Update, respectively. The Kalman filter acts to update the error state and its covariance. Different Kalman filters, developed on different navigation frames, have unique filter Pazopanib-d6 Technical Information states x and covariance matrices P, which have 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 higher latitudes, the integrated filter is designed 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,six ofThen, the transformation relationship on the filtering state and the covariance Tiaprofenic acid medchemexpress matrix have to be deduced. Comparing (24) and (25), it can be noticed that the states that remain unchanged ahead of and following the navigation frame adjust would be the gyroscope bias b as well as the accelerometer bias b . Thus, it is only necessary to establish a transformation connection among the attitude error , the velocity error v, along with the position error p. The transformation relationship between the attitude error n and G is determined as follows. G In line with the definition of Cb :G G Cb = -[G Cb G G G In the equation, Cb = Cn Cn , Cb may be 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 often described as: G G G = Cn n + nG G G exactly 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 connection in between 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 might be 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 partnership amongst the system 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 three a O3 3 O3 3 G O3 Cn b O3 3 O3 three = O3 three O3 3 c O3 three O3 3 O3 3 O3 three O3 3 I 3 three O3 3 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 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 with the polar region, xG and PG must 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 process on the covariance transformation system is shown in Figure 2. At middle and low latitudes, the program accomplishes the inertial navigation mechanization inside the n-frame. When the aircraft enters the polar regions, the program accomplishes the inertial navigation mechanization in the G-frame. Correspondingly, the navigation parameters are output within the G-frame. Through the navigation parameter conversion, the navigation outcomes and Kalman filter parameter might be established in accordance with the proposed system.Figure 2. two. The approach ofcovariance transformatio.