Sliced Pulley Error
FirePick Delta has a sliced drive pulley that reduces arm/pulley collisions. The slice interrupts the pulley circumference and introduces non-linearity. Consider the following diagram depicting the delta drive pulley and gear. The dotted circular outline represents a circular pulley arm. The sliced pulley is shown shaded light orange.
The motion of a circular pulley arm and a sliced pulley arm are identical until the arm rises up to the Critical Angle, shown as the vector MC. The point C is the critical point at which we start noticing Sliced Pulley Error (SPE). Notice that the angle of the slice itself is irrelevant and that C is at the intersection of the slice chord and the pulley circle.
SPE increases as the pulley arm continues up past the critical angle.
For circular pulley delta machine, if we move the arm an angle of CMQ
above the critical point, our belt (blue line) will travel a distance
||PC||, which is the circular arc length from C to Q.
For sliced pulley delta machines, the belt travels less to arrive
at the angle CMQ (red line).
We therefore define the
Sliced Pulley Error (SPE) as the difference, ||DQ|| - ||DP||, between
the circular pulley delta and sliced pulley delta traversals
for the angle CMQ. SPE will be negative, meaning that for an arm
angle in the SPE zone, we need to move
the stepper LESS for a sliced pulley than a circular pulley.
| Dimension | Type | Description |
|---|---|---|
| MS | Vector | Distance between drive gear and arm pulley |
| C | Point | Pulley arm critical point above which SPE is non-zero |
| MC | Angle | Critical angle at which non-linearity starts |
| D | Point | Drive belt tangent point at critical angle |
| P | Point | Circular pulley belt travel for angle CMQ |
| Q | Point | Sliced pulley location for angle CMQ |
| SPE | Distance |
The critical angle is related to the belt wrap angle, which is defined in several different ways. The critical angle formula is:
criticalAngle MC = arcsin( (||MC|| - ||SD||) / ||MS|| )
SPE kinematics are actually quite complicated, since the belt wrap angle at SD
changes as the sliced pulley moves above the critical point. This change affects
the calculation of ||DQ||, but we will assume a constant belt wrap angle at SD
for simplicity given that FPD doesn't go to far into the SPE zone.
For FirePick Delta, SPE becomes a factor when any arm rises above -52.3 degrees. This is a fairly extreme angle and is unlikely to affect normal FPD motion other than homing, which typically happens at 67.2 degrees:
The chart also displays a
good linear model
for SPE as having a fixed slope
of -0.383 mm/degree with a max SPE of -5 for FPD.
The x-intercept of the SPE linear model is the SPE linear model critical point
(-54.6 degrees),
which is different than the actual critical point
critical point of the SPE curve (-52.3 degrees).
The slope of the SPE linear model is heuristically derived from
a least-squares fit of the SPE curve between
-54 and -72 degrees. Since the SPE curve itself is approximate (and neglects the changing SD belt wrap
angle), we choose the linear model to coincide with the SPE curve in the range of interest
for auto-calibration (i.e., near the typical 67.2 homing angle).
The Arm Critical Angle is the arm angle at the critical point C. For 3DLC0002S, the SPE linear model Arm Critical Angle is:
{"dimspa":-54.617}If you use a different pulley, you may need to use a copy of the SPE calculation spreadsheet to recalculate this angle so that FireStep can model your machine accurately.
The SPE Ratio is used to calculate the angular error from the SPE belt travel error.
SPERatio = SPESlope * beltTravelPerDegree
FireStep uses SPE Ratio to calculate the change in pulse count or arm angle caused by SPE:
{"dimspr":-0.38296}The SPE Ratio is a function of:
- drive gear diameter GT2 16-mm drive gear is default
- pulley radius to bearing surface 47.5mm is the default. See 3DLC0002S,
- arm position w/r to critical point (MC:Arm angle) -24.17 degrees is the default. See 3DLC0002S,
- stepper/pulley distance 92.114mm is the default
- drive plate thickness affects stepper/pulley distance (6mm is assumed)
- belt type drive belt dimensions are for GT2 6mm belt
If you customize any of the above components, you will need to recompute or measure your new SPE Ratio. The SPE calculation spreadsheet may help you recalculate the ratio.
The SPE Ratio is required for [auto-calibration](Delta Calibration). You can also just set the SPE Ratio to zero, ignore SPE and calibrate FPD manually.
FPD Z-probe auto-calibration is accurate and stable to within about 0.5mm. The following chart shows the individual hex probes from 15 successive auto-calibrations, where each auto calibration consists of 6 hex probes:
Notice that:
- It takes about 3 hex probes to converge into the +/- 0.25mm stable zone.
- There is a slight upwards drift in the calibration over many iterations, which may be caused by inaccuracies in the kinematic model or the dimensions used for calculation.