In this article, we introduce an approach for planning minimum-control optimal trajectories for a two-wheeled mo bile robot in the absence of obstacles. The trajectory planning problem is first formulated as a minimum-control, fixed-time, optimal control problem with no terminal cost. This results in a two-point boundary value problem that is numerically solved using the relaxation method. Some simulation results of the relaxation method are presented. While we cannot claim rigor ously that the trajectories obtained here are globally optimal, there is strong evidence from numerical robustness tests that they are well-isolated extrema, and are possibly the globally optimal solutions. The two-point boundary value problem is also solved analytically. It is shown that the optimal motion of the two-wheeled mobile robot is similiar to the motion of a pendulum in a gravitational field, and as such, is described by four constants of motion. A procedure for solving the constants of motion is discussed. At the end, it is possible to obtain a closed-form solution to the optimal trajectories in terms of Jacobian elliptic functions.