The integrated Singular Value Decomposition (SVD) and Unscented Kalman Filter (UKF) method can recursively estimate the attitude and attitude rates of a nanosatellite. At first, Wahba's loss function is minimized using the SVD and the optimal attitude angles are determined on the basis of the magnetometer and Sun sensor measurements. Then, the UKF makes use of the SVD's attitude estimates as measurement results and provides more accurate attitude information as well as the attitude rate estimates. The elements of "Rotation angle error covariance matrix" calculated for the SVD estimations are used in the UKF as the measurement noise covariance values. The algorithm is compared with the SVD and UKF only methods for estimating the attitude from vector measurements. Possible algorithm switching ideas are discussed especially for the eclipse period, when the Sun sensor measurements are not available.