Update, respectively. The Kalman filter acts to update the error state and its covariance. Distinct Kalman filters, developed on diverse navigation frames, have distinctive filter states x and covariance matrices P, which really need 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 high latitudes, the integrated filter is created inside 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, 11,6 ofThen, the transformation partnership of the filtering state and also the covariance matrix ought to be deduced. Comparing (24) and (25), it may be noticed that the states that remain unchanged before and after the navigation frame change are the gyroscope bias b plus the accelerometer bias b . Thus, it is actually only necessary to establish a transformation connection amongst the attitude error , the velocity error v, plus the position error p. The transformation relationship between 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 might be described as: G G G = Cn n + nG G G where nG will 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 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 can 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 relationship among the system 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 3 = O3 three O3 3 c O3 three O3 three O3 3 O3 3 O3 3 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 of 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 area, xG and PG must be converted to xn and Pn , which is usually described as: xn ( t ) = -1 x G ( t ) Pn ( t ) = -1 P G ( t ) – T (35)Appl. Sci. 2021, 11,The course of action from the covariance transformation strategy is shown in Figure two. At middle and low latitudes, the technique Methyl aminolevulinate medchemexpress accomplishes the inertial navigation mechanization in the n-frame. When the aircraft enters the polar regions, the technique accomplishes the inertial navigation mechanization within the G-frame. Correspondingly, the navigation parameters are output inside the G-frame. During the navigation Cilastatin (sodium) Anti-infection parameter conversion, the navigation final results and Kalman filter parameter is usually established in line with the proposed strategy.Figure 2. 2. The procedure ofcovariance transformatio.
