The GGL variational principle for constrained mechanical systems

L., Kinon, Philipp; Betsch, Peter; Schneider, Simeon

doi:10.1007/s11044-023-09889-6

Cited by 5 publications

(9 citation statements)

References 26 publications

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“…Moreover, a generalized total energy function depending on position, velocity and momentum is introduced, which serves as a conserved quantity of the equations of motion pertaining to Livens' principle. The present work can be linked to previous works [30,29] and shows how to overcome the restrictions therein to constant and non-singular mass matrices.…”

Section: Discussionmentioning

confidence: 72%

“…In contrast to existing literature ( [7,44,58]), no sophisticated augmentation of the mass matrix with additional inertia terms is needed here. Lastly, as a byproduct, the current work extends Livens' principle with respect to more general mass matrices, thus paving the way for a more general GGL principle, see previous works [30,29].…”

Section: Present Contributionsmentioning

confidence: 72%

“…In the previous works [30,29], the usage of Livens' principle has been restricted to constrained mechanical systems with constant mass matrix. However, in a multitude of frameworks, the mass matrix can even become singular or does depend on the coordinates themselves (cf.…”

Section: Non-standard Mass Matricesmentioning

confidence: 99%

“…In the present work, we propose a new approach to multibody systems with singular and/or statedependent mass matrix based on Livens' principle ( [39,11,25,30,29]). As we show, the proposed approach circumvents the need to invert the mass matrix and offers a new possibility for the design of structure-preserving time integration methods.…”

Section: Present Contributionsmentioning

confidence: 99%

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Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics

Kinon,

Betsch

2024

Preprint

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Mechanical systems with singular and/or configuration-dependent mass matrix can pose difficulties to Hamiltonian formulations, which are the standard choice for the design of energy-momentum conserving time integrators. In this work, we derive a structure-preserving time integrator for constrained mechanical systems, based on a mixed variational approach. Livens’ principle (or sometimes called Hamilton-Pontryagin principle) features independent velocity and momentum quantities and circumvents the need to invert the mass matrix. In particular, we take up the description of rigid body rotations using unit quaternions. Using Livens’ principle, a new and comparatively easy approach to the simulation of these problems is presented. The equations of motion are approximated by using (partitioned) midpoint discrete gradients, thus generating a new energy-momentum conserving integration scheme for mechanical systems with singular and/or configuration-dependent mass matrix. The derived method is second-order accurate and algorithmically preserves a generalized energy function as well as the holonomic constraints and momentum maps corresponding to symmetries of the system. We study the numerical performance of the newly devised scheme in representative examples for multibody dynamics and rigid body rotations.

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Section: Discussionmentioning

confidence: 72%

Section: Present Contributionsmentioning

confidence: 72%

Section: Non-standard Mass Matricesmentioning

confidence: 99%

Section: Present Contributionsmentioning

confidence: 99%

See 2 more Smart Citations

Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics

Kinon,

Betsch

2024

Preprint

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“…From relation (4c), it becomes visible that the Lagrange multipliers 𝑝 emerge as the conjugate momenta. For further details on Livens principle and an extension to constraint mechanical systems, see [9,10]. Furthermore, (4d) exposes that 𝑆 can be viewed as the work-conjugated stresses corresponding to the strains, which are computed as the gradient of the internal energy.…”

Section: Governing Equationsmentioning

confidence: 99%

Discrete nonlinear elastodynamics in a port‐Hamiltonian framework

Kinon,

Thoma,

Betsch

et al. 2023

Proc Appl Math and Mech

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We provide a fully nonlinear port‐Hamiltonian formulation for discrete elastodynamical systems as well as a structure‐preserving time discretization. The governing equations are obtained in a variational manner and represent index‐1 differential algebraic equations. Performing an index reduction, one obtains the port‐Hamiltonian state space model, which features the nonlinear strains as an independent state next to position and velocity. Moreover, hyperelastic material behavior is captured in terms of a nonlinear stored energy function. The model exhibits passivity and losslessness and has an underlying symmetry yielding the conservation of angular momentum. We perform temporal discretization using the midpoint discrete gradient, such that the beneficial properties are inherited by the developed time stepping scheme in a discrete sense. The numerical results obtained in a representative example are demonstrated to validate the findings.

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Variational integrators and graph-based solvers for multibody dynamics in maximal coordinates

Brüdigam,

Sosnowski,

Manchester

et al. 2023

Multibody Syst Dyn

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Multibody dynamics simulators are an important tool in many fields, including learning and control in robotics. However, many existing dynamics simulators suffer from inaccuracies when dealing with constrained mechanical systems due to unsuitable integrators with bad energy behavior and problematic constraint violations, for example in contact interactions. Variational integrators are numerical discretization methods that can reduce physical inaccuracies when simulating mechanical systems, and formulating the dynamics in maximal coordinates allows for easy and numerically robust incorporation of constraints such as kinematic loops or contacts. Therefore, this article derives a variational integrator for mechanical systems with equality and inequality constraints in maximal coordinates. Additionally, efficient graph-based sparsity-exploiting algorithms for solving the integrator are provided and implemented as an open-source simulator. The evaluation of the simulator shows improved physical accuracy due to the variational integrator and the advantages of the sparse solvers. Comparisons to minimal-coordinate algorithms show improved numerical robustness, and application examples of a walking robot and an exoskeleton with explicit constraints demonstrate the necessity and capabilities of maximal coordinates.

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The GGL variational principle for constrained mechanical systems

Cited by 5 publications

References 26 publications

Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics

Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics

Discrete nonlinear elastodynamics in a port‐Hamiltonian framework

Variational integrators and graph-based solvers for multibody dynamics in maximal coordinates

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