A Non-singular Horizontal Position Representation | The Journal of Navigation | Cambridge Core (original) (raw)

Abstract

Position calculations, e.g. adding, subtracting, interpolating, and averaging positions, depend on the representation used, both with respect to simplicity of the written code and accuracy of the result. The latitude/longitude representation is widely used, but near the pole singularities, this representation has several complex properties, such as error in latitude leading to error in longitude. Longitude also has a discontinuity at ±180°. These properties may lead to large errors in many standard algorithms. Using an ellipsoidal Earth model also makes latitude/longitude calculations complex or approximate. Other common representations of horizontal position include UTM and local Cartesian ‘flat Earth’ approximations, but these usually only give approximate answers, and are complex to use over larger distances. The normal vector to the Earth ellipsoid (called n-vector) is a non-singular position representation that turns out to be very convenient for practical position calculations. This paper presents this representation, and compares it with other alternatives, showing that n-vector is simpler to use and gives exact answers for all global positions, and all distances, for both ellipsoidal and spherical Earth models. In addition, two functions based on n-vector are presented, that further simplify most practical position calculations, while ensuring full accuracy.

References

Aeronautical Systems Div Wright-Patterson AFB OH (1986). Specification for USAF Standard Form, Fit and Function Medium Accuracy Inertial Navigation Unit (SNU-84.1).Google Scholar

Britting, K.R. (1971). Inertial Navigation Systems Analysis. Wiley Interscience.Google Scholar

Craig, J.J. (1989). Introduction to Robotics. Addison-Wesley Publishing Company, Boston, 2nd edn.Google Scholar

Fortescue, P.W., Stark, J. and Swinerd, G. (2003). Spacecraft Systems Engineering. John Wiley and Sons, 3rd edn.Google Scholar

Gade, B.H., and Gade, K. (2007). n-vector – formulas with derivations. FFI/RAPPORT 2007/00633, Norwegian Defence Research Establishment (FFI).Google Scholar

Gade, K. (2004). NavLab, a Generic Simulation and Post-processing Tool for Navigation. European Journal of Navigation, 2, 51–59.Google Scholar

Hagen, P.E., Storkersen, N., and Vestgard, K. (2003). The HUGIN AUVs – multi-role capability for challenging underwater survey operations. EEZ International.Google Scholar

Hager, J.W., Behensky, J.F., and Drew, B.W. (1989). The Universal Grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS). DMA Technical Manual 8358.2, Defence Mapping Agency.Google Scholar

Hofmann-Wellenhof, B., Wieser, M., and Lega, K. (2003). Navigation: Principles of Positioning and Guidance. Springer.CrossRefGoogle Scholar

Jalving, B., Gade, K., Hagen, O.K., and Vestgard, K. (2004). A Toolbox of Aiding Techniques for the HUGIN AUV Integrated Inertial Navigation System. Modeling, Identification and Control, 25, 173–190.Google Scholar

Levine, W.S. (2000). Control System Applications. CRC Press.Google Scholar

Longley, P.A., Goodchild, M.F., Maguire, D.J. and Rhind, D.W. (2005). Geographic Information Systems and Science. John Wiley and Sons, 2nd edn.Google Scholar

Marthiniussen, R, Faugstadmo, J. E. and Jakobsen, H. P. (2004). HAIN an integrated acoustic positioning and inertial navigation system. Proceedings from MTS/IEEE Oceans 2004, Kobe, Japan.Google Scholar

McGill, D.J., and King, W.W. (1995). Engineering Mechanics. PWS-KENT, Boston, 3rd edn.Google Scholar

Moore, J.R. and Blair., W.D. (2000). Practical Aspects of Multisensor Tracking, in Multitarget-Multisensor Tracking: Applications and Advances, Volume III, Eds: Bar-Shalom, Y. and Blair, W.D., Artech House.Google Scholar

National Imagery and Mapping Agency (2000). Department of Defense World Geodetic System 1984: Its Definition and Relationships With Local Geodetic Systems. NIMA Technical Report TR8350.2, 3rd edn.Google Scholar

Obaidat, M.S. and Papadimitriou, G.I. (2003). Applied System Simulation: Methodologies and Applications. Springer.Google Scholar

Phillips, W.F. (2004). Mechanics of Flight. John Wiley and Sons.Google Scholar

Savage, P.G. (2000). Strapdown Analytics. Strapdown Associates, Inc., Maple Plain.Google Scholar

Sinnott, R.W. (1984). Virtues of the Haversine. Sky and Telescope, 68, 159.Google Scholar

Snyder, J.P. (1987). Map Projections – A Working Manual. U. S. Geological Survey Professional Paper 1395. U. S. Government Printing Office.Google Scholar

Strang, G., and Borre, K. (1997). Linear Algebra, Geodesy, and GPS. Wellesley-Cambridge Press, Wellesley.Google Scholar

Stuelpnagel, J. (1964). On the Parametrization of the Three-Dimensional Rotation Group, SIAM Review, 6, 422–430.CrossRefGoogle Scholar

Vermeille, H. (2004). Computing geodetic coordinates from geocentric coordinates. Journal of Geodesy, 78, 94–95.CrossRefGoogle Scholar

Weisstein, E.W. (2003). CRC Concise Encyclopedia of Mathematics. CRC Press.Google Scholar

Zipfel, P.H. (2000). Modeling and Simulation of Aerospace Vehicle Dynamics. AIAA Education Series, Reston.Google Scholar