卡尔曼滤波..docx

3.The Kalman filter is a powerful recursive data fusion tool based on estimation theory. It works as an estimator for a state vector containing one or more random variables that are assumed to be Gaussian distributed.The underlying principle of a Kalman filter is a predictor-corrector structure. In the prediction cycle, the state vector dynamics are described by a state space model. The transition from one state to the next one at a time is expressed by a state transition matrix or, in the nonlinear case, as a nonlinear function. Assigned to the state vector is a covariance matrix, which describes the statistical dispersion of the random state vector components. The covariance matrix also runs through the prediction cycle, where the quality of the state space model is described by an additive noise component Q.An important precondition for using a Kalman filter is that at least one of the elements in the state vector is observable and that there is additional information available for the observed element. This additional information can consist of e.g. measurements or known constraints and is combined in a measurement vector. The observation is used for the correction cycle (or also: filter update). Left multiplication of the predicted state vector with the observation matrix C yields a vector that is subtracted from the measurement vector to obtain a residual. After weighting by the Kalman gain, the residual is added to the predicted state vector to obtain a filter estimate.The Kalman gain includes the prediction error covariance of the observed elements in the state vector and the measurement noise R. From the formula for the calculation of the Kalman gain it can be seen that the residual will be weighted less the higher the measurement noise is. In other words, if the accuracy of the prediction is high (low covariance values) compared to the measurement noise, the Kalman filter will “follow” rather the state space model than the measurements. If the accuracy of