| #!/usr/bin/python |
| |
| import control_loop |
| import controls |
| import polytope |
| import polydrivetrain |
| import numpy |
| import sys |
| from matplotlib import pylab |
| |
| class Claw(control_loop.ControlLoop): |
| def __init__(self, name="RawClaw"): |
| super(Claw, self).__init__(name) |
| # Stall Torque in N m |
| self.stall_torque = 2.42 |
| # Stall Current in Amps |
| self.stall_current = 133 |
| # Free Speed in RPM |
| self.free_speed = 5500.0 |
| # Free Current in Amps |
| self.free_current = 2.7 |
| # Moment of inertia of the claw in kg m^2 |
| # measured from CAD |
| self.J_top = 0.3 |
| self.J_bottom = 0.9 |
| # Resistance of the motor |
| self.R = 12.0 / self.stall_current |
| # Motor velocity constant |
| self.Kv = ((self.free_speed / 60.0 * 2.0 * numpy.pi) / |
| (13.5 - self.R * self.free_current)) |
| # Torque constant |
| self.Kt = self.stall_torque / self.stall_current |
| # Gear ratio |
| self.G = 14.0 / 48.0 * 18.0 / 32.0 * 18.0 / 66.0 * 12.0 / 60.0 |
| # Control loop time step |
| self.dt = 0.01 |
| |
| # State is [bottom position, bottom velocity, top - bottom position, |
| # top - bottom velocity] |
| # Input is [bottom power, top power - bottom power * J_top / J_bottom] |
| # Motor time constants. difference_bottom refers to the constant for how the |
| # bottom velocity affects the difference of the top and bottom velocities. |
| self.common_motor_constant = -self.Kt / self.Kv / (self.G * self.G * self.R) |
| self.bottom_bottom = self.common_motor_constant / self.J_bottom |
| self.difference_bottom = -self.common_motor_constant * (1 / self.J_bottom |
| - 1 / self.J_top) |
| self.difference_difference = self.common_motor_constant / self.J_top |
| # State feedback matrices |
| |
| self.A_continuous = numpy.matrix( |
| [[0, 0, 1, 0], |
| [0, 0, 0, 1], |
| [0, 0, self.bottom_bottom, 0], |
| [0, 0, self.difference_bottom, self.difference_difference]]) |
| |
| self.A_bottom_cont = numpy.matrix( |
| [[0, 1], |
| [0, self.bottom_bottom]]) |
| |
| self.A_diff_cont = numpy.matrix( |
| [[0, 1], |
| [0, self.difference_difference]]) |
| |
| self.motor_feedforward = self.Kt / (self.G * self.R) |
| self.motor_feedforward_bottom = self.motor_feedforward / self.J_bottom |
| self.motor_feedforward_difference = self.motor_feedforward / self.J_top |
| self.motor_feedforward_difference_bottom = ( |
| self.motor_feedforward * (1 / self.J_bottom - 1 / self.J_top)) |
| self.B_continuous = numpy.matrix( |
| [[0, 0], |
| [0, 0], |
| [self.motor_feedforward_bottom, 0], |
| [-self.motor_feedforward_bottom, self.motor_feedforward_difference]]) |
| |
| print "Cont X_ss", self.motor_feedforward / -self.common_motor_constant |
| |
| self.B_bottom_cont = numpy.matrix( |
| [[0], |
| [self.motor_feedforward_bottom]]) |
| |
| self.B_diff_cont = numpy.matrix( |
