David Baraff

Large steps in cloth simulation

The bottle-neck in most cloth simulation systems is that time steps must be small to avoid numerical instability. This paper describes a cloth simulation system that can stably take large time steps. The simulation system couples a new technique for enforcing constraints on individual cloth particles with an implicit integration method. The simulator models cloth as a triangular mesh, with internal cloth forces derived using a simple continuum formulation that supports modeling operations such as local anisotropic stretch or compression; a unified treatment of damping forces is included as well. The implicit integration method generates a large, unbanded sparse linear system at each time step which is solved using a modified conjugate gradient method that simultaneously enforces particles’ constraints. The constraints are always maintained exactly, independent of the number of conjugate gradient iterations, which is typically small. The resulting simulation system is significantly faster than previous accounts of cloth simulation systems in the literature.

Fast contact force computation for nonpenetrating rigid bodies

A new algorithm for computing contact forces between solid objects with friction is presented. The algorithm allows a mix of contact points with static and dynamic friction. In contrast to previous approaches, the problem of computing contact forces is not transformed into an optimization problem. Because of this, the need for sophisticated optimization software packages is eliminated. For both systems with and without friction, the algorithm has proven to be considerably faster, simple, and more reliable than previous approaches to the problem. In particular, implementation of the algorithm by nonspecialists in numerical programming is quite feasible.

Analytical methods for dynamic simulation of non-penetrating rigid bodies

A method for analytically calculating the forces between systems of rigid bodies in resting (non-colliding) contact is presented. The systems of bodies may either be in motion or static equilibrium and adjacent bodies may touch at multiple points. The analytic formulation of the forces between bodies in non-colliding contact can be modified to deal with colliding bodies. Accordingly, an improved method for analytically calculating the forces between systems of rigid bodies in colliding contact is also presented. Both methods can be applied to systems with arbitrary holonomic geometric constraints, such as linked figures. The analytical formulations used treat both holonomic and non-holonomic constraints in a consistent manner.

Linear-time dynamics using Lagrange multipliers

Current linear-time simulation methods for articulated figures are based exclusively on reduced-coordinate formulations. This paper describes a general, non-iterative linear-time simulation method based instead on Lagrange multipliers. Lagrange multiplier methods are important for computer graphics applications because they bypass the difficult (and often intractable) problem of parameterizing a system’s degrees of freedom. Given a loop-free set of n equality constraints acting between pairs of bodies, the method takes O.n/ time to compute the system’s dynamics. The method does not rely on matrix bandwidth, so no assumptions about the constraints’ topology are needed. Bodies need not be rigid, constraints can be of various dimensions, and unlike reduced-coordinate approaches, nonholonomic (e.g. velocity-dependent) constraints are allowed. An additional set of k one-dimensional constraints which induce loops and/or handle inequalities can be accommodated with cost O.kn/. This makes it practical to simulate complicated, closedloop articulated figures with joint-limits and contact at interactive rates. A complete description of a sample implementation is provided in pseudocode.

Curved surfaces and coherence for non-penetrating rigid body simulation

A formulation for the contact forces between curved surfaces in resting (non-colliding) contact is presented. In contrast to previous formulations, constraints on the allowable tangential movement between contacting surfaces are not required. Surfaces are restricted to be twice-differentiable surfaces without boundary. Only finitely many contact points between surfaces are allowed; however, the surfaces need not be convex. The formulation yields the contact forces between curved surfaces and polyhedra as well. Algorithms for performing collision detection during simulation on bodies composed of both polyhedra and strictly convex curved surfaces are also presented. The collision detection algorithms exploit the geometric coherence between successive time steps of the simulation to achieve efficient running times.

