Summer CD - Kinematics problem

[efoote868]

Got a difficult math calculus / differential equation problem for you to chew on.

Hypothetically, (Problem background) I’ve got a customer that wants line segments of glue applied to a surface with a constant width. When the glue machine dispenses, it dispenses at a constant rate (either on or off). The glue nozzle is on a gantry that can be commanded to move linearly from one coordinate to another. The gantry can be tuned to operate with S-Curve parameters: A maximum velocity (V), a maximum acceleration (A), and a maximum jerk (J - derivative of acceleration).

Question - what’s the distance D required for the gantry to travel to ramp up to its max velocity from stop, as a function of maximum jerk, maximum acceleration, and maximum velocity?
Hint: There are two closed form solutions, Under what conditions will the gantry reach maximum acceleration?

[efoote868]

Case 1: Full Acceleration is reached
Case 2: Full Acceleration is not reached (“Jerk / Velocity limited”)

Two things that were amazing to me. 1st was how the equation defining t2 to t3 blew up. And 2nd was that plugging t3 into the distance formula simplified so nicely.

Solution

Case 1: D(J,A,V) = [ VA / 2J ] + [ [V^2] / 2A ]
Case 2: D(J,A,V) = Sqrt[ [V^3] / J ]