03 Dec 2019
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A realistic simulation requires the estimation of friction forces.

## Friction Forces

Colliding as well as resting contacts cause constraint impulses as well as friction impulses.
So far we have only estimated the constraint impulses of colliding and resting contacts.
Constraint impulses are in the direction of the contact's normal vector.
Friction impulses act tangential to the contact plane (orthogonal to the normal vector).
The inequality constraint is
*J* and *b* have two rows.
The vectors *t1* and *t2* are chosen so that they are orthogonal to the normal vector *n* and orthogonal to each other.
Friction forces try to reduce tangential velocities to zero.
I.e. *b* is

The linear components of *J* are
and the angular components of *J* are

As before *λ* is computed
Here *λ* has two elements.
*λ1* and *λ2* are scaled with the same positive factor if necessary so that the length of the 2D vector consisting of *λ1* and *λ2* is smaller than the normal impulse *Pn* multiplied with the friction constant *μ*.
Basically the overall force vector needs to reside in the friction cone.
In a similar fashion as for constraint impulses, friction impulses are subtracted, recomputed, and then added to the accumulated impulses *P* of the two objects, when iterating.

If you have made it this far, you know how to build a small physics engine!

Any feedback, comments, and suggestions are welcome.

## Further reading

01 Dec 2019
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In order to handle collisions and resting contacts, it is necessary to determine contact points.
One can achieve this using the separating plane algorithm (also see Baraff).

## Separating Planes Algorithm

The separating plane algorithm only applies to convex polyhedra.
However non-convex objects can in many cases be approximated by a union of convex polyhedra.
A separating plane is a plane such that all points of the polyhedra A and B lie on opposite sides of the plane.
If two polyhedra are not intersecting, a separating plane can either be defined using one of the faces of one of the polyhedra or
be constructed from one edge of A and one edge of B.

The distance *d* of a point *x* from the plane *P(p,n)* defined by the point *p* and the normal vector *n* is:

In the case of using a face from polyhedron A, the point *p* is a corner of the face.
The normal vector *n* is the normalized cross product of two edges of the face and pointing outward from the polyhedron.
The point *x* of polyhedron B with the smallest distance *d* defines the distance of B to the face of A.
If this distance is positive, the face of polyhedron A is a separating plane.

In the case of using two edges, the point *p* is the corner of an edge from polyhedron A.
The normal vector *n* is the positive or negative normalized cross product of the two edge vectors.
Again the point *x* of polyhedron B with the smallest distance *d* defines the distance of B to the face of A.

From all the planes, the one with the largest distance *d* is selected.
If the distance is zero or negative, the two polyhedra are in contact.
One can place a plane in the "middle" of the contact by moving it by half the distance *d*, i.e.

In the next step all points from A and B, which have a distance less than *ε* to the plane, are selected.
Two vectors *u* and *v* orthogonal to the normal *n* and orthogonal to each other are determined.
The points are projected onto the plane and represented using the vectors *u* and *v*:
The convex hull of the each of the two polygons is determined.
Then the intersection of the resulting two 2D polygons is computed.
Finally the convex hull of the intersection is taken.
The points are projected back into 3D space.
These are the contact points used for simulating the colliding and resting contacts.

29 Nov 2019
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The following article is partially based on Hubert Eichner's article on inequality constraints.

## Inequality Constraints for Colliding Contacts

At a colliding contact the relative speed of the two objects is negative.
I.e. a contact is a colliding (and not a resting) contact if the normal speed at the contact point is below a small negative threshold.
Colliding contacts have restitution, i.e. the collision is partially elastic and the two objects will separate after the collision.
The colliding contacts and affected joints are handled using a special time step of zero duration.
Again the inequality constraint is
The linear and angular components of *J* are the same as for a resting contact.
The correction term *b* however depends on the initial normal speed at the contact point *vn* and the restitution coefficient *ε*:

## Sequential Impulses for Collisions

Again the contact and joint impulses are estimated iteratively.
Note that the normal speed *vn* at each contact is determined beforehand and stays constant during the iterations.

- determine
*vn* for all colliding contacts
- for each iteration
- for each joint
- compute Jacobian
*J* and correction vector *b*
- predict speed
*u*
- compute
*λ*
- compute the impulse
*P*
- add
*P* to accumulated impulses of the two objects

- for each colliding contact
- subtract old impulses
*P* from previous iteration from accumulated impulses of the two objects
- compute Jacobian
*J* and correction vector *b*
- predict speed
*u*
- compute new
*λ* and clip it to be greater or equal to zero
- compute the impulse
*P*
- add
*P* to accumulated impulses of the two objects

- apply impulses to objects

The value *λ* is stored as *Pn* for limiting friction impulses lateron.
The time step here is zero!
Therefore external forces do not need to be considered while handling collisions.
An object falling to the floor will experience several collisions until the linear and angular speed has decreased sufficiently for the contacts to become resting contacts.

25 Nov 2019
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part 3
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The following article is based on Hubert Eichner's article on inequality constraints.

## Inequality Constraints for Resting Contacts

Contact points (resting contacts) are represented as inequality constraints.
In contrast to a joint, a resting contact can only create repellent forces and no attracting forces.
Similar to a joint constraint, the resting contact is represented using a function *C(y(t))*.
Here *C* is one-dimensional and it is basically the distance of the two objects at the contact point.
The inequality constraint of the resting contact is
with the matrix *J* having one row and twelve columns.
Instead of anchor points, one uses vectors *ri* and *rj* from the center of each object to the contact point.
Using the contact normal *n* the inequality constraint becomes:
The rotational component can be simplified as shown below:
Thus the linear components of *J* are
And the angular components of *J* are
The correction term depends on the distance *d*.
I.e. if the distance is negative, a correction is required so that the objects do not penetrate each other any more.

## Sequential Impulses (updated)

The impulses generated by contact constraints and joint constraints are accumulated in a loop.
The (cumulative) *λ* obtained for contact constraint furthermore is clipped to be greater or equal to zero.
Note that it has to be the cumulative value which is clipped.
The algorithm for solving the joint and contact constraints becomes:

- for each iteration
- for each joint
- compute Jacobian
*J* and correction vector *b*
- predict speed
*u*
- compute
*λ*
- compute the impulse
*P*
- add
*P* to accumulated impulses of the two objects

- for each resting contact
- subtract old impulses
*P* from previous iteration from accumulated impulses of the two objects
- compute Jacobian
*J* and correction vector *b*
- predict speed
*u*
- compute new
*λ* and clip it to be greater or equal to zero
- compute the impulse
*P*
- add
*P* to accumulated impulses of the two objects

- use impulses and external forces to perform Runge Kutta integration step

The sequential impulse iterations have to be performed four times when using the Runge-Kutta method.
The value *λ* is stored as *Pn* for limiting friction impulses lateron.