Time to travel between two states with acceleration and deceleration

[zero_sum]

Hi everyone :)

Starting off with a really simple question I'm seemingly too stupid to solve:

For a heuristic measurement I need to estimate the time it takes to get from one state (position and velocity) to another (different position and velocity). Neglecting things like speed limits, orientation, and rotational velocity (which I intend to bring in later) I came up with this basic first problem:

Given:
- a distance between two points (p0 and p1) in space, s.
- a velocity at p0, v0
- a velocity at p1, v1
- a maximum acceleration force a and a maximum deceleration force d

So from this, one can generate the time/velocity diagram in the attached file.
Now I'm looking for the time it takes to travel s, t1-t0, or, as t0 = 0, just t1.

I started off with splitting the area into an acceleration phase (duration tA) and a deceleration phase (duration tD), and stating that the velocities at the point when acceleration switches to deceleration must be equal, i.e. v0 + a*tA = v1 + d * tD, solving for tA.

So next, as I need one more variable, I tried inserting tA into the general distance/velocity/acceleration equation

s(t) = a/2 * t² + v0 * t

but then I lost it, somehow. I got a formula, but it generates different results if I swap (v0,a) and (v1,d) - though this should be symmetrical, as far as I understand...

Anyone there to enlighten me?
Thanks :)

 

[nmf77]

Are you trying to find the acceleration/deceleration profile that minimises the time of travel, or are there some other constraints perhaps?

 

[zero_sum]

Hi nmf77,

sooner or later, yes. Right now having acceleration and deceleration fixed at their maximum values will ensure the quickest traversal. So everything I listed as "given" is a constant.