Aerospace Sensor Systems and Applications

probability density function.- Expectation.- Generalized moments.- Stationarity.- Power and variance.- 2.3 Joint distribution, correlation.- Correlation.- Orthogonality.- The distribution law.- Autocorrelation.- 2.4 Correlation coefficient and functions.- The correlation coefficient.- The autocorrelation function.- The cross-correlation function.- 2.5 Time and ensemble averages, ergodicity.- Conditions for equivalence of time and ensemble averages.- Ergodicity.- 2.6 Mathematical operations on random processes.- Autocorrelation of the sum of random functions.- Cross-correlation between a random function and its time derivatives ?.- Filtering of white noise, Markov processes.- Further properties of correlation functions.- Elementary prediction ?.- 2.7 Input-output relationships.- Convolution.- Nonstationary processes.- Output power in the steady state.- Response of linear systems to white noise.- Response of linear systems to slowly varying input.- Single, double, and triple integration.- 2.8 Spectral analysis.- The self-spectrum.- Parseval's theorem.- Spectral decomposition.- Cross-spectrum.- Problems.- Appendix A2: Integration of power density spectra.- References.- 3. Inertial Force Sensors-Accelerometers.- 3.0 Introduction.- 3.1 Specific force readings on moving platforms.- Elementary strapdown mechanization for vehicle guidance.- 3.2 Leveling the supporting platform.- Mathematical Schuler pendulum.- Physical Schuler pendulum.- 3.3 Schuler frequency on other planets.- 3.4 Force balance accelerometers.- Pickoff considerations.- Spring-mass accelerometer model.- The capacitive detector.- Implementation of loop closure.- The Q-flex accelerometer.- 3.5 Measurement of angular acceleration.- Possible approaches.- Angular accelerometers.- Accelerometer pairs.- Differentiating angular rates.- 3.6 Integrating accelerometers.- 3.7 Vibrating beam accelerometers.- Background.- Principle of operation.- Acceleration-to-frequency conversion.- Double ended tuning fork.- Crystal controlled oscillation.- Signal processing system.- Frequency-to-acceleration conversion algorithm.- Resolution and dynamic range.- Sensitivity to clock frequency variations.- Frequency response and sensitivity to extraneous vibration.- 3.8 Piezo and capacitive transducers.- Piezo sensors/transducers.- Piezoelectric accelerometers.- Piezoresistive sensors.- Variable capacitance accelerometers.- Problems.- References.- 4. Inertial Rotation Sensors.- 4.0 Introduction.- 4.1 The free gyroscope.- Basic mechanization and dynamics.- Torque and precession relationships.- Effects of mechanical imperfections.- 4.2 The vertical gyroscope.- Description and imposing the vertical.- Dynamical model and erection process.- 4.3 Error sources in the vertical.- Error due to gyro drift rate.- Elimination of drift rate errors by integral control.- Errors induced by off-great-circle motion.- Coriolis acceleration.- The cut-out mechanism.- Errors due to atmospheric turbulence.- 4.4 The directional gyroscope.- Description.- Imposing the horizontal.- Effects of motion and drift rate.- Effects of aircraft angular rates.- Slaving to a compass.- 4.5 Gyrocompassing.- Geometry and motion.- Directional error in the steady state.- 4.6 The single axis deflection rate gyro.- Description.- Dynamical model.- Imperfections and limitations.- Bandwidth.- Drift rate.- Scale factor errors.- Nonlinearity and hysteresis.- Saturation.- 4.7 The floated rate integrating gyro (RIG).- Description.- Mathematical model.- Summary of properties.- Pendulous integrating gyro accelerometer (PIGA) ?.- 4.8 The dynamically tuned gyro (DTG).- Description of the dry tuned rotor gyro.- Principle of operation.- The analytical model: error sources ?.- Rotor tuning.- Performance as a free gyro.- Operation in closed loop.- Performance characteristics.- Summary of main properties.- 4.9 Very high-precision free gyroscopes.- The gas-bearing gyroscope ?.- The electrostatically supported gyroscope (ESG).- Problems.- Appendix A4: Euler angle transformation.- Appendix B4: Electrostatic flotation ?.- References.- 5. Applications of Rate Gyros.- 5.0 Introduction.- 5.1 Two-axis platform.- Description.- Angular stabilization by rate gyros.- Control loops and disturbances.- Summary of principal properties.- Application to target tracking.