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

(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]:

(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

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

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