| [[0], |
| [self.motor_feedforward_difference]]) |
| |
| self.C = numpy.matrix([[1, 0, 0, 0], |
| [1, 1, 0, 0]]) |
| self.D = numpy.matrix([[0, 0], |
| [0, 0]]) |
| |
| self.A, self.B = self.ContinuousToDiscrete( |
| self.A_continuous, self.B_continuous, self.dt) |
| |
| self.A_bottom, self.B_bottom = controls.c2d( |
| self.A_bottom_cont, self.B_bottom_cont, self.dt) |
| self.A_diff, self.B_diff = controls.c2d( |
| self.A_diff_cont, self.B_diff_cont, self.dt) |
| |
| self.K_bottom = controls.dplace(self.A_bottom, self.B_bottom, [.65, .45]) |
| self.K_diff = controls.dplace(self.A_diff, self.B_diff, [.40, .28]) |
| |
| print "K_diff", self.K_diff |
| print "K_bottom", self.K_bottom |
| |
| print "A" |
| print self.A |
| print "B" |
| print self.B |
| |
| |
| self.Q = numpy.matrix([[(1.0 / (0.10 ** 2.0)), 0.0, 0.0, 0.0], |
| [0.0, (1.0 / (0.06 ** 2.0)), 0.0, 0.0], |
| [0.0, 0.0, 0.10, 0.0], |
| [0.0, 0.0, 0.0, 0.1]]) |
| |
| self.R = numpy.matrix([[(1.0 / (40.0 ** 2.0)), 0.0], |
| [0.0, (1.0 / (5.0 ** 2.0))]]) |
| #self.K = controls.dlqr(self.A, self.B, self.Q, self.R) |
| |
| self.K = numpy.matrix([[self.K_bottom[0, 0], 0.0, self.K_bottom[0, 1], 0.0], |
| [0.0, self.K_diff[0, 0], 0.0, self.K_diff[0, 1]]]) |
| |
| # Compute the feed forwards aceleration term. |
| self.K[1, 0] = -self.B[1, 0] * self.K[0, 0] / self.B[1, 1] |
| |
| lstsq_A = numpy.identity(2) |
| lstsq_A[0, :] = self.B[1, :] |
| lstsq_A[1, :] = self.B[3, :] |
| print "System of Equations coefficients:" |
| print lstsq_A |
| print "det", numpy.linalg.det(lstsq_A) |
| |
| out_x = numpy.linalg.lstsq( |
| lstsq_A, |
| numpy.matrix([[self.A[1, 2]], [self.A[3, 2]]]))[0] |
| self.K[1, 2] = -lstsq_A[0, 0] * (self.K[0, 2] - out_x[0]) / lstsq_A[0, 1] + out_x[1] |
| |
| print "K unaugmented" |
| print self.K |
| print "B * K unaugmented" |
| print self.B * self.K |
| F = self.A - self.B * self.K |
| print "A - B * K unaugmented" |
| print F |
| print "eigenvalues" |
| print numpy.linalg.eig(F)[0] |
| |
| self.rpl = .05 |
| self.ipl = 0.008 |
| self.PlaceObserverPoles([self.rpl + 1j * self.ipl, |
| self.rpl + 1j * self.ipl, |
| self.rpl - 1j * self.ipl, |
| self.rpl - 1j * self.ipl]) |
| |
| # The box formed by U_min and U_max must encompass all possible values, |
| # or else Austin's code gets angry. |
| self.U_max = numpy.matrix([[12.0], [12.0]]) |
| self.U_min = numpy.matrix([[-12.0], [-12.0]]) |
| |
| # Compute the steady state velocities for a given applied voltage. |
| # The top and bottom of the claw should spin at the same rate if the |
| # physics is right. |
| X_ss = numpy.matrix([[0], [0], [0.0], [0]]) |
| |
| U = numpy.matrix([[1.0],[1.0]]) |
| A = self.A |
| B = self.B |
| #X_ss[2, 0] = X_ss[2, 0] * A[2, 2] + B[2, 0] * U[0, 0] |
| X_ss[2, 0] = 1 / (1 - A[2, 2]) * B[2, 0] * U[0, 0] |
