EKF_Code_Reading

1. Code show

import math
import numpy as np
import matplotlib.pyplot as plt

# Covariance for EKF simulation
Q = np.diag([
    0.1, # variance of location on x-axis
    0.1, # variance of location on y-axis
    np.deg2rad(1.0), # variance of yaw angle
    1.0 # variance of velocity
    ])**2  # predict state covariance
R = np.diag([1.0, 1.0])**2  # Observation x,y position covariance

#  Simulation parameter
INPUT_NOISE = np.diag([1.0, np.deg2rad(30.0)])**2
GPS_NOISE = np.diag([0.5, 0.5])**2

DT = 0.1  # time tick [s]
SIM_TIME = 50.0  # simulation time [s]
show_animation = True

def calc_input():
    v = 1.0  # [m/s]
    yawrate = 0.1  # [rad/s]
    u = np.array([[v], [yawrate]])
    return u


def observation(xTrue, xd, u):
    xTrue = motion_model(xTrue, u)
    # add noise to gps x-y
    z = observation_model(xTrue) + GPS_NOISE @ np.random.randn(2, 1)
    # add noise to input
    ud = u + INPUT_NOISE @ np.random.randn(2, 1)
    xd = motion_model(xd, ud)

    return xTrue, z, xd, ud


def motion_model(x, u):
    F = np.array([[1.0, 0, 0, 0],
                  [0, 1.0, 0, 0],
                  [0, 0, 1.0, 0],
                  [0, 0, 0, 0]])
    B = np.array([[DT * math.cos(x[2, 0]), 0],
                  [DT * math.sin(x[2, 0]), 0],
                  [0.0, DT],
                  [1.0, 0.0]])
    x = F @ x + B @ u
    return x


def observation_model(x):
    H = np.array([
        [1, 0, 0, 0],
        [0, 1, 0, 0]])
    z = H @ x

    return z


def jacobF(x, u):
    """
    Jacobian of Motion Model
    motion model
    x_{t+1} = x_t+v*dt*cos(yaw)
    y_{t+1} = y_t+v*dt*sin(yaw)
    yaw_{t+1} = yaw_t+omega*dt
    v_{t+1} = v{t}
    so
    dx/dyaw = -v*dt*sin(yaw)
    dx/dv = dt*cos(yaw)
    dy/dyaw = v*dt*cos(yaw)
    dy/dv = dt*sin(yaw)
    """
    yaw = x[2, 0]
    v = u[0, 0]
    jF = np.array([
        [1.0, 0.0, -DT * v * math.sin(yaw), DT * math.cos(yaw)],
        [0.0, 1.0, DT * v * math.cos(yaw), DT * math.sin(yaw)],
        [0.0, 0.0, 1.0, 0.0],
        [0.0, 0.0, 0.0, 1.0]])

    return jF


def jacobH(x):
    # Jacobian of Observation Model
    jH = np.array([
        [1, 0, 0, 0],
        [0, 1, 0, 0]
    ])

    return jH


def ekf_estimation(xEst, PEst, z, u):

    #  Predict
    xPred = motion_model(xEst, u)
    jF = jacobF(xPred, u)
    PPred = jF@PEst@jF.T + Q

    #  Update
    jH = jacobH(xPred)
    zPred = observation_model(xPred)
    y = z - zPred
    S = jH@PPred@jH.T + R
    K = PPred@jH.T@np.linalg.inv(S)
    xEst = xPred + K@y
    PEst = (np.eye(len(xEst)) - K@jH)@PPred

    return xEst, PEst


def plot_covariance_ellipse(xEst, PEst):  # pragma: no cover
    Pxy = PEst[0:2, 0:2]
    eigval, eigvec = np.linalg.eig(Pxy)

    if eigval[0] >= eigval[1]:
        bigind = 0
        smallind = 1
    else:
        bigind = 1
        smallind = 0

