles has lent new interest to the topic of obstacle and collision avoidance in motion planning strategies. One of the major issues is that avoidance constraints lead to a nonconvex feasible domain. It is worth underlining that such formulations are intrinsic to the problem and cannot be avoided. [1][2][3] This paper concentrates on the "corner cutting" issue, ie, avoidance constraints are checked at each sampling time, but the control input is ultimately applied (eg, via a zero-order hold block) to continuous dynamics. Hence, the intrasample behavior of the agent (the autonomous vehicle to be steered) cannot be overlooked. While alluring, obstacle enlargement techniques 4 or sampling time reduction are not always appropriate. The former increases the conservatism of the formulation (potentially leading to infeasibility), and the latter reduces the time available for computing the input.The computational limitations are particularly troublesome since modeling the associated optimization problem is often done via the mixed-integer (MI) programming framework, 5,6 which scales badly with problem size and complexity (ie, number of obstacles). 4528

Robust intersample crossing of target sets with mixed‐integer linear programming

“…The discrete‐time model is given by the state difference equation x ( k + 1) = A x ( k ) + B u ( k ) + w ( k ) and the output algebraic equation s ( k ) = C x ( k ), with the state, control, and output vectors x T = [x v x y v y ], u T = [ a x a y ], and s T = [x y], respectively, and disturbances acting on the state, with a (bounded) polytope. The model matrices for a sample period normalized to one are: 8 …”

“…However, the usage of the shadow regions required the solution of a more complex nonlinear optimization problem, thus the authors considered a more conservative approximation to obtain a MILP formulation. Later, Stoican et al (2018) 8 detailed this formulation and also compared the solution of the more restrictive MILP formulation with the exact mixed‐integer nonlinear program (MILNP), obtaining a much (two orders of magnitude) greater computational effort for the latter. This field of research has become known as intersample collision avoidance or “corner cutting” avoidance.…”