Update, respectively. The Kalman filter acts to update the error state and its covariance. Distinct Kalman filters, designed on distinctive navigation frames, have distinct filter states x and covariance matrices P, which really need to be transformed. The filtering state at low and middle latitudes is normally 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 high latitudes, the integrated filter is designed within 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 from the filtering state and also the covariance matrix have to be deduced. Comparing (24) and (25), it may be noticed that the states that remain unchanged prior to and just after the navigation frame adjust will be the gyroscope bias b plus the accelerometer bias b . Consequently, it truly is only essential to establish a transformation connection involving the attitude error , the velocity error v, and also the position error p. The transformation relationship amongst 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 From the equation, Cb = Cn Cn , Cb is usually 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 definitely 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 partnership 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 often written as:-( R N + h) sin L cos -( R N + h) sin L sin y = R N (1 – f )two + 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 method error state xn (t) and is as follows: xG (t) = xn (t) (32)where is determined by Equations (28)31), and is given by: G Cn O3 3 a O3 3 O3 three G O3 Cn b O3 three O3 three = O3 three O3 3 c O3 3 O3 3 O3 three O3 three 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 )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 in the polar area, xG and PG ought 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 approach of your covariance transformation method is shown in Figure two. At middle and low latitudes, the system accomplishes the inertial navigation mechanization inside the n-frame. When the aircraft enters the polar regions, the system accomplishes the inertial navigation mechanization within the G-frame. Correspondingly, the navigation parameters are output inside the G-frame. In the course of the navigation parameter conversion, the navigation final results and Kalman filter parameter might be established as outlined by the proposed Metalaxyl manufacturer strategy.Figure 2. two. The course of action ofcovariance transformatio.