Leapfrog Algorithm incorporates Pontryagin's Maximum Principle (PMP) to generate a control law for a control system which optimizes a given cost function. The Leapfrog algorithm begins with a feasible trajectory, which is divided into smaller subdivisions where the local optimal control problems are solved. The locally optimal trajectories are concatenated and following a certain scheme of updating the number of subdivisions, the algorithm ends with the generation of the optimal trajectory along with the corresponding control law. We use the Leapfrog Algorithm for the case of a differential drive wheeled mobile robot in an unobstructed plane environment. To illustrate the algorithm numerically, we optimize a quadratic cost with the control objective of steering the mobile robot from an initial state to the origin in the plane with given final orientation. The results of the numerical simulation have been used to conduct experiments for a mobile robot platform to traverse an optimal trajectory using trajectory tracking control. © 2017 IEEE.