docs: Update Kinematics.md document with "smooth pressure advance"
Signed-off-by: Kevin O'Connor <kevin@koconnor.net>
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Kevin O'Connor
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@ -264,35 +264,36 @@ through the nozzle orifice (as in
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key idea is that the relationship between filament, pressure, and flow
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rate can be modeled using a linear coefficient:
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```
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stepper_position = requested_e_position + pressure_advance_coefficient * nominal_extruder_velocity
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pa_position = nominal_position + pressure_advance_coefficient * nominal_velocity
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```
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See the [pressure advance](Pressure_Advance.md) document for
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information on how to find this pressure advance coefficient.
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Once configured, Klipper will push in an additional amount of filament
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during acceleration. The higher the desired filament flow rate, the
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more filament must be pushed in during acceleration to account for
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pressure. During head deceleration the extra filament is retracted
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(the extruder will have a negative velocity).
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The basic pressure advance formula can cause the extruder motor to
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make sudden velocity changes. Klipper implements "smoothing" of the
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extruder movement to avoid this.
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One may notice that the pressure advance algorithm can cause the
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extruder motor to make sudden velocity changes. This is tolerated
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based on the idea that the majority of the inertia in the system is in
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changing the extruder pressure. As long as the extruder pressure does
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not change rapidly the sudden changes in extruder motor velocity are
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tolerated.
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The above graph shows an example of two extrusion moves with a
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non-zero cornering velocity between them. Note that the pressure
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advance system causes additional filament to be pushed into the
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extruder during acceleration. The higher the desired filament flow
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rate, the more filament must be pushed in during acceleration to
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account for pressure. During head deceleration the extra filament is
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retracted (the extruder will have a negative velocity).
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One area where sudden velocity changes become problematic is during
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small changes in head speed due to cornering.
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The "smoothing" is implemented by averaging the extruder position over
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a small time period (as specified by the
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`pressure_advance_smooth_time` config parameter). This averaging can
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span multiple g-code moves. Note how the extruder motor will start
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moving prior to the nominal start of the first extrusion move and will
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continue to move after the nominal end of the last extrusion move.
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To prevent this, the Klipper pressure advance code utilizes the move
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look-ahead queue to detect intermittent speed changes. During a
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deceleration event the code finds the maximum upcoming head speed
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within a configurable time window. The pressure is then only adjusted
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to this found maximum. This can greatly reduce (or even completely
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eliminate) pressure changes during cornering.
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Key formula for "smoothed pressure advance":
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```
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smooth_pa_position(t) =
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( definitive_integral(pa_position, from=t-smooth_time/2, to=t+smooth_time/2)
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/ smooth_time )
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```
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@ -0,0 +1,145 @@
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#!/usr/bin/env python2
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# Generate extruder pressure advance motion graphs
|
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#
|
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# Copyright (C) 2019 Kevin O'Connor <kevin@koconnor.net>
|
||||
#
|
||||
# This file may be distributed under the terms of the GNU GPLv3 license.
|
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import math, optparse, datetime
|
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import matplotlib
|
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|
||||
SEG_TIME = .000100
|
||||
INV_SEG_TIME = 1. / SEG_TIME
|
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|
||||
|
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######################################################################
|
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# Basic trapezoid motion
|
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######################################################################
|
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# List of moves: [(start_v, end_v, move_t), ...]
|
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Moves = [
|
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(0., 0., .200),
|
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(0., 100., None), (100., 100., .200), (100., 60., None),
|
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(60., 100., None), (100., 100., .200), (100., 0., None),
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(0., 0., .300)
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]
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EXTRUDE_R = (.4 * .4 * .75) / (math.pi * (1.75 / 2.)**2)
|
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ACCEL = 3000. * EXTRUDE_R
|
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|
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def gen_positions():
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out = []
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start_d = start_t = t = 0.
|
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for start_v, end_v, move_t in Moves:
|
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start_v *= EXTRUDE_R
|
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end_v *= EXTRUDE_R
|
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if move_t is None:
|
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move_t = abs(end_v - start_v) / ACCEL
|
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half_accel = 0.
|
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if end_v > start_v:
|
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half_accel = .5 * ACCEL
|
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elif start_v > end_v:
|
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half_accel = -.5 * ACCEL
|
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end_t = start_t + move_t
|
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while t <= end_t:
|
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rel_t = t - start_t
|
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out.append(start_d + (start_v + half_accel * rel_t) * rel_t)
|
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t += SEG_TIME
|
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start_d += (start_v + half_accel * move_t) * move_t
|
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start_t = end_t
|
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return out
|
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|
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def gen_deriv(data):
|
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return [0.] + [(data[i+1] - data[i]) * INV_SEG_TIME
|
||||
for i in range(len(data)-1)]
|
||||
|
||||
def time_to_index(t):
|
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return int(t * INV_SEG_TIME + .5)
|
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|
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|
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######################################################################
|
||||
# Pressure advance
|
||||
######################################################################
|
||||
|
||||
PA_HALF_SMOOTH_T = .040 / 2.
