The extremum trajectories of wheeled robot on pipeline surfaces was studied. The dynamic model of the wheeled mobile robot on pipeline surfaces was presented, and then the controllability of robots and the existence of time optimal trajectories were proved. By applying Pontryagin's maximum principle, the necessary conditions of time optimal control were obtained, and the structure equations of extremal trajectories were given. The extremum trajectories were studied with the analysis of structure equations. It was notified that control variables of the extremum trajectories were defined by structure equations. The extremum trajectories were combined finitely by two basic types of trajectories. Moreover, the exact values for the control variables of this two basic types were given.