https://inria.hal.science/inria-00073475Boissonnat, Jean-DanielJean-DanielBoissonnatPRISME - Geometry, Algorithms and Robotics - CRISAM - Inria Sophia Antipolis - Méditerranée - Inria - Institut National de Recherche en Informatique et en AutomatiqueDevillers, OlivierOlivierDevillersLazard, SylvainSylvainLazardMotion Planning of Legged RobotsHAL CCSD1997MOTION PLANNINGLEGGED ROBOTSMOBILE ROBOTCOMPUTATIONAL GEOMETRY[INFO.INFO-OH] Computer Science [cs]/Other [cs.OH]Inria, Rapport De Recherche2006-05-24 12:57:042023-03-15 08:58:092006-05-31 14:24:27enReportsapplication/pdf1We study the problem of computing the free space \Fsimple 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: 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). Moreover, the robot can only put its feet on some regions, called the foothold regions. The free space $\calf$ is the set of positions of the body of the robot such that there exists a set of accessible footholds for which the robot is stable. We present an efficient algorithm that computes \F(n^2\log n)$ time and $O(n^2\alpha(n))$ space for point footholds where $\alpha(n)$ is an extremely slowly growing function ($\alpha(n)\leq 3$ for any practical value of $n$). We also present an algorithm to compute \Fthe foothold regions are pairwise disjoint polygons with $n$ edges in total. This algorithm computes \F(n^2\alpha_8(n)\log n)$ time using $O(n^2\alpha_8(n))$ space ($\alpha_8(n)$ is also an extremely slowly growing function). These results are close to optimal since $Ømega(n^2)$ is a lower bound for the size of~\F