| #!/usr/bin/python |
| |
| from __future__ import print_function |
| |
| from matplotlib import pylab |
| import gflags |
| import glog |
| import numpy |
| import scipy |
| import scipy.integrate |
| import sys |
| |
| from frc971.control_loops.python import polydrivetrain |
| import y2018.control_loops.python.drivetrain |
| |
| """This file is my playground for implementing spline following. |
| |
| All splines here are cubic bezier splines. See |
| https://en.wikipedia.org/wiki/B%C3%A9zier_curve for more details. |
| """ |
| |
| FLAGS = gflags.FLAGS |
| |
| |
| def spline(alpha, control_points): |
| """Computes a Cubic Bezier curve. |
| |
| Args: |
| alpha: scalar or list of spline parameters to calculate the curve at. |
| control_points: n x 4 matrix of control points. n[:, 0] is the |
| starting point, and n[:, 3] is the ending point. |
| |
| Returns: |
| n x m matrix of spline points. n is the dimension of the control |
| points, and m is the number of points in 'alpha'. |
| """ |
| if numpy.isscalar(alpha): |
| alpha = [alpha] |
| alpha_matrix = [[(1.0 - a)**3.0, 3.0 * (1.0 - a)**2.0 * a, |
| 3.0 * (1.0 - a) * a**2.0, a**3.0] for a in alpha] |
| |
| return control_points * numpy.matrix(alpha_matrix).T |
| |
| |
| def dspline(alpha, control_points): |
| """Computes the derivative of a Cubic Bezier curve wrt alpha. |
| |
| Args: |
| alpha: scalar or list of spline parameters to calculate the curve at. |
| control_points: n x 4 matrix of control points. n[:, 0] is the |
| starting point, and n[:, 3] is the ending point. |
| |
| Returns: |
| n x m matrix of spline point derivatives. n is the dimension of the |
| control points, and m is the number of points in 'alpha'. |
| """ |
| if numpy.isscalar(alpha): |
| alpha = [alpha] |
| dalpha_matrix = [[ |
| -3.0 * (1.0 - a)**2.0, 3.0 * (1.0 - a)**2.0 + -2.0 * 3.0 * |
| (1.0 - a) * a, -3.0 * a**2.0 + 2.0 * 3.0 * (1.0 - a) * a, 3.0 * a**2.0 |
| ] for a in alpha] |
| |
| return control_points * numpy.matrix(dalpha_matrix).T |
| |
| |
| def ddspline(alpha, control_points): |
| """Computes the second derivative of a Cubic Bezier curve wrt alpha. |
| |
| Args: |
| alpha: scalar or list of spline parameters to calculate the curve at. |
| control_points: n x 4 matrix of control points. n[:, 0] is the |
| starting point, and n[:, 3] is the ending point. |
| |
| Returns: |
| n x m matrix of spline point second derivatives. n is the dimension of |
| the control points, and m is the number of points in 'alpha'. |
| """ |
| if numpy.isscalar(alpha): |
| alpha = [alpha] |
| ddalpha_matrix = [[ |
| 2.0 * 3.0 * (1.0 - a), |
| -2.0 * 3.0 * (1.0 - a) + -2.0 * 3.0 * (1.0 - a) + 2.0 * 3.0 * a, |
| -2.0 * 3.0 * a + 2.0 * 3.0 * (1.0 - a) - 2.0 * 3.0 * a, |
| 2.0 * 3.0 * a |
| ] for a in alpha] |
| |
| return control_points * numpy.matrix(ddalpha_matrix).T |
| |
| |
| def dddspline(alpha, control_points): |
| """Computes the third derivative of a Cubic Bezier curve wrt alpha. |
| |
| Args: |
| alpha: scalar or list of spline parameters to calculate the curve at. |
| control_points: n x 4 matrix of control points. n[:, 0] is the |
| starting point, and n[:, 3] is the ending point. |
| |
| Returns: |
| n x m matrix of spline point second derivatives. n is the dimension of |
| the control points, and m is the number of points in 'alpha'. |
| """ |
| if numpy.isscalar(alpha): |
| alpha = [alpha] |
| ddalpha_matrix = [[ |
| -2.0 * 3.0, |
| 2.0 * 3.0 + 2.0 * 3.0 + 2.0 * 3.0, |
| -2.0 * 3.0 - 2.0 * 3.0 - 2.0 * 3.0, |
| 2.0 * 3.0 |
