Added simulation for the brushless motor.
This is the motor we used in powerwheels.
Change-Id: I33a1d43d8c6374e289873c774ba5c16147619ff9
diff --git a/motors/python/phase_current.py b/motors/python/phase_current.py
new file mode 100755
index 0000000..2e88fae
--- /dev/null
+++ b/motors/python/phase_current.py
@@ -0,0 +1,563 @@
+#!/usr/bin/python3
+
+import numpy
+from matplotlib import pylab
+import scipy.integrate
+from frc971.control_loops.python import controls
+import time
+import operator
+
+K1 = 1.81e04
+K2 = -2.65e03
+
+# Make the amplitude of the fundamental 1 for ease of playing with.
+K2 /= K1
+K1 = 1
+
+vcc = 30.0 # volts
+R_motor = 0.0055 # ohms for the motor
+R = 0.008 # ohms for system
+
+L = 10.0 * 1e-6 # Henries
+M = L / 10.0
+
+Kv = 22000.0 * 2.0 * numpy.pi / 60.0 / vcc * 2.0
+J = 0.0000007
+
+R_shunt = 0.0003
+
+# RC circuit for current sense filtering.
+R_sense1 = 768.0
+R_sense2 = 1470.0
+C_sense = 10.0 * 1e-9
+
+# So, we measured the inductance by switching between ~5 and ~20 amps through
+# the motor.
+# We then looked at the change in voltage that should give us (assuming duty
+# cycle * vin), and divided it by the corresponding change in current.
+
+# We then looked at the amount of time it took to decay the current to 1/e
+# That gave us the inductance.
+
+# Overrides for experiments
+J = J * 10.0
+
+# Firing phase A -> 0.0
+# Firing phase B -> - numpy.pi * 2.0 / 3.0
+# Firing phase C -> + numpy.pi * 2.0 / 3.0
+
+hz = 20000.0
+
+#switching_pattern = 'front'
+switching_pattern = 'centered'
+#switching_pattern = 'rear'
+#switching_pattern = 'centered front shifted'
+#switching_pattern = 'anticentered'
+
+Vconv = numpy.matrix([[2.0, -1.0, -1.0],
+ [-1.0, 2.0, -1.0],
+ [-1.0, -1.0, 2.0]]) / 3.0
+
+def f_single(theta):
+ return K1 * numpy.sin(theta) + K2 * numpy.sin(theta * 5)
+
+def g_single(theta):
+ return K1 * numpy.sin(theta) - K2 * numpy.sin(theta * 5)
+
+def gdot_single(theta):
+ """Derivitive of the current.
+
+ Must be multiplied by omega externally.
+ """
+ return K1 * numpy.cos(theta) - 5.0 * K2 * numpy.cos(theta * 5.0)
+
+f = numpy.vectorize(f_single, otypes=(numpy.float,))
+g = numpy.vectorize(g_single, otypes=(numpy.float,))
+gdot = numpy.vectorize(gdot_single, otypes=(numpy.float,))
+
+def torque(theta):
+ return f(theta) * g(theta)
+
+def phase_a(function, theta):
+ return function(theta)
+
+def phase_b(function, theta):
+ return function(theta + 2 * numpy.pi / 3)
+
+def phase_c(function, theta):
+ return function(theta + 4 * numpy.pi / 3)
+
+def phases(function, theta):
+ return numpy.matrix([[phase_a(function, theta)],
+ [phase_b(function, theta)],
+ [phase_c(function, theta)]])
+
+def all_phases(function, theta_range):
+ return (phase_a(function, theta_range) +
+ phase_b(function, theta_range) +
+ phase_c(function, theta_range))
+
+theta_range = numpy.linspace(start=0, stop=4 * numpy.pi, num=10000)
+one_amp_driving_voltage = R * g(theta_range) + (L * gdot(theta_range) + M * gdot(theta_range + 2.0 / 3.0 * numpy.pi) + M * gdot(theta_range - 2.0 / 3.0 * numpy.pi)) * Kv * vcc / 2.0
+
+max_one_amp_driving_voltage = max(one_amp_driving_voltage)
+
+# The number to divide the product of the unit BEMF and the per phase current
+# by to get motor current.
