thesis - Computer Graphics Group - Charles University - Univerzita ...

108 CHAPTER 8. FIGURE LIBRARY
subtleties are considered the last (such as the animation of the hands and head). Systems working
on the dynamics level allow to automatically handle most bottom level animation tasks, that are
laborious to handle on the kinematics level, such as fixing the head to prevent it from tilting and
leaning or make the hands swing naturally in accordance to the animation of arms, prevent feet
from sliding, etc. All these tasks can be handled by imposing soft unbounded or soft bounded
equality constraints, abstracted by lower level controllers or the simulator library 7 .
While unbounded equality constraints do not limit the magnitudes of the restoring constraint
forces due to the constraints, bounded equality constraints impose limits on the constraint DOF
multipliers (limiting the constraint force magnitude) and instruct the constraint solver to “work
as hard as possible to enforce the constraint, but allow the constraint to be violated when it can
not be satisfied”. This is a key distinction. For example if we position an articulated figure to the
front of a wall and tell it to walk against the wall, following a kinematic animation by imposing
unbounded equality constraints and actuating the figure’s root site against the wall, then the
non-penetration constraints due to the wall and the contacting figure segments will “fight” with
the animation constraints and there will be no feasible solution — no constraint force will be
computed and all constraints will break (non-penetration will not be enforced, figure limbs will
be torn off the torso, etc).
The remedy is to impose bounded equality constraints with finite λ lo and λ hi limits instead
of unbounded constraints where appropriate. In the case of motion control, where lower level
controllers formulate constraints to affect the figure motion, our controllers provide an option
to impose bounded equality constraints instead of unbounded equality constraints, allowing the
constraint solver to limit the constraint force (motor force) magnitude. Recalling the example
with a walking figure, if the root joint’s motor force was limited, the non-penetration constraints
due to the wall would “win” and so the solution would exist 8 . However there is one thing that
must be kept in mind — bounded equality constraints can not be primary and so are expensive
to handle.
7 These constraints are described in sections (6.6), (7.4.2) and (6.5.3).
8 The mechanism of dragging rigid bodies by the mouse cursor in the demo application is implemented by
imposing bounded equality constraints as well. This ensures that constraints are feasible even when the body is
dragged against a fixed body.