Determining Resting contact between sphere and pla

问题:

This question has one major question, and one minor question. I believe I am right in either question from my research, but not both.

For my physics loop, the first thing I do is apply a gravitational force to my TotalForce for a rigid body object. I then check for collisions using my TotalForce and my Velocity. My TotalForce is reset to (0, 0, 0) after every physics loop, although I will keep my velocity.

I am familiar with doing a collision check between a moving sphere and a static plane when using only velocity. However, what if I have other forces besides velocity, such as gravity? I put the other forces into TotalForces (right now I only have gravity). To compensate for that, when I determine that the sphere is not currently overlapping the plane, I do

    Vector3 forces = (sphereTotalForces + sphereVelocity);
    Vector3 forcesDT = forces * fElapsedTime;
    float denom = Vec3Dot(&plane->GetNormal(), &forces);

However, this can be problematic for how I thought was suppose to be resting contact. I thought resting contact was computed by

denom * dist == 0.0f

Where dist is

float dist = Vec3Dot(&plane->GetNormal(), &spherePosition) - plane->d;

(For reference, the obvious denom * dist > 0.0f meaning the sphere is moving away from the plane)

However, this can never be true. Even when there appears to be "resting contact". This is due to my forces calculation above always having at least a .y of -9.8 (my gravity). When when moving towards a plane with a normal of (0, 1, 0) will produce a y of denom of -9.8.

My question is

1) Am I calculating resting contact correctly with how I mentioned with my first two code snippets?

If so,

2) How should my "other forces" such as gravity be used? Is my use of TotalForces incorrect?

For reference, my timestep is

  mAcceleration = mTotalForces / mMass;
  mVelocity += mAcceleration * fElapsedTime;
  Vector3 translation = (mVelocity * fElapsedTime);

EDIT

Since it appears that some suggested changes will change my collision code, here is how i detect my collision states

if(fabs(dist) <= sphereRadius)
{ // There already is a collision }
else
{
    Vector3 forces = (sphereTotalForces + sphereVelocity);
    float denom = Vec3Dot(&plane->GetNormal(), &forces);

    // Resting contact
    if(dist == 0) { }
    // Sphere is moving away from plane
    else if(denom * dist > 0.0f) { }
    // There will eventually be a collision
    else
    {
        float fIntersectionTime = (sphereRadius - dist) / denom;
        float r;
        if(dist > 0.0f)
            r = sphereRadius;
        else
            r = -sphereRadius;

        Vector3 collisionPosition = spherePosition + fIntersectionTime * sphereVelocity - r * planeNormal;
    }
}

回答1:

1. You should use if(fabs(dist) < 0.0001f) { /* collided */ } This is to acocunt for floating point accuracies. You most certainly would not get an exact 0.0f at most angles or contact.

2. the value of dist if negative, is in fact the actual amount you need to shift the body back onto the surface of the plane in case it goes through the plane surface. sphere.position = sphere.position - plane.Normal * fabs(dist);

3. Once you have moved it back to the surface, you can optionally make it bounce in the opposite direction about the plane normal; or just stay on the plane.

   parallel_vec = Vec3.dot(plane.normal, -sphere.velocity);

   perpendicular_vec = sphere.velocity - parallel_vec;

   bounce_velocity = parallel - perpendicular_vec;

4. you cannot blindly do totalforce = external_force + velocity unless everything has unit mass.

EDIT:

1. To fully define a plane in 3D space, you plane structure should store a plane normal vector and a point on the plane. http://en.wikipedia.org/wiki/Plane_(geometry) .

Vector3 planeToSphere = sphere.point - plane.point;

float dist = Vector3.dot(plane.normal, planeToSphere) - plane.radius;

if(dist < 0) { // collided. }

I suggest you study more Maths first if this is the part you do not know.

NB: Sorry, the formatting is messed up... I cannot mark it as code block.

EDIT 2: Based on my understanding on your code, either you are naming your variables badly or as I mentioned earlier, you need to revise your maths and physics theory. This line does not do anything useful.

float denom = Vec3Dot(&plane->GetNormal(), &forces);

A at any instance of time, a force on the sphere can be in any direction at all unrelated to the direction of travel. so denom essentially calculates the amount of force in the direction of the plane surface, but tells you nothing about whether the ball will hit the plane. e.g. gravity is downwards, but a ball can have upward velocity and hit a plane above. With that, you need to Vec3Dot(plane.normal, velocity) instead.

Alternatively, Mark Phariss and Gerhard Powell had already give you the physics equation for linear kinematics, you can use those to directly calculate future positions, velocity and time of impact.

e.g. s = 0.5 * (u + v) * t; gives the displacement after future time t. compare that displacement with distance from plane and you get whether the sphere will hit the plane. So again, I suggest you read up on http://en.wikipedia.org/wiki/Linear_motion and the easy stuff first then http://en.wikipedia.org/wiki/Kinematics .

Yet another method, if you expect or assume no other forces to act on the sphere, then you do a ray / plane collision test to find the time t at which it will hit the plane, in that case, read http://en.wikipedia.org/wiki/Line-plane_intersection .

回答2:

There will always be -9.8y of gravity acting on the sphere. In the case of a suspended sphere this will result in downwards acceleration (net force is non-zero). In the case of the sphere resting on the plane this will result in the plane exerting a normal force on the sphere. If the plane was perfectly horizontal with the sphere at rest this normal force would be exactly +9.8y which would perfectly cancel the force of gravity. For a sphere at rest on a non-horizontal plane the normal force is 9.8y * cos(angle) (angle is between -90 and +90 degrees).

Things get more complicated when a moving sphere hits a plane as the normal force will depend on the velocity and the plane/sphere material properties. Depending what your application requirements are you could either ignore this or try some things with the normal forces and see how it works.

