In this paper, we study the trajectory planning problem for planar underactuated robot manipulators with two revolute joints in the absence of gravity. This problem is studied as an optimal control problem in which, given the dynamic model of a planar horizontal robot manipulator with two revolute joints one of which is not actuated, the initial state, and some specifications about the final state of the system, we find the available control input and the resulting trajectory that minimize the energy consumption during the motion. Our method consists in a numerical resolution of a reformulation of the optimal control problem as an unconstrained calculus of variations problem in which the dynamic equations of the mechanical system are regarded as constraints and treated using special derivative multipliers. We solve the resulting calculus of variations problem using a numerical approach based on the Euler-Lagrange necessary condition in integral form in which time is discretized and admissible variations for each variable are approximated using a linear combination of piecewise continuous basis functions of time. The use of the Euler-Lagrange necessary condition in integral form avoids the need for numerical corner conditions and the necessity of patching together solutions between corners.
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