For three satellites, Eq. (2.43) becomes
(2.44)
Equation (2.44) becomes
(2.45)
(2.46)
where
| D | = | known vector |
| N | = | known matrix |
| U v | = | unknown user velocity vector |
However, if the rank of N is <3, N will not be invertible.
Problems
Refer to Appendix B for coordinate system definitions and to Section B.3.10 for satellite orbit equations.
1 2.1 Which of the following coordinate systems is not rotating?North–east–down (NED)East–north–up (ENU)Earth‐centered, Earth‐fixed (ECEF)Earth‐centered inertial (ECI)Moon‐centered, moon fixed
2 2.2 Show that the 3 × 3 identity matrix. (Hint: ).
3 2.3 Rank VDOP, HDOP, and PDOP from smallest (best) to largest (worst) under normal conditions:VDOP ≤ HDOP ≤ PDOPVDOP ≤ PDOP ≤ HDOPHDOP ≤ VDOP≤PDOPHDOP ≤ PDOP ≤ VDOPPDOP ≤ HDOP ≤ VDOPPDOP ≤ VDOP ≤ HDOP
4 2.4 UTC time and the GPS time are offset by an integer number of seconds (e.g. 16 seconds as of June 2012) as well as a fraction of a second. The fractional part is approximately.0.1–0.5 s1–2 ms100–200 ns10–20 ns
5 2.5 Derive equations (2.41) and (2.42).
6 2.6 For the following GPS satellites, find the satellite position in ECEF coordinates at t = 3 seconds. (Hint: See Appendix B.) Ω0 and θ0 are given below at time t0 = 0:Ω0 (°)θ0 (°)(a)32668(b)2634
7 2.7 Using the results of the previous problem, find the satellite positions in the local reference frame. Reference should be to the COMSAT facility in Santa Paula, California, located at 32.4° latitude, −119.2° longitude. Use coordinate shift matrix S = 0. (Refer to Section B.3.10.)
8 2.8 Given the following GPS satellite coordinates and pseudoranges:SatelliteΩ0 (°)θ0 (°)ρ (m)1326682.324 × 1072263402.0755 × 10731461982.1103 × 1074862712.3491 × 107Find the user's antenna position in ECEF coordinates.Find the user's antenna position in locally level coordinates referenced to 0° latitude, 0° longitude. Coordinate shift matrix S = 0.Find the various DOPs.
9 2.9 Given two satellites in north and east coordinateswith pseudorangesand starting with an initial guess of xest, yest, find the user's antenna position.
10 2.10 A satellite position at time t = 0 is specified by its orbital parameters as Ω0 = 92.847°, θ0 = 135.226°, α = 55°, R = 26 560 000 m.Find the satellite position at one second, in ECEF coordinates.Convert the satellite position from (a) with user atfrom WGS84 (ECEF) to ENU coordinates with origin at
References
1 1 Department of Transportation (1990). LORAN‐C User's Handbook, Commandant Instruction M12562.3. Washington, DC: U.S. Coast Guard.
2 2 Logsdon, T. (1992). The NAVSTAR Global Positioning System. New York: Van Nostrand Reinhold.
3 3 Grewal, M.S. and Andrews, A.P. (2019). Application of Kalman Filtering to GPS, INS, & Navigation, Short Course Notes. Anaheim, CA: Kalman Filtering Consultant Associates.
4 4 Grewal, M.S. and Andrews, A.P. (2015). Kalman Filtering: Theory and Practice Using MATLAB®, 4e. New York: Wiley.
5 5 Allan, D.W., Ashby, N., and Hodge, C.C. (1997). The Science of Timekeeping, Hewlett Packard, Application Note 1289. Palo Alto, CA: Hewlett‐Packard.
3 Fundamentals of Inertial Navigation
An inertial system does for geometry…what a watch does for time.1
Charles Stark Draper (1901–1987)
Charles Stark Draper was the American pioneer in inertial navigation who founded the Instrumentation Laboratory at MIT in 1932 to develop aircraft instrumentation technologies. In his analogy previously quoted, watches keep track of time by being set to the correct time, then incrementing that time according to the inputs from a “time sensor” (a frequency source) to update that initial value.
An inertial navigation system (INS) does something similar, only with different variables – and it increments doubly. An INS needs to be set to the correct position and velocity. Thereafter, they use measured accelerations to increment that initial velocity, and use the resulting velocities to increment position.
3.1 Chapter Focus
The overview of inertial navigation in Section 1.3 alluded to the history and terminology of the technology. The focus here is on how inertial sensors function and how they are integrated into navigation systems, including the following:
1 Terminology for the phenomenology and apparatus of inertial navigation
2 Technologies used for sensing rotation and acceleration
3 Error characteristics of inertial sensors
4 Sensor error compensation methods
5 How to compensate for unsensed gravitational accelerations
6 Initializing and propagating navigation solutions for attitude (rotational orientation), velocity, and position
7 Carouseling and indexing as methods for mitigating the effects of sensor errors
8 System‐level testing and evaluation
9 INS performance metrics and standards
How this all affects navigation performance is discussed in Chapter 11.
Scope.
The technology of inertial navigation has been evolving for nearly a century, its diversity and sophistication have grown enormously, and the scale of inertial systems has shrunk by orders of magnitude – from the unbearable to the wearable. As a result, we cannot