Game Development Reference
In-Depth Information
Spin and Orbital Angular Velocity
The spin angular velocity of a rigid body is equal to the orbital angular
velocity of every point on the rigid body.
When talking about orbital angular velocity, we must be clear about
what point o we are measuring the angular velocity relative to. We do not
measure the orbital angular velocity relative to the center of mass c ! We
measure the orbital angular velocity relative to the point that is actually
being orbited, and only those points on the “equator” of the object are
actually orbiting around c . Given any other arbitrary point, it will orbit a
point o that lies on the axis of rotation, as shown in Figure 12.16.
Figure 12.16
The spin angular velocity of the robot
is the same as the orbital velocity of
every point on the robot, provided
that we choose o carefully.
Now, the astute reader may have noticed some circularity in the defini-
tions just given. We said that the spin angular velocity of the rigid body is
equal to the orbital angular velocity of each and every point, provided that
the orbital velocity is measured relative to the closest point on the axis of
rotation. But how did we know the axis of rotation in the first place? The
question is typically moot because, both in analytical kinematics equations
and in a digital simulation in a computer, the angular velocity vector ω is
simply one of the fundamental state variables that we track, so we do not
need to infer it from the point velocities. Still, it is worth pointing out that
this axis is uniquely determined (up to the reversal of signs). Remember
that the axis of rotation is perpendicular to the velocity an orbiting parti-
cle. (We must measure the velocity of the particle relative to the center of
Search WWH ::




Custom Search