Game Engine Architecture

648 12. Collision and Rigid Body Dynamics

() ()

() ().

() ()

xxxxyxzx

yyxyyyzy

zzxzyzzz

Lt I I I t

Lt I I I t

Lt I I I t

⎡ ⎤ ⎡ ⎤⎡ω ⎤

⎢ ⎥ ⎢ ⎥⎢ ⎥

⎢ ⎥ ⎢=ω⎥⎢ ⎥

⎢⎣ ⎥ ⎢⎦ ⎣ ⎥⎢⎦⎣ω ⎥⎦

Because the angular velocity ω is not conserved, we do not treat it as

a primary quantity in our dynamics simulations the way we do the linear

velocity v. Instead, we treat angular momentum L as the primary quantity.

The angular velocity is a secondary quantity, determined only aft er we have

determined the value of L at each time step of the simulation.

12.4.6.4. Torque in Three Dimensions

In three dimensions, we still calculate torque as the cross product between

the radial position vector of the point of force application and the force vector

itself (N = r × F). Equation (12.8) still holds, but we always write it in terms of

the angular momentum because angular velocity is not a conserved quantity:

NI I I

L

== =()

=

()

()

()

().

t

dt

dt

d

dt

t

dt

dt

ω

α ω

12.4.6.5. Solving the Equations of Angular Motion in Three Dimensions

When solving the equations of angular motion in three dimensions, we might

be tempted to take exactly the same approach we used for linear motion and

two-dimensional angular motion. We might guess that the diff erential equa-

tions of motion should be writt en

A3D(?):

NInet

() ();

() ();

t

t

=

=





ωt

ω θt

L:

Fv

vr

net()^ ();

() (),

t mt

t

=

=



t

and using the explicit Euler method, we might guess that the approximate

solutions to these ODEs would look something like this:

A3D(?):

() ()IN() ;

() () () ;

tt

tt

211 1

21 1

=+

=+

−

net^

Δ

θ Δ

ω

ω

ω t

θ t t

L:

vv

F

r rv

() ()

()

;

() () ().

t

t

m t

tt

21 1

211

=+

=+

net

Δ

Δ

t

t t

However, this is not actually correct. The diff erential equations of angular mo-

tion diff er from their linear and two-dimensional angular counterparts in two

important ways:

1. Instead of solving for the angular velocity ω, we solve for the angular
   momentum L directly. We then calculate the angular velocity vector as a
   secondary quantity using I and L. We do this because angular momen-
   tum is conserved , while angular velocity is not.