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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

Embedded Optimal Control of Robot Manipulators with Passive Joints