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36
Stochastic variational integrators
 IMA Journal of Numerical Analysis Advance
, 2008
"... This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. The main result is to derive stochastic governing equations for such systems from a critical point of a stochastic action. Using this result the paper derives Langevintype equations for ..."
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Cited by 26 (1 self)
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This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. The main result is to derive stochastic governing equations for such systems from a critical point of a stochastic action. Using this result the paper derives Langevintype equations for constrained mechanical systems and implements a stochastic analog of Lagrangian reduction. These are easy consequences of the fact that the stochastic action is intrinsically defined. Stochastic variational integrators (SVIs) are developed using a discretized stochastic variational principle. The paper shows that the discrete flow of an SVI is a.s. symplectic and in the presence of symmetry a.s. momentummap preserving. A firstorder meansquare convergent SVI for mechanical systems on Lie groups is introduced. As an application of the theory, SVIs are exhibited for multiple, randomly forced and torqued rigidbodies interacting via a potential. 1
Error analysis of variational integrators of unconstrained Lagrangian systems
, 2008
"... Due to a singularity at zero timestep, existence and uniqueness, and accuracy, of variational integrators, cannot be established by straightforward use of the implicit function theorem. We show existence and uniqueness for variational discretizations by blowing up the variational principle. The sin ..."
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Cited by 16 (3 self)
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Due to a singularity at zero timestep, existence and uniqueness, and accuracy, of variational integrators, cannot be established by straightforward use of the implicit function theorem. We show existence and uniqueness for variational discretizations by blowing up the variational principle. The singularity implies an accuracy one less than is observed in simulations, a deficit that is recovered by a past–future symmetry at zero timestep. 1
Energy Stability and Fracture for Frame Rate Rigid Body Simulations
"... Our goal is to design robust algorithms that can be used for building realtime systems, but rather than starting with overly simplistic particlebased methods, we aim to modify higherend visual effects algorithms. A major stumbling block in utilizing these visual effects algorithms for realtime s ..."
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Cited by 12 (2 self)
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Our goal is to design robust algorithms that can be used for building realtime systems, but rather than starting with overly simplistic particlebased methods, we aim to modify higherend visual effects algorithms. A major stumbling block in utilizing these visual effects algorithms for realtime simulation is their computational intensity. Physics engines struggle to fully exploit available resources to handle high scene complexity due to their need to divide those resources among many smaller time steps, and thus to obtain the maximum spatial complexity we design our algorithms to take only one time step per frame. This requires addressing both accuracy and stability issues for collisions, contact, and evolution in a manner significantly different from a typical simulation in which one can rely on shrinking the time step to ameliorate accuracy and stability issues. In this paper we present a novel algorithm for conserving both energy and momentum when advancing rigid body orientations, as well as a novel technique for clamping energy gain during contact and collisions. We also introduce a technique for fast and realistic fracture of rigid bodies using a novel collisioncentered prescoring algorithm.
Lie group integrators for animation and control of vehicles
 ACM TRANSACTIONS ON GRAPHICS
, 2009
"... This paper is concerned with the animation and control of vehicles with complex dynamics such as helicopters, boats, and cars. Motivated by recent developments in discrete geometric mechanics we develop a general framework for integrating the dynamics of holonomic and nonholonomic vehicles by preser ..."
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Cited by 12 (3 self)
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This paper is concerned with the animation and control of vehicles with complex dynamics such as helicopters, boats, and cars. Motivated by recent developments in discrete geometric mechanics we develop a general framework for integrating the dynamics of holonomic and nonholonomic vehicles by preserving their statespace geometry and motion invariants. We demonstrate that the resulting integration schemes are superior to standard methods in numerical robustness and efficiency, and can be applied to many types of vehicles. In addition, we show how to use this framework in an optimal control setting to automatically compute accurate and realistic motions for arbitrary userspecified constraints.
Spectral variational integrators
"... In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving. Furthermore, we prove that und ..."
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Cited by 5 (3 self)
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In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving. Furthermore, we prove that under appropriate assumptions, variational integrators constructed using Galerkin techniques will yield numerical methods that are in a certain sense optimal, converging at the same rate as the best possible approximation in a certain function space. We further prove that certain geometric invariants also converge at an optimal rate, and that the error associated with these geometric invariants is independent of the number of steps taken. We close with several numerical examples that demonstrate the predicted rates of convergence.
