# Particle moving at varying speed along a path

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 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:
$$L = \int_{0}^{t}\sigma(t) dt$$

I know the value of L. I know that my function will be some form of
$$\sigma_{1}(t) = 1/(1+e^{-t})$$
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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mfb
Mentor
s0 and s1 are given, and s1 is the maximum of the sigmoid function?
How much are you allowed to modify this function?

The maximal derivative of the sigmoid function can be calculated, and can be set equal to A. That will always reduce the number of unknowns by one.

At u=0, my speed is $$s_{0}$$ and by the time I get to the end of the curve (u=1), I may be traveling at my maximum speed $$s_{m}$$ or I may only get up to some speed $$s_{1} < s_{m}$$

I suppose this depends on the length of the curve, the maximum speed, the maximum (magnitude of) acceleration, etc.

I'm allowed to modify this curve but I would like to have the (magnitude of) acceleration curve (σ'(t)) be as symmetrical as possible.

Thanks!

mfb
Mentor
Well, you can write down the corresponding equations to those requirements, and then see how to solve the equations.