Optimal Time and Energy Efficiency in Legged Robotics

Although bio-inspired legged robots have advantageous mobility, they can be very inefficient. Their intrinsic walking mobility is sometimes outweighed by the inefficiency of their drive-train. Some of these inefficiencies are due to collision losses, but they are also due to sub-optimal powering schemes. This paper addresses the powering schemes and seeks to clearly delineate an optimal solution to powering the walking motion of a two-legged or biped walker. We examine a simplified model of locomotion called the "rocket car" to extract the meaningful parameters that affect time and energy cost. Using Pontryagin's Maximum Principle, we dissect the cost function, the state equation, co-state equation, and control input constraints to describe the optimal control. The result of the paper shows a "bang-off" control, and we describe the "coasting line" between these extremes. It is not possible to find a complete closed-form solution for the problem, and numerical methods, such as dynamic programming must be used for future simulation and visualization of the results.
Presented at the Ground Vehicle Systems Engineering and Technology Symposium (GVSETS), held in Troy, Michigan, on 17-22 August 2009. Prepared in cooperation with Oakland University, Rochester, MI. The original document contains color images.