Graphics Reference
In-Depth Information
Of course, not all possible state vectors are physically feasible. In the cloth
example, self-intersection of the cloth and deformations that imply the cloth is
shredded are excluded. Constraints on the deformation dynamics have to be in-
cluded that limit the set of possible states to those that are reasonable. In fact,
the set of state vectors attainable in an particular simulation is much smaller than
the set of feasible states simply because the system starts out in a particular initial
state and there are limits to the possible deformation forces applied in the simu-
lation. For example, if only the door is allowed to move, the cloth attached to the
door can flap around but will not tie itself into knots.
Dynamic simulation is done in a set of discrete time steps. For simplicity, the
time between adjacent time steps is constant. A basic deformation describes a
discrete change of state from time
t
to time
t
1. The factors that affect the defor-
mation, which includes such things as forces on the object at various points, are
collected into a vector of system parameters
+
t
. The change of state is described
α
x
t
+
1
x
t
t
=
(
,
α
)
abstractly by a function
f
,as
have to
be defined specifically in relation to the deformation model. The function
f
is an
abstract representation of an algorithm used to compute the change in state given
a particular set of parameters. But because it is a true function, its values can be
precomputed for a collection of representative states, and values at other states
can be interpolated from these precomputed values.
For precomputed values, the authors define a
data-driven state space
consist-
ing of a set of state vectors arising from certain specific deformations. The state
space is modeled as a set of state vector nodes together with a set of directional
connections (arcs or edges, in graph-theory terminology) joining the nodes that
represent transitions. A node
f
. The system parameters
α
x
2
is connected to another node
x
1
if, for some set
of parameters (forces)
. That is, each node is connected to all
other nodes that it can reach in a single time step by a feasible action. The full
state space is vastly too large to be precomputed or stored, so the authors consider
a subspace of precomputed state vectors that arise from basic deformations. A
sequence of connected state vectors is called an
orbit
. An orbit can be regarded
as a “motion clip” because it records the shape of the object under a specific
deformation sequence across a sequence of time steps. A set of precomputed rep-
resentative orbits can be reused during the dynamic simulation; “nearby” orbits
can be used if the state vectors and forces are sufficiently close.
α
,
x
2
=
f
(
x
1
,
α
)
10.3.2 Impulse Response Function
A set of arbitrary precomputed orbits are not likely to be sufficient in itself to be
useful in dynamic simulation. Without a way of handling user control at run time,
the simulation would not be truly dynamic—it would reduce to simply replaying
