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Mathematical Problems in Engineering
Volume 2015, Article ID 348178, 21 pages
Research Article

Embedded Optimal Control of Robot Manipulators with Passive Joints

School of Communication Engineering, Rey Juan Carlos University, Fuenlabrada, 28943 Madrid, Spain

Received 12 March 2015; Accepted 19 August 2015

Academic Editor: Sivaguru Ravindran

Copyright © 2015 Alberto Olivares and Ernesto Staffetti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper studies the optimal control problem for planar underactuated robot manipulators with two revolute joints and brakes at the unactuated joints in the presence of gravity. The presence of a brake at an unactuated joint gives rise to two operating modes for that joint: free and braked. As a consequence, there exist two operating modes for a robot manipulator with one unactuated joint and four operating modes for a manipulator with two unactuated joints. Since these systems can change dynamics, they can be regarded as switched dynamical systems. The optimal control problem for these systems is solved using the so-called embedding approach. The main advantages of this technique are that assumptions about the number of switches are not necessary, integer or binary variables do not have to be introduced to model switching decisions between modes, and the optimal switching times between modes are not unknowns of the optimal control problem. As a consequence, the resulting problem is a classical continuous optimal control problem. In this way, a general method for the solution of optimal control problems for switched dynamical systems is obtained. It is shown in this paper that it can deal with nonintegrable differential constraints.