# Floating point errors in collision routines

D

#### Dave

Hi folks,

I am trying to develop a routine that will handle sphere-sphere and
sphere-triangle collisions and interactions. My aim is to develop a
quake style collision engine where a player can interact with a rich
3D environment. Seem to be 90% of the way there! My problems are
related to calculations where the result tends to zero (or another
defined limit.)

Have loads of cases where this kind of interaction occurs but this one
is as tricky (?) as any...

When a moving sphere approaches a triangulated plane I am using a
sphere-plane collision routine to see if a collision will occur. All
goes well until I start to have near parallel interactions (e.g. when
the sphere has sufficient energy to collide with, and then slide along
the plane.) During the next pass of the collision routine the velocity
of the sphere is perpendicular (in a single plane) to the face normal
of the previous colliding plane. A dot product between the velocity of
the sphere and the normal of the colliding plane should be zero. It's
close to zero but how close is close enough to warrant no further
collision? A "need to collide" case occurs where the sphere is just
above the plane with a velocity almost parallel to the plane BUT just
on a collision course. The above dot product will be very close to
zero again, but this time there should be a collision with this plane!

Since I need to account for glancing off faces, approaching faces at
right angles, interacting with steps, falling off edges etc the number
of vector operations to determine the final position and velocity of
the sphere need to be as accurate as possible.

I guess my question (sorry for the waffle) is how to minimise rounding
errors in floating point operations?

So far I've tried to account for the errors (or eliminate them) by:

- Using double precision decelerations and calculations
- Using the epsilon error on the variables to account for known
internal storage errors

Unable to eliminate or account for the errors I then set about trying
to use flags to ignore colliding objects from future collision during
the current recursive collision routine call. Helped in some cases so
started to try to account for all near critical iterations using
flags.

Getting close to a general solution now but my once clear simple
routine has turned into a logical nightmare!

When working with numbers where accuracy is paramount should I be
using a different approach? I've considered the following:

- Using a quantised system where all values have to fall along a
virtual 3D grid
- Using sin and cos tables to support the above goal of quantisation
- Using integer mathematics
- Use a high precision custom storage class for calculations (speed is
off the essence so might not be an option?)

Perhaps I am worrying too much! My current "effort" seems to handle
interactions in a similar, if much more shaky, way to Unreal
Tournament 2004. Kind of clunky at times, stops occasionally and seems
to have problems jumping on steps!

I'm working on Windows 2000 using Visual Studio 6 and .net enterprise
architect 2003 (c++). It's a DirectX application if that makes any
difference?

Any advice would be really appreciated.

Many thanks,

Dave

G

#### Greg Schmidt

I guess my question (sorry for the waffle) is how to minimise rounding
errors in floating point operations?

So far I've tried to account for the errors (or eliminate them) by:

- Using double precision decelerations and calculations

This will certainly decrease your rounding errors. You'll have to
measure whether any speed decrease is worth the mathematical
improvement.
- Using the epsilon error on the variables to account for known
internal storage errors

Do you mean using something like
if(result < epsilon) result = 0;
That's something I've had to do on occasion. (epsilon usually around
10^-8 I think, but don't quote me on that.)
Unable to eliminate or account for the errors I then set about trying
to use flags to ignore colliding objects from future collision during
the current recursive collision routine call. Helped in some cases so
started to try to account for all near critical iterations using
flags.

This might also be a useful optimization trick, if it can often replace
a bunch of floating point calculations with a single boolean comparison.
Getting close to a general solution now but my once clear simple
routine has turned into a logical nightmare!

That does tend to be the way of these things. 99% of all situations are
handled very simply, and then you need a mass of special cases for the
last 1%.
When working with numbers where accuracy is paramount should I be
using a different approach? I've considered the following:

- Using a quantised system where all values have to fall along a
virtual 3D grid
- Using sin and cos tables to support the above goal of quantisation
- Using integer mathematics

These three will probably improve speed but decrease accuracy.
- Use a high precision custom storage class for calculations (speed is
off the essence so might not be an option?)

This could increase accuracy but decrease speed.
I'm working on Windows 2000 using Visual Studio 6 and .net enterprise
architect 2003 (c++).

Have you investigated the Visual Studio "improve floating point
consistency" compilation option? In a couple of projects I've worked
on, it has made a significant improvement. I didn't notice any speed
degradation, but my apps were not as computationally intense as yours
and didn't have real-time requirements.

J

#### John Nagle

Trying to converge a floating point calculation all the way to
zero is fundamentally futile. You get characteristic underflow
before you get zero.

In general, bear in mind that subtracting two large numbers
to get a tiny result is inherently prone to roundoff error. You
must design your algorithms to avoid that.

Some collision detection algorithms, especially early versions
of GJK, are badly behaved when two faces are very close to
parallel. Of course, if you're doing physics right, objects
in contact come to rest in face-parallel situations, so the
near-parallel situation gets explored very thoroughly.
Newer collision engines, like (I think) Gino's SOLID, get
it right.

