For mechanical systems, it is straight forward to compute the dynamic equations of motion which govern the evolution of the systems' configuration variables as the system is subjected to set of input forces. Solving the "reverse" problem or the motion planning problem, that is, finding the input forces that cause the system variables to move from a start to a goal configuration is a more challenging problem.; Researchers have successfully tackled particular flavors of this challenging problem. In fact, trajectory tracking controllers were defined in a coordinate free way for fully actuated mechanical systems, that is, systems that have as many input forces as the systems' degrees of freedom. On the other hand, motion planning for underactuated mechanical systems is still an ongoing research. In this dissertation, we approach this specific problem and present our preliminary results.; Motion planning for underactuated mechanical systems is particularly hard due to the nonlinearity of equations of motion and additionally due to their complex expressions. We express the same governing equations of motion in a simplified and reduced form for a large family of locomotion systems. Utilizing these reduced forms we develop intuitive evaluation tools that relate the evolution of the unactuated degrees of freedom to the actuated ones. Then, we utilize these evaluation tools to actually generate gaits that locomote a large class of mechanical systems along a desired direction. In other words, we design curves in the actuated subspace of the configuration space that will cause a desired change in unactuated degrees of freedom.
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