An efficient high-precision recursive dynamic algorithm for closed-loop multibody systems

As most closed-loop multibody systems do not have independent generalized coordinates, their dynamic equations are differential/algebraic equations (DAEs). In order to accurately solve DAEs, a usual method is using generalized a-class numerical methods to convert DAEs into difference equations by differential discretization and solve them by the Newton iteration method. However, the complexity of this method is O(n(2)) or more in each iteration, since it requires calculating the complex Jacobian matrix. Therefore, how to improve computational efficiency is an urgent problem. In this paper, we modify this method to make it more efficient. The first change is in the phase of building dynamic equations. We use the spatial vector note and the recursive method to establish dynamic equations (DAEs) of closed-loop multibody systems, which makes the Jacobian matrix have a special sparse structure. The second change is in the phase of solving difference equations. On the basis of the topology information of the system, we simplify this Jacobian matrix by proper matrix processing and solve the difference equations recursively. After these changes, the algorithm complexity can reach O(n) in each iteration. The algorithm proposed in this paper is not only accurate, which can control well the position/velocity constraint errors, but also efficient. It is suitable for chain systems, tree systems, and closed-loop systems.