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Geometric, Variational Integrators for Computer Animation

Eurographics/ACM SIGGRAPH Symposium on Computer Animation (2006) Cani, J. O Brien (Editors) geometric , Variational Integrators for Computer AnimationL. Kharevych Weiwei Y. Tong E. Kanso J. E. Marsden P. Schr der M. DesbrunCaltech - USCA bstractWe present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems an im-portant computational tool at the core of most physics-based Animation techniques. Several features make thisparticular time integrator highly desirable for Computer Animation : it numerically preserves important invariants,such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energybehavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quitesimply; finally, our simple methodology allows for the design of high-order accurate schemes if needed.

Kharevych et al. / Geometric, Variational Integrators for Computer Animation tional formulation of mechanics we mentioned above, pro-viding a solution for most ordinary and partial differential

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Transcription of Geometric, Variational Integrators for Computer Animation

1 Eurographics/ACM SIGGRAPH Symposium on Computer Animation (2006) Cani, J. O Brien (Editors) geometric , Variational Integrators for Computer AnimationL. Kharevych Weiwei Y. Tong E. Kanso J. E. Marsden P. Schr der M. DesbrunCaltech - USCA bstractWe present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems an im-portant computational tool at the core of most physics-based Animation techniques. Several features make thisparticular time integrator highly desirable for Computer Animation : it numerically preserves important invariants,such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energybehavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quitesimply; finally, our simple methodology allows for the design of high-order accurate schemes if needed.

2 Two keyproperties set the method apart from earlier approaches. First, the nonlinear equations that must be solved duringan update step are replaced by a minimization of a novel functional, speeding up time stepping by more than afactor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in theimplementation of the method. These properties are achieved using a discrete form of a general Variational princi-ple called the Pontryagin-Hamilton principle, expressing time integration in a geometric manner. We demonstratethe applicability of our Integrators to the simulation of non-linear elasticity with implementation IntroductionMathematical models of the evolution in time of dynamicalsystems (whether in biology, economics, or Computer ani-mation) generally involve systems of differential physical system means figuring out how to movethe system forward in time from a set of initial conditions,allowing the computation of, for instance, the trajectory ofa ball ( , its position as a function of time) thrown upin the air.

3 Although this example can easily be solved an-alytically, direct solutions of the differential equations gov-erning a system are generally hard or impossible we needto resort to numerical techniques to find a discrete tempo-ral description of a motion. Consequently, there has beena significant amount of research in applied mathematics onhow to deal with some of the most useful systems of equa-tions, leading to a plethora of numerical schemes with var-ious properties, orders of accuracy, and levels of complex-ity of implementation [PFTV92]. In Computer Animation ,these Integrators are crucial computational tools at the coreof most physics-based Animation techniques, and classicalmethods (such as fourth-order Runge-Kutta, implicit Euler,and more recently the Newmark scheme) have been meth-ods of choice in practice [Par01].

4 Surprisingly, developingbetter ( , faster and/or more reliable) Integrators receivedvery little attention in our community, even if the few papersdedicated to this goal showed encouraging results [HES03].In this paper, we follow ageometric instead of a tradi-tional numerical-analytic approach to the problem of timeintegration. Motivated by the success of discrete variationalapproaches in geometric modeling and discrete differentialgeometry, we will consider mechanics from a variationalpoint of view. The very essence of a mechanical system isindeed characterized by itssymmetriesandinvariants( ,momenta), thus preserving these geometric notions into thediscrete computational setting is of paramount importance ifone wants discrete time integration to properly capture theunderlying continuous motion.

5 Consequently, we advocatethe use ofdiscrete Variational principles as a way to derivesimple, robust, and accurate time Integrators . In particular,we derive a novel, simple geometric integrator based on thevery general Hamilton-Pontryagin BackgroundDynamics as a Variational ProblemConsidering mechan-ics from a Variational point of view goes back to Euler, La-grange and Hamilton. The form of the Variational principlemost important for continuous mechanics is due to Hamil-ton, and is often calledHamilton s principleor theleast ac-tion principle(as we will see later, this is a bit of a mis-nomer: stationary action principle would be more correct):it states that a dynamical system always finds an optimalcourse from one position to another (a more formal defini-tion will be presented in Section2).

