The approach developed for the nonholonomic integrator generalizes to
systems of the form
and
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 action variables. If is even, then the trajectory
for each variable is a sum of sinusoids at frequencies ,
, ,
. If is odd, there are
instead sinusoids; the sequence of frequencies remains the
same. Suppose is even (the odd case is similar). Each action is
selected as
|
(15.157) |
The other state variables evolve as
|
(15.158) |
which provides a constraint similar to (15.153). The
periodic behavior of these action trajectories causes the
variables to return to their original values while steering the
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:
- Apply any action trajectory that brings every to its goal
value. The evolution of the states is ignored in this stage.
- Apply sinusoids of integrally related frequencies to the action
variables. Choose each so that reaches its goal
value. In this stage, the 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:
in which is the set of all unique triples formed from distinct
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 ,
the second handles , and the third resolves . See
[727,846] for more details.
Steven M LaValle
2020-08-14