Cusp Points in the Parameter Space of Degenerate 3-RPR Planar Parallel Manipulators

This paper investigates the conditions in the design parameter space for the existence and distribution of the cusp locus for planar parallel manipulators. Cusp points make possible non-singular assembly-mode changing motion, which increases the maximum singularity-free workspace. An accurate algorithm for the determination is proposed amending some imprecisions done by previous existing algorithms. This is combined with methods of Cylindric Algebraic Decomposition, Gr\"obner bases and Discriminant Varieties in order to partition the parameter space into cells with constant number of cusp points. These algorithms will allow us to classify a family of degenerate 3-RPR manipulators.Comment: ASME Journal of Mechanisms and Robotics (2012) 1-1

Generation of the global workspace roadmap of the 3-RPR using rotary disk search

Path planning for parallel manipulators in the configuration space can be a challenging task due to the existence of multiple direct kinematic solutions. Hence the aim of this paper is to define a generalised hierarchical path planning scheme for trajectory generation between two configurations in the configuration space for manipulators that exhibit more than one solution in their direct kinematics. This process is applied to the 3-RPR mechanism, constrained to a 2-DOF system by setting active joint parameter ρ1 to a constant. The overall reachable workspace is discretised and deconstructed into smaller patches, which are then stitched together creating a global workspace roadmap. Using the roadmap, path feasibility is obtained and local path planning is used to generate a complete trajectory. This method can determine a singularity-free path between any two connectible points in the configuration space, including assembly mode changes. © 2014 Elsevier Ltd

Kinetostatic Analysis and Solution Classification of a Planar Tensegrity Mechanism

Tensegrity mechanisms have several interesting properties that make them suitable for a number of applications. Their analysis is generally challenging because the static equilibrium conditions often result in complex equations. A class of planar one-degree-of-freedom (dof) tensegrity mechanisms with three linear springs is analyzed in detail in this paper. The kinetostatic equations are derived and solved under several loading and geometric conditions. It is shown that these mechanisms exhibit up to six equilibrium configurations, of which one or two are stable. Discriminant varieties and cylindrical algebraic decomposition combined with Groebner base elimination are used to classify solutions as function of the input parameters.Comment: 7th IFToMM International Workshop on Computational Kinematics, May 2017, Poitiers, France. 201

Kinematics, workspace and singularity analysis of a multi-mode parallel robot

A family of reconfigurable parallel robots can change motion modes by passing through constraint singularities by locking and releasing some passive joints of the robot. This paper is about the kinematics, the workspace and singularity analysis of a 3-PRPiR parallel robot involving lockable Pi and R (revolute) joints. Here a Pi joint may act as a 1-DOF planar parallelogram if its lock-able P (prismatic) joint is locked or a 2-DOF RR serial chain if its lockable P joint is released. The operation modes of the robot include a 3T operation modes to three 2T1R operation modes with two different directions of the rotation axis of the moving platform. The inverse kinematics and forward kinematics of the robot in each operation modes are dealt with in detail. The workspace analysis of the robot allow us to know the regions of the workspace that the robot can reach in each operation mode. A prototype built at Heriot-Watt University is used to illustrate the results of this work.Comment: International Design Engineering Technical Conferences \& Computers and Information in Engineering Conference, Aug 2017, Cleveland, United States. 201

The 3-PPPS parallel robot with U-shape Base, a 6-DOF parallel robot with simple kinematics

International audienceOne of the main problems associated with the use of 6 DOF parallel robots remains the solving of their kinematic models. This is rarely possible to analytically solve their models thereby justifying the application of numerical methods. These methods are difficult to implement in an industrial controller and can cause solution bifurcations close to singularities resulting in following an unplanned trajectory. Recently, a 3-PPPS robot with U-shaped base was introduced where an analytical kinematic model can be derived. Previously, quaternion parameters were used to represent the orientation of the mobile platform. To allow for simpler model handling, this article introduces the use of Euler angles which have a physical meaning for the users. Compact writing of the direct and inverse kinematic model is thus obtained. Using algebraic and cylindrical decomposition for the workspace, this provides a simpler representation of the largest domain without singularity around the " home " configuration

