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 non-negative 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 non-flat 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

Preliminary results of a new computational approach to generate aggressive trajectories in real-time 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 real-time 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: Real-time optimization, nonlinear control design, optimal control, constrained trajectory generation, guidance. 1

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 real-time motion planning of small autonomous fixed-wing UAVs. The approach divides the trajectory generation into four tasks: waypoint path planning, dynamic trajectory smoothing, trajectory tracking, and low-level 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 pop-up 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. Monte-Carlo simulations were done to analyze the performance and feasibility of the approach and determine real-time computation requirements. A planar version of the algorithm has also been implemented and tested in a low-cost micro-controller. The paper describes a custom UAV built to test the algorithms.

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.

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

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 non-genericity 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...

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 2-trailers case is treated in details and illustrated through parking simulations. Key words Nonholonomic motion planning, dynamic feedback linearization, linearizing output, flatness, mobile robots. 1

: 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 Lie-Backlund transformations. It is proved in this framework that dynamic equivalence of single-input systems is the same as static equivalence. Key-words: Dynamic feedback equivalence, dynamic feedback linearization, flat systems, infinite jet bundles, contact transformations, Lie-Backlund 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. E-mail: pomet@sophia.inria.fr, Fax: (+33) 93 65 78 58. Unite de recherche INR...

Flat systems, an important subclass of nonlinear control systems introduced via differential-algebraic 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 infinite-dimensional manifold equipped with a privileged vector field. After recalling the definition of aLie-Bäcklund mapping, we say that two systems are equivalent if they are related by a Lie-Bäcklund 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.

A solution of the motion planning without obstacles for the standard n-trailer system is proposed. This solution relies basically on the fact that the system is flat with the Cartesian coordinates of the last trailer as a linearizing output. The Frénet formulae are used to simplify the calculations and permit to deal with angle constraints. The general 1-trailer system, where the trailer is not directly hitched to the car at the center of the rear axle, is also flat. The geometric construction used for the standard 1-trailer system can be extended to this more realistic system. MATLAB simulations illustrate this method. 1