0
Technical Briefs

# Error Analysis of a 3-DOF Parallel Mechanism for Milling Applications

[+] Author and Article Information
G. E. E. Gojtan

UNINOVE - Associação
Educacional Nove de Julho,
Sao Paulo 01005-010, Brazil

T. A. Hess-Coelho

e-mail: tarchess@.usp.br
Department of Mechatronics and Mechanical Systems Engineering,
University of Sao Paulo,
Sao Paulo 01005-010, Brazil

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received December 17, 2010; final manuscript received January 30, 2013; published online June 10, 2013. Assoc. Editor: Vijay Kumar.

J. Mechanisms Robotics 5(3), 034501 (Jun 24, 2013) (9 pages) Paper No: JMR-10-1178; doi: 10.1115/1.4024235 History: Received December 17, 2010; Revised January 30, 2013

## Abstract

Parallel mechanisms have been investigated during the last two decades, due to the fact that they present some advantages in a comparison with serial structures. This work deals with the error analysis of a 3-dof asymmetric parallel mechanism, purposely conceived for milling applications. In a comparison with the previous proposed concepts, this type of kinematic structure demonstrates a promising behavior. Topologically, the architecture is simpler and lighter than Tricept because it has no central passive limb. In addition, only the constraining active limb needs to satisfy the parallelism and orthogonality conditions. Furthermore, one degree of freedom, associated to the third actuator, is decoupled from the other two. Important issues, related to this type of kinematic structure, such as the mappings of the tool positioning error throughout the available workspace, due to the actuators imprecisions and manufacturing tolerances, are discussed in detail.

<>

## References

Merlet, J. P., 2006, Parallel Robots, 2nd ed., Springer, Dordrecht.
Tsai, L. W., 1999, Robot Analysis: The Mechanics of Serial and Parallel Robots, John Wiley & Sons, New York.
Adept Technology, Inc., “Adept Quattro s 650H,” Livermore, last accessed April 30, 2010, CA, Available at
Khol, R., 1994, “A Machine Tool Built From Mathematics,” American Machinist, pp. 53–55.
Neumann, K. E., 2004, “Next Generation Tricept—A True Revolution in Parallel Kinematics,” Parallel Kinematic Machines in Research and Practice, Neugebauer, R., ed., Verlag Wissenschaftliche Scripten, Zwickau, 24, pp. 591–594.
Gosselin, C. M., Kong, X., Foucault, S., and Bonev, I. A., 2004, “A Fully-Decoupled 3-DOF Translational Parallel Mechanism,” Parallel Kinematic Machines in Research and Practice, Vol. 24, Neugebauer, R., ed., Verlag Wissenschaftliche Scripten, Zwickau, 24, pp. 595–610.
Caro, S., Wenger, P., Bennis, F., and Chablat, D., 2006, “Sensitivity Analysis of the Orthoglide, A 3-DOF Translational Parallel Kinematic Machine,” ASME J. Mech. Des., 128, pp. 392–402.
Huang, T., Li, M., Zhao, X. M., Mei, J. P., Chetwynd, D. G., and Hu, S. J., 2005, “Conceptual Design and Dimensional Synthesis for a 3-DOF Module of the TriVariant—A Novel 5-DOF Reconfigurable Hybrid Robot,” IEEE Trans. Rob., 21, pp. 449–456.
DS Technologie Wekzeugmaschinenbau GmbH, 2006, “Ecospeed F,” Möschengladbach.
Metrom Mechatronische Maschinen GmbH, 2004, Parallel Kinematics Five Axis Milling Machine: Metrom P2000, Chemnitz, Germany, 6 p.
Hong, K.-S., Kim, J.-G., 2000, “Manipulability Analysis of a Parallel Machine Tool: Application to Optimal Link Length Design,” J. Rob. Syst., 17(8), pp 403–415.
Hervé, J. M., 1999, “The Lie Group of Rigid Body Displacements, A Fundamental Tool for Mechanism Design,” Mech. Mach. Theory, 34, pp. 719–730.
Kong, X., and Gosselin, C., 2007, Type Synthesis of Parallel Mechanisms, Springer-Verlag, Berlin-Heidelberg.
Gogu, G., 2009, Structural Synthesis of Parallel Robots: Part 2—Translational Topologies With Two and Three Degrees of Freedom, Kluwer, Dordrecht.
Hess-Coelho, T. A., 2006, “Topological Synthesis of a Parallel Wrist Mechanism,” ASME J. Mech. Des., 128(1), pp. 230–235.
Wang, S. M., and Ehmann, K. F., 2002, “Error Model and Accuracy Analysis of a Six-DOF Stewart Platform,” ASME J. Manuf. Sci. Eng., 124(2), pp. 286–295.
Briot, S., and Bonev, I. A., 2008, “Accuracy Analysis of 3-DOF Planar Parallel Robots,” Mech. Mach. Theory, 43, pp. 445–458.
Liu, H., Huang, T., and Chetwynd, D. G., 2011, “A General Approach for Geometric Error Modeling of Lower Lobility Parallel Manipulators,” ASME J. Mech. Rob., 3(2), p. 021013.
Yang., N., 2005, “Dynamic Neural Network Modeling for Nonlinear, Nonstationary Machine Tool Thermally Induced Error,” Int. J. Mach. Tools Manuf., 45(4), pp. 455–465.
Xi, F., Zhang, D., Mechefske, C. M., and Lang, S. Y. T., 2004, “Global Kinetostatic Modelling of Tripod-Based Parallel Kinematic Machine,” Mech. Mach. Theory, 39(4), pp. 357–377.
Rizk, R., Munteanu, M. G. H., Fauroux, J.-C., and Gogu, G., 2007, “Semi-Analytical Stiffness Model of Parallel Robots From the Isoglide Family via the Sub-Structuring Principle,” 12th IFToMM World Congress. Besançon, France.
Pritschow, G., Eppler, C., and Garber, T., 2002, “Influence of the Dynamic Stiffness on the Accuracy of PKM,” 3rd Chemnitzer Parallelkinematik Seminar, Chemnitz, April 23–25, pp. 313–333.
Wu, G., Bai, S., Kepler, J. A., and Caro, S., 2012, “Error Modeling and Experimental Validation of a Planar 3-PPR Parallel Manipulator With Joint Clearances,” ASME J. Mech. Rob., 4(11), p. 041008.
Sika, Z., Hamrle, V., Valasek, M., and Benes, P., 2012, “Calibrability as Additional Design Criterion of Parallel Kinematic Machines,” Mech. Mach. Theory, 50, pp. 48–63.
Hess-Coelho, T. A., 2007, “An Alternative Procedure for Type Synthesis of Parallel Mechanisms,” 12th IFToMM World Congress, Besançon, France, Paper No. A612.
Angeles, J., 2004, “Qualitative Synthesis of Parallel Manipulators,” ASME J. Mech. Des., 126, pp. 617–624.
Baradat, C., Nabat, V., Company, O., Krut, S., and Pierrot, F., 2008, “Par2: A Spatial Mechanism For Fast Planar, 2-Dof, Pick-And-Place Applications,” Proceedings of the 2nd International Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Montpellier, September 21–22, pp. 261–270.
Kumazawa, V. D., Hess-Coelho, T. A., Rinaudi, D., Carbone, G., and Ceccarelli, M., 2009, “Kinematic Analysis and Operation Feasibility of a 3-DOF Asymmetric Parallel Mechanism,” 20th COBEM, Gramado, Brazil, Paper No. COB09-0744.
Hess-Coelho, T. A., Gojtan, G. E. E., and Furtado, G. P., 2010, “Kinematic Analysis of a 3-DOF Parallel Mechanism for Milling Applications,” Open Mech. Eng. J., Special Issue on Kinematic Design of Manipulators, 4, pp. 48–55.
International Organization for Standardization, 2010, “ISO 286-1:2010—Geometrical Product Specifications (GPS)—ISO Code System for Tolerances on Linear Sizes—Part 1: Basis of Tolerances, Deviations and Fits,” p. 38.

