Higher-order integration of index-3 DAE with friction using time finite elements on Lie groups

Abstract In multibody dynamics, Euler angles and Euler parameters (unit-quaternions) are common choices of global orientation parameters. Both approaches have their limitations, e.g., gimbal lock and an extra normalization constraint, respectively. To circumvent these, the Lie-group based frameworks methods employ SO(3) or SE(3) representation of rotations with corresponding higher order Lie-group integration methods. However, quaternions represent a much more compact representation of rotations and offer a computational advantage over SO(3) or SE(3) representations. Implicit integration inevitably requires computation of Jacobian matrices for Newton-Raphson iterations. However, deriving the component of Newton-Raphson matrix corresponding to body-frame rotations within SO(3) formalism is not limited to calculation of partial derivatives. In fact, the process of deriving these matrices involves a sequence of algebraic manipulations. Subsequently, they often need to be derived for the acting forces and torques, on a case-by-case basis. It is this limitation that emerges as a bottle-neck for modeling friction within Lie-group formalism, as friction loads are intricate nonlinear functions of generalized coordinates (and hence quaternions), velocities as well as Lagrange multipliers. To address this limitation, we propose a systematic partial-derivative based approach for computing the sub-element coefficient matrices with respect to body-frame Euler-rotations. This is achieved by defining general-purpose operators, as presented in this work, that support convenient computer application due to accessibility to a wide range of numerical and auto-differentiation tools. Further, recent advances in time-finite element based methods have demonstrated higher-order accurate solutions for nonlinear and chaotic systems with relatively simple computer implementation. This work implements a family of time-finite element integrators in conjunction with exponential integration of rotations using Lie-group properties of quaternions. Further, numerical experiments showed that the Lie-group time-finite element integrators compare favorably with Lie-group Newmark- $\beta $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> and trapezoidal schemes.