Workspace, Joint space and Singularities of a family of Delta-Like Robot

International audienceThis paper presents the workspace, the joint space and the singularities of a family of delta-like parallel robots by using algebraic tools. The different functions of SIROPA library are introduced, which is used to induce an estimation about the complexity in representing the singularities in the workspace and the joint space. A Groebner based elimination is used to compute the singularities of the manipulator and a Cylindrical Algebraic Decomposition algorithm is used to study the workspace and the joint space. From these algebraic objects, we propose some certified three-dimensional plotting describing the shape of workspace and of the joint space which will help the engineers or researchers to decide the most suited configuration of the manipulator they should use for a given task. Also, the different parameters associated with the complexity of the serial and parallel singularities are tabulated, which further enhance the selection of the different configuration of the manipulator by comparing the complexity of the singularity equations

Distance-based formulations for the position analysis of kinematic chains

This thesis addresses the kinematic analysis of mechanisms, in particular, the position analysis of kinematic chains, or linkages, that is, mechanisms with rigid bodies (links) interconnected by kinematic pairs (joints). This problem, of completely geometrical nature, consists in finding the feasible assembly modes that a kinematic chain can adopt. An assembly mode is a possible relative transformation between the links of a kinematic chain. When an assignment of positions and orientations is made for all links with respect to a given reference frame, an assembly mode is called a configuration. The methods reported in the literature for solving the position analysis of kinematic chains can be classified as graphical, analytical, or numerical. The graphical approaches are mostly geometrical and designed to solve particular problems. The analytical and numerical methods deal, in general, with kinematic chains of any topology and translate the original geometric problem into a system of kinematic analysis of all the Assur kinematic chains resulting from replacing some of its revolute joints by slider joints. Thus, it is concluded that the polynomials of all fully-parallel planar robots can be derived directly from that of the widely known 3-RPR robot. In addition to these results, this thesis also presents an efficient procedure, based on distance and oriented area constraints, and geometrical arguments, to trace coupler curves of pin-jointed Gr¨ubler kinematic chains. All these techniques and results together are contributions to theoretical kinematics of mechanisms, robot kinematics, and distance plane geometry. equations that defines the location of each link based, mainly, on independent loop equations. In the analytical approaches, the system of kinematic equations is reduced to a polynomial, known as the characteristic polynomial of the linkage, using different elimination methods —e.g., Gr¨obner bases or resultant techniques. In the numerical approaches, the system of kinematic equations is solved using, for instance, polynomial continuation or interval-based procedures. In any case, the use of independent loop equations to solve the position analysis of kinematic chains, almost a standard in kinematics of mechanisms, has seldom been questioned despite the resulting system of kinematic equations becomes quite involved even for simple linkages. Moreover, stating the position analysis of kinematic chains directly in terms of poses, with or without using independent loop equations, introduces two major disadvantages: arbitrary reference frames has to be included, and all formulas involve translations and rotations simultaneously. This thesis departs from this standard approach by, instead of directly computing Cartesian locations, expressing the original position problem as a system of distance-based constraints that are then solved using analytical and numerical procedures adapted to their particularities. In favor of developing the basics and theory of the proposed approach, this thesis focuses on the study of the most fundamental planar kinematic chains, namely, Baranov trusses, Assur kinematic chains, and pin-jointed Gr¨ubler kinematic chains. The results obtained have shown that the novel developed techniques are promising tools for the position analysis of kinematic chains and related problems. For example, using these techniques, the characteristic polynomials of most of the cataloged Baranov trusses can be obtained without relying on variable eliminations or trigonometric substitutions and using no other tools than elementary algebra. An outcome in clear contrast with the complex variable eliminations require when independent loop equations are used to tackle the problem. The impact of the above result is actually greater because it is shown that the characteristic polynomial of a Baranov truss, derived using the proposed distance-based techniques, contains all the necessary and sufficient information for solving the positionEsta tesis aborda el problema de análisis de posición de cadenas cinemáticas, mecanismos con cuerpos rígidos (enlaces) interconectados por pares cinemáticos (articulaciones). Este problema, de naturaleza