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This paper presents a dynamically smooth nonlinear observer for satellite attitude determination. The proposed observer uses a 3-axis gyro and a single vector measurement to estimate the attitude of a satellite. The proposed observer preserves orthogonality of the estimated attitude matrix for all time. Almost global and exponential convergence of the estimated attitude to its true value is proven without persistency of excitation conditions. The convergence rate is shown to depend on properties of certain time varying reference vectors expressed in inertial frame. A procedure for maximizing a lower bound of the convergence rate is also presented. Performance of the proposed observer is illustrated in a realistic case study by simulation where a magnetometer is used to provide the single vector measurement.