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t ie
t ve
124 ERROR STATE DYNAMIC MODEL =
t t ve ie
t ie
t ve I
The simpli ed expression appears as the last two terms in the second row of eqn (1236)
Attitude Error Model
A dynamic model for the attitude error between the body and geographic frames is derived in Sections 1052 and 1142 This section uses an alternative derivation approach [113] to nd the dynamic model for the attitude error between the platform and tangent frames The derivative of the actual and computed rotation matrices are Rt = Rt p p p tp and Rt = Rt p p p tp
Therefore, the derivative of Rt = Rt Rt is p p p Rt p Rt p = Rt Rt p p = (I + [ ]) Rt p Rt p p tp p tp = Rt p p + [ ]Rt p tp tp p p tp (1237)
Using eqn (1233) the error matrix Rt = Rt Rt satis es p p p Rt = [ ]Rt p p (1238)
A second equations for Rt can be found by di erentiation of eqn (1238): p Rt p p = [ ]Rt + [ ]Rt p p tp (1239)
Combining eqns (1237) and (1239) and solving for [ ], we have p [ ]Rt [ ] = [ ]Rt p + Rt p p + [ ]Rt p tp tp p tp p p tp = [ ] t + Rt p p Rp + [ ]Rt p Rp tp tp t tp p p tp t (1240)
= Rt p p Rp t p ip it
where second order terms have been dropped and eqn (223) has been used In vector form (using eqns (223) and (B15)), the equivalent equation is = Rt p p p ip it (1241)
which is analogous to eqn (1171)
CHAPTER 12
LBL AND DOPPLER AIDED INS
The gyro instrumentation error p model is ip p ip Eqn (128) de nes it p t it + p it p it p it = = = = bg g p it (1242) An expression for Rp t t it p (I Rt can be derived as follows:
it [ ]) t + t it (1243)
t Rp t + Rp it t t it
to rst order By linearization of eqn (128), sin( ) p 0 t = ie it p cos( ) Combining eqns (1241), (1242), and (1243), we obtain = t it t Rt bg Rt g it p p
(1244)
(1245)
Calibration Parameter Error Models
The previous discussion of this chapter has introduced three instrument calibration parameters: the accelerometer bias ba , the gyro bias bg , and speed of sound in water c Each of these is modeled as a random constant plus a random walk; therefore, the dynamic models for the error in these calibration parameters are (1246) ba = a (1247) bg = g c = c c + c (1248) The characteristics of the driving noise terms a and g are de ned in Section 1221 while c and c are discussed in Section 1225
Error Model Summary
x = F x + G (1249) 0 I 0 0 0 0 0 0
Based on the derivations of the above sections, the dynamic error model is
where
F =
0 Fvp F p 0 0 0
Rt p p + p ie ip 0 0 0 0
[ e ] vt Fv t it 0 0 0
[ e ] vp 0 t Rp 0 0 0 0 0 0 c
125 AIDING MEASUREMENT MODELS = = 0 0 I [ e ] vp 0 Rt p 0 0 0 0 0 0 a g a 0 0 0 I 0 0 g 0 0 0 0 I 0 c 0 0 0 0 0 1 ,
where
Fvp Fv F p
= Rp t
t gt + [ e ] ie vt p p
t = Rp [gt ] + t ve t ie sin( ) 0 = ie p cos( )
t ie
t ve I ,
The matrix Fvp is small, but the vertical term due to gravity is destabilizing See Section 11511 In this application, the unstable vertical error dynamics are stabilized by the depth sensor aiding Matrix F p is derived from eqn (1244) In the subsequent analysis, due to their small size, F p , the last two t ie terms in Fv , and p will be approximated as zero It should also be noted that ie = it because te = 0
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