The algorithm for the systematic detection of an embedded rigid structure in enumerating a basic kinematic chain has been developed. The main idea in this algorithm is the sequential removal of binary link strings from the given kinematic chain to simplify the remaining chain. The proposed algorithm is implemented in a program using the LISP language and applied to several kinds of kinematic chains, such as those with 1, 2, 3, and 4 degrees of freedom and having 2, 3, 4, and 5 independent loops, respectively. In most cases, the results are in good agreement with those presented by other investigators. However, there are some discrepancies for several cases in the number of kinematic chains. The proposed algorithm is fairly efficient in that it can be applied to some unexplored cases as listed in the Results section. The structural data of each kinematic chain are also obtainable if needed.