Sensitivity analysis for multibody systems formulated on a Lie group

A direct differentiation method and an adjoint variable method are proposed for the efficient semi-analytical evaluation of the sensitivities of multibody systems formulated in a matrix Lie group framework. These methods rely on the linearization of the equations of motion and/or of the time integration procedure. The simpler structure of the equations of motion in the Lie group formalism appears as an advantage for that purpose. Lie bracket contributions and the non-linearity of the exponential map need to be taken into account in the sensitivity algorithms. Nevertheless, essential characteristics of formulations of the direct differentiation method and the adjoint variable method on linear spaces are recovered. Some implementation issues are discussed and two relevant examples illustrate the properties of these methods.

[1]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[2]  Phailaung Phohomsiri,et al.  Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[3]  J. P. Dias,et al.  Sensitivity Analysis of Rigid-Flexible Multibody Systems , 1997 .

[4]  Jorge Ambrósio,et al.  Optimization of a complex flexible multibody systems with composite materials , 2007 .

[5]  D. Bestle,et al.  Analyzing and Optimizing Multibody Systems , 1992 .

[6]  Tamer M. Wasfy,et al.  Modeling and sensitivity analysis of multibody systems using new solid, shell and beam elements , 1996 .

[7]  Edward J. Haug,et al.  Design Sensitivity Analysis of Structural Systems , 1986 .

[8]  Peter Eberhard,et al.  Automatic differentiation of numerical integration algorithms , 1999, Math. Comput..

[9]  Raphael T. Haftka,et al.  Recent developments in structural sensitivity analysis , 1989 .

[10]  Peter Eberhard,et al.  Sensitivity analysis for dynamic mechanical systems with finite rotations , 2008 .

[11]  P. Kowalczyk.  Design sensitivity analysis in large deformation elasto‐plastic and elasto‐viscoplastic problems , 2006 .

[12]  Daniel A. Tortorelli,et al.  Sensitivity analysis for non‐linear constrained elastostatic systems , 1992 .

[13]  D. Bestle,et al.  Sensitivity analysis of constrained multibody systems , 1992, Archive of Applied Mechanics.

[14]  Olivier Bruls,et al.  Lie group generalized-α time integration of constrained flexible multibody systems , 2012 .

[15]  Sotirios Natsiavas,et al.  A new look into the kinematics and dynamics of finite rigid body rotations using Lie group theory , 2013 .

[16]  Shengtai Li,et al.  Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution , 2002, SIAM J. Sci. Comput..

[17]  C. Bottasso,et al.  On the Solution of Inverse Dynamics and Trajectory Optimization Problems for Multibody Systems , 2004 .

[18]  Andrei Serban Schaffer,et al.  On the adjoint formulation of design sensitivity analysis of multibody dynamics cs , 2005 .

[19]  Jorge Ambrósio,et al.  Sensitivity analysis of flexible multibody systems using composite materials components , 2009 .

[20]  M. Géradin,et al.  Flexible Multibody Dynamics: A Finite Element Approach , 2001 .

[21]  Steen Krenk,et al.  Conservative integration of rigid body motion by quaternion parameters with implicit constraints , 2012 .

[22]  Olivier Bruls,et al.  Two Lie Group Formulations for Dynamic Multibody Systems With Large Rotations , 2011 .

[23]  Darryl D. Holm,et al.  Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions , 2009 .

[24]  Olivier Bruls,et al.  On the Use of Lie Group Time Integrators in Multibody Dynamics , 2010 .

[25]  Javier Cuadrado,et al.  Multibody Dynamics: Computational Methods and Applications , 2007 .

[26]  D. Tortorelli,et al.  Tangent operators and design sensitivity formulations for transient non‐linear coupled problems with applications to elastoplasticity , 1994 .

[27]  Peter Eberhard,et al.  Optimization of Multibody Systems and Their Structural Components , 2011 .