| ] for a in alpha] |
| |
| return control_points * numpy.matrix(ddalpha_matrix).T |
| |
| |
| def spline_theta(alpha, control_points, dspline_points=None): |
| """Computes the heading of a robot following a Cubic Bezier curve at alpha. |
| |
| Args: |
| alpha: scalar or list of spline parameters to calculate the heading at. |
| control_points: n x 4 matrix of control points. n[:, 0] is the |
| starting point, and n[:, 3] is the ending point. |
| |
| Returns: |
| m array of spline point headings. m is the number of points in 'alpha'. |
| """ |
| if dspline_points is None: |
| dspline_points = dspline(alpha, control_points) |
| |
| return numpy.arctan2( |
| numpy.array(dspline_points)[1, :], numpy.array(dspline_points)[0, :]) |
| |
| |
| def dspline_theta(alpha, |
| control_points, |
| dspline_points=None, |
| ddspline_points=None): |
| """Computes the derivative of the heading at alpha. |
| |
| This is the derivative of spline_theta wrt alpha. |
| |
| Args: |
| alpha: scalar or list of spline parameters to calculate the derivative |
| of the heading at. |
| control_points: n x 4 matrix of control points. n[:, 0] is the |
| starting point, and n[:, 3] is the ending point. |
| |
| Returns: |
| m array of spline point heading derivatives. m is the number of points |
| in 'alpha'. |
| """ |
| if dspline_points is None: |
| dspline_points = dspline(alpha, control_points) |
| |
| if ddspline_points is None: |
| ddspline_points = ddspline(alpha, control_points) |
| |
| dx = numpy.array(dspline_points)[0, :] |
| dy = numpy.array(dspline_points)[1, :] |
| |
| ddx = numpy.array(ddspline_points)[0, :] |
| ddy = numpy.array(ddspline_points)[1, :] |
| |
| return 1.0 / (dx**2.0 + dy**2.0) * (dx * ddy - dy * ddx) |
| |
| |
| def ddspline_theta(alpha, |
| control_points, |
| dspline_points=None, |
| ddspline_points=None, |
| dddspline_points=None): |
| """Computes the second derivative of the heading at alpha. |
| |
| This is the second derivative of spline_theta wrt alpha. |
| |
| Args: |
| alpha: scalar or list of spline parameters to calculate the second |
| derivative of the heading at. |
| control_points: n x 4 matrix of control points. n[:, 0] is the |
| starting point, and n[:, 3] is the ending point. |
| |
| Returns: |
| m array of spline point heading second derivatives. m is the number of |
| points in 'alpha'. |
| """ |
| if dspline_points is None: |
| dspline_points = dspline(alpha, control_points) |
| |
| if ddspline_points is None: |
| ddspline_points = ddspline(alpha, control_points) |
| |
| if dddspline_points is None: |
| dddspline_points = dddspline(alpha, control_points) |
| |
| dddspline_points = dddspline(alpha, control_points) |
| |
| dx = numpy.array(dspline_points)[0, :] |
| dy = numpy.array(dspline_points)[1, :] |
| |
| ddx = numpy.array(ddspline_points)[0, :] |
| ddy = numpy.array(ddspline_points)[1, :] |
| |
| dddx = numpy.array(dddspline_points)[0, :] |
| dddy = numpy.array(dddspline_points)[1, :] |
| |
| return -1.0 / ((dx**2.0 + dy**2.0)**2.0) * (dx * ddy - dy * ddx) * 2.0 * ( |
| dy * ddy + dx * ddx) + 1.0 / (dx**2.0 + dy**2.0) * (dx * dddy - dy * |
| dddx) |
| |
| |
| class Path(object): |
| """Represents a path to follow.""" |
| def __init__(self, control_points): |
| """Constructs a path given the control points.""" |
| self._control_points = control_points |
| |
| def spline_velocity(alpha): |
| return numpy.linalg.norm(dspline(alpha, self._control_points), axis=0) |
| |
| self._point_distances = [0.0] |
| num_alpha = 100 |
