A Forward, Inverse Kinematics and Workspace Analysis of 3RPS and 3RPS-R Parallel Manipulators

Abstract

Parallel mechanisms are finding wide applications in manufacturing. The design methodology of parallel mechanisms is generally considered as a complex procedure. Many times they are custom designed to suit a specific task. In this work, a generalized analytical approach to forward, inverse kinematics and workspace analysis of three-degrees-of-freedom 3RPS (revolute–prismatic–spherical) and 3RPS-R (revolute–prismatic–spherical–revolute) parallel manipulator is presented. Methodology for obtaining various position and orientation of the moving platform for the provided actuation by varying the distance between the base platform and the moving platform has been discussed in detail. The actuation is given to the prismatic joint of the manipulator, which results in a variation of limb lengths, thereby altering the position and orientation of the moving platform. Sylvester dialytic elimination method is used to solve the nonlinear polynomial expressions originally obtained from the loop–closure equations. Various numerical solutions have been obtained for a different combination of limb lengths and by changing the height of the manipulator. The workspace obtained is not evenly distributed, and hence, an additional degree of freedom (DOF) of base rotation has been suggested. With the addition of one DOF, the volume of workspace obtained is enhanced without any discontinuities which would enable the manipulator to generate circular trajectories. The methodology presented in this work is general and will be suitable for similar geometries by altering the loop closure equations suitably.

Appendix

$$ \begin{aligned} A_{2} & = - 3a^{2} + 3b^{2} + 3bl_{2} - 3bl_{3} \cos \theta_{3} + l_{2}^{2} - l_{2} l_{3} \cos \theta_{3} + l_{3}^{2} \\ B_{2} & = - 4l_{2} l_{3} \sin \theta_{3} \\ C_{2} & = - 3a^{2} + 3b^{2} - 3bl_{2} - 3bl_{3} \cos \theta_{3} + l_{2}^{2} + l_{2} l_{3} \cos \theta_{3} + l_{3}^{2} \\ \end{aligned} $$

