DOI: 10.7274/28792619.v1
title: Accelerated MPC Algorithms and Augmented System Design for Legged Robots

description: In the coming years, robots are expected to become key companions in our daily lives, act as labor-force multipliers in industry, and assist astronauts in exploring planetary environments. However, several challenges must be addressed before robots can become ubiquitous. These include insufficient battery capacity for extended performance, limited sensing capabilities for reliable operation, and significant safety concerns. The unstructured nature of the world exacerbates these issues, particularly for legged robots, which require a large stability region and rapid control solutions to handle unexpected events. This dissertation aims to address these fundamental challenges by enabling fast numerical control decisions and exploring design augmentations to expand control authority.
We make multiple advances to Differential Dynamic Programming (DDP), a classical control approach that solves for optimal trajectories based on a cost function and the robot's dynamics. Starting with an initial guess trajectory, DDP iteratively refines it based on local approximations of the system's dynamics. This dissertation explores higher-fidelity approximations to the dynamics and their impact on controller performance. Several contributions are proposed to ensure that these approximations can be computed quickly. The approach is powered by a newly introduced algorithm, named the modified Recursive Newton Euler Algorithm~(mRNEA), which is inspired by classical RNEA, and computes the dynamics information pre-multiplied with a fixed vector. This subtle but important change mirrors how the dynamics information is incorporated into DDP and allows us to leverage reverse-mode automatic differentiation to compute the necessary sensitivity information efficiently. Using this approach, we demonstrate that the needed second-order dynamics sensitivities can be computed with the same computational complexity as the first-order counterpart of DDP. Further, we extend the mRNEA to robots making contact with the environment, introducing the mRNEA for contacts (mRNEAc). This algorithm enables efficiently computing second-order sensitivities, with results similar to mRNEA approaches before. Overall, these advances lead to the first experimental demonstration of legged locomotion via second-order DDP with a full-body model.
Supplemental contributions focus on addressing a broad class of integration methods that incorporate the nonlinear continuous-time robot dynamics within DDP. Specifically, we derive the necessary sensitivities of explicit and implicit Runge-Kutta (RK) integration approaches, as applied to solving optimal control problems. We show that implicit integrators and multi-stage integrators offer more accurate integration but are more computationally intensive when included in DDP.
Beyond improving DDP's numerical efficiency, we enhance the Mini Cheetah’s postural stability through mechanical design, inspired by animals that use specialized limbs (e.g., tails) to regulate angular momentum. Herein, we design a flywheel system that can `absorb' angular momentum from the robot's main body, enhancing postural stability. We validate this design both in simulation and hardware. Overall, this dissertation aims to bridge some of the gaps that exist before legged robots can become ubiquitous. Collectively, the results of this dissertation advance legged robot stability, offering better handling of disturbances and accelerating control computations for quick responses to unexpected events...