Update, respectively. The Kalman filter acts to update the error state and its covariance. Distinctive Kalman filters, designed on unique navigation frames, have different filter states x and covariance matrices P, which should 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 within the grid frame. The filtering state is normally 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 connection on the filtering state and the covariance matrix should be deduced. Comparing (24) and (25), it might be seen that the states that remain unchanged before and soon after the navigation frame Ombitasvir site transform will be the gyroscope bias b and the accelerometer bias b . Consequently, it truly is only essential to establish a transformation relationship between the attitude error , the velocity error v, along with the position error p. The transformation connection 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 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 often described as: G G G = Cn n + nG G G exactly where nG could 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 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 amongst 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 provided by: G Cn O3 three a O3 three O3 3 G O3 Cn b O3 three O3 3 = O3 3 O3 three c O3 three O3 3 O3 three O3 three O3 three I 3 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 on 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 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 method of the covariance transformation approach 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 technique accomplishes the inertial navigation mechanization inside the G-frame. Correspondingly, the navigation parameters are output in the G-frame. Through the navigation parameter conversion, the navigation outcomes and Kalman filter parameter can be established based on the proposed approach.Figure 2. two. The course of action ofcovariance transformatio.
