First-order controllable systems

The approach developed for the nonholonomic integrator generalizes to systems of the form

$\displaystyle {\dot x}_i$ $\displaystyle = u_i$ $\displaystyle \qquad$ $\displaystyle \mbox{for $i$ from $1$ to $m$}$    
$\displaystyle {\dot x}_{ij}$ $\displaystyle = x_i u_j - x_j u_i$ $\displaystyle \qquad$ $\displaystyle \mbox{for all $i,j$ so that $i < j$ and $1 \leq j \leq m$}$ (15.155)


$\displaystyle {\dot x}_i$ $\displaystyle = u_i$ $\displaystyle \qquad$ $\displaystyle \mbox{for $i$ from $1$ to $m$}$    
$\displaystyle {\dot x}_{ij}$ $\displaystyle = x_i u_j$ $\displaystyle \qquad$ $\displaystyle \mbox{for all $i,j$ such that $i < j$ and $1 \leq j \leq m$}$$\displaystyle .$ (15.156)

Brockett showed in [142] that for such first-order controllable systems, the optimal action trajectory is obtained by applying a sum of sinusoids with integrally related frequencies for each of the $ m$ action variables. If $ m$ is even, then the trajectory for each variable is a sum of $ m/2$ sinusoids at frequencies $ 2 \pi$, $ 2 \cdot 2\pi$, $ \ldots $, $ (m/2)\cdot 2\pi$. If $ m$ is odd, there are instead $ (m-1)/2$ sinusoids; the sequence of frequencies remains the same. Suppose $ m$ is even (the odd case is similar). Each action is selected as

$\displaystyle u_i = \sum_{k=1}^{m/2} a_{ik} \sin 2 \pi k t + b_{ik} \cos 2 \pi k t .$ (15.157)

The other state variables evolve as

$\displaystyle x_{ij} = x_{ij}(0) + \frac{1}{2} \sum_{k=1}^{m/2} \frac{1}{k} (a_{jk} b_{ik} - a_{ik} b_{jk}) ,$ (15.158)

which provides a constraint similar to (15.153). The periodic behavior of these action trajectories causes the $ x_i$ variables to return to their original values while steering the $ x_{ij}$ to their desired values. In a sense this is a vector-based generalization in which the scalar case was the nonholonomic integrator.

Once again, a two-phase steering approach is obtained:

  1. Apply any action trajectory that brings every $ x_i$ to its goal value. The evolution of the $ x_{ij}$ states is ignored in this stage.
  2. Apply sinusoids of integrally related frequencies to the action variables. Choose each $ u_i(0)$ so that $ x_{ij}$ reaches its goal value. In this stage, the $ x_i$ variables are ignored because they will return to their values obtained in the first stage.

This method has been generalized even further to second-order controllable systems:

$\displaystyle {\dot x}_i$ $\displaystyle = u_i$ $\displaystyle \qquad$ $\displaystyle \mbox{for $i$ from $1$ to $m$}$    
$\displaystyle {\dot x}_{ij}$ $\displaystyle = x_i u_j$ $\displaystyle \qquad$ $\displaystyle \mbox{for all $i,j$ such that $i < j$ and $1 \leq j \leq m$}$ (15.159)
$\displaystyle {\dot x}_{ijk}$ $\displaystyle = x_{ij} u_k$ $\displaystyle \qquad$ $\displaystyle \mbox{for all $(i,j,k) \in J$,}$    

in which $ J$ is the set of all unique triples formed from distinct $ i,j,k \in \{1,\ldots,m\}$ and removing unnecessary permutations due to the Jacobi identity for Lie brackets. For this problem, a three-phase steering method can be developed by using ideas similar to the first-order controllable case. The first phase determines $ x_i$, the second handles $ x_{ij}$, and the third resolves $ x_{ijk}$. See [727,846] for more details.

Steven M LaValle 2012-04-20