An optimal motion planning scheme for a three-link planar under-actuated manipulator moving between two stationary states was considered. The manipulator has two prismatic joints that are actuated, while the third joint of the manipulator is a revolute passive joint which has neither an actuator nor a holding brake. The dynamic constraint on the free link is shown to be second-order nonholonomic. The states and inputs of the system were expressed in terms of higher order derivatives of a subset of states by using coordinate and input transformations. The explicit expressions for the states and inputs were used to change a constrained dynamic optimization problem into an unconstrained one. The necessary conditions for optimality were derived by calculus of variations.The scheme was applied to a planar PPR under-actuated manipulator and numerical results were reported.
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