y2018/control_loops/python/extended_lqr.py

#!/usr/bin/python
# This is an initial, hacky implementation of the extended LQR paper. It's just
# a proof of concept, so don't trust it too much.

import numpy
import scipy.optimize
from matplotlib import pylab
import sys

from frc971.control_loops.python import controls


class ArmDynamics(object):
  def __init__(self, dt):
    self.dt = dt

    self.l1 = 1.0
    self.l2 = 0.8
    self.num_states = 4
    self.num_inputs = 2

  def dynamics(self, X, U):
    """Calculates the dynamics for a double jointed arm.

    Args:
      X, numpy.matrix(4, 1), The state.  [theta1, omega1, theta2, omega2]
      U, numpy.matrix(2, 1), The input.  [torque1, torque2]

    Returns:
      numpy.matrix(4, 1), The derivative of the dynamics.
    """
    return numpy.matrix([[X[1, 0]],
                         [U[0, 0]],
                         [X[3, 0]],
                         [U[1, 0]]])

  def discrete_dynamics(self, X, U):
    return RungeKutta(lambda startingX: self.dynamics(startingX, U), X, dt)

  def inverse_discrete_dynamics(self, X, U):
    return RungeKutta(lambda startingX: -self.dynamics(startingX, U), X, dt)

# Simple implementation for a quadratic cost function.
class ArmCostFunction:
  def __init__(self, dt, dynamics):
    self.num_states = 4
    self.num_inputs = 2
    self.dt = dt
    self.dynamics = dynamics

    q_pos = 0.5
    q_vel = 1.65
    self.Q = numpy.matrix(numpy.diag([
        1.0 / (q_pos ** 2.0), 1.0 / (q_vel ** 2.0),
        1.0 / (q_pos ** 2.0), 1.0 / (q_vel ** 2.0)]))

    self.R = numpy.matrix(numpy.diag([1.0 / (12.0 ** 2.0),
                                      1.0 / (12.0 ** 2.0)]))

    final_A = numerical_jacobian_x(self.dynamics.discrete_dynamics,
                                   numpy.matrix(numpy.zeros((4, 1))),
                                   numpy.matrix(numpy.zeros((2, 1))))
    final_B = numerical_jacobian_u(self.dynamics.discrete_dynamics,
                                   numpy.matrix(numpy.zeros((4, 1))),
                                   numpy.matrix(numpy.zeros((2, 1))))
    print 'Final A', final_A
    print 'Final B', final_B
    K, self.S = controls.dlqr(
        final_A, final_B, self.Q, self.R, optimal_cost_function=True) 
    print 'Final eig:', numpy.linalg.eig(final_A - final_B * K)

  def final_cost(self, X, U):
    """Computes the final cost of being at X

    Args:
      X: numpy.matrix(self.num_states, 1)
      U: numpy.matrix(self.num_inputs, 1), ignored

    Returns:
      numpy.matrix(1, 1), The quadratic cost of being at X
    """
    return 0.5 * X.T * self.S * X

  def cost(self, X, U):
    """Computes the incremental cost given a position and U.

    Args:
      X: numpy.matrix(self.num_states, 1)
      U: numpy.matrix(self.num_inputs, 1)

    Returns:
      numpy.matrix(1, 1), The quadratic cost of evaluating U.
    """
    return U.T * self.R * U + X.T * self.Q * X

  def estimate_Q_final(self, X_hat):
    """Returns the quadraticized final Q around X_hat.

    This is calculated by evaluating partial^2 cost(X_hat) / (partial X * partial X)

    Args:
      X_hat: numpy.matrix(self.num_states, 1), The state to quadraticize around.

    Result:
      numpy.matrix(self.num_states, self.num_states)
    """
    zero_U = numpy.matrix(numpy.zeros((self.num_inputs, 1)))
    print 'S', self.S
    print 'Q_final', numerical_jacobian_x_x(self.final_cost, X_hat, zero_U)
    return numerical_jacobian_x_x(self.final_cost, X_hat, zero_U)

  def estimate_partial_cost_partial_x_final(self, X_hat):
    """Returns \frac{\partial cost}{\partial X}(X_hat) for the final cost.

