#### The exponential map

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 2012-04-20