## Figures

Fig. 1

Structural synthesis: (a) the constraining limb and (b) the CAD model of the 2 UPS + PRP

Fig. 2

The notation for the position analysis

Fig. 3

An example of singular configuration when det JP is null: points U1, U2, S1, and S2 belong to the same plane

Fig. 4

First mapping of the kinematic error in the x-direction, assuming that Δh1 = Δh2 = Δh3 = 5 μm, for (a) H = 240 mm and (b) H = 480 mm

Fig. 5

First mapping of the kinematic error in the y-direction, assuming that Δh1 = Δh2 = Δh3 = 5 μm, for (a) H = 240 mm and (b) H = 480 mm

Fig. 6

Second mapping of the kinematic error in the x-direction, assuming that Δh1 = Δh3 = 5 μm and Δh2 = −5 μm, for (a) H = 240 mm and (b) H = 480 mm

Fig. 7

Second mapping of the kinematic error in the y-direction, assuming that Δh1 = Δh3 = 5 μm and Δh2 = −5 μm, for (a) H = 240 mm (b) H = 480 mm

Fig. 8

First mapping of the geometric error in the x-direction, assuming the symmetrical condition, for (a) H = 240 mm and (b) H = 480 mm

Fig. 9

First mapping of the geometric error in the y-direction, assuming the symmetrical condition, for (a) H = 240 mm and (b) H = 480 mm

Fig. 10

Second mapping of the geometric error in the x-direction, assuming the asymmetric condition, for (a) H = 240 mm and (b) H = 480 mm

Fig. 11

Second mapping of the geometric error in the y-direction, assuming the asymmetric condition, for (a) H = 240 mm and (b) H = 480 mm

Fig. 12

Sensitivity of variable θ with respect to the 12 parameters, for (a) H = 240 mm and (b) H = 480 mm

Fig. 13

Sensitivity of variable R with respect to the 12 parameters, for (a) H = 240 mm and (b) H = 480 mm

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections