Controlling 3 stepper to move on xyz direction

Question

I am working on coding CNC. I can move x and y axes simultaneously to draw lines (Bresenham's algorithm). Now I want to add a 3rd axis but I do not have any idea how to move from (x0,y0,z0) to (x1,y1,z1).

Is there any algorithm for that purpose?

Answer 1

Use the Bresenham's again.

Let DX = X1 - X0; DY = Y1 - Y0; DZ = Y1 - Y0; ... DЪ = Ъ1 - Ъ0;

Let Dmax = max(|DX|, |DY|, |DX|, ..., |DЪ|); where |x| = modulus of x

Let's init X = X0; Y = Y0; Z = Z0; ... Ъ = Ъ0;

Let's init EX = 0; EY = 0; EZ = 0; .... EЪ = 0;

for i := 1 to Dmax do 
  EX = EX + |DX|;
  if EX >= Dmax then
    EX = EX -Dmax;
    X = X + sign(DX);
    MOTOR_X_ONE_STEP(DX > 0 ? FORWARD : BACKWARD);
  end if 

  EY = EY + |DY|;
  if EY >= Dmax then
    EY = EY -Dmax;
    Y = Y + sign(DY);
    MOTOR_Y_ONE_STEP(DY > 0 ? FORWARD : BACKWARD);
  end if 

  EZ = EZ + |DZ|;
  if EZ >= Dmax then
    EZ = EZ -Dmax;
    Z = Z + sign(DZ);
    MOTOR_Z_ONE_STEP(DZ > 0 ? FORWARD : BACKWARD);
  end if 

  ...

  EЪ = EЪ + |DЪ|;
  if EЪ >= Dmax then
    EЪ = EЪ -Dmax;
    Ъ = Ъ + sign(DЪ);
    MOTOR_Ъ_ONE_STEP(DЪ > 0 ? FORWARD : BACKWARD);
  end if 

  // Now we are in the point (X, Y, Z, ..., Ъ);
end for

where sign(x) = (x < 0) ? -1 : 1;

doesn't matter how many axis you have, Bresenham's algorithm still the same for all of them.

As you can see the axis, which have the maximal |Dn| will increase each step.

By the way. You can make the line looks better, if init all En values with half of Dmax:

EX = Dmax / 2; EY = Dmax / 2; EZ = Dmax / 2; .... EЪ = Dmax /2;