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Research Papers

# Polynomial Solution to the Position Analysis of Two Assur Kinematic Chains With Four Loops and the Same Topology

[+] Author and Article Information
Júlia Borràs

Institut de Robòtica i Informàtica Industrial (UPC-CSIC), Barcelona 08028, Spainjborras@iri.upc.eduDepartment of Engineering (EnDiF), University of Ferrara, Ferrara 44100, Italyjborras@iri.upc.edu

Raffaele Di Gregorio1

Institut de Robòtica i Informàtica Industrial (UPC-CSIC), Barcelona 08028, Spainrdigregorio@ing.unife.itDepartment of Engineering (EnDiF), University of Ferrara, Ferrara 44100, Italyrdigregorio@ing.unife.it

The motion plane is a plane surface perpendicular to all the revolute-pair axes.

The unit sphere is a sphere surface with unit radius and center coincident with the center of the spherical motion. It is worth noting that the unit sphere is perpendicular to all the revolute-pair axes since all the revolute-pair axes converge toward the center of the spherical motion.

The distance between two points on a sphere surface is the length of the shortest great-circle arc joining the two points. On the unit sphere, this distance coincides with the convex central angle delimited by the two radii passing through the two points if the angle is measured in radians.

The measure of the convex central angle between two radius vectors gives the distance, on the unit sphere, between the two points located on the sphere by the two radius vectors.

Remind that radius vectors of the unit sphere coincide with position vectors of the unit-sphere points, located by the radius vectors, in Cartesian reference systems with origin at the unit-sphere center $O$.

1

Corresponding author.

J. Mechanisms Robotics 1(2), 021003 (Jan 06, 2009) (11 pages) doi:10.1115/1.3046134 History: Received March 30, 2008; Revised August 30, 2008; Published January 06, 2009

## Abstract

The direct position analysis (DPA) of a manipulator is the computation of the end-effector poses (positions and orientations) compatible with assigned values of the actuated-joint variables. Assigning the actuated-joint variables corresponds to considering the actuated joints locked, which makes the manipulator a structure. The solutions of the DPA of a manipulator one to one correspond to the assembly modes of the structure that is generated by locking the actuated-joint variables of that manipulator. Determining the assembly modes of a structure means solving the DPA of a large family of manipulators since the same structure can be generated from different manipulators. This paper provides an algorithm that determines all the assembly modes of two structures with the same topology that are generated from two families of mechanisms: one planar and the other spherical. The topology of these structures is constituted of nine links (one quaternary link, four ternary links, and four binary links) connected through 12 revolute pairs to form four closed loops.

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## Figures

Figure 1

Topology of the studied structures: graph vertices represent links and graph edges represent joints (R stands for revolute pair)

Figure 2

Four-loop PS with the topology of Fig. 1

Figure 3

ith loop of the PS: notation (i=1,…,4; k=(i+1) modulo 4)

Figure 4

Four-loop SS with the topology of Fig. 1

Figure 5

ith loop of the SS: notation (i=1,…,4; k=(i+1) modulo 4)

Figure 6

Planar structure: assembly modes corresponding to the real solutions reported in Table 1

Figure 7

Spherical structure: assembly modes corresponding to the real solutions reported in Table 2

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