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This paper presents a new method using UKF(unscented Kalman filtering) for estimating the attitude-quaternion as well as rate with magnetic field data from Three-Axis-Magnetometer(TAM). UKF is a new filtering method suitable to nonlinear systems. The magnetic field components are used to update as measurement is used to propagate the state vector. A globally self-initializing attitude determination filter has been developed to estimate three-axis attitude and attitude rate from magnetometer data only. This filter can serve as the main attitude determination system for a low-weight, low-budget, low-accuracy micro-satellite mission. In this paper, we present a new algorithm without gyro to determine the attitude and attitude rate simultaneously. Numerical simulations are given at last, and the algorithm is confirmed to be feasible and effective.