Kinematics and statics of cable-driven parallel robots by interval-analysis-based methods

In the past two decades the work of a growing portion of researchers in robotics focused on a particular group of machines, belonging to the family of parallel manipulators: the cable robots. Although these robots share several theoretical elements with the better known parallel robots, they still present completely (or partly) unsolved issues. In particular, the study of their kinematic, already a difficult subject for conventional parallel manipulators, is further complicated by the non-linear nature of cables, which can exert only efforts of pure traction. The work presented in this thesis therefore focuses on the study of the kinematics of these robots and on the development of numerical techniques able to address some of the problems related to it. Most of the work is focused on the development of an interval-analysis based procedure for the solution of the direct geometric problem of a generic cable manipulator. This technique, as well as allowing for a rapid solution of the problem, also guarantees the results obtained against rounding and elimination errors and can take into account any uncertainties in the model of the problem. The developed code has been tested with the help of a small manipulator whose realization is described in this dissertation together with the auxiliary work done during its design and simulation phases.Negli ultimi decenni il lavoro di una parte sempre maggiore di ricercatori che si occupano di robotica si è concentrato su un particolare gruppo di robot appartenenti alla famiglia dei manipolatori paralleli: i robot a cavi. Nonostante i numerosi studi al riguardo, questi robot presentano ancora oggi numerose problematiche del tutto (o in parte) irrisolte. Lo studio della loro cinematica nello specifico, già complesso per i manipolatori paralleli tradizionali, è ulteriormente complicato dalla natura non lineare dei cavi, i quali possono esercitare sforzi di sola trazione. Il lavoro presentato in questa tesi si concentra dunque sullo studio della cinematica dei robot a cavi e sulla messa a punto di tecniche numeriche in grado di affrontare parte delle problematiche ad essa legate. La maggior parte del lavoro è incentrata sullo sviluppo di una procedura per la soluzione del problema geometrico diretto di un generico manipolatore a cavi basata sull'analisi per intervalli. Questa tecnica di analisi numeirica, oltre a consentire una rapida soluzione del problema, permette di garantire i risultati ottenuti in caso di errori di cancellazione e arrotondamento e consente di considerare eventuali incertezze presenti nel modello del problema. Il codice sviluppato è stato testato attraverso un piccolo prototipo di manipolatore a cavi la cui realizzazione, avvenuta durante il percorso di dottrato, è descritta all'interno dell'elaborato unitamente al lavoro collaterale svolto durante la fase di progettazione e simulazione.Pendant les dernières décennies, le travail d'une partie toujours croissante de chercheurs qui s'occupent de robotique s'est focalisé sur un groupe spécifique de robots qui fait partie de la famille des manipulateurs parallèles: les robots à câbles. Malgré les nombreux études que l'on a consacré à ce sujet, ces robots présentent encore aujourd'hui plusieurs problématiques complètement ou partiellement irrésolues. En particulier l'étude de leur cinématique, qui se révèle déjà complexe pour les manipulateurs parallèles traditionnels, est rendu encore plus compliqué par la nature non linéaire des câbles qui peuvent seulement exercer des efforts de traction. Le travail présenté dans ma thèse concentre donc son attention sur l'étude de la cinématique des robots à câbles et sur la mise au point de techniques numériques capables d'aborder une partie des problématiques liées à cela. La plupart du travail se concentre sur l'élaboration d'un algorithme pour la résolution du problème géométrique direct d'un manipulateur à câbles général qui se fonde sur l'analyse par intervalles. Cette technique d'analyse permet non seulement de résoudre rapidement le problème mais également de garantir les résultats obtenus en cas d'erreur de cancellation et d'arrondi et de prendre en considération les incertitudes éventuellement presentes dans le modèle du problème. Le code développé a été testé grâce à un petit prototype de manipulateur à câbles dont la réalisation, qui a eu lieu pendant le parcours de doctorat, est décrite à l'intérieur du devoir en accord avec la phase de conception du projet et de simulation

Inverse geometrico-static problem of underconstrained . . .

