By Frédéric Jean
Nonholonomic platforms are regulate structures which count linearly at the regulate. Their underlying geometry is the sub-Riemannian geometry, which performs for those platforms a similar position as Euclidean geometry does for linear structures. 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 software to the movement making plans challenge for nonholonomic systems.
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Extra info for Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning
2)”, since the values at p of an adapted frame contain a basis Y1 ( p), . . , Yn s ( p) of each subspace ωs ( p) of the flag. By continuity, at a point q close enough to p the values of Y1 , . . , 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 , .
2 3! and X 2(−1) = X 2 = δσ . 1 First-Order Approximations 27 ⎡ X 1 = δx + σ δ y , ⎡ X 2 = δσ . We easily check that the Lie brackets of length 3 of these vectors are zero, that X 1, ⎡ X 2 ]] = [ ⎡ X 2, [ ⎡ X 1, ⎡ X 2 ]] = 0, and so the Lie algebra Lie( ⎡ X 1, ⎡ X 2 ) is is, [ ⎡ X 1, [ ⎡ nilpotent of step 2. The homogeneous nilpotent approximation is not uniquely defined by the m-tuple (X 1 , . . , X m ), since it depends on the chosen system of privileged coordinates. X m and ⎡ X 1, . . , ⎡ X m are the nilpotent approximations associated However, if ⎡ X 1, .
Yn−1 ), where each pol is a polynomial function without constant nor linear terms. The inverse change of coordinates takes the same triangular form, which makes the use of these coordinates easy for computations. 3 Nilpotent Approximation Fix a system of privileged coordinates (z 1 , . . , z n ) at p. Every vector field X i is of order ≥ −1, hence it has, in z coordinates, a Taylor expansion a∂, j z ∂ δz j , X i (z) ⊂ ∂, j where w(∂) ≥ w j − 1 if a∂, j = 0. Grouping together the monomial vector fields of same weighted degree we express X i as a series (−1) Xi = Xi (s) where X i (0) + Xi (1) + Xi + ··· , is a homogeneous vector field of degree s.