| # Integrate the xy velocity as a function of alpha for each step in the |
| # table to get an alpha -> distance calculation. Gaussian Quadrature |
| # is quite accurate, so we can get away with fewer points here than we |
| # might think. |
| for alpha in numpy.linspace(0.0, 1.0, num_alpha)[:-1]: |
| self._point_distances.append( |
| scipy.integrate.fixed_quad(spline_velocity, alpha, alpha + 1.0 |
| / (num_alpha - 1.0))[0] + |
| self._point_distances[-1]) |
| |
| def distance_to_alpha(self, distance): |
| """Converts distances along the spline to alphas. |
| |
| Args: |
| distance: A scalar or array of distances to convert |
| |
| Returns: |
| An array of distances, (1 big if the input was a scalar) |
| """ |
| if numpy.isscalar(distance): |
| return numpy.array([self._distance_to_alpha_scalar(distance)]) |
| else: |
| return numpy.array([self._distance_to_alpha_scalar(d) for d in distance]) |
| |
| def _distance_to_alpha_scalar(self, distance): |
| """Helper to compute alpha for a distance for a single scalar.""" |
| if distance <= 0.0: |
| return 0.0 |
| elif distance >= self.length(): |
| return 1.0 |
| after_index = numpy.searchsorted( |
| self._point_distances, distance, side='right') |
| before_index = after_index - 1 |
| |
| # Linearly interpolate alpha from our (sorted) distance table. |
| return (distance - self._point_distances[before_index]) / ( |
| self._point_distances[after_index] - |
| self._point_distances[before_index]) * (1.0 / ( |
| len(self._point_distances) - 1.0)) + float(before_index) / ( |
| len(self._point_distances) - 1.0) |
| |
| def length(self): |
| """Returns the length of the spline (in meters)""" |
| return self._point_distances[-1] |
| |
| # TODO(austin): need a better name... |
| def xy(self, distance): |
| """Returns the xy position as a function of distance.""" |
| return spline(self.distance_to_alpha(distance), self._control_points) |
| |
| # TODO(austin): need a better name... |
| def dxy(self, distance): |
| """Returns the xy velocity as a function of distance.""" |
| dspline_point = dspline( |
| self.distance_to_alpha(distance), self._control_points) |
| return dspline_point / numpy.linalg.norm(dspline_point, axis=0) |
| |
| # TODO(austin): need a better name... |
| def ddxy(self, distance): |
| """Returns the xy acceleration as a function of distance.""" |
| alpha = self.distance_to_alpha(distance) |
| dspline_points = dspline(alpha, self._control_points) |
| ddspline_points = ddspline(alpha, self._control_points) |
| |
| norm = numpy.linalg.norm( |
| dspline_points, axis=0)**2.0 |
| |
| return ddspline_points / norm - numpy.multiply( |
| dspline_points, (numpy.array(dspline_points)[0, :] * |
| numpy.array(ddspline_points)[0, :] + |
| numpy.array(dspline_points)[1, :] * |
| numpy.array(ddspline_points)[1, :]) / (norm**2.0)) |
| |
| def theta(self, distance, dspline_points=None): |
| """Returns the heading as a function of distance.""" |
| return spline_theta( |
| self.distance_to_alpha(distance), |
| self._control_points, |
| dspline_points=dspline_points) |
| |
| def dtheta(self, distance, dspline_points=None, ddspline_points=None): |
| """Returns the angular velocity as a function of distance.""" |
| alpha = self.distance_to_alpha(distance) |
| if dspline_points is None: |
| dspline_points = dspline(alpha, self._control_points) |
| if ddspline_points is None: |
| ddspline_points = ddspline(alpha, self._control_points) |
| |
| dtheta_points = dspline_theta(alpha, self._control_points, |
| dspline_points, ddspline_points) |
