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

Suppose that a system is given in the form

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

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

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

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

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

\begin{displaymath}\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}\end{displaymath} (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}$:

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

This yields

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

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

which can also be expressed as

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

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

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

Steven M LaValle 2012-04-20