scripts: scripts to simulate input_shaper response and toolhead movement (#3063)

Signed-off-by: Dmitry Butyugin <dmbutyugin@google.com>

1 files changed, 427 insertions, 0 deletions

diff --git a/scripts/graph_motion.py b/scripts/graph_motion.py
new file mode 100755
index 00000000..29a5410b
--- /dev/null
+++ b/scripts/graph_motion.py
@@ -0,0 +1,427 @@
+#!/usr/bin/env python2
+# Script to graph motion results
+#
+# Copyright (C) 2019-2020  Kevin O'Connor <kevin@koconnor.net>
+# Copyright (C) 2020  Dmitry Butyugin <dmbutyugin@google.com>
+#
+# This file may be distributed under the terms of the GNU GPLv3 license.
+import optparse, datetime, math
+import matplotlib
+
+SEG_TIME = .000100
+INV_SEG_TIME = 1. / SEG_TIME
+
+SPRING_FREQ=35.0
+DAMPING_RATIO=0.05
+
+CONFIG_FREQ=40.0
+CONFIG_DAMPING_RATIO=0.1
+
+######################################################################
+# Basic trapezoid motion
+######################################################################
+
+# List of moves: [(start_v, end_v, move_t), ...]
+Moves = [
+    (0., 0., .100),
+    (6.869, 89.443, None), (89.443, 89.443, .120), (89.443, 17.361, None),
+    (19.410, 120., None), (120., 120., .130), (120., 5., None),
+    (0., 0., 0.01),
+    (-5., -100., None), (-100., -100., .100), (-100., -.5, None),
+    (0., 0., .200)
+]
+ACCEL = 3000.
+MAX_JERK = ACCEL * 0.6 * SPRING_FREQ
+
+def get_accel(start_v, end_v):
+    return ACCEL
+
+def get_accel_jerk_limit(start_v, end_v):
+    effective_accel = math.sqrt(MAX_JERK * abs(end_v - start_v) / 6.)
+    return min(effective_accel, ACCEL)
+
+# Standard constant acceleration generator
+def get_acc_pos_ao2(rel_t, start_v, accel, move_t):
+    return (start_v + 0.5 * accel * rel_t) * rel_t
+
+# Bezier curve "accel_order=4" generator
+def get_acc_pos_ao4(rel_t, start_v, accel, move_t):
+    inv_accel_t = 1. / move_t
+    accel_div_accel_t = accel * inv_accel_t
+    accel_div_accel_t2 = accel_div_accel_t * inv_accel_t
+
+    c4 = -.5 * accel_div_accel_t2;
+    c3 = accel_div_accel_t;
+    c1 = start_v
+    return ((c4 * rel_t + c3) * rel_t * rel_t + c1) * rel_t
+
+# Bezier curve "accel_order=6" generator
+def get_acc_pos_ao6(rel_t, start_v, accel, move_t):
+    inv_accel_t = 1. / move_t
+    accel_div_accel_t = accel * inv_accel_t
+    accel_div_accel_t2 = accel_div_accel_t * inv_accel_t
+    accel_div_accel_t3 = accel_div_accel_t2 * inv_accel_t
+    accel_div_accel_t4 = accel_div_accel_t3 * inv_accel_t
+
+    c6 = accel_div_accel_t4;
+    c5 = -3. * accel_div_accel_t3;
+    c4 = 2.5 * accel_div_accel_t2;
+    c1 = start_v;
+    return (((c6 * rel_t + c5) * rel_t + c4)
+            * rel_t * rel_t * rel_t + c1) * rel_t
+
+get_acc_pos = get_acc_pos_ao2
+get_acc = get_accel
+
+# Calculate positions based on 'Moves' list
+def gen_positions():
+    out = []
+    start_d = start_t = t = 0.
+    for start_v, end_v, move_t in Moves:
+        if move_t is None:
+            move_t = abs(end_v - start_v) / get_acc(start_v, end_v)
+        accel = (end_v - start_v) / move_t
+        end_t = start_t + move_t
+        while t <= end_t:
+            rel_t = t - start_t
+            out.append(start_d + get_acc_pos(rel_t, start_v, accel, move_t))
+            t += SEG_TIME
+        start_d += get_acc_pos(move_t, start_v, accel, move_t)
+        start_t = end_t
+    return out
+
+
+######################################################################
+# Estimated motion with belt as spring
+######################################################################
+
+def estimate_spring(positions):
+    ang_freq2 = (SPRING_FREQ * 2. * math.pi)**2
+    damping_factor = 4. * math.pi * DAMPING_RATIO * SPRING_FREQ
+    head_pos = head_v = 0.