    Args:
      X_hat: numpy.matrix(self.num_states, 1), The state to quadraticize around.

    Result:
      numpy.matrix(self.num_states, 1)
    """
    return numerical_jacobian_x(self.final_cost, X_hat,
                                numpy.matrix(numpy.zeros((self.num_inputs, 1)))).T

  def estimate_q_final(self, X_hat):
    """Returns q evaluated at X_hat for the final cost function."""
    return self.estimate_partial_cost_partial_x_final(X_hat) - self.estimate_Q_final(X_hat) * X_hat



class SkidSteerDynamics(object):
  def __init__(self, dt):
    self.width = 0.2
    self.dt = dt
    self.num_states = 3
    self.num_inputs = 2

  def dynamics(self, X, U):
    """Calculates the dynamics for a 2 wheeled robot.

    Args:
      X, numpy.matrix(3, 1), The state.  [x, y, theta]
      U, numpy.matrix(2, 1), The input.  [left velocity, right velocity]

    Returns:
      numpy.matrix(3, 1), The derivative of the dynamics.
    """
    #return numpy.matrix([[X[1, 0]],
    #                     [X[2, 0]],
    #                     [U[0, 0]]])
    return numpy.matrix([[(U[0, 0] + U[1, 0]) * numpy.cos(X[2, 0]) / 2.0],
                         [(U[0, 0] + U[1, 0]) * numpy.sin(X[2, 0]) / 2.0],
                         [(U[1, 0] - U[0, 0]) / self.width]])

  def discrete_dynamics(self, X, U):
    return RungeKutta(lambda startingX: self.dynamics(startingX, U), X, dt)

  def inverse_discrete_dynamics(self, X, U):
    return RungeKutta(lambda startingX: -self.dynamics(startingX, U), X, dt)


# Simple implementation for a quadratic cost function.
class CostFunction:
  def __init__(self, dt):
    self.num_states = 3
    self.num_inputs = 2
    self.dt = dt
    self.Q = numpy.matrix([[0.1, 0, 0],
                           [0, 0.6, 0],
                           [0, 0, 0.1]]) / self.dt / self.dt
    self.R = numpy.matrix([[0.40, 0],
                           [0, 0.40]]) / self.dt / self.dt

  def final_cost(self, X, U):
    """Computes the final cost of being at X

    Args:
      X: numpy.matrix(self.num_states, 1)
      U: numpy.matrix(self.num_inputs, 1), ignored

    Returns:
      numpy.matrix(1, 1), The quadratic cost of being at X
    """
    return X.T * self.Q * X * 1000

  def cost(self, X, U):
    """Computes the incremental cost given a position and U.

    Args:
      X: numpy.matrix(self.num_states, 1)
      U: numpy.matrix(self.num_inputs, 1)

    Returns:
      numpy.matrix(1, 1), The quadratic cost of evaluating U.
    """
    return U.T * self.R * U + X.T * self.Q * X

  def estimate_Q_final(self, X_hat):
    """Returns the quadraticized final Q around X_hat.

    This is calculated by evaluating partial^2 cost(X_hat) / (partial X * partial X)

    Args:
      X_hat: numpy.matrix(self.num_states, 1), The state to quadraticize around.

    Result:
      numpy.matrix(self.num_states, self.num_states)
    """
    zero_U = numpy.matrix(numpy.zeros((self.num_inputs, 1)))
    return numerical_jacobian_x_x(self.final_cost, X_hat, zero_U)

  def estimate_partial_cost_partial_x_final(self, X_hat):
    """Returns \frac{\partial cost}{\partial X}(X_hat) for the final cost.

    Args:
      X_hat: numpy.matrix(self.num_states, 1), The state to quadraticize around.

