Optimal Average Case Strategy for Looking Around a Corner

A robot is free to move in a non-convex polygonal region, starting against an edge on the boundary. Ahead of the robot at one unit of distance is a corner, i.e. a reflex vertex (see Figure 1). The angle θ is unknown to the robot. The robot’s task is to look at the region around the corner, which it initially cannot see. We let φ = π − θ. If φ ≥ π/2, the robot is best off moving straight to the vertex. However, if φ is close to 0, much shorter paths exist, making this solution suboptimal. Therefore we ask: What is the best path for the robot to follow? In this paper, we look into the problem of finding an optimal average-case strategy under a homogeneous probability distribution for φ. The average-case performance of a strategy is measured by its average cost, defined as the expected value of the strategy’s competitive function. Given a value for φ, the competitive function of a strategy gives the ratio of the distance the robot travels to look around the corner (as prescribed by the strategy) to the shortest distance it must travel to do so. We give strong evidence that an optimal average-case strategy exists and achieves an average cost of ∼ 1.189.