+    out = []
+    for stepper_pos in positions:
+        head_pos += head_v * SEG_TIME
+        head_a = (stepper_pos - head_pos) * ang_freq2
+        head_v += head_a * SEG_TIME
+        head_v -= head_v * damping_factor * SEG_TIME
+        out.append(head_pos)
+    return out
+
+
+######################################################################
+# List helper functions
+######################################################################
+
+MARGIN_TIME = 0.050
+
+def time_to_index(t):
+    return int(t * INV_SEG_TIME + .5)
+
+def indexes(positions):
+    drop = time_to_index(MARGIN_TIME)
+    return range(drop, len(positions)-drop)
+
+def trim_lists(*lists):
+    keep = len(lists[0]) - time_to_index(2. * MARGIN_TIME)
+    for l in lists:
+        del l[keep:]
+
+
+######################################################################
+# Common data filters
+######################################################################
+
+# Generate estimated first order derivative
+def gen_deriv(data):
+    return [0.] + [(data[i+1] - data[i]) * INV_SEG_TIME
+                   for i in range(len(data)-1)]
+
+# Simple average between two points smooth_time away
+def calc_average(positions, smooth_time):
+    offset = time_to_index(smooth_time * .5)
+    out = [0.] * len(positions)
+    for i in indexes(positions):
+        out[i] = .5 * (positions[i-offset] + positions[i+offset])
+    return out
+
+# Average (via integration) of smooth_time range
+def calc_smooth(positions, smooth_time):
+    offset = time_to_index(smooth_time * .5)
+    weight = 1. / (2*offset - 1)
+    out = [0.] * len(positions)
+    for i in indexes(positions):
+        out[i] = sum(positions[i-offset+1:i+offset]) * weight
+    return out
+
+# Time weighted average (via integration) of smooth_time range
+def calc_weighted(positions, smooth_time):
+    offset = time_to_index(smooth_time * .5)
+    weight = 1. / offset**2
+    out = [0.] * len(positions)
+    for i in indexes(positions):
+        weighted_data = [positions[j] * (offset - abs(j-i))
+                         for j in range(i-offset, i+offset)]
+        out[i] = sum(weighted_data) * weight
+    return out
+
+# Weighted average (`h**2 - (t-T)**2`) of smooth_time range
+def calc_weighted2(positions, smooth_time):
+    offset = time_to_index(smooth_time * .5)
+    weight = .75 / offset**3
+    out = [0.] * len(positions)
+    for i in indexes(positions):
+        weighted_data = [positions[j] * (offset**2 - (j-i)**2)
+                         for j in range(i-offset, i+offset)]
+        out[i] = sum(weighted_data) * weight
+    return out
+
+# Weighted average (`(h**2 - (t-T)**2)**2`) of smooth_time range
+def calc_weighted4(positions, smooth_time):
+    offset = time_to_index(smooth_time * .5)
+    weight = 15 / (16. * offset**5)
+    out = [0.] * len(positions)
+    for i in indexes(positions):
+        weighted_data = [positions[j] * ((offset**2 - (j-i)**2))**2
+                         for j in range(i-offset, i+offset)]
+        out[i] = sum(weighted_data) * weight
+    return out
+
+# Weighted average (`(h - abs(t-T))**2 * (2 * abs(t-T) + h)`) of range
+def calc_weighted3(positions, smooth_time):
+    offset = time_to_index(smooth_time * .5)
+    weight = 1. / offset**4
+    out = [0.] * len(positions)
+    for i in indexes(positions):
+        weighted_data = [positions[j] * (offset - abs(j-i))**2
+                         * (2. * abs(j-i) + offset)
+                         for j in range(i-offset, i+offset)]
+        out[i] = sum(weighted_data) * weight
+    return out
+
+
+######################################################################