| |
| return dtheta_points / numpy.linalg.norm(dspline_points, axis=0) |
| |
| def ddtheta(self, |
| distance, |
| dspline_points=None, |
| ddspline_points=None, |
| dddspline_points=None): |
| """Returns the angular acceleration as a function of distance.""" |
| alpha = self.distance_to_alpha(distance) |
| if dspline_points is None: |
| dspline_points = dspline(alpha, self._control_points) |
| if ddspline_points is None: |
| ddspline_points = ddspline(alpha, self._control_points) |
| if dddspline_points is None: |
| dddspline_points = dddspline(alpha, self._control_points) |
| |
| dtheta_points = dspline_theta(alpha, self._control_points, |
| dspline_points, ddspline_points) |
| ddtheta_points = ddspline_theta(alpha, self._control_points, |
| dspline_points, ddspline_points, |
| dddspline_points) |
| |
| # TODO(austin): Factor out the d^alpha/dd^2. |
| return ddtheta_points / numpy.linalg.norm( |
| dspline_points, axis=0)**2.0 - numpy.multiply( |
| dtheta_points, (numpy.array(dspline_points)[0, :] * |
| numpy.array(ddspline_points)[0, :] + |
| numpy.array(dspline_points)[1, :] * |
| numpy.array(ddspline_points)[1, :]) / |
| ((numpy.array(dspline_points)[0, :]**2.0 + |
| numpy.array(dspline_points)[1, :]**2.0)**2.0)) |
| |
| |
| |
| def main(argv): |
| # Build up the control point matrix |
| start = numpy.matrix([[0.0, 0.0]]).T |
| c1 = numpy.matrix([[0.5, 0.0]]).T |
| c2 = numpy.matrix([[0.5, 1.0]]).T |
| end = numpy.matrix([[1.0, 1.0]]).T |
| control_points = numpy.hstack((start, c1, c2, end)) |
| |
| # The alphas to plot |
| alphas = numpy.linspace(0.0, 1.0, 1000) |
| |
| # Compute x, y and the 3 derivatives |
| spline_points = spline(alphas, control_points) |
| dspline_points = dspline(alphas, control_points) |
| ddspline_points = ddspline(alphas, control_points) |
| dddspline_points = dddspline(alphas, control_points) |
| |
| # Compute theta and the two derivatives |
| theta = spline_theta(alphas, control_points, dspline_points=dspline_points) |
| dtheta = dspline_theta( |
| alphas, control_points, dspline_points=dspline_points) |
| ddtheta = ddspline_theta( |
| alphas, |
| control_points, |
| dspline_points=dspline_points, |
| dddspline_points=dddspline_points) |
| |
| # Plot the control points and the spline. |
| pylab.figure() |
| pylab.plot( |
| numpy.array(control_points)[0, :], |
| numpy.array(control_points)[1, :], |
| '-o', |
| label='control') |
| pylab.plot( |
| numpy.array(spline_points)[0, :], |
| numpy.array(spline_points)[1, :], |
| label='spline') |
| pylab.legend() |
| |
| # For grins, confirm that the double integral of the acceleration (with |
| # respect to the spline parameter) matches the position. This lets us |
| # confirm that the derivatives are consistent. |
| xint_plot = numpy.matrix(numpy.zeros((2, len(alphas)))) |
| dxint_plot = xint_plot.copy() |
| xint = spline_points[:, 0].copy() |
| dxint = dspline_points[:, 0].copy() |
| xint_plot[:, 0] = xint |
| dxint_plot[:, 0] = dxint |
| for i in range(len(alphas) - 1): |
| xint += (alphas[i + 1] - alphas[i]) * dxint |
| dxint += (alphas[i + 1] - alphas[i]) * ddspline_points[:, i] |
| xint_plot[:, i + 1] = xint |
| dxint_plot[:, i + 1] = dxint |
| |
| # Integrate up the spline velocity and heading to confirm that given a |
| # velocity (as a function of the spline parameter) and angle, we will move |
| # from the starting point to the ending point. |
| thetaint_plot = numpy.zeros((len(alphas),)) |
| thetaint = theta[0] |
| dthetaint_plot = numpy.zeros((len(alphas),)) |
