Game Development Reference
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chapter up with several showing you how to implement both 2D and 3D rigid-body
simulations.
Integrating the Equations of Motion
By now you should have a thorough understanding of the dynamic equations of motion
for particles and rigid bodies. If not, you may want to go back and review Chapter 1
through Chapter 4 before reading this one. The next aspect of dealing with the equations
of motion is actually solving them in your simulation. The equations of motion that
we've been discussing can be classified as ordinary differential equations. In Chap‐
ter 2 and Chapter 4 , you were able to solve these differential equations explicitly since
you were dealing with simple functions for acceleration, velocity, and displacement.
This won't be the case for your simulations. As you'll see in later chapters, force and
moment calculations for your system can get pretty complicated and may even rely on
tabulated empirical data, which will prevent you from writing simple mathematical
functions that can be easily integrated. This means that you have to use numerical in‐
tegration techniques to approximately integrate the equations of motion. We say ap‐
proximately because solutions based on numerical integration won't be exact and will
have a certain amount of error depending on the chosen method.
We're going to start with a rather informal explanation of how we'll apply numerical
integration because it will be easier to grasp. Later we'll get into some of the formal
mathematics. Take a look at the differential equation of linear motion for a particle (or
rigid body's center of mass):
F = m dv/dt
Recall that this equation is a statement of force equals mass times acceleration, where F
is force, m is mass, and dv / dt is the time derivative of velocity, which is acceleration. In
the simple examples of the earlier chapters of this topic, we rewrote this equation in the
following form so it could be integrated explicitly:
dv/dt = F/m
dv = (F/m) dt
One way to interpret this equation is that an infinitesimally small change in velocity,
dv , is equal to ( F / m ) times an infinitesimally small change in time. In earlier examples,
we integrated explicitly by taking the definite integral of the left side of this equation
with respect to velocity and the right side with respect to time. In numerical integration
you have to take finite steps in time, thus dt goes from being infinitely small to some
discrete time increment, ∆ t , and you end up with a discrete change in velocity, ∆ v :
∆v = (F/m) ∆t
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