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The motion of free-floating space robots is characterized by nonholonomic, i.e., non-integrable rate constraint equations. These constraints originate from principles of conservation of linear and angular momentum. It is well known that these rate constraints can also be written as input-affine drift-less control systems. Trajectory planning of these systems is extremely challenging and computation intensive since the motion must satisfy differential constraints. However, under certain conditions, these drift-less control systems can be shown to be differentially flat. The property of flatness allows a computationally in-expensive way to plan trajectories for the dynamic system between two configurations as well as develop feedback controllers. Nonholonomic rate constraints for free-floating planar open-chain robots are systematically studied to determine the design conditions under which the system exhibits differential flatness. Under these design conditions, the property of flatness is used for trajectory planning and feedback control under perturbations in the initial state.