. 2022 Nov 21;22(22):9002. doi: 10.3390/s22229002

Table 2.

EKF methodology for the Star Tracker using the state error approach.

States	$x_{k} = {\{\begin{matrix} q_{0} & q_{1} & q_{2} & q_{3} & ω_{x} & ω_{y} & ω_{z} \end{matrix}\}}^{T}$
	$Δ x_{k} = {\{\begin{matrix} δ θ_{x} & δ θ_{y} & δ θ_{z} & δ ω_{x} & δ ω_{y} & δ ω_{z} \end{matrix}\}}^{T}$
	$z_{k} = {\{\begin{matrix} b_{1}^{T} & \dots & b_{n}^{T} \end{matrix}\}}^{T}$
Propagation Update	${\hat{q}}_{k + 1}^{-} = Φ_{q q} q_{k}^{+}$
	${\hat{ω}}_{k + 1}^{-} = {\hat{ω}}_{k}^{+}$
	$P_{k + 1}^{-} = Φ P_{k}^{+} Φ^{T} + Q$
	$Φ = [\begin{matrix} Φ_{δ θ δ θ} & Φ_{δ θ δ ω} \\ 0_{3} & I_{3} \end{matrix}]$
Measurement Update	$K_{k} = P_{k}^{-} H_{k}^{T} ({\hat{x}}_{k}^{-} {[H_{k} ({\hat{x}}_{k}^{-}) P_{k}^{-} H_{k}^{T} ({\hat{x}}_{k}^{-}) + R_{k}]}^{- 1}$
	$Δ {\hat{x}}_{k}^{+} = K_{k} [{\hat{z}}_{k} - h_{k} ({\hat{x}}_{k}^{-})]$
	$P_{k}^{+} = [I_{7} - K_{k} H_{k} ({\hat{x}}_{k}^{-})] P_{k}^{-}$
	$H_{k} ({\hat{x}}_{k}^{-}) = [\begin{matrix} [A ({\hat{q}}_{k}^{-}) r_{1} \times] & 0_{3} \\ ⋮ & ⋮ \\ {\hat{q}}_{k}^{-}) r_{N} \times] & 0_{3} \end{matrix}]$
	$h_{k} ({\hat{x}}_{k}^{-}) = \{\begin{matrix} A ({\hat{q}}_{k}^{-}) r_{1} \\ ⋮ \\ A ({\hat{q}}_{k}^{-}) r_{N} \end{matrix}\}$
	$q_{k}^{* +} = {\hat{q}}_{k}^{-} + \frac{1}{2} Ξ ({\hat{q}}_{k}^{-}) δ θ_{k}^{+}$
	${\hat{q}}_{k}^{+} = \frac{{\hat{q}}_{k}^{* +}}{\| {\hat{q}}_{k}^{* +} \|}$
	${\hat{ω}}_{k}^{+} = {\hat{ω}}_{k}^{-} + δ {\hat{ω}}_{k}^{+}$