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roboticskalman-filterslam

Why calculate jacobians in ekf-slam

I know it is a very basic question but I want to know why do we calculate the Jacobian matrices in EKF-SLAM, I have tried so hard to understand this, well it won't be that hard but I want to know it. I was wondering if anyone could help me on this.

Solution

  • The Kalman filter operates on linear systems. The steps update two parts in parallel: The state x, and the error covariances P. In a linear system we predict the next x by Fx. It turns out that you can compute the exact covariance of Fx as FPF^T. In a non-linear system, we can update x as f(x), but how do we update P? There are two popular approaches:

    1. In the EKF, we choose a linear approximation of f() at x, and then use the usual method FPF^T.
    2. In the UKF, we build an approximation of the distribution of x with covariance P. The approximation is a set of points called sigma points. Then we propagate those states through our real f(sigma_point) and we measure the resulting distribution's variance.

    You are concerned with the EKF (case 1). What is a good linear approximation of a function? If you zoom way in on a curve, it starts to look like a straight line, with a slope that's the derivative of the function at that point. If that sounds strange, look at Taylor series. The multi-variate equivalent is called the Jacobian. So we evaluate the Jacobian of f() at x to give us an F. Now Fx != f(x), but that's okay as long as the changes we're making to x are small (small enough that our approximated F wouldn't change much from before to after).

    The main problem with the EKF approximation is that when we use the approximation to update the distributions after the measurement step, it tends to make the resulting covariance P too low. It acts like corrections "work" in a linear way. The actual update will depart slightly from the linear approximation and not be quite as good. These small amounts of overconfidence build up as the KF iterates and have to be offset by adding some fictitious process noise to Q.