+one_amp_scalar = (phases(f_single, 0.0).T * phases(g_single, 0.0))[0, 0]
+
+print 'Max BEMF', max(f(theta_range))
+print 'Max current', max(g(theta_range))
+print 'Max drive voltage (one_amp_driving_voltage)', max(one_amp_driving_voltage)
+print 'one_amp_scalar', one_amp_scalar
+
+pylab.figure()
+pylab.subplot(1, 1, 1)
+pylab.plot(theta_range, f(theta_range), label='bemf')
+pylab.plot(theta_range, g(theta_range), label='phase_current')
+pylab.plot(theta_range, torque(theta_range), label='phase_torque')
+pylab.plot(theta_range, all_phases(torque, theta_range), label='sum_torque/current')
+pylab.legend()
+
+
+def full_sample_times(Ton, Toff, dt, n, start_time):
+ """Returns n + 4 samples for the provided switching times.
+
+ We need the timesteps and Us to integrate.
+
+ Returns:
+ array of [t, U matrix]
+ """
+
+ assert((Toff <= 1.0).all())
+
+ if (Ton <= Toff).all():
+ # Verify that they are all ordered correctly.
+ on_before_off = True
+ else:
+ on_before_off = False
+
+ Toff = Toff.copy() * dt
+ Toff[Toff < 100e-9] = -1.0
+ Toff[Toff > dt] = dt
+
+ Ton = Ton.copy() * dt
+ Ton[Ton < 100e-9] = -1.0
+ Ton[Ton > dt - 100e-9] = dt + 1.0
+
+ result = []
+ t = 0
+
+ result_times = numpy.concatenate(
+ (numpy.linspace(0, dt, num=n),
+ numpy.reshape(numpy.asarray(Ton[numpy.logical_and(Ton < dt, Ton > 0.0)]), (-1,)),
+ numpy.reshape(numpy.asarray(Toff[numpy.logical_and(Toff < dt, Toff > 0.0)]), (-1,))
+ ))
+ result_times.sort()
+
+ for t in numpy.nditer(result_times):
+ if on_before_off:
+ U = numpy.matrix([[vcc], [vcc], [vcc]])
+ U[t <= Ton] = 0.0
+ U[Toff < t] = 0.0
+ else:
+ U = numpy.matrix([[0.0], [0.0], [0.0]])
+ U[t > Ton] = vcc
+ U[t <= Toff] = vcc
+ result.append((float(t + start_time), U.copy()))
+
+ return result
+
+def sample_times(T, dt, n, start_time):
+ if switching_pattern == 'rear':
+ T = 1.0 - T
+ ans = full_sample_times(T, numpy.matrix(numpy.ones((3, 1))) * 1.0, dt, n, start_time)
+ elif switching_pattern == 'centered front shifted':
+ # Centered, but shifted to the beginning of the cycle.
+ Ton = 0.5 - T / 2.0
+ Toff = 0.5 + T / 2.0
+
+ tn = min(Ton)[0, 0]
+ Ton -= tn
+ Toff -= tn
+
+ ans = full_sample_times(Ton, Toff, dt, n, start_time)
+ elif switching_pattern == 'centered':
+ # Centered, looks waaay better.
+ Ton = 0.5 - T / 2.0
+ Toff = 0.5 + T / 2.0
+
+ ans = full_sample_times(Ton, Toff, dt, n, start_time)
+ elif switching_pattern == 'anticentered':
+ # Centered, looks waaay better.
