From spider robots to half disk robots

Studies the problem of computing the set F of accessible and stable placements of a spider robot. The body of this robot is a single point and the legs are line segments attached to the body. The robot can only put its feet on some regions, called the foothold regions. Moreover, the robot is subject to two constraints: each leg has a maximal extension R (accessibility constraint) and the body of the robot must lie above the convex hull of its feet (stability constraint). The authors present an efficient algorithm to compute F. If the foothold regions are polygons with n edges in total, the authors´ algorithm computes F in O(n² log n) time and O(n²α(n)) space where α is the inverse of Ackerman´s function. Ω(n²) is a lower bound for the size of F