Motion planning of legged robots

We study the problem of computing the free space F of a simple legged robot called the spider robot. The body of this robot is a single point and the legs are attached to the body. The robot is subject to two constraints: eac h 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) . Moreover, the robot can only put its feet on some regions, called the foo thold regions. The free space F is the set of positions of the body of the robot such that there exists a set of accessible footholds for which the ro bot is stable. We present an efficient algorithm that computes F in O(n(2) log n) time using O(n(2)alpha(n)) space for n discrete point footholds wher e alpha(n) is an extremely slowly growing function (alpha(n) less than or e qual to 3 for any practical value of n). We also present an algorithm for c omputing F when the foothold regions are pairwise disjoint polygons with n edges in total. This algorithm computes F in O(n(2)alpha(8)(n) log n) time using O(n(2)alpha(8)(n)) space. (alpha(8)(n) is also an extremely slowly gr owing function.) These results are close to optimal since Omega(n(2)) is a lower bound for the size of F.