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Dynamic Bipedal Locomotion: From Hybrid Zero Dynamics to Control Lyapunov Functions via Experimentally Realizable Methods


Reher, Jenna (2021) Dynamic Bipedal Locomotion: From Hybrid Zero Dynamics to Control Lyapunov Functions via Experimentally Realizable Methods. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/h8v0-vd47.


Robotic bipedal locomotion has become a rapidly growing field of research as humans increasingly look to augment their natural environments with intelligent machines. In order for these robotic systems to navigate the often unstructured environments of the world and perform tasks, they must first have the capability to dynamically, reliably, and efficiently locomote. Due to the inherently hybrid and underactuated nature of dynamic bipedal walking, the greatest experimental successes in the field have often been achieved by considering all aspects of the problem; with explicit consideration of the interplay between modeling, trajectory planning, and feedback control.

The methodology and developments presented in this thesis begin with the modeling and design of dynamic walking gaits on bipedal robots through hybrid zero dynamics (HZD), a mathematical framework that utilizes hybrid system models coupled with nonlinear controllers that results in stable locomotion. This will form the first half of the thesis, and will be used to develop a solid foundation of HZD trajectory optimization tools and algorithms for efficient synthesis of accurate hybrid motion plans for locomotion on two underactuated and compliant 3D bipeds. While HZD and the associated trajectory optimization are an existing framework, the resulting behaviors shown in these preliminary experiments will extend the limits of what HZD has demonstrated is possible thus far in the literature. Specifically, the core results of this thesis demonstrate the first experimental multi-contact humanoid walking with HZD on the DURUS robot and then through the first compliant HZD motion library for walking over a continuum of walking speeds on the Cassie robot.

On the theoretical front, a novel formulation of an optimization-based control framework is introduced that couples convergence constraints from control Lyapunov functions (CLF)s with desirable formulations existing in other areas of the bipedal locomotion field that have proven successful in practice, such as inverse dynamics control and quadratic programming approaches. The theoretical analysis and experimental validation of this controller thus forms the second half of this thesis. First, a theoretical analysis is developed which demonstrates several useful properties of the approach for tuning and implementation, and the stability of the controller for HZD locomotion is proven. This is then extended to a relaxed version of the CLF controller, which removes a convergence inequality constraint in lieu of a conservative CLF cost within a quadratic program to achieve tracking. It is then explored how this new CLF formulation can fully leverage the planned HZD walking gaits to achieve the target performance on physical hardware. Towards this goal, an experimental implementation of the CLF controller is derived for the Cassie robot, with the resulting experiments demonstrating the first successful realization of a CLF controller for a 3D biped on hardware in the literature. The accuracy of the robot model and synthesized HZD motion library allow the real-time control implementation to regularize the CLF optimization cost about the nominal walking gait. This drives the controller to choose smooth input torques and anticipated spring torques, as well as regulate an optimal distribution of feasible ground reaction forces on hardware while reliably tracking the planned virtual constraints. These final results demonstrate how each component of this thesis were brought together to form an effective end-to-end implementation of a nonlinear control framework for underactuated locomotion on a bipedal robot through modeling, trajectory optimization, and then ultimately real-time control.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Robotics; control lyapunov functions; legged locomotion; underactuated robotics;
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Mechanical Engineering
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Ames, Aaron D.
Thesis Committee:
  • Burdick, Joel Wakeman (chair)
  • Murray, Richard M.
  • Chung, Soon-Jo
  • Ames, Aaron D.
Defense Date:22 February 2021
Non-Caltech Author Email:jenna.reher (AT)
Record Number:CaltechTHESIS:05282021-062435188
Persistent URL:
Related URLs:
URLURL TypeDescription into part of Chapters 1, 2 and 3. into part of Chapter 6. into part of Chapter 6. into part of Chapter 4 and 5. into part of Chapter 4 and 5. VideoDynamic walking on DURUS (aka PROXI) at the 2015 DRC finals. VideoAlgorithmic foundations of realizing multi-contact locomotion on the humanoid robot DURUS. VideoDURUS outdoor experiments at Georgia Tech. VideoPassive dynamic balancing and walking in actuated environments. VideoDynamic walking with compliance on a Cassie bipedal robot. VideoAn inverse dynamics approach to control Lyapunov functions. Simulated walking and dynamic crouching with CLFs on Cassie. VideoInverse dynamics control of compliant hybrid zero dynamic walking on Cassie. VideoControl Lyapunov functions for compliant hybrid zero dynamic walking.
Reher, Jenna0000-0002-8297-3847
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14188
Deposited By: Jenna Reher
Deposited On:02 Jun 2021 23:52
Last Modified:02 Jun 2021 23:54

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