Absolute Triangulation Algorithms for Space Exploration

Images are an important source of information for spacecraft navigation and for three-dimensional (3-D) reconstruction of observed space objects. Both of these applications take the form of a triangulation problem when the camera has a known attitude and the measurements extracted from the image are line-of-sight (LOS) directions. This work provides a comprehensive review of the history and theoretical foundations of triangulation. A variety of classical triangulation algorithms are reviewed, including a number of suboptimal linear methods (many LOS measurements) and the optimal method of Hartley and Sturm (only two LOS measurements). It is shown that the optimal many-measurement case may be solved without iteration as a linear system using the new linear optimal sine triangulation (LOST) method. Both LOST and the polynomial method of Hartley and Sturm provide the same result in the case of only two measurements. The various triangulation algorithms are assessed with a few numerical examples, including planetary terrain relative navigation, angles-only optical navigation at Uranus, 3-D reconstruction of Notre-Dame de Paris, and angles-only relative navigation.