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Kalman filter is an effective recursive filter used to estimate the state of a linear dynamic system. In the standard Kalman filter, both the system and the observation model are linear, while in the extended Kalman filter (EKF) and iterative Kalman filter (IKF), the system or the observation model can be nonlinear. The following is the difference in the use of the two update formulas:

Standard Kalman filter update formula:

This formula is the standard update procedure for Kalman filtering, where residual is the direct difference between the observation and the current state prediction. K is the Kalman gain calculated based on the current estimated covariance and the observation model. This formula assumes that the observation model is linear, that is, the observation residual can be used directly to update the state.

Extended/iterative Kalman filter update formula:

This formula appears in the extended Kalman filter (EKF) or iterative Kalman filter (IKF) to handle situations with nonlinear observation models. Here H * dx represents the prediction residual obtained by linearizing the observation model H (usually a first-order Taylor expansion) the predicted state change dx under the current state estimate. This amount is subtracted from the residual to eliminate the prediction error caused by nonlinear characteristics.

Differences in the use of formulas:

线性 vs. 非线性：
第一个公式假设观测模型是线性的，而第二个公式适用于处理非线性观测模型。
预测误差的校正：
第二个公式通过减去 H * dx 来校正由于非线性特性导致的预测误差，使得状态更新更加准确。
计算复杂度：
第二个公式由于需要计算 H * dx，可能会比第一个公式有更高的计算复杂度，特别是在 H 的维度较高或 dx 的维度较大时。

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