Collision detection is a critical functionality for robotics. The degree to which objects collide cannot be represented as a continuously differentiable function for any shapes other than spheres. This paper proposes a framework for handling collision detection between polyhedral shapes. We frame the signed distance between two polyhedral bodies as the optimal value of a convex optimization, and consider constraining the signed distance in a bilevel optimization problem. To avoid relying on specialized bilevel solvers, our method exploits the fact that the signed distance is the minimal point of a convex region related to the two bodies. Our method enumerates the values obtained at all extreme points of this region and lists them as constraints in the higher-level problem. We compare our formulation to existing methods in terms of reliability and speed when solved using the same mixed complementarity problem solver. We demonstrate that our approach more reliably solves difficult collision detection problems with multiple obstacles than other methods, and is faster than existing methods in some cases.

Unmanned Aerial Vehicles (UAVs), although adept at aerial surveillance, are often constrained by limited battery capacity. By refueling on slow-moving Unmanned Ground Vehicles (UGVs), their operational endurance can be significantly enhanced. This paper explores the computationally complex problem of cooperative UAV-UGV routing for vast area surveillance within the speed and fuel constraints, presenting a sequential multi-agent planning framework for achieving feasible and optimally satisfactory solutions. By considering the UAV fuel limits and utilizing a minimum set cover algorithm, we determine UGV refueling stops, which in turn facilitate UGV route planning at the first step and through a task allocation technique and energy constrained vehicle routing problem modeling with time windows (E-VRPTW) we achieve the UAV route at the second step of the framework. The effectiveness of our multi-agent strategy is demonstrated through the implementation on 30 different task scenarios across 3 different scales. This work offers significant insight into the collaborative advantages of UAV-UGV systems and introduces heuristic approaches to bypass computational challenges and swiftly reach high-quality solutions.