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Research Papers

# A Unified Approach to Direct Kinematics of Some Reduced Motion Parallel Manipulators

[+] Author and Article Information
P. J. Zsombor-Murray

Centre for Intelligent Machines, McGill University, Montreal, QC, H3A 2K6, Canadapaul@cim.mcgill.ca

A. Gfrerrer

Institut für Geometrie, TU-Graz, Graz A8010, Austriagfrerrer@tugraz.at

Husty (7) was first to apply this technique to formulate the DK algorithm for the general Stuart–Gough platform manipulator where six points in EE are displaced onto six spheres in FF.

Only six of the bilinear terms $xiyj$ occur.

Vogler (16) recently gave an alternative proof of this fact.

As one can easily check $k2−1≥0$ is equivalent with $dist(P1,P2)≥dist(l1,l2)$. Clearly a solution to the DK problem exists only if the latter condition holds.

The factor $1/2l$ can be omitted since we deal with homogeneous equations.

Strictly speaking, Husty (7) introduced the technique in his notable DK solution of the general Stewart–Gough platform. Here we have reintroduced the technique in the context of parallel Schönflies robots.

J. Mechanisms Robotics 2(2), 021006 (Apr 19, 2010) (10 pages) doi:10.1115/1.4001095 History: Received March 29, 2009; Revised November 19, 2009; Published April 19, 2010; Online April 19, 2010

## Abstract

After discussing the Study point transformation operator, a unified way to formulate kinematic problems, using “points moving on planes or spheres” constraint equations, is introduced. Application to the direct kinematics problem solution of a number of different parallel Schönflies motion robots is then developed. Certain not widely used but useful tools of algebraic geometry are explained and applied for this purpose. These constraints and tools are also applied to some special parallel robots called “double triangular” to show that the approach is flexible and universally pertinent to manipulator kinematics in reducing the complexity of some previously achieved solutions. Finally a novel two-legged Schönflies architecture is revealed to emphasize that good design is not only essential to good performance but also to easily solve kinematic models. In this example architecture, with double basally actuated legs so as to minimize moving mass, the univariate polynomial solution turns out to be simplest, i.e., of degree 2.

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## Figures

Figure 1

Various leg architectures in a variety of Schönflies parallel manipulator contexts

Figure 2

Two-legged Schönflies manipulator

Figure 3

Two screw actuators for double basal actuation

Figure 4

Spherical double triangular manipulator

Figure 5

Spatial double triangular manipulator

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