    Result:
      numpy.matrix(self.num_states, 1)
    """
    return numerical_jacobian_x(self.final_cost, X_hat, numpy.matrix(numpy.zeros((self.num_inputs, 1)))).T

  def estimate_q_final(self, X_hat):
    """Returns q evaluated at X_hat for the final cost function."""
    return self.estimate_partial_cost_partial_x_final(X_hat) - self.estimate_Q_final(X_hat) * X_hat


def RungeKutta(f, x, dt):
  """4th order RungeKutta integration of F starting at X."""
  a = f(x)
  b = f(x + dt / 2.0 * a)
  c = f(x + dt / 2.0 * b)
  d = f(x + dt * c)
  return x + dt * (a + 2.0 * b + 2.0 * c + d) / 6.0

def numerical_jacobian_x(fn, X, U, epsilon=1e-4):
  """Numerically estimates the jacobian around X, U in X.

  Args:
    fn: A function of X, U.
    X: numpy.matrix(num_states, 1), The state vector to take the jacobian
      around.
    U: numpy.matrix(num_inputs, 1), The input vector to take the jacobian
      around.

  Returns:
    numpy.matrix(num_states, num_states), The jacobian of fn with X as the
      variable.
  """
  num_states = X.shape[0]
  nominal = fn(X, U)
  answer = numpy.matrix(numpy.zeros((nominal.shape[0], num_states)))
  # It's more expensive, but +- epsilon will be more reliable
  for i in range(0, num_states):
    dX_plus = X.copy()
    dX_plus[i] += epsilon
    dX_minus = X.copy()
    dX_minus[i] -= epsilon
    answer[:, i] = (fn(dX_plus, U) - fn(dX_minus, U)) / epsilon / 2.0
  return answer

def numerical_jacobian_u(fn, X, U, epsilon=1e-4):
  """Numerically estimates the jacobian around X, U in U.

  Args:
    fn: A function of X, U.
    X: numpy.matrix(num_states, 1), The state vector to take the jacobian
      around.
    U: numpy.matrix(num_inputs, 1), The input vector to take the jacobian
      around.

  Returns:
    numpy.matrix(num_states, num_inputs), The jacobian of fn with U as the
      variable.
  """
  num_states = X.shape[0]
  num_inputs = U.shape[0]
  nominal = fn(X, U)
  answer = numpy.matrix(numpy.zeros((nominal.shape[0], num_inputs)))
  for i in range(0, num_inputs):
    dU_plus = U.copy()
    dU_plus[i] += epsilon
    dU_minus = U.copy()
    dU_minus[i] -= epsilon
    answer[:, i] = (fn(X, dU_plus) - fn(X, dU_minus)) / epsilon / 2.0
  return answer

def numerical_jacobian_x_x(fn, X, U):
  return numerical_jacobian_x(
      lambda X_inner, U_inner: numerical_jacobian_x(fn, X_inner, U_inner).T, X, U)

def numerical_jacobian_x_u(fn, X, U):
  return numerical_jacobian_x(
      lambda X_inner, U_inner: numerical_jacobian_u(fn, X_inner, U_inner).T, X, U)

def numerical_jacobian_u_x(fn, X, U):
  return numerical_jacobian_u(
      lambda X_inner, U_inner: numerical_jacobian_x(fn, X_inner, U_inner).T, X, U)

def numerical_jacobian_u_u(fn, X, U):
  return numerical_jacobian_u(
      lambda X_inner, U_inner: numerical_jacobian_u(fn, X_inner, U_inner).T, X, U)


class ELQR(object):
  def __init__(self, dynamics, cost):
    self.dynamics = dynamics
    self.cost = cost
  
  def Solve(self, x_hat_initial, horizon, iterations):
    l = horizon
    num_states = self.dynamics.num_states
    num_inputs = self.dynamics.num_inputs
    self.S_bar_t = [numpy.matrix(numpy.zeros((num_states, num_states))) for _ in range(l + 1)]
    self.s_bar_t = [numpy.matrix(numpy.zeros((num_states, 1))) for _ in range(l + 1)]
    self.s_scalar_bar_t = [numpy.matrix(numpy.zeros((1, 1))) for _ in range(l + 1)]

