The steering problem will be solved by performing calculations on
. The formal power series of
is the set of all linear combinations of
monomials, including those that have an infinite number of terms.
Similarly, the formal Lie series of
can be
defined.
The formal exponential map is defined for any
as
|
(15.116) |
In the nilpotent case, the formal exponential map is defined
for any
as
|
(15.117) |
The formal series is truncated because all terms with exponents larger
than vanish.
A formal Lie group is constructed as
|
(15.118) |
If the formal Lie algebra is not nilpotent, then a formal Lie group
can be defined as the set of all , in which
is represented using a formal Lie series.
The following example is taken from [574]:
Example 15..22 (Formal Lie Groups)
Suppose that the generators
and
are given. Some elements of
the formal Lie group
are
|
(15.119) |
|
(15.120) |
and
|
(15.121) |
in which
is the formal Lie group identity.
Some elements of the formal Lie group
are
|
(15.122) |
|
(15.123) |
and
|
(15.124) |
To be a group, the axioms given in Section 4.2.1
must be satisfied. The identity is , and associativity clearly
follows from the series representations. Each has an inverse,
, because
. The only remaining axiom to
satisfy is closure. This is given by the
Campbell-Baker-Hausdorff-Dynkin formula (or CBHD formula),
for which the first terms for any
are
|
(15.125) |
in which
alternatively denotes for any . The
formula also applies to
, but it becomes truncated
into a finite series. This fact will be utilized later. Note that
, which differs from the standard definition
of exponentiation.
The CBHD formula is often expressed as
|
(15.126) |
in which
, and
.
The operator
provides a compact way to express some nested Lie
bracket operations. Additional terms of (15.125) can be
obtained using (15.126).
Steven M LaValle
2020-08-14