The motion of free-floating space robots is characterized by non-holonomic constraints, i.e., non-integrable rate constraint equations. These constraints originate from the principles of conservation of angular momentum. It is well known that these rate constraints can also be written to form an input-affine drift-less control systems. Trajectory planning of these systems is extremely challenging and computation intensive since the motion must satisfy differential constraints. Under certain conditions, these drift-less control systems can be shown to be differentially flat. The property of flatness allows a computationally inexpensive way to plan trajectories for a dynamic system between two configurations as well as develop feedback controllers. The key contribution of this paper is to systematically study the non-holonomic rate constraints for free-floating planar open-chain robots and determine the design conditions under which the system exhibits differential flatness. A design is then proposed that can exploit the property effectively for trajectory planning and feedback control.