How do you tune the process and measurement noise covariances in a Kalman filter?

1 What are process and measurement noise covariances?

The process noise covariance, denoted by Q, is a matrix that describes the uncertainty in the state transition model of the system. It represents the random disturbances that affect the evolution of the system state over time. The measurement noise covariance, denoted by R, is a matrix that describes the uncertainty in the observation model of the system. It represents the random errors that affect the measurements of the system state. Both Q and R are assumed to be positive definite and constant.

2 Why are they important for the Kalman filter?

The Kalman filter is an algorithm that combines the state transition and observation models with the process and measurement noise covariances to produce an optimal estimate of the system state. The optimal estimate is the one that minimizes the mean squared error between the true state and the estimate. The process and measurement noise covariances affect how much the filter trusts the model predictions and the measurements, respectively. A high Q means that the filter expects a lot of variation in the state transition, and a high R means that the filter expects a lot of noise in the measurements. The filter updates the estimate by weighting the model prediction and the measurement according to their respective covariances.

4 How do you tune Q and R iteratively?

Once you have chosen some initial values for Q and R, you can tune them iteratively to improve the filter performance. One common approach is to use a trial-and-error method based on criteria or metrics, such as the innovation sequence, the residual sequence, or the trace of the error covariance matrix. For example, you can plot the innovation sequence over time to check if it has zero mean and constant variance. If not, you can adjust Q and R accordingly. Similarly, you can monitor the trace of the error covariance matrix over time to see if it converges to a small value. If not, you can adjust Q and R accordingly.

5 What are some pitfalls to avoid when tuning Q and R?

Tuning Q and R is not a straightforward task, and there are a few potential issues that should be avoided. These include overfitting or underfitting the noise covariances, which can lead to a biased or noisy estimate, respectively. Additionally, the cross-correlation between Q and R should be taken into account, as ignoring it can result in an inadequate estimate. Finally, if the system or sensors are nonlinear, then a nonlinear Kalman filter extension such as the extended Kalman filter or the unscented Kalman filter should be used to ensure an accurate estimate. Neglecting the nonlinearities can lead to an inadequate estimate that does not reflect the true state behavior.