1、建立仿真模型

（1）假设有一辆小车在一平面运动，起始坐标为[0，0]，运动速度为1m/s，加速度为0.1 m / s 2 m/s^2 m/s2，则可以建立如下的状态方程：
Y = A ∗ X + B ∗ U Y=A*X+B*U Y=A∗X+B∗U
U为速度和加速度的的矩阵
U = [ 1 0.1 ] U= \begin{bmatrix} 1 \\ 0.1\\ \end{bmatrix} U=[10.1​]
X为当前时刻的坐标，速度，加速度
X = [ x y y a w V ] X= \begin{bmatrix} x \\ y \\ yaw \\ V \end{bmatrix} X=⎣⎢⎢⎡​xyyawV​⎦⎥⎥⎤​
Y为下一时刻的状态
则观察矩阵A为：
A = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 ] A= \begin{bmatrix} 1&0 & 0 &0 \\ 0 & 1 & 0&0 \\ 0 & 0 &1 &0 \\ 0&0 & 0 &0 \end{bmatrix} A=⎣⎢⎢⎡​1000​0100​0010​0000​⎦⎥⎥⎤​
矩阵B则决定小车的运动规矩，这里取B为：
B = [ c o s ( x ) ∗ t 0 s i n ( x ) ∗ t 0 0 t 1 0 ] B= \begin{bmatrix} cos(x)*t &0\\ sin(x)*t &0\\ 0&t\\ 1&0 \end{bmatrix} B=⎣⎢⎢⎡​cos(x)∗tsin(x)∗t01​00t0​⎦⎥⎥⎤​
python编程实现小车的运动轨迹:

"""Particle Filter localization sampleauthor: Atsushi Sakai (@Atsushi_twi)"""import mathimport matplotlib.pyplot as pltimport numpy as npfrom scipy.spatial.transform import Rotation as RotDT = 0.1 # time tick [s]SIM_TIME = 50.0 # simulation time [s]MAX_RANGE = 20.0 # maximum observation range# Particle filter parameterNP = 100 # Number of ParticleNTh = NP / 2.0 # Number of particle for re-samplingdef calc_input(): v = 1.0 # [m/s] yaw_rate = 0.1 # [rad/s] u = np.array([[v, yaw_rate]]).T return udef 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.dot(x) + B.dot(u) return xdef main(): print(__file__ + " start!!") time = 0.0 # State Vector [x y yaw v]' x_true = np.zeros((4, 1)) x = [] y = [] while SIM_TIME >= time: time += DT u = calc_input() x_true = motion_model(x_true, u) x.append(x_true[0]) y.append(x_true[1]) plt.plot(x,y, "-b") if __name__ == '__main__': main()

运行结果：

2、生成观测数据

实际运用中，我们需要对小车的位置进行定位，假设坐标系上有4个观测点，在小车运动过程中，需要定时将小车距离这4个观测点的位置距离记录下来，这样，当小车下一次寻迹时就有了参考点；

def observation(x_true, xd, u, rf_id): x_true = motion_model(x_true, u) # add noise to gps x-y z = np.zeros((0, 3)) for i in range(len(rf_id[:, 0])): dx = x_true[0, 0] - rf_id[i, 0] dy = x_true[1, 0] - rf_id[i, 1] d = math.hypot(dx, dy) if d <= MAX_RANGE: dn = d + np.random.randn() * Q_sim[0, 0] ** 0.5 # add noise zi = np.array([[dn, rf_id[i, 0], rf_id[i, 1]]]) z = np.vstack((z, zi)) # add noise to input ud1 = u[0, 0] + np.random.randn() * R_sim[0, 0] ** 0.5 ud2 = u[1, 0] + np.random.randn() * R_sim[1, 1] ** 0.5 ud = np.array([[ud1, ud2]]).T xd = motion_model(xd, ud) return x_true, z, xd, ud

