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A short course in state estimation

Three Ways to Know Where You Are

The extended Kalman filter, the unscented Kalman filter, and the factor graph — built up from first principles, then set against each other.

Written for engineering students
Prerequisites: linear algebra · probability · calculus
Six chapters · three plates

Contents

  1. IThe problem, before any algorithmYou have a robot. It carries sensors, and every one of them lies a little. The wheel encoders slip, the IMU drifts, the GPS wanders by a few metres, the camera sees…Chapter one
  2. IIThe Kalman filter: the exact caseSuppose the world is kind. Suppose f and h are linear, and every noise is Gaussian. Then something remarkable happens: a Gaussian pushed through a linear map is still…Chapter two
  3. IIIThe EKF: pretend it is a straight lineReality curves. Rotations compose multiplicatively, a camera projects by dividing by depth, range to a beacon is a square root of a sum of squares. None of these is…Chapter three
  4. IVThe UKF: do not differentiate, sampleJulier and Uhlmann began from an observation that sounds almost too simple: it is easier to approximate a probability distribution than to approximate an arbitrary…Chapter four
  5. VFactor graphs: keep everything, solve it allA filter is a machine that forgets. It compresses the entire past into one mean and one covariance, and once that compression has happened the information is gone for…Chapter five
  6. VIThe three compared, and unifiedNow that each method has been built from its own starting point, the differences are easy to state, because they all reduce to two decisions: how do you handle the…Chapter six