+ Toff = T / 2.0
+ Ton = 1.0 - T / 2.0
+
+ ans = full_sample_times(Ton, Toff, dt, n, start_time)
+ elif switching_pattern == 'front':
+ ans = full_sample_times(numpy.matrix(numpy.zeros((3, 1))), T, dt, n, start_time)
+ else:
+ assert(False)
+
+ return ans
+
+class DataLogger(object):
+ def __init__(self, title=None):
+ self.title = title
+ self.ia = []
+ self.ib = []
+ self.ic = []
+ self.ia_goal = []
+ self.ib_goal = []
+ self.ic_goal = []
+ self.ia_controls = []
+ self.ib_controls = []
+ self.ic_controls = []
+ self.isensea = []
+ self.isenseb = []
+ self.isensec = []
+
+ self.va = []
+ self.vb = []
+ self.vc = []
+ self.van = []
+ self.vbn = []
+ self.vcn = []
+
+ self.ea = []
+ self.eb = []
+ self.ec = []
+
+ self.theta = []
+ self.omega = []
+
+ self.i_goal = []
+
+ self.time = []
+ self.controls_time = []
+ self.predicted_time = []
+
+ self.ia_pred = []
+ self.ib_pred = []
+ self.ic_pred = []
+
+ self.voltage_time = []
+ self.estimated_velocity = []
+ self.U_last = numpy.matrix(numpy.zeros((3, 1)))
+
+ def log_predicted(self, current_time, p):
+ self.predicted_time.append(current_time)
+ self.ia_pred.append(p[0, 0])
+ self.ib_pred.append(p[1, 0])
+ self.ic_pred.append(p[2, 0])
+
+ def log_controls(self, current_time, measured_current, In, E, estimated_velocity):
+ self.controls_time.append(current_time)
+ self.ia_controls.append(measured_current[0, 0])
+ self.ib_controls.append(measured_current[1, 0])
+ self.ic_controls.append(measured_current[2, 0])
+
+ self.ea.append(E[0, 0])
+ self.eb.append(E[1, 0])
+ self.ec.append(E[2, 0])
+
+ self.ia_goal.append(In[0, 0])
+ self.ib_goal.append(In[1, 0])
+ self.ic_goal.append(In[2, 0])
+ self.estimated_velocity.append(estimated_velocity)
+
+ def log_data(self, X, U, current_time, Vn, i_goal):
+ self.ia.append(X[0, 0])
+ self.ib.append(X[1, 0])
+ self.ic.append(X[2, 0])
+
+ self.i_goal.append(i_goal)
+
+ self.isensea.append(X[5, 0])
+ self.isenseb.append(X[6, 0])
+ self.isensec.append(X[7, 0])
+
+ self.theta.append(X[3, 0])
+ self.omega.append(X[4, 0])
+
+ self.time.append(current_time)
+
+ self.van.append(Vn[0, 0])
+ self.vbn.append(Vn[1, 0])
+ self.vcn.append(Vn[2, 0])
+
+ if (self.U_last != U).any():
+ self.va.append(self.U_last[0, 0])
+ self.vb.append(self.U_last[1, 0])
+ self.vc.append(self.U_last[2, 0])
+ self.voltage_time.append(current_time)
+
+ self.va.append(U[0, 0])
+ self.vb.append(U[1, 0])
+ self.vc.append(U[2, 0])
+ self.voltage_time.append(current_time)
+ self.U_last = U.copy()
+
+ def plot(self):
+ fig = pylab.figure()
+ pylab.subplot(3, 1, 1)
+ pylab.plot(self.controls_time, self.ia_controls, 'ro', label='ia_controls')
+ pylab.plot(self.controls_time, self.ib_controls, 'go', label='ib_controls')
+ pylab.plot(self.controls_time, self.ic_controls, 'bo', label='ic_controls')
+ pylab.plot(self.controls_time, self.ia_goal, 'r--', label='ia_goal')
+ pylab.plot(self.controls_time, self.ib_goal, 'g--', label='ib_goal')
+ pylab.plot(self.controls_time, self.ic_goal, 'b--', label='ic_goal')
+
+ #pylab.plot(self.controls_time, self.ia_pred, 'r*', label='ia_pred')
+ #pylab.plot(self.controls_time, self.ib_pred, 'g*', label='ib_pred')