This paper studies underconstrained cable-driven parallel robots (CDPRs) with three cables. A major challenge in the study of these robots is the intrinsic coupling between kinematics and statics, which must be tackled simultaneously. Effective elimination procedures are presented which provide the complete solution sets of the inverse geometrico-static problems (IGPs) with assigned orientation or position. In the former case, the platform orientation is given, whereas the platform position and the cable lengths and tensions must be computed. In the latter case, the platform position is known, whereas the platform orientation and the cable lengths and tensions are to be calculated. The described problems are proven to admit at the most 1 and 24 real solutions, respectively

A panorama of methods for dealing with sagging cables in cable-driven parallel robots

International audienceWe are considering cable-driven parallel robot (CDPR), where the legs of the robot are constituted of cables that can be independently coiled/uncoiled. We show that whatever the size of the CDPR is we may have slack cables so that using a sagging cable model that takes into account both the mass and elasticity of the cables will improve the positioning accuracy.Being able to solve the inverse and direct kinematics (IK/DK) with sagging cables is crucial for kinematic analysis while being quite complex as both IK/DK may have multiple solutions. We present a panorama of solving methods for the IK/DK with their advantages and drawbacks

Dynamic Control of a Novel Planar Cable-Driven Parallel Robot with a Large Wrench Feasible Workspace

Cable-Driven Parallel Robots (CDPRs) are special manipulators where rigid links are replaced with cables. The use of cables offers several advantages over the conventional rigid manipulators, one of the most interesting being their ability to cover large workspaces since cables are easily winded. However, this workspace coverage has its limitations due to the maximum permissible cable tensions, i.e., tension limitations cause a decrease in the Wrench Feasible Workspace (WFW) of these robots. To solve this issue, a novel design based in the addition of passive carriages to the robot frame of three degrees-of-freedom (3DOF) fully-constrained CDPRs is used. The novelty of the design allows reducing the variation in the cable directions and forces increasing the robot WFW; nevertheless, it presents a low stiffness along the x direction. This paper presents the dynamic model of the novel proposal together with a new dynamic control technique, which rejects the vibrations caused by the stiffness loss while ensuring an accurate trajectory tracking. The simulation results show that the controlled system presents a larger WFW than the conventional scheme of the CDPR, maintaining a good performance in the trajectory tracking of the end-effector. The novel proposal presented here can be applied in multiple planar applications

Simulation of Discrete-Time Controlled Cable-Driven Parallel Robots on a Trajectory

International audienceThis paper addresses the simulation of the state of a discrete-time controlled cable-driven parallel robot (CDPR) with nondeformable or elastic cables over a given trajectory. Being given a CDPR, an arbitrary model for the coiling system and for the control strategy, we exhibit a simulation algorithm that allows one to determine, in a guaranteed way, the platform pose and the cable tensions at any time. We show that such a simulation may require a computing accuracy that imposes to use extended arithmetic and that discrete-time control may lead to drastic differences in the cable tensions as compared to usual continuous time simulation. Hence, the proposed simulation tool allows for a better estimation of the positioning accuracy together with safer estimation of the maximum of the cable tensions

Direct kinematics of CDPR with extra cable orientation sensors: the 2 and 3 cables case with perfect measurement and ideal or elastic cables

International audienceDirect kinematics (DK) of cable-driven parallel robots (CDPR) based only on cable lengths measurements is a complex issue even with ideal cables and consequently even harder for more realistic cable models. A natural way to simplify the DK solving is to add sensors. We consider here sensors that give a partial or complete measurement of the cable direction at the anchor points and spatial CDPR with 2/3 cables and we assume that these measurements are exact. We provide a solving procedure and maximal number of DK solutions for an extensive combination of sensors while considering two different cables models: ideal and linearly elastic without deformation

Rest-to-Rest Trajectory Planning for Underactuated Cable-Driven Parallel Robots

This article studies the trajectory planning for underactuated cable-driven parallel robots (CDPRs) in the case of rest-to-rest motions, when both the motion time and the path geometry are prescribed. For underactuated manipulators, it is possible to prescribe a control law only for a subset of the generalized coordinates of the system. However, if an arbitrary trajectory is prescribed for a suitable subset of these coordinates, the constraint deficiency on the end-effector leads to the impossibility of bringing the system at rest in a prescribed time. In addition, the behavior of the system may not be stable, that is, unbounded oscillatory motions of the end-effector may arise. In this article, we propose a novel trajectory-planning technique that allows the end effector to track a constrained geometric path in a specified time, and allows it to transition between stable static poses. The design of such a motion is based on the solution of a boundary value problem, aimed at a finding solution to the differential equations of motion with constraints on position and velocity at start and end times. To prove the effectiveness of such a method, the trajectory planning of a six-degrees-of-freedom spatial CDPR suspended by three cables is investigated. Trajectories of a reference point on the moving platform are designed so as to ensure that the assigned path is tracked accurately, and the system is brought to a static condition in a prescribed time. Experimental validation is presented and discussed

