126 An index reduction method in holonomic system dynamics Marco Borri ∗ , Lorenzo Trainelli, Alessandro Croce Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, 20156 Milan, Italy Abstract We present the application of the Embedded Projection Method, a general methodology to reduce the DAE index to 1, to holonomically constrained mechanical systems, which are naturally formulated in terms of index-3 DAE problems. This procedure represents an extension to higher index of that already presented in Ref. [5] for non-holonomic systems. The holonomic case is more complex, requiring the introduction of modified coordinates in addition to modified momenta and differentiated multipliers. Eventually we recover the same qualitative result as in the non-holonomic case: a complete uncoupling of the algebraic and differential parts of the problem which implies a gentler numerical behavior with enhanced accuracy and stability. As a consequence, the numerical integration of a index-higher-than-2 DAE can be performed by any suitable ODE integrator, by-passing the need for a specialized DAE solver. Keywords: Differential-algebraic equations; Embedded Projection Method; Index reduction; Constraint stabilization; Multibody dynamics; Holonomic systems; Constrained systems 1. Introduction Constrained dynamical systems of arbitrary topology are generally modeled through a set of differential-algebraic equations (DAEs), i.e. systems composed of ordinary dif- ferential equations (ODEs) and purely algebraic equations that explicitly enforce the constraints. Solving general DAE systems still represents an open field of research, since their intrinsic numerical difficulty has prevented to date from reaching the same degree of maturity achieved in the numerical treatment of standard ODEs. The difficulty, both in terms of accuracy and stability, in the numerical treatment of these problems can be measured qualitatively through the concept of index (see e.g. Refs. [1–3]): the higher the index, the harder an efficient and reliable solu- tion. This applies in particular when mechanical systems with holonomic constraints, yielding an index-3 DAE sys- tem, are concerned. Widely adopted approaches for high index problems basically resort either to a reformulation to get a more tractable lower index system or to keeping the origi- nal index-3 formulation and dealing with the numerical difficulties through projections, stiff integrators, etc. (see Ref. [4] for a quick account of several ‘good’ techniques). With respect to the first approach, the ideal reformulation ∗ Corresponding author. Tel.: +39 (2) 2399-8342; Fax: +39 (2) 2399-8334; E-mail: marco.borri@polimi.it consists in eliminating the redundant degrees of freedom of the system to obtain a minimal set formulation, i.e. a pure ODE (an index-0 problem). This approach has several drawbacks, ranging from the difficulty inherent to the elim- ination process when complex topologies with loops are examined, to the lack of flexibility of the same process with respect to local design changes, to the bad numerical con- ditioning that the process may induce when finite element models with large number of degrees of freedom are dealt with. Nevertheless, the appeal of a minimal set formula- tion is undeniable when considering that the corresponding governing equations can be integrated by employing any suitable off-the-shelf ODE integrator, without concern for accuracy and stability issues related to constraint imposi- tion. Indeed, in such a case, the solution inherently satisfies the constraints at all levels of differentiation. The present work is intended as a companion paper to Ref. [5], where non-holonomic systems were addressed. Presently, we are concerned with the application of similar ideas to the case of holonomic systems. Indeed, both the non-holonomic case and the holonomic case fall into a more general class of applications of the Embedded Pro- jection Method (EPM), recently devised as a contribution to a better numerical treatment of general constrained sys- tems, including systems that do not belong to the field of mechanics, such as chemical and control systems. The case of proper non-holonomic system (index-2 DAEs) can be considered preliminary to the understanding of the EPM  2003 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics 2003 K.J. Bathe (Editor)