| #X_ss[3, 0] = X_ss[3, 0] * A[3, 3] + X_ss[2, 0] * A[3, 2] + B[3, 0] * U[0, 0] + B[3, 1] * U[1, 0] |
| #X_ss[3, 0] * (1 - A[3, 3]) = X_ss[2, 0] * A[3, 2] + B[3, 0] * U[0, 0] + B[3, 1] * U[1, 0] |
| X_ss[3, 0] = 1 / (1 - A[3, 3]) * (X_ss[2, 0] * A[3, 2] + B[3, 1] * U[1, 0] + B[3, 0] * U[0, 0]) |
| #X_ss[3, 0] = 1 / (1 - A[3, 3]) / (1 - A[2, 2]) * B[2, 0] * U[0, 0] * A[3, 2] + B[3, 0] * U[0, 0] + B[3, 1] * U[1, 0] |
| X_ss[0, 0] = A[0, 2] * X_ss[2, 0] + B[0, 0] * U[0, 0] |
| X_ss[1, 0] = A[1, 2] * X_ss[2, 0] + A[1, 3] * X_ss[3, 0] + B[1, 0] * U[0, 0] + B[1, 1] * U[1, 0] |
| |
| print "X_ss", X_ss |
| |
| self.U_poly = polytope.HPolytope( |
| numpy.matrix([[1, 0], |
| [-1, 0], |
| [0, 1], |
| [0, -1]]), |
| numpy.matrix([[12], |
| [12], |
| [12], |
| [12]])) |
| |
| self.InitializeState() |
| |
| |
| class ClawDeltaU(Claw): |
| def __init__(self, name="Claw"): |
| super(ClawDeltaU, self).__init__(name) |
| A_unaugmented = self.A |
| B_unaugmented = self.B |
| C_unaugmented = self.C |
| |
| self.A = numpy.matrix([[0.0, 0.0, 0.0, 0.0, 0.0], |
| [0.0, 0.0, 0.0, 0.0, 0.0], |
| [0.0, 0.0, 0.0, 0.0, 0.0], |
| [0.0, 0.0, 0.0, 0.0, 0.0], |
| [0.0, 0.0, 0.0, 0.0, 1.0]]) |
| self.A[0:4, 0:4] = A_unaugmented |
| self.A[0:4, 4] = B_unaugmented[0:4, 0] |
| |
| self.B = numpy.matrix([[0.0, 0.0], |
| [0.0, 0.0], |
| [0.0, 0.0], |
| [0.0, 0.0], |
| [1.0, 0.0]]) |
| self.B[0:4, 1] = B_unaugmented[0:4, 1] |
| |
| self.C = numpy.concatenate((C_unaugmented, numpy.matrix([[0.0], [0.0]])), |
| axis=1) |
| self.D = numpy.matrix([[0.0, 0.0], |
| [0.0, 0.0]]) |
| |
| #self.PlaceControllerPoles([0.55, 0.35, 0.55, 0.35, 0.80]) |
| self.Q = numpy.matrix([[(1.0 / (0.04 ** 2.0)), 0.0, 0.0, 0.0, 0.0], |
| [0.0, (1.0 / (0.01 ** 2)), 0.0, 0.0, 0.0], |
| [0.0, 0.0, 0.01, 0.0, 0.0], |
| [0.0, 0.0, 0.0, 0.08, 0.0], |
| [0.0, 0.0, 0.0, 0.0, (1.0 / (10.0 ** 2))]]) |
| |
| self.R = numpy.matrix([[0.000001, 0.0], |
| [0.0, 1.0 / (10.0 ** 2.0)]]) |
| self.K = controls.dlqr(self.A, self.B, self.Q, self.R) |
| |
| self.K = numpy.matrix([[50.0, 0.0, 10.0, 0.0, 1.0], |
| [50.0, 0.0, 10.0, 0.0, 1.0]]) |
| #self.K = numpy.matrix([[50.0, -100.0, 0, -10, 0], |
| # [50.0, 100.0, 0, 10, 0]]) |
| |
| controlability = controls.ctrb(self.A, self.B); |
| print "Rank of augmented controlability matrix.", numpy.linalg.matrix_rank(controlability) |
| |
| print "K" |
| print self.K |
| print "Placed controller poles are" |
| print numpy.linalg.eig(self.A - self.B * self.K)[0] |
| print [numpy.abs(x) for x in numpy.linalg.eig(self.A - self.B * self.K)[0]] |
| |
| self.rpl = .05 |
| self.ipl = 0.008 |
| self.PlaceObserverPoles([self.rpl + 1j * self.ipl, 0.10, 0.09, |
| self.rpl - 1j * self.ipl, 0.90]) |
| #print "A is" |
| #print self.A |
| #print "L is" |
| #print self.L |
| #print "C is" |
| #print self.C |
| #print "A - LC is" |
