Run yapf on all python files in the repo
Signed-off-by: Ravago Jones <ravagojones@gmail.com>
Change-Id: I221e04c3f517fab8535b22551553799e0fee7a80
diff --git a/y2014/control_loops/python/extended_lqr.py b/y2014/control_loops/python/extended_lqr.py
index b3f2372..699dd30 100755
--- a/y2014/control_loops/python/extended_lqr.py
+++ b/y2014/control_loops/python/extended_lqr.py
@@ -17,8 +17,9 @@
num_inputs = 2
x_hat_initial = numpy.matrix([[0.10], [1.0], [0.0]])
+
def dynamics(X, U):
- """Calculates the dynamics for a 2 wheeled robot.
+ """Calculates the dynamics for a 2 wheeled robot.
Args:
X, numpy.matrix(3, 1), The state. [x, y, theta]
@@ -27,29 +28,33 @@
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]) / width]])
+ #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]) / width]])
+
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
+ """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 discrete_dynamics(X, U):
- return RungeKutta(lambda startingX: dynamics(startingX, U), X, dt)
+ return RungeKutta(lambda startingX: dynamics(startingX, U), X, dt)
+
def inverse_discrete_dynamics(X, U):
- return RungeKutta(lambda startingX: -dynamics(startingX, U), X, dt)
+ return RungeKutta(lambda startingX: -dynamics(startingX, U), X, dt)
+
def numerical_jacobian_x(fn, X, U, epsilon=1e-4):
- """Numerically estimates the jacobian around X, U in X.
+ """Numerically estimates the jacobian around X, U in X.
Args:
fn: A function of X, U.
@@ -62,20 +67,21 @@
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
+ 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.
+ """Numerically estimates the jacobian around X, U in U.
Args:
fn: A function of X, U.
@@ -88,48 +94,56 @@
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
+ 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)
+ 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)
+ 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)
+ 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)
+ return numerical_jacobian_u(
+ lambda X_inner, U_inner: numerical_jacobian_u(fn, X_inner, U_inner).T,
+ X, U)
+
# Simple implementation for a quadratic cost function.
class CostFunction:
- def __init__(self):
- self.num_states = num_states
- self.num_inputs = num_inputs
- self.dt = dt
- self.Q = numpy.matrix([[0.1, 0, 0],
- [0, 0.6, 0],
- [0, 0, 0.1]]) / dt / dt
- self.R = numpy.matrix([[0.40, 0],
- [0, 0.40]]) / dt / dt
- def estimate_Q_final(self, X_hat):
- """Returns the quadraticized final Q around X_hat.
+ def __init__(self):
+ self.num_states = num_states
+ self.num_inputs = num_inputs
+ self.dt = dt
+ self.Q = numpy.matrix([[0.1, 0, 0], [0, 0.6, 0], [0, 0, 0.1]
+ ]) / dt / dt
+ self.R = numpy.matrix([[0.40, 0], [0, 0.40]]) / dt / dt
+
+ 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)
@@ -139,11 +153,11 @@
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)
+ 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.
+ 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.
@@ -151,14 +165,17 @@
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
+ 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 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 final_cost(self, X, U):
- """Computes the final cost of being at X
+ def final_cost(self, X, U):
+ """Computes the final cost of being at X
Args:
X: numpy.matrix(self.num_states, 1)
@@ -167,10 +184,10 @@
Returns:
numpy.matrix(1, 1), The quadratic cost of being at X
"""
- return X.T * self.Q * X * 1000
+ return X.T * self.Q * X * 1000
- def cost(self, X, U):
- """Computes the incremental cost given a position and U.
+ def cost(self, X, U):
+ """Computes the incremental cost given a position and U.
Args:
X: numpy.matrix(self.num_states, 1)
@@ -179,250 +196,334 @@
Returns:
numpy.matrix(1, 1), The quadratic cost of evaluating U.
