The Lie Bracket

The Lie bracket attempts to generate velocities that are not directly permitted by the state transition equation. For the car-like robot, it will produce a vector field that can move the car sideways (it is achieved through combinations of vector fields, and therefore does not violate the nonholonomic constraint). This operation is called the Lie bracket (pronounced as ``Lee''), and for given vector fields ${\vec V}$ and ${\vec W}$, it is denote by $[{\vec V},{\vec W}]$. The Lie bracket is computed by

$$[{\vec V},{\vec W}] = D{\vec W}\cdot {\vec V}- D{\vec V}\cdot {\vec W}$$

in which $\cdot$ denotes a matrix-vector multiplication,

$$D{\vec V}= \pmatrix{ \displaystyle\strut\frac{\partial V_1}... ... \displaystyle\strut\frac{\partial V_n}{\partial x_n} \cr } ,$$

and

$$D{\vec W}= \pmatrix{ \displaystyle\strut\frac{\partial W_1}... ... \displaystyle\strut\frac{\partial W_n}{\partial x_n} \cr } .$$

In the expressions above, V_i and W_i denote the i^th components of ${\vec V}$ and ${\vec W}$, respectively.

It is sometimes convenient for computation of the Lie bracket to directly use the expression for each component of the new vector field (obtained by performing the multiplication indicated above). The i^th component of the Lie bracket is given by

$$\displaystyle\strut\sum_{j=1}^{n} \left( V_j \frac{\partial ... ...partial x_j} - W_j \frac{\partial V_i}{\partial x_j} \right) .$$

Two well-known properties of the Lie bracket are:

1.

(skew-symmetry) $[{\vec V},{\vec W}] = - [{\vec W},{\vec V}]$ for any two vector fields, ${\vec V}$ and ${\vec W}$

2.

(Jacobi identity) $[[{\vec V},{\vec W}],{\vec U}] + [[{\vec W},{\vec U}],{\vec V}] + [[{\vec U},{\vec V}],{\vec W}] = 0$