Energy-optimal trajectory planning for the Pendubot and the Acrobot

In this paper, we study the trajectory-planning problem for planar underactuated robot manipulators with two revolute joints in the presence of gravity. This problem is studied as an optimal control problem. We consider both possible models of planar underactuated robot manipulators, namely with the elbow joint not actuated, called Pendubot, and with the shoulder joint not actuated, called Acrobot. Our method consists of 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 by means of special derivative multipliers. We solve the resulting calculus of variations problem by using a numerical approach based on the EulerLagrange necessary condition in integral form. In which, time is discretized, and admissible variations for each variable are approximated using a linear combination of the piecewise continuous basis functions of time. In this way, a general method for the solution of optimal control problems with constraints is obtained, which, in particular, can deal with nonintegrable differential constraints arising from the dynamic models of underactuated planar robot manipulators with two revolute joints in the presence of gravity. Copyright (c) 2012 John Wiley & Sons, Ltd.