- 5.2 Gyroscopic seeker head.- Description and definition of variables.- Dynamics and control.- Detection of angular deviation-amplitude modulation.- Detection of angular deviation-phase modulation.- 5.3 Application to missile homing.- Homing equations.- The effect of gyro drift rate-stationary target.- The effect of gyro drift rate-moving target.- Accelerating target.- 5.4 Beam riding guidance.- Geometry and motion.- Guidance equations and effect of target maneuver.- The effect of seeker head noise.- 5.5 Three-axis platform for inertial navigation.- Reference axes.- Gimbaled mechanization.- Mathematical model-velocity Schuler tuned platform.- Summary of principal properties.- Positional Schuler tuning.- Velocity and positional error propagation.- Strapdown mechanization and the analytical platform.- 5.6 Stability augmentation-effect of gyro bandwidth.- Background.- Effect on roll rate control.- Problems.- Appendix A5: Direction cosines and quaternions.- Direction cosines.- Incremental Euler transformations.- Quaternions.- References.- 6. Coriolis Angular Rate Sensors.- 6.0 Introduction.- 6.1 Rotating Coriolis angular rate sensors.- Description.- General equations for specific force.- Accelerometer triad readings.- 6.2 Combined angular rate and acceleration sensing.- Signal separation by demodulation.- Imperfections.- 6.3 Rockwell-Collins rotating Coriolis sensor.- Schematic description.- Principle of operation and signal processing.- Technical data.- 6.4 Dithered accelerometers.- Geometry.- Signal separation by demodulation.- 6.5 Dithered accelerometer pairs.- Geometry.- Signal preprocessing.- Readings of accelerometer pairs.- Force and angular rate components.- 6.6 Silicon mechanization of dither.- The shuttle mechanism.- Silicon micromechanization ?.- 6.7 Sensor output signal processing.- Conversion of frequency variations to pulse counts.- Signal description.- 6.8 Projected performance characteristics.- Concluding comments.- Problems.- References.- 7. The Interferometric Fiber-Optic Gyro.- 7.0 Introduction.- 7.1 The Sagnac interferometer.- Mechanization by discrete optical components.- Fiber-optic mechanization.- 7.2 Effect of angular rate on Sagnac phase shift.- Effect in vacuum.- Effect in optical medium.- 7.3 Relationship between power output and phase shift.- Phase characteristics-biasing and modulation.- The need for a90 deg phase shift.- Extraction of angular rate by phase dither.- Implementing nonreciprocity by phase modulation.- Implementation of a 90 deg phase shift.- 7.4 Implementing the IFOG in a closed loop.- Phase nulling laser gyro (PNLG).- Implementation by Serrodyne shifter.- Technical realization of the PNLG ?.- Sensitivity and resolution.- Discrete phase nulling.- Limitations and imperfections.- 7.5 Effect of photon shot noise.- Effect on Sagnac phase uncertainty.- Sources of bias errors.- Recent IFOG test results.- Problems.- References.- 8. The Ring Laser Gyro.- 8.0 Introduction.- 8.1 Operating principle.- Description.- The interference fringe pattern.- 8.2 Technical description.- Spontaneous oscillation.- Path length control.- 8.3 The lock-in phenomenon.- The effect of lock-in on the RLG characteristic.- Prevention of lock-in by mechanical dither.- Removing the dither signal from the output.- Alternative methods for lock-in compensation.- The differential laser gyro-performance data.- Typical performance data.- Scale and drift error maps of angular rate sensors.- Concluding remarks.- Problems.- References.- 9. Filtering, Estimation, and Aiding.- 9.0 Introduction.- 9.1 Complementary filtering.- Basic formulation.- Compass-aided heading gyro.- Doppler-inertial ground speed estimation.- The baro-inertial altimeter.- Complementary filtering for discrete measurements ?.- 9.2 Equivalence of the CF and the stationary KF.- Formulation for inertial navigation.- The CF as the Luenberger observer ?.- Inclusion of noise.- The optimum observer.- The steady-state Kalman filter.- The augmented system model ?.- 9.3 Aircraft attitude angle estimation.- Background.- Gravity aiding.- Magnetic aiding.- Aiding by aircraft kinematics.- Aiding by the aircraft dynamical model ?.- Problems.- Appendix A9: Equations of aircraft dynamics.- Appendix B9: Extended Kalman filter formulation.- Appendix C9: Aircraft aerodynamic coefficients.- References.