# innovation step ekf localization?

Let's say we have a bunch of observations $z^{i}$ from sensor and we have a map in which we can get the predicted measurements $\hat{z}^{i}$ for landmarks. In EKF localization in correction step, should we compare each observation $z^{i}$ with the entire predicted measurement $\hat{z}^{i}$?, so in this case we have two loops? Or we just compare each observation with each predicted measurement?, so in this case we have one loop. I assume the sensor can give all observations for all landmarks every scan. The following picture depicts the scenario. Now every time I execute the EKF-Localization I get $z^{i} = \{ z^{1}, z^{2}, z^{3}, z^{4}\}$ and I have $m$, so I can get $\hat{z}^{i} = \{ \hat{z}^{1}, \hat{z}^{2}, \hat{z}^{3}, \hat{z}^{4}\}$. To get the innovation step, this is what I did $$Z^{1} = z^{1} - \hat{z}^{1} \\ Z^{2} = z^{2} - \hat{z}^{2} \\ Z^{3} = z^{3} - \hat{z}^{3} \\ Z^{4} = z^{4} - \hat{z}^{4} \\$$ where $Z$ is the innovation. For each iteration I get four innovations. Is this correct? I'm using EKF-Localization in this book Probabilistic Robotics page 204.

• As a matter of fact, I was right about my assumption. I got good results. – CroCo May 31 '14 at 14:44

1. Each measurement is independent (i.e., the (Gaussian) distribution of observation $z_i$ is uncorrelated with $z_j$). Usually this is a fair assumption (e.g., measuring the position of landmarks with a laser scanner).