# Collision response forces in rigid body simulation

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Basically, its the same project I'm working on, 2D driving game. I have 2 (or more, but 2 for now) vehicles using rigid body based physics, all in 2D. right now there are no other obsticles to collide with, just the 2 vehicles.

I think I can detect collisions perfectly (based on bounding boxes only), and those same collision detection functions return the point of contact relative to both vehicles. I can easily determine the velocity of those points on each vehicle.

So, what I did was take the difference of those 2 speeds, and multiply that by the other vehicle's mass to get the force, and then apply that to the vehicle at the point of contact, do that for both vehicles. The result looks ok for stuff like head on collisions, or high speed collisions, if one vehicle is stationary, etc... probably not too realistic, but doesn't need to be perfect, just not too absurd.

However, the real problem occurs when I try to push another vehicle with the players vehicle, the result isn't bad for the first few frames, but then the player starts to push its way into the NPC vehicle, so the images overlap.

So, what is the problem here? I'm a bit confused about how this is happening, at this point, the vehicles are even the same mass, so if I mix up which vehicle's mass I am multiplying, the result should be the same.

This seems like something that should have a well known solution, I'm just too dumb to figure it out on my own.

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I made a multi-rigid-body game (a load of boxes falling about) in javascript and found that trying to solve the explicit impulse equations is basically a total nightmare when there are multiple collisions and bodies in contact with each other.

My approach was to let them overlap a bit and then say that the reactive force is some suitable function of the overlap.  The reactive force acts in equal and opposite directions on each body (Newton three), this is crucial and means that the momentum and angular momentum work out right.  You need the function to have a hysterisis so that energy gets swallowed up in each collision or your objects go haywire.  Basically, the force function must make the force smaller as the bodies push apart but the exact details are not so important.

I have let the overlap effect be quite noticeable to show what is going on, but you could reduce it by making the bodies more rigid and also having a bit of a buffer around each object.

Realistic friction is a whole other question though.

Feel free to grab and modify the javascript.  I'm learning about websites and things and have put up a site as a project and some of my silly javascript games are on it.  The site works in Mozilla but looks poo in IE8 and the Canvas object the game uses needs at least IE9 anyway.

Go here:

http://www.moonsch.com/

Go to 'Games Links' and look at 'Boxes and Cat'.  Double click the boxes to make them vanish.  Reply if you have trouble, I'll try to respond to any questions.

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I made a multi-rigid-body game (a load of boxes falling about) in javascript and found that trying to solve the explicit impulse equations is basically a total nightmare when there are multiple collisions and bodies in contact with each other.

My approach was to let them overlap a bit and then say that the reactive force is some suitable function of the overlap.  The reactive force acts in equal and opposite directions on each body (Newton three), this is crucial and means that the momentum and angular momentum work out right.  You need the function to have a hysterisis so that energy gets swallowed up in each collision or your objects go haywire.  Basically, the force function must make the force smaller as the bodies push apart but the exact details are not so important.

I have let the overlap effect be quite noticeable to show what is going on, but you could reduce it by making the bodies more rigid and also having a bit of a buffer around each object.

Realistic friction is a whole other question though.

Feel free to grab and modify the javascript.  I'm learning about websites and things and have put up a site as a project and some of my silly javascript games are on it.  The site works in Mozilla but looks poo in IE8 and the Canvas object the game uses needs at least IE9 anyway.

Go here:

http://www.moonsch.com/

Go to 'Games Links' and look at 'Boxes and Cat'.  Double click the boxes to make them vanish.  Reply if you have trouble, I'll try to respond to any questions.

Cool, thanks. Awesome game, I played it far too many times before even looking at the code.

Actually, still haven't gotten around to looking at it really, although I at least opened it. I don't care too much about realistic friction, your game reminds me a lot of the physics in FreeGish, which I also played an embarrassing number of times. I don't know if it will work for my game or not, but I'll check it out and see.

Long answer: You need to look up impulse equations for rigid bodies, try the link below.

http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec23.pdf

Yes, I have been reading about impulse. Whenever I see a site with all those big equations my mind just goes blank and I can't focus enough to make it make sense, if they would just write it all on one line like an equation in C I could understand it no problem, but instead they gotta be all mathy.