A Lie Group Variational Integrator for Rigid Body Motion in SE(3) with Applications to Underwater Vehicle Dynamics
"... Abstract — The topic of variational integrators for mechanical systems whose dynamics evolve on nonlinear spaces has seen strong growth recently. Within this class of variational integrators is the subclass of Lie group variational integrators that can be used for mechanical systems whose dynamics e ..."
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Cited by 4 (4 self)
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Abstract — The topic of variational integrators for mechanical systems whose dynamics evolve on nonlinear spaces has seen strong growth recently. Within this class of variational integrators is the subclass of Lie group variational integrators that can be used for mechanical systems whose dynamics evolve on Lie groups. This class of mechanical systems includes all systems that can be modeled as rigid bodies or connections of rigid bodies. In this paper, we present a Lie group variational integrator for the full (translation and orientation) motion of a rigid body under the possible influence of nonconservative forces and torques. We use a discretization scheme for such systems which is based on the discrete Lagranged’Alembert principle to obtain the Lie group variational integrator. We apply the composition of the Lie group variational integrator with its adjoint and a CrouchGrossman method to the example of a conservative underwater system. We show numerically that with respect to energy these manifold methods, as expected, behave as a symplectic integrator and a nonsymplectic integrator, respectively. I.
Geometric discrete analogues of tangent bundles and constrained Lagrangian systems
, 2008
"... We develop discretizations of the variational principles of holonomic Lagrangian systems, to the generality of discretizing nonholonomic mechanical systems with nonlinear constraints. The development is based on geometric discrete analogues of tangent bundles, systematically obtained by extending ta ..."
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Cited by 4 (2 self)
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We develop discretizations of the variational principles of holonomic Lagrangian systems, to the generality of discretizing nonholonomic mechanical systems with nonlinear constraints. The development is based on geometric discrete analogues of tangent bundles, systematically obtained by extending tangent vectors to finite curve segments. 1
GEOMETRIC DISCRETIZATION OF NONHOLONOMIC SYSTEMS WITH SYMMETRIES
"... Abstract. The paper develops discretization schemes for mechanical systems for integration and optimization purposes through a discrete geometric approach. We focus on systems with symmetries, controllable shape (internal variables), and nonholonomic constraints. Motivated by the abundance of import ..."
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Cited by 4 (0 self)
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Abstract. The paper develops discretization schemes for mechanical systems for integration and optimization purposes through a discrete geometric approach. We focus on systems with symmetries, controllable shape (internal variables), and nonholonomic constraints. Motivated by the abundance of important models from science and engineering with such properties, we propose numerical methods specifically designed to account for their special geometric structure. At the core of the formulation lies a discrete variational principle that respects the structure of the state space and provides a framework for constructing accurate and numerically stable integrators. The dynamics of the systems we study is derived by vertical and horizontal splitting of the variational principle with respect to a nonholonomic connection that encodes the kinematic constraints and symmetries. We formulate a discrete analog of this principle by evaluating the Lagrangian and the connection at selected points along a discretized trajectory and derive discrete momentum equation and discrete reduced EulerLagrange equations resulting from the splitting of this principle. A family of nonholonomic integrators that are general, yet simple and easy to implement, are then obtained and applied to two examplesthe steered robotic car and the snakeboard. Their numerical advantages are confirmed through comparisons with standard methods. 1. Introduction. The
DISCRETE DIRAC STRUCTURES AND VARIATIONAL DISCRETE DIRAC MECHANICS
"... Abstract. We construct discrete analogues of Dirac structures by considering the geometry of symplectic maps and their associated generating functions, in a manner analogous to the construction of continuous Dirac structures in terms of the geometry of symplectic vector fields and their associated H ..."
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Cited by 3 (3 self)
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Abstract. We construct discrete analogues of Dirac structures by considering the geometry of symplectic maps and their associated generating functions, in a manner analogous to the construction of continuous Dirac structures in terms of the geometry of symplectic vector fields and their associated Hamiltonians. We demonstrate that this framework provides a means of deriving implicit discrete Lagrangian and Hamiltonian systems, while incorporating discrete Dirac constraints. In particular, this yields implicit nonholonomic Lagrangian and Hamiltonian integrators. We also introduce a discrete Hamilton–Pontryagin variational principle on the discrete Pontryagin bundle, which provides an alternative derivation of the same set of integration algorithms. In so doing, we explicitly characterize the discrete Dirac structures that are preserved by Hamilton–Pontryagin integrators. In addition to providing a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of Dirac mechanics, it provides a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators. Since discrete Lagrangians and discrete Hamiltonians are essentially generating functions of different types, the theoretical framework described in this paper is sufficiently general to encompass all possible Dirac integrators through an appropriate