Suboptimal Filter Stability
When a stochastic system is partially controllable, such as in the case of SLAM, the Gaussian noise sources v k do not affect all of the elements of the state space. The diagonal elements of P corresponding to these incorruptible states will be driven to zero by the Kalman filter, and once this happens, these estimates will remain fixed and no further observations will alter their values. The dynamics of the model assume the landmarks are fixed elements, for which no process noise is considered. Therefore, their associated noise covariance (its determinant) will asymptotically tend to zero . The filter gain for the landmark states will also tend to zero. Figure 3.1 shows two new simulations for a linear SLAM case, a monobot under Brownian motion with one and two landmarks. The simulations show the evolution of the localization errors for both the monobot and the landmarks, and the reduction of the landmark part of the Kalman gain, due to the uncontrollability of the system. The only way to remedy this situation is to add a positive definite pseudo-noise covariance to those incorruptible states .
KeywordsCovariance Estimate Kalman Gain Landmark State State Error Covariance Nonlinear Vehicle
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