    t = np.arange(0, 2 * math.pi + 0.1, 0.1)
    a = math.sqrt(eigval[bigind])
    b = math.sqrt(eigval[smallind])
    x = [a * math.cos(it) for it in t]
    y = [b * math.sin(it) for it in t]
    angle = math.atan2(eigvec[bigind, 1], eigvec[bigind, 0])
    R = np.array([[math.cos(angle), math.sin(angle)],
                  [-math.sin(angle), math.cos(angle)]])
    fx = R@(np.array([x, y]))
    px = np.array(fx[0, :] + xEst[0, 0]).flatten()
    py = np.array(fx[1, :] + xEst[1, 0]).flatten()
    plt.plot(px, py, "--r")


def main():
    print(__file__ + " start!!")

    time = 0.0

    # State Vector [x y yaw v]'
    xEst = np.zeros((4, 1))
    xTrue = np.zeros((4, 1))
    PEst = np.eye(4)

    xDR = np.zeros((4, 1))  # Dead reckoning

    # history
    hxEst = xEst
    hxTrue = xTrue
    hxDR = xTrue
    hz = np.zeros((2, 1))

    while SIM_TIME >= time:
        time += DT
        u = calc_input()

        xTrue, z, xDR, ud = observation(xTrue, xDR, u)

        xEst, PEst = ekf_estimation(xEst, PEst, z, ud)

        # store data history
        hxEst = np.hstack((hxEst, xEst))
        hxDR = np.hstack((hxDR, xDR))
        hxTrue = np.hstack((hxTrue, xTrue))
        hz = np.hstack((hz, z))

        if show_animation:
            plt.cla()
            plt.plot(hz[0, :], hz[1, :], ".g")
            plt.plot(hxTrue[0, :].flatten(),
                     hxTrue[1, :].flatten(), "-b")
            plt.plot(hxDR[0, :].flatten(),
                     hxDR[1, :].flatten(), "-k")
            plt.plot(hxEst[0, :].flatten(),
                     hxEst[1, :].flatten(), "-r")
            plot_covariance_ellipse(xEst, PEst)
            plt.axis("equal")
            plt.grid(True)
            plt.pause(0.001)


if __name__ == '__main__':
    main()

2. Filter design

in this simulaton, the robot has a state vector includes 4 states at time $t$

X_{t} = [x_{t}, y_{t}, ϕ_{t}, v_{t}]

$X_t = [x_t, y_t, \phi_t, v_t]$

$x$ , $y$ are a 2D x-y position, $\phi$ is orientation, and v isvelociyy.

in the code, “xEst” means the state vector.

$P_t$ is covariace matrix of the state,

$Q$ is covariance matrix of process noise,

$R$ is covariance matrix of observation noise at time $t$

The robot has a speed sensor and a gyro sensor.

So, the input vector can be used as each time step

u_{t} = [v_{t}, w_{t}]

$u_t = [v_t, w_t]$

Also, the robot has a GNSS sensor, it means that the robot can observe x-y position at each time.

z_{t} = [x_{t}, y_{t}]

$z_t = [x_t, y_t]$

The input and observation vector includes sensor noise. In the code, “observation” function generates the input and observation vector with noise.

3. Motion Model

The robot model is

\dot{ϕ} = w

$\dot{\phi}=w$

So, the motion model is

x_{t + 1} = F x_{t} + B u_{t}

$x_{t+1}=Fx_t+Bu_t$

where

F = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

$F = \begin{bmatrix} {1}&{0}&{0}&{0}\\ {0}&{1}&{0}&{0}\\ {0}&{0}&{1}&{0}\\ {0}&{0}&{0}&{1}\\ \end{bmatrix}$

B = [\begin{matrix} c o s (ϕ) d t & 0 \\ s i n (ϕ) d t & 0 \\ 0 & d t \\ 1 & 0 \end{matrix}]

$B = \begin{bmatrix} {cos(\phi)dt}&{0}&\\ {sin(\phi)dt}&{0}&\\ {0}&{dt}&\\ {1}&{0}&\\ \end{bmatrix}$

