Download Control of Nonholonomic Systems: from Sub-Riemannian by Frédéric Jean PDF

By Frédéric Jean

ISBN-10: 3319086898

ISBN-13: 9783319086897

ISBN-10: 3319086901

ISBN-13: 9783319086903

Nonholonomic platforms are regulate platforms which rely linearly at the keep watch over. Their underlying geometry is the sub-Riemannian geometry, which performs for those platforms an identical position as Euclidean geometry does for linear platforms. particularly the standard notions of approximations on the first order, which are crucial for keep an eye on reasons, need to be outlined by way of this geometry. the purpose of those notes is to offer those notions of approximation and their software to the movement making plans challenge for nonholonomic systems.

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Additional info for Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning

Example text

But they are not privileged: indeed, the coordinate z is of order 2 at 0 since (X 2 X 2 z)(0) = 1. 1 Let us define privileged functions at p to be smooth functions f on U such that ord p ( f ) = max{s ∈ N : d f (ωs ( p)/ωs−1 ( p)) = 0}, (a similar notion has been suggested by Kupka [17]). It results from the discussion above that some local coordinates (z 1 , . . , z n ) are privileged at p if and only if each z i is a privileged function at p. Let us now show how to compute orders using privileged coordinates.

Let us consider a singular point p and a radius r > 0. Since regular points form an open set, B( p, r ) contains small balls centered at regular points, and thus dimH B( p, r ) ≥ 4. 3. We choose as a volume form Ω = d x ∧ dy ∧ dz, the associated measure volΩ being the Lebesgue measure μ on R3 . 3 Application to Carnot-Carathéodory Spaces 45 takes only two values, f q,ε (X) = |x|ε4 if X = (X 1 , X 2 , X [1,2] ) and f q,ε (X) = ε5 if X = (X 1 , X 2 , X [1,[1,2]] ). Thus, for q = (x, y, z) close enough to p and for ε > 0 small enough, 1 4 ε max(|x|, ε) → μ(B(q, ε)) → Cε4 max(|x|, ε).

M be m vector fields on a manifold M, and r be a positive integer. The Lie algebra Lie(ξ1 , . . , ξm ) is said to be free up to step r if, for every x ∈ M, the elements n 1 (x), . . , nr (x) of the growth vector are equal to n 1 , . . , nr . 5 Consider a manifold M of dimension nr . If Lie(ξ1 , . . , ξm ) is free up to step r , then every point in M is regular. It is in particular the case when the degree of nonholonomy of (ξ1 , . . , ξm ) equals r at every point of M. Let L (s) be the subspace of L generated by elements of length equal to s.

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Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning by Frédéric Jean

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