class EKFSLAM(object):
# Construct an EKF instance with the following set of variables
# mu: The initial mean vector
# Sigma: The initial covariance matrix
# R: The process noise covariance
# Q: The measurement noise covariance
# visualize: Boolean variable indicating whether to visualize
# the filter
def __init__(self, mu, Sigma, R, Q, visualize=True):
self.mu = mu
self.Sigma = Sigma
self.R = R
self.Q = Q
self.nfeatures = 0
# step 6
# expand matrix to 3,3
#self.Q = np.pad(self.Q, ((0,1),(0,1)), mode = 'constant',
# constant_values = 0.0)
# You may find it useful to keep a dictionary that maps a
# a feature ID to the corresponding index in the mean_pose_handle
# vector and covariance matrix
self.mapLUT = {}
self.visualize = visualize
if self.visualize == True:
self.vis = Visualization()
else:
self.vis = None
# Visualize filter strategies
# deltat: Step size
# XGT: Array with ground-truth pose
def render(self, XGT=None):
deltat = 0.1
self.vis.drawEstimates(self.mu, self.Sigma)
if XGT is not None:
#print XGT
self.vis.drawGroundTruthPose(XGT[0], XGT[1], XGT[2])
plt.pause(deltat)
def mu_extended(self, mu, n):
placeholders = np.array([[0.0] * n]).ravel()
return np.concatenate([mu, placeholders])
# Perform the prediction step to determine the mean and covariance
# of the posterior belief given the current estimate for the mean
# and covariance, the control data, and the process model
# u: The forward distance and change in heading
def prediction(self, u):
u1, u2 = u
v1, v2 = np.random.multivariate_normal([0.0, 0.0], self.R)
# 10.2.3
F_x = np.concatenate(
[np.eye(3), np.zeros((3, self.nfeatures * 3))], axis=1)
# put mu in higher dimensional space
#mu = self.mu_extended(self.mu)
mu = self.mu.ravel()[:3]
x, y, theta = mu
noise_jac = np.array([[cos(theta), 0], [sin(theta), 0], [0, 1]])
mot = [] # column vector of len 3
# apply motion model (eq 8 in pset), 10.2.4 in book
mot.append(x + (u1 + v1) * cos(theta))
mot.append(y + (u1 + v1) * sin(theta))
mot.append(theta + u2 + v2)
mot = np.array(mot)
# jacobian of motion model f(x,y,d,t)
#F = np.array([[1.0, 0.0, np.cos(self.mu[2]), -(u1 + v1) * np.sin(self.mu[2])],
# [0.0, 1.0, np.sin(self.mu[2]), (u1 + v1) * np.cos(self.mu[2])],
# [0.0, 0.0, 1.0, 1.0]])
# 10.16
g_t = np.array([[1.0, 0.0, cos(theta)], [0.0, 1.0,
sin(theta)], [0.0, 0.0, 1.0]])
#sigma_new = np.matmul(F, self.Sigma)
#sigma_new = np.matmul(sigma_new, F.T)
#self.Sigma = sigma_new + self.R
# noise jacobian
# following from pp 314 in prob. robotics, s10.2
mubar_t = mu + np.matmul(F_x.T, mot) # 10.2.4
I = np.eye(((3 * self.nfeatures) + 3))
# step 4
G_t = I + np.matmul(np.matmul(F_x.T, g_t), F_x)
R_t = np.matmul(noise_jac, np.matmul(self.R, noise_jac.T))
sigmabar_t = (np.matmul(np.matmul(G_t, self.Sigma), G_t.T) +
np.matmul(np.matmul(F_x.T, R_t), F_x))
self.Sigma = sigmabar_t
self.mu = mubar_t
print self.mu.shape
# Perform the measurement update step to compute the posterior
# belief given the predictive posterior (mean and covariance) and
# the measurement data
# z: The (x,y) position of the landmark relative to the robot, and
# i: The ID of the observed landmark
def update(self, z, i):