"""
- return U.T * self.R * U + X.T * self.Q * X
+ return U.T * self.R * U + X.T * self.Q * X
+
cost_fn_obj = CostFunction()
-S_bar_t = [numpy.matrix(numpy.zeros((num_states, num_states))) for _ in range(l + 1)]
+S_bar_t = [
+ numpy.matrix(numpy.zeros((num_states, num_states))) for _ in range(l + 1)
+]
s_bar_t = [numpy.matrix(numpy.zeros((num_states, 1))) for _ in range(l + 1)]
s_scalar_bar_t = [numpy.matrix(numpy.zeros((1, 1))) for _ in range(l + 1)]
-L_t = [numpy.matrix(numpy.zeros((num_inputs, num_states))) for _ in range(l + 1)]
+L_t = [
+ numpy.matrix(numpy.zeros((num_inputs, num_states))) for _ in range(l + 1)
+]
l_t = [numpy.matrix(numpy.zeros((num_inputs, 1))) for _ in range(l + 1)]
-L_bar_t = [numpy.matrix(numpy.zeros((num_inputs, num_states))) for _ in range(l + 1)]
+L_bar_t = [
+ numpy.matrix(numpy.zeros((num_inputs, num_states))) for _ in range(l + 1)
+]
l_bar_t = [numpy.matrix(numpy.zeros((num_inputs, 1))) for _ in range(l + 1)]
-S_t = [numpy.matrix(numpy.zeros((num_states, num_states))) for _ in range(l + 1)]
+S_t = [
+ numpy.matrix(numpy.zeros((num_states, num_states))) for _ in range(l + 1)
+]
s_t = [numpy.matrix(numpy.zeros((num_states, 1))) for _ in range(l + 1)]
s_scalar_t = [numpy.matrix(numpy.zeros((1, 1))) for _ in range(l + 1)]
-
-last_x_hat_t = [numpy.matrix(numpy.zeros((num_states, 1))) for _ in range(l + 1)]
+last_x_hat_t = [
+ numpy.matrix(numpy.zeros((num_states, 1))) for _ in range(l + 1)
+]
for a in range(15):
- x_hat = x_hat_initial
- u_t = L_t[0] * x_hat + l_t[0]
- S_bar_t[0] = numpy.matrix(numpy.zeros((num_states, num_states)))
- s_bar_t[0] = numpy.matrix(numpy.zeros((num_states, 1)))
- s_scalar_bar_t[0] = numpy.matrix([[0]])
+ x_hat = x_hat_initial
+ u_t = L_t[0] * x_hat + l_t[0]
+ S_bar_t[0] = numpy.matrix(numpy.zeros((num_states, num_states)))
+ s_bar_t[0] = numpy.matrix(numpy.zeros((num_states, 1)))
+ s_scalar_bar_t[0] = numpy.matrix([[0]])
- last_x_hat_t[0] = x_hat_initial
+ last_x_hat_t[0] = x_hat_initial
- Q_t = numerical_jacobian_x_x(cost_fn_obj.cost, x_hat_initial, u_t)
- P_t = numerical_jacobian_x_u(cost_fn_obj.cost, x_hat_initial, u_t)
- R_t = numerical_jacobian_u_u(cost_fn_obj.cost, x_hat_initial, u_t)
+ Q_t = numerical_jacobian_x_x(cost_fn_obj.cost, x_hat_initial, u_t)
+ P_t = numerical_jacobian_x_u(cost_fn_obj.cost, x_hat_initial, u_t)
+ R_t = numerical_jacobian_u_u(cost_fn_obj.cost, x_hat_initial, u_t)
- q_t = numerical_jacobian_x(cost_fn_obj.cost, x_hat_initial, u_t).T - Q_t * x_hat_initial - P_t.T * u_t
- r_t = numerical_jacobian_u(cost_fn_obj.cost, x_hat_initial, u_t).T - P_t * x_hat_initial - R_t * u_t
+ q_t = numerical_jacobian_x(cost_fn_obj.cost, x_hat_initial,
+ u_t).T - Q_t * x_hat_initial - P_t.T * u_t
+ r_t = numerical_jacobian_u(cost_fn_obj.cost, x_hat_initial,
+ u_t).T - P_t * x_hat_initial - R_t * u_t