    self.L_t = [numpy.matrix(numpy.zeros((num_inputs, num_states))) for _ in range(l + 1)]
    self.l_t = [numpy.matrix(numpy.zeros((num_inputs, 1))) for _ in range(l + 1)]
    self.L_bar_t = [numpy.matrix(numpy.zeros((num_inputs, num_states))) for _ in range(l + 1)]
    self.l_bar_t = [numpy.matrix(numpy.zeros((num_inputs, 1))) for _ in range(l + 1)]

    self.S_t = [numpy.matrix(numpy.zeros((num_states, num_states))) for _ in range(l + 1)]
    self.s_t = [numpy.matrix(numpy.zeros((num_states, 1))) for _ in range(l + 1)]
    self.s_scalar_t = [numpy.matrix(numpy.zeros((1, 1))) for _ in range(l + 1)]

    self.last_x_hat_t = [numpy.matrix(numpy.zeros((num_states, 1))) for _ in range(l + 1)]

    # Iterate the solver
    for a in range(iterations):
      x_hat = x_hat_initial
      u_t = self.L_t[0] * x_hat + self.l_t[0]
      self.S_bar_t[0] = numpy.matrix(numpy.zeros((num_states, num_states)))
      self.s_bar_t[0] = numpy.matrix(numpy.zeros((num_states, 1)))
      self.s_scalar_bar_t[0] = numpy.matrix([[0]])

      self.last_x_hat_t[0] = x_hat_initial

      Q_t = numerical_jacobian_x_x(self.cost.cost, x_hat_initial, u_t)
      P_t = numerical_jacobian_x_u(self.cost.cost, x_hat_initial, u_t)
      R_t = numerical_jacobian_u_u(self.cost.cost, x_hat_initial, u_t)

      q_t = numerical_jacobian_x(self.cost.cost, x_hat_initial, u_t).T \
          - Q_t * x_hat_initial - P_t.T * u_t
      r_t = numerical_jacobian_u(self.cost.cost, x_hat_initial, u_t).T \
          - P_t * x_hat_initial - R_t * u_t

      q_scalar_t = self.cost.cost(x_hat_initial, u_t) \
          - 0.5 * (x_hat_initial.T * (Q_t * x_hat_initial + P_t.T * u_t) \
          + u_t.T * (P_t * x_hat_initial + R_t * u_t)) \
          - x_hat_initial.T * q_t - u_t.T * r_t

      start_A_t = numerical_jacobian_x(self.dynamics.discrete_dynamics, x_hat_initial, u_t)
      start_B_t = numerical_jacobian_u(self.dynamics.discrete_dynamics, x_hat_initial, u_t)
      x_hat_next = self.dynamics.discrete_dynamics(x_hat_initial, u_t)
      start_c_t = x_hat_next - start_A_t * x_hat_initial - start_B_t * u_t

      B_svd_u, B_svd_sigma_diag, B_svd_v = numpy.linalg.svd(start_B_t)
      B_svd_sigma = numpy.matrix(numpy.zeros(start_B_t.shape))
      B_svd_sigma[0:B_svd_sigma_diag.shape[0], 0:B_svd_sigma_diag.shape[0]] = \
          numpy.diag(B_svd_sigma_diag)

      B_svd_sigma_inv = numpy.matrix(numpy.zeros(start_B_t.shape)).T
      B_svd_sigma_inv[0:B_svd_sigma_diag.shape[0],
                      0:B_svd_sigma_diag.shape[0]] = \
                          numpy.linalg.inv(numpy.diag(B_svd_sigma_diag))
      B_svd_inv = B_svd_v.T * B_svd_sigma_inv * B_svd_u.T

      self.L_bar_t[1] = B_svd_inv
      self.l_bar_t[1] = -B_svd_inv * (start_A_t * x_hat_initial + start_c_t)

      self.S_bar_t[1] = self.L_bar_t[1].T * R_t * self.L_bar_t[1]