z_x, z_y = z
x, y, theta = self.mu.ravel()[:3]
if i not in self.mapLUT:
w1, w2 = np.random.multivariate_normal([0.0, 0.0], self.Q)
i_loc = self.mu.shape[0]
self.mapLUT[i] = {'x': i_loc, 'y': i_loc + 1}
self.mu = self.mu_extended(self.mu, 2)
self.nfeatures += 2
self.mu[i_loc] = cos(theta) * z_x - sin(theta) * z_y - x + w1
self.mu[i_loc + 1] = sin(theta) * z_x - cos(theta) * z_y - y + w2
#self.mu = np.pad(self.mu, ((0,1),(0,1)), mode = 'constant',
# constant_values = 0.0)
#mu[i_loc,:2] = (np.matmul(np.array([[cos(theta), sin(theta)],
# -sin(theta), cos(theta)]),
# np.array([[z_x - x, z_y - y]])) +
# np.random.multivariate_normal([0.0,0.0], self.Q))
#else:
# i_loc = self.mapLUT[i]
x_idx = self.mapLUT[i]['x']
y_idx = self.mapLUT[i]['y']
j = x_idx + 1
N = self.mu.shape[0] + 1
delta = np.array([self.mu[x_idx] - x, self.mu[y_idx] - y])
q = np.matmul(delta.T, delta)
# measurement jacobian
H = np.array([[
-1.0, 0.0, -z_x * cos(theta) - z_x * sin(theta),
cos(theta), -sin(theta)
],
[
0.0, -1.0, -z_y * sin(theta) + z_x * cos(theta),
sin(theta), -cos(theta)
]])
# matrix to map to higher dimensions
F_l = np.pad(np.eye(3), ((0, 2), (0, 2 * j - 2)),
'constant',
constant_values=0.0)
F_r = np.pad(np.eye(3), ((2, 0), (0, 2 * N - 2 * j)),
'constant',
constant_values=0.0)
F = np.concatenate([F_l, F_r], axis=1)
H = np.matmul(H, F)
right = np.linalg.inv(
np.matmul(np.matmul(H, self.Sigma), H.T) + self.Q)
left = np.matmul(self.Sigma, H.T)
K = np.matmul(left, right)
# measurement model + measure jac
#z_i = np.matmul(np.array([[cos(theta), sin(theta)],
# -sin(theta), cos(theta)]),
#x,y, theta = self.mu[i_loc:i_loc + 3,]
# expand mu
#self.mu[i_loc,]
#print H.shape
#print self.Sigma.shape
#print self.Q.shape
#print H.T.shape
#left = np.matmul(self.Sigma, H.T)
#right = np.linalg.inv(np.matmul(np.matmul(H,self.Sigma), H.T) + self.Q)
#raise TypeError
#K = np.matmul(left, right)
#w = np.random.multivariate_normal([0.0, 0.0], self.Q)
#measure_model = np.array([[cos(theta), sin(theta)],
# -sin(theta), cos(theta)])
#measure_model = np.matmul(measure_model,
# np.array([[z_x - x, z_y - y]])) + w
#self.mu = self.mu + np.matmul(K, z - measure_model)
#self.Sigma = np.matmul((np.eye(K.shape) - np.matmul(K,H)),
# self.Sigma)
# Augment the state vector to include the new landmark
# z: The (x,y) position of the landmark relative to the robot
# i: The ID of the observed landmark
def augmentState(self, z, i):
# define h
#jacob = np.array([[-1, -np.cos(self.mu[2]), -sin(self.mu[2])],
# [-1, np.sin(self.mu[2]), -np.cos(self.mu[2])]]) d
x, y, theta = self.mu
z1, z2 = z
G = np.array([
[
-1.0, 0.0, -z2 * np.cos(theta) - z1 * np.sin(theta),
np.cos(theta), -np.sin(theta)
],
[
0.0, -1.0, -z2 * np.sin(theta) + z1 * np.cos(theta),
np.sin(theta), -np.cos(theta)
],
])
if i not in self.mapLUT:
mu_new = np.array(
[[self.mu],
[
np.cos(theta) * z1 - np.sin(theta * z2 - x1 + w1),
np.sin(theta) * z1 - np.cos(theta * z2 - x2 + w2)
]])
sigma_new = np.array(
[[self.Sigma, np.matmul(self.Sigma, G.T)],
[
np.matmul(G, self.Sigma),