|
||||
PRESSURE_ADVANCE = .045
|
||||
|
||||
def calc_pa_raw(t, positions):
|
||||
pa = PRESSURE_ADVANCE * INV_SEG_TIME
|
||||
i = time_to_index(t)
|
||||
return positions[i] + pa * (positions[i+1] - positions[i])
|
||||
|
||||
def calc_pa_smooth(t, positions):
|
||||
start_index = time_to_index(t - PA_HALF_SMOOTH_T) + 1
|
||||
end_index = time_to_index(t + PA_HALF_SMOOTH_T)
|
||||
pa = PRESSURE_ADVANCE * INV_SEG_TIME
|
||||
pa_data = [positions[i] + pa * (positions[i+1] - positions[i])
|
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for i in range(start_index, end_index)]
|
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return sum(pa_data) / (end_index - start_index)
|
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|
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|
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######################################################################
|
||||
# Plotting and startup
|
||||
######################################################################
|
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|
||||
MARGIN_TIME = 0.100
|
||||
|
||||
def plot_motion():
|
||||
# Nominal motion
|
||||
positions = gen_positions()
|
||||
drop = int(MARGIN_TIME * INV_SEG_TIME)
|
||||
times = [SEG_TIME * t for t in range(len(positions))][drop:-drop]
|
||||
velocities = gen_deriv(positions[drop:-drop])
|
||||
# Motion with pressure advance
|
||||
pa_positions = [calc_pa_raw(t, positions) for t in times]
|
||||
pa_velocities = gen_deriv(pa_positions)
|
||||
# Smoothed motion
|
||||
sm_positions = [calc_pa_smooth(t, positions) for t in times]
|
||||
sm_velocities = gen_deriv(sm_positions)
|
||||
# Build plot
|
||||
shift_times = [t - MARGIN_TIME for t in times]
|
||||
fig, ax1 = matplotlib.pyplot.subplots(nrows=1, sharex=True)
|
||||
ax1.set_title("Extruder Velocity")
|
||||
ax1.set_ylabel('Velocity (mm/s)')
|
||||
pa_plot, = ax1.plot(shift_times, pa_velocities, 'r',
|
||||
label='Pressure Advance', alpha=0.3)
|
||||
nom_plot, = ax1.plot(shift_times, velocities, 'black', label='Nominal')
|
||||
sm_plot, = ax1.plot(shift_times, sm_velocities, 'g', label='Smooth PA',
|
||||
alpha=0.9)
|
||||
fontP = matplotlib.font_manager.FontProperties()
|
||||
fontP.set_size('x-small')
|
||||
ax1.legend(handles=[nom_plot, pa_plot, sm_plot], loc='best', prop=fontP)
|
||||
ax1.set_xlabel('Time (s)')
|
||||
ax1.grid(True)
|
||||
fig.tight_layout()
|
||||
return fig
|
||||
|
||||
def setup_matplotlib(output_to_file):
|
||||
global matplotlib
|
||||
if output_to_file:
|
||||
matplotlib.rcParams.update({'figure.autolayout': True})
|
||||
matplotlib.use('Agg')
|
||||
import matplotlib.pyplot, matplotlib.dates, matplotlib.font_manager
|
||||
import matplotlib.ticker
|
||||
|
||||
def main():
|
||||
# Parse command-line arguments
|
||||
usage = "%prog [options]"
|
||||
opts = optparse.OptionParser(usage)
|
||||
opts.add_option("-o", "--output", type="string", dest="output",
|
||||
default=None, help="filename of output graph")
|
||||
options, args = opts.parse_args()
|
||||
if len(args) != 0:
|
||||
opts.error("Incorrect number of arguments")
|
||||
|
||||
# Draw graph
|
||||
setup_matplotlib(options.output is not None)
|
||||
fig = plot_motion()
|
||||
|
||||
# Show graph
|
||||
if options.output is None:
|
||||
matplotlib.pyplot.show()
|
||||
else:
|
||||
fig.set_size_inches(6, 2.5)
|
||||
fig.savefig(options.output)
|
||||
|
||||
if __name__ == '__main__':
|
||||
main()
|
||||
Loading…
Reference in New Issue