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32
Flatness and defect of nonlinear systems: Introductory theory and examples
 International Journal of Control
, 1995
"... We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman’s controllability. The distance to flatness is ..."
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Cited by 184 (16 self)
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We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman’s controllability. The distance to flatness is measured by a nonnegative integer, the defect. We utilize differential algebra which suits well to the fact that, in accordance with Willems ’ standpoint, flatness and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of planar curves. The three nonflat examples, the simple, double and variable length pendulums, are borrowed from nonlinear physics. A high frequency control strategy is proposed such that the averaged systems become flat. ∗This work was partially supported by the G.R. “Automatique ” of the CNRS and by the D.R.E.D. of the “Ministère de l’Éducation Nationale”. 1 1
A New Computational Approach to RealTime Trajectory Generation for Constrained Mechanical Systems
, 2000
"... Preliminary results of a new computational approach to generate aggressive trajectories in realtime for constrained mechanical systems are presented. The algorithm is based on a combination of nonlinear control theory, spline theory, and sequential quadratic programming. It is demonstrated that rea ..."
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Cited by 78 (19 self)
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Preliminary results of a new computational approach to generate aggressive trajectories in realtime for constrained mechanical systems are presented. The algorithm is based on a combination of nonlinear control theory, spline theory, and sequential quadratic programming. It is demonstrated that realtime trajectory generation for constrained mechanical systems is possible by mapping the problem to one of finding trajectory curves in a lower dimensional space. Performance of the algorithm is compared with existing optimal trajectory generation techniques. Numerical results are reported using the NTG software package. Keywords: Realtime optimization, nonlinear control design, optimal control, constrained trajectory generation, guidance. 1
Autonomous Vehicle Technologies for Small Fixed Wing UAVs
 AIAA JOURNAL OF AEROSPACE COMPUTING, INFORMATION, AND COMMUNICATION
, 2003
"... Autonomous unmanned air vehicle flight control systems require robust path generation to account for terrain obstructions, weather, and moving threats such as radar, jammers, and unfriendly aircraft. In this paper, we outline a feasible, hierarchal approach for realtime motion planning of small aut ..."
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Cited by 46 (15 self)
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Autonomous unmanned air vehicle flight control systems require robust path generation to account for terrain obstructions, weather, and moving threats such as radar, jammers, and unfriendly aircraft. In this paper, we outline a feasible, hierarchal approach for realtime motion planning of small autonomous fixedwing UAVs. The approach divides the trajectory generation into four tasks: waypoint path planning, dynamic trajectory smoothing, trajectory tracking, and lowlevel autopilot compensation. The waypoint path planner determines the vehicle 's route without regard for the dynamic constraints of the vehicle. This results in a significant reduction in the path search space, enabling the generation of complicated paths that account for popup and dynamically moving threats. Kinematic constraints are satisfied using a trajectory smoother which has the same kinematic structure as the physical vehicle. The third step of the approach uses a novel tracking algorithm to generate a feasible state trajectory that can be followed by a standard autopilot. MonteCarlo simulations were done to analyze the performance and feasibility of the approach and determine realtime computation requirements. A planar version of the algorithm has also been implemented and tested in a lowcost microcontroller. The paper describes a custom UAV built to test the algorithms.
Configuration Flatness of Lagrangian Systems Underactuated by One Control
, 1996
"... Lagrangian control systems that are differentially flat with flat outputs that only depend on configuration variables are said to be configuration flat. We provide a complete characterisation of configuration flatness for systems with n degrees of freedom and n  1 controls whose range of control fo ..."
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Cited by 27 (6 self)
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Lagrangian control systems that are differentially flat with flat outputs that only depend on configuration variables are said to be configuration flat. We provide a complete characterisation of configuration flatness for systems with n degrees of freedom and n  1 controls whose range of control forces only depends on configuration and whose Lagrangian has the form of kinetic energy minus potential. The method presented allows us to determine if such a system is configuration flat and, if so provides a constructive method for finding all possible configuration flat outputs. Our characterisation relates configuration flatness to Riemannian geometry. We illustrate the method by two examples.
Oscillations, SE(2)snakes and motion control: a study of the Roller Racer
 Dynamical Systems
, 2001
"... This report is concerned with the problem of motion generation via cyclic variations in selected degrees of freedom (usually referred to as shape variables) in mechanical systems subject to nonholonomic constraints (here the classical one of a disk rolling without sliding on a at surface). In earlie ..."
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Cited by 25 (15 self)
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This report is concerned with the problem of motion generation via cyclic variations in selected degrees of freedom (usually referred to as shape variables) in mechanical systems subject to nonholonomic constraints (here the classical one of a disk rolling without sliding on a at surface). In earlier work, we identi ed an interesting class of such problems arising in the setting of Lie groups, and investigated these under a hypothesis on constraints, that naturally led to a purely kinematic approach. In the present work, the hypothesis on constraints does not hold, and as a consequence, it is necessary to take into account certain dynamical phenomena. Speci cally we concern ourselves with the group SE(2) of rigid motions in the plane and a concrete mechanical realization dubbed the 2{node, 1{module SE(2){snake. In a restricted version, it is also known as the Roller Racer (a patented ride/toy). Based on the work of Bloch, Krishnaprasad, Marsden and Murray, one recognizes in the example of this report a balance law called the momentum equation, which is a direct consequence of the interaction of the SE(2){symmetry of the problem with the
Necessary Condition and Genericity of Dynamic Feedback Linearization
 J. Math. Systems Estim. Control
, 1994
"... A new necessary condition for dynamic feedback linearization in the sense of [3] is proposed. This condition concerns control systems x = f(x;u) with strictly less control variables than state variables. This necessary condition allows to prove the nongenericity of dynamic feedback linearizability ..."
