inertialnav_v2 - Inertial Navigation Mechanization and...

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Unformatted text preview: Inertial Navigation Mechanization and Error Equations 1 Navigation in Earth-centered coordinates Coordinate systems: • i – i nertial coordinate system; ECI. • e – e arth fixed coordinate system; ECEF. • n – n avigation coordinate system; local level, NED. Fundamental equation of inertial navigation: ¨ r i = f i + G i Navigation in a coordinate system fixed to the Earth is more practical: r i = C i e r e First derivative: ˙ r i = ˙ C i e r e + C i e ˙ r e = C i e Ω e i/e r e + C i e ˙ r e Second derivative: ¨ r i = ˙ C i e Ω e i/e r e + C i e ˙ Ω e i/e r e + C i e Ω e i/e ˙ r e + ˙ C i e ˙ r e + C i e ¨ r e = C i e Ω e i/e Ω e i/e r e + C i e ˙ Ω e i/e r e + C i e Ω e i/e ˙ r e + C i e Ω e i/e ˙ r e + C i e ¨ r e = C i e Ω e i/e Ω e i/e r e + ˙ Ω e i/e r e + 2Ω e i/e ˙ r e + ¨ r e Combining with the fundamental equation: f e + G e = ω e i/e × ω e i/e × r e | {z } Centripetal + ˙ ω e i/e × r e | {z } Euler + 2 ω e i/e × ˙ r e | {z } Coriolis +¨ r e 1 2 Navigation in local-level coordinates It is even more practical to calculate velocities in local level coordinates (velocity east, velocity north), and calculate position in geodetic coordinates (latitude, longitude). 2.1 Velocity equation By definition: v n , C n e ˙ r e First derivative: ˙ v n = ˙ C n e ˙ r e + C n e ¨ r e = C n e Ω e n/e ˙ r e + C n e ¨ r e Solve for ¨ r e : ¨ r e = C e n ˙ v n- Ω e n/e ˙ r e Now, repeating the Earth centered navigation equation from the previous section with the Euler term neglected: f e + G e = Ω e i/e Ω e i/e r e | {z } First term + 2Ω e i/e ˙ r e | {z } Second term + ¨ r e |{z} Third term We will convert the above equation to navigation coordinates term by term. First term: Ω e i/e Ω e i/e r e = C e n ( C n e Ω e i/e C e n ) ( C n e Ω e i/e C e n ) C n e r e = C e n Ω n i/e Ω n i/e r n Second term: 2Ω e i/e ˙ r e = 2Ω e i/e C e n v n = 2 C e n ( C n e Ω e i/e C e n ) v n = 2 C e n Ω n i/e v n Third term: ¨ r e = C e n ˙ v n- Ω e n/e ˙ r e = C e n ˙ v n- C e n ( C n e Ω e n/e C e n ) C n e ˙ r e = C e n ˙ v n- C e n Ω n n/e v n 2 Recombining: f e + G e = Ω e i/e Ω e i/e r e + 2Ω e i/e ˙ r e + ¨ r e = C e n Ω n i/e Ω n i/e r n + 2 C e n Ω n i/e v n + C e n ˙ v n- C e n Ω n n/e v n = C e n ( Ω n i/e Ω n i/e r n + 2Ω n i/e v n + ˙ v n- Ω n n/e v n ) = C...
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inertialnav_v2 - Inertial Navigation Mechanization and...

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