A variety of stabilization methods for nonlinear systems in chained form can be found in the literature. However, very few results exist in the area of systematically converting a general nonlinear system with two inputs to a chained form. This paper presents an algorithm for the conversion of a class of nonlinear systems with two inputs to a high-order chained, or generalized chained, form. The feedback transformation accomplishing this conversion is derived, provided certain conditions hold, by solving a system of partial differential equations. The proposed algorithm is illustrated by means of a physically motivated example, namely an under-actuated surface vessel.
Issue Section:
Technical Briefs
1.
Baillieul
, J.
, 1992, “Anholonomy in the Control of Kinematically Redundant Mechanisms With Flexible Components
,” In IFAC Symposium on Nonlinear Control Systems Design
, pp. 168
–173
, Bordeaux, France.2.
Baillieul
, J.
, 1993, “Kinematically Redundant Robots With Flexible Components
,” IEEE Control Syst. Mag.
0272-1708, 13
, pp. 15
–21
.3.
De Luca
, A.
, Mattone
, R.
, and Oriolo
, G.
, 1996 “Control of Underactuated Mechanical Systems: Application to the Planar 2R Robot
,” In 35th Conference on Decision and Control
, pp. 1455
–1460
, Kobe, Japan.4.
Oriolo
, G.
, and Nakamura
, Y.
, 1991, “Control of Mechanical Systems With Second-Order Nonholonomic Constraints: Underactuated Manipulators
,” In 35th Conference on Decision and Control
, pp. 2398
–2403
, Brighton, England, December.5.
Wichlund
, K. Y.
, Sørdalen
, J. O.
, and Egeland
, O.
, 1995, “Control of Vehicles With Second-Order Nonholonomic Constraints: Underactuated Vehicles
,” in European Control Conference
, pp. 3086
–3091
, Rome, Italy.6.
Ailon
, A.
, Lozano-Leal
, R.
, and Gil
, M.
, 1997, “Point-To-Point Regulation of a Robot With Flexible Joints Including Electrical Effects of Actuator Dynamics
,” IEEE Trans. Autom. Control
0018-9286, 42
(4
), pp. 559
–564
.7.
Kolmanovsky
, I.
, and McClamroch
, N. H.
, 1995, “Developments in Nonholonomic Control Problems
,” IEEE Control Syst. Mag.
0272-1708, 15
(6
), pp. 20
–36
.8.
Morin
, P.
, Pomet
, J. B.
, and Samson
, C.
, 1998, “Developments in Time-Varying Feedback Stabilization of Nonlinear Systems
,” In IFAC Symposium on Nonlinear Control Systems Design
, pp. 587
–594
, Enschede, The Netherlands.9.
Murray
, R. M.
and Sastry
, S. S.
, 1993, “Nonholonomic Motion Planning: Steering Using Sinusoids
,” IEEE Trans. Autom. Control
0018-9286, 38
, pp. 700
–716
.10.
Bushnell
, L. G.
, Tilbury
, D. M.
, and Sastry
, S. S.
, 1993, “Steering Three-Input Chained Form Nonholonomic Systems Using Sinusoids: The Fire Truck Example
,” In European Control Conference
, pp. 1432
–1437
, Groningen, The Netherlands.11.
Tilbury
, D.
, Laumond
, J.-P.
, Murray
, R.
, Sastry
, S.
, and Walsh
, G. C.
, 1992, “Steering Car-Like Systems With Trailers Using Sinusoids
,” in IEEE International Conference on Robotics and Automation
, pp. 1993
–1998
.12.
Murray
, R. M.
, 1995, “Nilpotent Bases for a Class of Non-Integrable Distributions With Applications to Trajectory Generation for Nonholonomic Systems
,” Math. Control, Signals, Syst.
0932-4194, 7
(1
), pp. 58
–75
.13.
Tilbury
, D.
, Murray
, R. M.
, and Sastry
, S. S.
, 1995, “Trajectory Generation for the n -Trailer Problem Using Goursat Normal Form
,” IEEE Trans. Autom. Control
0018-9286, 40
(5
), pp. 802
–819
.14.
Tilbury
, D.
and Sastry
, S. S.
, 1994, “On Goursat Normal Forms, Prolongations and Control Systems
,” In 33th Conference on Decision and Control
, pp. 1797
–1802
, Orlando.15.
Laiou
, M. C.
, and Astolfi
, A.
, 1998, “Exponential Stabilization of High-Order Chained Systems
,” In IFAC Symposium on Nonlinear Control Systems Design
, 3
, pp. 649
–654
, Enschede, The Netherlands.16.
Laiou
, M. C.
and Astolfi
, A.
, 1999, “Exponential Stabilization of High-Order Generalized Chained Systems
,” Syst. Control Lett.
0167-6911, 37
, pp. 309
–322
.17.
Isidori
, A.
, Nonlinear Control Systems
, 2nd ed., Springer-Verlag
, Berlin, 1989.Copyright © 2005
by American Society of Mechanical Engineers
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