A Trilateration Scheme for Relative Positioning
We introduce a trilateration scheme that evaluates the 3-dimensional (3-D) relative position between a reference spacecraft and a target spacecraft using raw-range measurements from a distance baseline of known locations, which we call “anchors”. The anchors can be antennas of a ground-based network (e.g., Deep Space Network (DSN) or Near Earth Network (NEN) stations), or satellites of a spacebased network (e.g., global positioning system (GPS) or tracking and data relay satellite (TDRS)). We define raw-range as the range that includes all the systematic errors that occur during range measurements. A unique feature of this approach is that accurate relative position is derived from a “differencing function” of raw-range measurements of the reference spacecraft and target spacecraft, thereby eliminating most of the systematic errors, such as media effects, ephemeris errors, instrument delays, clock bias, etc. There can be an arbitrary number of target spacecraft, and relative positioning of target spacecraft with respect to the reference spacecraft can be done simultaneously. In this paper, we first assume an idealized system in which clocks on the reference and target spacecraft are synchronized, with clocks of the anchors synchronized as well. We develop a novel iterative algorithm that computes the relative position of the target spacecraft with respect to the reference spacecraft. We illustrate the relative positioning method using the scenario of a network of three ground stations (i.e., the anchors) at Goldstone, California, USA, Madrid, Spain, and Marlargue, Argentina tracking two spacecraft at geosynchronous orbit distance. We demonstrate that the algorithm converges to submeter accuracy in estimating the relative position, in the presence of random errors and systematic errors in raw-range measurements, and in the presence of angular errors in estimating the pointing vectors between the anchors and the reference spacecraft. Next, we relax the requirement of perfect time synchronization between spacecraft, and show that by using an additional anchor, one can estimate and remove the clock biases between the reference and target spacecraft. We add a ground station at Kourou to the above example of three ground stations of Goldstone, Madrid, and Marlargue, and demonstrate that the updated algorithm also converges to meter-level accuracy (submeter in some cases) in the presence of clock biases in addition to the random errors, systematic errors, and angular errors as shown in the above case. We compare this scheme with a similar trilateration scheme for relative positioning scheme first proposed by Montenbruck in 2002.