2D Physics Engine collision response rotation of objects

Question

I'm writing my own basic physic engine and now I come to a problem I can't solve. Probably because I don't how to google this problem.

So here is my problem. I hope this image can explain it:

Collision response

I have two objects. The gray one is fixed and don't move and the green one which falls from the top. The green object has three vectors: a force, the acceleration and the velocity. It collides with the fixed gray object.

The real question is how can I get the rotation of the green object when it falls down?

Answer 1

It sounds like you may not have an understanding of the fundamental physics underlying rigid body dynamics. I say that only because you don't mention any of the terminology commonly used when talking about this kind of problem. You'll need to introduce the idea of orientation and angular velocity (the rotational analogs of position and linear velocity) to each dynamic body in the system, and compute all kinds of intermediate quantities like moment of inertia, angular acceleration, and torque.

Perhaps the best introductory reference for this is Chris Hecker's series of articles for Game Developer Magazine. Assuming you already have non-rotational dynamics (covered in part 1) and collision detection (not covered by this series) solved, you should begin with part 2 and proceed to part 3. They'll give you a solid foundation in the physics and mathematics necessary for implementing rotational collision response.

Answer 2

You do as described below once, when the objects collide.

Let us call the green rectangle "a", and the other one "b".

1.

First you need the rectangles "rotational mass", mass of inertia.

a.i = 4/3 * width * height * (width^2 + height^2) * a.density

2.

Then you need the vector pointing from the rectangle's center of mass (average position of all corners) to the contact position (where the rectangles collide), let us call it "r".

3.

Then you need to find the collision normal. This normal is the direction of an impulse being applied to a from b. The normal is a vector with length 1 unit. In your example the normal would probably point upwards. Let us call the normal vector "n".

4.

Now you will need the velocity of the contact point on a. If a is not rotating, the formula would be:

vp = a.vel

If a is rotating the formula would be:

vp = a.vel + cross(a.r_vel, r)

a.r_vel is a's rotational velocity given in radians and positive direction is counter clockwise.

cross() means cross product, the function is:

cross (v,i) = [-i * v.y , i * v.x]

The expanded formula would be:

vp = a.v + [-r * a.r_vel.y , r * a.r_vel.x]

5.

Now you need to calculate whether the objects are moving towards each other. Project the vp onto n.

vp_p = dot(vp, n)

dot (v1, v2) = v1.x * v2.x + v1.y * v2.y

vp_p is a scalar (a value, not a vector).

If vp_p is negative the obejcts are moving towards each other, if it is > 0 they are moving apart.

6.

Now you need to calculate the impulse to stop a from moving into b, the impulse is:

j = -vp_p / ( 1/a.mass + cross(r,n)^2 / a.i )

The cross product between two vectors are:

cross(v1,v2) = v1.x * v2.y - v1.y * v2.x

It returns a scalar.

Multiply the impulse with the normal to get the impulse vector:

jn = j * n

7.

Now you need to apply the impulse to a:

a.new_vel = a.old_vel + jn / a.mass;

a.new_r_vel = a.old_r_vel + cross(r,jn) / a.i;

If you want the collision to be fully elastic, you must multiply the impulse by 2. Let us call this multiplier "e". e needs to be between 1 and 2. 1 means no energy is conserved, 2 means all energy is conserved.

Example code:

var vp = a.vel + cross(a.r_vel, r);
var vp_p = dot(vp,n); // negative val = moving towards each other
if (vp_p >= 0) { // do they move apart?
    return false;
}

// normal impulse
var j = - e * vp_p / (
            1/a.mass + cross(r,n)^2 / a.i
        );
var jn = j * n;
//
a.vel = a.vel + jn / a.mass;
a.r_vel = a.r_vel + cross(r,jn) / a.i;

If b is not static the algorithm will be slightly different:

a.r = vector pointing from a's center of mass to the contact position

var vp = a.vel + cross(a.r_vel, a.r) - b.vel - cross(b.r_vel, b.r);
var vp_p = dot(vp,n); // negative val = moving towards each other
if (vp_p >= 0) { // do they move apart?
    return false;
}

// normal impulse
var j = - e * vp_p / (
            1/a.mass + cross(a.r,n)^2 / a.i +
            1/b.mass + cross(b.r,n)^2 / b.i
        );
var jn = j * n;
//
a.vel = a.vel + jp / a.mass;
a.r_vel = a.r_vel + cross(a.r,jn) / a.i;
b.vel = b.vel - jp / b.mass;
b.r_vel = b.r_vel - cross(b.r,jn) / b.i;

How the formulas work / sources:

http://www.myphysicslab.com/collision.html#resting_contact