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Cited by 25 (9 self)
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A new necessary condition for dynamic feedback linearization in the sense of [3] is proposed. This condition concerns control systems x = f(x;u) with strictly less control variables than state variables. This necessary condition allows to prove the nongenericity of dynamic feedback linearizability, for the Whitney C 1 topology on mappings (x; u) ! f(x; u). However, this topology reveals to be too coarse to capture the nature of practical uncertainties: the polymerization reactor studied in [17] is shown to be linearizable via dynamic feedback for generic kinetic and thermal laws. Key words: dynamic feedback linearization, flatness, elimination theory, structural stability, chemical reactor control 1 Introduction In [18] the genericity and structural stability of affine control systems which are linearizable via dynamic feedback [3] is investigated. We address here a similar problem for general control systems where the dependence with respect to the control variables is not suppo...
A Differential Geometric Setting for Dynamic Equivalence and Dynamic Linearization
 in Proceedings Banach Center Publications
, 1994
"... : This paper presents an (infinite dimensional) geometric framework for control system, based on infinite jet bundles, where a system is represented by a single vector field and dynamic equivalence (to be precise : equivalence by endogenous dynamic feedback) is conjugation by diffeomorphisms. These ..."
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Cited by 19 (3 self)
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: This paper presents an (infinite dimensional) geometric framework for control system, based on infinite jet bundles, where a system is represented by a single vector field and dynamic equivalence (to be precise : equivalence by endogenous dynamic feedback) is conjugation by diffeomorphisms. These diffeomorphisms are very much related to LieBacklund transformations. It is proved in this framework that dynamic equivalence of singleinput systems is the same as static equivalence. Keywords: Dynamic feedback equivalence, dynamic feedback linearization, flat systems, infinite jet bundles, contact transformations, LieBacklund transformations. (R'esum'e : tsvp) This is the final version of the text of a presentation at the "Extended Workshop on Geometry in Nonlinear Control", Warsaw, june, 1993, Banach Center Semester on Control Theory, to appear in the proceedings, Banach Center Publications, 1994 or 1995. Email: pomet@sophia.inria.fr, Fax: (+33) 93 65 78 58. Unite de recherche INR...
Flatness and motion planning: the car with n trailers
 In Proc. European Control Conference
, 1993
"... Abstract A solution of the motion planning without obstacles for the nonholonomic system describing a car with n trailers is proposed. This solution relies basically on the fact that the system is flat with the cartesian coordinates of the last trailer as linearizing output. The Frnet formulas are u ..."
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Cited by 18 (0 self)
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Abstract A solution of the motion planning without obstacles for the nonholonomic system describing a car with n trailers is proposed. This solution relies basically on the fact that the system is flat with the cartesian coordinates of the last trailer as linearizing output. The Frnet formulas are used to simplify the calculation. The 2trailers case is treated in details and illustrated through parking simulations. Key words Nonholonomic motion planning, dynamic feedback linearization, linearizing output, flatness, mobile robots. 1
Flat systems, equivalence and trajectory generation
, 2003
"... Flat systems, an important subclass of nonlinear control systems introduced via differentialalgebraic methods, are defined in a differential geometric framework. We utilize the infinite dimensional geometry developed by Vinogradov and coworkers: a control system is a diffiety, or more precisely, an ..."
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Cited by 10 (4 self)
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Flat systems, an important subclass of nonlinear control systems introduced via differentialalgebraic methods, are defined in a differential geometric framework. We utilize the infinite dimensional geometry developed by Vinogradov and coworkers: a control system is a diffiety, or more precisely, an ordinary diffiety, i.e. a smooth infinitedimensional manifold equipped with a privileged vector field. After recalling the definition of aLieBäcklund mapping, we say that two systems are equivalent if they are related by a LieBäcklund isomorphism. Flat systems are those systems which are equivalent to a controllable linear one. The interest of such an abstract setting relies mainly on the fact that the above system equivalence is interpreted in terms of endogenous dynamic feedback. The presentation is as elementary as possible and illustrated by the VTOL aircraft.
Extended Controller Form and Invariants of Nonlinear Control Systems with a Single Input
 J. Math. Systems Estim. Control
, 1994
"... In this paper, a normal form for nonlinear control systems with a singleinput by studying its kjet is found. It is a generalization of linear Brunovsky form to nonlinear control systems. A set of invariants which characterizes the homogeneous parts of a control system is also found. Then the probl ..."
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Cited by 7 (1 self)
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In this paper, a normal form for nonlinear control systems with a singleinput by studying its kjet is found. It is a generalization of linear Brunovsky form to nonlinear control systems. A set of invariants which characterizes the homogeneous parts of a control system is also found. Then the problem of convergence and uniqueness of the normal forms is studied. Key words: nonlinear systems, extended controller form, invariants AMS Subject Classifications: 93C10, 93C15 1 Introduction A fruitful technique for many dynamic systems and control systems consists of transforming them into a simple form. Normal forms for nonlinear dynamic systems have been studied since the end of last century, which was first addressed by H. Poincar'e. Since then, Poincar'e's technique has been successfully applied to the research in the area of nonlinear vector fields, Hamilton dynamic systems, nonlinear mappings and bifurcation phenomenon (see, for instance, Arnold [1], Siegel [19], Bruno [6] and Baider...