3、实现粒子滤波

#def gauss_likelihood(x, sigma): p = 1.0 / math.sqrt(2.0 * math.pi * sigma ** 2) * \ math.exp(-x ** 2 / (2 * sigma ** 2)) return pdef pf_localization(px, pw, z, u): """ Localization with Particle filter """ for ip in range(NP): x = np.array([px[:, ip]]).T w = pw[0, ip] # 预测输入 ud1 = u[0, 0] + np.random.randn() * R[0, 0] ** 0.5 ud2 = u[1, 0] + np.random.randn() * R[1, 1] ** 0.5 ud = np.array([[ud1, ud2]]).T x = motion_model(x, ud) # 计算权重 for i in range(len(z[:, 0])): dx = x[0, 0] - z[i, 1] dy = x[1, 0] - z[i, 2] pre_z = math.hypot(dx, dy) dz = pre_z - z[i, 0] w = w * gauss_likelihood(dz, math.sqrt(Q[0, 0])) px[:, ip] = x[:, 0] pw[0, ip] = w pw = pw / pw.sum() # 归一化 x_est = px.dot(pw.T) p_est = calc_covariance(x_est, px, pw) #计算有效粒子数 N_eff = 1.0 / (pw.dot(pw.T))[0, 0] #重采样 if N_eff < NTh: px, pw = re_sampling(px, pw) return x_est, p_est, px, pwdef re_sampling(px, pw): """ low variance re-sampling """ w_cum = np.cumsum(pw) base = np.arange(0.0, 1.0, 1 / NP) re_sample_id = base + np.random.uniform(0, 1 / NP) indexes = [] ind = 0 for ip in range(NP): while re_sample_id[ip] > w_cum[ind]: ind += 1 indexes.append(ind) px = px[:, indexes] pw = np.zeros((1, NP)) + 1.0 / NP # init weight return px, pw

