Synthesis of 3-PRS Manipulator Based on Revolute and Cone Angle of Spherical Constraints on Range of Rotational Motion of Spherical Joints

Abstract

This paper presents synthesis of 3-prismatic-revolute-spherical parallel manipulator according to the spherical constraints such as limitation of cone angle on the range of rotational motion of spherical joint because the range of motion of mobile platform of manipulator will be influenced by the spherical constraints such as the range of cone angle of rotational motion of spherical joints. Therefore, while determining the parameters of the 3-PRS manipulator, the limitations on the cone angle of rotational motion of spherical joints are considered. The results of cone angle of rotational motion of each spherical joint in each position $\left( {{\varvec{n}}_{{{\varvec{ji}}}} } \right)$ are determined and compared with the spherical constraint condition of spherical joint of the manipulator. The synthesis of the manipulator is carried out by the limitation on range of motion of cone angle of spherical constraints and is solved by using least-square method, and the synthesized multi-positional 3-PRS parallel manipulator is suitable in surgical and space applications.

Appendices

Appendix 1: An Appendix “The Constraint Condition for the Positions of Revolute Joint of the 3-PRS Manipulator” Section

Equations (2) and (3) contain the terms $\overrightarrow {{P_{i} }} , \overrightarrow {{P_{i} }}^{^{\prime}} , o_{{R_{{P_{i} }} }} , o_{{R_{{P_{i} }}^{^{\prime}} }} , \tilde{j}$ which are represented by

$$\overrightarrow {{P_{i} }} = \left[ {\begin{array}{*{20}c} {P_{xi} } \\ {P_{yi} } \\ {P_{zi} } \\ \end{array} } \right]$$

$$\overrightarrow {{P_{i} }}^{^{\prime}} = \left[ {\begin{array}{*{20}c} {P_{xi} } \\ {P_{yi} } \\ {P_{zi} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {P_{x1} } \\ {P_{y1} } \\ {P_{z1} } \\ \end{array} } \right]$$

$$o_{{R_{{P_{i} }}^{^{\prime}} }} = o_{{R_{{P_{i} }} }} - o_{{R_{{P_{1} }} }}$$

$${o}_{{R}_{{p}_{i}}}=\left[\begin{array}{ccc}{u}_{xi}& {v}_{xi}& {w}_{xi}\\ {u}_{yi}& {v}_{yi}& {w}_{yi}\\ {u}_{zi}& {v}_{zi}& {w}_{zi}\end{array}\right]=\left[\begin{array}{ccc}{s\psi }_{i}{s\theta }_{i}{s\phi }_{i}+{c\theta }_{i}{c\phi }_{i}& {s\psi }_{i}{s\theta }_{i}{c\phi }_{i}{-c\theta }_{i}{s\phi }_{i}& {s\theta }_{i}{c\psi }_{i}\\ {c\psi }_{i}{s\phi }_{i}& {c\psi }_{i}{c\phi }_{i}& -{s\psi }_{i}\\ {s\psi }_{i}{c\theta }_{i}{s\phi }_{i}{-s\theta }_{i}{c\phi }_{i}& {s\psi }_{i}{c\theta }_{i}{c\phi }_{i}+{s\theta }_{i}{s\phi }_{i}& {c\theta }_{i}{c\psi }_{i}\end{array}\right]$$

where $\phi_{i} , \theta_{i} , \psi_{i}$ denotes orientations of moving platform w. r. t. base reference frame

$$\tilde{j} = \left[ {\begin{array}{*{20}c} {\mathop j\limits_{xj} } \\ {\mathop j\limits_{yj} } \\ 1 \\ \end{array} } \right]$$

Appendix 2: An Appendix “Determination of Cone Angle of Each Spherical Joint in Each Position on Range of Rotational Motion of Spherical Constraint of a 3-PRS Manipulator” section

Equation (28) contains the constants $\left( {G_{ji} , H_{ji} ,K_{ji} , L_{ji} , M_{ji} , N_{ji} } \right)$ which are calculated by substituting the synthesized values $\left( {h_{xj} , h_{yj} , h_{zj} } \right),$ the positions of spherical joints values $\left( {q_{xji} , q_{yji} , q_{zji} } \right)$ and the prismatic actuator position values $\left( {P_{{R_{xji} }} , P_{{R_{yji} }} , P_{{R_{zji} }} } \right) .$ The constants $\left( {G_{ji} , H_{ji} ,K_{ji} , L_{ji} , M_{ji} , N_{ji} } \right)$ in range of rotational motion of successive spherical joints in Eq. (28) and used to determined the initial values of $\eta_{ji}$ and $d_{asi}$ are

$$G_{ji} = A_{ji} E_{ji}$$

$$H_{ji} = A_{ji} F_{ji}$$

$$K_{ji} = A_{ji} D_{ji}$$

$$L_{ji} = - C_{ji}^{2}$$

$$M_{ji} = - 2B_{ji} C_{ji}$$

$$N_{ji} = - B_{ji}^{2}$$

$$A_{ji} = a_{xji}^{2} + a_{yji}^{2} + a_{zji}^{2}$$

$$a_{xji} = q_{xji} - P_{{R_{xji} }}$$

$$a_{yji} = q_{yji} - P_{{R_{yji} }}$$

$$a_{zji} = q_{zji} - P_{{R_{zji} }}$$

$$B_{ji} = q_{xci} a_{xji} + q_{yci} a_{yji} + q_{zci} a_{zji} + b_{ji}$$

$$b_{ji} = - \left( {q_{xji} a_{xji} + q_{yji} a_{yji} + q_{zji} a_{zji} } \right)$$

$$C_{{ji}} = n_{{1i}} a_{{xji}} + n_{{2i}} a_{{yji}} + n_{{3i}} a_{{zji}}$$

$$D_{{ji}} = q_{{xci}}^{2} + q_{{yci}}^{2} + q_{{zci}}^{2} - 2\left( {q_{{xji}} q_{{xci}} + q_{{yji}} q_{{yci}} + q_{{zji}} q_{{zci}} } \right) + q_{{xji}}^{2} + q_{{yji}}^{2} + q_{{zji}}^{2}$$

$$E_{{ji}} = n_{{1i}}^{2} + n_{{2i}}^{2} + n_{{3i}}^{2}$$

$$F_{{ji}} = 2\left( {q_{{xci}} n_{{1i}} + q_{{yci}} n_{{2i}} + q_{{zci}} n_{{3i}} - q_{{xji}} n_{{1i}} - q_{{yji}} n_{{2i}} - q_{{zji}} n_{{3i}} } \right)$$