A Tutorial on Horizon-Based Optical Navigation and Attitude Determination With Space Imaging Systems

Abstract

Images of a nearby celestial body collected by a camera on an exploration spacecraft contain a wealth of actionable information. This work considers how the apparent location of the observed body's horizon in a digital image may be used to infer the relative position, attitude, or both. When the celestial body is a sphere, spheroid, or ellipsoid (as is the case for most large bodies in the Solar System), the projected horizon in an image is a conic-usually an ellipse at large distances and a hyperbola at small distances. This work develops non-iterative and analytically exact methods for every case (all combinations of unknown state parameters and quadric shapes), completely superseding older horizon-based methods that are iterative, approximate, or both. Some of the analytic methods presented in this work are new. Recognizing that these developments build on techniques that may be unfamiliar to many spacecraft navigators, this work is fashioned as a tutorial. Descriptive illustrations and numerical examples are provided to make concepts clear and to validate the proposed algorithms.

Lunar Crater Identification in Digital Images

“…As discussed in Ref. [26], we can construct the pinhole camera by letting xC ∝ ξ C , where xC is the projected image plane coordinate of the 3D point ξ C . With a digital camera, however, we never observe a point xC , but instead we observe the 2D pixel coordinate…”

“…Assuming the camera frame convention from Refs. [24] and [26], we place the +z axis out of the camera and along the optical axis, the +x axis to the image right and in the direction of increasing pixel column number, and the y axis completing the right-handed system. Furthermore, let the origin of the pixel u-v system be in the upper left-hand corner of the image, the u axis be to the right (increasing pixel column count), the v axis be down (increasing pixel row count), and integer values of [u, v] occurring at the pixel centers.…”

“…How to best use such images of the Moon for navigation depends on many factors-with distance and lighting often being the two most important considerations. It is usually best to use horizon-based OPNAV [21,26,108] when far away from the Moon and when a large portion of the lunar disk is contained within the image. As the vehicle gets closer to the Moon (e.g., low lunar orbit, lunar lander descent), it becomes more appropriate to observe specific landmarks on the lunar surface.…”

“…Fig 26. Example synthetic images along a reference trajectory as the spacecraft moves away from the Moon (top to bottom).…”

Absolute Triangulation Algorithms for Space Exploration

“…Cameras, telescopes, and other optical instruments are an important source of information for modern space vehicles [1]. Information extracted from the images produced by these sensors may be used for many different purposes, with shape modeling [2,3] and spacecraft navigation [4,5] being two of the most common. In both of these applications, the objective is achieved by exploiting the geometry that relates the three-dimensional (3-D) location of objects in the observed scene to their apparent pixel locations in a digital image.…”