E grid frameare expressed as:.G G G 0 G v = C fb 0 2ie + eG 1vG + gG – b.CC G b n. G cos = CG = sinb- sin 0 b G ib – iG cosG Cb(four)(five) (six) (7)e The Melperone medchemexpress updated equations of your attitude, the velocity, and the position within the grid frame R = Ce v G G are expressed as:Appl. Sci. 2021, 11,four ofG where iG will be the turn price from the G-frame with respect to the i-frame. G e G G G G iG = ie + eG = Ce ie + eG 1 1 – Ry -ie sin cos L f G 1 1 G ie = ie cos cos L , eG = Rx – f ie sin L – RyfvG E vG N (eight)where R x would be the radius of curvature in the grid east, Ry may be the radius of curvature of your grid north, and f may be the distorted radius. Because the meridian converges quickly inside the polar area, the position from the aircraft inside the polar region is usually expressed inside the ECEF frame. The relationship among the coordinates x, y, z plus the latitude L and also the longitude is provided by: x = ( R N + h) cos L cos y = ( R N + h) cos L sin (9) z = R N (1 – f )2 + h sin L two.two. Dynamic Model of the Grid SINS The mechanization of the grid SINS is achieved in Section 2.1. Next, the Kalman filter, based on the G-frame, desires to become made. So that you can design the Kalman filter, the dynamic model in the G-frame, which includes 3 differential equations, is offered beneath, as place forward in [10]. The attitude error is defined as:G Cb = I – G Cb G(ten) (11)G = -Cb Cb G exactly where Cb is the estimated attitude, expressed with regards to the direction cosine matrix. Differentiating Equation (11) provides: = -Cb Cb – Cb G.G .G .G .G G GGCb.GT(12)Substituting Cb and Cb from Equation (5) gives: .G b G b G = -Cb ib Cb + iG Cb Cb + Cb ib Cb – Cb Cb iG G G G G b G G = -Cb ib Cb + iG Cb Cb – Cb Cb iG G G G G G G G G G G(13)Substituting Cb from Equation (ten) offers: .G G b G G = – I – G Cb ib Cb + iG I – G – I – G iG GG(14)=G -Cbb ib Cb G+G iG -G iG G +G G iG Based on Equation (12), the attitude error equation is expressed by:G G G b = -iG G + iG – Cb ib .G(15)Appl. Sci. 2021, 11,five ofThe velocity error is defined as: vG = vG – vG As outlined by Equation (6), the velocity error equation can be written as: v.G G G G G G = Cb f – 2ie + eG vG + gG – Cb fb + 2ie + eG vG – gG G G G G G G = Cb – Cb fb + Cb fb – 2ie + eG vG – 2ie + eG vG – gG G G G G G = fG G + vG (2ie + eG ) – (2ie + eG ) vG + Cb fb G G b(16)(17)Substituting Cb from Equation (ten) and ignoring the error of gravity vector gives:G G G G G v = fG G + vG (2ie + eG ) – (2ie + eG ) vG + Cb fb .GG(18)From Equation (7), the position error equation is as follows: R = Ce vG + Ce vG G G exactly where:G G Ce = Cn Cn + Cn Cn e e G G In accordance with Equation (2), Cn and Cn may be written as: e .e(19)(20)- cos – sin 0 Cn = – cos L cos + sin L sin – cos L sin – sin L cos – sin L e – sin L cos – cos L sin – sin L sin + cos L cos cos L – sin – cos 0 G Cn = cos – sin 0 0 0(21)(22)where is the grid angle error, and its dynamic equation could be obtained by differentiating Equation (1): sin cos cos L 1 – cos2 cos2 L L + (23) = sin L sin L three. Style of an INS/GNSS integrated Navigation Filter Model with Covariance Transformation When an aircraft flies inside the polar region, it can be necessary to transform navigation frames from the n-frame to G-frame, and vice versa. As well as the transformation of navigation parameters, the integrated navigation filter also wants to transform. The Kalman filter includes the state equation and also the observation equation, and its update procedure consists of a prediction update and measure.
