We consider the problem of planning motions of a simple legged robot called the spider robot. The robot is modelled as a point where all its legs are attached, and the footholds where the robot can securely place its feet consist of a set of n points in the plane. We show that the space F of admissible and stable placements of such robots has size Θ(n^2) and can be constructed in O(n^2 log n) time and O(n^2) space. Once F has been constructed, we can efficiently solve several problems related to motion planning.