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A novel nonlinear filter called square-root adaptive cubature Kalman filter is proposed to estimate the spacecraft attitude from vector measurements. The algorithm combines an adaptive process noise estimation with the square-root cubature Kalman filter, which has a consistently improved numerical stability because all the resulting covariance matrices are guaranteed to stay positive semi-definite. The process noise estimate for efficient square-root implementation is derived. The quaternion is used to describe the spacecraft attitude kinematics, while a three-dimensional generalized Rodrigues parameter is used to maintain the quaternion normalization constraint in the filter formulation. The simulation results indicate that the proposed filter provides lower attitude estimation errors with faster convergence rate than the square-root cubature Kalman filter.