4、完整源码

该代码来源于https://github.com/AtsushiSakai/PythonRobotics

"""Particle Filter localization sampleauthor: Atsushi Sakai (@Atsushi_twi)"""import mathimport matplotlib.pyplot as pltimport numpy as npfrom scipy.spatial.transform import Rotation as Rot# Estimation parameter of PFQ = np.diag([0.2]) ** 2 # range errorR = np.diag([2.0, np.deg2rad(40.0)]) ** 2 # input error# Simulation parameterQ_sim = np.diag([0.2]) ** 2R_sim = np.diag([1.0, np.deg2rad(30.0)]) ** 2DT = 0.1 # time tick [s]SIM_TIME = 50.0 # simulation time [s]MAX_RANGE = 20.0 # maximum observation range# Particle filter parameterNP = 100 # Number of ParticleNTh = NP / 2.0 # Number of particle for re-samplingshow_animation = Truedef calc_input(): v = 1.0 # [m/s] yaw_rate = 0.1 # [rad/s] u = np.array([[v, yaw_rate]]).T return udef observation(x_true, xd, u, rf_id): x_true = motion_model(x_true, u) # add noise to gps x-y z = np.zeros((0, 3)) for i in range(len(rf_id[:, 0])): dx = x_true[0, 0] - rf_id[i, 0] dy = x_true[1, 0] - rf_id[i, 1] d = math.hypot(dx, dy) if d <= MAX_RANGE: dn = d + np.random.randn() * Q_sim[0, 0] ** 0.5 # add noise zi = np.array([[dn, rf_id[i, 0], rf_id[i, 1]]]) z = np.vstack((z, zi)) # add noise to input ud1 = u[0, 0] + np.random.randn() * R_sim[0, 0] ** 0.5 ud2 = u[1, 0] + np.random.randn() * R_sim[1, 1] ** 0.5 ud = np.array([[ud1, ud2]]).T xd = motion_model(xd, ud) return x_true, z, xd, uddef 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.dot(x) + B.dot(u) return xdef gauss_likelihood(x, sigma): p = 1.0 / math.sqrt(2.0 * math.pi * sigma ** 2) * \ math.exp(-x ** 2 / (2 * sigma ** 2)) return pdef calc_covariance(x_est, px, pw): """ calculate covariance matrix see ipynb doc """ cov = np.zeros((3, 3)) n_particle = px.shape[1] for i in range(n_particle): dx = (px[:, i:i + 1] - x_est)[0:3] cov += pw[0, i] * dx @ dx.T cov *= 1.0 / (1.0 - pw @ pw.T) return covdef pf_localization(px, pw, z, u): """ Localization with Particle filter """ for ip in range(NP): x = np.array([px[:, ip]]).T w = pw[0, ip] # Predict with random input sampling ud1 = u[0, 0] + np.random.randn() * R[0, 0] ** 0.5 ud2 = u[1, 0] + np.random.randn() * R[1, 1] ** 0.5 ud = np.array([[ud1, ud2]]).T x = motion_model(x, ud) # Calc Importance Weight for i in range(len(z[:, 0])): dx = x[0, 0] - z[i, 1] dy = x[1, 0] - z[i, 2] pre_z = math.hypot(dx, dy) dz = pre_z - z[i, 0] w = w * gauss_likelihood(dz, math.sqrt(Q[0, 0])) px[:, ip] = x[:, 0] pw[0, ip] = w pw = pw / pw.sum() # normalize x_est = px.dot(pw.T) p_est = calc_covariance(x_est, px, pw) N_eff = 1.0 / (pw.dot(pw.T))[0, 0] # Effective particle number if N_eff < NTh: px, pw = re_sampling(px, pw) return x_est, p_est, px, pwdef re_sampling(px, pw): """ low variance re-sampling """ w_cum = np.cumsum(pw) base = np.arange(0.0, 1.0, 1 / NP) re_sample_id = base + np.random.uniform(0, 1 / NP) indexes = [] ind = 0 for ip in range(NP): while re_sample_id[ip] > w_cum[ind]: ind += 1 indexes.append(ind) px = px[:, indexes] pw = np.zeros((1, NP)) + 1.0 / NP # init weight return px, pwdef plot_covariance_ellipse(x_est, p_est): # pragma: no cover p_xy = p_est[0:2, 0:2] eig_val, eig_vec = np.linalg.eig(p_xy) if eig_val[0] >= eig_val[1]: big_ind = 0 small_ind = 1 else: big_ind = 1 small_ind = 0 t = np.arange(0, 2 * math.pi + 0.1, 0.1) # eig_val[big_ind] or eiq_val[small_ind] were occasionally negative # numbers extremely close to 0 (~10^-20), catch these cases and set the # respective variable to 0 try: a = math.sqrt(eig_val[big_ind]) except ValueError: a = 0 try: b = math.sqrt(eig_val[small_ind]) except ValueError: b = 0 x = [a * math.cos(it) for it in t] y = [b * math.sin(it) for it in t] angle = math.atan2(eig_vec[1, big_ind], eig_vec[0, big_ind]) rot = Rot.from_euler('z', angle).as_matrix()[0:2, 0:2] fx = rot.dot(np.array([[x, y]])) px = np.array(fx[0, :] + x_est[0, 0]).flatten() py = np.array(fx[1, :] + x_est[1, 0]).flatten() plt.plot(px, py, "--r")def main(): print(__file__ + " start!!") time = 0.0 # RF_ID positions [x, y] rf_id = np.array([[10.0, 0.0], [10.0, 10.0], [0.0, 15.0], [-5.0, 20.0]]) # State Vector [x y yaw v]' x_est = np.zeros((4, 1)) x_true = np.zeros((4, 1)) px = np.zeros((4, NP)) # Particle store pw = np.zeros((1, NP)) + 1.0 / NP # Particle weight x_dr = np.zeros((4, 1)) # Dead reckoning # history h_x_est = x_est h_x_true = x_true h_x_dr = x_true while SIM_TIME >= time: time += DT u = calc_input() x_true, z, x_dr, ud = observation(x_true, x_dr, u, rf_id) x_est, PEst, px, pw = pf_localization(px, pw, z, ud) # store data history h_x_est = np.hstack((h_x_est, x_est)) h_x_dr = np.hstack((h_x_dr, x_dr)) h_x_true = np.hstack((h_x_true, x_true)) if show_animation: plt.cla() # for stopping simulation with the esc key. plt.gcf().canvas.mpl_connect( 'key_release_event', lambda event: [exit(0) if event.key == 'escape' else None]) for i in range(len(z[:, 0])): plt.plot([x_true[0, 0], z[i, 1]], [x_true[1, 0], z[i, 2]], "-k") plt.plot(rf_id[:, 0], rf_id[:, 1], "*k") plt.plot(px[0, :], px[1, :], ".r") plt.plot(np.array(h_x_true[0, :]).flatten(), np.array(h_x_true[1, :]).flatten(), "-b") plt.plot(np.array(h_x_dr[0, :]).flatten(), np.array(h_x_dr[1, :]).flatten(), "-k") plt.plot(np.array(h_x_est[0, :]).flatten(), np.array(h_x_est[1, :]).flatten(), "-r") plot_covariance_ellipse(x_est, PEst) plt.axis("equal") plt.grid(True) plt.pause(0.001)if __name__ == '__main__': main()

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