In this paper, we study the singularities of differentially flat systems, in the perspective of providing global or semi-global motion planning solutions for such systems: flat outputs may fail to be globally defined, thus potentially preventing from planning trajectories leaving their domain of definition, the complement of which we call singular. Such singular subsets are classified into two types: apparent and intrinsic. A rigorous definition of these singularities is introduced in terms of atlas and local charts in the framework of the differential geometry of jets of infinite order and Lie-Bäcklund isomorphisms. We then give an inclusion result allowing to effectively compute all or part of the intrinsic singularities. Finally, we show how our results apply to the global motion planning of the celebrated example of non holonomic car.
The ENGINO R Toy System introduced two challenging problems. The first was to get bounds on the number of possible models/toys which can be constructed using a given package of building blocks. And the second is to generate automatically the assembly instructions for a given toy. In this report we summarize our insights and provide preliminary results for the two challenges.
This paper is a theoretical study of the following Non-Rigid Structure from Motion problem. What can be computed from a monocular view of a parametrically deforming set of points? We treat various variations of this problem for affine and polynomial deformations with calibrated and uncalibrated cameras. We show that in general at least three images with quasi-identical two deformations are needed in order to have a finite set of solutions of the points' structure and calculate some simple examples.
We investigate apparent and intrinsic singularities of a flat model of aircrafts, illustrated with numerical simulations using Python and Maple. We consider failure situations and maneuvers for which apparent singularities of the previously known flat outputs may appear, making necessary to use some of the new flat outputs we consider. Basically, the aircraft flat outputs are x, y, z, the coordinates of the gravity center, completed with any function of the sideslip angle β, the angle of attack α, the bank angle µ and the thrust F . The choice of β was previously used, but does not allow gravity-free flight, for which µ is the best choice, as well as for decrabe maneuver. The choice of F is adapted for dead-stick landing conditions with β = 0, such as forward slip maneuver. This approach also allows to replace usual control with new controls in case of failures, e.g. differential thrust can be used in case of rudder failure. Our results are illustrated by numerical simulations, using realistic non linear aerodynamics models. In a first stage, we investigate the ability of the flatness based control to reject perturbations. Since flatness in that case requires some model simplification, in a second stage, we focus on model errors and show that a suitable feed-back allows to keep trajectories with the complete real model close to the trajectories planned with the simplified one.
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