Under-actuated systems are unavoidable in certain applications. For example, a biped can not have an actuator between the foot and the ground. For industrial robots, underactuation is preferable due to cost considerations. A fully actuated system can execute any joint trajectory. However, if the system is under-actuated, not all joint trajectories are attainable. For such systems, it is difficult to characterize attainable joint trajectories analytically and numerical methods are generally used to characterize them. This paper investigates the property of differential flatness for under-actuated planar open chain robots and study its dependence on inertia distribution within the system. Once this property is established, trajectory between any two points in its differentially flat output space is feasible and can be shown to be consistent with the dynamics of the under-actuated system. It is shown that certain choices of inertia distributions make an under-actuated open-chain planar robot with revolute joints feedback linearizable, I.e., also differentially flat. Hence, both cyclic and point to point trajectories can be guaranteed with these under-actuated systems. The methodology proposed is demonstrated with an under-actuated three degree-of-freedom planar robot.
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