+ #pylab.plot(self.controls_time, self.ic_pred, 'b*', label='ic_pred')
+ pylab.plot(self.time, self.isensea, 'r:', label='ia_sense')
+ pylab.plot(self.time, self.isenseb, 'g:', label='ib_sense')
+ pylab.plot(self.time, self.isensec, 'b:', label='ic_sense')
+ pylab.plot(self.time, self.ia, 'r', label='ia')
+ pylab.plot(self.time, self.ib, 'g', label='ib')
+ pylab.plot(self.time, self.ic, 'b', label='ic')
+ pylab.plot(self.time, self.i_goal, label='i_goal')
+ if self.title is not None:
+ fig.canvas.set_window_title(self.title)
+ pylab.legend()
+
+ pylab.subplot(3, 1, 2)
+ pylab.plot(self.voltage_time, self.va, label='va')
+ pylab.plot(self.voltage_time, self.vb, label='vb')
+ pylab.plot(self.voltage_time, self.vc, label='vc')
+ pylab.plot(self.time, self.van, label='van')
+ pylab.plot(self.time, self.vbn, label='vbn')
+ pylab.plot(self.time, self.vcn, label='vcn')
+ pylab.plot(self.controls_time, self.ea, label='ea')
+ pylab.plot(self.controls_time, self.eb, label='eb')
+ pylab.plot(self.controls_time, self.ec, label='ec')
+ pylab.legend()
+
+ pylab.subplot(3, 1, 3)
+ pylab.plot(self.time, self.theta, label='theta')
+ pylab.plot(self.time, self.omega, label='omega')
+ pylab.plot(self.controls_time, self.estimated_velocity, label='estimated omega')
+
+ pylab.legend()
+
+ fig = pylab.figure()
+ pylab.plot(self.controls_time,
+ map(operator.sub, self.ia_goal, self.ia_controls), 'r', label='ia_error')
+ pylab.plot(self.controls_time,
+ map(operator.sub, self.ib_goal, self.ib_controls), 'g', label='ib_error')
+ pylab.plot(self.controls_time,
+ map(operator.sub, self.ic_goal, self.ic_controls), 'b', label='ic_error')
+ if self.title is not None:
+ fig.canvas.set_window_title(self.title)
+ pylab.legend()
+ pylab.show()
+
+
+# So, from running a bunch of math, we know the following:
+# Van + Vbn + Vcn = 0
+# ia + ib + ic = 0
+# ea + eb + ec = 0
+# d ia/dt + d ib/dt + d ic/dt = 0
+#
+# We also have:
+# [ Van ] [ 2/3 -1/3 -1/3] [Va]
+# [ Vbn ] = [ -1/3 2/3 -1/3] [Vb]
+# [ Vcn ] [ -1/3 -1/3 2/3] [Vc]
+#
+# or,
+#
+# Vabcn = Vconv * V
+#
+# The base equation is:
+#
+# [ Van ] [ R 0 0 ] [ ia ] [ L M M ] [ dia/dt ] [ ea ]
+# [ Vbn ] = [ 0 R 0 ] [ ib ] + [ M L M ] [ dib/dt ] + [ eb ]
+# [ Vbn ] [ 0 0 R ] [ ic ] [ M M L ] [ dic/dt ] [ ec ]
+#
+# or
+#
+# Vabcn = R_matrix * I + L_matrix * I_dot + E
+#
+# We can re-arrange this as:
+#
+# inv(L_matrix) * (Vconv * V - E - R_matrix * I) = I_dot
+# B * V - inv(L_matrix) * E - A * I = I_dot
+class Simulation(object):
+ def __init__(self):
+ self.R_matrix = numpy.matrix(numpy.eye(3)) * R
+ self.L_matrix = numpy.matrix([[L, M, M], [M, L, M], [M, M, L]])
+ self.L_matrix_inv = numpy.linalg.inv(self.L_matrix)
+ self.A = self.L_matrix_inv * self.R_matrix
+ self.B = self.L_matrix_inv * Vconv
+ self.A_discrete, self.B_discrete = controls.c2d(-self.A, self.B, 1.0 / hz)
+ self.B_discrete_inverse = numpy.matrix(numpy.eye(3)) / (self.B_discrete[0, 0] - self.B_discrete[1, 0])
+
+ self.R_model = R * 1.0
+ self.L_model = L * 1.0
+ self.M_model = M * 1.0
+ self.R_matrix_model = numpy.matrix(numpy.eye(3)) * self.R_model
