Dangers of two-point holonomic constraints for variational integrators

Johnson, Elliot R.; Murphey, Todd D.

doi:10.1109/acc.2009.5160488

Cited by 3 publications

(4 citation statements)

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“…A set of holonomic constraints of the form h i (q) = 0, is incorporated in the integrator at the endpoints as h i (q k+1 ) = 0 [18]. As the DEL equations are only in terms of configurations, the constrained DEL equations ensure that holonomic constraints are exactly satisfied at every time step.…”

Section: A Numerical Properties Of the Integratormentioning

confidence: 99%

Extending filter performance through structured integration

Schultz

Murphey

2014

2014 American Control Conference

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Estimation and filtering form an important component of most modern control systems. Techniques such as extended Kalman filters and particle filters have been successfully utilized for estimation in many different applications. Integrators derived from discrete mechanics possess desirable numerical properties such as stable long-time energy behavior, exact constraint satisfaction, and accurate statistical calculations. In the present work, we leverage these features by utilizing a variational integrator derived from discrete mechanics within extended Kalman filters and particle filters. By filtering real experimental data from the nonlinear, underactuated planar crane problem we demonstrate that the linearizations available through the discrete mechanics framework increase the accuracy of uncertainty estimates provided by an extended Kalman filter, especially when operating at low frequencies. Additionally, we illustrate situations where particle filter performance is increased through the statistics-preserving properties provided by the variational integrator.

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Section: A Numerical Properties Of the Integratormentioning

confidence: 99%

Extending filter performance through structured integration

Schultz

Murphey

2014

2014 American Control Conference

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“…The nice part of the story is that the integrators derived in this way naturally preserve (or nearly preserve) the quantities that are preserved in the continuous framework, the symplectic form, the total energy (for conservative systems) and, in presence of symmetries, the linear and/or angular momentum (for more details, see [36]). Furthermore, other aspects of the continuous theory can be "easily" adapted, symmetry reduction [7,10,22], constraints [23,25], control forces [8,39], etc.…”

Section: Discrete Mechanics and Variationalmentioning

confidence: 99%

“…In the simplest of the cases, that is, the case where one considers a Lagrangian of the form kinetic minus potential energy, L(q, q) = 1 2 qT M q − U (q), with M a constant mass matrix; s = 2 micro-nodes (inner-stages); and Lobatto's quadrature, c 1 = 0, c 2 = 1; one may show that both schemes, spRK (23) and sG (29), reduce to the well-known leap-frog or Verlet method. They will differ when the previous main assumptions are violated, for instance if M is not constant or the quadrature is other than Lobatto's.…”

Section: High Order Variational Integrators High Order Variational In...mentioning

confidence: 99%

“…Example 3.1. We consider a Lagrangian with a scalar mass matrix dependent on the configuration, that is, a Lagrangian of the form L(q, q) = 1 2 λ(q) q 2 − V (q), with λ : Q → R. Under this assumption and noting λ 1/2 := λ0+λ1 2 , (∇)λ i := (∇)λ(q i ), (∇)V i := (∇)V (q i ), i = 0, 1, the spRK scheme (23) as well as the the sG scheme (29) reduce to…”

Section: High Order Variational Integrators High Order Variational In...mentioning

confidence: 99%

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High order variational integrators in the optimal control of mechanical systems

Campos¹,

Ober‐Blöbaum²,

Trélat³

2015

Discrete &Amp; Continuous Dynamical Systems - A

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International audienceIn recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimensional optimal control problems posed with mechanical systems. The convergence of state and control variables of the approximated problem is shown. Furthermore, by analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that, for these particular integrators, dualization and discretization commute

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High Order Variational Integrators: A Polynomial Approach

Campos

2014

Advances in Differential Equations and Applications

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We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the structural properties of these systems, like the symplectic form, the evolution of the momentum maps or the energy behaviour. Also they are easily applicable to optimal control problems based on mechanical systems as proposed in Ober-Blöbaum et al. [2011].Following the same approach, we develop a family of variational integrators to which we refer as symplectic Galerkin schemes in contrast to symplectic partitioned Runge-Kutta. These two families of integrators are, in principle and by construction, different one from the other. Furthermore, the symplectic Galerkin family can as easily be applied in optimal control problems, for which Campos et al.[2012b] is a particular case.2010 Mathematics Subject Classification. Primary 65P10, Secondary 65L06, 65K10, 49Mxx.

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Dangers of two-point holonomic constraints for variational integrators

Cited by 3 publications

References 5 publications

Extending filter performance through structured integration

Extending filter performance through structured integration

High order variational integrators in the optimal control of mechanical systems

High Order Variational Integrators: A Polynomial Approach

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