Mohinder S. Grewal - Global Navigation Satellite Systems, Inertial Navigation, and Integration

5 The Mathworks file exchange at https://www.mathworks.com/matlabcentral/fileexchange/includes many m‐file implementations of navigation procedures, including a complete WGS84 geoid model.

6 In addition, the World Wide Web includes many surveys and reports on inertial sensors and systems.

Refer to Appendix B for coordinate system definitions, and satellite orbit equations.

1 3.1 Which, if any, of the following coordinate systems is not rotating?Northeast–down (NED)East–north–up (ENU)Earth‐centered Earth‐fixed (ECEF)Earth‐centered inertial (ECI)Moon‐centered moon‐fixed

2 3.2 What is the minimum number of two‐axis gyroscopes (i.e. gyroscopes with two, independent, orthogonal input axes) required for inertial navigation?123Not determined.

3 3.3 What is the minimum number of gimbal axes required for gimbaled inertial navigators in fully maneuverable host vehicles? Explain your answer.1234

4 3.4 Define specific force.

5 3.5 An inertial sensor assembly (ISA) operating at a fixed location on the surface of the Earth would measureNo acceleration1 acceleration downward1 acceleration upward

6 3.6 Explain why an inertial navigation system is not a good altimeter.

7 3.7 The inertial rotation rate of the Earth is1 revolution per day15 deg/h15 arc‐seconds per secondNone of the above

8 3.8 Define CEP and CEP rate for an INS.

9 3.9 The CEP rate for a medium accuracy INS is in the order of2 m/s200 m/h2000 m/h20 km/h

10 3.10 Derive the equivalent formulas in terms of (yaw angle), (pitch angle), and (roll angle) for unit vectors 1, 1, 1 in NED coordinates and 1, 1, 1 in RPY coordinates.

11 3.11 Explain why accelerometers cannot sense gravitational accelerations.

12 3.12 Show that the matrix defined in Eq. (3.35)is orthogonal by showing that the identity matrix. (Hint: Use .)

13 3.13 Calculate the numbers of computer multiplies and adds required forgyroscope scale factor/misalignment/bias compensation (Eq. ( 3.4with )accelerometer scale factor/misalignment/bias compensation (Eq. ( 3.4with ) andtransformation of accelerations to navigation coordinates ( Figure 3.22) using quaternion rotations (see Appendix B on quaternion algebra)If the INS performs these 100 times per second, how many operations per second will be required?

1 1 IEEE Standard 528‐2001 (2001). IEEE Standard for Inertial Sensor Terminology. New York: Institute of Electrical and Electronics Engineers.

2 2 IEEE Standard 1559‐2009 (2009). IEEE Standard for Inertial System Terminology. New York: Institute of Electrical and Electronics Engineers.

3 3 Mueller, F.K. (1985). A history of inertial navigation. Journal of the British Interplanetary Society 38: 180–192.

4 4 Everitt, C.W.F., DeBra, D.B., Parkinson, B.W. et al. (2011). Gravity probe B: final results of a space experiment to test general relativity. Physical Review Letters 106: 221101.

5 5 Bernstein, J., Cho, S., King, A.T. et al. (1993). A micromachined comb‐drive tuning fork rate gyroscope. Proceedings IEEE Micro Electro Mechanical Systems. IEEE, pp. 143–148.

6 6 von Laue, M. (1920). Zum Versuch von F. Harress. Annalen der Physik 367 (13): 448–463.

7 7 Collin, J., Kirkko‐Jaakkola, M., and Takala, J. (2015). Effect of carouseling on angular rate sensor error processes. IEEE Transactions on Instrumentation and Measurement 64 (1): 230–240.

8 8 Renkoski, B.M. (2008). The effect of carouseling on MEMS IMU performance for gyrocompassing applications. MS thesis. Massachusetts Institute of Technology.

9 9 Bortz, J.E. (1971). A new mathematical formulation for strapdown inertial navigation. IEEE Transactions on Aerospace and Electronic Systems AES‐7: 61–66.

10 10 Draper, C.S. (1981). Origins of inertial navigation AIAA Journal of Guidance and Control 4 (5): 449–456.

11 11 Chairman of Joint Chiefs of Staff, US Department of Defense (2003). 2003 CJCS Master Positioning, Navigation and Timing Plan. Rept. CJCSI 6130.01C.

12 12 Titterton, D.H. and Weston, J.L. (2004). Strapdown Inertial Navigation Technology, 2e. Stevenage, UK: Institution of Electrical Engineers.

13 13 Savage, P.G. (1996). Introduction to Strapdown Inertial Navigation Systems, Vols. 1 & 2. Maple Plain, MN: Strapdown Associates.

14 14 Groves, P.D. (2013). Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, 2e. Artech House.

1 1 Quoted by author Tom Pickens in “Doc Gyro and His Wonderful ‘Where Am I?’ Machine,” American Way Magazine, 1972.

2 2Named after the Italian physician, inventor, and polymath Girolamo Cardano (1501–1576), who also invented what Americans call a “universal joint” and Europeans call a “Cardan shaft.”

3 3The inertial rotation rate of the Earth was already quite well known, thanks to astronomers. Foucault was only using it to demonstrate gyroscopic physics.

4 4An effect discovered by George Darwin, second son of Charles Darwin.

5 5The vehicle dynamic model used for gyrocompass alignment filtering can be “tuned” to include the major resonance modes of the vehicle suspension.

6 6The nautical mile was originally defined in the seventeenth century as the surface distance covered by one arc‐minute of latitude at sea level. However, because the Earth is not exactly spherical and latitude is measured with respect to the local vertical, this number varies from equator to pole from about 1843 m to about 1861 m.

7 7Defined in Chapter 10.

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