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Research Papers

# Position Analysis, Workspace, and Optimization of a 3-$P̱$PS Spatial Manipulator

[+] Author and Article Information
M. Ruggiu

Department of Mechanical Engineering, University of Cagliari, Cagliari 09123, Italyruggiu@dimeca.unica.it

J. Mech. Des 131(5), 051010 (Apr 15, 2009) (9 pages) doi:10.1115/1.3116257 History: Received June 09, 2008; Revised February 19, 2009; Published April 15, 2009

## Abstract

The present paper describes the analytical solution of position kinematics for a three degree-of-freedom parallel manipulator. It also provides a numeric example of workspace calculation and a procedure for its optimization. The manipulator consists of a base and a moving platform connected to the base by three identical legs; each leg is provided with a $P̱$PS chain, where $P̱$ designates an actuated prismatic pair, P stands for a passive prismatic pair, and S a spherical pair. The direct analysis yields a nonlinear system with eight solutions at the most. The inverse analysis is solved in three relevant cases: (i) the orientation of the moving platform is given, (ii) the position of a reference point of the moving platform is given, and (iii) two rotations (pointing) and one translation (focusing) are given. In the present paper it is proved that case (i) yields an inverse singularity condition of the mechanism; case (ii) provides a nonlinear system with four distinct solutions at the most; case (iii) allows the finding of some geometrical configurations of the actuated pairs for minimizing parasitic movements in the case of a pointing/focusing operation of the manipulator.

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## Figures

Figure 1

Geometry of the 3-P̱PS

Figure 2

Kinematic description of the manipulator

Figure 3

Inverse kinematic singularity: sequence of motion

Figure 4

Euler angles of the moving platform

Figure 5

ψ and ϑ pointing Euler angles

Figure 6

Position of the reference point of the moving platform

Figure 7

Motion of the reference point in the z=0 plane

Figure 8

Definition of β

Figure 9

Comparison between δmin and δ(β=π/3): ● δmin and +δ(β=π/3)

Figure 10

Occurrence of optimal β values

Figure 11

Comparison between δs calculated by different βs: ● β=0, +β=π/4, and ◻ β=π/2

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