## 一、简介 著名学者Julier等提出近似非线性函数的均值和方差远比近似非线性函数本身更容易，因此提出了基于确定性采样的UKF算法。 该算法的核心思想是：采用UT变换，利用一组Sigma采样点来描述随机变量的高斯分布，然后通过非线性函数的传递，再利用加权统计线性回归技术来近似非x线性函数的后验均值和方差。 相比于EKF，UKF的估计精度能够达到泰勒级数展开的二阶精度。 **1 UT变换**  **2 采样策略** 根据Sigma点采样策略不同，相应的Sigma点以及均值权值和方差权值也不尽相同，因此UT变换的估计精度也会有差异，但总体来说，其估计精度能够达到泰勒级数展开的二阶精度。 为保证随机变量x经过采样之后得到的Sigma采样点仍具有原变量的必要特性，所以采样点的选取应满足：  下面介绍两种经常使用的采样策略：比例采样和比例修正对称采样     **3 UKF算法流程**  ## 二、源代码 ```c %% UKF bicycle test clear all close all % load params from file load('bicycle_data.mat') use_laser = 1; use_radar = 1; stop_for_sigmavis = false; %% Data Initialization x_pred_all = []; % predicted state history x_est_all = []; % estimated state history with time at row number 6 NIS_radar_all = []; % estimated state history with time at row number 6 NIS_laser_all = []; % estimated state history with time at row number 6 est_pos_error_squared_all = []; laser_pos_error_squared_all = []; P_est = 0.2*eye(n_x); % initial uncertainty matrix P_est(4,4) = 0.3; % initial uncertainty P_est(5,5) = 0.3; % initial uncertainty %% process noise acc_per_sec = 0.3; % acc in m/s^2 per sec yaw_acc_per_sec = 0.3; % yaw acc in rad/s^2 per sec Z_l_read = []; std_las1 = 0.15; std_las2 = 0.15; std_radr = 0.3; std_radphi = 0.03; std_radrd = 0.3; % UKF params n_aug = 7; kappa = 3-n_aug; w = zeros(2*n_aug+1,1); w(1) = kappa/(kappa+n_aug); for i=2:(2*n_aug+1) w(i) = 0.5/(n_aug+kappa); end %% UKF filter recursion %x_est_all(:,1) = GT(:,1); Xi_pred_all = []; Xi_aug_all = []; x_est = [0.1 0.1 0.1 0.1 0.01]; last_time = 0; % load measurement data from file fid = fopen('obj_pose-laser-radar-synthetic-ukf-input.txt'); %% State Initialization tline = fgets(fid); % read first line % find first laser measurement while tline(1) ~= 'L' % laser measurement tline = fgets(fid); % go to next line end line_vector = textscan(tline,'%s %f %f %f %f %f %f %f %f %f'); last_time = line_vector{4};% get timestamp x_est(1) = line_vector{2}; % initialize position p_x x_est(2) = line_vector{3}; % initialize position p_y tline = fgets(fid); % go to next line % counter k = 1; while ischar(tline) % go through lines of data file % find time of measurement if tline(1) == 'L' % laser measurement if use_laser == false tline = fgets(fid); % skip this line and go to next line continue; else % read laser meas time line_vector = textscan(tline,'%s %f %f %f %f %f %f %f %f %f'); meas_time = line_vector{1,4}; end elseif tline(1) == 'R' % radar measurement if use_radar == false tline = fgets(fid); % skip this line and go to next line continue; else % read radar meas time line_vector = textscan(tline,'%s %f %f %f %f %f %f %f %f %f %f'); meas_time = line_vector{5}; end else % neither laser nor radar