DOI: 10.7274/29578508.v1
title: From Contact Feasibility to Gait Sequence Optimization for Legged Robot Locomotion

description: Humans have always dreamed of versatile robotic assistants, and to navigate freely within spaces designed for humans, legged robots are indispensable. Thanks to significant advancements in robot hardware and control methodologies, humanoid and quadruped robots have made substantial progress in performing diverse tasks. Nevertheless, they have yet to achieve the same level of maneuverability as their biological counterparts, such as humans and dogs. The main objective of this dissertation is to contribute to the development of more stable, efficient, and versatile legged robots, capable of performing a wide range of tasks in various environments. Three levels of legged robot control are studied: low-level contact maintenance, intermediate-level planning and balance control, and high-level gait optimization.

A fundamental problem in legged locomotion is to verify whether a desired motion satisfies all physical constraints, especially those for maintaining contacts. Although foot tipping can be avoided via the Zero Moment Point (ZMP) condition, preventing foot sliding and twisting leads to the more complex Contact Wrench Cone (CWC) constraints. An efficient algorithm is presented to verify the feasibility of a net contact wrench on level ground with uniform friction. This algorithm achieves a speedup of 139 times with 7.13% error compared to the CWC. The key step in the algorithm is a novel exact geometric characterization of the yaw moment limits in the case when the support polygon is approximated by a single supporting line. Unlike the conventional ZMP condition, this method provides a sufficient condition for contact wrench feasibility, which offers a guarantee of a legged robot's contact stability.


Another key issue of legged robots is their maneuverability compared to animals. A planning and control strategy that decouples sagittal and lateral/rotational motion is presented to push the boundary of robots' capabilities by controlling a quadruped robot's two-legged hopping. 
The motion in the sagittal plane is controlled by re-planning via a template spring-loaded inverted pendulum (SLIP) model.
A control Lyapunov function-based quadratic programming (CLF-QP) controller is used to modulate nominal ground reaction forces (GRFs) to balance the torso while tracking the sagittal motion. 
The CLF construction is guided by a variational-based linearization (VBL) applied to a reduced-order single-rigid-body (SRB) model along the planned motion and treats underactuation via solving a Riccati equation to obtain the CLF. The method is verified in a full dynamics simulator, demonstrating in-place hopping at heights from 0.52 m to 0.71 m.

Finally, this dissertation studies the optimization of gait sequences for quadruped robots, aiming to enhance their locomotion capabilities and adaptability to disturbances. Unlike conventional methods, a decision tree is constructed for the contact sequence backward in time to separate discrete and continuous states. The approach employs a relaxed dynamics model to bound the cost-to-here and enforces rigid contact when computing the cost-to-go, both resolved through a continuous Differential Dynamic Programming (DDP) solver. The framework uses a hybrid search algorithm combining depth-first and A*, using the lower bound of cost to identify promising gait sequences. When growing the search tree, the relaxed model can be projected to any contact mode for efficient warm starting. The proposed approach generates emergent contact sequences that adapt well to speed changes and external disturbances for which the fixed gait controller failed.