Sufficient conditions for penalty formulation methods in analytical dynamics

The axiomatic derivation of constraints in analytical mechanics as the limiting case of motion in a potential field that grows asymptotically in strength has been known for years. However, with the emergence of the need for robust simulation methods for complex, multi-degree-of-freedom, nonlinear mechanical systems, researchers have shown renewed interest in these `penalty formulations' as practical computational schemes. While much empirical evidence has been collected regarding the efficiency of these methods, relatively few convergence results are available for a wide class of the nonlinear simulation methods. This paper derives sufficient conditions for the convergence of a class of penalty methods by extending the Rubin-Ungar theorem. One advantage of the approach taken in this paper is that considerable simplification of the original Rubin-Ungar derivation is achieved for the convergence of transverse constraint velocities. This paper also emphasizes the importance of maintaining a rank condition on the Jacobian of the constraint matrix. This is of particular importance in that one claimed benefit of certain penalty methods is that they are effective in cases in which the constraint Jacobian loses rank. For the class of penalty methods considered in this paper, if the Jacobian does not meet the specified rank conditions, a diverse collection of spurious, pathological responses can be obtained using this method. In one sense, this type of pathological response is worse than encountering a `configuration-singular' generalized mass matrix and having a simulation diverge; indeed, the regularized solution procedure can proceed along some incorrect trajectory with little to no indication that something is amiss.