Physically Based Animation - Computer Animation: Algorithms and Techniques - page 230

have to be strong enough to prevent interpenetration; (2) must only push objects apart, not together; and
(3) have to go to zero at the point of contact at the moment that objects begin to separate.
To analyze what is happening at a point of contact, use a distance function, d i ( t ), which evaluates
to the distance between the objects at the i th point of contact. Assuming objects A and B are involved
in the i th contact and the points involved in the i th contact are p A and p B from the respective objects,
then the distance function is given by Equation 7.67 .

d i ðtÞ¼ð p A ðtÞ p B ðtÞÞ N i (7.67)

If d i ( t ) is zero, then the objects involved are still in contact. Whenever d i ( t ) > 0, the objects are

separated. One of the objectives is to avoid d i ( t ) 0, which would indicate penetration.

Assume at some time t 0 , the distance between objects is zero. To prevent object penetration from
time t 0 onward, the relative velocity of the two objects must be greater than or equal to zero,

<

d i ðt 0 Þ

0
for t > t 0 . The equation for relative velocity is produced by differentiating Equation 7.67 and is shown
in Equation 7.68 . At time t 0 , the objects are touching, so p A ( t 0 )

¼ p B ( t 0 ). This results in Equation 7.69 .In

d i ðt 0 Þ¼

addition, for resting contact, 0.

d i ðtÞ¼ N i ðtÞ ð p A ðtÞ p B ðtÞÞ þ N i ð p A ðtÞ p B ðtÞÞ

(7.68)

¼ d i ( t 0 )

Since d i ( t 0 ) ¼ 0, penetration will be avoided if the second derivative is greater than or equal

to zero, d ¨ i ( t )

0. The second derivative is produced by differentiating Equation 7.68 as shown in
Equation 7.69 . At t 0 , remembering that p A ( t 0 ) ¼ p B ( t 0 ), one finds that the second derivative simplifies

as shown in Equation 7.70 . Notice that Equation 7.70 further simplifies if the normal to the surface of
contact does not change ( n ˙ i ( t 0 ) ¼ 0).

d i ðtÞ¼ð

p B ðtÞÞ N i þð p A ðtÞ p B ðtÞÞ N i þð p A ðtÞ p B ðtÞÞ p A ðtÞ N i (7.69)

p B ðt 0 ÞÞ N i þð p A ðt 0 Þ p B ðt 0 ÞÞ

d i ðtÞ¼

2 ð _ p A ðt 0 Þ_ N i (7.70)

The constraints on the forces as itemized at the beginning of this section can now be written as
equations: the forces must prevent penetration (Eq. 7.71 ) ; the forces must push objects apart, not
together ( Eq. 7.72 ) ; and either the objects are not separating or, if the objects are separating, then
the contact force is zero ( Eq. 7.73 ) .

d i ðtÞ 0 (7.71)

f i 0 (7.72)

d i ðtÞf i ¼

0 (7.73)
The relative acceleration of the two objects at the i th contact point, d ¨ i ( t 0 ), is written as a linear
combination of all of the unknown f ij s( Eq. 7.74 ). For a given contact, j , its effect on the relative
acceleration of the bodies involved in the i th contact point needs to be determined. Referring to
Equation 7.70 , the component of the relative acceleration that is dependent on the velocities of the
points, 2

p B ( t 0 )), is not dependent on the force f j and is therefore part of the b i term.
The component of the relative acceleration dependent on the accelerations of the points involved in the
contact, n i ( t 0 )(

n i ( t 0 )( p A ( t 0 )

p B ( t 0 )), is dependent on f j if object A or object B is involved in the j th contact.
The acceleration at a point on an object arising from a force can be determined by differentiating
Equation 7.59 , producing Equation 7.75 where r A ( t )

p A ( t 0 )

¼ p A ( t ) x A ( t ).