geométrica, consiste en encontrar los modos de ensamblaje factibles que una cadena cinemática puede adoptar. Un modo de ensamblaje es una transformación relativa posible entre los enlaces de una cadena cinemática. Los métodos reportados en la literatura para la solución del análisis de posición de cadenas cinemáticas se pueden clasificar como gráficos, analíticos o numéricos. Los enfoques gráficos son geométricos y se diseñan para resolver problemas particulares. Los métodos analíticos y numéricos tratan con cadenas cinemáticas de cualquier topología y traducen el problema geométrico original en un sistema de ecuaciones cinemáticas que define la ubicación de cada enlace, basado generalmente en ecuaciones de bucle independientes. En los enfoques analíticos, el sistema de ecuaciones cinemáticas se reduce a un polinomio, conocido como el polinomio característico de la cadena cinemática, utilizando diferentes métodos de eliminación. En los métodos numéricos, el sistema se resuelve utilizando, por ejemplo, la continuación polinomial o procedimientos basados en intervalos. En cualquier caso, el uso de ecuaciones de bucle independientes, un estándar en cinemática de mecanismos, rara vez ha sido cuestionado a pesar de que el sistema resultante de ecuaciones es bastante complicado, incluso para cadenas simples. Por otra parte, establecer el análisis de la posición de cadenas cinemáticas directamente en términos de poses, con o sin el uso de ecuaciones de bucle independientes, presenta dos inconvenientes: sistemas de referencia arbitrarios deben ser introducidos, y todas las fórmulas implican traslaciones y rotaciones de forma simultánea. Esta tesis se aparta de este enfoque estándar expresando el problema de posición original como un sistema de restricciones basadas en distancias, en lugar de directamente calcular posiciones cartesianas. Estas restricciones son posteriormente resueltas con procedimientos analíticos y numéricos adaptados a sus particularidades. Con el propósito de desarrollar los conceptos básicos y la teoría del enfoque propuesto, esta tesis se centra en el estudio de las cadenas cinemáticas planas más fundamentales, a saber, estructuras de Baranov, cadenas cinemáticas de Assur, y cadenas cinemáticas de Grübler. Los resultados obtenidos han demostrado que las técnicas desarrolladas son herramientas prometedoras para el análisis de posición de cadenas cinemáticas y problemas relacionados. Por ejemplo, usando dichas técnicas, los polinomios característicos de la mayoría de las estructuras de Baranov catalogadas se puede obtener sin realizar eliminaciones de variables o sustituciones trigonométricas, y utilizando solo álgebra elemental. Un resultado en claro contraste con las complejas eliminaciones de variables que se requieren cuando se utilizan ecuaciones de bucle independientes. El impacto del resultado anterior es mayor porque se demuestra que el polinomio característico de una estructura de Baranov, derivado con las técnicas propuestas, contiene toda la información necesaria y suficiente para resolver el análisis de posición de las cadenas cinemáticas de Assur que resultan de la sustitución de algunas de sus articulaciones de revolución por articulaciones prismáticas. De esta forma, se concluye que los polinomios de todos los robots planares totalmente paralelos se pueden derivar directamente del polinomio característico del conocido robot 3-RPR. Adicionalmente, se presenta un procedimiento eficaz, basado en restricciones de distancias y áreas orientadas, y argumentos geométricos, para trazar curvas de acoplador de cadenas cinemáticas de Grübler. En conjunto, todas estas técnicas y resultados constituyen contribuciones a la cinemática teórica de mecanismos, la cinemática de robots, y la geometría plana de distancias. Barcelona 13

Advances in Robot Kinematics : Proceedings of the 15th international conference on Advances in Robot Kinematics

International audienceThe motion of mechanisms, kinematics, is one of the most fundamental aspect of robot design, analysis and control but is also relevant to other scientific domains such as biome- chanics, molecular biology, . . . . The series of books on Advances in Robot Kinematics (ARK) report the latest achievement in this field. ARK has a long history as the first book was published in 1991 and since then new issues have been published every 2 years. Each book is the follow-up of a single-track symposium in which the participants exchange their results and opinions in a meeting that bring together the best of world’s researchers and scientists together with young students. Since 1992 the ARK symposia have come under the patronage of the International Federation for the Promotion of Machine Science-IFToMM.This book is the 13th in the series and is the result of peer-review process intended to select the newest and most original achievements in this field. For the first time the articles of this symposium will be published in a green open-access archive to favor free dissemination of the results. However the book will also be o↵ered as a on-demand printed book.The papers proposed in this book show that robot kinematics is an exciting domain with an immense number of research challenges that go well beyond the field of robotics.The last symposium related with this book was organized by the French National Re- search Institute in Computer Science and Control Theory (INRIA) in Grasse, France

