By Frédéric Jean
Nonholonomic platforms are regulate platforms which rely linearly at the regulate. Their underlying geometry is the sub-Riemannian geometry, which performs for those structures a similar position as Euclidean geometry does for linear platforms. specifically the standard notions of approximations on the first order, which are crucial for keep an eye on reasons, need to be outlined when it comes to this geometry. the purpose of those notes is to offer those notions of approximation and their program 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
X m generate a sub-Riemannian distance on Rn which is homogeneous with respect to the dilation αt . 1 (i) The family ( ⎡ X 1, . . , ⎡ X m ) satisfies Chow’s Condition on Rn . (ii) The growth vector at 0 of ( ⎡ X 1, . . , ⎡ X m ) is equal to the one at p of (X 1 , . . , X m ). (iii) Let d⎡ be the sub-Riemannian distance on Rn associated with ( ⎡ X 1, . . , ⎡ X m ). The distance d⎡ is homogeneous of degree 1, ⎡ y). 5). Proof Through the coordinates z we identify the neighbourhood U of p in M with a neighbourhood of 0 in Rn .
Z n ) defined on an open neighbourhood U of the point p. We define first the one-parameter family of dilations αt : (z 1 , . . , z n ) ∗ (t w1 z 1 , . . , t wn z n ), t ≥ 0. Each dilation αt is a map from Rn to Rn . By abuse of notations, for q ∈ U and t small enough we write αt (q) instead of αt (z(q)), where z(q) are the coordinates of q. A dilation αt acts also on functions and vector fields by pull-back: αt∗ f = f ◦ αt and αt∗ X is the vector field such that (αt∗ X )(αt∗ f ) = αt∗ (X f ).
Yn still form a basis of Tq M. 2) at q. Let us explain now the relation between weights and orders. We write first the tangent space as a direct sum, T p M = ω1 ( p) ⊕ ω2 ( p)/ω1 ( p) ⊕ · · · ⊕ ωr ( p)/ωr −1 ( p), where ωs ( p)/ωs−1 ( p) denotes a supplementary of ωs−1 ( p) in ωs ( p). Let us choose local coordinates (y1 , . . , yn ). The dimension of each space ωs ( p)/ωs−1 ( p) is equal to n s − n s−1 , so we can assume that, up to a reordering, the local coordinates satisfy dy j (ωs ( p)/ωs−1 ( p)) = 0 for n s−1 < j → n s .