Game Engine Architecture

650 12. Collision and Rigid Body Dynamics

The orientation quaternion q(t) should be renormalized periodically to reverse

the eff ects of the inevitable accumulation of fl oating-point error.

As always, the explicit Euler method is being used here just as an ex-

ample. In a real engine, we would employ velocity Verlet, RK4, or some other

more-stable and more-accurate numerical method.

12.4.7. Collision Response

Everything we’ve discussed so far assumes that our rigid bodies are neither

colliding with anything, nor is their motion constrained in any other way.

When bodies collide with one another, the dynamics simulation must take

steps to ensure that they respond realistically to the collision and that they

are never left in a state of interpenetration aft er the simulation step has been

completed. This is known as collision response.

12.4.7.1. Energy

Before we discuss collision response, we must understand the concept of en-

ergy. When a force moves a body over a distance, we say that the force does

work. Work represents a change in energy—that is, a force either adds energy

to a system of rigid bodies (e.g., an explosion) or it removes energy from the

system (e.g., friction). Energy comes in two forms. The potential energy V of a

body is the energy it has simply because of where it is relative to a force fi eld

such as a gravitational or a magnetic fi eld. (For example, the higher up a body

is above the surface of the Earth, the more gravitational potential energy it

has.) The kinetic energy of a body T represents the energy arising from the

fact that it is moving relative to other bodies in a system. The total energy

E = V + T of an isolated system of bodies is a conserved quantity, meaning that

it remains constant unless energy is being drained from the system or added

from outside the system.

The kinetic energy arising from linear motion can be writt en

Tlinear=^12 mv^2 ,

or in terms of the linear momentum and velocity vectors:

Tlinear=⋅^12 pv.

Analogously, the kinetic energy arising from a body’s rotational motion is as

follows:

Energy and its conservation can be extremely useful concepts when solving

all sorts of physics problems. We’ll see the role that energy plays in the deter-

mination of collision responses in the following section.

Tangular=⋅^12 Lω.