Extended Kalman Filter in Orekit

Covariance/noise matrix 在 body frame 和 inertial frame 之间的转换

这部分代码出现在 UnivariateProcessNoise 中，see
#537: `FIXME` in codes
#403: Add a provider generating process noise increasing in time, for better Kalman filtering
for more discussions.

RSW === LVLH in Orekit.

lofCartesianProcessNoiseMatrix Local orbital frame (RSW, or LVLH) 中的 covariance $Q^{RSW}$

• GP-prediction covariance, a diagonal 6 x 6 matrix. (in my codes)

jacLofToInertial $J_{R2I} = J_{RSW\rightarrow Iner.} = \frac{\partial X^{Iner.}}{\partial X^{RSW}}$

• Question? Is there a loss of information at this step?

jacParametersWrtCartesian $J_{P2C} = J_{Par.\rightarrow Cart.} = \frac{\partial Par.}{\partial Cart.} = \frac{\partial Par.}{\partial X^{Iner.}}$

• If the parameters used to represent the orbit is inertial Cartesian, then this should be $I_6$.
• Otherwise, it’s not.

Finial conversion:

$$\begin{aligned} Q^{Iner.} &= Q^{Par.} \\ &= \frac{\partial Par.}{\partial X^{Iner.}} \frac{\partial X^{Iner.}}{\partial X^{RSW}} Q^{RSW} \left(\frac{\partial X^{Iner.}}{\partial X^{RSW}}\right)^T \left(\frac{\partial Par.}{\partial X^{Iner.}}\right)^T \\ &= J_{P2C} \cdot J_{R2I} \cdot Q^{RSW} J_{R2I}^T \cdot J_{P2C}^T \end{aligned}$$

inverse: from Inertial to RSW

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