  The benefits of legged locomotion shown in nature overcome challenges such as
obstacles or terrain smoothness typically encountered with wheeled vehicles.
This paper evaluates the benefits of using optimal control on a single leg
hopper during the entire hopping motion. Basic control without considering
physical constraints is implemented through hand-tuned PD controllers following
the Raibert control framework. The differential flatness of the first-order
equations of motion and the Liouvillian property for the second-order equations
for the hopper system are proved, enabling flat outputs for control. A
two-point boundary value problem (BVP) is then used to minimize jerk in the
flat system to gain implicit smoothness in the output controls. This smoothness
ensures that the planned trajectories are feasible, allowing for given
waypoints to be reached.