+ self.L_matrix_model = numpy.matrix([[self.L_model, self.M_model, self.M_model],
+ [self.M_model, self.L_model, self.M_model],
+ [self.M_model, self.M_model, self.L_model]])
+ self.L_matrix_inv_model = numpy.linalg.inv(self.L_matrix_model)
+ self.A_model = self.L_matrix_inv_model * self.R_matrix_model
+ self.B_model = self.L_matrix_inv_model * Vconv
+ self.A_discrete_model, self.B_discrete_model = \
+ controls.c2d(-self.A_model, self.B_model, 1.0 / hz)
+ self.B_discrete_inverse_model = numpy.matrix(numpy.eye(3)) / (self.B_discrete_model[0, 0] - self.B_discrete_model[1, 0])
+
+ print 'constexpr double kL = %g;' % self.L_model
+ print 'constexpr double kM = %g;' % self.M_model
+ print 'constexpr double kR = %g;' % self.R_model
+ print 'constexpr float kAdiscrete_diagonal = %gf;' % self.A_discrete_model[0, 0]
+ print 'constexpr float kAdiscrete_offdiagonal = %gf;' % self.A_discrete_model[1, 0]
+ print 'constexpr float kBdiscrete_inv_diagonal = %gf;' % self.B_discrete_inverse_model[0, 0]
+ print 'constexpr float kBdiscrete_inv_offdiagonal = %gf;' % self.B_discrete_inverse_model[1, 0]
+ print 'constexpr double kOneAmpScalar = %g;' % one_amp_scalar
+ print 'constexpr double kMaxOneAmpDrivingVoltage = %g;' % max_one_amp_driving_voltage
+ print('A_discrete', self.A_discrete)
+ print('B_discrete', self.B_discrete)
+ print('B_discrete_sub', numpy.linalg.inv(self.B_discrete[0:2, 0:2]))
+ print('B_discrete_inv', self.B_discrete_inverse)
+
+ # Xdot[5:, :] = (R_sense2 + R_sense1) / R_sense2 * (
+ # (1.0 / (R_sense1 * C_sense)) * (-Isense * R_sense2 / (R_sense1 + R_sense2) * (R_sense1 / R_sense2 + 1.0) + I))
+ self.mk1 = (R_sense2 + R_sense1) / R_sense2 * (1.0 / (R_sense1 * C_sense))
+ self.mk2 = -self.mk1 * R_sense2 / (R_sense1 + R_sense2) * (R_sense1 / R_sense2 + 1.0)
+
+ # ia, ib, ic, theta, omega, isensea, isenseb, isensec
+ self.X = numpy.matrix([[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]])
+
+ self.K = 0.05 * Vconv
+ print('A %s' % repr(self.A))
+ print('B %s' % repr(self.B))
+ print('K %s' % repr(self.K))
+
+ print('System poles are %s' % repr(numpy.linalg.eig(self.A)[0]))
+ print('Poles are %s' % repr(numpy.linalg.eig(self.A - self.B * self.K)[0]))
+
+ controllability = controls.ctrb(self.A, self.B)
+ print('Rank of augmented controlability matrix. %d' % numpy.linalg.matrix_rank(
+ controllability))
+
+ self.data_logger = DataLogger(switching_pattern)
+ self.current_time = 0.0
+
+ self.estimated_velocity = self.X[4, 0]
+
+ def motor_diffeq(self, x, t, U):
+ I = numpy.matrix(x[0:3]).T
+ theta = x[3]
+ omega = x[4]
+ Isense = numpy.matrix(x[5:]).T
+
+ dflux = phases(f_single, theta) / Kv
+
+ Xdot = numpy.matrix(numpy.zeros((8, 1)))
+ di_dt = -self.A_model * I + self.B_model * U - self.L_matrix_inv_model * dflux * omega
+ torque = I.T * dflux
+ Xdot[0:3, :] = di_dt
+ Xdot[3, :] = omega
+ Xdot[4, :] = torque / J
+
+ Xdot[5:, :] = self.mk1 * I + self.mk2 * Isense
+ return numpy.squeeze(numpy.asarray(Xdot))
+
+ def DoControls(self, goal_current):
+ theta = self.X[3, 0]
+ # Use the actual angular velocity.
+ omega = self.X[4, 0]
+
+ measured_current = self.X[5:, :].copy()
+
+ # Ok, lets now fake it.
