Graphics Reference
In-Depth Information
FIGURE 7.27
Frames from a sequence of constrained dynamics. The large sphere's motion is driven by outside forces; the
small spheres are repositioned to satisfy distance constraints.
reestablish the distance constraint with respect to a particle already processed.
Figure 7.27
shows
frames from a simple example.
When an impulse force is applied to the torso of a figure, the torso reacts to the force as an inde-
pendent rigid body. The appendages react to the force by enforcing distance constraints one link at a
time traversing the hierarch from the root outwards as shown in
Figure 7.28
.
In reality, however, as the torso reacts to the applied force, the appendages exert forces and torques
on the torso as it tries to move away from them. The full equations of motion must account for the
dynamic interactions among all the links. The
Featherstone equations
do this.
The Featherstone equations
The actual forces of one link acting on adjacent links can be explicitly calculated using basic principles of
physics. The effect of one link on any attached links must be computed. The equations presented here are
referred to as the Featherstone equations of motion [
5
]
[
14
]
[
15
]. In a nutshell, the algorithm uses four
hierarchy-traversal loops: initializing link velocities (from root outward), initializing link values (from root
outward), updating values (from end effector inward), and computing accelerations (from root outward).
The notation to relate one link to the next is shown in
Figure 7.29
.
A
spatial notation
, which combines the angular and linear components, has been developed to make
v
v
ΒΌ
(7.79)
Similarly,
the spatial acceleration vector is made of angular and linear acceleration terms
(
Eq. 7.80
).
FIGURE 7.28
Sequence of impulse force applied to linked figure. Motion is generated by constrained dynamics.
(Image courtesy of Domin Lee.)
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