| #print self.A - self.L * self.C |
| |
| #print "Placed observer poles are" |
| #print numpy.linalg.eig(self.A - self.L * self.C)[0] |
| |
| self.U_max = numpy.matrix([[12.0], [12.0]]) |
| self.U_min = numpy.matrix([[-12.0], [-12.0]]) |
| |
| self.InitializeState() |
| |
| |
| def FullSeparationPriority(claw, U): |
| bottom_u = U[0, 0] |
| top_u = U[1, 0] |
| |
| if bottom_u > claw.U_max[0, 0]: |
| top_u -= bottom_u - claw.U_max[0, 0] |
| if top_u < claw.U_min[1, 0]: |
| top_u = claw.U_min[1, 0] |
| |
| bottom_u = claw.U_max[0, 0] |
| if top_u > claw.U_max[1, 0]: |
| bottom_u -= top_u - claw.U_max[1, 0] |
| if bottom_u < claw.U_min[0, 0]: |
| bottom_u = claw.U_min[0, 0] |
| |
| top_u = claw.U_max[1, 0] |
| if top_u < claw.U_min[1, 0]: |
| bottom_u -= top_u - claw.U_min[1, 0] |
| if bottom_u > claw.U_max[0, 0]: |
| bottom_u = claw.U_max[0, 0] |
| |
| top_u = claw.U_min[1, 0] |
| if bottom_u < claw.U_min[0, 0]: |
| top_u -= bottom_u - claw.U_min[0, 0] |
| if top_u > claw.U_max[1, 0]: |
| top_u = claw.U_max[1, 0] |
| |
| bottom_u = claw.U_min[0, 0] |
| |
| return numpy.matrix([[bottom_u], [top_u]]) |
| |
| def ScaleU(claw, U, K, error): |
| """Clips U as necessary. |
| |
| Args: |
| claw: claw object containing moments of inertia and U limits. |
| U: Input matrix to clip as necessary. |
| """ |
| bottom_u = U[0, 0] |
| top_u = U[1, 0] |
| |
| if (bottom_u > claw.U_max[0, 0] or bottom_u < claw.U_min[0, 0] or |
| top_u > claw.U_max[0, 0] or top_u < claw.U_min[0, 0]): |
| |
| position_K = K[:, 0:2] |
| velocity_K = K[:, 2:] |
| position_error = error[0:2, 0] |
| velocity_error = error[2:, 0] |
| |
| # H * U <= k |
| # U = UPos + UVel |
| # H * (UPos + UVel) <= k |
| # H * UPos <= k - H * UVel |
| # |
| # Now, we can do a coordinate transformation and say the following. |
| # |
| # UPos = position_K * position_error |
| # (H * position_K) * position_error <= k - H * UVel |
| # |
| # Add in the constraint that 0 <= t <= 1 |
| # Now, there are 2 ways this can go. Either we have a region, or we don't |
| # have a region. If we have a region, then pick the largest t and go for it. |
| # If we don't have a region, we need to pick a good comprimise. |
| |
| # First prototype! -> Only cap the position power, leave the velocity power active. |
| |
| #u_velocity = velocity_K * velocity_error |
| #u_position = position_K * position_error |
| #scalar = min(1.0, claw.U_max[0, 0] / max(numpy.abs(u_position[0, 0]), numpy.abs(u_position[1, 0]))) |
| #return u_velocity + scalar * u_position |
| |
| pos_poly = polytope.HPolytope( |
| claw.U_poly.H * position_K, |
| claw.U_poly.k - claw.U_poly.H * velocity_K * velocity_error) |
| |
| adjusted_pos_error = polydrivetrain.CoerceGoal(pos_poly, numpy.matrix([[position_error[1, 0], -position_error[0, 0]]]), 0.0, position_error) |
| |
