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From Bipedal to Quadrupedal Locomotion, Experimental Realization of Lyapunov Approaches


Ma, Wen-Loong (2021) From Bipedal to Quadrupedal Locomotion, Experimental Realization of Lyapunov Approaches. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/j1ty-zb28.


Possibly one of the most significant innovations of the past decade is the hybrid zero dynamics (HZD) framework, which formally and rigorously designs a control algorithm for robotic walking. In this methodology, Lyapunov stability, which is often used to certificate a dynamical system's stability, was introduced to the control law design for a hybrid control system. However, the prerequisites of precise modeling to apply the HZD methodology can often be too restrictive to design controllers for uncertain and complex real-world hardware experiments. This thesis addresses the problem raised by noisy measurements and the intricate hybrid structure of locomotion dynamics.

First, the HZD methodology's construction is based on the full-order, hybrid dynamics of legged locomotion, which can be intractable for control synthesis for high-dimensional systems. This thesis studies the general structure of hybrid control systems for walking systems, ranging from 1D hopping, 2D walking, 2D running, and 3D quadrupedal locomotion on rough terrains. Further, we characterize a walking behavior--gait--as a solution (execution) to a hybrid control system. To find these solutions, which represent a "gait," we employed advanced numerical methods such as collocation methods to parse the solution-finding problem into the open- and closed-loop trajectory optimization problems. The result is that we can find versatile gaits for ten different robotic platforms efficiently. This includes bipedal running, bipedal walking on slippery surfaces, and quadrupedal robots walking on sloped terrains. The numerous solution-finding examples expand the applicability of the HZD framework towards more complex dynamical systems.

Further, for the uncertain and noisy real-world implementation, the exponential stability of the continuous dynamics is an ideal but restrictive condition for hybrid stability. This condition is especially challenging to satisfy for highly dynamical behaviors such as bipedal running, which loses ground support for a short period. This thesis observes the destabilizing effect of the noisy measurements of the phasing variable. By reformulating the traditional input-to-state stability (ISS) concept into phase-uncertainty to state stability, we are able to synthesize a robust controller for bipedal running on DURUS-2D. This time+state-based controller formally guarantees stability under noisy measurements and stabilizes the 1.75 m/s running experiments.

Lastly, robotic dynamics have long been characterized as the interconnection of rigid-body dynamics. We take this perspective one step further and incorporate controller design into the formulation of coupled control systems (CCS). We first view a quadrupedal robot as two bipedal robots connected via some holonomic constraints. In a dimensional reduction manner, we develop a novel optimization framework, and the computational performance is reduced to a few seconds for gait generation. Furthermore, we can design local controllers for each bipedal subsystem and still guarantee the overall system's stability. This is done by combining the HZD framework and the ISS properties to contain the disturbance induced by the other subsystems' inputs. Utilizing the proposed CCS methods, we will experimentally realize quadrupedal walking on various outdoor rough terrains.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Nonlinear control, nonconvex optimization, legged locomotion
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)
  • Ames, Aaron D.
  • Murray, Richard M.
  • Chung, Soon-Jo
Defense Date:5 May 2021
Non-Caltech Author (AT)
Funding AgencyGrant Number
Record Number:CaltechTHESIS:05042021-155258800
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for ch.4 adapted for ch.4 adapted for ch.4 adapted for ch.4 adapted for ch.4 adapted for ch.2 and ch.4 adapted for ch.2 and ch.3 adapted for ch.2 adapted for ch.2 adapted for ch.3 adapted for ch.2 2016.7759856DOIArticle adapted for ch.3 adapted for ch.2 adapted for ch.1 and ch.2 adapted for ch.2
Ma, Wen-Loong0000-0002-0115-5632
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14132
Deposited By: Wenlong Ma
Deposited On:04 Jun 2021 00:16
Last Modified:02 Nov 2021 17:58

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