I found this site: http://hyperphysics.phy-astr.gsu.edu/hbase/truckc.html

Which talks about impulse here: http://hyperphysics.phy-astr.gsu.edu/hbase/impulse.html

Which seems to tell me what I want, but I think I am only thinking of the head on collision which is basically 1D, and 2D collisions are more complicated than that.

I looked at the one you linked to also. Scary stuff. But thanks, I gotta sit down and work all that out and figure out how to do it in my code.

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There is one thing that I am really really confused about.

Force = Mass * Acceleration

Right? This is basic stuff. If a vehicle collides with a brick wall, I can use this equation to figure out how much force gets applied to the vehicle to make it stop moving. However, in a collision, there are TWO objects colliding. In the F=ma , which object's mass do I use? If every action has an equal and opposite reaction, it must be the same amount of force in both directions.

Ok. I guess I am starting to see my problem. acceleration is not known, right? If I use the velocity of the player it applies enough force to stop the player, which won't be the case if the vehicles have different masses.

So the answer is much more compicated than I thought, I see now. However, since the vehicles that I am testing with DO have the same mass, shouldn't what I am doing (difference of velocity times mass (of either vehicle, since they are the same)) work correctly?

Ok, also confused about impulse. I understand what it is, however, in my simulation, it shouldn't apply because the collision takes place instantly instead of over a period of time. So I only need to calculate force, right? If the collision is continuous, then it will count as multiple collisions, isn't that sufficient?

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I hope I'm not teaching you to suck eggs but I would seriously recommend simplifying everything down to one object moving under gravity and getting that going first.

I haven't used impulses at all, just forces.  I am just playing around with things myself with no ambition beyond my own amusement so I've no idea if there's a standard way of laying these things out.   Obviously this won't compile and the numbers are made up, but the sort of framework I've used is:

Object MyThing = new Object();

double MyThing.mass=5;

double MyThing.x_position=100;

double MyThing.y_position=100;

double MyThing.x_velocity=0;

double MyThing.y_velocity=0;

double MyThing.x_force=0;

double MyThing.y_force=0;

controlLoop(RunMeEvery TimeInterval)

{

GetBulkForces(MyThing);

GetCollisionForces(MyThing,YourThing);

//velocity is change in postion per unit time

MyThing.x_position += MyThing.x_velocity;

MyThing.y_position += MyThing.y_velocity;

//acceleration is change in velocity per unit time (F=ma, rearranged into a=F/m)

MyThing.x_velocity += (MyThing.x_force/MyThing.mass);

MyThing.y_velocity += (MyThing.y_force/MyThing.mass);

DrawOnScreen(MyThing);

}

GetBulkForces(Thing)

{

//Let's just have it falling under gravity, F=m*g, on a planet where gravity is -0.1 in some units

double g=-0.1;

Thing.yforce = Thing.mass*g;

}

GetCollisionForces(Thing1,Thing2)

{

if(CollisionHappened(Thing1,Thing2))

{

double x_force=ForceOfTheCollision(Thing1,Thing2,"X");

Thing1.x_force += x_force;

Thing2.x_force -= x_force; //i.e. opposite to Thing1!

double y_force=ForceOfTheCollision(Thing1,Thing2,"Y");

Thing1.y_force += y_force;

Thing2.y_force -= y_force; //i.e. opposite to Thing1!

}

}

ForceOfTheCollision(Thing1,Thing2,XYdirection)

{

//Lots of trig and geometry to work out how far the Things overlapped and in what direction the collision should rebound

}

CollisionHappened(Thing1,Thing2)

{

//Did a collision happen?

}

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In a collision, you can treat the change in velocity as instantaneous. In that case, you will not be considering forces and accelerations but impulses.

