Geometric, Variational Integrators for Computer Animation

We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems—an important computational tool at the core of most physics-based animation techniques. Several features make this particular 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 energy behavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quite simply; finally, our simple methodology allows for the design of high-order accurate schemes if needed. Two key properties set the method apart from earlier approaches. First, the nonlinear equations that must be solved during an update step are replaced by a minimization of a novel functional, speeding up time stepping by more than a factor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in the implementation of the method. These properties are achieved using a discrete form of a general variational principle called the Pontryagin-Hamilton principle, expressing time integration in a geometric manner. We demonstrate the applicability of our integrators to the simulation of non-linear elasticity with implementation details

Phase-fitted Discrete Lagrangian Integrators

Phase fitting has been extensively used during the last years to improve the behaviour of numerical integrators on oscillatory problems. In this work, the benefits of the phase fitting technique are embedded in discrete Lagrangian integrators. The results show improved accuracy and total energy behaviour in Hamiltonian systems. Numerical tests on the long term integration (100000 periods) of the 2-body problem with eccentricity even up to 0.95 show the efficiency of the proposed approach. Finally, based on a geometrical evaluation of the frequency of the problem, a new technique for adaptive error control is presented

Formulation and performance of variational integrators for rotating bodies

Variational integrators are obtained for two mechanical systems whose configuration spaces are, respectively, the rotation group and the unit sphere. In the first case, an integration algorithm is presented for Euler’s equations of the free rigid body, following the ideas of Marsden et al. (Nonlinearity 12:1647–1662, 1999). In the second example, a variational time integrator is formulated for the rigid dumbbell. Both methods are formulated directly on their nonlinear configuration spaces, without using Lagrange multipliers. They are one-step, second order methods which show exact conservation of a discrete angular momentum which is identified in each case. Numerical examples illustrate their properties and compare them with existing integrators of the literature

A Discrete Geometric Optimal Control Framework for Systems with Symmetries

This paper studies the optimal motion control of mechanical systems through a discrete geometric approach. At the core of our formulation is a discrete Lagrange-d’Alembert- Pontryagin variational principle, from which are derived discrete equations of motion that serve as constraints in our optimization framework. We apply this discrete mechanical approach to holonomic systems with symmetries and, as a result, geometric structure and motion invariants are preserved. We illustrate our method by computing optimal trajectories for a simple model of an air vehicle flying through a digital terrain elevation map, and point out some of the numerical benefits that ensue

Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete LQR problem to find a locally stabilizing controller for a 40 DOF system with 6 constraints.Comment: 13 page

Asynchronous Variational Contact Mechanics

An asynchronous, variational method for simulating elastica in complex contact and impact scenarios is developed. Asynchronous Variational Integrators (AVIs) are extended to handle contact forces by associating different time steps to forces instead of to spatial elements. By discretizing a barrier potential by an infinite sum of nested quadratic potentials, these extended AVIs are used to resolve contact while obeying momentum- and energy-conservation laws. A series of two- and three-dimensional examples illustrate the robustness and good energy behavior of the method

Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete LQR problem to find a locally stabilizing controller for a 40 DOF system with 6 constraints.Comment: 13 page