Manipulators with free joints fall in a class of second-order nonholonomic systems whose dynamical constraints are nonintegrable. Control of such systems is one of major research topics in robotics and control engineering. In the literature, the authors analyzed nonlinear behaviors of a two joint manipulator with a free joint in response to a periodic input. The manipulator showed chaotic behaviors for larger input-amplitudes. A control method to position both joints via modulation of the input-amplitude was also proposed. Although the strategy was effective, it sufferes from heuristics and difficulty for generalization. In this paper, behaviors of the free-joint manipulator with a periodic input are approximated via the 1st-order and 2nd-order averaging methods. The invariant manifolds of the averaged behavior are analyzed and identified by defining a “pseudo-Hamiltonian” on an “input-amplitude normalized phase plane”. A control method to reach a desired invariant manifold via modulation of the input-amplitude is also proposed and a termination control at the destination is formulated. The effectiveness of the proposed control methods is verified by computer simulations.