Update, respectively. The Kalman filter acts to update the error state and its covariance. Distinctive Kalman filters, designed on different navigation frames, have unique filter states x and covariance matrices P, which need to 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 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 partnership of your filtering state as well as the covariance matrix should be deduced. Comparing (24) and (25), it could be observed that the states that stay unchanged ahead of and following the navigation frame transform would be the gyroscope bias b and the accelerometer bias b . Consequently, it really is only necessary to establish a transformation partnership involving the attitude error , the velocity error v, and the position error p. The transformation relationship in between the attitude error n and G is determined as follows. G In accordance with 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 Equation (26), G could be 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 among 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 relationship between the system error state xn (t) and is as follows: xG (t) = xn (t) (32)exactly where is determined by Equations (28)31), and is given by: G Cn O3 three a O3 three O3 3 G O3 Cn b O3 3 O3 three = O3 three O3 3 c O3 3 O3 3 O3 three O3 3 O3 three I three 3 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 )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 from 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 of the polar area, xG and PG ought to be converted to xn and Pn , which could be described as: xn ( t ) = -1 x G ( t ) Pn ( t ) = -1 P G ( t ) – T (35)Appl. Sci. 2021, 11,The process of the covariance transformation process is shown in Figure two. At middle and low latitudes, the method accomplishes the inertial navigation mechanization within the n-frame. When the aircraft enters the polar regions, the technique accomplishes the inertial navigation mechanization in the G-frame. DBCO-PEG4-Maleimide Antibody-drug Conjugate/ADC Related Correspondingly, the navigation parameters are output inside the G-frame. Throughout the navigation parameter conversion, the navigation benefits and Kalman filter parameter can be established based on the proposed method.Figure 2. two. The procedure ofcovariance transformatio.