      TotalS_1 = start_B_t.T * self.S_t[1] * start_B_t + R_t
      Totals_1 = start_B_t.T * self.S_t[1] * (start_c_t + start_A_t * x_hat_initial) \
          + start_B_t.T * self.s_t[1] + P_t * x_hat_initial + r_t
      Totals_scalar_1 = 0.5 * (start_c_t.T + x_hat_initial.T * start_A_t.T) * self.S_t[1] * (start_c_t + start_A_t * x_hat_initial) \
          + self.s_scalar_t[1] + x_hat_initial.T * q_t + q_scalar_t \
          + 0.5 * x_hat_initial.T * Q_t * x_hat_initial \
          + (start_c_t.T + x_hat_initial.T * start_A_t.T) * self.s_t[1]

      optimal_u_1 = -numpy.linalg.solve(TotalS_1, Totals_1)
      optimal_x_1 = start_A_t * x_hat_initial \
          + start_B_t * optimal_u_1 + start_c_t

      # TODO(austin): Disable this if we are controlable.  It should not be needed then.
      S_bar_1_eigh_eigenvalues, S_bar_1_eigh_eigenvectors = \
          numpy.linalg.eigh(self.S_bar_t[1])
      S_bar_1_eigh = numpy.matrix(numpy.diag(S_bar_1_eigh_eigenvalues))
      S_bar_1_eigh_eigenvalues_stiff = S_bar_1_eigh_eigenvalues.copy()
      for i in range(S_bar_1_eigh_eigenvalues_stiff.shape[0]):
        if abs(S_bar_1_eigh_eigenvalues_stiff[i]) < 1e-8:
          S_bar_1_eigh_eigenvalues_stiff[i] = max(S_bar_1_eigh_eigenvalues_stiff) * 1.0

      S_bar_stiff = S_bar_1_eigh_eigenvectors * numpy.matrix(numpy.diag(S_bar_1_eigh_eigenvalues_stiff)) * S_bar_1_eigh_eigenvectors.T

      print 'Min u', -numpy.linalg.solve(TotalS_1, Totals_1)
      print 'Min x_hat', optimal_x_1
      self.s_bar_t[1] = -self.s_t[1] - (S_bar_stiff + self.S_t[1]) * optimal_x_1
      self.s_scalar_bar_t[1] = 0.5 * (optimal_u_1.T * TotalS_1 * optimal_u_1 \
          - optimal_x_1.T * (S_bar_stiff + self.S_t[1]) * optimal_x_1) \
          + optimal_u_1.T * Totals_1 \
          - optimal_x_1.T * (self.s_bar_t[1] + self.s_t[1]) \
          - self.s_scalar_t[1] + Totals_scalar_1

      print 'optimal_u_1', optimal_u_1
      print 'TotalS_1', TotalS_1
      print 'Totals_1', Totals_1
      print 'Totals_scalar_1', Totals_scalar_1
      print 'overall cost 1', 0.5 * (optimal_u_1.T * TotalS_1 * optimal_u_1) \
                               + optimal_u_1.T * Totals_1 + Totals_scalar_1
      print 'overall cost 0', 0.5 * (x_hat_initial.T * self.S_t[0] * x_hat_initial) \
                              + x_hat_initial.T * self.s_t[0] + self.s_scalar_t[0]

      print 't forward 0'
      print 'x_hat_initial[ 0]: %s' % (x_hat_initial)
      print 'x_hat[%2d]: %s' % (0, x_hat.T)
      print 'x_hat_next[%2d]: %s' % (0, x_hat_next.T)
      print 'u[%2d]: %s' % (0, u_t.T)
      print ('L[ 0]: %s' % (self.L_t[0],)).replace('\n', '\n        ')
      print ('l[ 0]: %s' % (self.l_t[0],)).replace('\n', '\n        ')

      print ('A_t[%2d]: %s' % (0, start_A_t)).replace('\n', '\n         ')
      print ('B_t[%2d]: %s' % (0, start_B_t)).replace('\n', '\n         ')
      print ('c_t[%2d]: %s' % (0, start_c_t)).replace('\n', '\n         ')