- q_scalar_t = cost_fn_obj.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
+ q_scalar_t = cost_fn_obj.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(discrete_dynamics, x_hat_initial, u_t)
- start_B_t = numerical_jacobian_u(discrete_dynamics, x_hat_initial, u_t)
- x_hat_next = discrete_dynamics(x_hat_initial, u_t)
- start_c_t = x_hat_next - start_A_t * x_hat_initial - start_B_t * u_t
+ start_A_t = numerical_jacobian_x(discrete_dynamics, x_hat_initial, u_t)
+ start_B_t = numerical_jacobian_u(discrete_dynamics, x_hat_initial, u_t)
+ x_hat_next = 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_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
+ 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
- L_bar_t[1] = B_svd_inv
- l_bar_t[1] = -B_svd_inv * (start_A_t * x_hat_initial + start_c_t)
+ L_bar_t[1] = B_svd_inv
+ l_bar_t[1] = -B_svd_inv * (start_A_t * x_hat_initial + start_c_t)
- S_bar_t[1] = L_bar_t[1].T * R_t * L_bar_t[1]
+ S_bar_t[1] = L_bar_t[1].T * R_t * L_bar_t[1]
- TotalS_1 = start_B_t.T * S_t[1] * start_B_t + R_t
- Totals_1 = start_B_t.T * S_t[1] * (start_c_t + start_A_t * x_hat_initial) + start_B_t.T * 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) * S_t[1] * (start_c_t + start_A_t * x_hat_initial) + 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) * s_t[1]
+ TotalS_1 = start_B_t.T * S_t[1] * start_B_t + R_t
+ Totals_1 = start_B_t.T * S_t[1] * (
+ start_c_t + start_A_t *
+ x_hat_initial) + start_B_t.T * 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
+ ) * S_t[1] * (start_c_t + start_A_t * x_hat_initial) + 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) * 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
+ 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
- S_bar_1_eigh_eigenvalues, S_bar_1_eigh_eigenvectors = numpy.linalg.eigh(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_1_eigh_eigenvalues, S_bar_1_eigh_eigenvectors = numpy.linalg.eigh(
+ 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
- #print 'eigh eigenvalues of S bar', S_bar_1_eigh_eigenvalues
- #print 'eigh eigenvectors of S bar', S_bar_1_eigh_eigenvectors.T
+ #print 'eigh eigenvalues of S bar', S_bar_1_eigh_eigenvalues
+ #print 'eigh eigenvectors of S bar', S_bar_1_eigh_eigenvectors.T
- #print 'S bar eig recreate', S_bar_1_eigh_eigenvectors * numpy.matrix(numpy.diag(S_bar_1_eigh_eigenvalues)) * S_bar_1_eigh_eigenvectors.T
- #print 'S bar eig recreate error', (S_bar_1_eigh_eigenvectors * numpy.matrix(numpy.diag(S_bar_1_eigh_eigenvalues)) * S_bar_1_eigh_eigenvectors.T - S_bar_t[1])
+ #print 'S bar eig recreate', S_bar_1_eigh_eigenvectors * numpy.matrix(numpy.diag(S_bar_1_eigh_eigenvalues)) * S_bar_1_eigh_eigenvectors.T
+ #print 'S bar eig recreate error', (S_bar_1_eigh_eigenvectors * numpy.matrix(numpy.diag(S_bar_1_eigh_eigenvalues)) * S_bar_1_eigh_eigenvectors.T - S_bar_t[1])
