Two Degree of Freedom Design for Robust Nonlinear Control of Mechanical Systems

Richard M. Murray
California Institute of Technology

National Science Foundation

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Project Overview

Motivated by applications in flight control and robotics, this project is focused on the use of two degree of freedom design techniques to generate nonlinear controllers for mechanical systems performing motion control tasks. Sample applications include high-performance control of piloted aircraft using vectored thrust propulsion, navigation and control of unmanned flight vehicles performing surveillance and other tasks, motion control and stabilization of underwater vehicles and ships, and control of land-based robotic locomotion systems. This project makes use of experimental facilities currently available at Caltech as well as interaction with industrial projects to insure that research results in this area are relevant to physical systems and existing applications.

The basic approach of two degree of freedom design is to initially separate the controller synthesis problem into design of a feasible trajectory for the nominal model of the system followed by regulation around that trajectory using controllers which have guaranteed performance in the presence of uncertainties. In the applications which are considered, it is important that the trajectory generation phase be done in real-time and hence traditional techniques such as optimal control cannot be applied directly. Recent techniques in nonlinear control theory allow certain high-dimensional dynamic problems to be reduced to algebraic problems which are amenable to computationally efficient algorithms for trajectory generation. By exploiting these ideas, new methods are begin developed for quickly generating (suboptimal) trajectories which accomplish a desired control objective.

Many existing nonlinear control strategies (such as feedback linearization and nonlinear regulator theory) implicitly generate feasible trajectories for the nominal system and then regulate the system to follow those trajectories. Unfortunately, these techniques either feedback linearize the dynamics of the system, destroying the nonlinear nature of the system instead of exploiting it, or use the linearization about a single point for doing regulation, leading to poor performance for motions which do not remain near that single equilibrium point. Since most of the applications motivating this work involve rapid motion over a wide operating envelope, these techniques often do not perform adequately. By splitting the trajectory generation from the regulation phases, one is not forced to feedback linearize the system about a single operating point and can therefore more fully exploit the nonlinear nature of the system.

By restricting attention to mechanical systems, it is possible to exploit the rich structure which is available for this class of applications. In particular, driven by new theoretical results in Lagrangian control systems, a new understanding of the role of symmetries, constraints, and external forces is emerging which has important implications for the specific systems considered in this project. By properly accounting for the second order nature of mechanical systems, it is possible to analyze the control of these systems in a way which remains true to the underlying geometry and allows the nonlinear nature of the mechanical system to be exploited to a fuller extent that previously possible. This structure is being used to understand the notion of robustness in the presence of physically meaningful model uncertainty and disturbances, such as uncertainty in dynamic and kinematic parameters and disturbance forces due to wind, fuel slosh, and actuator noise.

Finally, the problem of interaction between the two phases of design is also considered. That is, how does the trajectory generation affect one's ability to achieve robust performance and vice-versa? There are many simple examples where optimal trajectories are very difficult to follow because the optimization criteria did not properly account for the controllability properties of the system. By exploiting the extra structure which is implicit in mechanical systems, techniques are being developed for avoiding these problems. Initial work concentrates on specific experimental systems and uses this as motivation for understanding the more general framework.

Students, Postdocs, and Visitors


Configuration controllability of simple mechanical control systems
Andrew D. Lewis and Richard M. Murray
SIAM Review, 41(3):555-574, 1999

Trajectory Generation for Mechanical Systems with Application to Robotic Locomotion
Richard M. Murray, Joel W. Burdick, Scott D. Kelly, James Radford
Proceedings, 1998 Workshop on Algorithmic Foundations of Robotics

The Mechanics and Control of Robotic Locomotion with Applications to Aquatic Vehicles
Scott D. Kelly
PhD Dissertation, Caltech, June 1998

Nonlinear Control of Mechanical Systems: A Reimannian Geometry Approach
Francesco Bullo
PhD Dissertation, Caltech, August 1998

An Experimental Comparison of Tradeoffs in Using Compliant Manipulators for Robotic Grasping Tasks
Sudipto Sur and Richard M. Murray
1997 International Conference on Robotics and Automation

Tracking for Fully Actuated Mechanical Systems: A Geometric Framework
Francesco Bullo and Richard M. Murray
Automatica 35: (1) 17-34, 1999

Robotic Manipulation with Flexible Link Fingers
Sudipto Sur
PhD Dissertation, Caltech, January 1997

Differentially Flat Nonlinear Control Systems
Muruhan Rathinam
PhD Dissertation, Caltech, May 1997

Autonomous Reorientation of a Manuever-Limited Spacecraft Under Simple Pointing Constraints
Charles A. Vannelli
Engineer's Thesis, Caltech, May 1997

Simultaneous Force-Position Control for Grasping Using Flexible Link Manipulators
Sudipto Sur and Richard M. Murray
1997 American Control Conference

Differential Flatness of Two One-Forms in Arbitrary Number of Variables
Muruhan Rathinam and Richard M. Murray
Systems and Control Letters, 36:317-326, 1999.

Trajectory tracking for fully actuated mechanical systems
Francesco Bullo and Richard M. Murray
1997 European Control Conference

Real-Time Control Experiments for Instruction and Research at Caltech
Richard M. Murray
1996 Conference on Decision and Control

Decompositions for Control Systems on Manifolds with an Affine Connection
Andrew Lewis and Richard M. Murray
Systems & Control Letters 31:199-205, 1997

Configuration Flatness of Lagrangian Systems Underactuated by One Control
Muruhan Rathinam and Richard M. Murray
SIAM J. Control and Optimization, 36(1):164-179, 1998

Trajectory Generation for a Towed Cable System using Differential Flatness
Richard M. Murray
1996 IFAC World Congress

A Homotopy Algorithm for Approximating Geometric Distributions by Integrable Systems
Willem M. Sluis, Andzrej Banaszuk, John Hauser, Richard M. Murray
System and Control Letters, 27: (5) 285-291, 1996

Proportional Derivative (PD) Control on the Euclidean Group
Francesco Bullo and Richard Murray
CDS Technical Report 95-010

Aspects of Geometric Mechanics and Control of Mechanical Systems
Andrew D. Lewis
PhD Dissertation, Caltech, Jun 1995

Exponential Stabilization of Driftless Nonlinear Control Systems
Robert T. M'Closkey
PhD Dissertation, Caltech, Dec 1994

Additional Information

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Last modified: 07/09/00