Extending the Modified Inertia Representation to Constrained Rigid Multibody Systems

The inertia representation of a constrained system includes the formulation of the kinetic energy and its corresponding mass matrix, given the coordinates of the system. The way to find a proper inertia representation achieving better numerical performance is largely unexplored. This paper extends the modified inertia representation (MIR) to the constrained rigid multibody systems. By using the orthogonal projection, we show the possibility to derive the MIR for many types of non-minimal coordinates. We present examples of the derivation of the MIR for both planar and spatial rigid body systems. Error estimation shows that the MIR is different from the traditional inertia representation (TIR) in that its parameter $γ$ can be used to reduce the kinetic energy error. With preconditioned $γ$, numerical results show that the MIR consistently presents significantly higher numerical accuracy and faster convergence speed than the TIR for the given variational integrator. The idea of using different inertia representations to improve the numerical performance may go beyond constrained rigid multibody systems to other systems governed by differential algebraic equations.