- S_bar_stiff = S_bar_1_eigh_eigenvectors * numpy.matrix(numpy.diag(S_bar_1_eigh_eigenvalues_stiff)) * S_bar_1_eigh_eigenvectors.T
+ 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
- s_bar_t[1] = -s_t[1] - (S_bar_stiff + S_t[1]) * optimal_x_1
- s_scalar_bar_t[1] = 0.5 * (optimal_u_1.T * TotalS_1 * optimal_u_1 - optimal_x_1.T * (S_bar_stiff + S_t[1]) * optimal_x_1) + optimal_u_1.T * Totals_1 - optimal_x_1.T * (s_bar_t[1] + s_t[1]) - s_scalar_t[1] + Totals_scalar_1
+ print 'Min u', -numpy.linalg.solve(TotalS_1, Totals_1)
+ print 'Min x_hat', optimal_x_1
+ s_bar_t[1] = -s_t[1] - (S_bar_stiff + S_t[1]) * optimal_x_1
+ s_scalar_bar_t[1] = 0.5 * (
+ optimal_u_1.T * TotalS_1 * optimal_u_1 - optimal_x_1.T *
+ (S_bar_stiff + S_t[1]) *
+ optimal_x_1) + optimal_u_1.T * Totals_1 - optimal_x_1.T * (
+ s_bar_t[1] + s_t[1]) - 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 * S_t[0] * x_hat_initial) + x_hat_initial.T * s_t[0] + s_scalar_t[0]
+ 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 * S_t[0] * x_hat_initial
+ ) + x_hat_initial.T * s_t[0] + 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' % (L_t[0],)).replace('\n', '\n ')
- print ('l[ 0]: %s' % (l_t[0],)).replace('\n', '\n ')
+ 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' % (L_t[0], )).replace('\n', '\n ')
+ print('l[ 0]: %s' % (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 ')
+ 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((S_t[1] + S_bar_stiff), (s_t[1] + s_bar_t[1]))
- print 'new xhat', x_hat
+ # TODO(austin): optimal_x_1 is x_hat
+ x_hat = -numpy.linalg.solve((S_t[1] + S_bar_stiff), (s_t[1] + s_bar_t[1]))
+ print 'new xhat', x_hat
- S_bar_t[1] = S_bar_stiff
+ S_bar_t[1] = S_bar_stiff
- last_x_hat_t[1] = x_hat
+ last_x_hat_t[1] = x_hat
- for t in range(1, l):
- print 't forward', t
- u_t = L_t[t] * x_hat + l_t[t]
+ for t in range(1, l):
+ print 't forward', t
+ u_t = L_t[t] * x_hat + l_t[t]
- x_hat_next = discrete_dynamics(x_hat, u_t)
- A_bar_t = numerical_jacobian_x(inverse_discrete_dynamics, x_hat_next, u_t)
- B_bar_t = numerical_jacobian_u(inverse_discrete_dynamics, x_hat_next, u_t)
- c_bar_t = x_hat - A_bar_t * x_hat_next - B_bar_t * u_t
+ x_hat_next = discrete_dynamics(x_hat, u_t)
+ A_bar_t = numerical_jacobian_x(inverse_discrete_dynamics, x_hat_next,
+ u_t)
+ B_bar_t = numerical_jacobian_u(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, L_t[t],)).replace('\n', '\n ')
- print ('l[%2d]: %s' % (t, l_t[t],)).replace('\n', '\n ')
- print 'u[%2d]: %s' % (t, u_t.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,
+ L_t[t],
+ )).replace('\n', '\n ')
+ print('l[%2d]: %s' % (
+ t,
+ 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 ')
+ 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(cost_fn_obj.cost, x_hat, u_t)
- P_t = numerical_jacobian_x_u(cost_fn_obj.cost, x_hat, u_t)
