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ddp.py
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import numpy as np
import jax.numpy as jnp
class DDP:
def __init__(self, n, m, f, f_x, f_u, f_xx, f_uu, f_xu, f_ux, l_x, l_u, l_xx, l_uu, lf_x, lf_xx, cost):
self.n = n # Number of states
self.m = m # Number of controls
self.f = f # Discrete dynamics
self.f_x = f_x
self.f_u = f_u
self.f_xx = f_xx
self.f_uu = f_uu
self.f_xu = f_xu
self.f_ux = f_ux
self.l_x = l_x
self.l_u = l_u
self.l_xx = l_xx
self.l_uu = l_uu
self.lf_x = lf_x
self.lf_xx = lf_xx
self.cost = cost # Objective function cost
self.N = None # Number of timesteps
self.eps = 1e-3 # Tolerance
self.max_iter = 800 # Maximum iterations for DDP
self.max_iter_reg = 10 # Maximum iterations for regularization
def run_DDP(self, x0, u_traj, N):
"""Runs the iLQR or DDP algorithm.
Inputs:
- x0(np.ndarray): The initial state [nx1]
- u_traj(np.ndarray): The initial control solution [mxN]
- N(int): Simulation timesteps
Returns:
- x_traj(np.ndarray): The states trajectory [nxN]
- u_traj(np.ndarray): The controls trajectory [mxN]
- J(float): The optimal cost
"""
print("DDP beginning..")
self.N = N
# Initial Rollout
x_traj = np.zeros((self.n, self.N))
x_traj[:, 0] = x0
for k in range(self.N-1):
x_traj[:, k+1] = self.f(x_traj[:, k], u_traj[:, k])
J = self.cost(x_traj, u_traj)
# Initialize DDP matrices
p = np.ones((self.n, self.N))
P = np.zeros((self.n, self.n, self.N))
d = np.ones(self.N-1)
K = np.zeros((self.m, self.n, self.N-1))
itr = 0
prev_err = np.inf
err_diff = np.inf
while np.linalg.norm(d, np.inf) > self.eps and itr < self.max_iter and err_diff > 1e-6:
# Backward Pass
DJ, p, P, d, K = self.backward_pass(p, P, d, K, x_traj, u_traj)
# Forward rollout with line search
x_new, u_new, Jn = self.rollout_with_linesearch(x_traj, u_traj, d, K, J, DJ)
# Update values
J = Jn
x_traj = x_new
u_traj = u_new
itr += 1
err_diff = abs(np.linalg.norm(d, np.inf) - prev_err)
prev_err = np.linalg.norm(d, np.inf)
if itr%10 == 0:
print("Iteration: {}, J = {}".format(itr, J))
print("\nDDP took: {} iterations".format(itr))
return x_traj, u_traj, J
def backward_pass(self, p, P, d, K, x_traj, u_traj):
"""Performs a backward pass to update values."""
