Game Development Reference
In-Depth Information
12.5
Rotational Dynamics
We are now ready to extend the ideas we have learned about particles to
rigid bodies. Particles have position, but until now we have not concerned
ourselves with orientation. Likewise, particles have mass, but until now
we have not thought about the size or shape of a particle and how that
mass was distributed. The key linear quantities and laws each have ro-
tational analogs, and there is a certain beautiful correspondence between
them. This correspondence is certainly pedagogically convenient and will
be leveraged in our discussion. As we did for linear dynamics, we first
define the basic kinematics quantities and consider those issues related to
describing the rotation without worrying about the causes of the rotation.
We then examine the rotational analogs to mass, force, and momentum,
although we will discuss these topics in a different order.
You might notice that this section is surprisingly brief, both compared
to our discussion of linear matters, and also similar presentations of other
sources. There are two reasons for this. First, we spent considerable time
in the previous chapter building up intuition about derivatives and linear
dynamics, and these ideas need not be repeated here—though there will be
some important differences concerning integration of angular displacement.
Second, there are certain prerequisites that are usually bundled in this
discussion in traditional physics topics; in this topic, it has been more
appropriate to place these prerequisites elsewhere. You should make sure
you have read and understood these prerequisites before reading the rest of
this section. In particular, we use the cross product, which we covered in
Section 2.12, and basic methods for describing rotation in three dimensions,
which were the subject of Chapter 8.
12.5.1
Rotational Kinematics
Chapter 11 was about linear kinematics: we considered a function
p
(t) that
described the position of a particle as a function of time. We also considered
its first and second derivatives, the velocity and acceleration functions,
which we denoted
v
(t) =
p
(t) and
a
(t) =
p
(t), respectively. The rotational
analog of position is, of course, orientation. Several methods can be used
to describe the orientation of a body. A considerable number of pages
were spent explaining and comparing these methods in Chapter 8, and in
this chapter we assume you are familiar with the basics. In a rigid body
simulator, it's common to keep on hand redundant copies of the orientation
in alternate formats. Typically, both a quaternion and rotation matrix are
maintained. We will adopt a similar policy here with our notation. We let
R
(t) be the object-to-upright rotation matrix at time t; it is the orientation
of the body expressed in matrix form. We also use
q
(t) to refer to that same
Search WWH ::
Custom Search