disp('Error: not laser nor radar') return; end delta_t_sec = ( meas_time - last_time ) / 1e6; % us to sec last_time = meas_time; %% Prediction part p1 = x_est(1); p2 = x_est(2); v = x_est(3); yaw = x_est(4); yaw_dot = x_est(5); % yaw_dot: yaw velocity x = [p1; p2; v; yaw; yaw_dot]; % state vector std_a = acc_per_sec; % process noise long. acceleration std_ydd = yaw_acc_per_sec; % process noise yaw acceleration if std_a == 0 std_a = 0.0001; end if std_ydd == 0 std_ydd = 0.0001; end % Create sigma points x_aug = [x ; 0 ; 0]; P_aug = [P_est zeros(n_x,2) ; zeros(2,n_x) [std_a^2 0 ; 0 std_ydd^2 ]]; %P_aug = nearestSPD(P_aug); Xi_aug = zeros(n_aug,2*n_aug+1); sP_aug = chol(P_aug,'lower'); % Cholesky factorization. Xi_aug(:,1) = x_aug; for i=1:n_aug Xi_aug(:,i+1) = x_aug + sqrt(n_aug+kappa) * sP_aug(:,i); Xi_aug(:,i+1+n_aug) = x_aug - sqrt(n_aug+kappa) * sP_aug(:,i); end % Predict sigma points Xi_pred = zeros(n_x,2*n_aug+1); for i=1:(2*n_aug+1) p1 = Xi_aug(1,i); p2 = Xi_aug(2,i); v = Xi_aug(3,i); yaw = Xi_aug(4,i); yaw_dot = Xi_aug(5,i); nu_a = Xi_aug(6,i); nu_yaw_dd = Xi_aug(7,i); if abs(yaw_dot) > 0.001 %turn around p1_p = p1 + v/yaw_dot * ( sin (yaw + yaw_dot*delta_t_sec) - sin(yaw)); p2_p = p2 + v/yaw_dot * ( cos(yaw) - cos(yaw+yaw_dot*delta_t_sec) ); else %not turn around p1_p = p1 + v*delta_t_sec*cos(yaw); p2_p = p2 + v*delta_t_sec*sin(yaw); end v_p = v; yaw_p = yaw + yaw_dot*delta_t_sec; yaw_dot_p = yaw_dot; % add noise p1_p = p1_p + 0.5*nu_a*delta_t_sec^2 * cos(yaw); p2_p = p2_p + 0.5*nu_a*delta_t_sec^2 * sin(yaw); v_p = v_p + nu_a*delta_t_sec; yaw_p = yaw_p + 0.5*nu_yaw_dd*delta_t_sec^2; yaw_dot_p = yaw_dot_p + nu_yaw_dd*delta_t_sec; Xi_pred(1,i) = p1_p; Xi_pred(2,i) = p2_p; Xi_pred(3,i) = v_p; Xi_pred(4,i) = yaw_p; Xi_pred(5,i) = yaw_dot_p; end % average and covar of sigma points x_pred = 0; P_pred = zeros(5,5); for i=1:2*n_aug+1 x_pred = x_pred + w(i)* Xi_pred(:,i); end for i=1:2*n_aug+1 P_pred = P_pred + w(i)* (Xi_pred(:,i) - x_pred)*(Xi_pred(:,i) - x_pred)'; end %% visualize sigma point examples if stop_for_sigmavis && k == 25 disp('Stopping for sigma point visualization'); % 2d example P_s = P_est (1:2,1:2); x_s = x(1:2); Xi_s = zeros(2,5); A = chol(P_s,'lower'); Xi_s(:,1) = x_s; for i=1:2 Xi_s(:,i+1) = x_s + sqrt(3) * A(:,i); Xi_s(:,i+1+2) = x_s - sqrt(3) * A(:,i); end error_ellipse(P_s,x_s,'conf', 0.4, 'style', 'k-'); Xi_aug_p1 = squeeze(Xi_s(1,:,:)); Xi_aug_p2 = squeeze(Xi_s(2,:,:)); hold on; plot(Xi_aug_p1, Xi_aug_p2, 'or'); legend('P', 'sigma points') axis equal xlabel('p_x in m'); ylabel('p_y in m'); save('sigma_visualization.mat', 'x_s','P_s','A','Xi_s', 'Xi_aug', 'Xi_pred'); %return; end k=k+1; ``` ## 三、运行结果     ## 四、备注 版本：2014a