Design of a differentially flat 3R planar under-actuated manipulator with a single input at the second joint

In contrast to a fully actuated system, an under-actuated system cannot execute all joint trajectories. However, under-actuation may be unavoidable at times, such as during joint failure. In recent literature, an under-actuated open-chain planar manipulator with revolute joints has been shown to be feedback linearizable with specific choices of mass and inertia distribution. It has been shown that the dynamic model of an n-joint manipulator with center of mass located at joint 2 and with only one actuator is static feedback linearizable if and only if the input is at the first or the last joint. In our pursuit for novel designs of under-actuated arms, which are both controllable and feedback linearizable, we investigate the feasibility of designing a 3R planar under-actuated manipulator which has a different distribution of compliance. This study shows that this new architecture of 3R planar under-actuated manipulator with one input at the second joint can still be designed to be differentially flat. With this design, a simple stabilizing controller in the flat output space enables the system to perform point-to-point motion and mitigate initial errors.