Exploiting Constraints, Geometry, and Passivity for the Control of Bipedal Walking

Eric R. Westervelt*
Department of Mechanical Engineering,
The Ohio State University

Mark W. Spong
Department of Electrical and Computer Engineering,
University of Illinois at Urbana-Champaign

Benjamin Morris
Department of Electrical Engineering and Computer Science,
University of Michigan

Bipedal walking-walking on two legs-has become a popular area of research because of the strong interest in the development of humanoid robots and the advantages of walking in environments with discontinuous support. Another important reason is the potential impact on the development of human assistive device, such as prosthetics, orthotics, and devices for rehabilitation.

The control of bipedal walking is a source of challenging control problems. The challenges arise from the models that describe walking and limited control authority. Models of bipedal walking are typically nonlinear, hybrid, high degree of freedom, and contain unilateral constraints. Each of these difficulties' effects can be mitigated by allowing only slow gaits, which are typically inefficient. However, when fast and efficient gaits are desired, these difficulties must be addressed in controller design.

This tutorial session will cover recent research developments in the control of fast and efficient robotic bipedal walking that use geometric and energy-based methods. In addition to these approaches, the session will cover an important, practical issue: the synthesis of controllers using numerical methods. Specifically, this session will address the following questions/topics in three presentations.

1. Motivation: What is the interest in the study of the control of bipedal walking?
2. Challenges: What are the challenges associated with the control of bipedal walking?
3. Modeling: How are bipedal robots modeled?
4. Constraint-based control: controller design using constraints imposed via feedback.
5. Geometry and Passivity-based control: the design of controllers using controlled symmetries, Routhian reduction, and passivity-based control.
6. Controller synthesis: What techniques may be used for computationally-efficient controller design?

The three presentations will last 40, 30, and 30 minutes and cover items 1 through 4, 5, and 6, respectively. The materials from the presentations will be made available to the tutorial participants via the web.

List of Presentations

Details of the session's presentations are described below.

1. Title: An Overview of Modeling and Control of Bipedal Walking
   Authors: Eric R. Westervelt, Jessy W. Grizzle, Christine Chevallereau, Jun Ho Choi, Benjamin Morris
   Abstract: This presentation is a tutorial on the basics of the modeling, analysis, and constraint-based control of planar bipedal walking. The material covered in the presentation will be taken from the recently-published book Feedback Control of Dynamic Bipedal Robot Locomotion by Westervelt et al. Specific topics that will be
   covered include: the motivations for studying bipedal walking; challenges that are inherent to bipedal walking; the modeling of walking with point feet; and the design of constraint-based controllers with emphasis given to the practical aspects of controller
   implementation. Simulation examples and the results of experiments will be used extensively to illustrate the developments.
2. Title: Geometry and Passivity in the Control of Bipedal Locomotion
   Authors: Mark W. Spong
   Abstract: We will present an overview of several ideas from geometric nonlinear control that can be used to exploit the existence of 2-D and 3-D passive limit cycles for the control of biped robots. These ideas include so-called controlled symmetries for eliminating sensitivity to ground slope, Routhian reduction for generating 3-D limit controlled limit cycles from 2-D passive limit cycles, and passivity-based control in the form of total energy shaping for robustness against disturbances. We will also present
   some new results on the application of Interconnection and Damping Assignment (IDA) control, which is a method of total energy shaping based on the Hamiltonian formulation of the robot dynamic equations.
3. Title: Complexity Issues in Controller Synthesis
   Authors: Benjamin Morris
   Abstract: The derivation of constraint-based walking controllers relies heavily on the designer's ability to take derivatives both of the constraint functions and of the robot's equations of motion. Such derivatives are trivial for bipeds with low numbers of links, but for robots with five or more links, the complexity of the required computations can prohibit closed-form computation or can result in expressions whose subsequent numerical evaluation are unnecessarily computationally expensive. This presentation
   describes three controller synthesis routines that emphasize the tradeoffs between runtime efficiency and code complexity. These examples show how the different approaches to controller synthesis can affect simulation times by an order of magnitude or more.