| return velocity_K * velocity_error + position_K * adjusted_pos_error |
| |
| #U = Kpos * poserror + Kvel * velerror |
| |
| #scalar = claw.U_max[0, 0] / max(numpy.abs(top_u), numpy.abs(bottom_u)) |
| |
| #top_u *= scalar |
| #bottom_u *= scalar |
| |
| return numpy.matrix([[bottom_u], [top_u]]) |
| |
| def AverageUFix(claw, U, preserve_v3=False): |
| """Clips U as necessary. |
| |
| Args: |
| claw: claw object containing moments of inertia and U limits. |
| U: Input matrix to clip as necessary. |
| preserve_v3: There are two ways to attempt to clip the voltages: |
| -If you preserve the imaginary v3, then it will attempt to |
| preserve the effect on the separation of the two claws. |
| If it is not able to do this, then it calls itself with preserve_v3 |
| set to False. |
| -If you preserve the ratio of the voltage of the bottom and the top, |
| then it will attempt to preserve the ratio of those two. This is |
| equivalent to preserving the ratio of v3 and the bottom voltage. |
| """ |
| bottom_u = U[0, 0] |
| top_u = U[1, 0] |
| seperation_u = top_u - bottom_u * claw.J_top / claw.J_bottom |
| |
| bottom_bad = bottom_u > claw.U_max[0, 0] or bottom_u < claw.U_min[0, 0] |
| top_bad = top_u > claw.U_max[0, 0] or top_u < claw.U_min[0, 0] |
| |
| scalar = claw.U_max[0, 0] / max(numpy.abs(top_u), numpy.abs(bottom_u)) |
| if bottom_bad and preserve_v3: |
| bottom_u *= scalar |
| top_u = seperation_u + bottom_u * claw.J_top / claw.J_bottom |
| if abs(top_u) > claw.U_max[0, 0]: |
| return AverageUFix(claw, U, preserve_v3=False) |
| elif top_bad and preserve_v3: |
| top_u *= scalar |
| bottom_u = (top_u - seperation_u) * claw.J_bottom / claw.J_top |
| if abs(bottom_u) > claw.U_max[0, 0]: |
| return AverageUFix(claw, U, preserve_v3=False) |
| elif (bottom_bad or top_bad) and not preserve_v3: |
| top_u *= scalar |
| bottom_u *= scalar |
| print "Scaling" |
| |
| return numpy.matrix([[bottom_u], [top_u]]) |
| |
| def ClipDeltaU(claw, U): |
| delta_u = U[0, 0] |
| top_u = U[1, 0] |
| old_bottom_u = claw.X[4, 0] |
| |
| # TODO(austin): Preserve the difference between the top and bottom power. |
| new_unclipped_bottom_u = old_bottom_u + delta_u |
| |
| #print "Bottom is", new_unclipped_bottom_u, "Top is", top_u |
| if new_unclipped_bottom_u > claw.U_max[0, 0]: |
| top_u -= new_unclipped_bottom_u - claw.U_max[0, 0] |
| new_unclipped_bottom_u = claw.U_max[0, 0] |
| if top_u > claw.U_max[1, 0]: |
| new_unclipped_bottom_u -= top_u - claw.U_max[1, 0] |
| top_u = claw.U_max[1, 0] |
| if top_u < claw.U_min[1, 0]: |
| new_unclipped_bottom_u -= top_u - claw.U_min[1, 0] |
| top_u = claw.U_min[1, 0] |
| if new_unclipped_bottom_u < claw.U_min[0, 0]: |
| top_u -= new_unclipped_bottom_u - claw.U_min[0, 0] |
| new_unclipped_bottom_u = claw.U_min[0, 0] |
| |