Dynamic simulation of non-penetrating flexible bodies

A model for the dynamic simulation of flexible bodies subject to non-penetration constraints is presented. Flexible bodies are described in terms of global deformations of a rest shape. The dynamical behavior of these bodies that most closely matches the behavior of ideal continuum bodies is derived, and subsumes the results of earl ier Lagrangian dynamics-based models. The dynamics derived for the flexible-body model allows the unification of previous work on flexible body simulation and previous work on non-penetrating rigid body simulation. The non-penetration constraints for a system of bodies that contact at multiple points are maintained by analytically calculated contact forces. An implementation for first- and second-order polynomially deformable bodies is described. The simulation of second-order or higher deformations currently involves a polyhedral boundary approximation for collision detection purposes,

Issues in computing contact forces for non-penetrating rigid bodies

In rigid-body simulation it is necessary to compute the forces that arise between contacting bodies to prevent interpenetration. This paper studies the problem of rigid-body simulation when the bodies being simulated are restricted to contact at only finitely many points. Some theoretical and practical issues in computing contact forces for systems with large numbers of contact points are considered. Both systems of rigid bodies with and without Coulomb friction are studied. Complexity results are derived for certain classes of configurations and numerical methods for computing contact forces are discussed.

Coping with friction for non-penetrating rigid body simulation

Algorithms and computational complexity measures for simulating the motion of contacting bodies with friction are presented. The bodies are restricted to be perfectly rigid bodies that contact at finitely many points. Contact forces between bodies must satisfy the Coulomb model of friction. A traditional principle of mechanics is that contact forces are impulsive if and only if non-impulsive contact forces are insufficient to maintain the non-penetration constraints between bodies. When friction is allowed, it is known that impulsive contact forces can be necessary even in the absence of collisions between bodies. This paper shows that computing contact forces according to this traditional principle is likely to require exponential time. An analysis of this result reveals that the principle for when impulses can occur is too restrictive, and a natural reformulation of the principle is proposed. Using the reformulated principle, an algorithm with expected polynomial time behaviour for computing contact forces is presented.

Interactive simulation of solid rigid bodies

The article describes the implementation of an interactive system for simulating rigid bodies with contact and friction. The system can simulate moderately complex mechanical systems at interactive rates (20-30 Hz on low end Silicon Graphics workstations). New objects and user specified constraints can be added into the simulation environment on the fly. The system uses analytical methods to compute contact forces, as opposed to the penalty methods common in other interactive systems. Currently, the systems weakest feature is that it can fail to detect high speed collisions. Objects that move at high speeds (relative to the step size of the simulation) are subject to a form of aliasing and may tunnel through other objects without causing a collision. Other simplifications of the system involve approximating collision times and locations by interpolation methods, and periodic error correction adjustments of geometric tolerances. It is difficult to try to quantify the error incurred by a given approximation or tradeoff. Some of the design choices will likely curtail the systems use for highly predictive applications. However, they do not seriously affect simulating the basic dynamics of a mechanism like a feeder. In general, the system performs with sufficient accuracy and realism to be considered a viable interactive simulation environment. >

Interactive physically-based manipulation of discrete/continuous models

Physically-based modeling has been used in the past to support a variety of interactive modeling tasks including free-form surface design, mechanism design, constrained drawing, and interactive camera control. In these systems, the user interacts with the model by exerting virtual forces, to which the system responds subject to the active constraints. In the past, this kind of interaction has been applicable only to models that are governed by continuous parameters. In this paper we present an extension to mixed continuous/discrete models, emphasizing constrained layout problems that arise in architecture and other domains. When the object being dragged is blocked from further motion by geometric constraints, a local discrete search is triggered, during which transformations such as swapping of adjacent objects may be performed. The result of the search is a “nearby” state in which the target object has been moved in the indicated direction and in which all constraints are satisfied. The transition to this state is portrayed using simple but effective animated visual effects. Following the transition, continuous dragging is resumed. The resulting seamless transitions between discrete and continuous manipulation allow the user to easily explore the mixed design space just by dragging objects. We demonstrate the method in application to architectural floor plan design, circuit board layout, art analysis, and page layout.