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Trying to solve a problem for the acceleration of an automated shuttle car at my work, been a while since I studied this stuff so thought I'd reach out for help.

I have a shuttle car that is tasked with moving a certain distance to pick up a load. I need to calculate the time that this movement will take. The complication is that there are four different accelerations the shuttle will use for different speeds.

Initially, the shuttle accelerates at $a_1$ for some time until it reaches a velocity threshold $v_1$. Once it reaches $v_1$, it begins accelerating at $a_2$ until it reaches speed $v_2$. It continues in this fashion until it reaches max speed $v_4$ (although, for some shorter distances it may never reach some of these thresholds). Deceleration happens in the same fashion.

So to summarize, given a distance and a table of acceleration constants (which depend on velocity), calculate the time required to move said distance. Any and all help much appreciated.

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  • $\begingroup$ Basically you need to calculate a sum $$2\sum_i \frac {\sqrt{2 a_i d_i + v_i^2} - v_i}{a_i}.$$ Factor $2$ before sum is accounting for decelerations. $\endgroup$
    – Agnius Vasiliauskas
    Commented Aug 8 at 10:38

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If you are familiar with VLOOKUP in Excel, I think that would be ideal.

Basically you would create a $dt$ column where $dt$ is the time interval your robot uses to adjust velocity. Maybe 0.1 s? Generically you could just choose a smallish number like 0.1 s, but it's best to use the actual robot parameter.

The next columns would be updating the equations:

$$v = v_{previous}+adt$$

$$s=\frac 1 2 a dt^2 + vdt + s_{previous}$$

Your value for $a$ will have to be chosen via VLOOKUP (or equivalent) based on your current velocity.

The $s$ column will be your position at a given time $t=n\,dt$.

Keep in mind, you haven't discussed how the robot knows when to start slowing down. When it reaches 1/2 its destination? This can have a big effect on any answer. Does it use active feedback to determine when to apply gas/brakes? If so the previous logic might be totally invalid.

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  • $\begingroup$ Thanks very much for your answer. As you say, slowing down is a bit of a problem. The shuttle will dynamically calculate the correct distance to begin decelerating at based on its speed and the given acceleration parameters. But we are unsure how to make this calculation! $\endgroup$
    – potassiumcyanide
    Commented Oct 26, 2022 at 4:26
  • $\begingroup$ I should probably also clarify that we are only interested in the total time taken for the movement. Ideally we would like to pass a single function the acceleration parameters and the move distance and get out the total time. $\endgroup$
    – potassiumcyanide
    Commented Oct 26, 2022 at 4:40

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