Contact Joint Jacobian and Friction Pyramid Facet Count

[pink_fish]

Hello,

this might be a stupid question but I just cant figure it out:

in Erleben's Thesis pdf, (Page 83 Section 4.9.7)

he says that

From previously we know that the contact Jacobian has 1 + η constraints, so it is a
(1 + η)-by-12 dimensional matrix,

* isnt η the number of direction vectors for the discretized friction cone?

* what puzzles me is the bullet and opentissue implementation of this:
they both set only 3 rows maximum of the jacobian matrix, and for the rows concerned with friction they use the tangential directions t1 and t2 directly, what happens to the other rows?!?! i must have undestood sthg in a wrong way!

* what i was expecting is that t1 and t2 are used to make η directions and these equations are then used to fill η jacobian rows, instead only t1 and t2 are used to fill only 2 rows maximum.

* now intuitivly i undestand that 2 directions t1 and t2 are enough to span the whole tangential plane, if yes then y this whole η story? y not just 2 vectors t1 and t2?

can someone explain that to me please? and excuse my stupidity!!

[Erin Catto]

Using more friction directions can lead to greater accuracy. Most games don't need the additional accuracy.

[pink_fish]

thx for the reply erin,

heres the email reply of Kenny Erleben himself:

Chapter 4 derives the LCP formulation in the spirit of Stewart and Trinkle. It is quite sometime ago that I wrote this chapter, but as I recall the friction directions make a positive span of the friction cone and eta is defined as 2 times the number of friction directions use in the friction cone approximation.

The implementation used in OpenTissue is explained in Chapter 6, here only two friction directions are used, the reason is speed over accuracy. To get some accuracy t1 is aligned with the direction of relative sliding, thus providing an ``exact'' modelling of the friction force in case of dynamic friction. Static friction could be subject to a square root 2 error. Note however that the projection onto the friction cone is completely general and you can even make your own extentions to a completely general limit surface, I just picked the simplest possible solution, a square:-)

/Kenny