Celestial Navigation Algorithms

Three papers, listed below, published in the (U.S.) Institute of Navigation journal Navigation, provide a set of new algorithms for celestial navigation that incorporate a moving observer as part of the basic construction. It assumes a set of observations of the altitudes of stars above the horizon, either from a sextant or some sort of automated star tracker. This approach, based on a least-squares analysis of the observations, is closely analogous to "orbit correction" problems familiar to astronomers who deal with the dynamics of bodies (natural or artificial) in the solar system. Although more complex mathematically than previous sight-reduction schemes (a computer is definitely required), the new procedure provides the course and speed of the vessel along with the fix — if enough observations are available, of course. When only a few observations are available, the procedure still provides a good fix, but cannot provide course and speed information.

The second paper listed below, "Determining the Position and Motion..." describes the method in detail. The first paper, "Practical Sailing Formulas..." provides a piece of the necessary mathematical foundation — relatively simple but precise formulas that describe a vessel's motion in longitude and latitude as a function of time, as it sails along a rhumb-line track. (It was surprising that such a gap existed in the literature.) The third paper, "A Navigation Solution Involving Changes..." extends the method to multi-leg tracks. These three papers assume that the reader has a knowledge of basic calculus and statistical analysis and they contain a fair number of equations.

The fourth paper in the list, published in the Navigator's Newsletter, is a less technical review of how the motion of the observer has been previously dealt with in celestial navigation. It assumes knowledge of standard celestial navigation practice.

The fifth paper is a readable overview of modern navigational technology (as of 1999) and how automated celestial observing systems developed for space systems could be more widely used for surface and air navigation. It includes a non-technical description (with figures) of the new algorithms, which are well suited to automated observing systems.

The sixth paper describes a particular algorithm for what is generically called angles-only navigation (because no distances are involved). Angles-only navigation is a field in itself and is quite general; its applications often use fixed landmarks on Earth. However, it can also be applied to images of artifical Earth satellites against a star background, and how the algorithm presented in the paper might work in such an application is discussed.

Reprints of any of these papers (single copies) are available from the author or from the USNO Astronomical Applications Department; please include a complete mailing address.

  1. Kaplan, G. H., 1995: "Practical Sailing Formulas for Rhumb-Line Tracks on an Oblate Earth", Navigation, Vol. 42, No. 2, pp. 313-326.   (Abstract and download from ION)

  2. Kaplan, G. H., 1995. "Determining the Position and Motion of a Vessel from Celestial Observations", Navigation, Vol. 42, No. 4, pp. 631-648.   (Abstract and download from ION)

  3. Kaplan, G. H., 1996: "A Navigation Solution Involving Changes to Course and Speed", Navigation, Vol. 43, No. 4, pp. 469-482.   (Abstract and download from ION)

  4. Kaplan, G. H., 1996: "The Motion of the Observer in Celestial Navigation", Navigator's Newsletter, Issue 51 (Spring 1996), pp. 10-14.   (PDF file)   Note: the Navigator's Newsletter was published by the Navigation Foundation, which is now inactive.

  5. Kaplan, G. H., 1999: "New Technology for Celestial Navigation", in Proceedings, Nautical Almanac Office Sesquicentennial Symposium, U.S. Naval Observatory, March 3–4, 1999, ed. A. D. Fiala and S. J. Dick (USNO, Washington, 1999), pp. 239-254.   (HTML file)

  6. Kaplan, G. H., 2011: "Angles-Only Navigation: Position and Velocity Solution from Absolute Triangulation", Navigation, Vol. 58, No. 3, pp. 187-201.   (Abstract and download from ION; see also U.S. Patent No. 8,260,567.)