The differential geometric filter is implemented to estimate the absolute and relative positions of the spacecraft in a formation. The extended Kalman Filter is also implemented as a benchmark for the differential geometric estimation. Only relative positions between the spacecraft are measured. Relative positions are measured using a wireless local positioning system (WLPS) installed in all spacecraft. Two different scenarios are studied: (1) the observations include WLPS measurements only and (2) the observations include WLPS measurements in addition to measurements for the absolute position of one spacecraft made by a radar that takes measurements from the earth’s surface. Results show that the differential geometric estimation has better stability performance and a faster convergence rate compared to the extended Kalman filter. 1. Introduction The relative and absolute positions estimation of spacecraft formations is a fundamental task in many space missions. Relative position estimation plays an important role in Spacecraft Formation Flying (SFF) missions, a subject that has been the focus of many researches during last decade. Some SFF missions require that multiple spacecraft, in different orbits, communicate without interruption, for example, satellites in the CITRIS-COSMIC system are required to communicate with each other to monitor the ionospheric irregularities [1]. Relative positions between satellites, such as Cluster and Cluster-II satellites launched by the European Space Agency, are estimated and controlled to support many collaborative tasks where satellites are required to maintain a specific formation in a continuous manner within the mission period [2]. Several relative position estimation methods have been developed. A GPS-like technology that can be applied to SFF have been introduced [3, 4], where each spacecraft is equipped with a communication system to localize other spacecraft in the formation. This system provides high precision estimates for relative positions. Yet, complex hardware is required on each spacecraft. This system does not stand alone and requires a GPS system. A relative position and attitude estimation through a Vision-Based Navigation System (VISNAV) has been addressed extensively in the literature [5–8]. The VISNAV consists of a Position Sensing Diode (PSD) sensor and an array of LED source beacons installed on a plate. The PSD detects the energy of the light source emitted by the beacons. Then the PSD generates a current flow to four terminals that are installed on the PSD plate. At that point, the centroid
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