In this paper,a new method is presented for planning the motion of a mobile robot moving on general terrains.The terrains are represented by NURBS surfaces and the terrains with obstacles or collisional regions are represented by Trimmed NURBS surfaces. The concept of a geodesic and its computation method, as well as A* search algorithm are used to obtain the path of shortest distance and time optimal path between given any two given points on general terrains.It considers robot vehicle dynamics,terrain topography and surface features.The paths are represented by NURBS curves.The adaptability and efficiency of this method are shown by experimental results.