  It is often overlooked by roboticists when designing locomotion controllers
for their legged machines, that energy consumption plays an important role in
selecting the best gaits for locomotion at high speeds or over long distances.
The purpose of this study is to examine four similar asymmetrical quadrupedal
gaits that are frequently observed in legged animals in nature. To understand
how a specific footfall pattern will change the energetics of a legged system,
we first developed a full body model of a quadrupedal robot called A1. And for
each gait we created a hybrid system with desired footfall sequence and rigid
impacts. In order to find the most energy efficient gait, we used optimal
control methods to formulate the problem as a trajectory optimization problem
with proper constraints and objective function. This problem was implemented
and solved in a nonlinear programming framework called FROST. Based on the
optimized trajectories for each gait, we investigated the values of cost of
transport and the work done by all joints. Moreover, we analyzed the exchange
of angular momentum in different components of the system during the whole
stride cycle. According to the simulation results, bounding with two flight
phases is likely to be the most energy efficient gait for A1 across a wide
range of speed.
