SciELO - Scientific Electronic Library Online

 
vol.8 issue2Routing with Wavelet-Based Self-Similarity EstimationSynchronizing Hyperchaotic Maps to Encode/Decode Information author indexsubject indexsearch form
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • Have no similar articlesSimilars in SciELO

Share


Computación y Sistemas

Print version ISSN 1405-5546

Comp. y Sist. vol.8 n.2 México Oct./Dec. 2004

 

Un Algoritmo para Resolver la Cinemática Directa de Plataformas Gough–Stewart Tipo 6–3

 

An Algorithm to Solve Forward Kinematics Gough Stewart 6–3 Platforms

 

J. Gallardo–Alvarado, J.M. Rico–Martínez y H. Orozco–Mendoza

 

Departamento de Ingeniería Mecánica Instituto Tecnológico de Celaya 38010 Celaya, Gto., México Tel: +52 (461)6117575, Fax: +52 (461)6117979. e–mails: gjaime@itc.mx ; mrico@itc.mx ; horacio@itc.mx

 

Article received on June 06, 2002
Accepted on July 30, 2004

 

Resumen

Un algoritmo para resolver la cinemática directa, hasta el análisis de aceleración, de una plataforma Gough–Stewart con una topología especial, conocida como tipo 6–3, es introducido en este trabajo. El análisis directo de posición se lleva a efecto aplicando simple conceptos geométricos que conducen a un sistema no lineal de tres ecuaciones con tres incógnitas, el cual se resuelve por medio del método de Newton–Raphson. Las propiedades de la forma de Klein, una forma simétrica bilineal o producto interno del álgebra de Lie $e(3)$, permiten obtener expresiones simples y compactas para el cálculo de la velocidad angular y de la aceleración angular de la plataforma móvil con respecto a la plataforma fija. Para este fin, el estado de velocidad, o el giro sobre un tornillo (Ball 1900), y el estado de aceleración reducida de la plataforma móvil se expresan en forma de tornillos a través de cada una de las seis cadenas serie del manipulador paralelo. Con la ayuda del programa de computadora Maple© se resuelve un ejemplo numérico, y los resultados numéricos así generados se validan con el programa de análisis ADAMS©.

Palabras Clave: Plataforma paralela, Análisis de aceleración, Forma de Klein, Teoría de tornillos, Cinemática.

 

Abstract

An algorithm for solving the forward kinematics, up to the acceleration analysis, of a Gough Stewart platform with a special topology, namely type 6–3, is introduced in this work. The forward position analysis is carried out by applying simple geometric procedures that leads to a non–linear system of three equations with three unknowns, which is solved by means of the Newton–Raphson method. Afterwards, the properties of the Klein form, a bilinear symmetric form or inner product of the Lie algebra e(3), allow to obtain simple and compact expressions for the computation of the angular velocity and the angular acceleration of the moving platform with respect to the fixed platform. To this end, the velocity state, or the twist about a screw (Ball 1900), and the reduced acceleration state of the moving platform are expressed in screw form through each one of the six limbs of the parallel manipulator. With the aid of special software like Maple a numerical example is solved, and the numerical results so obtained are validated with the software of analysis ADAMS ©.

Keywords: Parallel platform, Acceleration Analysis, Klein form, Screw Theory, Kinematics.

 

DESCARGAR ARTÍCULO EN FORMATO PDF

 

Reconocimientos

Los autores agradecen al Consejo de Ciencia y Tecnología del Estado de Guanajuato, Concyteg, y al Consejo del Sistema Nacional de Educación Tecnológica, Cosnet, el apoyo económico otorgado para la realización de la presente investigación.

De igual forma se agradece al grupo SSC, campus San Miguel de Allende Guanajuato, las facilidades otorgadas para la utilización del programa de análisis de mecanismos por computadora ADAMS©.

 

Referencias

1. Ball, R.S., A Treatise on the Theory of Screws, Cambridge University Press: Cambridge U. K., 1900 (Reedición 1998).        [ Links ]

2. Baron, L. y Angeles, J., "The direct kinematics of parallel manipulators under joint–sensor redundancy", IEEE Transactions on Robotoics and Automation, Vol. 16, No. 1, 2000, pp. 12–19.        [ Links ]

3. Brand, L., Vector and Tensor Analysis, John Wiley & Sons, New York, 1947.        [ Links ]

4. Di Gregorio, R., "Singularity–locus expression of a class of parallel mechanisms", Robotica, Vol. 20, 2002, pp. 323–328.        [ Links ]

5. Gallardo, J. y Rico, J.M., "Screw theory and helicoidal fields", Proc. 1998 ASME Design Engineering Technical Conferences, CD–ROM, Paper DETC98/MECH–5893.        [ Links ]

6. Gosselin, C, y Angeles, J., "Singularity analysis of closed–loop kinematic chains", IEEE Transactions on Robotics and Automation, Vol. 6, No. 32, 1990, pp. 281–290.        [ Links ]

7. Husty, M.L., "An algorithm for solving the direct kinematics of general Stewart–Gough platforms", Mechanism and Machine Theory, Vol. 31, No. 4, 1996, pp. 365–380.        [ Links ]

8. Innocenti, C., "Forward kinematics in polynomial form of the general Stewart platform", Proc. 1998 ASME Design Engineering Technical Conference, CD–ROM, Paper DETC98/MECH–5894.        [ Links ]

9. Ku, D.–M., "Forward kinematic analysis of a 6–3 type Stewart platform mechanism", IMechE Part K Journal of Multibody Dynamics, Vol. 214, No. K4, pp. 233–241.        [ Links ]

10. Merlet, J.–P., Parallel Robots, Kluwer Academic Publishers, 1999.        [ Links ]

11. Merlet, J.–P., "Direct kinematics of parallel manipulators", IEEE transactions on Robotics and Automation, Vol. 9, No. 6, 1993, pp. 842–846.        [ Links ]

12. Raghavan, M., "The Stewart platform of general geometry has 40 configurations", ASME Journal of Mechanical Design, Vol. 115, No. 2, 1993, pp. 277–282.        [ Links ]

13. Rico, J.M. y Duffy, J., "An application of screw algebra to the acceleration analysis of aerial chains", Mechanism and Machine Theory, Vol. 31, No. 4, 1996, pp. 445–457.        [ Links ]

14. Rico, J.M. y Duffy, J., "Forward and inverse acceleration analyses of in–parallel manipulators", ASME Journal of Mechanical Design, Vol. 122, 2000, pp. 299–303.        [ Links ]

15. St–Onge, B.M. y Gosselin, C., "Singularity analysis and representation of the general Gough–Stewart platform", Int. J. Robotics Research, Vol. 19, No. 3, 2000, pp. 271–288.        [ Links ]

16. Tsai, L.W., Enumeration of Kinematic Structures According to Function, CRC Press, 2001.        [ Links ]

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License