This letter presents a state estimation algorithm for the legged robot by defining the problem as a Maximum A Posteriori (MAP) estimation problem and solving the problem with the Gauss-Newton algorithm. Moreover, marginalization by the Schur Complement method is adopted to make a fixed size problem. Each component of the cost function and its Jacobian are derived utilizing the SO(3) manifold structure, while we reparameterize the state with nominal state and variation to make linear algebra and vector calculus applied properly. Furthermore, a slip rejection method is proposed to reduce the erroneous effect of fault modeling of kinematics models. The proposed algorithm is verified by comparison with the Invariant Extended Kalman Filter (IEKF) in real robot experiments on various environments.