      # TODO(austin): optimal_x_1 is x_hat
      x_hat = -numpy.linalg.solve((self.S_t[1] + S_bar_stiff),
                                  (self.s_t[1] + self.s_bar_t[1]))
      print 'new xhat', x_hat

      self.S_bar_t[1] = S_bar_stiff

      self.last_x_hat_t[1] = x_hat

      for t in range(1, l):
        print 't forward', t
        u_t = self.L_t[t] * x_hat + self.l_t[t]

        x_hat_next = self.dynamics.discrete_dynamics(x_hat, u_t)
        A_bar_t = numerical_jacobian_x(self.dynamics.inverse_discrete_dynamics,
                                       x_hat_next, u_t)
        B_bar_t = numerical_jacobian_u(self.dynamics.inverse_discrete_dynamics,
                                       x_hat_next, u_t)
        c_bar_t = x_hat - A_bar_t * x_hat_next - B_bar_t * u_t

        print 'x_hat[%2d]: %s' % (t, x_hat.T)
        print 'x_hat_next[%2d]: %s' % (t, x_hat_next.T)
        print ('L[%2d]: %s' % (t, self.L_t[t],)).replace('\n', '\n        ')
        print ('l[%2d]: %s' % (t, self.l_t[t],)).replace('\n', '\n        ')
        print 'u[%2d]: %s' % (t, u_t.T)

        print ('A_bar_t[%2d]: %s' % (t, A_bar_t)).replace('\n', '\n             ')
        print ('B_bar_t[%2d]: %s' % (t, B_bar_t)).replace('\n', '\n             ')
        print ('c_bar_t[%2d]: %s' % (t, c_bar_t)).replace('\n', '\n             ')

        Q_t = numerical_jacobian_x_x(self.cost.cost, x_hat, u_t)
        P_t = numerical_jacobian_x_u(self.cost.cost, x_hat, u_t)
        R_t = numerical_jacobian_u_u(self.cost.cost, x_hat, u_t)

        q_t = numerical_jacobian_x(self.cost.cost, x_hat, u_t).T \
            - Q_t * x_hat - P_t.T * u_t
        r_t = numerical_jacobian_u(self.cost.cost, x_hat, u_t).T \
            - P_t * x_hat - R_t * u_t

        q_scalar_t = self.cost.cost(x_hat, u_t) \
            - 0.5 * (x_hat.T * (Q_t * x_hat + P_t.T * u_t) \
            + u_t.T * (P_t * x_hat + R_t * u_t)) - x_hat.T * q_t - u_t.T * r_t

        C_bar_t = B_bar_t.T * (self.S_bar_t[t] + Q_t) * A_bar_t + P_t * A_bar_t
        D_bar_t = A_bar_t.T * (self.S_bar_t[t] + Q_t) * A_bar_t
        E_bar_t = B_bar_t.T * (self.S_bar_t[t] + Q_t) * B_bar_t + R_t \
            + P_t * B_bar_t + B_bar_t.T * P_t.T
        d_bar_t = A_bar_t.T * (self.s_bar_t[t] + q_t) \
            + A_bar_t.T * (self.S_bar_t[t] + Q_t) * c_bar_t
        e_bar_t = r_t + P_t * c_bar_t + B_bar_t.T * self.s_bar_t[t] \
            + B_bar_t.T * (self.S_bar_t[t] + Q_t) * c_bar_t

        self.L_bar_t[t + 1] = -numpy.linalg.inv(E_bar_t) * C_bar_t
        self.l_bar_t[t + 1] = -numpy.linalg.inv(E_bar_t) * e_bar_t

        self.S_bar_t[t + 1] = D_bar_t + C_bar_t.T * self.L_bar_t[t + 1]
        self.s_bar_t[t + 1] = d_bar_t + C_bar_t.T * self.l_bar_t[t + 1]
        self.s_scalar_bar_t[t + 1] = \
            -0.5 * e_bar_t.T * numpy.linalg.inv(E_bar_t) * e_bar_t \
            + 0.5 * c_bar_t.T * (self.S_bar_t[t] + Q_t) * c_bar_t \
            + c_bar_t.T * self.s_bar_t[t] + c_bar_t.T * q_t \
            + self.s_scalar_bar_t[t] + q_scalar_t

