Sensor Fusion - Example Sensor Fusion Calculations

Example Sensor Fusion Calculations

Two example sensor fusion calculations are illustrated below.

Let and denote two sensor measurements with noise variances and, respectively. One way of obtaining a combined measurement is to apply the Central Limit Theorem, which is also employed within the Fraser-Potter fixed-interval smoother, namely

,

where is the variance of the combined estimate. It can be seen that the fused result is simply a linear combination of the two measurements weighted by their respective noise variances.

Another method to fuse together two measurements is to use the optimal Kalman filter. Suppose that the data is generated by a first-order system and let denote the solution of the filter's Riccati equation. By applying Cramer's rule within the gain calculation it can be found that the filter gain is given by

{\textbf{L}}_k = \begin{bmatrix} \tfrac{\scriptstyle\sigma_2^{2}{\textbf{P}}_k}{\scriptstyle\sigma_2^{2}{\textbf{P}}_k + \scriptstyle\sigma_1^{2}{\textbf{P}}_k + \scriptstyle\sigma_1^{2} \scriptstyle\sigma_2^{2}} & \tfrac{\scriptstyle\sigma_1^{2}{\textbf{P}}_k}{\scriptstyle\sigma_2^{2}{\textbf{P}}_k + \scriptstyle\sigma_1^{2}{\textbf{P}}_k + \scriptstyle\sigma_1^{2} \scriptstyle\sigma_2^{2}} \end{bmatrix}.

By inspection, when the first measurement is noise free, the filter ignores the second measurement and vice versa. That is, the combined estimate is weighted by the quality of the measurements.

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