Absolute Triangulation Algorithms for Space Exploration

Images are an important source of information for spacecraft navigation and for three-dimensional 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). Two new optimal non-iterative triangulation algorithms are introduced that provide the same solution as Hartley and Sturm. The optimal two-measurement case can be solved as a quadratic equation in many common situations. The optimal many-measurement case may be solved without iteration as a linear system using the new Linear Optimal Sine Triangulation (LOST) method. 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.