Dynamic Walking: Toward Agile and Efficient Bipedal Robots

Dynamic walking on bipedal robots has evolved from an idea in science fiction to a practical reality. This is due to continued progress in three key areas: a mathematical understanding of locomotion, the computational ability to encode this mathematics through optimization, and the hardware capable of realizing this understanding in practice. In this context, this review article outlines the end-to-end process of methods which have proven effective in the literature for achieving dynamic walking on bipedal robots. We begin by introducing mathematical models of locomotion, from reduced order models that capture essential walking behaviors to hybrid dynamical systems that encode the full order continuous dynamics along with discrete footstrike dynamics. These models form the basis for gait generation via (nonlinear) optimization problems. Finally, models and their generated gaits merge in the context of real-time control, wherein walking behaviors are translated to hardware. The concepts presented are illustrated throughout in simulation, and experimental instantiation on multiple walking platforms are highlighted to demonstrate the ability to realize dynamic walking on bipedal robots that is agile and efficient

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

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.</p

Control of Separable Subsystems with Application to Prostheses

Nonlinear control methodologies have successfully realized stable human-like walking on powered prostheses. However, these methods are typically restricted to model independent controllers due to the unknown human dynamics acting on the prosthesis. This paper overcomes this restriction by introducing the notion of a separable subsystem control law, independent of the full system dynamics. By constructing an equivalent subsystem, we calculate the control law with local information. We build a subsystem model of a general open-chain manipulator to demonstrate the control method's applicability. Employing these methods for an amputee-prosthesis model, we develop a model dependent prosthesis controller that relies solely on measurable states and inputs but is equivalent to a controller developed with knowledge of the human dynamics and states.Comment: 8 pages, 6 figure

Control of Separable Subsystems with Application to Prostheses

Nonlinear control methodologies have successfully realized stable human-like walking on powered prostheses. However, these methods are typically restricted to model independent controllers due to the unknown human dynamics acting on the prosthesis. This paper overcomes this restriction by introducing the notion of a separable subsystem control law, independent of the full system dynamics. By constructing an equivalent subsystem, we calculate the control law with local information. We build a subsystem model of a general open-chain manipulator to demonstrate the control method's applicability. Employing these methods for an amputee-prosthesis model, we develop a model dependent prosthesis controller that relies solely on measurable states and inputs but is equivalent to a controller developed with knowledge of the human dynamics and states

Dynamic Walking: Toward Agile and Efficient Bipedal Robots

Dynamic walking on bipedal robots has evolved from an idea in science fiction to a practical reality. This is due to continued progress in three key areas: a mathematical understanding of locomotion, the computational ability to encode this mathematics through optimization, and the hardware capable of realizing this understanding in practice. In this context, this review article outlines the end-to-end process of methods which have proven effective in the literature for achieving dynamic walking on bipedal robots. We begin by introducing mathematical models of locomotion, from reduced order models that capture essential walking behaviors to hybrid dynamical systems that encode the full order continuous dynamics along with discrete footstrike dynamics. These models form the basis for gait generation via (nonlinear) optimization problems. Finally, models and their generated gaits merge in the context of real-time control, wherein walking behaviors are translated to hardware. The concepts presented are illustrated throughout in simulation, and experimental instantiation on multiple walking platforms are highlighted to demonstrate the ability to realize dynamic walking on bipedal robots that is agile and efficient