Extremal Trajectories for CRS Robots on Surfaces

In order to solve the path planning problem of mobile CRS robots on surface,a kinematic model of CRS robots on surfaces is presented.By using the Lie group theory of differential geometry and Pontryagin's maximum principle,it is pointed out that the mobile CRS robot on surface is controllable,and the necessary conditions and the structure equations of extremal trajectories are proposed.It is noted that control varies of velocity and angular velocity are definied by structure equations and Gaussian curvature.The analysis of stucture equations indicates that the extremal trajectories are combinated finitely by five basic trajectories,and that the control varies of velocity for the basic trajectories are forward or backward moving.The exact distribution for the control varies of angular velocity is presented.