  We study an orbital compass model on a checkerboard lattice where orbital
degree of freedom is represented by the pseudo-spin operator. Competition
arises from an Ising interaction for the $z$ component of pseudo-spins along
the vertical/horizontal bonds and an Ising interaction for the $x$ component
along diagonal bonds. Classical and quantum compass models are analyzed by
utilizing several analytical methods and numerical simulations. At a fully
frustrated point where the two Ising interactions compete with each other, a
macroscopic number of orbital configurations are degenerate in a classical
ground state. This degeneracy is lifted by thermal and quantum fluctuations,
and a staggered long-range order of the $z$ component of the pseudo-spin is
realized. A tricritical point for this order appears due to competition between
the bond dependent Ising interactions. Roles of geometrical frustration on
excitation dynamics are also examined.
