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.

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Abbreviations

\(\overrightarrow {{P_{i} }}\) :

Position vector of \(\overrightarrow {OP}\)

\(\phi_{i} , \theta_{i} , \psi_{i}\) :

Orientations of moving platform w.r.t. base reference-frame

\(o_{{R_{{P_{i} }} }}\) :

Rotation of moving platform about \(\overrightarrow {OP}\)

\(\overrightarrow {{q_{ji} }}\) :

Spherical joint position w.r.t. base reference-frame

\(\overrightarrow {{h_{j} }}\) :

Spherical joint design parameter w.r.t. moving platform reference-frame

\(\tilde{j}\) :

Direction of revolute joint axis

\(\overrightarrow {{g_{j} }}\) :

Position vector of \(\overrightarrow {OB}\)

\(\overrightarrow {{l_{j} }}\) :

Dimensions of moving limb length in the form of vector

\(S_{ci}\) :

Position coordinates of centre of gravity of three spherical joints of moving platform

\(\hat{N}\) :

Normal vector of moving platform

\(\hat{N}_{{\mathop {1i}\limits^{{\smile }} }}\) :

General form of normal vector at spherical joint S1

\(\hat{n}_{{\mathop {1i}\limits^{\smile } }}\) :

Normal vector in normal form at spherical joint S1

\(d_{asi}\) :

Shortest distance between the intersection point of axis of symmetry of spherical joints and the moving platform

\(S_{asi}\) :

Coordinates of axis of symmetry of spherical joints in rotational position

\({\varvec{\eta}}_{{\max}}\) :

Prescribed range of rotational angle of spherical joint

\({\varvec{\eta}}_{{{\varvec{ji}}}}\) :

Obtained range of rotational angles of successive spherical joints

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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Pundru Srinivasa Rao. The first draft of the manuscript was written by Pundru Srinivasa Rao, and all authors commented on previous versions of the manuscript. All others read and approved the final manuscript.

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Correspondence to Srinivasa Rao Pundru.

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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

$$G_{ji} = A_{ji} E_{ji}$$
$$H_{ji} = A_{ji} F_{ji}$$
$$K_{ji} = A_{ji} D_{ji}$$
$$L_{ji} = - C_{ji}^{2}$$
$$M_{ji} = - 2B_{ji} C_{ji}$$
$$N_{ji} = - B_{ji}^{2}$$
$$A_{ji} = a_{xji}^{2} + a_{yji}^{2} + a_{zji}^{2}$$
$$a_{xji} = q_{xji} - P_{{R_{xji} }}$$
$$a_{yji} = q_{yji} - P_{{R_{yji} }}$$
$$a_{zji} = q_{zji} - P_{{R_{zji} }}$$
$$B_{ji} = q_{xci} a_{xji} + q_{yci} a_{yji} + q_{zci} a_{zji} + b_{ji}$$
$$b_{ji} = - \left( {q_{xji} a_{xji} + q_{yji} a_{yji} + q_{zji} a_{zji} } \right)$$
$$C_{{ji}} = n_{{1i}} a_{{xji}} + n_{{2i}} a_{{yji}} + n_{{3i}} a_{{zji}}$$
$$D_{{ji}} = q_{{xci}}^{2} + q_{{yci}}^{2} + q_{{zci}}^{2} - 2\left( {q_{{xji}} q_{{xci}} + q_{{yji}} q_{{yci}} + q_{{zji}} q_{{zci}} } \right) + q_{{xji}}^{2} + q_{{yji}}^{2} + q_{{zji}}^{2}$$
$$E_{{ji}} = n_{{1i}}^{2} + n_{{2i}}^{2} + n_{{3i}}^{2}$$
$$F_{{ji}} = 2\left( {q_{{xci}} n_{{1i}} + q_{{yci}} n_{{2i}} + q_{{zci}} n_{{3i}} - q_{{xji}} n_{{1i}} - q_{{yji}} n_{{2i}} - q_{{zji}} n_{{3i}} } \right)$$

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Pundru, S.R., Nalluri, M.R. Synthesis of 3-PRS Manipulator Based on Revolute and Cone Angle of Spherical Constraints on Range of Rotational Motion of Spherical Joints. J. Inst. Eng. India Ser. C 102, 209–219 (2021). https://doi.org/10.1007/s40032-020-00599-y

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Keywords

  • Spherical constraint
  • Revolute constraint
  • Multi-positions
  • Least-square method