Aiming at the case of the state vector with quaternion constraint in some nonlinear systems, a quaternion constrained cubature Kalman filter algorithm is presented. Based on a minimum constrained cost function, this algorithm uses three-degree spherical-radial cubature rule to approximate the posterior mean and covariance, and gives the QCCKF recursive formula. It can estimate the state effectively, and extends the applications of the cubature Kalman filter (CKF). By simulating for the spacecraft attitude estimation system, simulation results show that the proposed algorithm is superior to CKF and unscented quaternion estimator(USQUE) in precision, satisfies the quaternion constraint, and solves the problem that the nonlinear system has quaternion constraint, which show the validation of the proposed method.