Intrinsic and apparent singularities in differentially flat systems, and application to global motion planning

Abstract

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.

On singularities of flat affine systems with n states and n−1 controls

“…In this section, we briefly recall and adapt the main background and tools, introduced and defined in [12], to the present context of systems with n − 1 inputs.…”

“…The notion of a Lie-Bäcklund atlas for flat systems was initially introduced in [12] in the context of implicit systems. Our presentation here adapts this definition to the case of systems in explicit form.…”

“…To the authors knowledge, theorem 5 is the first one giving a practical condition for which the degenerating directions at a given point, producing a rank drop at this point, are replaced by those generated by another tower of constant rank and involutive subdistributions, namely the Γ a k 's. In contrast, the apparent singularities dealt with in [2,3,12] only concern singularities of the parameterization and are not related to a singularity of the distribution of control vector fields. Remark 12.…”

Finite determination of accessibility and singular points of nonlinear systems: An algebraic approach

“…Finite determination of accessibility and geometric structure of singular points for nonlinear systems Knowing the exact location of the set of accessibility singular points, denoted by ∞ in this paper, is equally important, since ∞ is an invariant set, and therefore it should be avoided for initialization of the system, or trapping the state within ∞ . Also ∞ may obstruct global controllability, if the set of regular points is disconnected by the set ∞ , just as singular points in flatness property can obstruct definition of global flat outputs and global motion planning [10].…”

“…Sarafrazi); 0000-0002-7697-769X (Ü. Kotta); 0000-0002-7250-4350 (Z. Bartosiewicz) M. A. Sarafrazi et al:Finite determination of accessibility and geometric structure of singular points for nonlinear systems Knowing the exact location of the set of accessibility singular points, denoted by ∞ in this paper, is equally important, since ∞ is an invariant set, and therefore it should be avoided for initialization of the system, or trapping the state within ∞ . Also ∞ may obstruct global controllability, if the set of regular points is disconnected by the set ∞ , just as singular points in flatness property can obstruct definition of global flat outputs and global motion planning [10].The approach presented in this paper, suggests that for deciding accessibility of a point in a finite number of steps, instead of examining accessibility property pointwise, one should look at the big picture of the entire set of singular points and the invariance relations between them. The main idea is the following: If the system is non-accessible from 0 , then any trajectory starting from 0 must evolve on the invariant set of non-accessible points.…”