UK method for solving multi-body system dynamics with singular problems

Abstract

In the process of solving differential algebraic equations of motion for constrained multi-body dynamic system, singular problems such as singular mass matrices, singular configurations, and redundant constraints may appear, which bring great difficulties to numerical simulation. The article presents a new equation of motion for multi-body dynamic system based on UK equations (explicit equations of motion proposed by Udwadia and Kalaba) and least square method, with which the dynamics problems are transformed into unconstrained optimization problems. Due to its pretty explicit and simple form, the new equation is very suitable for programming. Compared with the Lagrange method, the new equation does not require the introduction of Lagrange multipliers. Using this new equation, both the traditional optimization and modern optimization algorithms can be used in dynamics simulation for multi-body system. With the help of different optimization methods, approximate solutions can be found near the location of singular configurations to avoid instability of solution and numerical diffusion. In the end, several examples are given to illustrate that the new equation set is not only a practical modeling method for the above singular problems but also has faster solving speed and higher accuracy than the traditional methods.