Lecture 3: Holonomic Distributions

Differential Geometry

Manifold:

each point has a neighborhood homeomorphic to $ℝ^n$

Ex: union of lines $y = nx$: at each point $≠ (0,0)$, neighborhood homeomorphic to $ℝ$ ⟶ but not $(0, 0)$

Tangent space in $p$ (denoted by $T_p$):

linear space spanned by derivatives of $p_i$

Vector field:
X: p ⟼ X_p ∈ T_p
Distribution:

linear subspace of vector fields.

There is exactly one trajectory $γ$ going through a point $p$ and following a given vector field $X$ s.t.

γ(0) = p \\ \dot{γ}(t) = X_{γ(t)}

Notation: \exp(X) ≝ γ(1)

${\rm e}^{aX} \cdot p$:

starting from $p$, applying the vector field $aX$

Lie Brackets of Vector Fields

[X, Y] ≝ δXY - δYX
[X, Y] = -[Y, X]
Jacobi identity:
[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0
$i$-th coordinate:
[X, Y]_i = \sum\limits_{ j=1 }^n X_j\frac{δY_i}{δx_j} - Y_j \frac{δX_i}{δx_j}
\left[\begin{pmatrix} \cos θ \\ \sin θ \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\right] = \begin{pmatrix} \sin θ \\ - \cos θ\\ 0 \end{pmatrix}

Non holonomy degree: 2 (one spans all the space with this Lie bracket).

Campbell-Baker-Hausdorff-Dynkin Formula

{\rm e}^{tX} \cdot {\rm e}^{tY} = {\rm e}^{tX + tY - \frac 1 2 t^2 [X, Y] + t^2 ε(t)}

where $ε(t)$ is a formal series whose ceofficients are in the free Lee algebra $LA([X, Y])$.

Holonomic Distribution : Distribution integrable to a non-trivial manifold

Froebenius theorem:

$Δ_p$ is holonomic iff $rank(LA(Δ)_p) = rank(Δ_p)$

Maximally non-holonomic distribution:

distribution whihch does not reduce the dimension of the reachable space from $p$

Multibody car system

Multibody $(B_0, ⋯, B_n)$

  • $B_0$: car
  • $B_i$ ($i ≥ 1$): trailers
Placement of a body $B_i$:

$(x_i, y_i, θ_i)$

Distribution of the placement of all the bodies: $3(n+1)$

The “convoy” is defined by:
x_i - x_{i-1} = - \cos θ_i\\ y_i - y_{i-1} = - \sin θ_i

Each body is moving

\dot{z}_i \cos θ_i - \dot{y}_i \sin θ_i = 0

Ex:

\left[\begin{pmatrix} \cos θ \\ \sin θ \\ 0 \\ - \sin φ \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \\ 1 \end{pmatrix}\right] = \begin{pmatrix} \sin θ \\ - \cos θ\\ 0 \\ \cos φ \end{pmatrix}
\left[\begin{pmatrix} \cos θ \\ \sin θ \\ 0 \\ - \sin φ \end{pmatrix}, \begin{pmatrix} \sin θ \\ - \cos θ\\ 0 \\ \cos φ \end{pmatrix}\right] = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}

Non holonomy degree: 3 (one spans all the space with these two Lie brackets).

A Controllability Algorithm

Let $Δ$ be an nonholonomic distribution on a $n$-dimensional manifold and the filtration:

Δ_t = Δ_{t-1} + \sum\limits_{ j+k=t } [Δ_j, Δ_k]
Δ_0 ⊆ Δ_1 ⊆ ⋯
LA(Δ) = \bigcup_{n ∈ ℕ} Δ_n

Controllability theorem

The associated system is controllable at $c$ iff

∃ p_c; Δ_{p_c - 1} ≠ Δ_{p_c} = Δ_{p_c + 1} = ⋯ \text{ and } rank(Δ_{p_c}(c))=n

Philipp Hall family

Open problems

Canonical curves:

they keep the convoy angle (the relative angle between the trailer and the car) constant

⟶ Convex combinations between such curves, to reach the goal point

Flat systems:

a system such that there exists a subset of variables such that with these variables and their derivatives, on can determine all the other variables

The one third power law:

the velocity changes as a 1/3 power of the curvature

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