Control of nonlinear under-actuated systems is an area of ongoing research. In certain applications, under- actuated systems are unavoidable. For instance, a biped can not have an actuator between the foot and the ground. In industrial robots, under-actuation can minimize cost and dead weight. Differential flatness, if applicable, provides a systematic approach to plan and control feasible trajectories for such systems. Recently, the authors have formulated a philosophy to design under-actuated planar manipulators that are differentially flat [1]. The design philosophy has two sufficient conditions: (i) an inertia distribution scheme, and (ii) an actuator and torque spring placement scheme. The philosophy covers a broad range of n-DOF manipulator designs with the degree of under- actuation varying from 1 to n - 1 as opposed to under-actuation by one or two in most of the literature on under-actuated manipulators. This paper presents a 3-DOF planar manipulator designed on the basis of this philosophy and an experimental study of controllers based on its differential flatness property. It is demonstrated experimentally that the differential flatness based controllers are able to track the desired trajectories with small tracking errors even in the presence disturbances like friction, parameter uncertainty etc. It is shown via simulation that the errors observed in the trajectories can be attributed to the friction present at the unactuated joint.

A fully actuated system can execute any joint trajectory. However, if the system is underactuated, not all joint trajectories are attainable. For such systems, it is difficult to characterize attainable joint trajectories analytically. Numerical methods are generally used to characterize these. This paper investigates the property ofdifferentialflatnessfor underactuated planar open-chain robots and studies dependence on inertia distribution within the system. It is shown that certain choices of inertia distributions make an underactuated open-chain planar robot with revolute joints feedback linearizable, i.e., also differentially flat. Once this property is established, trajectory between any two points in the state space can be planned, and a controller can be developed to correct for errors. To demonstrate the proposed methodology in hardware, experiments with an underactuated 3-DOF planar robot are also presented.

In bipedal walking, a trajectory is acceptable as long as it is repetitive and allows the foot to clear the ground, while allowing the biped to move forward. Since the actual trajectory followed by a biped is not as important, a biped having more than one passive joints can also meet the motion requirements. Due to physical constraints, a biped is under-actuated at the ground contact with the feet. A biped should exhibit limit cycles when moving continuously in an environment. In general, it is difficult to prove existence of limit cycles for nonlinear systems. In this work, we generate limit cycles for a class of nonlinear underactuated bipeds using differential flatness. A specific inertia distribution renders the biped design differentially flat. Differential Flatness allows generation of a family of limit cycles amenable to numerical optimization. The results are illustrated by two DOF biped.