+# Spring motion estimation
+######################################################################
+
+def calc_spring_raw(positions):
+    sa = (INV_SEG_TIME / (CONFIG_FREQ * 2. * math.pi))**2
+    ra = 2. * CONFIG_DAMPING_RATIO * math.sqrt(sa)
+    out = [0.] * len(positions)
+    for i in indexes(positions):
+        out[i] = (positions[i]
+                  + sa * (positions[i-1] - 2.*positions[i] + positions[i+1])
+                  + ra * (positions[i+1] - positions[i]))
+    return out
+
+def calc_spring_double_weighted(positions, smooth_time):
+    offset = time_to_index(smooth_time * .25)
+    sa = (INV_SEG_TIME / (offset * CONFIG_FREQ * 2. * math.pi))**2
+    ra = 2. * CONFIG_DAMPING_RATIO * math.sqrt(sa)
+    out = [0.] * len(positions)
+    for i in indexes(positions):
+        out[i] = (positions[i]
+                  + sa * (positions[i-offset] - 2.*positions[i]
+                          + positions[i+offset])
+                  + ra * (positions[i+1] - positions[i]))
+    return calc_weighted(out, smooth_time=.5 * smooth_time)
+
+######################################################################
+# Input shapers
+######################################################################
+
+def get_zv_shaper():
+    df = math.sqrt(1. - CONFIG_DAMPING_RATIO**2)
+    K = math.exp(-CONFIG_DAMPING_RATIO * math.pi / df)
+    t_d = 1. / (CONFIG_FREQ * df)
+    A = [1., K]
+    T = [0., .5*t_d]
+    return (A, T, "ZV")
+
+def get_zvd_shaper():
+    df = math.sqrt(1. - CONFIG_DAMPING_RATIO**2)
+    K = math.exp(-CONFIG_DAMPING_RATIO * math.pi / df)
+    t_d = 1. / (CONFIG_FREQ * df)
+    A = [1., 2.*K, K**2]
+    T = [0., .5*t_d, t_d]
+    return (A, T, "ZVD")
+
+def get_mzv_shaper():
+    df = math.sqrt(1. - CONFIG_DAMPING_RATIO**2)
+    K = math.exp(-.75 * CONFIG_DAMPING_RATIO * math.pi / df)
+    t_d = 1. / (CONFIG_FREQ * df)
+
+    a1 = 1. - 1. / math.sqrt(2.)
+    a2 = (math.sqrt(2.) - 1.) * K
+    a3 = a1 * K * K
+
+    A = [a1, a2, a3]
+    T = [0., .375*t_d, .75*t_d]
+    return (A, T, "MZV")
+
+def get_ei_shaper():
+    v_tol = 0.05 # vibration tolerance
+    df = math.sqrt(1. - CONFIG_DAMPING_RATIO**2)
+    K = math.exp(-CONFIG_DAMPING_RATIO * math.pi / df)
+    t_d = 1. / (CONFIG_FREQ * df)
+
+    a1 = .25 * (1. + v_tol)
+    a2 = .5 * (1. - v_tol) * K
+    a3 = a1 * K * K
+
+    A = [a1, a2, a3]
+    T = [0., .5*t_d, t_d]
+    return (A, T, "EI")
+
+def get_2hump_ei_shaper():
+    v_tol = 0.05 # vibration tolerance
+    df = math.sqrt(1. - CONFIG_DAMPING_RATIO**2)
+    K = math.exp(-CONFIG_DAMPING_RATIO * math.pi / df)
+    t_d = 1. / (CONFIG_FREQ * df)
+
+    V2 = v_tol**2
+    X = pow(V2 * (math.sqrt(1. - V2) + 1.), 1./3.)
+    a1 = (3.*X*X + 2.*X + 3.*V2) / (16.*X)
+    a2 = (.5 - a1) * K
+    a3 = a2 * K
+    a4 = a1 * K * K * K
+
+    A = [a1, a2, a3, a4]
+    T = [0., .5*t_d, t_d, 1.5*t_d]
+    return (A, T, "2-hump EI")
+
+def get_3hump_ei_shaper():
+    v_tol = 0.05 # vibration tolerance
+    df = math.sqrt(1. - CONFIG_DAMPING_RATIO**2)
+    K = math.exp(-CONFIG_DAMPING_RATIO * math.pi / df)
+    t_d = 1. / (CONFIG_FREQ * df)
+
+    K2 = K*K
+    a1 = 0.0625 * (1. + 3. * v_tol + 2. * math.sqrt(2. * (v_tol + 1.) * v_tol))
+    a2 = 0.25 * (1. - v_tol) * K
+    a3 = (0.5 * (1. + v_tol) - 2. * a1) * K2
+    a4 = a2 * K2
+    a5 = a1 * K2 * K2
+
+    A = [a1, a2, a3, a4, a5]
+    T = [0., .5*t_d, t_d, 1.5*t_d, 2.*t_d]
+    return (A, T, "3-hump EI")
+
+
+def shift_pulses(shaper):
+    A, T, name = shaper
+    n = len(T)
+    ts = (sum([A[i] * T[i] for i in range(n)])) / sum(A)
+    for i in range(n):
+        T[i] -= ts
+
+def calc_shaper(shaper, positions):