- R_t = numerical_jacobian_u_u(cost_fn_obj.cost, x_hat, u_t)
+ Q_t = numerical_jacobian_x_x(cost_fn_obj.cost, x_hat, u_t)
+ P_t = numerical_jacobian_x_u(cost_fn_obj.cost, x_hat, u_t)
+ R_t = numerical_jacobian_u_u(cost_fn_obj.cost, x_hat, u_t)
- q_t = numerical_jacobian_x(cost_fn_obj.cost, x_hat, u_t).T - Q_t * x_hat - P_t.T * u_t
- r_t = numerical_jacobian_u(cost_fn_obj.cost, x_hat, u_t).T - P_t * x_hat - R_t * u_t
+ q_t = numerical_jacobian_x(cost_fn_obj.cost, x_hat,
+ u_t).T - Q_t * x_hat - P_t.T * u_t
+ r_t = numerical_jacobian_u(cost_fn_obj.cost, x_hat,
+ u_t).T - P_t * x_hat - R_t * u_t
- q_scalar_t = cost_fn_obj.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
+ q_scalar_t = cost_fn_obj.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 * (S_bar_t[t] + Q_t) * A_bar_t + P_t * A_bar_t
- D_bar_t = A_bar_t.T * (S_bar_t[t] + Q_t) * A_bar_t
- E_bar_t = B_bar_t.T * (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 * (s_bar_t[t] + q_t) + A_bar_t.T * (S_bar_t[t] + Q_t) * c_bar_t
- e_bar_t = r_t + P_t * c_bar_t + B_bar_t.T * s_bar_t[t] + B_bar_t.T * (S_bar_t[t] + Q_t) * c_bar_t
+ C_bar_t = B_bar_t.T * (S_bar_t[t] + Q_t) * A_bar_t + P_t * A_bar_t
+ D_bar_t = A_bar_t.T * (S_bar_t[t] + Q_t) * A_bar_t
+ E_bar_t = B_bar_t.T * (
+ 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 * (s_bar_t[t] + q_t) + A_bar_t.T * (S_bar_t[t] +
+ Q_t) * c_bar_t
+ e_bar_t = r_t + P_t * c_bar_t + B_bar_t.T * s_bar_t[t] + B_bar_t.T * (
+ S_bar_t[t] + Q_t) * c_bar_t
- L_bar_t[t + 1] = -numpy.linalg.inv(E_bar_t) * C_bar_t
- l_bar_t[t + 1] = -numpy.linalg.inv(E_bar_t) * e_bar_t
+ L_bar_t[t + 1] = -numpy.linalg.inv(E_bar_t) * C_bar_t
+ l_bar_t[t + 1] = -numpy.linalg.inv(E_bar_t) * e_bar_t
- S_bar_t[t + 1] = D_bar_t + C_bar_t.T * L_bar_t[t + 1]
- s_bar_t[t + 1] = d_bar_t + C_bar_t.T * l_bar_t[t + 1]
- 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 * (S_bar_t[t] + Q_t) * c_bar_t + c_bar_t.T * s_bar_t[t] + c_bar_t.T * q_t + s_scalar_bar_t[t] + q_scalar_t
+ S_bar_t[t + 1] = D_bar_t + C_bar_t.T * L_bar_t[t + 1]
+ s_bar_t[t + 1] = d_bar_t + C_bar_t.T * l_bar_t[t + 1]
+ 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 * (
+ S_bar_t[t] + Q_t) * c_bar_t + c_bar_t.T * s_bar_t[
+ t] + c_bar_t.T * q_t + s_scalar_bar_t[t] + q_scalar_t
- x_hat = -numpy.linalg.solve((S_t[t + 1] + S_bar_t[t + 1]), (s_t[t + 1] + s_bar_t[t + 1]))
+ x_hat = -numpy.linalg.solve((S_t[t + 1] + S_bar_t[t + 1]),
+ (s_t[t + 1] + s_bar_t[t + 1]))
- S_t[l] = cost_fn_obj.estimate_Q_final(x_hat)
- s_t[l] = cost_fn_obj.estimate_q_final(x_hat)
- x_hat = -numpy.linalg.inv(S_t[l] + S_bar_t[l]) * (s_t[l] + s_bar_t[l])
+ S_t[l] = cost_fn_obj.estimate_Q_final(x_hat)
+ s_t[l] = cost_fn_obj.estimate_q_final(x_hat)
+ x_hat = -numpy.linalg.inv(S_t[l] + S_bar_t[l]) * (s_t[l] + 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 = L_bar_t[t + 1] * x_hat + l_bar_t[t + 1]
