An Introduction To Kalman Filter

An Introduction to the Kalman Filter
Greg Welch1 and Gary Bishop2 TR 95-041 Department of Computer Science University of North Carolina at Chapel Hill Chapel Hill, NC 27599-3175 Updated: Monday, July 24, 2006

Abstract
In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem. Since that time, due in large part to advances indigital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error. The filter is very powerful inseveral aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown. The purpose of this paper is to provide a practical introduction to the discrete Kalman filter. This introduction includes a description and some discussion of the basic discrete Kalman filter, a derivation, description and some discussion ofthe extended Kalman filter, and a relatively simple (tangible) example with real numbers & results.

Definição

1. welch@cs.unc.edu, http://www.cs.unc.edu/~welch 2. gb@cs.unc.edu, http://www.cs.unc.edu/~gb

Welch & Bishop, An Introduction to the Kalman Filter

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The Discrete Kalman Filter

In 1960, R.E. Kalman published his famous paper describing a recursive solution to thediscretedata linear filtering problem [Kalman60]. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. A very “friendly” introduction to the general idea of the Kalman filter can be found in Chapter 1 of [Maybeck79], while a more completeintroductory discussion can be found in [Sorenson70], which also contains some interesting historical narrative. More extensive references include [Gelb74; Grewal93; Maybeck79; Lewis86; Brown92; Jacobs93]. The Process to be Estimated The Kalman filter addresses the general problem of trying to estimate the state x ∈ ℜ of a discrete-time controlled process that is governed by the linear stochasticdifference equation x k = Ax k – 1 + Bu k – 1 + w k – 1 , with a measurement z ∈ ℜ that is zk = H xk + vk . (1.2)
m n

(1.1)

The random variables w k and v k represent the process and measurement noise (respectively). They are assumed to be independent (of each other), white, and with normal probability distributions p ( w ) ∼ N ( 0, Q ) , p ( v ) ∼ N ( 0, R ) . (1.3) (1.4)

In practice, theprocess noise covariance Q and measurement noise covariance R matrices might change with each time step or measurement, however here we assume they are constant. The n × n matrix A in the difference equation (1.1) relates the state at the previous time step k – 1 to the state at the current step k , in the absence of either a driving function or process noise. Note that in practice A might changewith each time step, but here we assume it is constant. The n × l l matrix B relates the optional control input u ∈ ℜ to the state x. The m × n matrix H in the measurement equation (1.2) relates the state to the measurement zk. In practice H might change with each time step or measurement, but here we assume it is constant. The Computational Origins of the Filter ˆ We define x k ∈ ℜ (note the “superminus”) to be our a priori state estimate at step k given n ˆ knowledge of the process prior to step k, and x k ∈ ℜ to be our a posteriori state estimate at step k given measurement z k . We can then define a priori and a posteriori estimate errors as ˆ e k ≡ x k – x k , and ˆ ek ≡ xk – xk .
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UNC-Chapel Hill, TR 95-041, July 24, 2006

Welch & Bishop, An Introduction to the Kalman Filter...