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Inertial navigation systems integrate compensated inertial measurements to provide a position, velocity, and attitude reference trajectory Each integration introduces one constant of integration Therefore, initialization of a three dimensional INS requires speci cation of the nine initial conditions for position, velocity, and attitude plus initial conditions for the components of xa and xg This section is concerned with methods for determining these various initial conditions Three topics are of interest: Calibration The process of determining various factors to calibrate the inertial instruments Initialization The process of determining the INS initial position and velocity Alignment The process of determining the relative orientation of the inertial system platform and the reference navigation frame axes
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117 INITIALIZATION
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15 Time, t, hrs
Figure 116: Numeric solution of eqn (11108) showing the error response for a stationary system All instrument error states are zero with the exception of the east (or q) gyro bias that has a constant value of 0015 /hr The vertical states are assumed to be stabilized and have been removed The driving noise terms are zero
CHAPTER 11 AIDED INERTIAL NAVIGATION
Position, velocity, and attitude error estimation and correction are processes which occur at least at system start-up and may continue throughout system operation when aiding sources are available Instrument calibration may occur o -line in a laboratory and on-line through the use of state augmentation and state estimation techniques Since all three operations may occur simultaneously, the three topics are not completely distinct O -line instrument calibration techniques are discussed in [83, 126] Inaccurate position initialization a ects gravity and Earth rotation compensation Similarly, initial velocity error integrates into position error and results in alignment error Alignment error results in error in the transformation of platform measured quantities into the navigation frame Therefore, accurate system initialization, alignment, and calibration are critical to system accuracy The following sections present and analyze a few common initialization processes Some initialization methods can be implemented as a state estimator (eg, Kalman lter) Such methods are directly extendible to aiding during normal system operation In such situations, the same software can be used during initialization and normal (aided) operation
Self-Alignment Techniques
Self-alignment techniques use the inertial instruments together with knowledge of the navigation frame gravity vector and earth rate vector for alignment of the inertial platform relative to the navigation frame The system is assumed to be stationary at a known position With this information it is straightforward to initialize the position and velocity variables The main remaining issue is the initialization of attitude The methods discussed below are directly related to those discussed in Section 103 11711 Coarse Self-Initialization
For a stationary system, the accelerometer measurement vector will be related to the geographic frame gravity vector by fu 0 sin( ) fv = Rb 0 = cos( ) sin( ) g (11143) n g cos( ) cos( ) fw which can be solved for and as follows = arctan2( fv , fw ) arctan2(fu , f 2 + f 2 )
(11144)
The pitch and roll angles estimated by eqn (11144) are accurate to approximately the uncompensated accelerometer bias divided by gravity There-
117 INITIALIZATION
fore, the level error estimation sensitivity to accelerometer bias is one milliradian per milli-g The alternative method described below (see [26, 27, 125]), provides an initial estimate of Rn , from which roll, pitch, and yaw angles or the b quaternion parameters could be determined as necessary Let u and v be two independent vectors De ne w = u v Therefore, w is orthogonal to both u and v Assume that u and v are known in navigation frame and measured in platform frame Then the two sets of vectors are related according to [un , vn , wn ] = Rn ub , vb , wb b Rn b = [u , v , w ] u , v , w
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