In this paper, we consider the problem of collision-free motion planning for multiple nonholonomic planar vehicles. Each vehicle is capable of moving at constant speed along paths with bounded curvature, and is aware of the position and heading of other vehicles within a certain sensing radius. No other information exchange is required between vehicles. We propose a spatially decentralized, cooperative hybrid control policy that ensures safety for arbitrary numbers of vehicles. Furthermore, we show that under certain conditions, the policy avoids dead- and livelock, and eventually all vehicles reach their intended targets. Simulations and experimental results are presented and discussed.