A nonlinear feedback scheme for a gravity-assisted underactuated manipulator with second-order nonholonomic constraints is presented in this paper. The joints of the hyper articulated arm have no dedicated actuators but are activated by gravity. By tilting the base link appropriately, the gravitational torque drives the unactuated links to a desired angular position. With simple locking mechanisms, the hyperarticulated arm can change its configuration using only one actuator at the base. This underactuated arm design was motivated by the need for a compact snake-like robot that can go into aircraft wings and perform assembly operations using heavy end-effectors. The dynamics of the unactuated links are essentially second-order nonholonomic constraints for which there are no general methods to design closed-loop control. We propose a nonlinear closed-loop control law that is guaranteed to be stable in positioning one unactuated joint at a time. We synthesize a Lyapunov function to prove the convergence of this control scheme. The Lyapunov function also generates estimates of the domain of convergence of the control law for various control gains. The control algorithm is implemented on a prototype three-link system. Finally, we provide some experimental results to demonstrate the efficacy of the control scheme.
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