$$ \begin{aligned} A_{4} & = 2l_{1}^{2} (18a^{4} l_{2}^{2} + 45a^{4} l_{3}^{2} + 18b^{2} l_{2}^{4} + 72b^{3} l_{2}^{3} + 72b^{4} l_{2}^{2} + 18b^{2} l_{3}^{4} + 72b^{4} l_{3}^{2} + 2l_{1}^{4} l_{2}^{2} + 5l_{1}^{4} l_{3}^{2} + 2l_{2}^{2} l_{3}^{4} + 5l_{2}^{4} l_{3}^{2} \\ & \quad - 27a^{4} l_{3}^{2} \cos (2\theta_{3} ) - 3l_{1}^{4} l_{3}^{2} \cos (2\theta_{3} ) - 3l_{2}^{4} l_{3}^{2} \cos (2\theta_{3} ) + 36a^{2} bl_{2}^{3} - 12bl_{1}^{2} l_{2}^{3} + 12bl_{2}^{3} l_{3}^{2} + 72a^{2} b^{2} l_{2}^{2} \\ & \quad - 36a^{2} b^{2} l_{3}^{2} - 12a^{2} l_{1}^{2} l_{2}^{2} - 30a^{2} l_{1}^{2} l_{3}^{2} - 42a^{2} l_{2}^{2} l_{3}^{2} - 24b^{2} l_{1}^{2} l_{2}^{2} - 24b^{2} l_{1}^{2} l_{3}^{2} - 12b^{2} l_{2}^{2} l_{3}^{2} + 10l_{1}^{2} l_{2}^{2} l_{3}^{2} \\ & \quad - 72b^{3} l_{3}^{3} \cos \theta_{3} + 4l_{2}^{3} l_{3}^{3} \cos (\theta_{3} ) + 12bl_{2} l_{3}^{4} - 36a^{2} bl_{3}^{3} \cos (\theta_{3} ) - 12a^{2} l_{2} l_{3}^{3} \cos \theta_{3} - 12a^{2} l_{2}^{3} l_{3} \cos \theta_{3} \\ & \quad + 12bl_{1}^{2} l_{3}^{3} \cos \theta_{3} + 12bl_{2}^{2} l_{3}^{3} \cos \theta_{3} - 24b^{2} l_{2} l_{3}^{3} \cos \theta_{3} - 24b^{2} l_{2}^{3} l_{3} \cos \theta_{3} + 72b^{3} l_{2}^{2} l_{3} \cos \theta_{3} + 4l_{1}^{2} l_{2} l_{3}^{3} \cos \theta_{3} \\ & \quad + 4l_{1}^{2} l_{2}^{3} l_{3} \cos \theta_{3} - 24bl_{2}^{3} l_{3}^{2} \cos (2\theta_{3} ) - 72b^{3} l_{2} l_{3}^{2} \cos (2\theta_{3} ) - 108a^{2} bl_{2} l_{3}^{2} + 12bl_{1}^{2} l_{2} l_{3}^{2} + 36a^{4} l_{2} l_{3} \cos \theta_{3} \\ & \quad - 12bl_{2}^{4} l_{3} \cos (\theta_{3} ) + 144b^{4} l_{2} l_{3} \cos (\theta_{3} ) + 4l_{1}^{4} l_{2} l_{3} \cos (\theta_{3} ) + 108a^{2} b^{2} l_{3}^{2} \cos (2\theta_{3} ) + 18a^{2} l_{1}^{2} l_{3}^{2} \cos (2\theta_{3} ) \\ & \quad + 18a^{2} l_{2}^{2} l_{3}^{2} \cos (2\theta_{3} ) - 72b^{2} l_{2}^{2} l_{3}^{2} \cos (2\theta_{3} ) - 2l_{1}^{2} l_{2}^{2} l_{3}^{2} \cos (2\theta_{3} ) + 36a^{2} bl_{2}^{2} l_{3} \cos \theta_{3} + 144a^{2} b^{2} l_{2} l_{3} \cos \theta_{3} \\ & \quad - 24a^{2} l_{1}^{2} l_{2} l_{3} \cos \theta_{3} - 12bl_{1}^{2} l_{2}^{2} l_{3} \cos \theta_{3} - 48b^{2} l_{1}^{2} l_{2} l_{3} \cos \theta_{3} + 72a^{2} bl_{2} l_{3}^{2} \cos (2\theta_{3} )) \\ \end{aligned} $$

$$ \begin{aligned} B_{4} & = - 32l_{1}^{2} l_{2} l_{3} (18a^{4} \sin \theta_{3} - 36a^{2} b^{2} \sin \theta_{3} - 18a^{2} bl_{2} \sin \theta_{3} + 9a^{2} bl_{3} \sin (2\theta_{3} ) - 12a^{2} l_{1}^{2} \sin \theta_{3} \\ & \quad - 6a^{2} l_{2}^{2} \sin \theta_{3} - 6a^{2} l_{3}^{2} \sin \theta_{3} + 18b^{4} \sin \theta_{3} + 18b^{3} l_{2} \sin \theta_{3} - 9b^{3} l_{3} \sin (2\theta_{3} ) - 6b^{2} l_{1}^{2} \sin \theta_{3} \\ & \quad + 6b^{2} l_{2}^{2} \sin \theta_{3} - 9b^{2} l_{2} l_{3} \sin (2\theta_{3} ) + 6b^{2} l_{3}^{2} \sin \theta_{3} - 3bl_{2}^{2} l_{3} \sin (2\theta_{3} ) + 6bl_{2} l_{3}^{2} \sin \theta_{3} + 2l_{1}^{4} \sin \theta_{3} \\ & \quad + 2l_{1}^{2} l_{2}^{2} \sin \theta_{3} + l_{1}^{2} l_{2} l_{3} \sin (2\theta_{3} ) + 2l_{1}^{2} l_{3}^{2} \sin \theta_{3} + 2l_{2}^{2} l_{3}^{2} \sin \theta_{3} ) \\ \end{aligned} $$

