Physically Based Animation - Computer Animation: Algorithms and Techniques - page 243

Cloth response to collisions does not follow the rigid body paradigm. The collision can be viewed as
elastic in that the cloth will change shape as a result. It is really a damped local inelastic collision, but rigid
body computations are usually too expensive, especially for real-time applications, and even for off-line
simulations. The response to collision is typically to constrain the position and/or velocity of the colliding
vertices. This will constrain the motion of the colliding cloth to be on the surface of the body. This avoids
treating periods of extended contact with a collision detection-collision response cycle.
Folds are a particular visual feature of cloth that are important to handle in a believable way. In the
case of spring-damper models of cloth, there is typically no compression because the cloth easily bends,
producing folds and wrinkles. It is typical to incorporate some kind of bending spring in the cloth
model. Choi and Koh [ 31 ] have produced impressive cloth animation by special handling of what they
call the post-buckling instability together with the use of implicit numerical integration. A key feature
of their approach is the use of a different energy function for stretching than for compression.

l 3
ðl 1 þ l 2

l 3

F b ¼ k b

Þ

(7.91)

ðl 1 þ l 2

7.6 Enforcing soft and hard constraints
One of the main problems with using physically based animation is for the animator to get the object to
do what he or she wants while having it react to the forces modeled in the environment. One way to
solve this is to place constraints on the object that restrict some subset of the degrees of freedom (DOF)
of the object's motion. The remaining DOF are subject to the physically based animation system. Con-
straints are simply requirements placed on the object. For animation, constraints can take the form of
colocating points, maintaining a minimum distance between objects, or requiring an object to have a
certain orientation in space. The problem for the animation system is in enforcing the constraints while
the object's motion is controlled by some other method.
If the constraints are to be strictly enforced, then they are referred to as hard constraints . If the con-
straints are relations the system should only attempt to satisfy, then they are referred to as soft constraints .
Satisfying hard constraints requires more sophisticated numerical approaches than satisfying soft con-
straints. To satisfy hard constraints, computations are made that search for a motion that reacts to forces
in the system while satisfying all of the constraints. As more constraints are added to the system, this
becomes an increasingly difficult problem. Soft constraints are typically incorporated into a system as
additional forces that influence the final motion. One way to model flexible objects is to create soft dis-
tance constraints between the vertices of an object. These constraints, modeled by a mesh of intercon-
nected springs and dampers, react to other forces in the system such as gravity and collisions to create a
dynamic structure. Soft constraint satisfaction can also be phrased as an energy minimization problem in
which deviation from the constraints increases the system's energy. Forces are introduced into the system
that decrease the system's energy. These approaches are discussed in the following section.

7.6.1 Energy minimization
The concept of a system's energy can be used in a variety of ways to control the motion of objects. Used
in this sense, energy is not defined solely as physically realizable, but it is free to be defined in whatever
form serves the animator. Energy functions can be used to pin objects together, to restore the shape of
objects, or to minimize the curvature in a spline as it interpolates points in space.