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(11145) (11146)
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In particular, if un is the gravity vector and vn is the then 1 1 0 g tan( ) g 1 b b b b b 0 0 f Rn = , , f ie cos( ) 1 0 0 ie g cos( )
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earth rate vector, (11147)
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Non-benign environments (ie, environments having disturbances or vibration) may result in inaccurate initialization of the attitude, which may necessitate on-line, aided error calibration Implementation of the above approach requires high quality gyros with precision and accuracy su cient to measure earth rate The rotation matrix that results from the above process can be orthogonalized by the methods described in Appendix B When the gyros selected for a given application do not allow yaw initialization using the above methods, then alternative methods are required Eqn (11146) is valid for any three independent vectors Optical sightings are sometimes used to replace at least one of the three vectors in that equation Otherwise, the procedure is identical A few other common alternatives are magnetic direction sensing with local magnetic eld compensation or retrieval of the last stored yaw angle for a system which is known to have been stationary between periods of operation In either of these approaches, the accuracy will normally be low enough that on-line calibration of the azimuth error will be desirable via aiding sensors 11712 Fine Initialization: Physical Gyro Compassing
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For a mechanized system with accurately calibrated accelerometers and gyros, alignment can be achieved by a two step process known as gyrocompassing [27, 36, 69] In the rst step, referred to as leveling, the platform is torqued to null the outputs of the north and east accelerometers This step nominally aligns the z-axis with the local gravity vector The resulting
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CHAPTER 11 AIDED INERTIAL NAVIGATION
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leveling error is directly proportional to the uncompensated accelerometer bias Since the east component of the earth rotation rate is known to be zero, the second step of the procedure rotates the platform about the zaxis to null the output of the east gyro The second step is referred to as azimuth alignment At this point, at least theoretically, the north and down gyros could be used to estimate and correct the platform latitude as = atan2( D , N ) 11713 Fine Initialization: Analytic Gyro-Compassing
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In strap-down system applications, the platform cannot be mechanically torqued to cause alignment with the gravity and Earth rate vectors Instead, the misalignment is estimated and used to correct the variables required to compute the direction cosine matrix The computed speci c force vector in the body and navigation frames f are related according to n = Rn b The corresponding error model is f b n f = = (I [ ]) Rn f b + f b b f Pf +
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(11148) (11149)
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Rn f b b
to rst order, where f b represents the sum of the speci c force measurement errors For a system assumed to be stationary, the actual speci c force is the sum of gravity and disturbance accelerations In the geographic frame, fn = 0 0 g + fd
where fd represents the navigation frame disturbances These disturbances can be expected to be high frequency, but the amplitude and spectral content are not accurately known and may be time varying Disturbance motion can be decreased by mechanical isolation The residual navigation frame speci c force measurement is described by f f = f n n n = Pf Rn f b + fd b 0 D E 0 N = D E 0 N 0 g 0 N 0 0 E = g 0 0 0 D (11150) (11151)
0 0 Rn f b + fd (11152) b g Rn f b + fd b (11153)
Based on eqn (11153), two approaches are possible for estimating the alignment error In the rst approach, the measurement f = H x Rn a + fd b
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