An optimal control problem to find the fastest collision-free trajectory of a robot surrounded by obstacles is presented. The collision avoidance is based on linear programming arguments and expressed as state constraints. The optimal control problem is solved with a sequential programming method. In order to decrease the number of unknowns and constraints a backface culling active set strategy is added to the resolution technique. 2000 Mathematics Subject Classification. Primary: 49J15, 49M25, 49N90; Secondary: 90C30.