14.6.3.1 Expressing systems in terms of $s$, ${\dot s}$, and ${\ddot s}$

Suppose that a system is given in the form

$${\ddot q}= h(q,{\dot q},u) ,$$ (14.31)

in which there are $n$ action variables $u = (u_1,\ldots,u_n)$. It may be helpful to glance ahead to Example 14.6, which will illustrate the coming concepts for the simple case of double integrators ${\ddot q} = u$. The acceleration in ${\cal C}$ is determined from the state $x = (q,{\dot q})$ and action $u$. Assume $u \in U$, in which $U$ is an $n$-dimensional subset of ${\mathbb{R}}^n$. If $h$ is nonsingular at $x$, then an $n$-dimensional set of possible accelerations arises from choices of $u \in U$. This means it is fully actuated. If there were fewer than $n$ action variables, then there would generally not be enough freedom to follow a specified path. Therefore, $U$ must be $n$-dimensional. Which choices of $u$, however, constrain the motion to follow the given path $\tau$? To determine this, the $q$, ${\dot q}$, and ${\ddot q}$ variables need to be related to the path domain $s$ and its first and second time derivatives ${\dot s}$ and ${\ddot s}$, respectively. This leads to a subset of $U$ that corresponds to actions that follow the path.

Suppose that $s$, ${\dot s}$, ${\ddot s}$, and a path $\tau$ are given. The configuration $q \in {\cal C}_{free}$ is

$$q = \tau(s).$$ (14.32)

Assume that all first and second derivatives of $\tau$ exist. The velocity ${\dot q}$ can be determined by the chain rule as

$${\dot q}= \frac{d\tau}{ds} \frac{ds}{dt} = \frac{d\tau}{ds} \; {\dot s},$$ (14.33)

in which the derivative $d\tau/ds$ is evaluated at $s$. The acceleration is obtained by taking another derivative, which yields

$$\begin{split}{\ddot q}& = \frac{d}{dt} \left( \frac{d\tau}{ds... ...s^2} \; {\dot s}^2 + \frac{d\tau}{ds} \; {\ddot s}, \end{split}$$ (14.34)

by application of the product rule. The full state $x = (q,{\dot q})$ can be recovered from $(s,{\dot s})$ using (14.32) and (14.33).

The next step is to obtain an equation that looks similar to (14.31), but is expressed in terms of $s$, ${\dot s}$, and ${\ddot s}$. A function $h'(s,{\dot s},u)$ can be obtained from $h(q,{\dot q},u)$ by substituting $\tau(s)$ for $q$ and the right side of (14.33) for ${\dot q}$:

$$h'(s,{\dot s},u) = h(\tau(s),\frac{d\tau}{ds}\; {\dot s},u) .$$ (14.35)

This yields

$${\ddot q}= h'(s,{\dot s},u) .$$ (14.36)

For a given state $x$ (which can be obtained from $s$ and ${\dot s}$), the set of accelerations that can be obtained by a choice of $u$ in (14.36) is the same as that for the original system in (14.31). The only difference is that $x$ is now constrained to a 2D subset of $X$, which are the states that can be reached by selecting values for $s$ and ${\dot s}$.

Applying (14.34) to the left side of (14.36) constrains the accelerations to cause motions that follow $\tau$. This yields

$$\frac{d^2\tau}{ds^2} \; {\dot s}^2 + \frac{d\tau}{ds} \; {\ddot s}= h'(s,{\dot s},u) ,$$ (14.37)

which can also be expressed as

$$\frac{d\tau}{ds} \; {\ddot s}= h'(s,{\dot s},u) - \frac{d^2\tau}{ds^2} \; {\dot s}^2 ,$$ (14.38)

by moving the first term of (14.34) to the right. Note that $n$ equations are actually represented in (14.38). For each $i$ in which $d\tau_i/ds \not = 0$, a constraint of the form

$${\ddot s}= \frac{1}{d\tau_i/ds} h'_i(s,{\dot s},u_i) - \frac{d^2\tau_i}{ds^2} \; {\dot s}^2$$ (14.39)

is obtained by solving for ${\ddot s}$.