| dthetaint = dtheta[0] |
| thetaint_plot[0] = thetaint |
| dthetaint_plot[0] = dthetaint |
| |
| txint_plot = numpy.matrix(numpy.zeros((2, len(alphas)))) |
| txint = spline_points[:, 0].copy() |
| txint_plot[:, 0] = txint |
| for i in range(len(alphas) - 1): |
| dalpha = alphas[i + 1] - alphas[i] |
| txint += dalpha * numpy.linalg.norm( |
| dspline_points[:, i]) * numpy.matrix( |
| [[numpy.cos(theta[i])], [numpy.sin(theta[i])]]) |
| txint_plot[:, i + 1] = txint |
| thetaint += dalpha * dtheta[i] |
| dthetaint += dalpha * ddtheta[i] |
| thetaint_plot[i + 1] = thetaint |
| dthetaint_plot[i + 1] = dthetaint |
| |
| |
| # Now plot x, dx/dalpha, ddx/ddalpha, dddx/dddalpha, and integrals thereof |
| # to perform consistency checks. |
| pylab.figure() |
| pylab.plot(alphas, numpy.array(spline_points)[0, :], label='x') |
| pylab.plot(alphas, numpy.array(xint_plot)[0, :], label='ix') |
| pylab.plot(alphas, numpy.array(dspline_points)[0, :], label='dx') |
| pylab.plot(alphas, numpy.array(dxint_plot)[0, :], label='idx') |
| pylab.plot(alphas, numpy.array(txint_plot)[0, :], label='tix') |
| pylab.plot(alphas, numpy.array(ddspline_points)[0, :], label='ddx') |
| pylab.plot(alphas, numpy.array(dddspline_points)[0, :], label='dddx') |
| pylab.legend() |
| |
| # Now do the same for y. |
| pylab.figure() |
| pylab.plot(alphas, numpy.array(spline_points)[1, :], label='y') |
| pylab.plot(alphas, numpy.array(xint_plot)[1, :], label='iy') |
| pylab.plot(alphas, numpy.array(dspline_points)[1, :], label='dy') |
| pylab.plot(alphas, numpy.array(dxint_plot)[1, :], label='idy') |
| pylab.plot(alphas, numpy.array(txint_plot)[1, :], label='tiy') |
| pylab.plot(alphas, numpy.array(ddspline_points)[1, :], label='ddy') |
| pylab.plot(alphas, numpy.array(dddspline_points)[1, :], label='dddy') |
| pylab.legend() |
| |
| # And for theta. |
| pylab.figure() |
| pylab.plot(alphas, theta, label='theta') |
| pylab.plot(alphas, dtheta, label='dtheta') |
| pylab.plot(alphas, ddtheta, label='ddtheta') |
| pylab.plot(alphas, thetaint_plot, label='thetai') |
| pylab.plot(alphas, dthetaint_plot, label='dthetai') |
| pylab.plot( |
| alphas, |
| numpy.linalg.norm( |
| numpy.array(dspline_points), axis=0), |
| label='velocity') |
| |
| # Now, repeat as a function of path length as opposed to alpha |
| path = Path(control_points) |
| distance_count = 1000 |
| position = path.xy(0.0) |
| velocity = path.dxy(0.0) |
| theta = path.theta(0.0) |
| omega = path.dtheta(0.0) |
| |
| iposition_plot = numpy.matrix(numpy.zeros((2, distance_count))) |
| ivelocity_plot = numpy.matrix(numpy.zeros((2, distance_count))) |
| iposition_plot[:, 0] = position.copy() |
| ivelocity_plot[:, 0] = velocity.copy() |
| itheta_plot = numpy.zeros((distance_count,)) |
| iomega_plot = numpy.zeros((distance_count,)) |
| itheta_plot[0] = theta |
| iomega_plot[0] = omega |
| |
| distances = numpy.linspace(0.0, path.length(), distance_count) |
| |
| for i in xrange(len(distances) - 1): |
| position += velocity * (distances[i + 1] - distances[i]) |
| velocity += path.ddxy(distances[i]) * (distances[i + 1] - distances[i]) |
| iposition_plot[:, i + 1] = position |
| ivelocity_plot[:, i + 1] = velocity |
| |
| theta += omega * (distances[i + 1] - distances[i]) |
| omega += path.ddtheta(distances[i]) * (distances[i + 1] - distances[i]) |
| itheta_plot[i + 1] = theta |
| iomega_plot[i + 1] = omega |
| |
| pylab.figure() |
| pylab.plot(distances, numpy.array(path.xy(distances))[0, :], label='x') |
| pylab.plot(distances, numpy.array(iposition_plot)[0, :], label='ix') |