DJ = 0.0
p[:, self.N-1] = self.lf_x(x_traj[:, self.N-1])
P[:, :, self.N-1] = self.lf_xx()
for k in range(self.N-2, -1, -1):
# Compute derivatives
A = self.f_x(x_traj[:, k], u_traj[:, k])
B = self.f_u(x_traj[:, k], u_traj[:, k])
gx = self.l_x(x_traj[:, k]) + A.T@p[:, k+1] # (n,)
gu = self.l_u(u_traj[:, k]) + B.T@p[:, k+1] # (m,)
# iLQR (Gauss-Newton) version
# ------------------------------------------------------------------
Gxx = self.l_xx() + A.T@P[:, :, k+1]@A # nxn
Guu = self.l_uu() + B.T@P[:, :, k+1]@B # mxm
Gxu = A.T@P[:, :, k+1]@B # nxm
Gux = B.T@P[:, :, k+1]@A # mxn
# DDP (full Newton) version
# ------------------------------------------------------------------
# Ax = self.f_xx(x_traj[:, k], u_traj[:, k]) # nnxn
# Bx = self.f_ux(x_traj[:, k], u_traj[:, k]) # nxn
# Au = self.f_xu(x_traj[:, k], u_traj[:, k]) # nnxm
# Bu = self.f_uu(x_traj[:, k], u_traj[:, k]) # (n,)
#
# Gxx = self.l_xx() + A.T@P[:,:,k+1]@A + jnp.kron(p[:,k+1].T, jnp.eye(self.n))@self.comm_mat(self.n, self.n)@Ax # nxn
# Guu = self.l_uu() + B.T@P[:,:,k+1]@B + jnp.kron(p[:,k+1].T, jnp.eye(self.m))@self.comm_mat(self.n, self.m)@Bu # mxm
# Gxu = A.T@P[:,:,k+1]@B + jnp.kron(p[:,k+1].T, jnp.eye(self.n))@self.comm_mat(self.n, self.n)@Au # nxm
# Gux = B.T@P[:,:,k+1]@A + jnp.kron(p[:,k+1].T, jnp.eye(self.m))@self.comm_mat(self.n, self.m)@Bx # mxn
#
# # Regularization
# beta = 0.1
# G = np.block([[Gxx, Gxu],
# [Gux, Guu]])
# iter_reg = 0
# while not self.is_pos_def(G) and iter_reg < self.max_iter_reg:
# Gxx += beta*A.T@A
# Guu += beta*B.T@B
# Gxu += beta*A.T@B
# Gux += beta*B.T@A
# beta = 2*beta
# # print("regularizing G")
# iter_reg += 1
# # ------------------------------------------------------------------
d[k], _, _, _ = jnp.linalg.lstsq(Guu, gu, rcond=None)
K[:, :, k], _, _, _ = jnp.linalg.lstsq(Guu, Gux, rcond=None)
p[:, k] = gx - K[:, :, k].T@gu + (K[:, :, k].T@Guu*d[k]).reshape(4,) - (Gxu*d[k]).reshape(4,)
P[:, :, k] = Gxx + K[:, :, k].T@Guu@K[:, :, k] - Gxu@K[:, :, k] - K[:, :, k].T@Gux
DJ += gu.T*d[k]
return DJ, p, P, d, K
@staticmethod
def is_pos_def(A):
"""Check if matrix A is positive definite.
If symmetric and has Cholesky decomposition -> p.d.
"""
if np.allclose(A, A.T, rtol=1e-04, atol=1e-04): # Ensure it is symmetric
try:
np.linalg.cholesky(A)
return True
except np.linalg.LinAlgError:
return False
else:
return False
def rollout(self, x_traj, u_traj, d, K, a):
"""Forward rollout."""
x_new = np.zeros((self.n, self.N))
u_new = np.zeros((self.m, self.N-1))
x_new[:, 0] = x_traj[:, 0]
for k in range(self.N-1):
u_new[:, k] = u_traj[:, k] - a*d[k] - np.dot(K[:, :, k], x_new[:, k] - x_traj[:, k])
x_new[:, k+1] = self.f(x_new[:, k], u_new[:, k])
J_new = self.cost(x_new, u_new)
return x_new, u_new, J_new
def rollout_with_linesearch(self, x_traj, u_traj, d, K, J, DJ):
"""Forward rollout with linesearch to find best step size."""
a = 1.0 # Step size
b = 1e-2 # Armijo tolerance
x_new, u_new, Jn = self.rollout(x_traj, u_traj, d, K, a)
while Jn > (J - b*a*DJ):
a = 0.5*a
x_new, u_new, Jn = self.rollout(x_traj, u_traj, d, K, a)
return x_new, u_new, Jn
@staticmethod
def comm_mat(m, n):
"""Commutation matrix.
Used to transform the vectorized form of a matrix into the vectorized
form of its transpose.
Inputs:
- m(int): Number of rows
- n(int): Number of columns
"""
w = np.arange(m * n).reshape((m, n), order="F").T.ravel(order="F")
return np.eye(m * n)[w, :]