We propose a systematic scheme for computing the second- and higher-order infinitesimal mechanisms of a rigid origami modeled as a truss structure for approximating its folding path by a polynomial obtained from the series expansion of the nodal displacement with respect to the path parameter controlling the folding state. The first- to n-th-order terms of the nodal displacement are determined so that the first- to n-th-order terms of the series expansion of the compatibility equations are to be satisfied with identically zero. These terms can be systematically computed from the first-order infinitesimal mechanism satisfying the existence conditions of the infinitesimal mechanisms up to n-th-order. The approximation accuracy of the finite folding path is verified by varying the order of terms considered in the approximation.