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) |

This uses the Lie bracket that was computed in (15.82) to obtain a three-dimensional Lie algebra. No further Lie brackets are needed because the maximum number of independent vector fields has been already obtained.

Let the system be

This is a

The first Lie bracket produces

(15.103) |

Other vector fields that can be obtained by Lie brackets are

(15.104) |

and

(15.105) |

The resulting five vector fields are independent over all . This includes , , and the three obtained from Lie bracket operations. Independence can be established by placing them into a matrix,

(15.106) |

which has full rank for all . No additional vector fields can possibly be independent. Therefore, the five-dimensional Lie algebra is

(15.107) |

Steven M LaValle 2012-04-20