Singularity-free workspace analysis and geometric optimization of parallel mechanisms

Les mécanismes parallèles sont fréquemment utilisés comme robots manipulateurs, comme simulateurs de mouvement, comme machines parallèles, etc. Cependant, à cause des chaînes cinématiques fermées qui caractérisent leur architecture, le mouvement de leur plateforme est limité et des singularités cinématiques complexes peuvent apparaître à l'intérieur de leur espace de travail. Par conséquent, une maximisation l'espace de travail libre de singularité pour ce type de mécanismes est souhaitable dans un contexte de conception. Dans cette thèse, deux types de mécanismes parallèles sont étudiés: les mécanismes parallèles plans ?avec, en particulier le 3-RPR? et les mécanismes spatiaux ?avec, en particulier, la plateforme de Gough-Stewart. Pour chaque type de mécanisme parallèle, une forme simple d'équation de singularité est obtenue. Le principe consiste à séparer l'origine O' du repère mobile du point considéré P et de faire coïncider O' avec un point particulier de la plateforme. L'équation ainsi obtenue est l'équation de singularité du point P de la plateforme qui contient un ensemble minimal de paramètres géométriques. Par ailleurs, il est prouvé que les centres des cercles et sphères définissant l'espace de travail se trouvent exactement sur les lieux de singularité. Cette observation et l'équation de singularité simplifiée constituent les points de départ de l'analyse de l'espace de travail libre de singularité ainsi que de l'optimisation géométrique. Pour le mécanisme parallèle plan 3-RPR, l'espace de travail libre de singularité et les limites correspondantes pour la longueur des pattes dans une orientation prescrite sont déterminés. Ensuite l'architecture optimale qui permet d'obtenir un espace de travail maximal tout en étant libre de singularité est discutée. En ce qui concerne la plateforme de Gough-Stewart, cette thèse se concentre sur le manipulateur symétrique simplifié minimal (MSSM). Comme une plateforme de Gough- Stewart a 6 degrés de liberté, son espace de travail se divise en deux: l'espace de travail en position (ou simplement espace de travail) et l'espace de travail en orientation. A partir de l'équation de singularité simplifiée, une procédure générale est développée afin de déterminer l'espace de travail libre de singularité maximal autour d'un point particulier dans une orientation donnée, et afin de déterminer les limites correspondantes des longueurs de patte. Dans le but de maximiser l'espace de travail libre de singularité en orientation, un algorithme est présenté qui optimise les trois angles d'orientation. Sachant qu'une plateforme fonctionne habituellement pour une certaine gamme d'orientations, deux algorithmes qui calculent l'espace de travail en orientation libre de singularité maximal sont présentés. En utilisant les angles d'Euler en roulis, tangage et lacet, l'espace de travail en orientation pour une position prescrite peut être défini par 12 surfaces. Basé sur ce fait, un algorithme numérique est présenté qui évalue et représente l'espace de travail en orientation pour une position prescrite dans les limites données de longueur de patte. Ensuite, une procédure est proposée afin de déterminer l'espace de travail en orientation libre singularité maximal ainsi que les limites correspondantes des longueurs de patte. En pratique, une plateforme peut fonctionner dans un ensemble de positions. Ainsi, l'effet de la position de travail sur l'espace de travail en orientation libre de singularité maximal est analysé et deux algorithmes sont proposés pour calculer ce dernier pour tout un ensemble de positions particulières. Finalement, un algorithme qui optimise les paramètres géométriques est développé dans le but de déterminer l'architecture optimale qui permet à la plateforme de MSSM Gough-Stewart d'obtenir l'espace de travail libre singularité maximal autour d'une position particulière pour l'orientation de référence. Les résultats obtenus peuvent être utilisés pour la conception géométrique, la configuration des paramètres (longueur des pattes) ou la planification de trajectoires libres de singularité des mécanismes parallèles considérés. En outre, les algorithmes proposés peuvent également être appliqués à d'autres types de mécanismes parallèles

