## March 31, 2017

### Encoder Autocalibration for Brushless Motors: Offset and Eccentricity

I finally got around to writing a position sensor auto-calibration procedure, for measuring the orientation of the position sensor relative to the stator.  Until now, I've been manually measuring that position by setting my U phase high, and V and W low to, lock the rotor to the D-axis, looking at the position sensor reading about a few of these points, and hard-coding them into my firmware.

This process should work regardless of the order the motor phases are plugged in to the controller, or how the position sensor is initially oriented.

Step 1:  Determine Phase Ordering

The purpose of this step is so that commanding positive current on the q-axis produces torque in the direction that causes the encoder angle to increase.  Basically, the process is to apply a large, slowly rotating current to a "virtual" D-axis.  The rotor will closely follow this rotation.  This is basically equivalent to driving the motor like a stepper motor with microstepping.  As the current rotates around, if the encoder count is increasing in the positive direction, then everything's good.  If the encoder count is decreasing, then swap the voltage outputs and current sensor inputs on two phases.  Now the motor will rotate the correct direction.

In pseudo-code:

v_d = 1;                                                            // Volts on the D-Axis
v_q = 0;
reference_angle = 0;
start_angle = encoder.GetPosition();                                //starting position
while(reference_angle < 2*Pi){
[v_u, v_v, v_w] = abc_transform(reference_angle, v_d, v_q, 0)   //inverse dq0 transform to get phase voltages
wait();                                                         //give the rotor some time to settle into position
reference_angle += .001;
}
end_angle = encoder.GetPosition();                                  // final position
if(start_angle - end_angle > 0){                                    // if position decreased, swap phases
swap_phases();
}


Step 2:  Measure Encoder Offset

Now that the motor spins the right direction, you can measure the DC offset of the encoder.  Just like before, apply volts to the  D axis, and slowly rotate the axis through a whole mechanical rotation both backwards and forwards.

Looking at the position sensor output vs the reference angle, you'll see something like this.

Zooming in a bit, you can see some ripple in the tracking from cogging torque:

Looking at the error between the reference angle and the actual rotor angle, things get a little bit more confusing.  The error plot below has a few interesting features.

First, and most obviously, the mechanical offset of the encoder is just the average value of the error.  The electrical offset (which is what we actually care about for commutation) is just mod((mechanical error) * (number of pole pairs) , 2π).

The high frequency ripple (which in this zoomed-out view looks almost like noise), is from the cogging torque (the ripple that showed up in the red trace above).

Right in the middle of the plot, there's a jump downwards.  This is where the motor switches direction.  Since the motor has friction and inertia, it always lags slightly behind the reference angle, so that lag switches sides when the motor switches directions.

Then there's some low-frequency, much larger ripple on top of the signal.  This is from eccentricity of the position sensing IC/its magnet.  In my case, it's because I accidentally messed up the design of a PCB and placed the IC 0.5 mm off to one side.  The total peak-to-peak height of this ripple is only around 0.03 radians, or 1.7 degrees, which doesn't sound like a lot.  However, I'm using a 21 pole-pair motor, which means that 1.7 degrees of mechanical error translates to 36 degrees of electrical error.  Enough to seriously throw off commutation.  The next step will go through how I corrected for the error from eccentricity.  While this won't be nearly as much of a problem once I get the new round of boards in, there will always be a little error in the placement of the IC (especially when they're hand-soldered by me), so having this feature is still definitely useful.

Step 3:  Eccentricity Correction

The first trick for correcting for the eccentricity is separating out the error from cogging and friction from the eccentricity error.

Friction is easy.  Just average the samples rotating forwards with the samples rotating backwards.  Then plotting error you have something like this

Removing the ripple from cogging torque took some more thought, but it turns out can be done extremely effectively.  My initial though was to just low-pass filter the signal, and hope that cogging could be sufficiently attenuated without changing the eccentricity part of the signal too much.  Honestly, for this particular motor with lots of pole-pairs and many cogging steps per electrical cycle, that would have probably worked just fine.  But you can do better, and make the result more easily applicable to other motors.