An impulse is a change in momentum. Momentum is mass times velocity. The equations are not that complicated. For an impulse I, the new momentum is

m v + I = m v'

and the change in velocity is

dv = v'- v = I/m

So if you have two bodies receiving equal and opposite impulses in a collision, the change in relative velocity is

dv = I(1/m1 + 1/m2)

dv in an inelastic collision is the relative velocity, because the objects stick together. So the impulse would be

I = (v2 - v1)/(1/m1 + 1/m2)

If the second object is stationary with infinite mass (a wall), you can see that this reduces to

I = - v1 m1

Comparing the two, you can consider the two body system to have an effective mass of

1/(1/m1 + 1/m2) = m1 m2 / (m1 + m2)

Which should answer the question of which mass to use in the calculations.

Collisions in 2D are the same except that velocity is a vector. For collisions with rotation, the effective mass gets a couple of extra terms from the moments of inertia. the derivation is similar.

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I started out trying the impulse approach as in the mechanics textbooks, but a problem arises when you want the objects to eventually calm down and rest against each other in equilibrium; there is no velocity, so no collisions, so nothing stopping one falling through the other.  Also with more than two bodies you can get weird jittery effects.  I ended up just thinking about forces which I think are a more fundamental quantity.

How do you use impulses in a many-bodied system and in such a way as you don't need a special case to show one thing coming to rest on top of another?  I couldn't work it out!

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It's possible to resolve overlaps using impulses.

The simplest way is a penalty method; add a small factor to the required separation velocity proportional to the overlap distance.

But that can add energy to the system and potentially become unstable. So another way is split impulses, where you accumulate the penalty impulses separately, and apply to the positions directly.

The key to handling simultaneous collisions is to iterate.

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In a collision, you can treat the change in velocity as instantaneous. In that case, you will not be considering forces and accelerations but impulses.

An impulse is a change in momentum. Momentum is mass times velocity. The equations are not that complicated. For an impulse I, the new momentum is

m v + I = m v'

and the change in velocity is

dv = v'- v = I/m

So if you have two bodies receiving equal and opposite impulses in a collision, the change in relative velocity is

dv = I(1/m1 + 1/m2)

dv in an inelastic collision is the relative velocity, because the objects stick together. So the impulse would be

I = (v2 - v1)/(1/m1 + 1/m2)

If the second object is stationary with infinite mass (a wall), you can see that this reduces to

I = - v1 m1

Comparing the two, you can consider the two body system to have an effective mass of

1/(1/m1 + 1/m2) = m1 m2 / (m1 + m2)

Which should answer the question of which mass to use in the calculations.

Collisions in 2D are the same except that velocity is a vector. For collisions with rotation, the effective mass gets a couple of extra terms from the moments of inertia. the derivation is similar.

Wow, thats great. I think it answers all my questions about impulse. Except for 1 thing, my game deals with vehicles colliding with each other, now vehicles do compress somewhat on collisions to protect the driver, so the collisions will be almost inelastic, right? However, do I want a totally inelastic collision for my game? For a wall an inelastic collision is great, I never saw a car bounce off a wall before, but in a vehicle/vehicle collision should there be some elasticity? I'm not sure. an inelastic collision could result in problems because my collision detection actually checks if they are overlapping, I figured that would be sufficient, but maybe it isn't good enough, would it be better to check if the vehicles will overlap in the next step?

I started out trying the impulse approach as in the mechanics textbooks, but a problem arises when you want the objects to eventually calm down and rest against each other in equilibrium; there is no velocity, so no collisions, so nothing stopping one falling through the other.  Also with more than two bodies you can get weird jittery effects.  I ended up just thinking about forces which I think are a more fundamental quantity.

How do you use impulses in a many-bodied system and in such a way as you don't need a special case to show one thing coming to rest on top of another?  I couldn't work it out!

My game is top down, like blimp view, so I won't need to worry about gravity in this one. The cases that I am worried about are 1. One car constantly pushing against another. Which I am afraid the AI that I am going to write will do often... 2. cars at similar speeds and directions making contact with each other.

It's possible to resolve overlaps using impulses.

The simplest way is a penalty method; add a small factor to the required separation velocity proportional to the overlap distance.

But that can add energy to the system and potentially become unstable. So another way is split impulses, where you accumulate the penalty impulses separately, and apply to the positions directly.

The key to handling simultaneous collisions is to iterate.

I think I may need to do something like this too, since my collision detection allows them to overlap before it detects.

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