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This paper considers a robust strong tracking nonlinear filtering problem in the case there are model uncertainties including the model mismatch, unknown disturbance and status mutation in the spacecraft attitude estimation system with quaternion constraint. Two multiple fading factor matrices are employed to regulate the prediction error covariance matrix, which guarantees its symmetry. The spherical-radial cubature rule is developed to deal with the multi-dimensional integrals. The quaternion constraint is maintained by utilizing the gain correction method. Therefore a robust strong tracking cubature Kalman filter (RSTCKF) is formed for the spacecraft attitude estimation with quaternion constraint. Unlike adopting a single fading factor in the traditional strong tracking filter, the presented filter uses two multiple fading factor matrices to make different channels have respective filter adjustment capability, which improves the tracking performance of this algorithm. Simulation results show the effectiveness of the proposed RSTCKF.