The trick is to take advantage some properties of motors and FIR filters.  Since the motor is rotationally symmetric, the frequency of the cogging ripple must be some integer multiple of the electrical frequency.  FIR filters can be designed such that they have zero gain at certain frequencies and harmonics of those frequencies.  So you can easily design a filter that has zero gain at the electrical frequency, and all multiples of it, nearly perfectly cancelling out the cogging ripple without affecting lower frequencies.  I'm using basically the simplest possible FIR filter - I just average n samples around the point of interest, and choose n such that the samples I'm averaging exactly span one electrical cycle.  This should work pretty well on any motor, as long as the number of pole pairs is larger than the highest harmonic from eccentricity.  Here's what the the filtered version of the signal looks like:

Now, with that filtered signal, you can build a lookup-table to correct the encoder output.  It doesn't even need to be particularly high-resolution, since the signal varies slowly - I'm using 128 points, and it works great.  The lookup table is generated by subtracting the average value off of the error, and then converting the error back into raw encoder counts.  I'm cheating a little bit and using the reference position as the x-axis, rather than the actual encoder value, because it's nice and evenly spaced into a grid already.  This approximation works quite well though.

Here are three lookup tables generated independently, showing the consistency of this technique:

My latest round of motor control firmware can be found here, if you want to look through the autocalibration code.  It's fairly unpleasant to read, but it is well commented.  The code as a whole is very much a work in progress, so don't expect to put it on anything and have it work out-of-the-box.

At some point I'll throw a high-resolution optical encoder on the output to quantify the improvement in position error, but in terms of torque ripple, this process qualitatively improved things enormously.

## March 30, 2017

### Lathe Toolpost Spindle

I've been thinking for a while it would be nice to have live tooling for the MITERSlathe, for things like grinding and indexed drilling parts like hubs.

So far I've made everything out of random bits and pieces found at MITERS.  For the spindle, I started out with this old R8 to ER16 adapter:

I meticulously turned the shank down to 20mm.  I used sandpaper and Scotchbrite to bring the shaft to within a couple tenths down the whole length.

This was an incredibly satisfying measurement.

The bearings lightly press on the shaft - it's a tight enough fit that you just barely need an arbor press, but you can still easily un-press the bearings without damaging them.

Here's some of the housing machining.  I started out with this huge brick of mystery-steel I found on the floor of MITERS.  I got really lucky - it happened to be perfectly sized for machining a tool-holder,  and machined wonderfully.  It fly-cuts especially nicely, producing an incredibly shiny surface:

I squared up the block on the mill, cut the dovetail, and spot-drilled the spindle bore.

I did all the spindle boring on the lathe.  I started out by placing the housing on the toolpost, centering it vertically, and indicating the back face to be parallel to the travel of the lathe.  I put drills in the lathe's collet chuck, and use the lathe kind of like a horizontal  boring mill, to guarantee that the spindle bore was true to the  the lathe's travel:

To machine the bearing bores, I put the housing in the 4-jaw chuck and indicated in the drilled holes, as well as the faces, to make sure the housing was both square and centered.  I used this big indexable drill bit as a boring bar, because it has wonderfully sharp positive-rake inserts which produces a much better surface finish with low cutting pressure than the usual negative-rake turning inserts do.

I used a different bearing arrangement than I did with the tiny lathe, and I'm very satisfied with how it turned out.  I only had deep-groove ball  bearings to work with, rather than angular contact bearings, and I used a 3-bearing arrangement.  There's a pair of bearings at the front of the spindle, with a 0.002" shim between their inner races.  When the spindle housing is assembled, the outer races get squeezed together, so the shim provides a fixed preload.  This puts the bearings in a face-to-face arrangement, making the spindle somewhat tolerant to misalignment of the back bearing, if there is any.  The back bearing is a very close slip fit into the housing, and the outer race is lightly preloaded outwards by a wavy washer, to keep the balls in contact with both races.  Finally, there's a spacer between the inner races of the front and back bearings.  When the bolt on the back of the spindle is tightened, it squeezes down on the inner races and spacer, locking everything together.

Over all, I'm much happier with how this arrangement worked out compared to the lathe spindle.  Since preload is fixed, it takes no fussing to get the spindle to spin smoothly without slop, assuming the shim is sized correctly.  Also, since the front bearings are so close together, the preload will hardly change as the spindle heats up and expands.  For the lathe, I left the spindle on for a while, and pre-loaded the bearings with the spindle at normal operating temperature.

A plate on the front of the housing presses the outer races of the bearings together, to preload the bearings:

To test it out, I threw a Lovejoy coupling on the back, and a mating one on the output shaft of an old orbital sander motor, which was roughly appropriate speed.  Conclusion:  the spindle part works great, and, unsurprisingly, the orbital sander motor was terrible.  Needs more brushless servo.

Here are the two mounting configurations.  I tested it out with a 3/8" endmill, and the spindle was rock solid, producing an excellent surface finish without any signs of chattering.

Still to do:
- Find a better spindle motor, and make a motor attachment and height stop.
- Make an indexing system for the lathe spindle, so you can precisely lock the spindle to certain angles.