Abstract
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.
Highlights
In underactuated manipulators the number of available control inputs is strictly less than the number of the degrees of freedom of the robot
The results have shown that the Embedded Optimal Control (Eoc) approach is effective in solving optimal control problems for these underactuated mechanical systems, which are highly nonlinear and have nonholonomic constraints
The corresponding dynamical systems are described by vector fields that are affine in the continuous control variables, fulfilling the sufficient conditions for the existence of a solution stated in the previous section
Summary
In underactuated manipulators the number of available control inputs is strictly less than the number of the degrees of freedom of the robot. This paper studies the optimal control problem for planar underactuated robot manipulators with two revolute joints that move in a vertical plane and, are subject to the gravity force. The presence of two revolute joints in the mechanical system will be denoted by RR Both possible models of planar underactuated RR robot manipulators under gravity are considered, namely, the model in which only the shoulder joint is actuated, denoted by RR, which is usually called Pendubot, and the model in which only the elbow joint is actuated, denoted by RR, which is usually called Acrobot. The mechanical system obtained by removing both motors from the RR robot manipulator under gravity is considered, which is a double pendulum and will be denoted by R R
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