To address the redundant constraints and constraints violation issues arising from the stylized modeling of closed-loop multibody systems, the QR decomposition method with column permutation is employed to identify the independent constraint equations. This process yields the orthogonal basis vectors for both the constraint normal space and tangent space. Subsequently, by directly adjusting the generalized coordinates and generalized velocity, the constraint equation is satisfied prior to forming the motion equation, eliminating the need for corrections after the integration step. Concerning the issue of distorted acceleration in the multi-body system when entering a singular configuration, the direction of deviation amplification is determined to be orthogonal to the tangent space of the constraint manifold. The tangent space component of the generalized acceleration is then utilized for correction, significantly enhancing the numerical solution. Finally, the practicality and effectiveness of the proposed method are validated through numerical experiments.