We consider a model of a 24-degree-of-freedom monkey robot that is supposed to perform a brachiation locomotion, i.e. to swing from one row of a horizontal ladder to the next one using the arms. The robot hand is constructed as a planar hook so that the contact point, about which the robot swings, is a passive hinge.We identify the 10 most relevant degrees of freedom for this underactuated mechanical system and formulate a tractable search procedure consisting on the following steps: (a) to introduce a parametrized family of coordination patterns to be enforced on the dynamics with respect to a path coordinate; (b) to formulate geometric equality constraints that are necessary to achieve a periodic locomotion; (c) to generate trajectories from integrable reduced dynamics associated with the passive hinge; (d) to evaluate the energetic cost of transport. Moreover, we observe that a linear approximation of the reduced dynamics can be used for trajectory generation, which allows us to incorporate computation of