Update, respectively. The Kalman filter acts to update the error state and its covariance. Distinctive Kalman filters, developed on distinct navigation frames, have different filter states x and covariance matrices P, which have 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 created in 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 relationship of the filtering state plus the covariance matrix really need to be deduced. Comparing (24) and (25), it can be noticed that the states that stay unchanged ahead of and following the navigation frame change are the gyroscope bias b plus the accelerometer bias b . As a result, it is actually only essential to establish a transformation partnership between the attitude error , the velocity error v, plus the position error p. The transformation partnership between the attitude error n and G is determined as follows. G According to the definition of Cb :G G Cb = -[G Cb G G G From the equation, Cb = Cn Cn , Cb could 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 can be 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 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 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 method 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 3 a O3 3 O3 3 G O3 Cn b O3 three O3 3 = O3 3 O3 three c O3 3 O3 three O3 three O3 three O3 three I 3 3 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 of your 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 from the polar area, xG and PG needs to be converted to xn and Pn , which is often described as: xn ( t ) = -1 x G ( t ) Pn ( t ) = -1 P G ( t ) – T (35)Appl. Sci. 2021, 11,The method on the covariance transformation process is shown in Figure 2. At middle and low latitudes, the technique accomplishes the inertial navigation Phenthoate Inhibitor mechanization within the n-frame. When the aircraft enters the polar regions, the method accomplishes the inertial navigation mechanization within the G-frame. Correspondingly, the navigation parameters are output within the G-frame. In the course of the navigation parameter conversion, the navigation results and Kalman filter parameter can be established according to the proposed system.Figure 2. two. The procedure ofcovariance transformatio.
