The librating companions in HD 37124, HD 12661, HD 82943, 47 Ursa Majoris, and GJ 876: Alignment or antialignment?

We investigated the apsidal motion for the multiplanet systems. In the simulations, we found that the two planets of HD 37124, HD 12661, 47 UMa, and HD 82943 separately undergo apsidal alignment or antialignment. However, the companions of GJ 876 and v And are in apsidal lock only about 0degrees. Moreover, we obtained the criteria with Laplace-Lagrange secular theory to discern whether a pair of planets for a certain system are in libration or circulation.