Results
The results for the one dimentional case (no rotation) are as follows: (see this page for code and derivation)
|
General Case (between Perfectly elastic and Perfectly inelastic) |
|
| coefficient of restitution | e = ( -  )
/ ( -  ) |
| impulse = | (e+1)*(-
)*  |
| Final velocity of object a=
= |
*  +*(e+1)*
 |
| Final velocity of object b=
= |
*(e+1)*
+  |
The results for the two dimentional case are as follows: (see this page for code and derivation)
| General Case (between Perfectly elastic and Perfectly inelastic) | |
| impulse = |
Jx = (e+1)/k * (Vaix - Vbix)( 1/ma - rax2/Ia + 1/mb - rbx2/Ib)
- (e+1)/k * (Vaiy - Vbiy) (rax ray / Ia + rbx rby / Ib) where: k=1/ma2+ 2/ma mb +1/mb2 - rax2/maIa - rbx2/maIb - ray2/maIa - ray2/mbIa - rax2/mbIa - rbx2/mbIb - rby2/maIb - rby2/mbIb + ray2 rbx2/IaIb + rax2 rby2/IaIb - 2 rax ray rbx rby/IaIb |
Final velocity of object a= af
= |
Vaix - Jx/Ma Vaiy - Jy/Ma |
| Final velocity of object b=bf
= |
Vbix - Jx/Mb Vbiy - Jy/Mb |
| Final angular velocity of object a=waf = |
waix - (Jx ray - Jy rax) /I |
| Final angular velocity of object b=wbf = | wbix - (Jx rby - Jy rbx) /I wbiy - (Jx rby - Jy rbx) /I |
The results for the three dimentional case are as follows: (see this page for code and derivation)
| General Case (between Perfectly elastic and Perfectly inelastic) | |
| impulse = | |J| = (e+1) * (
- ) / (1/Ma +n•([Ia]-1(n
x ra)) x ra + 1/Mb +n•([Ib]-1(n x rb)) x rb) |
| Final velocity of object a=af
= |
Vai - J/Ma |
| Final velocity of object b=bf
= |
Vbi - J/Mb |
| Final angular velocity of object a=waf = | wai - [Ia]-1(J x ra) |
| Final angular velocity of object b=wbf = | wbi - [Ib]-1(J x rb) |
Collision Response for unconstrained rigid bodies
As already explained here, When unconstrained rigid bodies are floating round in free space, perhaps under the influence of forces, then each object might be represented by a state vector like the following:
Dynamics state vector for an object:
| θx |
| θy |
| θz |
| Px |
| Py |
| Pz |
| lx |
| ly |
| lz |
| mvx |
| mvy |
| mvz |
These values might be gradually changing with time in a way that can be described by simple linear equations. This would go on until two or more objects collide, this produces a step change in these variables, a discontinuity or singularity, when an impulse is transferred between the two objects in an infinitesimally small time. When such a collision occurs we need to calculate the new state vectors for each object. To do this we need to gather the following information which describes the collision:
Collision parameters:
| a | reference to object a | ||||||||||||||||||||||||||||||||||||
| b | reference to object b | ||||||||||||||||||||||||||||||||||||
| sva | state vector for object a - 12 dimensional vector as above | ||||||||||||||||||||||||||||||||||||
| svb | state vector for object b - 12 dimensional vector as above | ||||||||||||||||||||||||||||||||||||
| Newton-Euler matrix for object a |
This matrix is explaind here and combines
the following information:
|
||||||||||||||||||||||||||||||||||||
| Newton-Euler matrix for object b |
As above for object b |
||||||||||||||||||||||||||||||||||||
| n | A normal vector (dimention 3), this is a normal to the point of collision and shows the direction that the impulse is transferred. | ||||||||||||||||||||||||||||||||||||
| cfa, cfb | Optional coefficient of friction for each object |
Physics Assumptions
The following is an analytical approach to collisions, it assumes the collision occurs at a point contact between rigid objects. This means that the objects do not break or deform on collision. If these assumptions would cause too many errors, for instance modeling the deformations of a car in a crash, or if the analytical approach produces equations which are too complex then try a numerical approach to collisions.
