This tutorial is part of a series on Simultaneous Localization and Mapping (SLAM) using the extended Kalman filter. See the following for other tutorials in the series:

This tutorial will build incrementally on the previous tutorial: Localization With Known Correlation. We’re still just doing localization, which means the robot starts with a map of landmarks that does not change over the course of its run. The difference is this time when the robot gets a range and bearing measurement, it will not know to which landmark that measurement corresponds.

In this tutorial I will explain the process for estimating the correspondence between measurements and landmarks and present the correspondence algorithm on its own. Then I will show how the correspondence estimation algorithm can be integrated with the EKF localization algorithm introduced in the prior tutorial. As before, we’ll be using the UTIAS dataset for demonstration purposes.

Measurement Correspondence Intro

When our robot gets a measurement to a nearby landmark it can use that information to refine its estimate of its pose. In the last tutorial, each measurement received by our robot was in the form of $z_t^i = [r_t^i\ \phi_t^i\ s_t^i]^T$ where $z_t^i$ is the measurement to landmark $i$ at time $t$ . This vector contains the range $r_t^i$ , bearing $\phi_t^i$ , and, crucially, identity of the landmark being measured $s_t^i$ . This could be interpreted as “landmark $s$ is $r$ meters away, at a bearing of $\phi$ radians.”

In our current problem, the measurement vector will be $z_t^i=[r_t^i\ \phi_t^i]^T$ . So we are no longer directly telling the robot to which landmark each measurement corresponds. Without knowledge of this correspondence measurements are basically useless to our robot. Sure, it knows the location of a landmark relative to its own location. But without knowing where that landmark is in the world it can’t use that information to refine its estimate of its pose.

So our robot needs a way to estimate the correspondence between its measurements and the landmarks in its map. One way it can do this is through maximum likelihood data association.

Maximum Likelihood Data Association

In maximum likelihood data association, as presented in section 7.5.2 of Probabilistic Robotics, the robot computes the probability that its current measurement corresponds to every landmark in its map. It then chooses the landmark with the highest probability of correspondence and assumes it is the correct association. This can be written as

$\hat{c}_t = \text{argmax}_c\ p(z_t|c_{1:t},\ m,\ z_{1:t-1},\ u_{1:t})$

In other words, given the robot’s past measurement correspondences $c_{1:t}$ , map $m$ , past measurements $z_{1:t-1}$ , and past control inputs $u_{1:t}$ , which landmark has the highest probability of producing measurement $z_t$ ?

In practice we find this by calculating the expected measurement and measurement covariance for every landmark in the map. We can then use these to calculate the measurement probability for every landmark and then select the landmark with the highest probability. The algorithm goes as follows:

$\begin{align*} \text{for all}&\text{ measurements } z_t^i=[r_t^i\ \phi_t^i]^T \text{ do} \\ \text{for}&\text{ all landmarks } k \text{ in the map } m \text{ do} \\ &q = (m_{k,x} - \bar{\mu}_{t,x})^2 + (m_{k,y} - \bar{\mu}_{t,y})^2 \\ &\hat{z}_t^k = \begin{bmatrix} \sqrt{q} \\ \text{atan2}(m_{k,y}-\bar{\mu}_{t,y},\ m_{k,x}-\bar{\mu}_{t,x})-\bar{\mu}_{t,\theta} \end{bmatrix} \\ &H_t^k = \begin{bmatrix} -\frac{m_{k,x}-\bar{\mu}_{t,x}}{\sqrt{q}} & -\frac{m_{k,y}-\bar{\mu}_{t,y}}{\sqrt{q}} & 0 \\ \frac{m_{k,y}-\bar{\mu}_{t,y}}{q} & -\frac{m_{k,x}-\bar{\mu}_{t,x}}{q} & -1\end{bmatrix} \\ &S_t^k = H_t^k \bar{\Sigma}_t [H_t^k]^T + Q_t \\ \text{end}&\text{for} \\ j(i) &= \text{argmax}_k\ \text{det}(2\pi S_t^k)^{-\frac{1}{2}} \text{exp}\{-\frac{1}{2}(z_t^i-\hat{z}_t^k)[S_t^k]^{-1}(z_t^i-\hat{z}_t^k)^T\}\\ \text{endfor}& \end{align*}$

The end result $j(i)$ is the estimated correspondence $j$ for each measurement $i$ . It should be noted that the probability estimates are conditioned on past correspondences. However, this method assumes that the past estimates are correct. This means that the correspondence estimates are very sensitive to incorrect associations; incorrect estimates are likely to remain incorrect on subsequent measurements. This makes the maximum likelihood method brittle and likely to diverge when measurements are incorrectly associated.

There are alternative methods that are more robust to incorrect associations. Multi-Hypothesis Tracking is one of these, but the maximum likelihood method is the simplest. For an introduction I think the maximum likelihood method is sufficient.

The Complete Algorithm

The code for the complete algorithm is shown below. It is the same as the code for EKF localization with known correlation with the addition of the maximum likelihood algorithm for measurement correspondence (lines 85 to 120). Note also that the code rendering program I’m using doesn’t seem to handle MATLAB code very well, and some of the highlighting is a little messed up.

addpath ../common;

deltaT = .02;
alphas = [.2 .03 .09 .08 0 0]; % motion model noise parameters

% measurement model noise parameters
sigma_range = .43;
sigma_bearing = .6;

Q_t = [sigma_range^2 0;
       0 sigma_bearing^2];

n_robots = 1;
robot_num = 1;
n_landmarks = 15;
[Barcodes, Landmark_Groundtruth, Robots] = loadMRCLAMdataSet(n_robots);
[Robots, timesteps] = sampleMRCLAMdataSet(Robots, deltaT);

