EKF-SLAM中機器人的運動軌跡爲:X={x1,...xk},其中每個x表示機器人在第i時刻的位姿，其中機器人在平面運動，故使用2D的參數：xi=[x，y，θ]，路標的集合爲L={l1，...ln}，需要建立運動方程和觀測方程，由於上述兩個方式是非線性的，根據EKF理論，需要將他們線性化，進而求得幾個重要的雅可比矩陣，然後對EKF的預測部分兩個方程和更新部分三個方程進行求解。
EKF-SLAM有兩個重要的假設：誤差爲高斯分佈，以及工作點附近可以用導數線性化。當這兩個假設成立時，EKF-SLAM可以實時的更新機器人的當前位姿與路標。
EKF-SLAM程序如下：
%-------------------------------------------------------------------------
% FILE: slam.m
% DESC: Implements the SLAM algorithm based on a robot model and sensor
% data from Nebot.
%landmarks(真實路標)
%estbeac=estimation(估計)+beacon(路標)，將estbeac的值賦給landmarks然後將estbeac清除。
%xest 狀態矩陣
%Pest 協方差矩陣
%A W Jh 雅可比矩陣
%Q 過程激勵噪聲協方差矩陣
%k 迭代次數
%K 卡爾曼增益
%numStates 狀態的數量 初始化時候就三個，即位置(x，y)和姿態(phi)
%phi 姿態，即與x軸的夾角
%Steering 轉向角
%alpha 轉向角
%Ve = Velocity 後輪的轉速
%Vc 車水平平移速度
%zlaser=[rangeArray; bearingArray]; 機器人與路標的距離和方位
%--------------------------------------------------------------------------
clear all; clc;
load('data_set');
load('beac_juan3.mat'); %beac_juan3.mat中內容爲路標,命令whos –file查看該文件中的內容
landmarks=estbeac; clear estbeac; % true landmark positions (measured w/GPS sensor)
%--------------------- DATA INITIALIZATION -----------------------------
% Vehicle constants
L=2.83; % distance between front and rear axles(前後輪軸之間的距離)
h=0.76; % distance between center of rear axle and encoder(後輪軸中心與編碼器的距離)
b=0.5; % vertical distance from rear axle to laser(後輪軸中心到激光傳感器的垂直距離)
a=3.78; % horizontal distance from rear axle to laser(後輪軸中心到激光傳感器的水平距離)
% EKF components used in multiple functions
global xest Pest A Q W %定義全局變量
global numStates k
xest = zeros(3,1); % state matrix
Pest = zeros(3,3); % covariance matrix2
numStates=3; % initially, there are no landmarks(i.e. the number of value in the state matrix)
% matrixes initialization
A = zeros(3,3); % process model jacobian
W = eye(3); % process noise jacobian
Q = [.49 0 0; 0 .49 0; 0 0 (7*pi/180)^2];
Jh = zeros(2,3); % sensor model jacobian
K = zeros(3,3); % kalman gain
% initialize time
To = min([TimeGPS(1),Time_VS(1),TimeLaser(1)]); % data starts being collected at 45 secs for some reason??
Time = Time - To;
t=Time(1);
tfinal = 112; % there is 112 seconds worth of data
k=1; % loop iteration counter(循環迭代計數)
% initialize estimates
xest(1,1) = GPSLon(1); % x position of vehicle
xest(2,1) = GPSLat(1); % y position of vehicle
xest(3,1) = -126*pi/180; % phi of vehicle % LB(?):how to get this value?
Pest(:,:) = [0.1, 0, 0; 0, 0.1, 0; 0, 0, 15*pi/180];
% initial inputs
alpha = Steering(1); % measured steering angle(測量轉向角)
Ve = Velocity(1); % measured velocity at back wheel(測量後輪速率)
Vc = Velocity(1) / (1 - (h/L)*tan(alpha)); % measured velocity transformed to vehicle center
%--------------------- end of DATA INITIALIZATION ------------------------
disp('Running through Kalman filter...');
n_dsp = 0;
%------------------ EXTENDED KALMAN FILTER PROCEDURE --------------------
while t<tfinal
k=k+1;
t=Time(k);
dt=t-Time(k-1);
% Calculate time update matrices (prediction step of kalman filter)
% Vc and alpha are updated after prediction...therefore, Vc is Vc(k-1) & alpha is alpha(k-1)
%下面是公式（1）
phi = xest(3,k-1); % so I don't have to type in xest(3,k-1) for every phi below
xest(1,k) = xest(1,k-1) + dt*Vc*cos(phi) - dt*Vc/L*tan(alpha)*(a*sin(phi)+b*cos(phi)); % x prediction
xest(2,k) = xest(2,k-1) + dt*Vc*sin(phi) + dt*Vc/L*tan(alpha)*(a*cos(phi)-b*sin(phi)); % y prediction
xest(3,k) = xest(3,k-1) + dt*Vc/L*tan(alpha); % phi prediction
xest(3,k) = normalizeAngle(xest(3,k)); % keep phi between -180 and 180 degrees
% landmarks are assumed to be static, i.e. pi(k) = pi(k-1)
if(numStates>3)
for i=4:numStates
xest(i,k)= xest(i,k-1);
end
end
% Calculate Jacobian A based on vehicle model (df/dx)
A(1,1)=1; A(1,2)=0; A(1,3) = -dt*Vc*sin(phi) - dt*Vc/L*tan(alpha)*(a*cos(phi)-b*sin(phi));
