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.
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]]>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©.
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