By Frédéric Jean

Nonholonomic platforms are keep watch over platforms which rely 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 platforms. specifically the standard notions of approximations on the first order, which are crucial for keep watch over reasons, must be outlined by way of this geometry. the purpose of those notes is to give those notions of approximation and their program to the movement making plans challenge for nonholonomic systems.

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**Extra resources for Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning**

**Sample text**

Let K ∀ M be a compact set and rmax be the maximum of the degree of nonholonomy on K (as noticed in Sect. 2 rmax is finite). We assume that M is an oriented manifold with a volume form Ω. Let X be the set of n-tuples X = (X I1 , . . , X In ) of brackets of length |Ii | → rmax . It is a finite subset of Lie(X 1 , . . , X m )n . Given q ∈ K and ε > 0 we define a function f q,ε : X ∗ R by ⎪ ⎨ f q,ε (X) = Ωq X I1 (q)ε|I1 | , . . , X In (q)ε|In | . We say that X ∈ X is an adapted frame at (q, ε) if it achieves the maximum of f q,ε on X.

3 Set ⎡ X i = X i , i = 1, . . , m. The family of vector fields ⎡ ⎡ ( X 1 , . . , X m ) is a first-order approximation of (X 1 , . . , X m ) at p and generates a nilpotent Lie algebra of step r = wn . Recall that a Lie algebra Lie(X 1 , . . , X m ) is said to be nilpotent of step s if all brackets X I of length |I | greater than s are zero. Proof The fact that the vector fields ⎡ X 1, . . , ⎡ X m form a first-order approximation of X 1 , . . , X m results from their construction. Note further that any homogeneous vector field of degree smaller than −wn is zero, as it is easy to check in privileged coordinates.

M) be the free Lie algebra generated by {1, . . , m}. We use L s to denote the subspace generated by elements of L of length not greater than s, and n s to denote the dimension of L s . 13 Let ξ1 , . . , ξm 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 , .