% add pose estimate matrix to Robots
Robots{robot_num}.Est = zeros(size(Robots{robot_num}.G,1), 4);

start = 1; 
t = Robots{robot_num}.G(start, 1); 
% need mean and covariance for the initial pose estimate
poseMean = [Robots{robot_num}.G(start,2);
            Robots{robot_num}.G(start,3);
            Robots{robot_num}.G(start,4)];
poseCov = 0.01*eye(3);
       
measurementIndex = 1;

% set up map between barcodes and landmark IDs
codeDict = containers.Map(Barcodes(:,2),Barcodes(:,1));

while (Robots{robot_num}.M(measurementIndex, 1) < t - .05)
        measurementIndex = measurementIndex + 1;
end

% loop through all odometry and measurement samples
% updating the robot's pose estimate with each step
% reference table 7.3 in Probabilistic Robotics
for i = start:size(Robots{robot_num}.G, 1)
    theta = poseMean(3, 1);
    % update time
    t = Robots{robot_num}.G(i, 1);
    % update movement vector
    u_t = [Robots{robot_num}.O(i, 2); Robots{robot_num}.O(i, 3)];

    rot = deltaT * u_t(2);
    halfRot = rot / 2;
    trans = u_t(1) * deltaT;
    
    % calculate the movement Jacobian
    G_t = [1 0 trans * -sin(theta + halfRot);
           0 1 trans * cos(theta + halfRot);
           0 0 1];
    % calculate motion covariance in control space
    M_t = [(alphas(1) * abs(u_t(1)) + alphas(2) * abs(u_t(2)))^2 0;
           0 (alphas(3) * abs(u_t(1)) + alphas(4) * abs(u_t(2)))^2];
    % calculate Jacobian to transform motion covariance to state space
    V_t = [cos(theta + halfRot) -0.5 * sin(theta + halfRot);
           sin(theta + halfRot) 0.5 * cos(theta + halfRot);
           0 1];
    
    % calculate pose update from odometry
    poseUpdate = [trans * cos(theta + halfRot);
                  trans * sin(theta + halfRot);
                  rot];
    
    poseMeanBar = poseMean + poseUpdate;
    poseCovBar = G_t * poseCov * G_t' + V_t * M_t * V_t';
    
    % get measurements
    [z, measurementIndex] = getObservations(Robots, robot_num, t, measurementIndex, codeDict);
    
    % remove landmark ID from measurement because
    % we're trying to live without that
    z = z(1:2,:);
    
    % if features are observed
    % loop over all features and compute Kalman gain
    if z(1,1) > 0.1
        for k = 1:size(z, 2) % loop over every observed landmark
            
            % loop over all landmarks and compute MLE correspondence
            predZ = zeros(n_landmarks, 1, 2); % predicted measurements
            predS = zeros(n_landmarks, 2, 2); % predicted measurement covariances
            predH = zeros(n_landmarks, 2, 3); % predicted measurement Jacobians
            maxJ = 0; 
            landmarkIndex = 0;
            for j = 1:n_landmarks
                xDist = Landmark_Groundtruth(j, 2) - poseMeanBar(1);
                yDist = Landmark_Groundtruth(j, 3) - poseMeanBar(2);
                q = xDist^2 + yDist^2;
                
                % calculate predicted measurement
                predZ(j,:,:) = [sqrt(q);
                            conBear(atan2(yDist, xDist) - poseMeanBar(3))];
                        
                % calculate predicted measurement Jacobian
                predH(j,:,:) = [-xDist/sqrt(q) -yDist/sqrt(q) 0;
                                yDist/q        -xDist/q      -1];
                            
                % calculate predicted measurement covariance
                predS(j,:,:) = squeeze(predH(j,:,:)) * poseCovBar ...
                               * squeeze(predH(j,:,:))' + Q_t;
                
                % calculate probability of measurement correspondence          
                thisJ = det(2 * pi * squeeze(predS(j,:,:)))^(-0.5) * ...
                        exp(-0.5 * (z(:,k) - squeeze(predZ(j,:,:)))' ...
                        * inv(squeeze(predS(j,:,:))) ...
                        * (z(:,k) - squeeze(predZ(j,:,:))));
                
                % update correspondence if the probability is
                % higher than the previous maximum
                if thisJ > maxJ
                    maxJ = thisJ;
                    landmarkIndex = j;
                end
            end

            % compute Kalman gain
            K = poseCovBar * squeeze(predH(landmarkIndex,:,:))' ...
                * inv(squeeze(predS(landmarkIndex,:,:)));

            % update pose mean and covariance estimates
            poseMeanBar = poseMeanBar + K * (z(:,k) - squeeze(predZ(landmarkIndex,:,:)));
            poseCovBar = (eye(3) - (K * squeeze(predH(landmarkIndex,:,:)))) * poseCovBar;
        end
    end
    
    % update pose mean and covariance
    poseMean = poseMeanBar;
    poseMean(3) = conBear(poseMean(3));
    poseCov = poseCovBar;
    
    % add pose mean to estimated position vector
    Robots{robot_num}.Est(i,:) = [t poseMean(1) poseMean(2) poseMean(3)];
    
    % calculate error between mean pose and groundtruth 
    % for testing only
    %{
    groundtruth = [Robots{robot_num}.G(i, 2); 
                   Robots{robot_num}.G(i, 3); 
                   Robots{robot_num}.G(i, 4)];
    error = groundtruth - poseMean;
    %}
   
end
animateMRCLAMdataSet(Robots, Landmark_Groundtruth, timesteps, deltaT);

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Tag Archives: Probabilistic Robotics

Intro to the EKF Step 2: Localization With Unknown Correspondence

Measurement Correspondence Intro

Maximum Likelihood Data Association

The Complete Algorithm