Auto bed leveling

One thing I find tedious is leveling the bed of my machines, so I decided to make use of the Z-probe on HydraRaptor to measure the incline of the bed and compensate for it in software.

First I had to increase the resolution of my Z axis. When I first built the machine I did not realise that I would need fine resolution on Z, so I used an old 24V unipolar motor that I had in my junk collection. With half stepping it gave me 0.05mm resolution. I thought that it was a 200 step motor and that the lead screw had a pitch of 20mm. It turns out it must have been a 250 step motor because the pitch is actually 25mm. I replaced it with a Keling KL23H251-24-8B NEMA23 left over from my Darwin and I now drive it with a x10 microstepping controller the same as I use on X and Y. That gives me a resolution of 0.0125mm and also makes it the fastest axis on my machine. It can easily do 150 mm/s but seeing the nozzle approaching the bed at that speed and then stopping within 0.3mm is very unnerving, so I limit the speed to 50mm/s!

I no longer need the heat sink and fan because the new motor is more efficient and is directly mounted on the axis, which takes the heat away.

I use 6mm aluminium tooling plate so I make the assumption that the bed is a flat plane (rotated slightly around the X and Y axes and offset a little in Z). That means I only need to measure the height at three arbitrary points in order to characterise that plane. I then use the method here to calculate the equation of the plane in the form ax + by +cz +d = 0. The method puts two vectors through the three points and takes the cross product to get a vector at right angles to both of them. That is the normal to the plane and its components are the coefficients a, b and c. Substituting the first point into the equation gives d.

It is important that the three points are ordered anti-clockwise, otherwise the normal vector would point downwards and the machine would try to build the object upside down under the surface of the bed!

Given the bed's plane I then have to make the model coordinates relative to that inclined plane and transform them to the coordinate system of the machine. To do that I calculate two orthonormal basis vectors on the plane using it's equation and use the normalised normal vector for the third. I then multiply the model coordinates by those vectors and add the origin to find where they are in the machine's coordinates. Here is the Python code I used: -

class Plane:
    "A plane in 3D."

    def __init__(self, p0, p1, p2):
        "Construct from three anti-clockwise points"
        #
        # Calcluate the normal vector
        #
        v1 = p1.minus(p0)
        v2 = p2.minus(p0)
        normal = v1.cross(v2)
        if normal.z < 0:
            raise Exception, "Probe points must be anti-clockwise"
        #
        # Coefficients of the plane equation ax + by + cz + d = 0
        #
        a = normal.x
        b = normal.y
        c = normal.z
        d = -a * p0.x -b * p0.y -c * p0.z
        #
        # Generate three basis vectors aligned with the plane
        #
        self.origin = vector( 0, 0, -d / c)

        self.k = normal                      # k axis is simply the normalised normal
        self.k.normalize()

        px = vector( 1.0, 0.0, -(a + d) / c) # an arbitrary point on the x axis: x = 1, y = 0
        self.i = px.minus(self.origin)       # find direction to it from origin
        self.i.normalize()                   # make a unit vector

        self.j = self.k.cross(self.i)        # make a third vector mutually at right angles to the other two
        self.j.normalize()                   # make a unit vector, probably is already

    def transform(self, p):
        "Transform a point to be relative to the plane"
        return self.origin.plus(self.i.times(p.x)).plus(self.j.times(p.y)).plus(self.k.times(p.z))

To test the principal I put a 1mm thick washer under one corner support of the bed to give it an extreme slant compared to normal misalignment. I then built a 100 x 100 x 5mm cube with 0.35mm layers. This would normally be impossible without the bed being level to a small proportion of the layer height. The result was that it came out fine.

As the nozzle traverses the object in XY the Z axis moves a few microsteps. It is barely visible but I can hear it and feel it if I hold the stepper motor shaft. The object is built perpendicular to the plane of the bed, so the sides are very slightly slanted with respect to the machine axis and the nozzle. I am not sure how well it would work on Mendel as the z-axis is geared down so much. It would probably still work as the movement required is so small when the bed is reasonably level. I can't test it as there isn't room for a z-probe on my carriage due to the large heat sink.