Controllable velocity projection for constraint stabilization in multibody dynamics

Orden, Juan Carlos García; Martín, Sergio Conde

doi:10.1007/s11071-011-0224-y

Cited by 14 publications

(8 citation statements)

References 20 publications

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“…Cuadrado et al [29] developed a more efficient implementation of the mass-orthogonal projection which requires only successive forward reductions and back-substitutions, and then Blajer [30] gave a correcting formulation which doesn't need to update the Lagrange multipliers. The energy consideration with velocity projection was studied by Orden et al [31,32], providing an alternative interpretation of its effect on the stability and a practical criterion for the mass-orthogonal projection matrix selection. Blajer [30] also gave a geometrical interpretation to the augmented Lagrangian formulation which is capable of treating systems with changing topologies, redundant constraints, singular positions and some singularity of the mass matrix where the mass-orthogonal projection method can still be applied but the geometric elimination method cannot [20,21,23,24].…”

Section: Introductionmentioning

confidence: 99%

“…(2.18) and (2.19) in [36]. It had been interpreted by Blajer that the mass-orthogonal projection method can deal with some singularity of the mass matrix [28][29][30][31][32], and it is also capable of treating systems with changing topologies, redundant constraints and singular positions which are not considered in this paper. Although we have concluded that large penalty values adversely affect the numerical conditioning of the algebraic linear system and the mass-orthogonal method shows less physical meaning than the geometrical projection method with a mass weight matrix, it had been shown that the leading matrix applied in the mass-orthogonal projection method, which is derived from the augmented Lagrangian formulation [13,14], is always positive definite, i.e., invertible, even in singular positions and/or with linearly dependent constraints and some singularity of the mass matrix.…”

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%

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

Zhang

Liu

2015

Multibody Syst Dyn

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“…The TLM in Eqs. (25) features four sets of unknowns, namely q ρ ,q * ρ ,q * ρ , and λ * ρ . This TLM requires 2n c p initial conditions in the form…”

Section: Algorithm Implementationmentioning

confidence: 99%

“…It represents the extension of the formulations presented in [6,13] to nonholonomic systems and their generalization to more advanced integration schemes. This formulation constitutes a good exponent of what it should be expected from a modern multibody method: it is efficient, accurate, and robust; it exactly satisfies the constraints at position, velocity, and acceleration levels; it has predictable and controllable energy decaying properties [26,24,25]; it can deal with redundant constraints and singular configurations without special considerations [29] and it can handle both holonomic (scleronomic and rheonomic) and nonholonomic constraints.…”

Section: Introductionmentioning

confidence: 99%

Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections

et al. 2018

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Optimizing the dynamic response of mechanical systems is often a necessary step during the early stages of product development cycle. This is a complex problem that requires to carry out the sensitivity analysis of the system dynamics equations if gradient-based optimization tools are used. These dynamics equations are often expressed as a highly nonlinear system of Ordinary Differential Equations (ODEs) or Differential-Algebraic Equations (DAEs), if a dependent set of generalized coordinates with its corresponding kinematic constraints is used to describe the motion. Two main techniques are currently available to perform the sensitivity analysis of a multibody system, namely the direct differentiation and the adjoint variable methods.In this paper, we derive the equations that correspond to the direct sensitivity analysis of the index-3 augmented Lagrangian formulation with velocity and acceleration projections. Mechanical systems with both holonomic and nonholonomic constraints are considered. The evaluation of the system sensitivities requires the solution of a Tangent Linear Model (TLM) that corresponds to the Newton-Raphson iterative solution of the dynamics at configuration level, plus two additional nonlinear systems of equations for the velocity and acceleration projections. The method was validated in the sensitivity analysis of a set of examples, including a five-bar linkage with spring elements, which had been used in the literature as 1 D. Dopico et al.benchmark problem for similar multibody dynamics formulations, a point-mass system subjected to nonholonomic constraints, and a full-scale vehicle model.

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“…This prompted many research efforts focusing towards eliminating or reducing these undesirable consequences. Besides consideration of dissipative time integration schemes [1,3], this included efforts to properly scale the governing equations and constraints or to perform appropriate velocity projections [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Numerical integration of multibody dynamic systems involving nonholonomic equality constraints

2021

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This work considers a class of multibody dynamic systems involving bilateral nonholonomic constraints. An appropriate set of equations of motion is employed first. This set is derived by application of Newton's second law and appears as a coupled system of strongly nonlinear second order ordinary differential equations in both the generalized coordinates and the Lagrange multipliers associated to the motion constraints. Next, these equations are manipulated properly and converted to a weak form. Furthermore, the position, velocity and momentum type quantities are subsequently treated as independent. This yields a three-field set of equations of motion, which is then used as a basis for performing a suitable temporal discretization, leading to a complete time integration scheme. In order to test and validate its accuracy and numerical efficiency, this scheme is applied next to challenging mechanical examples, exhibiting rich dynamics. In all cases, the emphasis is put on highlighting the advantages of the new method by direct comparison with existing analytical solutions as well as with results of current state of the art numerical methods. Finally, a comparison is also performed with results available for a benchmark problem.

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Controllable velocity projection for constraint stabilization in multibody dynamics

Cited by 14 publications

References 20 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

Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections

Numerical integration of multibody dynamic systems involving nonholonomic equality constraints

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