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Kinematics of a three-legged 1R2T decoupled parallel manipulator

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Abstract

In this work, the kinematics of a three degrees of freedom parallel manipulator able to execute two independent translations and one independent decoupled rotation is approached by means of the theory of screws. The topology of the proposed mechanism is so simple that it is possible to generate in a few steps closed-form solutions for both the inverse and direct position analysis. Later on, the infinitesimal kinematics of the robot is approached by resorting to the theory of screws. In that concern, the analysis of the Jacobian matrices shows that the proposed parallel manipulator is isotropic and practically free of singular configurations. A case study is included with the purpose to numerically exemplify the inverse-forward kinematics, isotropy and workspace of the decoupled parallel manipulator.

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Abbreviations

\(A \; {[}\text {mm}{]}\) :

Center of universal joint

\({\pmb A}_*\) :

Reduced acceleration state

\({\pmb a}_* \; {[}\text {mm} \; \text {s}^{-2}{]}\) :

Acceleration vector

\(B \; {[}\text {mm}{]}\) :

Center of spherical joint

\({\pmb b} \; {[}\text {mm}{]}\) :

Position vector of point B

C   :

Cylindrical joint

EE  :

End-effector

\(e \; {[}\text {mm}{]}\) :

Separation between points \(B_2\) and \(B_3\)

\(f_*\) :

Individual connectivity

\(g \; {[}\text {mm}{]}\) :

Offset between the moving platform and the end-effector

\(h \; {[}\text {mm}{]}\) :

Length of the central limb

\(\hat{\pmb i} \; \) :

Unit vector associated to the \(X-\)axis

\({ J} \;\) :

Jacobian matrix

\(\hat{\pmb j} \; \) :

Unit vector associated to the \(Y-\)axis

\(k \;\) :

Number of kinematic pairs

\(\hat{\pmb k} \;\) :

Unit vector associated to the \(Z-\)axis

\({\pmb L} \;\) :

Lie screw of acceleration of the robot

\({\pmb L}_* \;\) :

Lie screw of acceleration of limb

\(M \;\) :

Mobility of parallel manipulator

\(m \;\) :

Moving platform

\(n \;\) :

Number of links

\(\mathrm{O}\_\mathrm{XYZ} \;\) :

Fixed reference frame

\(\mathrm{o}\_\mathrm{xyz} \;\) :

Moving reference frame

P  :

Prismatic joint

\({\mathrm Qa} \;\) :

Second-order driver matrix

\({\mathrm Qv} \;\) :

First-order driver matrix

\(q_* \;\) :

Generalized coordinate

R  :

Revolute joint, rotational freedom

\({^*{\mathrm {R}}^*} \;\) :

Rotation matrix

\({\pmb r}_{*/*} \; {[}\text {mm}{]}\) :

Position vector

S  :

Spherical joint

T  :

Translational freedom

\(t \; {[}\text {s}{]}\) :

Time

U  :

Universal joint

\(\hat{\pmb u} \;\) :

Unit vector between spherical joints

\({\pmb V}_* \; \) :

Velocity state

\({\pmb v}_* \; {[}\text {mm} \; \text {s}^{-1}{]}\) :

Velocity vector

\({\alpha } \; {[}\text {rad} \; \text {s}^{-2}{]}\) :

Joint acceleration rate

\({\pmb \alpha } \; {[}\text {rad} \; \text {s}^{-2}{]}\) :

Angular acceleration vector

\(\Delta \;\) :

Operator of polarity

\(\phi \; {}[\text {rad}{]}\) :

Orientation of the end-effector

\(\lambda \;\) :

Mobility of rigid body

\(\nu \;\) :

Number of over-constraints

\({\omega } \; {[}\text {mm} \; \text {s}^{-1}{]}\) :

Joint velocity rate

\({\pmb \omega } \; {[}\text {rad} \; \text {s}^{-1}{]}\) :

Angular velocity vector

\(\psi \;\) :

Manipulability index of Yoshikawa

\(\zeta \;\) :

Number of local degrees of freedom

\(\begin{bmatrix} * \; * \end{bmatrix}\) :

Lie product

\(\left\{ * ; * \right\} \) :

Klein form

SCARA :

Selective Compliant Assembly Robot Arm

\(\times \) :

Cross product

\(\cdot \) :

Inner product

\(\$\) :

Infinitesimal screw

0:

Fixed platform

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Authors and Affiliations

  1. Department of Mechanical Engineering, DICIS, Universidad de Guanajuato, Palo Blanco, 36885, Salamanca, GTO, Mexico

    Jaime Gallardo-Alvarado

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  1. Jaime Gallardo-Alvarado
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Correspondence to Jaime Gallardo-Alvarado.

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Gallardo-Alvarado, J. Kinematics of a three-legged 1R2T decoupled parallel manipulator. J Braz. Soc. Mech. Sci. Eng. 45, 109 (2023). https://doi.org/10.1007/s40430-023-04032-0

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