It is possible to impose nonlinear geometric (equality) constraints
on D-NURBS through Lagrange multiplier techniques [32].
This approach increases the number of degrees of freedom, hence the
computational cost, by adding unknowns 
--also known as
Lagrange multipliers--which determine the magnitudes of the
constraint forces. The method is applied to the B-spline model in
[6, 42]. The augmented Lagrangian method
[18] combines the Lagrange multipliers with the
simpler penalty method [26].
One of the best known techniques for applying constraints to dynamic models is the Baumgarte stabilization method [1] which solves constrained equations of motion through linear feedback control (see also [17, 25]). We augment (17) as follows:
where 
are generalized forces stemming from the
holonomic constraint equations. The term 
is the transpose
of the constraint Jacobian matrix and 
is a vector of Lagrange multipliers
that must be determined. We can obtain the same number of equations
as unknowns, by differentiating (24) twice
with respect to time: 
. Baumgart's
method replaces these additional equations with equations that have
similar solutions, but which are asymptotically stable; e.g., the
damped second-order differential equations 
, where a and b are stabilization factors.
For a given value of a, we can choose b = a to obtain the
critically damped solution 
which has the
quickest asymptotic decay towards constraint satisfaction
(24). Taking the second time derivative of
(24) and rearranging terms yields
We arrive at the following system of equations for the unknown constrained generalized accelerations and Lagrange multipliers:
This system can be solved for 
and 
using
standard direct or iterative techniques (or in the least squares sense
when it is overdetermined by conflicting constraints).
