Update, respectively. The Kalman filter acts to update the error state and its covariance. Unique Kalman filters, designed on distinctive navigation frames, have diverse filter states x and covariance matrices P, which have to be transformed. The filtering state at low and middle latitudes is usually 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 designed in the grid frame. The filtering state is generally 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,six ofThen, the transformation relationship in the filtering state and the covariance matrix really need to be deduced. Comparing (24) and (25), it could be observed that the states that stay unchanged prior to and after the navigation frame alter are the gyroscope bias b and the accelerometer bias b . Thus, it’s only necessary to establish a transformation relationship amongst the attitude error , the velocity error v, as well as the position error p. The transformation relationship among the attitude error n and G is determined as follows. G As outlined by the definition of Cb :G G Cb = -[G Cb G G G In 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 Elinogrel Biological Activity Equation (26), G is usually described as: G G G = Cn n + nG G G where nG would be 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 connection amongst 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 is usually 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 relationship amongst the technique error state xn (t) and is as follows: xG (t) = xn (t) (32)where is determined by Equations (28)31), and is provided by: G Cn O3 three a O3 3 O3 three G O3 Cn b O3 three O3 three = O3 three O3 three c O3 three O3 3 O3 3 O3 3 O3 3 I 3 3 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 )two + 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 with 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 on the polar region, xG and PG needs to be converted to xn and Pn , which may 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 on the covariance transformation strategy is shown in Figure two. At middle and low latitudes, the program accomplishes the inertial navigation mechanization within the n-frame. When the aircraft enters the polar regions, the technique accomplishes the inertial navigation mechanization inside the G-frame. Correspondingly, the navigation parameters are output within the G-frame. For the duration of the navigation parameter conversion, the navigation results and Kalman filter parameter may be established in line with the proposed process.Figure 2. two. The process ofcovariance transformatio.