| pylab.plot(distances, numpy.array(path.dxy(distances))[0, :], label='dx') |
| pylab.plot(distances, numpy.array(ivelocity_plot)[0, :], label='idx') |
| pylab.plot(distances, numpy.array(path.ddxy(distances))[0, :], label='ddx') |
| pylab.legend() |
| |
| pylab.figure() |
| pylab.plot(distances, numpy.array(path.xy(distances))[1, :], label='y') |
| pylab.plot(distances, numpy.array(iposition_plot)[1, :], label='iy') |
| pylab.plot(distances, numpy.array(path.dxy(distances))[1, :], label='dy') |
| pylab.plot(distances, numpy.array(ivelocity_plot)[1, :], label='idy') |
| pylab.plot(distances, numpy.array(path.ddxy(distances))[1, :], label='ddy') |
| pylab.legend() |
| |
| pylab.figure() |
| pylab.plot(distances, path.theta(distances), label='theta') |
| pylab.plot(distances, itheta_plot, label='itheta') |
| pylab.plot(distances, path.dtheta(distances), label='omega') |
| pylab.plot(distances, iomega_plot, label='iomega') |
| pylab.plot(distances, path.ddtheta(distances), label='alpha') |
| pylab.legend() |
| |
| # TODO(austin): Start creating a velocity plan now that we have all the |
| # derivitives of our spline. |
| |
| velocity_drivetrain = polydrivetrain.VelocityDrivetrainModel( |
| y2018.control_loops.python.drivetrain.kDrivetrain) |
| |
| vmax = numpy.inf |
| vmax = 10.0 |
| lateral_accel_plan = numpy.array(numpy.zeros((distance_count, ))) |
| lateral_accel_plan.fill(vmax) |
| curvature_plan = lateral_accel_plan.copy() |
| |
| longitudal_accel = 15.0 |
| lateral_accel = 20.0 |
| |
| for i, distance in enumerate(distances): |
| lateral_accel_plan[i] = min( |
| lateral_accel_plan[i], |
| numpy.sqrt(lateral_accel / numpy.linalg.norm(path.ddxy(distance)))) |
| |
| # We've now got the equation: K1 (dx/dt)^2 = A * K2 * dx/dt + B * U |
| # We need to find the feasible |
| current_ddtheta = path.ddtheta(distance)[0] |
| current_dtheta = path.dtheta(distance)[0] |
| K1 = numpy.matrix( |
| [[-velocity_drivetrain.robot_radius_l * current_ddtheta], |
| [velocity_drivetrain.robot_radius_r * current_ddtheta]]) |
| K2 = numpy.matrix( |
| [[1.0 - velocity_drivetrain.robot_radius_l * current_dtheta], |
| [1.0 + velocity_drivetrain.robot_radius_r * current_dtheta]]) |
| A = velocity_drivetrain.A_continuous |
| B = velocity_drivetrain.B_continuous |
| |
| # Now, rephrase it as K5 a + K3 v^2 + K4 v = U |
| # Now, rephrase it as K5 a + K3 v^2 + K4 v = U |
| K3 = numpy.linalg.inv(B) * K1 |
| K5 = numpy.linalg.inv(B) * K2 |
| K4 = -K5 |
| |
| x = [] |
| for a, b in [(K3[0, 0], K4[0, 0]), (K3[1, 0], K4[1, 0])]: |
| for c in [12.0, -12.0]: |
| middle = b * b - 4.0 * a * c |
| if middle >= 0.0: |
| x.append((-b + numpy.sqrt(middle)) / (2.0 * a)) |
| x.append((-b - numpy.sqrt(middle)) / (2.0 * a)) |
| |
| maxx = 0.0 |
| for newx in x: |
| if newx < 0.0: |
| continue |
| U = (K3 * newx * newx + K4 * newx).T |
| # TODO(austin): We know that one of these *will* be +-12.0. Only |
| # check the other one. |
| if not (numpy.abs(U) > 12.0 + 1e-6).any(): |
| maxx = max(newx, maxx) |
| |
| if maxx == 0.0: |
| print('Could not solve') |
| curvature_plan[i] = min(lateral_accel_plan[i], maxx) |
| |
| pylab.figure() |
| pylab.plot(distances, lateral_accel_plan, label='accel pass') |
| pylab.plot(distances, curvature_plan, label='voltage limit pass') |
| pylab.xlabel("distance along spline (m)") |
| pylab.ylabel("velocity (m/s)") |
| pylab.legend() |
| |
| pylab.show() |
| |
| |
| if __name__ == '__main__': |
| argv = FLAGS(sys.argv) |
| sys.exit(main(argv)) |