California Institute of Technology

National Science Foundation

CMS-9502224

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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.

- Kristi Morgansen - Postdoctoral Scholar in Control and Dynamical Systems, 99-00
- Francesco Bullo - Ph.D. in Control and Dynamical Systems, August 1998 [thesis]
- Scott Kelly - Ph.D. in Mechanical Engineering, June 1998 [thesis]
- Tony Vanelli - Engineer's Degree in Mechanical Engineering, May 1997 [thesis]
- Muruhan Rathinam - Ph.D. in Applied Mathematics, April 1997 [thesis]
- Sudipto Sur - Ph.D. in Mechanical Engineering, January 1997 [thesis]
- Andrew Lewis - Ph.D. in Applied Mechanics, June 1995 [thesis]
- Robert M'Closkey - Ph.D. in Mechanical Engineering, December 1994 [thesis]

- 99g/lm99-siamrev
- Configuration
controllability of simple mechanical control systems

Andrew D. Lewis and Richard M. Murray

SIAM Review, 41(3):555-574, 1999 - 98g/mur98-wafr
- 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 - 98/sdk98-phd
- The
Mechanics and Control of Robotic Locomotion with Applications to Aquatic
Vehicles

Scott D. Kelly

PhD Dissertation, Caltech, June 1998 - 98/fb98-phd
- Nonlinear
Control of Mechanical Systems: A Reimannian Geometry Approach

Francesco Bullo

PhD Dissertation, Caltech, August 1998 - 97g/sm97-icra
- 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 - 97a/bm97-cds
- Tracking
for Fully Actuated Mechanical Systems: A Geometric Framework

Francesco Bullo and Richard M. Murray

*Automatica*35: (1) 17-34, 1999 - 97/ss97-phd
- Robotic
Manipulation with Flexible Link Fingers

Sudipto Sur

PhD Dissertation, Caltech, January 1997 - 97/mr97-phd
- Differentially
Flat Nonlinear Control Systems

Muruhan Rathinam

PhD Dissertation, Caltech, May 1997 - 97/cav97-eng
- Autonomous
Reorientation of a Manuever-Limited Spacecraft Under Simple Pointing
Constraints

Charles A. Vannelli

Engineer's Thesis, Caltech, May 1997 - 96q/sm97-acc
- Simultaneous
Force-Position Control for Grasping Using Flexible Link Manipulators

Sudipto Sur and Richard M. Murray

1997 American Control Conference - 96n/rm97-ecc
- 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. - 96m/bm97-ecc
- Trajectory
tracking for fully actuated mechanical systems

Francesco Bullo and Richard M. Murray

1997 European Control Conference - 96k/mur96-cdc
- Real-Time
Control Experiments for Instruction and Research at Caltech

Richard M. Murray

1996 Conference on Decision and Control - 96j/lm96-scl
- Decompositions
for Control Systems on Manifolds with an Affine Connection

Andrew Lewis and Richard M. Murray

*Systems & Control Letters*31:199-205, 1997 - 96g/rm96-cdc
- 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 - 96b/mur96-ifac
- Trajectory
Generation for a Towed Cable System using Differential Flatness

Richard M. Murray

1996 IFAC World Congress - 95r/sbhm95-cds
- 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 - 95m/bm95b-cds
- Proportional
Derivative (PD) Control on the Euclidean Group

Francesco Bullo and Richard Murray

CDS Technical Report 95-010 - 95/adl95-phd
- Aspects
of Geometric Mechanics and Control of Mechanical Systems

Andrew D. Lewis

PhD Dissertation, Caltech, Jun 1995 - 94/rtm94-phd
- Exponential
Stabilization of Driftless Nonlinear Control Systems

Robert T. M'Closkey

PhD Dissertation, Caltech, Dec 1994

- Program presentations: October 1997 (PDF)
- Program summaries: September 1996 (Postscript)
- Original proposal: PDF

Richard Murray (murray@indra.caltech.edu) Last modified: 07/09/00