+ 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 = L_bar_t[t + 1] * x_hat + l_bar_t[t + 1]
- x_hat_prev = 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, L_bar_t[t + 1])).replace('\n', '\n ')
- print ('l_bar[%2d]: %s' % (t + 1, 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(discrete_dynamics, x_hat_prev, u_t)
- B_t = numerical_jacobian_u(discrete_dynamics, x_hat_prev, u_t)
- c_t = x_hat - A_t * x_hat_prev - B_t * u_t
+ x_hat_prev = 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, L_bar_t[t + 1])).replace(
+ '\n', '\n ')
+ print('l_bar[%2d]: %s' % (t + 1, 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(discrete_dynamics, x_hat_prev, u_t)
+ B_t = numerical_jacobian_u(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 ')
+ 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(cost_fn_obj.cost, x_hat_prev, u_t)
- P_t = numerical_jacobian_x_u(cost_fn_obj.cost, x_hat_prev, u_t)
- P_T_t = numerical_jacobian_u_x(cost_fn_obj.cost, x_hat_prev, u_t)
- R_t = numerical_jacobian_u_u(cost_fn_obj.cost, x_hat_prev, u_t)
+ Q_t = numerical_jacobian_x_x(cost_fn_obj.cost, x_hat_prev, u_t)
+ P_t = numerical_jacobian_x_u(cost_fn_obj.cost, x_hat_prev, u_t)
+ P_T_t = numerical_jacobian_u_x(cost_fn_obj.cost, x_hat_prev, u_t)
+ R_t = numerical_jacobian_u_u(cost_fn_obj.cost, x_hat_prev, u_t)
- q_t = numerical_jacobian_x(cost_fn_obj.cost, x_hat_prev, u_t).T - Q_t * x_hat_prev - P_T_t * u_t
- r_t = numerical_jacobian_u(cost_fn_obj.cost, x_hat_prev, u_t).T - P_t * x_hat_prev - R_t * u_t
+ q_t = numerical_jacobian_x(cost_fn_obj.cost, x_hat_prev,
+ u_t).T - Q_t * x_hat_prev - P_T_t * u_t
+ r_t = numerical_jacobian_u(cost_fn_obj.cost, x_hat_prev,
+ u_t).T - P_t * x_hat_prev - R_t * u_t
- q_scalar_t = cost_fn_obj.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
+ q_scalar_t = cost_fn_obj.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 * S_t[t + 1] * A_t
- D_t = Q_t + A_t.T * S_t[t + 1] * A_t
- E_t = R_t + B_t.T * S_t[t + 1] * B_t
- d_t = q_t + A_t.T * s_t[t + 1] + A_t.T * S_t[t + 1] * c_t
- e_t = r_t + B_t.T * s_t[t + 1] + B_t.T * S_t[t + 1] * c_t
- L_t[t] = -numpy.linalg.inv(E_t) * C_t
- l_t[t] = -numpy.linalg.inv(E_t) * e_t
- s_t[t] = d_t + C_t.T * l_t[t]
- S_t[t] = D_t + C_t.T * L_t[t]
- s_scalar_t[t] = q_scalar_t - 0.5 * e_t.T * numpy.linalg.inv(E_t) * e_t + 0.5 * c_t.T * S_t[t + 1] * c_t + c_t.T * s_t[t + 1] + s_scalar_t[t + 1]
+ C_t = P_t + B_t.T * S_t[t + 1] * A_t
+ D_t = Q_t + A_t.T * S_t[t + 1] * A_t
+ E_t = R_t + B_t.T * S_t[t + 1] * B_t
+ d_t = q_t + A_t.T * s_t[t + 1] + A_t.T * S_t[t + 1] * c_t
+ e_t = r_t + B_t.T * s_t[t + 1] + B_t.T * S_t[t + 1] * c_t
+ L_t[t] = -numpy.linalg.inv(E_t) * C_t
+ l_t[t] = -numpy.linalg.inv(E_t) * e_t
+ s_t[t] = d_t + C_t.T * l_t[t]
+ S_t[t] = D_t + C_t.T * L_t[t]
+ s_scalar_t[t] = q_scalar_t - 0.5 * e_t.T * numpy.linalg.inv(
