<p>Origami has inspired the modern design of flexible structures in science and engineering. However, the rigidity control of general origami structures beyond the well-studied Miura-ori pattern remains unclear. Here we show that the rigidity of a wide range of origami structures can be controlled by enforcing or relaxing the planarity condition of selected facets. Through numerical simulations on origami structures with different facet selection rules, we analyze how their geometry and topology affect their degrees of freedom. We also study the probabilistic properties of the rigidity change and identify key origami structural variables that govern the critical rigidity percolation transition. Moreover, we develop a unified model that describes the relationship between the critical percolation density, facet geometry and selection rules. Altogether, our work highlights the intricate similarities and differences in the rigidity control of general origami structures, shedding light on the design of flexible mechanical metamaterials for practical applications.</p>