We present a versatile framework for the computational co-design of legged robots and dynamic maneuvers. Current state-of-the-art approaches are typically based on random sampling or concurrent optimization. We propose a novel bilevel optimization approach that exploits the derivatives of the motion planning sub-problem (i.e., the lower level). These motion-planning derivatives allow us to incorporate arbitrary design constraints and costs in an general-purpose nonlinear program (i.e., the upper level). Our approach allows for the use of any differentiable motion planner in the lower level and also allows for an upper level that captures arbitrary design constraints and costs. It efficiently optimizes the robot's morphology, payload distribution and actuator parameters while considering its full dynamics, joint limits and physical constraints such as friction cones. We demonstrate these capabilities by designing quadruped robots that jump and trot. We show that our method is able to design a more energy-efficient Solo robot for these tasks.

Optimal control is a popular approach to synthesize highly dynamic motion. Commonly, L2regularization is used on the control inputs in order to minimize energy used and to ensure smoothness of the control inputs. However, for some systems, such as satellites, the control needs to be applied in sparse bursts due to how the propulsion system operates. In this paper, we study approaches to induce sparsity in optimal control solutions—namely via smooth L1and Huber regularization penalties. We apply these loss terms to state-of-the-art Differential Dynamic Programming (DDP)-based solvers to create a family of sparsity-inducing optimal control methods. We analyze and compare the effect of the different losses on inducing sparsity, their numerical conditioning, their impact on convergence, and discuss hyperparameter settings. We demonstrate our method in simulation and hardware experiments on canonical dynamics systems, control of satellites, and the NASA Valkyrie humanoid robot. We provide an implementation of our method and all examples for reproducibility on GitHub.

Shooting methods are an efficient approach to solving nonlinear optimal control problems. As they use local optimization, they exhibit favorable convergence when initialized with a good warm-start but may not converge at all if provided with a poor initial guess. Recent work has focused on providing an initial guess from a learned model trained on samples generated during an offline exploration of the problem space. However, in practice, the solutions contain discontinuities introduced by system dynamics or the environment. Additionally, in many cases, multiple equally suitable, i.e., multimodal, solutions exist to solve a problem. Classic learning approaches smooth across the boundary of these discontinuities and thus generalize poorly. In this work, we apply tools from algebraic topology to extract information on the underlying structure of the solution space. In particular, we introduce a method based on persistent homology to automatically cluster the dataset of precomputed solutions to obtain different candidate initial guesses. We then train a mixture-of-experts within each cluster to predict state and control trajectories to warm-start the optimal control solver and provide a comparison with modality-agnostic learning. We demonstrate our method on a cartpole toy problem and a quadrotor avoiding obstacles, and show that clustering samples based on inherent structure improves the warm-start quality.

In this paper, we study a wheeled robot with a prismatic extension joint. This allows the robot to build up momentum to perform jumps over obstacles and to swing up to the upright position after the loss of balance. We propose a template model for the class of such two-wheeled jumping robots. This model can be considered as the simplest wheeled-legged system. We provide an analytical derivation of the system dynamics which we use inside a model predictive controller (MPC). We study the behavior of the model and demonstrate highly dynamic motions such as swing-up and jumping. Furthermore, these motions are discovered through optimization from first principles. We evaluate the controller on a variety of tasks and uneven terrains in a simulator.