The dynamics equations for a single object were covered in the previous page. Here we extend this analysis to two objects so that we can calculate the result of a collision between the objects. As explained in the previous page, we need to translate all values (inertia, force, movement, etc.) to world or common coordinates.
Linear momentum, angular momentum and energy is passed between the colliding shapes at the point of collision by means of impulse. Assuming that we know the velocities before the impact, and the point of impact, then one or more of the following might be used to help us calculate the velocities after the collision:
- Conservation of momentum - The total momentum of the whole system will be conserved. The momentum is a vector quantity so in the 3D case it is conserved in whatever x,y and z axis we choose.
- Conservation of energy - The total energy of the whole system will be conserved, but energy may be converted between kinetic energy and other types of energy. So we need to make assumptions about this. In a perfectly elastic collision then the kinetic energy is conserved, in an inelastic collision some of the energy associated with the approaching objects may be converted to heat.
- Calculation of Impulse - The impulse could be calculated which would allow us to work out the change in velocity of both objects.
Also there are different outcomes depending on the properties of the objects colliding:
- If the objects are perfectly elastic, they will absorb the kinetic energy of the collision as potential energy (like a spring) and then instantly convert it back to kinetic energy, so that the objects bounce off each other with a separating velocity equal and opposite to the approaching velocity.
- If the objects are perfectly inelastic, they will stick together and the final velocities of the two objects will be equal. In this case the total kinetic energy will be reduced and some energy will be converted into heat or other forms of energy.
- The objects may be somewhere in-between these two extremes. So we could introduce an elasticity factor for each colliding object which would give final velocities somewhere between these two extremes. Another complication is that this factor might be different in different dimensions, for instance objects may collide and slide against each other, so friction may also need to be taken into account.
Other concepts:
| Impulse is the integral of force over time, it is measured in Newton-seconds. For instance a force of one Newton applied over one second will change the momentum, a force of two Newton's applied over half of a second will have the same effect. For rigid body collisions we take this to its limit and apply an infinite force over an infinitesimally small time. This impulse is equal to the change in momentum of the colliding objects. Because we are talking about forces here, Newton's third law applies, and the impulse on the colliding objects will be equal and opposite. |
|
Once we start working in two and three dimensions, there is the possibility of rotation to include. This makes the general solution of a collision very difficult to work out (too difficult for me so far - but I'm working on it, can you help?). Rotation has similar laws to linear collisions, there are equivalents to Newtons laws for rotations and the total angular momentum of a system is conserved in the same way as linear momentum is conserved. However the following issues make the calculation of collision response difficult:
|
|
In a complicated collision, then friction may be involved. However friction (as opposed to elasticity) is only involved if there is a force, such as gravity, pushing the colliding objects together. |
Collisions in 3 dimensions is quite a complex problem, I'll start with 1 dimension first, then try to build up to 2 and then 3 dimensions.
- Rigid Body Collisions - deriving general equations
- Rigid Body Collisions - numerical solution
- collisions with deformations
Implementing this in program
One issue I would like to keep in mind through these pages is, how do we use this theory in the program? It is proposed to include information about these physical properties in the scene graph.
For information about how to use this in the program see dynamics tutorial.
Determining the impact point
We not only need to detect that there has been a collision, we also need to find the point of impact relative to the centres of mass so that we can pass this information on to the collision responce system.
Next Step and Further Reading
This page assumes that the collision has been detected in some way, this is discussed here.
To see how an algorithm for implementing collision responce of a solid object see here.
To see a proposal for a structure to hold this collision responce information see here.
Here is an off topic question, there seems to be 1 way an body can rotate in 2 dimensions and 3 ways that a body can rotate in 3 dimensions. How many ways could a body rotate in 4 dimensions?
