Cusp points and assembly changing motions in the PRR-PR-PRR planar parallel manipulator

Abstract

Most parallel manipulators have multiple solutions to the direct kinematic problem. The ability to perform assembly changing motions has received the attention of a few researchers. Cusp points play an important role in the kinematic behavior. This study investigates the cusp points and assembly changing motions in a two degrees of freedom planar parallel manipulator. The direct kinematic problem of the manipulator yields a quartic polynomial equation. Each root in the equation determines the assembly configuration, and four solutions are obtained for a given set of actuated joint coordinates. By regarding the discriminant of the repeated roots of the quartic equation as an implicit function of two actuated joint coordinates, the direct kinematic singularity loci in the joint space are determined by the implicit function. Cusp points are then obtained by the intersection of a quadratic curve and a cubic curve. Two assembly changing motions by encircling different cusp points are highlighted, for each pair of solutions with the same sign of the determinants of the direct Jacobian matrices.

Key words： planar parallel manipulator; assembly changing motions; cusp points; quartic polynomial; discriminant of repeated roots

Nomenclature

b1, b2, l1, l2	Geometric parameters of the manipulator under study
Ci	Coefficient of the quartic polynomial equation
fi	Constraint equation of a non-redundant manipulator
g	Reduced configuration space of a non-redundant manipulator
h, φ	Pose coordinates of the moving platform of the manipulator under study
I, J	Redefined variables of the quartic polynomial equation
JDKP	Direct Jacobian matrix
JIKP	Inverse Jacobian matrix
p, α	Actuated joint coordinates of the manipulator under study
P, λ	Redefined unknown variables
q	Direct kinematic singularity loci in the joint space of a non-redundant manipulator
t1, t2	Output variables for a non-redundant manipulator
u	Tangent-half-angle of φ
ρ1, ρ2	Input variables for a non-redundant manipulator
\mathrm{\Delta }	Quartic discriminant

Acknowledgements

The author’s contributions are as follows: Xu Pei was in charge of the whole trial; Chengwei Shen wrote the manuscript; Jingjun Yu assisted with sampling. The first author thanks Prof. Lubin Hang of Shanghai University of Engineering Science for his previous introduction on cusp points and singularity-free transitions between DKP solutions in parallel manipulators, which stimulated interest in this study. This research was partially supported by the National Natural Science Foundation of China (Grant Nos. U1813221 and 52075015).

Conflict of Interests

The authors declare no competing financial interests.

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