Update, respectively. The Kalman filter acts to update the error state and its covariance. Different Kalman filters, designed on unique navigation frames, have unique filter states x and covariance matrices P, which should be transformed. The filtering state at low and middle latitudes is generally expressed by:n n n xn (t) = [E , n , U , vn , vn , vU , L, , h, b , b , b , x y z N E N b x, b y, b T z](24)At higher latitudes, the integrated filter is created inside the grid frame. The filtering state is usually expressed by:G G G G xG (t) = [E , N , U , vG , vG , vU , x, y, z, b , b , b , x y z E N b x, b y, b T z](25)Appl. Sci. 2021, 11,6 ofThen, the transformation connection in the filtering state plus the covariance matrix ought to be deduced. Comparing (24) and (25), it can be seen that the states that stay unchanged prior to and after the navigation frame transform are the gyroscope bias b as well as the accelerometer bias b . Therefore, it is only essential to establish a transformation partnership between the attitude error , the velocity error v, as well as the position error p. The transformation connection between the attitude error n and G is determined as 1-?Furfurylpyrrole custom synthesis follows. G In line with the definition of Cb :G G Cb = -[G Cb G G G From the equation, Cb = Cn Cn , Cb is often expressed as: b G G G G G G Cb = Cn Cn + Cn Cn = -[nG Cn Cn – Cn [n Cn b b b b G Substituting Cb from Equation (26), G is usually described as: G G G = Cn n + nG G G where nG is the error angle vector of Cn : G G G G G Cn = Cn – Cn = – nG Cn nG = G(26)(27)(28)-T(29)The transformation relationship involving the velocity error vn and vG is determined as follows: G G G G G vG = Cn vn + Cn vn = Cn vn – [nG Cn vn (30) From Equation (9), the position error may be written as:-( R N + h) sin L cos -( R N + h) sin L sin y = R N (1 – f )2 + h cos L zx xG ( t )-( R N + h) cos L sin cosL cos L ( R N + h) cos L cos cos L sin 0 sin L h(31)To sum up, the transformation connection between the program error state xn (t) and is as follows: xG (t) = xn (t) (32)exactly where is determined by Equations (28)31), and is offered by: G Cn O3 three a O3 three O3 3 G O3 Cn b O3 three O3 three = O3 three O3 three c O3 three O3 3 O3 three O3 3 O3 three I three three O3 three O3 O3 O3 O3 I3 0 0 0 0 0 0 a =cos L sin cos sin L0 G b = vU -vG N1-cos2 L cos2 0 sin L G – vU v G N 0 -vG a E vG 0 E(33)-( R N + h) sin L cos c = -( R N + h) sin L sin R N (1 – f )2 + h cos L-( R N + h) cos L sin cosL cos ( R N + h) cos L cos cos L sin 0 sin LAppl. Sci. 2021, 11,7 ofThe transformation relation in the covariance matrix is as follows: PG ( t )=ExG ( t ) – xG ( t )xG ( t ) – xG ( t )T= E (xn (t) – xn (t))(xn (t) – xn (t))T T = E (xn(34)(t) – xn (t))(xn (t) – xn (t))TT= Pn (t) TOnce the aircraft flies out in the polar region, xG and PG needs to be converted to xn and Pn , which can be described as: xn ( t ) = -1 x G ( t ) Pn ( t ) = -1 P G ( t ) – T (35)Appl. Sci. 2021, 11,The course of action of the covariance transformation system is shown in Figure two. At middle and low latitudes, the technique accomplishes the inertial navigation mechanization in the n-frame. When the aircraft enters the polar regions, the method accomplishes the inertial navigation mechanization within the G-frame. Correspondingly, the navigation parameters are output in the G-frame. For the duration of the navigation parameter conversion, the navigation results and Kalman filter parameter is usually established in accordance with the proposed technique.Figure 2. two. The process ofcovariance transformatio.