Numerical computation and avoidance of manipulator singularities

This thesis develops general solutions to two open problems of robot kinematics: the exhaustive computation of the singularity set of a manipulator, and the synthesis of singularity-free paths between given configurations. Obtaining proper solutions to these problems is crucial, because singularities generally pose problems to the normal operation of a robot and, thus, they should be taken into account before the actual construction of a prototype. The ability to compute the whole singularity set also provides rich information on the global motion capabilities of a manipulator. The projections onto the task and joint spaces delimit the working regions in such spaces, may inform on the various assembly modes of the manipulator, and highlight areas where control or dexterity losses can arise, among other anomalous behaviour. These projections also supply a fair view of the feasible movements of the system, but do not reveal all possible singularity-free motions. Automatic motion planners allowing to circumvent problematic singularities should thus be devised to assist the design and programming stages of a manipulator. The key role played by singular configurations has been thoroughly known for several years, but existing methods for singularity computation or avoidance still concentrate on specific classes of manipulators. The absence of methods able to tackle these problems on a sufficiently large class of manipulators is problematic because it hinders the analysis of more complex manipulators or the development of new robot topologies. A main reason for this absence has been the lack of computational tools suitable to the underlying mathematics that such problems conceal. However, recent advances in the field of numerical methods for polynomial system solving now permit to confront these issues with a very general intention in mind. The purpose of this thesis is to take advantage of this progress and to propose general robust methods for the computation and avoidance of singularities on non-redundant manipulators of arbitrary architecture. Overall, the work seeks to contribute to the general understanding on how the motions of complex multibody systems can be predicted, planned, or controlled in an efficient and reliable way.Aquesta tesi desenvolupa solucions generals per dos problemes oberts de la cinemàtica de robots: el càlcul exhaustiu del conjunt singular d'un manipulador, i la síntesi de camins lliures de singularitats entre configuracions donades. Obtenir solucions adequades per aquests problemes és crucial, ja que les singularitats plantegen problemes al funcionament normal del robot i, per tant, haurien de ser completament identificades abans de la construcció d'un prototipus. La habilitat de computar tot el conjunt singular també proporciona informació rica sobre les capacitats globals de moviment d'un manipulador. Les projeccions cap a l'espai de tasques o d'articulacions delimiten les regions de treball en aquests espais, poden informar sobre les diferents maneres de muntar el manipulador, i remarquen les àrees on poden sorgir pèrdues de control o destresa, entre d'altres comportaments anòmals. Aquestes projeccions també proporcionen una imatge fidel dels moviments factibles del sistema, però no revelen tots els possibles moviments lliures de singularitats. Planificadors de moviment automàtics que permetin evitar les singularitats problemàtiques haurien de ser ideats per tal d'assistir les etapes de disseny i programació d'un manipulador. El paper clau que juguen les configuracions singulars ha estat àmpliament conegut durant anys, però els mètodes existents pel càlcul o evitació de singularitats encara es concentren en classes específiques de manipuladors. L'absència de mètodes capaços de tractar aquests problemes en una classe suficientment gran de manipuladors és problemàtica, ja que dificulta l'anàlisi de manipuladors més complexes o el desenvolupament de noves topologies de robots. Una raó principal d'aquesta absència ha estat la manca d'eines computacionals adequades a les matemàtiques subjacents que aquests problemes amaguen. No obstant, avenços recents en el camp de mètodes numèrics per la solució de sistemes polinòmics permeten ara enfrontar-se a aquests temes amb una intenció molt general en ment. El propòsit d'aquesta tesi és aprofitar aquest progrés i proposar mètodes robustos i generals pel càlcul i evitació de singularitats per manipuladors no redundants d'arquitectura arbitrària. En global, el treball busca contribuir a la comprensió general sobre com els moviments de sistemes multicos complexos es poden predir, planificar o controlar d'una manera eficient i segur