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Kalman filter

noun/KAHL-mahn/Also: KF, extended Kalman filter, EKF, unscented Kalman filter, UKF, Kalman–Bucy filter, linear quadratic estimation

In plain wordsA method that keeps a running best guess of something you cannot measure exactly, such as a vehicle's position, by blending predictions from a motion model with noisy sensor readings.

  1. A recursive algorithm that estimates the hidden state of a dynamical system — position, velocity, orientation, sensor biases — from a sequence of noisy measurements by alternating two steps. The predict step propagates the current estimate and its covariance through a model of the system's dynamics, which increases uncertainty; the update step corrects the prediction with a new measurement, weighting the discrepancy (the innovation) by the Kalman gain, which balances prediction uncertainty against measurement noise. For linear systems with Gaussian noise it is the optimal estimator, computing the exact Bayesian posteriorposterior, n.: What you believe after weighing new evidence against what you believed before. It is the updated belief that Bayes' theorem produces. mean and covariance, and it is cheap, needing only small matrix operations per step. Nonlinear systems use the extended Kalman filter, which linearizes the models with JacobiansJacobian, n.: A table listing how each output of a process responds to a small change in each input. It describes how a transformation locally stretches, squeezes and turns space., or the unscented Kalman filter, which propagates a small set of deterministically chosen sample points. Kalman filters are ubiquitous in navigation (GPS/inertial fusion in aircraft, spacecraft and phones), sensor fusionsensor fusion, n.: Combining readings from several different sensors, such as cameras, radar, lidar, GPS and motion sensors, to get a more accurate and reliable picture than any one sensor gives alone., odometryodometry, n.: Estimating how far and in which direction a robot or vehicle has moved by adding up its own motion, such as wheel turns or how camera images shift, like counting your steps., object tracking, early SLAMsimultaneous localization and mapping, n.: How a robot, drone or headset builds a map of an unfamiliar place while at the same time working out where it is on that map, using its cameras or laser scanners., econometrics and signal processing.

    The Kalman filter fuses 200 Hz inertial readings with 10 Hz GPS fixes to track the drone.

x^k∣k=x^k∣k−1+Kk(zk−Hkx^k∣k−1),Kk=Pk∣k−1Hk⊤(HkPk∣k−1Hk⊤+Rk)−1\hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k\big(z_k - H_k \hat{x}_{k|k-1}\big), \quad K_k = P_{k|k-1} H_k^{\top}\big(H_k P_{k|k-1} H_k^{\top} + R_k\big)^{-1}

Origin

Named after Rudolf E. Kálmán, whose 1960 paper 'A New Approach to Linear Filtering and Prediction Problems' (Journal of Basic Engineering) introduced the discrete-time filter; Richard Bucy co-developed the continuous-time version (1961), and Peter Swerling (1958) and Ruslan Stratonovich derived related results. Stanley Schmidt at NASA Ames adapted it, in what became the extended Kalman filter, for the Apollo navigation computer.

Usage note

Despite its name, it is a state estimator rather than a filter in the signal-processing sense of removing frequencies. The formula shows the update step. The particle filterparticle filter, n.: A way to track something uncertain, such as a robot's location, by keeping thousands of random guesses, scoring each against what the sensors see, and copying the best ones. handles strongly nonlinear or multimodal problems that defeat it.

Related

See also
  • particle filterparticle filter, n.: A way to track something uncertain, such as a robot's location, by keeping thousands of random guesses, scoring each against what the sensors see, and copying the best ones.
  • sensor fusionsensor fusion, n.: Combining readings from several different sensors, such as cameras, radar, lidar, GPS and motion sensors, to get a more accurate and reliable picture than any one sensor gives alone.
  • odometryodometry, n.: Estimating how far and in which direction a robot or vehicle has moved by adding up its own motion, such as wheel turns or how camera images shift, like counting your steps.
  • simultaneous localization and mappingsimultaneous localization and mapping, n.: How a robot, drone or headset builds a map of an unfamiliar place while at the same time working out where it is on that map, using its cameras or laser scanners.
  • Bayesian inferenceBayesian inference, n.: Reasoning that treats unknowns as uncertain beliefs and updates them with evidence using Bayes' theorem. The result is not one answer but a range of answers, each with a probability.
  • hidden Markov modelhidden Markov model, n.: A model of a sequence, such as speech or DNA, in which hidden states — say, the words being spoken — change step by step at random, and each state gives off signals that can be observed, such as sounds.
  • control theorycontrol theory, n.: The mathematics of making machines behave as intended by constantly measuring what they are doing and correcting them, as a thermostat, cruise control or self-balancing robot does.
Contrast with
  • particle filterparticle filter, n.: A way to track something uncertain, such as a robot's location, by keeping thousands of random guesses, scoring each against what the sensors see, and copying the best ones.

Sources

  1. Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Journal of Basic Engineering 82(1), 35–45.
  2. Julier, S. J. & Uhlmann, J. K. (1997). A new extension of the Kalman filter to nonlinear systems. Proc. SPIE AeroSense 1997.

Mentioned in

Bayes' theoremBayes' theorem, n.: A formula for updating how much you believe something after seeing new evidence: weigh your earlier belief by how well that explanation predicts what you actually observed., control theorycontrol theory, n.: The mathematics of making machines behave as intended by constantly measuring what they are doing and correcting them, as a thermostat, cruise control or self-balancing robot does., mobile robotmobile robot, n.: A robot that travels around by itself — on wheels, tracks, legs, propellers or thrusters — such as a robot vacuum, warehouse cart, delivery robot or planetary rover., object trackingobject tracking, n.: Following objects from frame to frame in a video and keeping track of which is which, so a system knows that the car it sees now is the same car it saw a second ago., probabilistic graphical modelprobabilistic graphical model, n.: A way of describing how many uncertain quantities depend on one another by drawing them as a network, which makes it practical for a computer to reason about all of them together., sensor fusionsensor fusion, n.: Combining readings from several different sensors, such as cameras, radar, lidar, GPS and motion sensors, to get a more accurate and reliable picture than any one sensor gives alone., simultaneous localization and mappingsimultaneous localization and mapping, n.: How a robot, drone or headset builds a map of an unfamiliar place while at the same time working out where it is on that map, using its cameras or laser scanners., covariancecovariance, n.: A measure of whether two quantities tend to rise and fall together, move in opposite directions, or show no consistent relationship., odometryodometry, n.: Estimating how far and in which direction a robot or vehicle has moved by adding up its own motion, such as wheel turns or how camera images shift, like counting your steps., particle filterparticle filter, n.: A way to track something uncertain, such as a robot's location, by keeping thousands of random guesses, scoring each against what the sensors see, and copying the best ones.