Well, `uaux` represent the auxiliary speeds, which are equal to 0, except in the case where you are finding constraint forces (which are normally "hidden"; i.e. don't need to be solved for).

The way it works, is that the partial velocities (as calculated from the velocity terms) and partial derivatives of the acceleration terms (with respect to the u_dot terms) are the same; i.e. if ^N v ^B* = u1 * n_1 + u2 * n_2, and ^N a ^B* = u1_dot * n_1 + u2_dot * n_2, then the partials w.r.t. the u and udot terms for the velocity and acceleration respectively are n_1, n_2 and n_1, n_2.

However, when you need to solve for the constraint forces, you can't say the partial velocities w.r.t. auxiliary speeds are 0 - but the corresponding accelerations would be 0. Therefore, one of the partial terms gets it's auxiliary speeds 0'd and the other does not.

That's probably confusing, so let me know what I need to clarify.