Optimal Fusion of a Given Quaternion with Vector Measurements

Abstract

Introduction S EVERAL satellites that use devices called autonomous star trackers (ASTs) are presently operating. These devices put out the satellites’attitude in the form of a quaternion.Usually the satellites also carry other attitudemeasuringdevices, such as sun sensors and magnetometers that measure vectors in body coordinates. Although the accuracy of the AST surpasses that of the other sensors, due to the synergistic effect of sensor fusion, it is still desirable to incorporate the measurements of the less accurate sensors in the attitude determination process. The question is, then, how to blend optimally the AST-generated quaternion with vector measurements. This problem rose, for example, in the design of the attitude determinationalgorithm of the MicrowaveAnisotropyProbe (MAP) satellite, which was launched on 30 June 2001. MAP has two ASTs and two sun sensors, one of which is more accurate than the other.Althoughwe have two quaternion-generating devices and two vector-measuring sensors, we consider here only one of each. The extension of the solution, proposed here, to multiple devices and multiple sensors, is immediate. We note that the quaternion is a four-element vector that yields the whole attitude, whereas vector-measuring sensors yield threedimensional vectors each containing only partial information on the attitude. Therefore, we cannot cast the problem of optimal attitude determination in the form of Wahba’s problem. That is, we cannot blend quaternions with vector measurements using known algorithms. If we have more than one simultaneous vector measurement, we can use the vector measurements to, Ž rst, Ž nd attitude expressed in quaternionform and, then, blend this quaternionwith the given one. However, when we have only one vector measurement, this is not possible. Therefore, we need an algorithm that can blend the given quaternion even with one vector measurement. The algorithm presented here consists of two steps. In the Ž rst step, the quaternion is converted into a pair of pseudovector measurements that express the attitude, and then, in the second step, these pseudovector measurements, together with the given vector measurement(or measurements), are used as inputs to the q-method algorithm, which generates the optimal quaternion. The resultant quaternion is optimal in the sense that it is the best Ž t, in the leastsquares sense, to all of the vectors.