algorithmgeometrylinerasterizingcnc

Algorithm for evenly arranging steps in 2 directions

I am currently programming the controller for a CNC machine, and therefore I need to get the amount of stepper motor steps in each direction when I get from point A to B. For example point A's coordinates are x=0 and y=0 and B's coordinates are x=15 and y=3. So I have to go 15 steps on the x axis and 3 und the y axis. But how do I get those two values mixed up in a way that is smooth (aka not first x and then y, this results in really ugly lines)? In my example with x=15 and y=3 I want it arranged like that:

for 3 times do:    
    x:4 steps  y:0 steps
    x:1 steps  y:1 step

But how can I get these numbers from an algorithm? I hope you get what my problem is, thanks for your time, Luca

Solution

  • there are 2 major issues in here:

    1. trajectory

      this can be handled by any interpolation/rasterization like:

      • DDA
      • Bresenham

      the DDA is your best option as it can handle any number of dimensions easily and can be computed on both integer and floating arithmetics. Its also faster (was not true in the x386 days but nowadays CPU architecture changed all)

      and even if you got just 2D machine the interpolation itself will be most likely multidimensional as you will probably add another stuff like: holding force, tool rpm, preasures for what ever, etc... That has to be interpolated along your line in the same way.

    2. speed

      This one is much much more complicated. You need to drive your motors smoothly from start position to the end concerning with these:

      • line start/end speeds so you can smoothly connect more lines together
      • top speed (dependent on the manufactoring process usually constant for each tool)
      • motor/mechanics resonance
      • motor speed limits: start/stop and top

      When writing about speed I mean frequency [Hz] for the steps of the motor or physical speed of the tool [m/s] or [mm/2].

      Linear interpolation is not good for this I am using cubics instead as they can be smoothly connected and provide good shape for the speed change. See:

      The interpolation cubic (form of CATMUL ROM) is exactly what I use for tasks like this (and I derived it for this very purpose)

      The main problem is the startup of the motor. You need to drive from 0 Hz to some frequency but usual stepping motor has resonance in the lower frequencies and as they can not be avoided for multidimensional machines you need to spend as small time in such frequencies as possible. There are also another means of handling this shifting resonance of kinematics by adding weights or change of shape, and adding inertial dampeners on the motors itself (rotary motors only)

      So usual speed control for single start/stop line looks like this:

      speed

      So you should have 2 cubics one per start up and one per stopping dividing your line into 2 joined ones. You have to do it so start and stop frequency is configurable ...

      Now how to merge speed and time? I am using discrete non linear time for this:

      its the same process but instead of time there is angle. The frequency of sinwave is changing linearly so that part you need to change with the cubic. Also You have not a sinwave so instead of that use the resulting time as interpolation parameter for DDA ... or compare it with time of next step and if bigger or equal do step and compute the next one ...

      Here another example of this technique:

      This one actually does exactly what you should be doing ... interpolate DDA with Speed controled by cubic curve.

    When done you need to build another layer on top of this which will configure the speeds for each line of trajectory so the result is as fast as possible and matching your machine speed limits and also matching tool speed if possible. This part is the most complicated one...

    In order to show you what is ahead of you when I put all this together mine CNC interpolator has ~166KByte of pure C++ code not counting depending libs like vector math, dynamic lists, communication etc... The whole control code is ~2.2 MByte