$$ \begin{aligned} C_{4} & = 8l_{1}^{2} (63a^{4} l_{2}^{2} + 36a^{4} l_{3}^{2} + 9b^{2} l_{2}^{4} + 36b^{4} l_{2}^{2} + 9b^{2} l_{3}^{4} + 36b^{4} l_{3}^{2} + 7l_{1}^{4} l_{2}^{2} + 4l_{1}^{4} l_{3}^{2} + 7l_{2}^{2} l_{3}^{4} + 4l_{2}^{4} l_{3}^{2} \\ & \quad - 27a^{4} l_{3}^{2} \cos^{2} \theta_{3} - 3l_{1}^{4} l_{3}^{2} \cos^{2} \theta_{3} - 3l_{2}^{4} l_{3}^{2} \cos^{2} \theta_{3} - 180a^{2} b^{2} l_{2}^{2} - 72a^{2} b^{2} l_{3}^{2} - 42a^{2} l_{1}^{2} l_{2}^{2} \\ & \quad - 24a^{2} l_{1}^{2} l_{3}^{2} - 66a^{2} l_{2}^{2} l_{3}^{2} - 12b^{2} l_{1}^{2} l_{2}^{2} - 12b^{2} l_{1}^{2} l_{3}^{2} + \,30b^{2} l_{2}^{2} l_{3}^{2} + 26l_{1}^{2} l_{2}^{2} l_{3}^{2} - 36b^{3} l_{3}^{3} \cos \theta_{3} \\ & \quad - 18a^{2} bl_{3}^{3} \cos \theta_{3} + 6bl_{1}^{2} l_{3}^{3} \cos \theta_{3} - 42bl_{2}^{2} l_{3}^{3} \cos \theta_{3} - 108b^{3} l_{2}^{2} l_{3} \cos \theta_{3} - 6bl_{2}^{4} l_{3} \cos \theta_{3} \\ & \quad + 108a^{2} b^{2} l_{3}^{2} \cos^{2} \theta_{3} + 18a^{2} l_{1}^{2} l_{3}^{2} \cos^{2} \theta_{3} + 18a^{2} l_{2}^{2} l_{3}^{2} \cos^{2} \theta_{3} + 72b^{2} l_{2}^{2} l_{3}^{2} \cos^{2} \theta_{3} - 18l_{1}^{2} l_{2}^{2} l_{3}^{2} \cos^{2} \theta_{3} \\ & \quad + 162a^{2} bl_{2}^{2} l_{3} \cos \theta_{3} - 6bl_{1}^{2} l_{2}^{2} l_{3} \cos \theta_{3} ) \\ \end{aligned} $$

$$ \begin{aligned} D_{4} & = - 32l_{1}^{2} l_{2} l_{3} (18a^{4} \sin \theta_{3} - 36a^{2} b^{2} \sin \theta_{3} + 18a^{2} bl_{2} \sin \theta_{3} + 9a^{2} bl_{3} \sin (2\theta_{3} ) - 12a^{2} l_{1}^{2} \sin \theta_{3} \\ & \quad - 6a^{2} l_{2}^{2} \sin \theta_{3} - 6a^{2} l_{3}^{2} \sin \theta_{3} + 18b^{4} \sin \theta_{3} - 18b^{3} l_{2} \sin \theta_{3} - 9b^{3} l_{3} \sin (2\theta_{3} ) - 6b^{2} l_{1}^{2} \sin \theta_{3} \\ & \quad + 6b^{2} l_{2}^{2} \sin \theta_{3} + 9b^{2} l_{2} l_{3} \sin (2\theta_{3} ) + 6b^{2} l_{3}^{2} \sin \theta_{3} - 3bl_{2}^{2} l_{3} \sin (2\theta_{3} ) - 6bl_{2} l_{3}^{2} \sin \theta_{3} + 2l_{1}^{4} \sin \theta_{3} \\ & \quad + 2l_{1}^{2} l_{2}^{2} \sin \theta_{3} - l_{1}^{2} l_{2} l_{3} \sin (2\theta_{3} ) + 2l_{1}^{2} l_{3}^{2} \sin \theta_{3} + 2l_{2}^{2} l_{3}^{2} \sin \theta_{3} ) \\ \end{aligned} $$

