Computational Geometric Optimal Control of Rigid Bodies

Lee, Taeyoung; Leok, Melvin; McClamroch, N. Harris

doi:10.4310/cis.2008.v8.n4.a5

Cited by 57 publications

(107 citation statements)

References 30 publications

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“…These two methods have been unified to obtain a Lie group variational integrator for Lagrangian/ Hamiltonian systems evolving on a Lie group [19]. This geometric integrator preserves symplecticity and group structure of those systems concurrently.…”

Section: Lie Group Variational Integratormentioning

confidence: 99%

“…Therefore, the variation of the rotation matrix should be consistent with the geometry of the special orthogonal group. In [18,19], it is expressed in terms of the exponential map as…”

Section: Variation Of G S Frommentioning

confidence: 99%

“…The resulting discrete-time Lagrangian system, referred to as a variational integrator, approximates the Euler-Lagrange equations to the same order of accuracy as the discrete Lagrangian approximates the Jacobi solution [19,22]. We construct a discrete Lagrangian for the string pendulum using the trapezoidal rule.…”

Section: Discrete Lagrangianmentioning

confidence: 99%

“…The Lagrangian of the string pendulum is derived, and the corresponding action integral is defined. Due to the unique dynamic characteristics of the string pendulum, the variation of the action integral should be carefully developed: (i) since the unstretched length of the deployed portion of the string is not fixed, when deriving the variation of the corresponding part of the action integral, we need to apply Green's theorem; (ii) since the attitude of the rigid body is represented in the special orthogonal group, the variation of rotation matrices are carefully expressed by using the exponential map [18,19]; (iii) since the portion of the string on the guide way and the drum is inextensible, the velocity of the string is not continuous at the guide way entrance. To take account of the effect of this velocity discontinuity, an additional modification term, referred to as a Carnot energy loss term is incorporated [15].…”

Section: Lagrangianmentioning

confidence: 99%

“…When numerically simulating such systems, it is critical to preserve both the symplectic property of Hamiltonian flows and the Lie group structure for numerical accuracy and efficiency [17]. A geometric numerical integrator, referred to as a Lie group variational integrator, has been developed for a Hamiltonian system on an arbitrary Lie group and it has been applied to several multibody systems ranging from binary asteroids to articulated rigid bodies and magnetic systems in [18,19]. This paper develops a Lie group variational integrator for the proposed string pendulum model.…”

Section: Introductionmentioning

confidence: 99%

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Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

2010

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A high fidelity model is developed for an elastic string pendulum, one end of which is attached to a rigid body while the other end is attached to an inertially fixed reel mechanism which allows the unstretched length of the string to be dynamically varied. The string is assumed to have distributed mass and elasticity that permits axial deformations. The rigid body is attached to the string at an arbitrary point, and the resulting string pendulum system exhibits nontrivial coupling between the elastic wave propagation in the string and the rigid body dynamics. Variational methods are used to develop coupled ordinary and partial differential equations of motion. Computational methods, referred to as Lie group variational integrators, are then developed, based on a finite element approximation and the use of variational methods in a discrete-time setting to obtain discrete-time equations of motion. This approach preserves the geometry of the configurations, and leads to accurate and efficient algorithms that have guaranteed accuracy properties that make them suitable for many dynamic simulations, especially over long simulation times. Numerical results are presented for typical examples involving a constant length string, string deployment, and string retrieval. These demonstrate the complicated dynamics that arise in a string pendulum from the interaction of the rigid body motion, elastic wave dynamics in the string, and the disturbances introduced by the reeling mechanism. Such interactions are dynamically important in many engineering problems, but tend be obscured in lower fidelity models.

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Section: Lie Group Variational Integratormentioning

confidence: 99%

“…Therefore, the variation of the rotation matrix should be consistent with the geometry of the special orthogonal group. In [18,19], it is expressed in terms of the exponential map as…”

Section: Variation Of G S Frommentioning

confidence: 99%

Section: Discrete Lagrangianmentioning

confidence: 99%

Section: Lagrangianmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

2010

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Asynchronous variational Lie group integration for geometrically exact beam dynamics

Demoures

Gay–Balmaz

Leitz

et al. 2013

Proc Appl Math and Mech

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For the elastodynamic simulation of a spatially discretized beam, asynchronous variational integrators (AVI) offer the possibility to use different time steps for every element [1]. They are symplectic and conserve discrete momentum maps and since the presented integrator for geometrically exact beam dynamics [2] is derived in the Lie group setting (SO (3) for the representation of rotational degrees of freedom), it intrinsically preserves the group structure without the need for constraints [3]. A decrease of computational cost is to be expected in situations, where the time steps have to be very low in certain parts of the beam but not everywhere, e.g. if some regions of the beam are moving faster than others. The implementation allows synchronous as well as asynchronous time stepping and shows very good energy behaviour, i.e. there is no drift of the total energy for conservative systems. Discrete variational mechanicsThe theory of discrete variational mechanics has its roots in the optimal control literature of the 1960's. The past ten years have seen a major development of discrete variational mechanics and corresponding numerical integrators, largely due to pioneering work by Jerrold Marsden and his collaborators, e.g. in [4]. The discrete Lagrangian L d approximates the action in a time interval [t j−1 , t j ]. For linear vector spaces, i.e. q ∈ R n , this leads to the discrete action sumand the discrete variational principle δS d = 0 results in the discrete Euler-Lagrange equationswhich are symplectic and conserve discrete momentum maps due to the variational derivation [4]. Asynchronous variational integrator for the beamAsynchronous variational integrators (AVI), as described by Lew et. al. [1], offer the possibility to use different time steps for every element of the spatial discretization. The use of AVI promises less computational costs by increasing the time step sizes for slowly moving parts of the beam. We review the kinematic description of the geometrically exact beam model in the ambient space R 3 presented in [2]. The configuration of a beam is defined by specifying the position of its line of centroids by means of a map φ : [0, L] → R 3 , and the orientation of cross-sections at points φ(S) by means of a moving basis {d 1 (S), d 2 (S), d 3 (S)} attached to the cross section. The orientation of the moving basis is described with the help of an orthogonal transformation Λ : [0, L] → SO(3) such that d I (S) = Λ(S)E I , I = 1, 2, 3where {E 1 , E 2 , E 3 } is a fixed basis referred to as the material frame. The configuration of the beam is thus completely determined by the maps φ and Λ in the configuration spacewith the Lie group SE (3) being the Euclidean group. If we take into account that the thickness of the rod is small compared to its length and that the material is homogenous and isotropic, we can consider the stored energy to be given by a quadratic

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