I'm looking for an equation to find the time it will take for a position displacement to happen, given known $V_{max}$, a (known) constant acceleration, deceleration, jerk, and displacement… and an initial velocity of 0.
I am trying to estimate/calculate the time it will take for a servo motor axis to travel a certain distance. Once I know the estimated time of that move, I can solve for jerk for the other axis. That way, I choose an optimally "slow" speed for whichever axis has a shorter move distance to make the maximum use of my time.
FYI, this is a pick and place XY robot gantry.
Best Answer
I give you the result of my calculus without the details:
The result is :
$$ \Delta t = \frac{\Delta x}{V_{Max}} + \frac{1}{2}(\frac{V_{Max}}{a} + \frac{a}{j}) + + \frac{1}{2}(\frac{V_{Max}}{d} + \frac{d}{j})$$
where :
$\Delta t$ is the total time.
$\Delta x$ is the total displacement.
$a$ is the maximum acceleration.
$d$ is the maximum decceleration.
$j$ is the jerk.
$V_{max}$ is the maximum speed.
As a test, with your values :
$\Delta x = 2 m, a = 1m/s^2, d = 1m/s^2, j = 1 m/s^3, V_{max} = 1m/s$, I find :
$$ \Delta t = \frac{2}{1} + \frac{1}{2}(\frac{1}{1} + \frac{1}{1}) + + \frac{1}{2}(\frac{1}{1} + \frac{1}{1}) = 2 + 1 + 1 = 4$$
which is the correct result.
So I am quite confident in the formula.
[EDIT] The formula to obtain jerk is :
$$j = \frac{a + d}{2 (\Delta t - \large \frac{\Delta x}{\large V_{max}}) - V_{max}(\large \frac{1}{a} + \large\frac{1}{d})}$$
[EDIT 2]
The used model is :
Phase 1 : constant (positive) jerk $j$
Phase 2 : constant acceleration $a$
Phase 3 : constant (negative) jerk ($- j$)
Phase 4 : constant speed $V_{Max}$
Phase 5 : constant (negative) jerk ($- j$)
Phase 6 : constant decceleration ($d$)
Phase 7 : constant (positive) jerk ($j$)
In the formulas above, there are constraints, more precisely the duration of the phases 2, 4, 6 must be positive:
$$\Delta t_2 = \frac{V_{Max}}{a} - \frac{a}{j}\ge 0$$
$$\Delta t_4 = \frac{\Delta x}{V_{Max}} -\frac{1}{2}(\frac{V_{Max}}{a} + \frac{a}{j}) - \frac{1}{2}(\frac{V_{Max}}{d} + \frac{d}{j}) \ge 0$$
$$\Delta t_6 = \frac{V_{Max}}{d} - \frac{d}{j}\ge 0$$
If one of these constraints is not satisfyed, this means that the hypothesis taken for the model are incoherent, so we need another model.