$d_t$ is a time interval. This is implemented at below:

Its Jacobian matrix is

J_{F} = [\begin{matrix} \frac{d x}{d x} & \frac{d x}{d y} & \frac{d x}{d ϕ} & \frac{d x}{d ϕ} & \frac{d x}{d v} \\ \frac{d y}{d x} & \frac{d y}{d y} & \frac{d y}{d ϕ} & \frac{d y}{d ϕ} & \frac{d y}{d v} \\ \frac{d ϕ}{d x} & \frac{d ϕ}{d y} & \frac{d ϕ}{d ϕ} & \frac{d ϕ}{d ϕ} & \frac{d ϕ}{d v} \\ \frac{d v}{d x} & \frac{d v}{d y} & \frac{d v}{d ϕ} & \frac{d v}{d ϕ} & \frac{d v}{d v} \end{matrix}]

$J_F = \begin{bmatrix} {\frac{dx}{dx}}&{\frac{dx}{dy}}&{\frac{dx}{d\phi}}&{\frac{dx}{d\phi}}&{\frac{dx}{dv}}\\ {\frac{dy}{dx}}&{\frac{dy}{dy}}&{\frac{dy}{d\phi}}&{\frac{dy}{d\phi}}&{\frac{dy}{dv}}\\ {\frac{d\phi}{dx}}&{\frac{d\phi}{dy}}&{\frac{d\phi}{d\phi}}&{\frac{d\phi}{d\phi}}&{\frac{d\phi}{dv}}\\ {\frac{dv}{dx}}&{\frac{dv}{dy}}&{\frac{dv}{d\phi}}&{\frac{dv}{d\phi}}&{\frac{dv}{dv}}\\ \end{bmatrix}$

= [\begin{matrix} 1 & 0 & - v s i n (ϕ) d t & c o s (ϕ) d e t \\ 0 & 1 & v c o s (ϕ) d t & s i n (ϕ) d t \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

$=\begin{bmatrix} {1}&{0}&{-vsin(\phi)dt}&{cos(\phi)det}\\ {0}&{1}&{vcos(\phi)dt}&{sin(\phi)dt}\\ {0}&{0}&{1}&{0}\\ {0}&{0}&{0}&{1}\\ \end{bmatrix}$

4.Observation Model

The robot can get x-y position information from GPS. So GPS Observation model is

z_{t} = H x_{t}

$z_t=Hx_t$

where

B = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}]

$B=\begin{bmatrix} {1}&{0}&{0}&{0}\\ {0}&{1}&{0}&{0}\\ \end{bmatrix}$

Its jacobian matrix is

J_{H} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}]

$J_H=\begin{bmatrix} {1}&{0}&{0}&{0}\\ {0}&{1}&{0}&{0}\\ \end{bmatrix}$

5. Extented Kalman Filter

Localization porcess using Extended Kalman Filter: EKF is

1. Predict

x_{p r e d} = F_{x_{t}} + B u_{t}

$x_{pred} = F_{x_t}+Bu_t$

P_{p r e d} = J_{f} P_{t} J_{f}^{T} + Q

$P_{pred} = J_fP_tJ_f^{T} + Q$

2. Update

Z_{p r e d} = H x_{p r e d}

$Z_{pred}= Hx_{pred}$

y = z - z_{p r e d}

$y = z - z_{pred}$

S = J_{H} P_{p r e d} J_{H}^{T} + R

$S = J_HP_{pred}J_H^{T}+R$

K = P_{p r e d} J_{H}^{T} S^{- 1}

$K = P_{pred}J_H^{T}S^{-1}$

x_{t + 1} = x_{p r e d} + K y

$x_{t+1} = x_{pred}+ Ky$

P_{t + 1} = (I - K J_{H}) P_{p r e d}

$P_{t+1} = (I - KJ_H)P_{pred}$

ekf代码学习笔记