+ E_imag1 = numpy.exp(1j * theta) * K1 * numpy.matrix(
+ [[-1j],
+ [-1j * numpy.exp(1j * numpy.pi * 2.0 / 3.0)],
+ [-1j * numpy.exp(-1j * numpy.pi * 2.0 / 3.0)]])
+ E_imag2 = numpy.exp(1j * 5.0 * theta) * K2 * numpy.matrix(
+ [[-1j],
+ [-1j * numpy.exp(-1j * numpy.pi * 2.0 / 3.0)],
+ [-1j * numpy.exp(1j * numpy.pi * 2.0 / 3.0)]])
+
+ overall_measured_current = ((E_imag1 + E_imag2).real.T * measured_current / one_amp_scalar)[0, 0]
+
+ current_error = goal_current - overall_measured_current
+ #print(current_error)
+ self.estimated_velocity += current_error * 1.0
+ omega = self.estimated_velocity
+
+ # Now, apply the transfer function of the inductor.
+ # Use that to difference the current across the cycle.
+ Icurrent = self.Ilast
+ # No history:
+ #Icurrent = phases(g_single, theta) * goal_current
+ Inext = phases(g_single, theta + omega * 1.0 / hz) * goal_current
+
+ deltaI = Inext - Icurrent
+
+ H1 = -numpy.linalg.inv(1j * omega * self.L_matrix + self.R_matrix) * omega / Kv
+ H2 = -numpy.linalg.inv(1j * omega * 5.0 * self.L_matrix + self.R_matrix) * omega / Kv
+ p_imag = H1 * E_imag1 + H2 * E_imag2
+ p_next_imag = numpy.exp(1j * omega * 1.0 / hz) * H1 * E_imag1 + \
+ numpy.exp(1j * omega * 5.0 * 1.0 / hz) * H2 * E_imag2
+ p = p_imag.real
+
+ # So, we now know how much the change in current is due to changes in BEMF.
+ # Subtract that, and then run the stock statespace equation.
+ Vn_ff = self.B_discrete_inverse * (Inext - self.A_discrete * (Icurrent - p) - p_next_imag.real)
+ print 'Vn_ff', Vn_ff
+ print 'Inext', Inext
+ Vn = Vn_ff + self.K * (Icurrent - measured_current)
+
+ E = phases(f_single, self.X[3, 0]) / Kv * self.X[4, 0]
+ self.data_logger.log_controls(self.current_time, measured_current, Icurrent, E, self.estimated_velocity)
+
+ self.Ilast = Inext
+
+ return Vn
+
+ def Simulate(self):
+ start_wall_time = time.time()
+ self.Ilast = numpy.matrix(numpy.zeros((3, 1)))
+ for n in range(200):
+ goal_current = 10.0
+ max_current = (vcc - (self.X[4, 0] / Kv * 2.0)) / max_one_amp_driving_voltage
+ min_current = (-vcc - (self.X[4, 0] / Kv * 2.0)) / max_one_amp_driving_voltage
+ goal_current = max(min_current, min(max_current, goal_current))
+
+ Vn = self.DoControls(goal_current)
+
+ #Vn = numpy.matrix([[0.20], [0.0], [0.0]])
+ #Vn = numpy.matrix([[0.00], [0.20], [0.0]])
+ #Vn = numpy.matrix([[0.00], [0.0], [0.20]])
+
+ # T is the fractional rate.
+ T = Vn / vcc
+ tn = -numpy.min(T)
+ T += tn
+ if (T > 1.0).any():
+ T = T / numpy.max(T)
+
+ for t, U in sample_times(T = T,
+ dt = 1.0 / hz, n = 10,
+ start_time = self.current_time):
+ # Analog amplifier mode!
+ #U = Vn
+
+ self.data_logger.log_data(self.X, (U - min(U)), self.current_time, Vn, goal_current)
+ t_array = numpy.array([self.current_time, t])
+ self.X = numpy.matrix(scipy.integrate.odeint(
+ self.motor_diffeq,
+ numpy.squeeze(numpy.asarray(self.X)),
+ t_array, args=(U,)))[1, :].T
+
+ self.current_time = t
+
+ print 'Took %f to simulate' % (time.time() - start_wall_time)
+
+ self.data_logger.plot()
+
+simulation = Simulation()
+simulation.Simulate()