adding numpy implementation

ukf.py

@@ -0,0 +1,143 @@
+import numpy as np
+from scipy import linalg
+
+TWO_PI = 2 * np.pi
+
+class UKF(object):
+    def __init__(self, n, m, r_var=0.1, p_var=1e+3, q_var=0.1):
+        """
+        Creates an Unscented Kalman Filter with n states and m observations
+        """
+        # size of the state vector
+        self.n_x = n
+
+        # size of augmented covariance matrix
+        self.n_aug = n + m
+
+        # number of sigma points (2n+1)
+        self.n_sig = 2 * self.n_aug + 1
+
+        # predicted sigma points matrix
+        self.X_sigma = np.zeros((n, self.n_sig))
+
+        # sigma point spreading parameter
+        self._lambda_ = 3. - self.n_aug
+
+        # weight vector
+        self.weights = np.zeros((1, self.n_sig))
+        self.weights[:, :1] = self._lambda_ / (self._lambda_ + self.n_aug)
+        self.weights[:, 1:] = .5 / (self._lambda_ + self.n_aug)
+
+        # measurement noise covariance
+        self.R = np.eye(m) * r_var
+
+        # process noise covariance
+        self.Q = np.eye(m) * q_var
+
+        self.x = np.ones((n, 1))   # initial state
+        self.P = np.eye(n) * p_var # initial prediction uncertainty covariance
+
+        # Normalized Innovation Squared (calculated in update step)
+        # this value will help us evaluate the consistency of our predictions
+        # and determine whether our noise parameters are good
+        self.NIS = 0.
+
+    def predict(self, delta_t):
+        """
+        Predicts sigma points matrix for time step t+1
+        """
+        # create augmented mean vector
+        x_aug = np.zeros((self.n_aug, 1))
+        x_aug[:self.n_x] = self.x
+
+        # create augmented state covariance
+        P_aug = np.zeros((self.n_aug, self.n_aug))
+        P_aug[:self.n_x, :self.n_x] = self.P
+        P_aug[self.n_x:, self.n_x:] = self.Q
+
+        # take square root (cholesky) of P_aug and modify by spreading param
+        P_aug_root = linalg.sqrtm((self._lambda_ + self.n_aug) * P_aug)
+
+        # create augmented sigma point matrix
+        X_sigma_aug = np.zeros((self.n_aug, self.n_sig))
+        X_sigma_aug[:, :1] = x_aug
+        X_sigma_aug[:, 1:self.n_aug+1] = x_aug + P_aug_root
+        X_sigma_aug[:,  self.n_aug+1:] = x_aug - P_aug_root
+
+        # predict sigma points
+        for col in range(self.n_sig):
+            px, py, v, yaw, yawd, nu_a, nu_yawdd = X_sigma_aug[:, col]
+
+            # predicted state values for CTRV motion model
+            # division by zero or very small value implies a nearly straight line,
+            # so degenerate to simpler CV motion model when yawd is sufficiently small
+            if abs(yawd) > 0.001:
+                px_p = px + v / yawd * (np.sin(yaw + yawd * delta_t) - np.sin(yaw))
+                py_p = py + v / yawd * (np.cos(yaw) - np.cos(yaw + yawd * delta_t))
+            else:
+                px_p = px + v * delta_t * np.cos(yaw)
+                py_p = py + v * delta_t * np.sin(yaw)
+
+            v_p = v
+            yaw_p = yaw + yawd * delta_t
+            yawd_p = yawd
+
+            # add acceleration noise
+            px_p   += .5 * nu_a * delta_t**2 * np.cos(yaw)
+            py_p   += .5 * nu_a * delta_t**2 * np.sin(yaw)
+            v_p    += nu_a * delta_t
+            yaw_p  += .5 * nu_yawdd * delta_t**2
+            yawd_p += nu_yawdd * delta_t
+
+            self.X_sigma[:, col] = px_p, py_p, v_p, yaw_p, yawd_p
+
+        # predicted x state vector
+        self.x = np.sum(self.X_sigma * self.weights, axis=1).reshape(-1, 1)
+
+        # state residual
+        x_diff = self.X_sigma - self.x
+
+        # normalize the predicted yaw component to between -pi & pi
+        x_diff[3] -= TWO_PI * np.floor((x_diff[3] + np.pi) / TWO_PI)
+
+        # predicted P uncertainty covariance
+        self.P = (self.weights * x_diff).dot(x_diff.T)
+
+    def update(self, z):
+        """
+        Update predictions with observed measurement z
+        """
+        nz = len(z)         # measurement dimension
+        z = np.array([z]).T # convert measurement to single column matrix
+
+        # create matrix for sigma points in measurement space
+        # from the x and y sigma point predictions
+        Z_sigma = self.X_sigma[:nz]
+
+        # mean predicted measurement
+        z_pred = np.sum(Z_sigma * self.weights[:nz], axis=1).reshape(-1, 1)
+
+        # measurement sigma and state residuals
+        z_sig_diff = Z_sigma - z_pred
+        x_diff = self.X_sigma - self.x
+        # x_diff[3] -= TWO_PI * np.floor((x_diff[3] + np.pi) / TWO_PI)
+
+        # compute cross correlation matrix
+        Tc = (self.weights * x_diff).dot(z_sig_diff.T)
+
+        # compute measurement covariance matrix + measurement noise
+        S  = (self.weights * z_sig_diff).dot(z_sig_diff.T) + self.R
+        S_inv = np.linalg.inv(S)
+
+        # compute the Kalman gain
+        K = Tc.dot(S_inv)
+
+        # measurement residual
+        z_diff = z - z_pred
+
+        # update x and prediction uncertainty
+        self.x += K.dot(z_diff)
+        self.P -= K.dot(S).dot(K.T)
+
+        # update NIS calculation
+        self.NIS = z_diff.T.dot(S_inv).dot(z_diff)