| new_bottom_u = numpy.clip(new_unclipped_bottom_u, claw.U_min[0, 0], claw.U_max[0, 0]) |
| new_top_u = numpy.clip(top_u, claw.U_min[1, 0], claw.U_max[1, 0]) |
| |
| return numpy.matrix([[new_bottom_u - old_bottom_u], [new_top_u]]) |
| |
| def main(argv): |
| # Simulate the response of the system to a step input. |
| #claw = ClawDeltaU() |
| #simulated_x = [] |
| #for _ in xrange(100): |
| # claw.Update(numpy.matrix([[12.0]])) |
| # simulated_x.append(claw.X[0, 0]) |
| |
| #pylab.plot(range(100), simulated_x) |
| #pylab.show() |
| |
| # Simulate the closed loop response of the system. |
| claw = Claw("TopClaw") |
| t = [] |
| close_loop_x_bottom = [] |
| close_loop_x_sep = [] |
| actual_sep = [] |
| actual_x_bottom = [] |
| close_loop_x_top = [] |
| close_loop_u_bottom = [] |
| close_loop_u_top = [] |
| R = numpy.matrix([[0.0], [0.00], [0.0], [0.0]]) |
| claw.X[0, 0] = 1.0 |
| claw.X[1, 0] = 0.0 |
| claw.X[2, 0] = 0.0 |
| claw.X[3, 0] = 0.0 |
| claw.X_hat = claw.X |
| #X_actual = claw.X |
| for i in xrange(100): |
| #print "Error is", (R - claw.X_hat) |
| U = claw.K * (R - claw.X) |
| #U = numpy.clip(claw.K * (R - claw.X_hat), claw.U_min, claw.U_max) |
| #U = FullSeparationPriority(claw, U) |
| #U = AverageUFix(claw, U, preserve_v3=False) |
| #U = claw.K * (R - claw.X_hat) |
| #U = ClipDeltaU(claw, U) |
| # TODO(austin): This scales the velocity power as well, which is really bad. Need to just scale the position power. |
| U = ScaleU(claw, U, claw.K, R - claw.X) |
| claw.UpdateObserver(U) |
| claw.Update(U) |
| #X_actual = claw.A_actual * X_actual + claw.B_actual * U |
| #claw.Y = claw.C * X_actual |
| close_loop_x_bottom.append(claw.X[0, 0] * 10) |
| close_loop_u_bottom.append(U[0, 0]) |
| #actual_sep.append(X_actual[2, 0] * 100) |
| #actual_x_bottom.append(X_actual[0, 0] * 10) |
| close_loop_x_sep.append(claw.X[1, 0] * 10) |
| close_loop_x_top.append((claw.X[1, 0] + claw.X[0, 0]) * 10) |
| close_loop_u_top.append(U[1, 0]) |
| t.append(0.01 * i) |
| |
| pylab.plot(t, close_loop_x_bottom, label='x bottom * 10') |
| pylab.plot(t, close_loop_x_sep, label='separation * 10') |
| #pylab.plot(t, actual_x_bottom, label='true x bottom') |
| #pylab.plot(t, actual_sep, label='true separation') |
| pylab.plot(t, close_loop_x_top, label='x top * 10') |
| pylab.plot(t, close_loop_u_bottom, label='u bottom') |
| pylab.plot(t, close_loop_u_top, label='u top') |
| pylab.legend() |
| pylab.show() |
| |
| # Write the generated constants out to a file. |
| if len(argv) != 3: |
| print "Expected .h file name and .cc file name for the claw." |
| else: |
| claw = Claw("Claw") |
| loop_writer = control_loop.ControlLoopWriter("Claw", [claw]) |
| if argv[1][-3:] == '.cc': |
| loop_writer.Write(argv[2], argv[1]) |
| else: |
| loop_writer.Write(argv[1], argv[2]) |
| |
| if __name__ == '__main__': |
| sys.exit(main(sys.argv)) |