Force capability in general 3 DoF planar mechanisms

• A closed-form solution to obtain the Fapp and Fm in 3-DoF mechanisms is proposed.
• The proposed solution can be easily adapted to several manipulators with M = 3.
• The proposed solution can be applied in real-time applications.
• The proposed solution is faster and more direct compared with other solutions.

This paper presents a mathematical closed-form solution to obtain the maximum force with a prescribed moment in robotic mechanisms with 3 degrees of freedom. The proposed mathematical model is obtained using classical optimization methods, screw theory and the Davies method as primary mathematical tools. The novelty of this study lies in the fact that the main result is not a method or a numerical algorithm, but a mathematical closed-form solution to obtain the force capability of manipulators with 3 DoF. It represents a direct, easy, fast and versatile way to solve the force capability problem in manipulators with 3 DoF. In addition the closed-form solution is able to obtain the interval in which the desired moment at the manipulator end effector can be imposed. The solution is general and can be applied to any planar manipulator with 3 degrees of freedom, whether it be serial, parallel or hybrid. The proposed closed-form solution can be used in applications requiring a real-time response in terms of force capability, such as milling, polishing, and machining. Some force capability polygons and force capability polytopes are obtained for different manipulators in order to exemplify the use of this mathematical closed-form solution.