Dual-EKF-Based Real-Time Celestial Navigation for Lunar Rover

Table 1

Dual-EKF algorithm.

Initialization	Parameter: 𝑥𝑝(𝑡0)=𝑥𝑝,0, 𝑃𝑝(𝑡0)=𝑃𝑝,0 State: 𝑥𝑠(𝑡0)=𝑥𝑠,0, 𝑃𝑠(𝑡0)=𝑃𝑠,0
State measurement update	𝐾𝑠,𝑘=𝑃^−𝑠,𝑘𝐻^𝑇𝑠,𝑘[𝐻𝑠,𝑘𝑃^−𝑠,𝑘𝐻^𝑇𝑠,𝑘+𝑅]^−1
	𝜀𝑘=𝑏̃𝑠,𝑘−ℎ𝑘(𝑥^−𝑠,𝑘,𝑥^−𝑝,𝑘)
	Δ𝐱^+𝑠,𝑘=𝐾𝑠,𝑘𝜀𝑘
	𝐪^+2,𝑘=𝐪^−2,𝑘+12Ξ(𝐪^−2,𝑘)𝛿𝛼^+2,𝑘, normalization
	𝜷^+𝑘=𝜷^−𝑘+Δ𝜷^+𝑘
	𝑃^+𝑠,𝑘=[𝐼−𝐾𝑠,𝑘𝐻𝑠,𝑘]𝑃^−𝑠,𝑘
Parameter measurement update	𝐾𝑝,𝑘=𝑃^−𝑝,𝑘𝐻^𝑇𝑝,𝑘[𝐻𝑝,𝑘𝑃^−𝑝,𝑘𝐻^𝑇𝑝,𝑘+𝑅^′]^−1
	𝑅^′=diag([𝑅,𝑅𝑢])
	𝐱^+𝑝,𝑘=𝐱^−𝑝,𝑘+𝐾𝑝,𝑘[𝜀𝑘;(𝐕𝑘−𝐕^−𝑘)]
	𝑃^+𝑝,𝑘=[𝐼−𝐾𝑝,𝑘𝐻𝑝,𝑘]𝑃^−𝑝,𝑘+1
Parameter time update	𝑥^−𝑝,𝑘+1=Φ𝑝𝑥^+𝑝,𝑘
	𝑃^−𝑝,𝑘+1=Φ𝑝𝑃^+𝑝,𝑘Φ^𝑇𝑝+𝐐𝑝
State time update	𝜔^𝑏𝑛𝑏=(𝜔^𝑏𝑖𝑏−𝛽^+𝑘)−𝐴(𝐪^+2,𝑘)𝜔^𝑛𝑖𝑛(𝑥^−𝑝,𝑘+1)
	𝐪̇2=12Ω(𝝎^𝐛𝐧𝐛)𝐪2
	𝛽̇=0
	𝑃̇𝑠=𝐹𝑠𝑃𝑠+𝑃𝑠𝐹𝑠^𝑇+𝐺𝑄𝑠𝐺^𝑇