np.matmul(np.matmul(G, self.Sigma), G.T) + self.Q
]])
self.mapLUT[i] = [mu_new, sigma_new]
# Runs the EKF SLAM algorithm
# U: Array of control inputs, one column per time step
# Z: Array of landmark observations in which each column
# [t; id; x; y] denotes a separate measurement and is
# represented by the time step (t), feature id (id),
# and the observed (x, y) position relative to the robot
# XGT: Array of ground-truth poses (may be None)
def run(self, U, Z, XGT=None, MGT=None):
# Draws the ground-truth map
if MGT is not None:
self.vis.drawMap(MGT)
# Iterate over the data
for t in range(U.shape[1]):
u = U[:, t]
num_match = np.where(Z[0, :] == t)
z = Z[:, np.where(Z[0, :] == t)]
self.prediction(u)
for k in range(z.shape[2]):
k = z[:, :, k].ravel()
self.update(k[2:], k[1])
# You may want to call the visualization function
# between filter steps
if self.visualize:
if XGT is None:
self.render(None)
else:
self.render(XGT[:, t])
class EKFSLAM(object):
# Construct an EKF instance with the following set of variables
# mu: The initial mean vector
# Sigma: The initial covariance matrix
# R: The process noise covariance
# Q: The measurement noise covariance
# visualize: Boolean variable indicating whether to visualize
# the filter
def __init__(self, mu, Sigma, R, Q, visualize=True):
self.mu = mu
self.Sigma = Sigma
self.R = R
self.Q = Q
# You may find it useful to keep a dictionary that maps a
# a feature ID to the corresponding index in the mean_pose_handle
# vector and covariance matrix
self.mapLUT = {}
self.visualize = visualize
if self.visualize == True:
self.vis = Visualization()
else:
self.vis = None
# Visualize filter strategies
# deltat: Step size
# XGT: Array with ground-truth pose
def render(self, XGT=None):
deltat = 0.1
self.vis.drawEstimates(self.mu, self.Sigma)
if XGT is not None:
#print XGT
self.vis.drawGroundTruthPose (XGT[0], XGT[1], XGT[2])
plt.pause(deltat/10)
# Perform the prediction step to determine the mean and covariance
# of the posterior belief given the current estimate for the mean
# and covariance, the control data, and the process model
# u: The forward distance and change in heading
def prediction(self, u):
F = np.zeros((3, 3))
noise = np.random.normal(0, self.R[0, 0])
F[0, 0] = 1
F[1, 1] = 1
F[2, 2] = 1
F[0, 2] = -1 * (u[0]+noise) * np.sin(self.mu[2])
F[1, 2] = (u[0]+noise) * np.cos(self.mu[2])
u1 = u[0]
u2 = u[1]
self.mu[0] = self.mu[0] + u1*np.cos(self.mu[2])
self.mu[1] = self.mu[1] + u1*np.sin(self.mu[2])
self.mu[2] = self.mu[2] + u2
upper_left = self.Sigma[0:3, 0:3]
upper_right = self.Sigma[0:3, 3:]
bot_left = self.Sigma[3:, 0:3]
bot_right = self.Sigma[3:, 3:]
temp = np.matmul(F, upper_left)
temp = np.matmul(temp, np.transpose(F))
temp[0, 0] = temp[0, 0] + self.R[0, 0]
temp[1, 1] = temp[1, 1] + self.R[0, 0]
temp[2, 2] = temp[2, 2] + self.R[1, 1]
upper_left = temp
upper_right = np.matmul(F, upper_right)
bot_left = np.matmul(bot_left, np.transpose(F))
up = np.concatenate((upper_left, upper_right), axis=1)
bot = np.concatenate((bot_left, bot_right), axis=1)
if not bot.shape[1] == 0:
self.Sigma = np.concatenate((up, bot), axis=0)
