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Suppose I have a arbitrary path X(u) for u∈[0,1] of length L.

I want to traverse the path at a variable speed (I'm only really concerned about the magnitude of velocity). I want the graph of my speed function σ(t) to have the shape of a

From basic calculus and physics I know that:

[tex] L = \int_{0}^{t}\sigma(t) dt[/tex]

I know the value of L. I know that my function will be some form of

[tex]\sigma_{1}(t) = 1/(1+e^{-t})[/tex]

There's a lot that I don't know unfortunately.

Is there a way to build my function σ(t) other than guessing? If not, how to best go about making intelligent guesses so that I eventually converge to my desired function?

Any suggestions welcome. Thanks!

I want to traverse the path at a variable speed (I'm only really concerned about the magnitude of velocity). I want the graph of my speed function σ(t) to have the shape of a

**sigmoid**and I want it to start at an arbitrary value s0 and be bounded by an arbitrary value s1. There's a bound on the maximum magnitude of acceleration equal to A. I want to get to my maximum speed or s1 as fast as possible while not going over the acceleration limit.From basic calculus and physics I know that:

[tex] L = \int_{0}^{t}\sigma(t) dt[/tex]

I know the value of L. I know that my function will be some form of

[tex]\sigma_{1}(t) = 1/(1+e^{-t})[/tex]

There's a lot that I don't know unfortunately.

Is there a way to build my function σ(t) other than guessing? If not, how to best go about making intelligent guesses so that I eventually converge to my desired function?

Any suggestions welcome. Thanks!

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