  We present a novel algorithm for the computational co-design of legged robots
and dynamic maneuvers. Current state-of-the-art approaches are based on random
sampling or concurrent optimization. A few recently proposed methods explore
the relationship between the gradient of the optimal motion and robot design.
Inspired by these approaches, we propose a bilevel optimization approach that
exploits the derivatives of the motion planning sub-problem (the inner level)
without simplifying assumptions on its structure. Our approach can quickly
optimize the robot's morphology while considering its full dynamics, joint
limits and physical constraints such as friction cones. It has a faster
convergence rate and greater scalability for larger design problems than
state-of-the-art approaches based on sampling methods. It also allows us to
handle constraints such as the actuation limits, which are important for
co-designing dynamic maneuvers. We demonstrate these capabilities by studying
jumping and trotting gaits under different design metrics and verify our
results in a physics simulator. For these cases, our algorithm converges in
less than a third of the number of iterations needed for sampling approaches,
and the computation time scales linearly.