# Perform the measurement update step to compute the posterior
# belief given the predictive posterior (mean and covariance) and
# the measurement data
# z: The (x,y) position of the landmark relative to the robot
# i: The ID of the observed landmark
def update(self, z, i):
mIdx = self.mapLUT[i]
xt = self.mu[0]
yt = self.mu[1]
thetat = self.mu[2]
xm = self.mu[mIdx]
ym = self.mu[mIdx+1]
H = np.zeros((2, self.mu.shape[0]))
H[0, 0] = (-1)*np.cos(thetat)
H[0, 1] = (-1)*np.sin(thetat)
H[0, 2] = (-1)*xm*np.sin(thetat) + xt*np.sin(thetat) + ym*np.cos(thetat) - yt*np.cos(thetat)
H[0, mIdx] = np.cos(thetat)
H[0, mIdx+1] = np.sin(thetat)
H[1, 0] = np.sin(thetat)
H[1, 1] = (-1)*np.cos(thetat)
H[1, 2] = (-1)*xm*np.cos(thetat) + xt*np.cos(thetat) - ym*np.sin(thetat) + yt*np.sin(thetat)
H[1, mIdx] = (-1)*np.sin(thetat)
H[1, mIdx+1] = np.cos(thetat)
inv_temp = np.matmul(H, self.Sigma)
inv_temp = np.matmul(inv_temp, np.transpose(H))
inv_temp = inv_temp + self.Q
temp = np.matmul(self.Sigma, np.transpose(H))
k = np.matmul(temp, inv(inv_temp))
temp0 = xm*np.cos(thetat) - xt*np.cos(thetat) + ym*np.sin(thetat) - yt*np.sin(thetat)
temp1 = (-1)*xm*np.sin(thetat) + xt*np.sin(thetat) + ym*np.cos(thetat) - yt*np.cos(thetat)
hu = np.array([temp0[0], temp1[0]])
gain = np.matmul(k, z - hu)
gain = np.reshape(gain, (self.mu.shape[0], 1))
self.mu = self.mu + gain
I = np.eye(self.mu.shape[0])
temp = np.matmul(k, H)
temp = I - temp
self.Sigma = np.matmul(temp, self.Sigma)
# Augment the state vector to include the new landmark
# z: The (x,y) position of the landmark relative to the robot
# i: The ID of the observed landmark
def augmentState(self, z, i):
self.mapLUT[i] = 3 + 2*len(self.mapLUT)
G = np.zeros((2, self.mu.shape[0]))
G[0, 0] = 1
G[0, 2] = (-1)*z[0] * np.sin(self.mu[2]) - z[1] * np.cos(self.mu[2])
G[1, 1] = 1
G[1, 2] = z[0] * np.cos(self.mu[2]) - z[1] * np.sin(self.mu[2])
xm = np.array([self.mu[0] + z[0] * np.cos(self.mu[2]) - z[1] * np.sin(self.mu[2])])
ym = np.array([self.mu[1] + z[0] * np.sin(self.mu[2]) + z[1] * np.cos(self.mu[2])])
self.mu = np.concatenate((self.mu, xm), axis=0)
self.mu = np.concatenate((self.mu, ym), axis=0)
upper_left = self.Sigma
upper_right = np.matmul(self.Sigma, np.transpose(G))
bot_left = np.matmul(G, self.Sigma)
temp = np.matmul(G, self.Sigma)
temp = np.matmul(temp, np.transpose(G))
bot_right = temp + self.Q
up = np.concatenate((upper_left, upper_right), axis=1)
bot = np.concatenate((bot_left, bot_right), axis=1)
if not bot.shape[1] == 0:
self.Sigma = np.concatenate((up, bot), axis=0)
# Update mapLUT to include the new landmark
# Runs the EKF SLAM algorithm
# U: Array of control inputs, one column per time step
# Z: Array of landmark observations in which each column
# [t; id; x; y] denotes a separate measurement and is
# represented by the time step (t), feature id (id),
# and the observed (x, y) position relative to the robot
# XGT: Array of ground-truth poses (may be None)
def run(self, U, Z, XGT=None, MGT=None):
# Draws the ground-truth map
if MGT is not None:
self.vis.drawMap (MGT)
print("init")