        x_hat = -numpy.linalg.solve((self.S_t[t + 1] + self.S_bar_t[t + 1]),
                                    (self.s_t[t + 1] + self.s_bar_t[t + 1]))
        self.S_t[l] = self.cost.estimate_Q_final(x_hat)
      self.s_t[l] = self.cost.estimate_q_final(x_hat)
      x_hat = -numpy.linalg.inv(self.S_t[l] + self.S_bar_t[l]) \
          * (self.s_t[l] + self.s_bar_t[l])

      for t in reversed(range(l)):
        print 't backward', t
        # TODO(austin): I don't think we can use L_t like this here.
        # I think we are off by 1 somewhere...
        u_t = self.L_bar_t[t + 1] * x_hat + self.l_bar_t[t + 1]

        x_hat_prev = self.dynamics.inverse_discrete_dynamics(x_hat, u_t)
        print 'x_hat[%2d]: %s' % (t, x_hat.T)
        print 'x_hat_prev[%2d]: %s' % (t, x_hat_prev.T)
        print ('L_bar[%2d]: %s' % (t + 1, self.L_bar_t[t + 1])).replace('\n', '\n            ')
        print ('l_bar[%2d]: %s' % (t + 1, self.l_bar_t[t + 1])).replace('\n', '\n            ')
        print 'u[%2d]: %s' % (t, u_t.T)
        # Now compute the linearized A, B, and C
        # Start by doing it numerically, and then optimize.
        A_t = numerical_jacobian_x(self.dynamics.discrete_dynamics, x_hat_prev, u_t)
        B_t = numerical_jacobian_u(self.dynamics.discrete_dynamics, x_hat_prev, u_t)
        c_t = x_hat - A_t * x_hat_prev - B_t * u_t

        print ('A_t[%2d]: %s' % (t, A_t)).replace('\n', '\n         ')
        print ('B_t[%2d]: %s' % (t, B_t)).replace('\n', '\n         ')
        print ('c_t[%2d]: %s' % (t, c_t)).replace('\n', '\n         ')

        Q_t = numerical_jacobian_x_x(self.cost.cost, x_hat_prev, u_t)
        P_t = numerical_jacobian_x_u(self.cost.cost, x_hat_prev, u_t)
        P_T_t = numerical_jacobian_u_x(self.cost.cost, x_hat_prev, u_t)
        R_t = numerical_jacobian_u_u(self.cost.cost, x_hat_prev, u_t)

        q_t = numerical_jacobian_x(self.cost.cost, x_hat_prev, u_t).T \
            - Q_t * x_hat_prev - P_T_t * u_t
        r_t = numerical_jacobian_u(self.cost.cost, x_hat_prev, u_t).T \
            - P_t * x_hat_prev - R_t * u_t

        q_scalar_t = self.cost.cost(x_hat_prev, u_t) \
            - 0.5 * (x_hat_prev.T * (Q_t * x_hat_prev + P_t.T * u_t) \
            + u_t.T * (P_t * x_hat_prev + R_t * u_t)) \
            - x_hat_prev.T * q_t - u_t.T * r_t

        C_t = P_t + B_t.T * self.S_t[t + 1] * A_t
        D_t = Q_t + A_t.T * self.S_t[t + 1] * A_t
        E_t = R_t + B_t.T * self.S_t[t + 1] * B_t
        d_t = q_t + A_t.T * self.s_t[t + 1] + A_t.T * self.S_t[t + 1] * c_t
        e_t = r_t + B_t.T * self.s_t[t + 1] + B_t.T * self.S_t[t + 1] * c_t
        self.L_t[t] = -numpy.linalg.inv(E_t) * C_t
        self.l_t[t] = -numpy.linalg.inv(E_t) * e_t
        self.s_t[t] = d_t + C_t.T * self.l_t[t]
        self.S_t[t] = D_t + C_t.T * self.L_t[t]
        self.s_scalar_t[t] = q_scalar_t \
            - 0.5 * e_t.T * numpy.linalg.inv(E_t) * e_t \
            + 0.5 * c_t.T * self.S_t[t + 1] * c_t \
            + c_t.T * self.s_t[t + 1] \
            + self.s_scalar_t[t + 1]

