Uncertainty Matrix
Statistical measures of uncertainty in state estimates quantify the expected error of a recursive estimation algorithm at any given time step. The kalman filter covariance represents the estimated accuracy of the current state vector by tracking the variances and correlations between different parameters. The uncertainty matrix provides the weight used to balance the prediction from a physical model against the arrival of new sensor data.
Recursive Update
Each iteration of the filter involves a time update and a measurement update that modify the internal values of the matrix. During the time update, the kalman filter covariance grows as the model projects the state forward, following the accumulation of process noise and modeling errors. When a measurement arrives, the gain calculation uses the current uncertainty to determine how much the state should be corrected.
The process reduces the values in the covariance matrix, indicating increased confidence in the estimate. High quality sensors with low noise profiles allow the filter to maintain smaller covariance values over longer durations.
Noise Influence
Adjusting the input noise parameters directly influences how the uncertainty grows between measurements. If the process noise is set too low, the kalman filter covariance may become artificially small, causing the filter to ignore new data and diverge from the true state.
Stability Boundary
Numerical stability depends on the matrix remaining positive definite throughout the operation of the filter. Rounding errors in floating point arithmetic can lead to negative eigenvalues, which represent an impossible physical state and cause the algorithm to fail.