print(self.mu)
print(self.R)
print(self.Q)
# Iterate over the data
zIdx = 0
for t in range(U.shape[1]):
#for t in range(0, 1):
print(t)
u = U[:,t]
self.prediction(u)
if zIdx < U.shape[1]:
while Z[0, zIdx] == t:
if Z[1, zIdx] in self.mapLUT:
self.update(Z[2:, zIdx], Z[1, zIdx])
else:
self.augmentState(Z[2:, zIdx], Z[1, zIdx])
zIdx = zIdx + 1
# You may want to call the visualization function
# between filter steps
if self.visualize:
if XGT is None:
self.render (None)
else:
self.render (XGT[:,t])
plt.savefig("small_noise")
class EKFSLAM(object):
# Construct an EKF instance with the following set of variables
# mu: The initial mean vector
# Sigma: The initial covariance matrix
# R: The process noise covariance
# Q: The measurement noise covariance
# visualize: Boolean variable indicating whether to visualize
# the filter
def __init__(self, mu, Sigma, R, Q, visualize=True):
self.mu = mu
self.Sigma = Sigma
self.R = R
self.Q = Q
# Maps feature IDs to row indices in mean matrix
self.mapLUT = {}
self.visualize = visualize
if self.visualize == True:
self.vis = Visualization()
else:
self.vis = None
# Visualize filter strategies
# deltat: Step size
# XGT: Array with ground-truth pose
def render(self, XGT=None):
deltat = 0.1
self.vis.drawEstimates(self.mu, self.Sigma)
if XGT is not None:
self.vis.drawGroundTruthPose(XGT[0], XGT[1], XGT[2])
plt.pause(deltat)
plt.savefig("res.png")
# Compute z_t according to the measurement model
def compute_zt(self, x_t, y_t, theta_t, x_m, y_m):
sn = np.sin(theta_t)
cs = np.cos(theta_t)
return np.array([[cs * (x_m - x_t) + sn * (y_m - y_t)],
[-sn * (x_m - x_t) + cs * (y_m - y_t)]])
# Perform the prediction step to determine the mean and covariance
# of the posterior belief given the current estimate for the mean
# and covariance, the control data, and the process model
# u: The forward distance and change in heading
def prediction(self, u):
u1 = u[0]
u2 = u[1]
sn = np.sin(self.mu[2])
cs = np.cos(self.mu[2])
# Project state ahead
self.mu[0] = self.mu[0] + u1 * cs
self.mu[1] = self.mu[1] + u1 * sn
self.mu[2] = self.mu[2] + u2
sn = np.sin(self.mu[2])
cs = np.cos(self.mu[2])
F = np.array([[1., 0., -u1 * sn], [0., 1., u1 * cs], [0., 0., 1.]],
dtype=float)
G = np.array([[cs, 0.], [sn, 0.], [0., 1.]], dtype=float)
# Project error covariance ahead
self.Sigma[:STATE_LEN, :
STATE_LEN] = F @ self.Sigma[:STATE_LEN, :
STATE_LEN] @ F.T + G @ self.R @ G.T
self.Sigma[:STATE_LEN,
STATE_LEN:] = F @ self.Sigma[:STATE_LEN, STATE_LEN:]
self.Sigma[
STATE_LEN:, :STATE_LEN] = self.Sigma[STATE_LEN:, :STATE_LEN] @ F.T
# Perform the measurement update step to compute the posterior
# belief given the predictive posterior (mean and covariance) and
# the measurement data
# z: The (x,y) position of the landmark relative to the robot
# i: The ID of the observed landmark
def update(self, z, i):
z = np.array(z).reshape((2, 1))
x_t, y_t, theta_t = self.mu[0, 0], self.mu[1, 0], self.mu[2, 0]
# Augment the state if this was the first time seeing the landmark
if i not in self.mapLUT:
self.augmentState(z, i)
return
