##### Autonomy Artificial Intelligence Robotics (AAIR)

##### 2009

# OPTIMAL TIME AND ENERGY EFFICIENCY IN LEGGED ROBOTICS

### by Paul Muench; Ka C Cheok

#### Abstract

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 suboptimal 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.