Now suppose that a set , , of vector fields is given as a driftless control-affine system, as in (15.53). Its associated distribution is interpreted as a vector space with coefficients in , and the Lie bracket operation was given by (15.81). It can be verified that the Lie bracket operation in (15.81) satisfies the required axioms for a Lie algebra.
As observed in Examples 15.9 and 15.10, the Lie bracket may produce vector fields outside of . By defining the Lie algebra of to be all vector fields that can be obtained by applying Lie bracket operations, a potentially larger distribution is obtained. The Lie algebra can be expressed using the notation by including , , and all independent vector fields generated by Lie brackets. Note that no more than independent vector fields can possibly be produced.
(15.101) |
Let the system be
The first Lie bracket produces
(15.103) |
(15.104) |
(15.105) |
(15.106) |
(15.107) |
Steven M LaValle 2020-08-14