# Otherwise, we've seen the landmark before, so do an update
# Compute H
cs = np.cos(theta_t)
sn = np.sin(theta_t)
h_ind = self.mapLUT[i]
x_m, y_m = self.mu[3 + 2 * h_ind, 0], self.mu[3 + 2 * h_ind + 1, 0]
hl = np.array([[-cs, -sn, -x_m * sn + x_t * sn + y_m * cs - y_t * cs],
[sn, -cs, -x_m * cs + x_t * cs - y_m * sn + y_t * sn]])
H = np.hstack((hl, np.zeros((2, 2 * len(self.mapLUT.keys())))))
H[0, 3 + 2 * h_ind] = cs
H[1, 3 + 2 * h_ind] = -sn
H[0, 3 + 2 * h_ind + 1] = sn
H[1, 3 + 2 * h_ind + 1] = cs
K = self.Sigma @ H.T @ np.linalg.inv(H @ self.Sigma @ H.T + self.Q)
# TODO add noise
h = self.compute_zt(x_t, y_t, theta_t, x_m, y_m)
self.mu = self.mu + K @ (z - h)
self.Sigma = self.Sigma - K @ H @ self.Sigma
# Augment the state vector to include the new landmark
# z: The (x,y) position of the landmark relative to the robot
# i: The ID of the observed landmark
def augmentState(self, z, i):
# Update mapLUT to include the new landmark
new_ind = len(self.mapLUT.keys())
self.mapLUT[i] = new_ind
# Compute the absolute position of the new landmark and augment the mean vector
position = self.computeNewLandmarkPosition(z)
self.mu = np.vstack((self.mu, position))
assert 2 * (new_ind + 1) + 3 == self.mu.shape[0]
# Compute the Jacobian of the inverse measurement function
theta = self.mu[2, 0]
sn = np.sin(theta)
cs = np.cos(theta)
zx, zy = z[0], z[1]
G = np.hstack((np.array(
[[1., 0., -zx * sn - zy * cs], [0., 1., zx * cs - zy * sn]],
dtype=float), np.zeros((2, 2 * new_ind), dtype=float)))
# Augment the covariance matrix
tl = self.Sigma
tr = self.Sigma @ G.T
bl = G @ self.Sigma
br = G @ self.Sigma @ G.T + self.Q
self.Sigma = np.hstack((np.vstack((tl, bl)), np.vstack((tr, br))))
# Compute the new landmark position (absolute) from the current mean vector
# and observation. This is `g`.
def computeNewLandmarkPosition(self, z):
z = np.reshape(z, (2, 1))
pos = self.mu[:2]
theta_t = self.mu[2, 0]
sn = np.sin(theta_t)
cs = np.cos(theta_t)
# The inverse of a rotation matrix is it's transpose
unrotate = np.array([[cs, -sn], [sn, cs]], dtype=float)
return unrotate @ z + pos
# Runs the EKF SLAM algorithm
# U: Array of control inputs, one column per time step
# Z: Array of landmark observations in which each column
# [t; id; x; y] denotes a separate measurement and is
# represented by the time step (t), feature id (id),
# and the observed (x, y) position relative to the robot
# XGT: Array of ground-truth poses (may be None)
def run(self, U, Z, XGT=None, MGT=None):
# Draws the ground-truth map
if MGT is not None:
self.vis.drawMap(MGT)
else:
print("No ground truth map")
# Column index of the last observation (iterate forwards in time, but not past t)
z_ind = 0
# Iterate over the data
for t in range(U.shape[1]):
u = U[:, t]
self.vis.ax.plot(self.mu[0], self.mu[1], "ro")
self.prediction(u)
# _, z_max = Z.shape
# while z_ind < z_max and Z[0,z_ind] <= t:
# _, landmark_id, x, y = Z[:, z_ind]
# self.update((x, y), landmark_id)
# z_ind += 1
# You may want to call the visualization function
# between filter steps
if self.visualize:
if XGT is None:
self.render(None)
else:
self.render(XGT[:, t])