Singularity-Invariant Leg Rearrangements in Stewart–Gough Platforms

Abstract

A la meva família, pel seu recolzament incondicional. AgraïmentsM'agradaria expressar la meva sincera gratitud al professor Federico Thomas.La seva inestimable guia al llarg d'aquests 4 anys m'ha permès no només realitzar aquesta tesi, sinó aprendre queés la recerca i a estimar la professió a través del seu entusiasme en els projectes però també de la seva mirada crítica, sempre constructiva.També voldria agrair a la professora Carme Torras el temps que m'ha dedicat, al llarg de tots els treballs que hem fet junts, pels seus punts de vista sovint diferents que m'han donat més perspectiva i per deixar-me aprendre de la seva professionalitat i versatilitat.La Maria Alberich també mereix el meu agraïment, per haver-me presentat al Federico, i mostrar-me el camí cap a l'Institut i el món de la investigació, així com també pels seus consells al llarg d'aquest temps, des de la carrera fins avui. Finalment, també vull agrair a tots els companys de l'Institut de Robòtica i Informàtica Industrial, pel bon ambient, els bons moments, les opinions compartides i el recolzament moral i professional. I a la meva família i al Joan, per estar sempre al meu costat. AbstractThe Stewart-Gough platform was first introduced by E. Gough in 1954 and, since then, it has been used for many applications thanks to its great stiffness, accuracy and robustness in comparison with serial manipulators. It has triggered the research on parallel manipulators and continues to be the center of many researches because, despite its simple geometry, its analysis translates into challenging mathematical problems. One of the most challenging ones is the geometric interpretation of its singularities, that is, those positions where the platform loses stiffness. A complete geometric characterization of these unstable poses is still an open problem.The present thesis provides new insight into this problem from a completely new approach: finding singularity-invariant leg rearrangements.Finding all the transformations that leave the solution of a problem invariant does not solve it, but it provides a lot of information that contribute to its resolution. In the Stewart-Gough platform context, this indirect approach consists in the characterization of all the leg rearrangements that leave the platform singularity locus invariant. Such singularity-invariant leg rearrangements are shown to be a powerful tool to obtain kinematically equivalent manipulators, to help to visualize at a glance the complexity of its kinematics and to provide a common and original framework for the study of both pose-dependent singularities and architectural singularities of Stewart-Gough platforms.The thesis analyzes all the rigid components that a Stewart-Gough platform can contain on a case-by-case basis. Then, it is shown how some of the most simple components admit any leg rearrangement that preserves the lines and planes that their attachments define. On the contrary, other more complex components only admit rearrangements that preserve some extra geometric constrains. This appare...

Review and Recent Results on Stewart Gough Platforms with Self-Motions

“…As the solvability condition of this system is equivalent with the criterion given in Eq. (12) of [48], also the singularity surface of the manipulator does not change by adding legs of Λ. Moreover, it was shown in [47], that in the general case Λ is oneparametric and that the base anchor points as well as the corresponding platform anchor points of Λ are located on planar cubic curves C and c (see Fig.…”

“…The correspondence between the points of C and c in the complex projective extension 3 P of the Euclidean 3-space is determined by the geometry of the manipulator 6 1 ,..., M m and can be computed according to [47,48]. As this correspondence has not to be a bijection (see Fig.…”

Architectural singularities of a class of pentapods

“…The same implicit equation (5) was deduced [22] by imposing that the squared leg lengths before and after a leg rearrangement -for an arbitrary fixed pose of the moving platform-are affinely related. This kind of reconfigurations keeps not only the singularity locus unchanged, but also the resulting kinematic equations are essentially the same.…”

Basic Result on Type II DM Self-Motions of Planar Stewart Gough Platforms

“…. , M 6 is not architecturally singular, then at least a one-parametric set of legs exists, which can be attached to the given manipulator without changing the forward kinematics [5,9] and the singularity set [2] of the manipulator. Moreover, it was shown that in general the base anchor points M i as well as the corresponding platform anchor points m i are located on planar cubic curves C and c, respectively.…”