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De ne the auxiliary state composed of the gyro scale factor and misalignment matrix parameters to be xAg = [SFp , SFq , SFr , gpr , gpq , gqr , gqp , grq , grp ] Then, the partial derivative p 0 Tp = Rg 0 q b xAg 0 0 where p = [p, q, r] ip Acceleration Sensitivity (27 States) Finally, the error model accounting for the gyro acceleration sensitivity (ie, g-sensitivity) up to and including second order e ects is 2 2 kp1 fu + kp2 fv + kp3 fw + kp4 fu + kp5 fv 2 +kp6 fw + kp7 fu fv + kp8 fv fw + kp9 fw fu 2 2 kq1 fu + kq2 fv + kq3 fw + kq4 fu + kq5 fv R3 1 (11139) kg = 2 +kq6 fw + kq7 fu fv + kq8 fv fw + kq9 fw fu 2 2 kp1 fu + kr2 fv + kr3 fw + kr4 fu + kr5 fv 2 +kr6 fw + kr7 fu fv + kr8 fv fw + kr9 fw fu where lines have been used to clearly separate the components of the vector Organizing the parameters into the auxiliary state xkg yields xkg = [kp1 , kp2 , kp3 , kp4 , kp5 , kp6 , kp7 , kp8 , kp9 , kq1 , kq2 , kq3 , kq4 , kq5 , kq6 , kq7 , kq8 , kq9 , kr1 , kr2 , kr3 , kr4 , kr5 , kr6 , kr7 , kr8 , kr9 ] of T with respect to xAg is 0 q r 0 0 0 0 p r 0 0 0 0 r 0 0 0 p 0 0 q (11138) (11137)
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The partial derivative of T with respect to xkg is Fkg Z9 Z9 Tp = Z9 Fkg Z9 xkg Z9 Z9 Fkg where Z9 is a 1 9 vector of zeros and
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2 2 2 Fkg = [fu , fv , fw , fu , fv , fw , fu fv , fv fw , fw fu ]
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This section presents a single example of the INS forced error response Figure 116 is the response of the system described by eqn (1179) with all initial conditions set equal to zero (ie, x = 0), and all the accelerometer and gyro error terms set equal to zero, except for the east (or pitch rate q) gyro bias having a constant value of 0015 /hr (ie, f b = 0 and b = [0000, 0015, 0000] /hr) For the rst several minutes, this simuib lation corresponds to the analytic solution given by eqn (1187) for the reduced order system of eqn (1181) Inspection of the F matrix in eqn (1190) yields the following comments on the simulation results displayed in Figure 116 The east gyro bias causes growth of the east tilt error The east tilt error causes azimuth, north velocity, and north tilt error The azimuth error develops a bias of bg N radians (008 ) which cancels the e ect of the gyro bias on the east tilt error This azimuth bias does not propagate into the remaining states since the system is stationary and the bias is perfectly canceled in the east tilt channel Due to the azimuth error bias canceling the gyro bias, the north velocity error is unbiased Therefore the latitude error does not grow linearly with time for a stationary system If the system were in motion, then the azimuth error would result in errors in the remaining states The fact that azimuth error and east gyro bias are not separately observable is one of the limiting factors in the practical application of the gyro compassing technique The oscillations at the earth frequency in the north velocity and azimuth error are out of phase by 180 , so that the earth frequency oscillation does not a ect the east tilt dynamics
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