        x_hat = -numpy.linalg.solve((self.S_t[t] + self.S_bar_t[t]),
                                    (self.s_t[t] + self.s_bar_t[t]))
        if t == 0:
          self.last_x_hat_t[t] = x_hat_initial
        else:
          self.last_x_hat_t[t] = x_hat

      x_hat_t = [x_hat_initial]

      pylab.figure('states %d' % a)
      pylab.ion()
      for dim in range(num_states):
        pylab.plot(numpy.arange(len(self.last_x_hat_t)),
                   [x_hat_loop[dim, 0] for x_hat_loop in self.last_x_hat_t],
                   marker='o', label='Xhat[%d]' % dim)
      pylab.legend()
      pylab.draw()

      pylab.figure('xy %d' % a)
      pylab.ion()
      pylab.plot([x_hat_loop[0, 0] for x_hat_loop in self.last_x_hat_t],
                 [x_hat_loop[1, 0] for x_hat_loop in self.last_x_hat_t],
                 marker='o', label='trajectory')
      pylab.legend()
      pylab.draw()

    final_u_t = [numpy.matrix(numpy.zeros((num_inputs, 1))) for _ in range(l + 1)]
    cost_to_go = []
    cost_to_come = []
    cost = []
    for t in range(l):
      cost_to_go.append(
          (0.5 * self.last_x_hat_t[t].T * self.S_t[t] * self.last_x_hat_t[t] \
          + self.last_x_hat_t[t].T * self.s_t[t] + self.s_scalar_t[t])[0, 0])
      cost_to_come.append(
          (0.5 * self.last_x_hat_t[t].T * self.S_bar_t[t] * self.last_x_hat_t[t] \
           + self.last_x_hat_t[t].T * self.s_bar_t[t] + self.s_scalar_bar_t[t])[0, 0])
      cost.append(cost_to_go[-1] + cost_to_come[-1])
      final_u_t[t] = self.L_t[t] * self.last_x_hat_t[t] + self.l_t[t]

    for t in range(l):
      A_t = numerical_jacobian_x(self.dynamics.discrete_dynamics,
                                 self.last_x_hat_t[t], final_u_t[t])
      B_t = numerical_jacobian_u(self.dynamics.discrete_dynamics,
                                 self.last_x_hat_t[t], final_u_t[t])
      c_t = self.dynamics.discrete_dynamics(self.last_x_hat_t[t], final_u_t[t]) \
          - A_t * self.last_x_hat_t[t] - B_t * final_u_t[t]
      print("Infeasability at", t, "is",
            ((A_t * self.last_x_hat_t[t] + B_t * final_u_t[t] + c_t) \
            - self.last_x_hat_t[t + 1]).T)

    pylab.figure('u')
    samples = numpy.arange(len(final_u_t))
    for i in range(num_inputs):
      pylab.plot(samples, [u[i, 0] for u in final_u_t],
                 label='u[%d]' % i, marker='o')
      pylab.legend()

    pylab.figure('cost')
    cost_samples = numpy.arange(len(cost))
    pylab.plot(cost_samples, cost_to_go, label='cost to go', marker='o')
    pylab.plot(cost_samples, cost_to_come, label='cost to come', marker='o')
    pylab.plot(cost_samples, cost, label='cost', marker='o')
    pylab.legend()

    pylab.ioff()
    pylab.show()

if __name__ == '__main__':
  dt = 0.05
  #arm_dynamics = ArmDynamics(dt=dt)
  #elqr = ELQR(arm_dynamics, ArmCostFunction(dt=dt, dynamics=arm_dynamics))
  #x_hat_initial = numpy.matrix([[0.10], [1.0], [0.0], [0.0]])
  #elqr.Solve(x_hat_initial, 100, 3)

  elqr = ELQR(SkidSteerDynamics(dt=dt), CostFunction(dt=dt))
  x_hat_initial = numpy.matrix([[0.10], [1.0], [0.0]])
  elqr.Solve(x_hat_initial, 100, 15)
  sys.exit(1)