A control method that makes the head of a snake robot follow an arbitrary trajectory on two non-parallel planes, including coexisting sloped and flat planes, is presented. We clarify an appropriate condition of contact between the robot and planes and design a controller for the part of the robot connecting the two planes that satisfies the contact condition. Assuming that the contact condition is satisfied, we derive a simplified model of the robot and design a controller for trajectory tracking of the robot's head. The controller uses kinematic redundancy to avoid violating the limit of the joint angle and a collision between the robot and the edge of a plane. The effectiveness of the proposed method is demonstrated in experiments using an actual robot.