$$ \begin{aligned} E_{4} & = 2l_{1}^{2} (18a^{4} l_{2}^{2} + 45a^{4} l_{3}^{2} + 18b^{2} l_{2}^{4} - 72b^{3} l_{2}^{3} + 72b^{4} l_{2}^{2} + 18b^{2} l_{3}^{4} + 72b^{4} l_{3}^{2} + 2l_{1}^{4} l_{2}^{2} + 5l_{1}^{4} l_{3}^{2} + 2l_{2}^{2} l_{3}^{4} \\ & \quad + 5l_{2}^{4} l_{3}^{2} - 27a^{4} l_{3}^{2} \cos (2\theta_{3} ) - 3l_{1}^{4} l_{3}^{2} \cos (2\theta_{3} ) - 3l_{2}^{4} l_{3}^{2} \cos (2\theta_{3} ) - 36a^{2} bl_{2}^{3} + 12bl_{1}^{2} l_{2}^{3} - 12bl_{2}^{3} l_{3}^{2} \\ & \quad + 72a^{2} b^{2} l_{2}^{2} - 36a^{2} b^{2} l_{3}^{2} - 12a^{2} l_{1}^{2} l_{2}^{2} - 30a^{2} l_{1}^{2} l_{3}^{2} - 42a^{2} l_{2}^{2} l_{3}^{2} - 24b^{2} l_{1}^{2} l_{2}^{2} - 24b^{2} l_{1}^{2} l_{3}^{2} - 12b^{2} l_{2}^{2} l_{3}^{2} \\ & \quad + 10l_{1}^{2} l_{2}^{2} l_{3}^{2} - 72b^{3} l_{3}^{3} \cos \theta_{3} - 4l_{2}^{3} l_{3}^{3} \cos \theta_{3} - 12bl_{2} l_{3}^{4} - 36a^{2} bl_{3}^{3} \cos \theta_{3} + 12a^{2} l_{2} l_{3}^{3} \cos \theta_{3} + 12a^{2} l_{2}^{3} l_{3} \cos \theta_{3} \\ & \quad + 12bl_{1}^{2} l_{3}^{3} \cos \theta_{3} + 12bl_{2}^{2} l_{3}^{3} \cos \theta_{3} + 24b^{2} l_{2} l_{3}^{3} \cos \theta_{3} + 24b^{2} l_{2}^{3} l_{3} \cos \theta_{3} + 72b^{3} l_{2}^{2} l_{3} \cos \theta_{3} - 4l_{1}^{2} l_{2} l_{3}^{3} \cos \theta_{3} \\ & \quad - 4l_{1}^{2} l_{2}^{3} l_{3} \cos \theta_{3} + 24bl_{2}^{3} l_{3}^{2} \cos (2\theta_{3} ) + 72b^{3} l_{2} l_{3}^{2} \cos (2\theta_{3} ) + 108a^{2} bl_{2} l_{3}^{2} - 12bl_{1}^{2} l_{2} l_{3}^{2} - 36a^{4} l_{2} l_{3} \cos \theta_{3} \\ & \quad - 12bl_{2}^{4} l_{3} \cos \theta_{3} - 144b^{4} l_{2} l_{3} \cos \theta_{3} - 4l_{1}^{4} l_{2} l_{3} \cos \theta_{3} + 108a^{2} b^{2} l_{3}^{2} \cos (2\theta_{3} ) + 18a^{2} l_{1}^{2} l_{3}^{2} \cos (2\theta_{3} ) \\ & \quad + 18a^{2} l_{2}^{2} l_{3}^{2} \cos (2\theta_{3} ) - 72b^{2} l_{2}^{2} l_{3}^{2} \cos (2\theta_{3} ) - 2l_{1}^{2} l_{2}^{2} l_{3}^{2} \cos (2\theta_{3} ) + 36a^{2} b*l_{2}^{2} l_{3} \cos \theta_{3} - 144a^{2} b^{2} l_{2} l_{3} \cos \theta_{3} \\ & \quad + 24a^{2} l_{1}^{2} l_{2} l_{3} \cos \theta_{3} - 12bl_{1}^{2} l_{2}^{2} l_{3} \cos \theta_{3} + 48b^{2} l_{1}^{2} l_{2} l_{3} \cos \theta_{3} - 72a^{2} bl_{2} l_{3}^{2} \cos (2\theta_{3} )) \\ \end{aligned} $$