<p>In this paper, a new numerical solution method is proposed for dealing with differential-algebraic equations (DAEs) for dynamics of multibody systems with nonholonomic constraints. The nonholonomic constraints directly restrict the velocity coordinates, resulting in no corresponding position constraint equations. Therefore, the traditional state-space method is insufficient to solve such DAEs. In the proposed state-space method, direct integration of the ordinary differential equations obtained from the index-1 DAEs, ensures that the acceleration constraints are satisfied and provides initial values for the dependent variables. Subsequently, position and velocity constraint equations are solved to update dependent variables, strictly ensuring satisfaction of constraints at three levels. Currently, LU decomposition is the most used method to define the state-space method. However, in order to ensure the accuracy and stability of the algorithm, coordinate identification is required at every time step, which reduces the computational efficiency. Therefore, in this paper, the state-space method defined by singular value decomposition (SVD) is proposed, which does not require frequent coordinate identification and improves the computational efficiency. Numerical examples show that the state-space method based on SVD outperforms the LU decomposition in terms of computational efficiency and stability.</p>