Efficient Solution of Maggi’s Equations

According to a recent paper (Laulusa and Bauchau, 2008, “Review of Classical Approaches for Constraint Enforcement in Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 3(1), 011004), Maggi’s formulation is a simple and stable way to solve the dynamic equations of constrained multibody systems. Among the difficulties of Maggi’s formulation, Laulusa and Bauchau quoted the need for an appropriate choice (and change, when necessary) of independent coordinates, as well as the high cost of computing and updating the basis of the tangent null space of constraint equations. In this paper, index-1 Lagrange’s equations are first considered, including the not-so-rare case of having a singular mass matrix and redundant constraints. The existence and uniqueness of solution for acceleration vector and Lagrange multipliers vector is studied in a very simple way. Then, following Von Schwerin (Von Schwerin, Multibody System Simulation. Numerical Methods, Algorithms and Software, Springer, New York, 1999), Maggi’s formulation is described as the most efficient way (in general) to solve these index-1 equations. Next, an improved double-step method, which implements the matrix transformations of Maggi’s formulation in an efficient way, is described. Finally, two large real-life examples are presented.