# Multibody collisions

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I'm trying to find an equation for predicting the outcome velocities of a group of 3 objects, all touching each other, as the first step in a more general solution. The first has a velocity, the others are at rest. By positioning the other objects (spheres) at different angles relative to each other, I can change the resultant velocities of the objects. I'm colliding the first and second object, then the second and third, pairwise and iteratively, until the solution converges to an arbitrary exactness. The idea is that if the objects' velocities were integrated, there would be no penetration between any of the three objects. Obviously a generalized solution to this problem would be very useful in a physics engine. Here's my test source (Ruby script)
class Vector
def initialize(x, y)
@x = x; @y = y;
end

def x
@x
end

def y
@y
end

def to_s
"<#@x,#@y>"
end

def -(other)
Vector.new(@x-other.x, @y-other.y)
end

def +(other)
Vector.new(@x + other.x, @y + other.y)
end

def dot(other)
@x*other.x + @y*other.y
end
def *(other)
Vector.new(@x*other, @y*other)
end

def /(other)
Vector.new(@x/other, @y/other)
end

def length()
Math.sqrt(@x*@x + @y*@y)
end

end

class Body
def mass
@mass
end

def mass=(val)
@mass = val
end

def vel
@vel
end

def vel=(val)
@vel = val
end

def pos
@pos
end

def pos=(val)
@pos = val
end

def initialize()
@mass = 1
@vel = Vector.new(0,0)
@pos = Vector.new(0,0)
end

def applyImpulse(j, normal)
@vel = @vel + j / @mass * normal
end
end

def findCollisionImpulse(normal, a, b)
((a.vel - b.vel).dot(normal)) /
(normal.dot(normal) * (1.0 / a.mass + 1.0 / b.mass))
end

def collide(normal, a, b)
impulse = findCollisionImpulse(normal, a, b)

puts impulse

a.vel = a.vel - normal * (impulse / a.mass)
b.vel = b.vel  + normal * (impulse / b.mass)
end

def abs(a)
Math.sqrt(a*a)
end

def main()
bodyA = Body.new()
bodyB = Body.new()
bodyC = Body.new()

bodyA.mass = 1
bodyA.vel = Vector.new(100,0)
bodyA.pos = Vector.new(0,0)

bodyB.mass = 1
bodyB.vel = Vector.new(0,0)
bodyB.pos = Vector.new(1, 1)

bodyC.mass = 1
bodyC.vel = Vector.new(0,0)
bodyC.pos = Vector.new(2, 1)

epsilon = 1e-10

normalAB = bodyB.pos - bodyA.pos
normalBC = bodyC.pos - bodyB.pos

counter = 0

while(abs((bodyB.vel - bodyA.vel).dot(normalAB)) > epsilon ||
abs((bodyC.vel - bodyB.vel).dot(normalBC)) > epsilon)
counter = counter + 1
puts "Iterating..."

puts abs((bodyB.vel - bodyA.vel).dot(normalAB))

collide(normalAB, bodyA, bodyB)
collide(normalBC, bodyB, bodyC)
end

puts counter.to_s + " iterations"

puts "Body A:\n\tpos: " + bodyA.pos.to_s
puts "\tvel: " + bodyA.vel.to_s

puts "Body B:\n\tpos: " + bodyB.pos.to_s
puts "\tvel: " + bodyB.vel.to_s

puts "Body C:\n\tpos: " + bodyC.pos.to_s
puts "\tvel: " + bodyC.vel.to_s

end

main()


The previous source code produces this result: (objects are assumed to be spheres, the length of the vector between objects' position doesn't matter).
14 iterations
Body A:
pos: <0,0>
vel: <71.4285714285779,-28.5714285714221>
Body B:
pos: <1,1>
vel: <14.285714285711,28.5714285714221>
Body C:
pos: <2,1>
vel: <14.285714285711,0.0>

Clearly there's some sort of series work involved here, but I slept through series in Calc ;). I Would be greatly appreciative if someone could point me in the right direction.

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Here's my stab at it, perhaps it will work, or at least give you an idea.

*assuming a completely reflective relationship with no deformation, diffusion, absorption, or bonding*
have all objects move.
find which objects have collided.
for each group of objects add all masses, and add all forces.
resultant total force is applied to all masses in that group.
check each vector for penetration (if it is going to be moving faster in a way that would cause a second collision before the next frame)
perform sub group collisions if such a case exists.
once all vectors can occur without penetration, group has been solved.
after all groups have been calculated, next frame.

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I'm working on that last bit, resolving collisions. My idea is to resolve collisions before I integrate to change positions, so that large stacks of items resting on top of each other won't interpenetrate and cause problems for the collision solver in the next step.

We're talking about inelastic collisions for this prestep. The above ruby code is a simulation so I can try different configurations to see the end result.

I'm trying a system of equations approach at the moment, I'll see if it'll pan out. But I still think there's probably a reasonable way to distribute the momentum through the group to achieve the same result.

I'll try to upload a picture a bit later.

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Here's a picture of the above situation:

I'm able to get a generalized solution for the case of any 3 spheres colliding using a system of equations, but the result isn't in vector form and so is extremely unwieldly.

Basically, there are 6 equations and 6 unknowns (2 components for each vector):

(v2 - v1) * ab = 0 //balls do not move towards each other(v3 - v2) * bc = 0 //dittom1 * (v1 - v1i) + m2 * (v2 - v2i) + m3 * (v3 - v3i) = <0, 0> //conservation of momentumv1 cross ab = v1i cross ab //collisions do not effect tangent velocitiesv3 cross bc = v3i cross bc //ditto

The result is over a page long in Maple, but shows promising signs that there is a better way to condense it. Does anyone have a vector friendly way of solving these equations in terms of dot products, cross products, and norms?

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