+ E_t) * e_t + 0.5 * c_t.T * S_t[t + 1] * c_t + c_t.T * s_t[
+ t + 1] + s_scalar_t[t + 1]
- x_hat = -numpy.linalg.solve((S_t[t] + S_bar_t[t]), (s_t[t] + s_bar_t[t]))
- if t == 0:
- last_x_hat_t[t] = x_hat_initial
- else:
- last_x_hat_t[t] = x_hat
+ x_hat = -numpy.linalg.solve((S_t[t] + S_bar_t[t]),
+ (s_t[t] + s_bar_t[t]))
+ if t == 0:
+ last_x_hat_t[t] = x_hat_initial
+ else:
+ last_x_hat_t[t] = x_hat
- x_hat_t = [x_hat_initial]
+ x_hat_t = [x_hat_initial]
- pylab.figure('states %d' % a)
- pylab.ion()
- for dim in range(num_states):
- pylab.plot(numpy.arange(len(last_x_hat_t)),
- [x_hat_loop[dim, 0] for x_hat_loop in last_x_hat_t], marker='o', label='Xhat[%d]'%dim)
- pylab.legend()
- pylab.draw()
+ pylab.figure('states %d' % a)
+ pylab.ion()
+ for dim in range(num_states):
+ pylab.plot(numpy.arange(len(last_x_hat_t)),
+ [x_hat_loop[dim, 0] for x_hat_loop in 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 last_x_hat_t],
- [x_hat_loop[1, 0] for x_hat_loop in last_x_hat_t], marker='o', label='trajectory')
- pylab.legend()
- pylab.draw()
+ pylab.figure('xy %d' % a)
+ pylab.ion()
+ pylab.plot([x_hat_loop[0, 0] for x_hat_loop in last_x_hat_t],
+ [x_hat_loop[1, 0] for x_hat_loop in 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 * last_x_hat_t[t].T * S_t[t] * last_x_hat_t[t] + last_x_hat_t[t].T * s_t[t] + s_scalar_t[t])[0, 0])
- cost_to_come.append((0.5 * last_x_hat_t[t].T * S_bar_t[t] * last_x_hat_t[t] + last_x_hat_t[t].T * s_bar_t[t] + s_scalar_bar_t[t])[0, 0])
- cost.append(cost_to_go[-1] + cost_to_come[-1])
- final_u_t[t] = L_t[t] * last_x_hat_t[t] + l_t[t]
+ cost_to_go.append((0.5 * last_x_hat_t[t].T * S_t[t] * last_x_hat_t[t] +
+ last_x_hat_t[t].T * s_t[t] + s_scalar_t[t])[0, 0])
+ cost_to_come.append(
+ (0.5 * last_x_hat_t[t].T * S_bar_t[t] * last_x_hat_t[t] +
+ last_x_hat_t[t].T * s_bar_t[t] + s_scalar_bar_t[t])[0, 0])
+ cost.append(cost_to_go[-1] + cost_to_come[-1])
+ final_u_t[t] = L_t[t] * last_x_hat_t[t] + l_t[t]
for t in range(l):
- A_t = numerical_jacobian_x(discrete_dynamics, last_x_hat_t[t], final_u_t[t])
- B_t = numerical_jacobian_u(discrete_dynamics, last_x_hat_t[t], final_u_t[t])
- c_t = discrete_dynamics(last_x_hat_t[t], final_u_t[t]) - A_t * last_x_hat_t[t] - B_t * final_u_t[t]
- print("Infeasability at", t, "is", ((A_t * last_x_hat_t[t] + B_t * final_u_t[t] + c_t) - last_x_hat_t[t + 1]).T)
+ A_t = numerical_jacobian_x(discrete_dynamics, last_x_hat_t[t],
+ final_u_t[t])
+ B_t = numerical_jacobian_u(discrete_dynamics, last_x_hat_t[t],
+ final_u_t[t])
+ c_t = discrete_dynamics(
+ last_x_hat_t[t],
+ final_u_t[t]) - A_t * last_x_hat_t[t] - B_t * final_u_t[t]
+ print("Infeasability at", t, "is",
+ ((A_t * last_x_hat_t[t] + B_t * final_u_t[t] + c_t) -
+ 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.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))