  We study the phase transition between the high temperature algebraic liquid
phase and the low temperature ordered phase in several different types of
locally constrained O(N) spin systems, using a unified constrained
Ginzburg-Landau formalism. The models we will study include: 1, O(N) spin-ice
model with cubic symmetry; 2, O(N) spin-ice model with easy-plane and easy-axis
anisotropy; 3, a novel O(N) "spin-plaquette" model, with a very different local
constraint from the spin-ice. We calculate the renormalization group equations
and critical exponents using a systematic \epsilon = 4 - d expansion with
constant N, stable fixed points are found for large enough N. In the end we
will also study the situation with softened constraints, the defects of the
constraints will destroy the algebraic phase and play an important role at all
the transitions.
