On corner cutting in multi-obstacle avoidance problems

Abstract

Abstract:One challenging and not extensively studied issue in obstacle avoidance is the corner cutting problem. Avoidance constraints are usually imposed at the sampling time without regards to the intra-sample behavior of the dynamics. This paper improves upon state of the art by describing a multi-obstacle environment over a hyperplane arrangement scaffolding, provides a piecewise description of the "shadow" regions and represents them into a combined mixed integer and predictive control formulation. Furthermore, over-approximation constraints which reduce to strictly binary formulations are discussed in detail. Illustrative proofs of concept, comparisons with the state of the art and simulation results over a classical multi-obstacle avoidance problem validate the benefits of the proposed approach.

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

“…). Noting that the cone spanned from x and tangent to ( • ) is completely characterized by x and ( • , x), as per Lemma 1 and applying Corollary 1, we reach formulation (11), and thus conclude the proof.…”

“…Proposition 1 shows that (11) has a piecewise structure, and thus, for any x in a given cell ( • ), at runtime, we need only to input the current value of x into (11). While this reduces the computation burden (ie, the tangent points (10) and the separation hyperplanes are already known and can be introduced directly in (11), thus avoiding a full recomputation of the cone as in (9)), the formulation for the shadow area is still difficult due to Cone (x, ( • , • )).…”

“…Proof. The proof is straightforward and is based on de Morgan's laws, ie, A ∩ B = A ∪ B and A ∪ B = A ∩ B and uses definitions (11), (14), and the fact that …”

“…The current work is based and expands previous results of the authors. Preliminary results in the work of Stoican et al 11 discussed the corner cutting topic (expanded here with additional theoretical results, comparisons with the state of the art, and extensive illustrative examples). Strutu et al 12 and Stoican et al 13 discussed the dual problem of coverage in a multiobstacle environment.…”

“…Finally, by exploiting the structure provided by the hyperplane arrangement, we provide MI descriptions for the regions of interest. That is, using results from the works of Prodan et al 6 and Stoican et al, 11 together with codification methods found in the work of Vielma and Nemhauser 15 and the references therein, we provide MI representations, which describe both the shadow region and its complement in both the exact and overapproximated forms. This paper is organized as follows.…”