A(2,1)=0; A(2,2)=1; A(2,3) = dt*Vc*cos(phi) - dt*Vc/L*tan(alpha)*(a*sin(phi)+b*cos(phi));
A(3,1)=0; A(3,2)=0; A(3,3)=1;
% rest of A (which is just 2 null matrices and 1 identity) is added in new_state function
% Calculate prediction for covariance matrix
Pest = A*Pest*A' + W*Q*W'; %(公式2)
% Check to see if new velocity and steering measurements are available
if Sensor(k)==2 % encoders(編碼器) are sensor #2
Ve = Velocity(Index(k));
alpha = Steering(Index(k));
Vc = Ve / (1-(h/L)*tan(alpha));
end
% Check to see if new laser(激光) data is available
if (Sensor(k)==3 && t>1) % SICK is sensor #3
% from a single 180 degree scan, determine # of landmarks & center of each landmark
[rangeArray bearingArray] = getLandmarkCenter(Laser(Index(k),:), Intensity(Index(k),:));
zlaser=[rangeArray; bearingArray]; % range/bearing data in observation model format
numLandmarks = size(rangeArray,2); % number of LMs captured in a single scan
for i=1:numLandmarks % for i = 1 to # of landmakrs picked up in single scan
if(numStates==3) % if 1st observed landmark, update state and covariance
new_state(zlaser(:,i)); % add new state (i.e. increase all matrix sizes)
numStates = numStates + 2; % increase number of states by 2 (1 for xi, 1 for yi)
[xest,Pest] = updateNew(zlaser(:,i)); % update state and covariance
else % not 1st landmark -> check to make sure no LMs in range
[dist,index]=nearby_LMs(zlaser(:,i)); % dist from observation to already incorporated LMs (< 3m)
min_dist = 2; % minimum distance between landmark and observation
if dist>min_dist % if dist to nearby_LM is > "min_dist", add observation as new LM
new_state(zlaser(:,i)); % add new state (i.e. increase all matrix sizes)
numStates = numStates + 2; % increase number of states by 2 (1 for xi, 1 for yi)
[xest,Pest] = updateNew(zlaser(:,i)); % update state and covariance
else % else find closest LM and associate
closest = 4 + 2*(index-1); % find LM which is closest to observation
[xest,Pest] = updateExisting(zlaser(:,i),closest); % update state and covariance
end
end
end
end
% LB_add: display information
if( floor(t/tfinal*100) > n_dsp)
clc;
n_dsp = floor(t/tfinal*100);
str = sprintf('運行卡爾曼濾波... [%d%c]', n_dsp,'%');
disp(str);
end
end % end while
%-------------- end of EXTENDED KALMAN FILTER PROCEDURE -----------------
%--------------------------- PLOT RESULTS --------------------------------
figure(1); hold on;
plot(GPSLon(1:560),GPSLat(1:560)); % robot position (true)
plot(xest(1,1:k),xest(2,1:k),'g'); % robot position (model)
plot(landmarks(:,1),landmarks(:,2),'*'); % landmark position (true)
for i=4:2:numStates
plot(xest(i,k),xest(i+1,k),'r*'); % landmark position (model)
end
legend('機器人實際位置','機器人slam位置','路標實際位置','路標slam位置',2);
xlabel('x [meters]'); ylabel('y [meters]');
axis([-10 20 -25 20]);
%% Estimate error(估計誤差)
x_error1 = GPSLon(1:560)-xest(1,1:560);
x_error2 = GPSLat(1:560)-xest(2,1:560);
for i=4:2:numStates
landmarks_error1=landmarks(:,2)-xest(i+1,k);
end
for i=4:2:numStates
landmarks_error2=landmarks(:,1)-xest(i,k);
end
%% Graph 2
figure(2)
plot(x_error1),grid on;
title('機器人位置誤差 on X axis')
xlabel('時間 [sec]')
ylabel('位置均方根誤差 [m]')
axis([0 112 -25 25]);
hold off;
%% Graph 3
figure(3)
plot(x_error2),grid on;
title('機器人位置誤差 on Y axis')
xlabel('時間 [sec]')
ylabel('位置均方根誤差 [m]')
axis([0 112 -25 25]);
hold off;
%% Graph 4
figure(4)
plot(landmarks_error1),grid on;
title('路標位置誤差 on x axis')
xlabel('時間 [sec]')
ylabel('位置均方根誤差 [m]')
axis([0 18 -25 25]);
hold off;
%% Graph 5
figure(5)
plot(landmarks_error2),grid on;
title('路標位置誤差 on Y axis')
xlabel('時間 [sec]')
ylabel('位置均方根誤差 [m]')
axis([0 18 -25 25]);
hold off;
%------------------------ end of PLOT RESULTS -----------------------------其中用到的子函數在EKF-SLAM matlab仿真(1)中下載文檔裏。分別爲
運行EKF-SLAM matlab程序結果如下圖所示：