6 One consequence is thatwe can recast the traditional way of thinking about an objectaccelerating in response to applied forces, into a geometricviewpoint. There the path followed by the object hasoptimalgeometric properties analogous to the notion of geodesicson curved surfaces. This point of view is equivalent to New-ton s laws in the context of classical mechanics , but is broadenough to encompass areas ranging to E&M and Integratorsare a class of numerical time-stepping methods that exploit the geometric structure of me-chanical systems [HLW02]. Of particular interest within thisclass, Variational Integrators [MW01] discretize the varia-c The Eurographics Association geometric , Variational Integrators for Computer Animationtional formulation of mechanics we mentioned above, pro-viding a solution for most ordinary and partial differentialequations that arise in mechanics .

7 While the idea of dis-cretizing Variational formulations of mechanics is standardfor elliptic problems using Galerkin Finite Element methodsfor instance, only recently did it get used to derive variationaltime-stepping algorithms for mechanical systems. This ap-proach allows the construction of Integrators with any orderof accuracy [Wes03,Lew03], that can handle constraints aswell as external forcing. These Integrators have been shownremarkably powerful for simulations of physical phenom-ena when compared to traditional numerical time steppingmethods [KMOW00]. This discrete- geometric framework isthus versatile, powerful, and general. For example, the well-known symplectic variant of the Newmark scheme (veloc-ity Verlet) can best be elucidated by writing it as a varia-tional integrator [Wes03].

8 Of particular interest in computeranimation, the simplest Variational integrator can be imple-mented by taking two consecutive positionsq0=q(t0)andq1=q(t0+dt)of the system to compute the next positionq2. Repeating this process calculates an entire discrete (intime) vs. Qualitative IntegratorsWhile it is unavoid-able to make approximations in numerical algorithms ( , todiffer from the continuous equivalent), the matter becomeswhether the numerics can provide satisfactory of phenomena is often favored in com-puter Animation over absoluteaccuracy. We argue in thefollowing that one does not have to ask foreitherplausi-bilityoraccuracy. In fact, we seek a simple method robustenough to provide good, qualitative simulations that canalsobe easily rendered arbitrarily accurate.

9 The simplectic char-acter of Variational Integrators provides good foundationsfor the design of robust algorithms: this property guaran-teesgood statistical predictabilitythrough accurate preser-vation of thegeometricproperties of the exact flow of thedifferential equations. As a consequence, symplecticity of-fers long-time energy preservation a crucial property sincelarge energy increase is often synonymous with numericaldivergence while a large decrease dampens the motion, de-creasing visual plausibility. A well-known example wherethis property is crucial is the simple pendulum (particularlyrelevant in robotic applications for articulated figures), forwhich other (even high-order) Integrators can fail in keep-ing the amplitude of the oscillations (see Figure1).

10 Withthis in mind, we will pursue numerical schemes which offerqualitatively-correctas well asarbitrarily accurate ContributionsWe address the problem of discrete time integration as adiscrete geometric problemwhere the dynamics is obtainedfrom a (stationary action)Hamilton-Pontryagin principle, , as the stationary point of a discrete action. Using theHamilton-Pontryagin principle provides conceptual and al-gorithmic simplicity even for dissipative systems and inthe presence of constraints. Computationally, our novel ap-Figure 1:Advantages of symplecticity: for the equation of motion ofa pendulum of length L in a gravitation field g (left), the usual ex-plicit Euler integrator amplifies oscillations, the implicit one damp-ens the motion, while asymplecticintegrator perfectly captures theperiodic nature of the pendulum (see [SD06] for details).


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