GMM-Based Adaptive Extended Kalman Filter Design for Satellite Attitude Estimation under Thruster-Induced Disturbances

. 2023 Apr 23;23(9):4212. doi: 10.3390/s23094212

Algorithm 2: GMM-EKF algorithm

\begin{array}{l} Input : {\hat{q}}_{k - 1}^{+}, {\hat{β}}_{k - 1}^{+}, P_{k - 1}^{+}, {\hat{ω}}_{k - 1}^{+}, a_{ζ}, {\tilde{y}}_{k}, \tilde{ω}, μ_{ζ}, Q_{k - 1}, R_{k - 1}, G_{k - 1} \\ P r o p a g a t i o n : \\ {\hat{q}}_{k}^{-} = \bar{Ω} ({\hat{ω}}_{k - 1}^{+}) {\hat{q}}_{k - 1}^{+} \\ {\hat{β}}_{k}^{-} = {\hat{β}}_{k - 1}^{+} \\ P_{k}^{-} = Φ_{k - 1} P_{k - 1}^{+} Φ_{k - 1}^{T} + G_{k - 1} Q_{k - 1} G_{k - 1}^{T} \\ Gain: \\ Computation and renormalization of h_{k, ζ} as in (58) \\ Computation of H_{k, ζ} as in (61) \\ Computation of P_{z}_{k, ζ}^{-} as in (60) \\ Computation of c_{k}, γ_{k, ζ} as in (62) and (63) \\ Computation of h_{k, sum} as in (57) \\ Computation of P_{z}_{k}^{-} as in (59) \\ Computation of H_{k, sum} as in (65) \\ K_{k, sum} = P_{k}^{-} H_{k, sum}^{T} {(P_{z}_{k}^{-})}^{- 1} \\ Update: \\ Δ {\hat{\tilde{x}}}_{k, sum}^{+} = K_{k, sum} [{\tilde{y}}_{k} - h_{k, sum}] \\ {\hat{β}}_{k}^{+} = {\hat{β}}_{k}^{-} + Δ {\hat{β}}_{k, sum}^{+} \\ {\hat{ω}}_{k}^{+} = {\tilde{ω}}_{k} - {\hat{β}}_{k}^{+} \\ {\hat{q}}_{k}^{+} = {\hat{q}}_{k}^{-} + \frac{1}{2} Ξ ({\hat{q}}_{k}^{-}) δ {\hat{α}}_{k, sum}^{+} \\ Computation of H_{k} as in (40) \\ P_{k}^{+} = [I - K_{k, sum} H_{k}] P_{k}^{-} \\ Output : {\hat{q}}_{k}^{+}, {\hat{β}}_{k}^{+}, {\hat{ω}}_{k}^{+}, P_{k}^{+} \end{array}