Monday 25 April 2011

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

        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"

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.

Monday 4 April 2011

Auto z-probe

A niggling problem I have with Hydraraptor is that the z-axis calibration varies with the weather and how much it is used. This is because the frame is made from wood, which absorbs atmospheric moisture and expands. When the machine is running constantly the heat from the bed dries it out and it plateaus at a low z-value. If I don't use it for a while the z-axis gets higher by as much as 0.5mm in wet weather and the first few builds need large adjustments. When printing raft-less the initial layer height needs to be accurate to about 0.05mm for 0.3mm layers.

When it was configured as a milling machine I made a tool height sensor to solve the problem. It doesn't work for FFF though because the nozzle usually has hot plastic dribbling from it and it also wastes some of the build area.

To solve the problem I designed a z-probe that hangs below the nozzle at the start of the build but then retracts itself after the measurement. It consists of a weighted metal rod that slides through a couple of plastic guides. It has a plastic flange on the top that depresses the plunger of a light action micro switch. In measurement mode the rod protrudes about 10mm below the nozzle. When the measurement is completed the axis descends to place the nozzle close to the bed. The rod lifts until the attractive force of two Neodymium magnets causes it to be pulled about 5mm above the nozzle and held there until the start of the next build.

Here it is installed on the axis.

I used a Meccano worm gear as an improvised weight to ensure the micro switch is activated, much cheaper options exist! The actual weight is surprisingly not very critical. It must be enough to activate the switch reliably but not too heavy for the magnets to lift.

The operating procedure is as follows: The machine warms up the bed and the extruder and waits for a couple of minutes for the nylon pillars that support the bed to expand fully. It then extrudes a length of filament with the z-axis at the top and gives an audio prompt on my computer. I grab the filament and snap it off and then lower the z-probe, which closes the switch and instructs the machine to start.

The axis descends rapidly to place the rod 1mm above the centre of the bed. It then descends in 0.1mm steps until the switch opens. Then it ascends in 0.01mm steps until the switch closes again and that gives the Z calibration point, which is a known distance (about 10mm) above the bed. The nozzle then descends to 1mm above the bed to retract the probe before it moves to the start point.

Here is a video of the sequence.

It could also lower the probe automatically simply by having a bracket near the top of the z-axis to catch the flange as the axis rises past it. The reason I don't do that at the moment is because I use the act of manually lowering the probe as a cue to the machine that I have removed the start extrusion.

The design is on Thingiverse.

Sunday 13 March 2011

Spot on flow rate

I have been doing some fine tuning of flow rate recently. I had previously noticed that PLA appears to need a slightly lower flow flow rate than ABS. I didn't notice this with HydraRaptor but I did when I changed from PLA to ABS on my Mendel, which has a Wade's extruder. My theory was that PLA feeds faster than ABS for the same rotational speed of the pinch wheel because, being much harder, it sits on the crests of the teeth and hence is driven by a larger effective pinch wheel diameter than ABS, which sinks in further. This effect is more extreme with a smaller pinch wheel. HydraRaptor has a 13mm pinch wheel compared to just 5mm for the hobbed bolt in my Wade's.

Other people have claimed that ABS changes density when it is extruded. I didn't believe that so I did an experiment to investigate.

I programmed HydraRaptor to extrude 100mm of ABS. I put a mark on the feedstock about 120mm away from the top of the extruder and measured how far the mark moved. I also measured the length and diameter of the extruded filament and I also weighed it and a 100mm sample of the feedstock. These are the results: -
Filament input to the extruder: 105mm of 2.98mm ABS equals 732mm3, weighs 0.777g, density 1.06 g/cm3.
Filament extruded: 2.89m at 0.56mm diameter equals 712mm3, weighed 0.764g, density 1.07 g/cm3.
So on the face of it the volume has gone down by 3% and the weight by 2% giving a slight increase in density. This could be explained by some volatile compounds boiling off, which they do, but I think it is mainly measurement error. In particular the diameter measurements have a big effect because of the square law for area. I took four measurements and averaged them but that is not many along 3m of extruded filament. Also the electronic scale I used to weigh the filament does not have a very stable display as it is only a cheap instrument. It is certainly a lot less than the 15% I have seen reported though.

I also extruded "100mm" of PLA and that actually fed 110mm, showing that with a 13mm pinch wheel it feeds about 5% faster. With a 5mm hobbed bolt I would expect that to be about 12%, which starts to become very noticeable.

So I corrected the pinch wheel diameter in my software for the correct value for ABS and added a bodge factor for PLA. That left the flow rate a bit too low as it has previously been producing good looking objects with the overfeed, so I reviewed the maths I was using.

I have always extruded filament with a 1.5:1 width over height ratio and use a flow rate that would fill a circle 1.25 times the layer height. That was because I originally observed that you need to squash the filament to 0.8 times its diameter to get a good bond and that makes the width about 1.5 times the height. However, that only gives a packing density of 82%, which is a bit low. If you increase the flow rate so the infill is 100% then the outlines will be too wide. This is because the infill can occupy the full rectangular cross section of the filament road, but the outline, being unconstrained, will not have straight sides, so will be wider.

I reasoned that the outline will be extruded with a flat top and bottom where it is constrained between the nozzle and the bed but the sides will most likely be semicircular due to surface tension effects. This led me to a formula that gives the width from the notional extrudate diameter and the layer height.

Equating the two areas gives πd⁄ 4 = πh⁄ 4 + h (w - h). So w = h + π(d⁄ h - h) ⁄ 4 allowing the width to be predicted from the layer height and the flow rate.

Calling the aspect ratio a = w ⁄ h and re-arranging to get the flow rate to make the desired width gives: d = h√(1+ 4(a - 1) ⁄ π). For an aspect ratio of 1.5 d = 1.28h. I had previously been using 1.25h which is about 5% too low but was compensated for by the pinch wheel overfeed. I made a single walled box with the corrected pinch wheel diameter and the new formula and verified that the walls were 1.5 times the layer height.

I also used the same flow rate for the infill, but that can be increased up to the full area of the rectangle w×h. Because the outline and infill use different flow rates there is a small deficit of plastic where they meet, as this model shows: -

This can be fixed by using the infill perimeter overlap ratio setting in Skienforge, but how much? The deficit in area is a rectangle h  ⁄ 2 × h minus a semicircle of diameter h, i.e. h⁄ 2- πh⁄ 8. If the infill overlaps by a distance x then it contributes an area x × h. Equating these gives x = h (0.5 -π/8).

Converting to a ratio of w gives x/w = (0.5 -π / 8) / a. For a = 1.5 that gives an overlap of 0.07 leading to a "fully stuffed" model where the solid layers are 100% plastic.

In practice that leaves no room for error and requires the nozzle to force the plastic into the corners of the rectangular channels like an injection molding machine. I found I get a better looking object with the volume reduced to 90% of that value. So for the infill I use the formula d = h√(0.9 × 4a  ⁄ π) giving d = 1.31h for a = 1.5, making the optimum flow rate for the infill about 5% more than the outline. I also use an overlap value of 0.05 giving the theoretical packing arrangement below.

Running the new equations on my Mendel certainly produces nice looking objects:

At least four people I have sold parts to have commented they look as good or better than parts they have seen from a commercial machine. I use filament about twice the diameter that commercial machines use, which results in more visible layers and rounded corners, etc, but apart from that I must be close now.