Returning to the system vector fields

Now the formal algebra concepts can be applied to the steering problem. The variables become the system vector fields: $y_i = h_i$ for all $i$ from $1$ to $m$. For the P. Hall basis elements, each $B_i$ becomes $b_i$. The Lie group becomes the state space $X$, and the Lie algebra is the familiar Lie algebra over the vector fields, which was introduced in Section 15.4.3. Consider how an element of the Lie group must evolve over time. This can be expressed using the differential equation

$$\dot{S}(t) = S(t)(v_1 b_1 + v_2 b_2 + \cdots + v_s b_s),$$ (15.128)

which is initialized with $S(0) = I$. Here, $S$ can be interpreted as a matrix, which may, for example, belong to $SE(3)$.

The solution at every time $t > 0$ can be written using the Chen-Fliess series, (15.127):

$$S(t) = e^{z_s(t) b_s} e^{z_{s-1}(t) b_{s-1}} \cdots e^{z_2(t) b_2} e^{z_1(t) b_1} .$$ (15.129)

This indicates that $S(t)$ can be obtained by integrating $b_1$ for time $z_1(t)$, followed by $b_2$ for time $z_2(t)$, and so on until $b_s$ is integrated for time $z_s(t)$. Note that the backward P. Hall coordinates now vary over time. If we determine how they evolve over time, then the differential equation in (15.128) is solved.

The next step is to figure out how the backward P. Hall coordinates evolve. Differentiating (15.129) with respect to time yields

$$\dot{S}(t) = \sum_{j=1}^s e^{z_s b_s} \cdots e^{z_{j+1} b_{j+1}} {\dot z}_j b_j e^{z_j b_j} \cdots e^{z_1 b_1} .$$ (15.130)