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A sub-problem of some velocity planning problems is to determine the velocity curve where for some k: $$ \frac{d^kv}{dt^k}=\text{constant}\tag{1} $$ However, the velocity should be solved in the form $v=v(s)$, rather than $v=v(t)$. Where $s$ is the arc length, so that $$ \frac{ds}{dt}=v\tag{2} $$

Therefore we have a change-of-variable problem for an ODE.

The Problem

Given ODE (1) and (2), I want to transform equation (1) from an equation of v(t) to an equation of v(s).

Here is what I have tried:
$$\begin{align} \frac{d^kv}{dt^k}&=(\frac{d}{dt})^kv\\ &=(\frac{ds}{dt}\frac{d}{ds})^kv \end{align} $$ Therefore, it is natural to consider a general formula for $(\frac{ds}{dt}\frac{d}{ds})^k$.

I've calculated formulas for the first few values of k: $$ (\frac{ds}{dt}\frac{d}{ds})^2=\frac{d^2s}{dt^2}\frac{d}{ds}+(\frac{ds}{dt})^2\frac{d^2}{ds^2} $$ $$ (\frac{ds}{dt}\frac{d}{ds})^3=\frac{d^3s}{dt^3}\frac{d}{ds}+3\frac{ds}{dt}\frac{d^s}{dt^2}\frac{d^2}{ds^2}+(\frac{ds}{dt})^3\frac{d^3}{ds^3} $$

These formulas looks a little like Equation (11) to (13) in this article. And I've also seen this article on stackexchange.

These are similar problems, but they have different answers. And one answer seems not giving me an insight for another.

Is there a general formula for my problem? Is there a general method for these problems? BTW, is there a software which can calculate these formulas automatically?

Thank you!

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up vote 2 down vote accepted

Maybe this rule can help you. See also this page.

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Ah..That is exactly what I want. Thank you! – Roun Apr 28 '12 at 14:24

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