Calculating parameters of motion profile

[aand74]

I would like to refer to this document of which I used some data :
https://www.researchgate.net/profil...THIRD-ORDER-POINT-TO-POINT-MOTION-PROFILE.pdf


In my program I have to start motions at a certain moment and the motion has to be at a certain positon at a certain moment.
The motion can be parametrised with maximum speed, maximum acceleration and jerk.
The motion is third order as you see in the timechart of the document. Using the calculations from the document, my program works ok. But I must say up to now I made sure the maximum acceleration was Always reached, so the acceleration cuve looks like the trapezium in the document. So, I used the tj, ta and tv calculations from case 'V'.
1) What if the maximum acceleration is not reached, and the acceleration profile will be a triangle and not a trapezium, which case (I to VI) holds the right calculations for tj, ta and tv?
2)If you have proposed parameters for maximum acceleration, jerk and maximum velocity, what is the easiest, fastest way to determine if the maximum acceleration will be reached?

 

[Peter Nachtwey]

Welcome to the problem from hell. Read this

http://www.plctalk.net/qanda/showthread.php?t=119883


Cheeco solved this problem but in e-mails he claimed he got so frustrated that he put the problem down for a year. I have spent more time on it than he has. I posted this problem on sci.math.symbolic many years ago. We were able to solve it but the solution was so long it stretched over 180 pages. The solution wasn't usable. Later I manually optimized that solution. A few substitutions made the problem much smaller. There was a professor at Fordham university that has a program called Fermat that he used to solve the problem and it was a little shorter. I have used Mathcad, wxMaxima and Mathematica. This problem looks easy but it quite difficult. This professor used this problem in one of his presentations. I have kept his e-mail.



Obviously Cheeco and I have solved this problem but we are not going to share the solution. It was too damn difficult to solve. I don't know if Cheeco can solve the problem in real time on-the-fly. The motion controller must be able to go from any initial position, velocity, acceleration to and final position, velocity and acceleration optimally. I have not found any motion controllers or PLC that solve this problem correctly. Most controllers have a solution that works but it is not optimal.



This is what separates the very good from the best.


I need a replacement. I am going to retire soon. I could have retired long ago but...