+    shift_pulses(shaper)
+    A = shaper[0]
+    inv_D = 1. / sum(A)
+    n = len(A)
+    T = [time_to_index(-shaper[1][j]) for j in range(n)]
+    out = [0.] * len(positions)
+    for i in indexes(positions):
+        out[i] = sum([positions[i + T[j]] * A[j] for j in range(n)]) * inv_D
+    return out
+
+# Ideal values
+SMOOTH_TIME = (2./3.) / CONFIG_FREQ
+
+def gen_updated_position(positions):
+    #return calc_weighted(positions, 0.040)
+    #return calc_spring_double_weighted(positions, SMOOTH_TIME)
+    #return calc_weighted4(calc_spring_raw(positions), SMOOTH_TIME)
+    return calc_shaper(get_ei_shaper(), positions)
+
+
+######################################################################
+# Plotting and startup
+######################################################################
+
+def plot_motion():
+    # Nominal motion
+    positions = gen_positions()
+    velocities = gen_deriv(positions)
+    accels = gen_deriv(velocities)
+    # Updated motion
+    upd_positions = gen_updated_position(positions)
+    upd_velocities = gen_deriv(upd_positions)
+    upd_accels = gen_deriv(upd_velocities)
+    # Estimated position with model of belt as spring
+    spring_orig = estimate_spring(positions)
+    spring_upd = estimate_spring(upd_positions)
+    spring_diff_orig = [n-o for n, o in zip(spring_orig, positions)]
+    spring_diff_upd = [n-o for n, o in zip(spring_upd, positions)]
+    head_velocities = gen_deriv(spring_orig)
+    head_accels = gen_deriv(head_velocities)
+    head_upd_velocities = gen_deriv(spring_upd)
+    head_upd_accels = gen_deriv(head_upd_velocities)
+    # Build plot
+    times = [SEG_TIME * i for i in range(len(positions))]
+    trim_lists(times, velocities, accels,
+               upd_velocities, upd_velocities, upd_accels,
+               spring_diff_orig, spring_diff_upd,
+               head_velocities, head_upd_velocities,
+               head_accels, head_upd_accels)
+    fig, (ax1, ax2, ax3) = matplotlib.pyplot.subplots(nrows=3, sharex=True)
+    ax1.set_title("Simulation: resonance freq=%.1f Hz, damping_ratio=%.3f,\n"
+                  "configured freq=%.1f Hz, damping_ratio = %.3f"
+                  % (SPRING_FREQ, DAMPING_RATIO, CONFIG_FREQ
+                      , CONFIG_DAMPING_RATIO))
+    ax1.set_ylabel('Velocity (mm/s)')
+    ax1.plot(times, upd_velocities, 'r', label='New Velocity', alpha=0.8)
+    ax1.plot(times, velocities, 'g', label='Nominal Velocity', alpha=0.8)
+    ax1.plot(times, head_velocities, label='Head Velocity', alpha=0.4)
+    ax1.plot(times, head_upd_velocities, label='New Head Velocity', alpha=0.4)
+    fontP = matplotlib.font_manager.FontProperties()
+    fontP.set_size('x-small')
+    ax1.legend(loc='best', prop=fontP)
+    ax1.grid(True)
+    ax2.set_ylabel('Acceleration (mm/s^2)')
+    ax2.plot(times, upd_accels, 'r', label='New Accel', alpha=0.8)
+    ax2.plot(times, accels, 'g', label='Nominal Accel', alpha=0.8)
+    ax2.plot(times, head_accels, alpha=0.4)
+    ax2.plot(times, head_upd_accels, alpha=0.4)
+    ax2.set_ylim([-5. * ACCEL, 5. * ACCEL])
+    ax2.legend(loc='best', prop=fontP)
+    ax2.grid(True)
+    ax3.set_ylabel('Deviation (mm)')
+    ax3.plot(times, spring_diff_upd, 'r', label='New', alpha=0.8)
+    ax3.plot(times, spring_diff_orig, 'g', label='Nominal', alpha=0.8)
+    ax3.grid(True)
+    ax3.legend(loc='best', prop=fontP)
+    ax3.set_xlabel('Time (s)')
+    return fig
+
+def setup_matplotlib(output_to_file):
+    global matplotlib
+    if output_to_file:
+        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(8, 6)
+        fig.savefig(options.output)
+
+if __name__ == '__main__':
+    main()