Developing collision algorithms that don't have floating
point precision problems yet don't "cheat" on contact analysis
is hard, but quite possible. See my US patent #6,067,096, which
covers the first "ragdoll physics" that worked. It's necessary
to be very careful about the limitations of floating point.
But once you get it right, it works very well.

As I usually tell people, either buy a physics engine
(probably Havok), figure out some way to make gameplay
work with lousy physics, or expect to spend a few years
on the problem.

John Nagle
Animats

D

#### Dave

Greg Schmidt said:
This will certainly decrease your rounding errors. You'll have to
measure whether any speed decrease is worth the mathematical
improvement.

Do you mean using something like
if(result < epsilon) result = 0;
That's something I've had to do on occasion. (epsilon usually around
10^-8 I think, but don't quote me on that.)

That's exactly what I've been doing! Thinking on this a bit more I
guess it's not really useful since it's only accounting for the
internal errors in the floating point operations and not the actual
errors due to the limitation of the storage class?
This might also be a useful optimization trick, if it can often replace
a bunch of floating point calculations with a single boolean comparison.

Certainly did! My first attempt involved testing if the sphere
collided with everthing in the environment. About a 100 refinements
later and the routine now only does any involved maths for a handful
of triangles. Using flags to help control critical situations has also
been useful in terms of performance but not so useful to avoid my
objects slipping through the environment!
That does tend to be the way of these things. 99% of all situations are
handled very simply, and then you need a mass of special cases for the
last 1%.

Guess that's why there are only a handful of quick, robust collision
engines out there.
These three will probably improve speed but decrease accuracy.

Why so? Working with a quantised system should enable the position and
velocity of an object to be "forced" into allignment. For a simple
environment where objects are alligned with the quantised system I'm
sure such an approach would work (dull game though!) For a complex
environment operations like calculating the point of collision between
a sphere and an off-axis triangle face would be very tricky to
quantise though!
This could increase accuracy but decrease speed.

And I'm not really that sure of the benefits. I think I'm stuck trying
to represent what are ultimately irrational numbers. No storage class
is going to help here!
Have you investigated the Visual Studio "improve floating point
consistency" compilation option? In a couple of projects I've worked
on, it has made a significant improvement. I didn't notice any speed
degradation, but my apps were not as computationally intense as yours
and didn't have real-time requirements.

Did have a look at this under VS6 but it didn't help too much.
Accuracy marginally improved but still below what is required. Will
have a look to see if .net is any better.

D

#### Dave

John Nagle said:
Trying to converge a floating point calculation all the way to
zero is fundamentally futile. You get characteristic underflow
before you get zero.

In general, bear in mind that subtracting two large numbers
to get a tiny result is inherently prone to roundoff error. You
must design your algorithms to avoid that.

Have done this... up to a point! Within each function I have tried to
avoid any such calculations but my code is such a mess that I am
probably doing all sorts of dodgy calculations. Certainly lots of
vector normalisation which I really must tidy up. I can see lots of
scope for improvement here but also many cases where subracting two
large numbers is pretty much inevitable. When you say two large
numbers do you mean two numbers with many decimal places or just large
numbers? Would it help to change the scale of the environment? Do you
know if floating point operations are generally more accurate within a
certain range of numbers? Guessing it's not important where the
decimal place sits as every digit needs to be accounted for!

Need to look out those old maths notes about Nth degree of error on a
caculation?
Some collision detection algorithms, especially early versions
of GJK, are badly behaved when two faces are very close to
parallel. Of course, if you're doing physics right, objects
in contact come to rest in face-parallel situations, so the
near-parallel situation gets explored very thoroughly.
Newer collision engines, like (I think) Gino's SOLID, get
it right.

I think I'm doing the physics right! For objects that have specific
frictional resistance I'm assuming that there will be a deceleration
period where face-face contact must be maintained. This is the messy
bit! This is why I am trying to use flags to maintain stability for
these cases. Please let me know if this assumption is wrong, I studied
physics but am no expert on collision mechanics!
Developing collision algorithms that don't have floating
point precision problems yet don't "cheat" on contact analysis
is hard, but quite possible. See my US patent #6,067,096, which
covers the first "ragdoll physics" that worked. It's necessary
to be very careful about the limitations of floating point.
But once you get it right, it works very well.

You're patent is a bit too much for my head on a Sunday evening! Very
impressive though! Were you able to control the floating point errors
by clever algorithm design alone? Did you find it necessary, or
error on floating point operations during comparisons?
As I usually tell people, either buy a physics engine
(probably Havok), figure out some way to make gameplay
work with lousy physics, or expect to spend a few years
on the problem.

Think I'll keep bashing away for now! Have just spent 3 hours playing
Kill Zone (purely for research you understand!) Seems to suffer from
the same sort of issues I am having (and some that I am not!) Perhaps
mine will be finished in time for the PS9!

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