Example of integrability and controllability

As an example of controllability and integrability, recall the differential drive model. From the example in Section 8.1, the original vector fields are $\alpha^1(x) = [\cos \theta \;\; \sin \theta \;\; 0]$ and $\alpha^2(x) = [0 \;\; 0 \;\; 1]$.

Let ${\vec V}$ denote $\alpha^1(x)$, and let ${\vec W}$ denote $\alpha^2(x)$.To determine integrability and controllability, the first step is to compute the Lie bracket, ${\vec Z}= [{\vec V},{\vec W}]$. The components are

$$Z_1 = V_1 \frac{\partial W_1}{\partial x} - W_1 \frac{\part... ...heta} - W_3 \frac{\partial V_1}{\partial \theta} = \sin\theta ,$$

$$Z_2 = V_1 \frac{\partial W_2}{\partial x} - W_1 \frac{\part... ...eta} - W_3 \frac{\partial V_2}{\partial \theta} = -\cos\theta ,$$

and

$$Z_3 = V_1 \frac{\partial Y_3}{\partial x} - W_1 \frac{\part... ...artial \theta} - W_3 \frac{\partial V_2}{\partial \theta} = 0 .$$

The resulting vector field is ${\vec Z}= [ \sin\theta \;\; -\cos\theta \;\; 0]$.

We immediately observe that ${\vec Z}$ is linear independent from ${\vec V}$ and ${\vec W}$. This can be seen by noting that the determinant of the matrix

$$\pmatrix{ \cos\theta & \sin\theta & 0 \cr 0 & 0 & 1 \cr \sin\theta & -\cos\theta & 0 \cr }$$

in nonzero for all $(x,y,\theta)$. This implies that the dimension of $CLA(\triangle) = 3$. Using the Frobenius theorem, it can be inferred that the state transition equation is not integrable, and the system is nonholonomic. From Chow's theorem, it is known that the system is small-time controllable.