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We consider a 24-degrees-of-freedom monkey robot that is supposed to perform brachiation locomotion, i.e. swinging 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: (a) introduce a family of coordination patterns to be enforced on the dynamics with respect to a path coordinate; (b) formulate geometric equality constraints that are necessary for periodic locomotion; (c) generate trajectories from integrable reduced dynamics associated with the passive hinge; (d) 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 the gradient of the cost function into the search algorithm.