Displacement Analysis of Under-Constrained Cable-Driven Parallel Robots

This dissertation studies the geometric static problem of under-constrained cable-driven parallel robots (CDPRs) supported by n cables, with n ≤ 6. The task consists of determining the overall robot configuration when a set of n variables is assigned. When variables relating to the platform posture are assigned, an inverse geometric static problem (IGP) must be solved; whereas, when cable lengths are given, a direct geometric static problem (DGP) must be considered. Both problems are challenging, as the robot continues to preserve some degrees of freedom even after n variables are assigned, with the final configuration determined by the applied forces. Hence, kinematics and statics are coupled and must be resolved simultaneously. In this dissertation, a general methodology is presented for modelling the aforementioned scenario with a set of algebraic equations. An elimination procedure is provided, aimed at solving the governing equations analytically and obtaining a least-degree univariate polynomial in the corresponding ideal for any value of n. Although an analytical procedure based on elimination is important from a mathematical point of view, providing an upper bound on the number of solutions in the complex field, it is not practical to compute these solutions as it would be very time-consuming. Thus, for the efficient computation of the solution set, a numerical procedure based on homotopy continuation is implemented. A continuation algorithm is also applied to find a set of robot parameters with the maximum number of real assembly modes for a given DGP. Finally, the end-effector pose depends on the applied load and may change due to external disturbances. An investigation into equilibrium stability is therefore performed

Reconfigurable fully constrained cable-driven parallel mechanism for avoiding collision between cables with human

Productivity can be increased by manipulators tracking the desired trajectory with some constraints. Humans as moving obstacles in a shared workspace are one of the most challenging problems for cable-driven parallel mechanisms (CDPMs) that are considered in this research. One of the essential primary issues in CDPM is collision avoidance among cables and humans in the shared workspace. This paper presents a model and simulation of a reconfigurable, fully constrained CDPM enabling detection and avoidance of cable–human collision. In this method, unlike conventional CDPMs where the attachment points are fixed, the attachment points on the rails can be moved (up and down on their rails), and then the geometric configuration is adapted. Karush–Kuhn–Tucker method is proposed, which focuses on estimating the shortest distance among moving obstacles (human limbs) and all cables. When cable and limbs are close to colliding, the new idea of reconfiguration is presented by moving the cable’s attachment point on the rail to increase the distance between the cables and human limbs while they are both moving. Also, the trajectory of the end effector remains unchanged. Some simulation results of reconfiguration theory as a new approach are shown for the eight-cable-driven parallel manipulator, including the workspace boundary variation. The proposed method could find a collision-free predefined path, according to the simulation results

Direct Kinematics of Underactuated Cable-Driven Parallel Robots: Sensitivity to Redundant Measurements

Underactuated cable-driven parallel robots (UACDPRs) shift a 6-degree-of-freedom end-effector (EE) with fewer than 6 cables. This thesis proposes a new automatic calibration technique that is applicable for under-actuated cable-driven parallel robots. The purpose of this work is to develop a method that uses free motion as an exciting trajectory for the acquisition of calibration data. The key point of this approach is to find a relationship between the unknown parameters to be calibrated (the lengths of the cables) and the parameters that could be measured by sensors (the swivel pulley angles measured by the encoders and roll-and-pitch angles measured by inclinometers on the platform). The equations involved are the geometrical-closure equations and the finite-difference velocity equations, solved using the least-squares algorithm. Simulations are performed on a parallel robot driven by 4 cables for validation. The final purpose of the calibration method is, still, the determination of the platform initial pose. As a consequence of underactuation, the EE is underconstrained and, for assigned cable lengths, the EE pose cannot be obtained by means of forward kinematics only. Hence, a direct-kinematics algorithm for a 4-cable UACDPR using redundant sensor measurements is proposed. The proposed method measures two orientation parameters of the EE besides cable lengths, in order to determine the other four pose variables, namely 3 position coordinates and one additional orientation parameter. Then, we study the performance of the direct-kinematics algorithm through the computation of the sensitivity of the direct-kinematics solution to measurement errors. Furthermore, position and orientation error upper limits are computed for bounded cable lengths errors resulting from the calibration procedure, and roll and pitch angles errors which are due to inclinometer inaccuracies