For your specific questions:

1. I believe contact is more specifically just when dist == 0.0f, that is the sphere and plane are making contact. I assume your collision takes into account that the sphere may move past the plane in any physics time step.
2. Right now you don't appear to have any normal forces being put on the sphere from the plane when they are making contact. I would do this by checking for contact (dist == 0.0f) and if true adding the normal force to the sphere. In the simple case of a falling sphere onto a near horizontal plane (angle between -90 and +90 degrees) it would just be sphereTotalForces += Vector3D(0, 9.8 * cos(angle), 0).

Edit:

From here your equation for dist to compute the distance from the edge of sphere to the plane may not be correct depending on the details of your problem and code (which isn't given). Assuming your plane goes through the origin the correct equation is:

dist = Vec3Dot(&spherePosition, &plane->GetNormal()) - sphereRadius;

This is the same as your equation if plane->d == sphereRadius. Note that if the plane is not at the origin then use:

D3DXVECTOR3 vecTemp(spherePosition - pointOnPlane);
dist = Vec3Dot(&vecTemp, &plane->GetNormal()) - sphereRadius;

回答3:

The exact solution to this problem involves some pretty serious math. If you want an approximate solution I strongly recommend developing it in stages.

1) Make sure your sim works without gravity. The ball must travel through space and have inelastic (or partially elastic) collisions with angled frictionless surfaces.

2) Introduce gravity. This will change ballistic trajectories from straight lines to parabolae, and introduce sliding, but it won't have much effect on collisions.

3) Introduce static and kinetic friction (independently). These will change the dynamics of sliding. Don't worry about friction in collisions for now.

4) Give the ball angular velocity and a moment of inertia. This is a big step. Make sure you can apply torques to it and get realistic angular accelerations. Note that realistic behavior of a spinning mass can be counter-intuitive.

5) Try sliding the ball along a level surface, under gravity. If you've done everything right, its angular velocity will gradually increase and its linear velocity gradually decrease, until it breaks into a roll. Experiment with giving the ball some initial spin ("draw", "follow" or "english").

6) Try the same, but on a sloped surface. This is a relatively small step.

If you get this far you'll have a pretty realistic sim. Don't try to skip any of the steps, you'll only give yourself headaches.

回答4:

Answers to your physics problems:

f = mg + other_f; // m = mass, g = gravity (9.8)
a = f / m; // a = acceleration
v = u + at; // v = new speed, u = old speed, t = delta time
s = 0.5 * (u + v) *t;

When you have a collision, you change the both speeds to 0 (or v and u = -(u * 0.7) if you want it to bounce).

Because speed = 0, the ball is standing still.

If it is 2D or 3D, then you just change the speed in the direction of the normal of the surface to 0, and keep the parallel speed the same. That will result in the ball rolling on the surface.

You must move the ball to the surface if it cuts the surface. You can make collision distance to a small amount (for example 0.001) to make sure it stay still.

http://www.physicsforidiots.com/dynamics.html#vuat

Edit:

NeHe is an amazing source of game engine design: Here is a page on collision detection with very good descriptions: http://nehe.gamedev.net/tutorial/collision_detection/17005/

Edit 2: (From NeHe)

double DotProduct=direction.dot(plane._Normal); // Dot Product Between Plane Normal And Ray Direction
Dsc=(plane._Normal.dot(plane._Position-position))/DotProduct; // Find Distance To Collision Point
Tc= Dsc*T / Dst
Collision point= Start + Velocity*Tc

回答5:

I suggest after that to take a look at erin cato articles (the author of Box2D) and Glenn fiedler articles as well. Gravity is a strong acceleration and results in strong forces. It is easy to have faulty simulations because of floating imprecisions, variable timesteps and euler integration, very quickly. The repositionning of the sphere at the plane surface in case it starts to burry itself passed the plane is mandatory, I noticed myself that it is better to do it only if velocity of the sphere is in opposition to the plane normal (this can be compared to face culling in 3D rendering: do not take into account backfaced planes).

also, most physics engine stops simulation on idle bodies, and most games never take gravity into account while moving, only when falling. They use "navigation meshes", and custom systems as long as they are sure the simulated objet is sticking to its "ground".

I don't know of a flawless physics simulator out there, there will always be an integration explosion, a missed collision (look for "sweeped collision").... it takes a lot of empirical fine-tweaking.

Also I suggest you look for "impulses" which is a method to avoid to tweak manually the velocity when encountering a collision.

Also take a look to "what every computer scientist should know about floating points"

good luck, you entered a mine field, randomly un-understandable, finger biting area of numerical computer science :)

回答6:

For higher fidelity (wouldn't solve your main problem), I'd change your timestep to

mAcceleration = mTotalForces / mMass;
Vector3 translation = (mVelocity * fElapsedTime) + 0.5 * mAcceleration * pow(fElapsedTime, 2);
mVelocity += mAcceleration * fElapsedTime;

You mentioned that the sphere was a rigid body; are you also modeling the plane as rigid? If so, you'd have an infinite point force at the moment of contact & perfectly elastic collision without some explicit dissipation of momentum.

Force & velocity cannot be summed (incompatible units); if you're just trying to model the kinematics, you can disregard mass and work with acceleration & velocity only.

Assuming the sphere is simply dropped onto a horizontal plane with a perfectly inelastic collision (no bounce), you could do [N.B., I don't really know C syntax, so this'll be Pythonic]

mAcceleration = if isContacting then (0, 0, 0) else (0, -9.8, 0)

If you add some elasticity (say half momentum conserved) to the collision, it'd be more like

mAcceleration = (0, -9.8, 0) + if isContacting then (0, 4.9, 0)