Geometric properties of projective constraint violation stabilization method for generally constrained multibody systems on manifolds

Terze, Zdravko; Naudet, Joris

doi:10.1007/s11044-008-9107-5

Cited by 18 publications

(10 citation statements)

References 16 publications

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“…However, mathematical studies underline the numerical difficulties associated with the solution of these DAEs, mostly related to the stability of the available integration schemes. If governing equations are not turned into minimal form and dynamic simulation is based on the mathematical model expressed via redundant coordinates [8], progressive drifts of the computed solution from the position, velocity or acceleration constraint manifolds are likely to occur during the simulation. Two kinds of popular techniques to solve this "drift" problem are the constraint violation stabilization (CVS) techniques and the constraint violation elimination (CVE) techniques [2].…”

Section: Introductionmentioning

confidence: 99%

“…The geometrical projection method was also utilized by Nikravesh [25] to correct the initial conditions prior to performing kinematic or forward dynamic analysis of multibody systems. Terze et al [8,26,27] formulated a constraint elimination method within the framework of the null space formulation, which can correct the constraint violation regardless of the actual magnitude of violation. The projective criterion, defined in [23] and then optimized in [8,27], was used in the generalized coordinates (positions or velocities) partitioning to identify a set of independent variables.…”

Section: Introductionmentioning

confidence: 99%

“…Terze et al [8,26,27] formulated a constraint elimination method within the framework of the null space formulation, which can correct the constraint violation regardless of the actual magnitude of violation. The projective criterion, defined in [23] and then optimized in [8,27], was used in the generalized coordinates (positions or velocities) partitioning to identify a set of independent variables. Displacement and velocity constraint violations were then iteratively eliminated by successively adjusting the sole dependent variables to satisfy the constraint equations, where the independent displacements and velocities were kept unchanged.…”

Section: Introductionmentioning

confidence: 99%

“…However, for large matrices with a high percentage of zero-valued elements, the use of sparse matrix algorithms is recommended which can speed up the processing and avoid numerical problems and accuracy losses for badly conditioned matrices [8]. Based on the geometrical interpretation provided in [8,23,24,26], the manifold orthogonal directions in the physically consistent (energy norm) orthogonal projection method are actually found along the 'near-orthogonal' direction by using the state vector approximate numerical values, and this direction is not determined exactly. Also, the projection method is based on adding additional terms to system's dynamic equations, which may turn the correction procedure into more complex and numerically extensive [8].…”

Section: Introductionmentioning

confidence: 99%

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A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

Zhang

Liu

2015

Multibody Syst Dyn

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The constraint violation problem of spatial multibody systems is analyzed in this paper. The mass matrix is singular in the equations of motion when the Euler parameters with the normalization constraints are used to describe the orientation of the spatial rigid body. The constrained and weighted least-squares-based geometrical projection method is implemented to suppress the constraint violation during numerical integration, and the explicit correction formulation can be obtained by the block matrix inversion scheme. The mass matrix weighted correction formulation gives the physically consistent energy norm, but it needs the mass matrix to be positive definite. To extend the physically consistent correction formulation for solving spatial multibody systems' constraint violation problems with a singular mass matrix, a Modified Mass-Orthogonal Projection Method (MMOPM) and a Generalized Physical Orthogonal Projection Method (GPOPM) are proposed. MMOPM modifies the mass matrix directly by adding a penalty factor matrix which appears in the mass-orthogonal projection method and leads to a positive definite weight matrix that satisfies the block matrix inversion scheme condition. GPOPM is a generalization of the physical orthogonal projection method where the constrained least-squares method is weighted by the positive semi-definite mass matrix and the correction formulation is given by using the generalized block matrix inversion scheme. Numerical results show the feasibility and accuracy of the presented MMOPM and GPOPM. The constraints in position and velocity can reach machine precision during numerical integration. The elimination of violation of position constrains may require few iterations, while the violation of velocity constraints is removed in one step, and GPOPM is more accurate in velocity correction than MMOPM.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

Zhang

Liu

2015

Multibody Syst Dyn

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“…To reduce this numerical drift, constraint stabilization methods have been proposed that either amend the motion equations [3,20] or correct the numerical solution by projecting it to the constraint manifold h −1 (0) [2,4,11,32]. All these methods aim at minimizing or correcting, rather than avoiding, constraint violations.…”

Section: Constraint Satisfaction For a General C-space Lie Groupmentioning

confidence: 99%

A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body

Müller

2015

Bit Numer Math

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The dynamics of a holonomically constrained rigid body can be modeled by Newton-Euler equations subjected to geometric constraints. This is frequently formulated as a differential-algebraic equation (DAE) system of index 1. In multibody system (MBS) dynamics it is common (1) to numerically solve this system by means of integration schemes for ordinary differential equations, and (2) to treat the rigid body motion on the direct product Lie group SO (3)×R 3 , although rigid body motions form the semidirect product Lie group SE (3). It is has been observed that the constraint satisfaction depends on which Lie group is used as configuration space (c-space). In this paper the problem is considered from a geometric perspective. It is shown that the constraints are exactly satisfied by a numerical integration scheme if they define a subgroup of the c-space. The subgroups of SE (3) have a significance for modeling mechanical systems, including lower kinematic (Reuleaux) pairs and are implicitly used in MBS modeling. It is concluded that SE (3) is the appropriate cspace for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.

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On the use of two-dimensional Euler parameters for the dynamic simulation of planar rigid multibody systems

Pappalardo

Guida

2017

Arch Appl Mech

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This paper introduces a new coordinate formulation for the kinematic and dynamic analysis of planar multibody systems composed of rigid bodies. The methodology presented in this work is called planar reference point coordinate formulation (RPCF) with Euler parameters. In the planar RPCF with Euler parameters, the rotational coordinates used for describing the body orientation are the redundant components of a two-dimensional unit quaternion that identify a planar set of Euler parameters. It is shown in the paper that the planar RPCF with Euler parameters allows for obtaining consistent kinematic and dynamic descriptions of two-dimensional rigid bodies. In the numerical solution of the equations of motion, the well-known generalized coordinate partitioning method can be effectively utilized to stabilize the violation of the algebraic constraints at the position and velocity levels leading to physically correct and numerically stable dynamic simulations. Furthermore, a standard numerical integration procedure can be employed for calculating an approximate solution of the equations of motion resulting from the planar RPCF with Euler parameters. In the paper, the computer implementation of the proposed formulation approach is demonstrated considering four rigid multibody systems which serve as simple benchmark problems

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Geometric properties of projective constraint violation stabilization method for generally constrained multibody systems on manifolds

Cited by 18 publications

References 16 publications

A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body

On the use of two-dimensional Euler parameters for the dynamic simulation of planar rigid multibody systems

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