On the false positives and false negatives of the Jacobian Matrix in kinematically redundant parallel mechanisms

The Jacobian matrix is a highly popular tool for the control and performance analysis of closed-loop robots. Its usefulness in parallel mechanisms is certainly apparent, and its application to solve motion planning problems, or other higher level questions, has been seldom queried, or limited to non-redundant systems. In this paper, we discuss the shortcomings of the use of the Jacobian matrix under redundancy, in particular when applied to kinematically redundant parallel architectures with non-serially connected actuators. These architectures have become fairly popular recently as they allow the end-effector to achieve full rotations, which is an impossible task with traditional topologies. The problems with the Jacobian matrix in these novel systems arise from the need to eliminate redundant variables when forming it, resulting in both situations where the Jacobian incorrectly identifies singularities (false positive), and where it fails to identify singularities (false negative). These issues have thus far remained unaddressed in the literature. We highlight these limitations herein by demonstrating several cases using numerical examples of both planar and spatial architectures

Parallel Manipulators

In recent years, parallel kinematics mechanisms have attracted a lot of attention from the academic and industrial communities due to potential applications not only as robot manipulators but also as machine tools. Generally, the criteria used to compare the performance of traditional serial robots and parallel robots are the workspace, the ratio between the payload and the robot mass, accuracy, and dynamic behaviour. In addition to the reduced coupling effect between joints, parallel robots bring the benefits of much higher payload-robot mass ratios, superior accuracy and greater stiffness; qualities which lead to better dynamic performance. The main drawback with parallel robots is the relatively small workspace. A great deal of research on parallel robots has been carried out worldwide, and a large number of parallel mechanism systems have been built for various applications, such as remote handling, machine tools, medical robots, simulators, micro-robots, and humanoid robots. This book opens a window to exceptional research and development work on parallel mechanisms contributed by authors from around the world. Through this window the reader can get a good view of current parallel robot research and applications

A method for extending planar axis-symmetric parallel manipulators to spatial mechanisms

This paper investigates axis-symmetric parallel manipulators, composed of a central base column and an arm system able to rotate around this column. The arm system includes several actuated upper arms, each connected to a manipulated platform by one or more lower arm linkages. Such manipulators feature an extensive positional workspace in relation to the manipulator footprint and equal manipulator properties in all radial half-planes defined by the common rotation-axis of the upper arms. The similarities between planar manipulators exclusively employing 2-degrees-of-freedom (2-DOF) lower arm linkages and lower mobility spatial manipulators only utilising 5-DOF lower arm linkages are analysed. The 2-DOF linkages are composed of a link with a 1-DOF hinge on both ends whilst the 5-DOF linkages utilise 3-DOF spherical joints and 2-DOF universal joints. By employing a proposed linkage substitution scheme, it is shown how a wide range of spatial axis-symmetric parallel manipulators can be derived from a limited range of planar manipulators of the same type

How to Expand the Workspace of Parallel Robots

In this chapter, methods for expanding the workspace of parallel robots are introduced. Firstly, methods for expanding the translational workspace of the parallel robot are discussed. The parallel robot has multiple solutions of the inverse and forward displacement analysis. By changing its configurations from one solution to another, the parallel robot can expand its translational workspace. However, conventional nonredundant parallel robot encounters singularity during the mode change. Singularity-free mode changes of the parallel robot by redundant actuation are introduced. Next, methods for expanding the rotational workspace of the parallel robot are shown. In order to achieve the large rotation, some mechanical gimmicks by gears, pulleys, and helical joints have been embedded in the moving part. A novel differential screw-nut mechanism for expanding the rotational workspace of the parallel robot is introduced

A compliant and redundantly actuated 3-DOF 4RRR PKM: First step to full planar motion

The development and optimisation of compliant (or flexure-based) manipulators with redundant actuation have been considered before, showing that the redundancy can be exploited to increase the support stiffness and reduce actuator loads. However, so far only 2-DOF manipulators have been considered which enable translational motion in two directions. In this paper a third degree of freedom, the in-plane rotation of the end effector, is added. The goal is to design and evaluate a first prototype capable of full planar motion. The dynamic performance of the manipulator is analysed with a flexible multibody model. The links are assumed to be rigid. The SPACAR software is used as its flexible beam element can describe the non-linear behaviour of the flexure joints well at rather large deflections and accounts for constraint warping. In this prototype the end effector range of motion is limited such that the joint rotations do not exceed ±30 deg. Butterfly and cartwheel flexure joints can handle the specifications without violating stress constraints. In the final design the lowest natural frequencies are 3.6 Hz for both translations and 7.9 Hz for the rotation. The first parasitic frequency is expected at 76 Hz, which is sufficiently high. This prototype has been manufactured with 3D printing. The lowest translational frequencies appear to be somewhat higher than expected, which could arise from stiffness added by the flexible coupling between actuator and upper arm. A higher resonance frequency is found near 80 Hz, which agrees well with the expected first relevant parasitic mode

Static force capabilities and dynamic capabilities of parallel mechanisms equipped with safety clutches

Cette thèse étudie les forces potentielles des mécanismes parallèles plans à deux degrés de liberté équipés d'embrayages de sécurité (limiteur de couple). Les forces potentielles sont étudiées sur la base des matrices jacobienne. La force maximale qui peut être appliquée à l'effecteur en fonction des limiteurs de couple ainsi que la force maximale isotrope sont déterminées. Le rapport entre ces deux forces est appelé l'efficacité de la force et peut être considéré ; comme un indice de performance. Enfin, les résultats numériques proposés donnent un aperçu sur la conception de robots coopératifs reposant sur des architectures parallèles. En isolant chaque lien, les modèles dynamiques approximatifs sont obtenus à partir de l'approche Newton-Euler et des équations de Lagrange pour du tripteron et du quadrupteron. La plage de l'accélération de l'effecteur et de la force externe autorisée peut être trouvée pour une plage donnée de forces d'actionnement.This thesis investigates the force capabilities of two-degree-of-freedom planar parallel mechanisms that are equipped with safety clutches (torque limiters). The force capabilities are studied based on the Jacobian matrices. The maximum force that can be applied at the end-effector for given torque limits (safety index) is determined together with the maximum isotropic force that can be produced. The ratio between these two forces, referred to as the force effectiveness, can be considered as a performance index. Finally, some numerical results are proposed which can provide insight into the design of cooperation robots based on parallel architectures. Considering each link and slider system as a single body, approximate dynamic models are derived based on the Newton-Euler approach and Lagrange equations for the tripteron and the quadrupteron. The acceleration range or the external force range of the end-effector are determined and given as a safety consideration with the dynamic models

A concept for actuating and controlling a leg of a novel walking parallel kinematic machine tool

The scope of this paper is to present a novel method of actuating the legs of a walking parallel kinematic machine tool (WalkingHex) such that the upper spherical joint can be actively driven while walking and remain a free, passive joint while performing machining operations. Different concepts for the number of Degrees of Freedom (DoF) and methods for actuating the chosen concept are presented, leading to a description of a three-wire actuated spherical joint arrangement. The inverse kinematics for the actuation mechanism is defined and a control methodology that accounts for the redundantly actuated nature of the mechanism is explored. It is demonstrated that a prototype of the system is capable of achieving a motion position accuracy within 5.64% RMS. Utilising the concept presented in this paper, it is possible to develop a walking robot that is capable of manoeuvring into location and performing precision machining or inspection operations