Experimental Results on an Integrated GPS and Multisensor System for Land Vehicle Positioning

Table 1

Summary of Kalman filter equations.

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Summary of Kalman filter equations

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Linear perturbation prediction:

           Δ𝑥⁻_𝑘=Φ_𝑘−1Δ𝑥_𝑘−1.

Conditioning the predicted perturbation on the measurement:

         Δ𝑥_𝑘=Δ𝑥⁻_𝑘+𝐾_𝑘(𝑦_𝑘−𝑦⁻_𝑘).

Predicted measurement:

           𝑦⁻_𝑘=𝐻_𝑘Δ𝑥⁻_𝑘−1.

A priori covariance matrix:

      𝑃⁻_𝑘=Φ_𝑘−1𝑃_𝑘−1Φ^𝑇_𝑘+𝐺_𝑘−1𝑄_𝑘−1𝐺^𝑇_𝑘−1.

Kalman gain:

        𝐾_𝑘=(𝑃⁻_𝑘)𝐻^𝑇_𝑘(𝐻_𝑘𝑃⁻_𝑘𝐻^𝑇_𝑘+𝑅_𝑘)⁻¹.

A posteriori covariance matrix:

      𝑃_𝑘=(𝐼−𝐾_𝑘𝐻_𝑘)𝑃⁻_𝑘(𝐼−𝐾_𝑘𝐻_𝑘)^𝑇+𝐾_𝑘𝑅_𝑘𝐾^𝑇_𝑘.

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