Singularity loci of planar parallel manipulators with revolute actuators

The determination of the singularity loci of planar parallel manipulators is addressed in this paper. The inverse kinematics of two kinds of planar parallel manipulators (a two-degree-of-freedom manipulator and a three-degree-of-freedom manipulator) are first computed and their velocity equations are then derived. At the same time, the branches of the manipulators are distinguished by the introduction of a branch index K_i. Using the velocity equations, the singularity analysis of the manipulators is completed and expressions which represent the singularity of the manipulators are obtained. A polynomial form of the singularity loci is also derived. For the first type of singularity of parallel manipulators, the singularity locus is obtained by finding the workspace limits of the manipulators. For the second type of singularity, the loci are obtained through the solution of nonlinear algebraic equations obtained from the velocity analysis. Finally, the graphical representation of the complete singularity loci of the manipulators is illustrated with examples. The algorithm introduced in this paper allows the determination of the singularity loci of planar parallel manipulators with revolute actuators, which has been elusive to previous approaches.     

Kinematics of a new spherical parallel manipulator with three equal legs: The 3‐URC wrist

Parallel manipulators are constituted of two rigid bodies, one movable (platform) and the other fixed (base), connected by a number of kinematic chains (legs). Manipulators with all equal legs that have the actuators adjacent to the base allow the use of simplified design procedures and manufacturing processes. Moreover, they are able to reach high performances. Further manufacturing simplifications can be obtained if only revolute pairs and passive cylindrical pairs are used. This article presents a new spherical parallel manipulator, named the 3‐URC wrist. The 3‐URC wrist is not overconstrained. It has three equal legs having only passive cylindrical pairs and revolute pairs with the actuators adjacent to the base. The 3‐URC wrist mounting and manufacturing conditions making spherical motions of the platform possible are defined. Then the position analysis and the mobility analysis of the 3‐URC wrist are solved. Finally, the singularity conditions are written in explicit form and geometrically interpreted. © 2001 John Wiley & Sons, Inc.     

Kinematics and dynamics analyses of a parallel manipulator with three active legs and one passive leg by a virtual serial mechanism

A novel CAD variation geometry approach and a virtual serial mechanism approach are proposed for analyzing the kinematics and dynamics of a parallel manipulator with three SPS-type active legs and one PU-type constrained passive leg. First, a simulation mechanism of this parallel manipulator is created, and some kinematic characteristics are analyzed. Second, the inverse formulae for solving pose and Jocabian matrix are derived, and workspace and singularity are determined. Third, a virtual serial mechanism is created, and the analytic formulae for solving active forces and constrained wrench of these parallel manipulators are derived. The analytic results are verified by using its simulation mechanism.     

基于螺旋理论的新型三自由度移动并联机器人奇异性分析

采用螺旋理论及线性几何学方法对具有3支链5关节（P-4R）的三自由度纯移动并联机器人的奇异性进行了分析和总结．研究了该并联机器人的结构奇异性以及主动关节（驱动关节）的配置的奇异性,并采用线性几何学图解形式直观地给出其处于奇异时的形位,为进一步研究其动力学、运动学性质奠定了基础．     

Management of parallel-manipulator singularities using joint-coupling

Joint-coupling (JC) is introduced in the design and control of parallel manipulators for managing manipulator singularities. The underlying idea consists of coupling the motions of several manipulator joints by driving the joints with a single actuator. This can be done by mechanical or electronic means, while preserving the mobility of the manipulator. Such JC is intended to alter the direct singularity condition of parallel manipulators upon changing the coupling coefficients. Both 2- and 3-d.o.f. planar parallel manipulator examples are provided to illustrate the concept of JC and singularity management. The prototype of a JC device and its associated 2-d.o.f. parallel manipulator intended for a feasibility study is introduced.     

Translational Parallel Manipulators: New Proposals

Translational parallel manipulators are parallel manipulators wherein the end‐effector performs only spatial translations. This paper presents a new family of translational parallel manipulators. The manipulators of this family are independent constraint manipulators. They have three limbs that are topologically identical and have no rotation singularity. The limbs of these manipulators feature five one‐degree‐of‐freedom kinematic pairs in series. Four joints are revolute pairs and the remaining one, called T‐pair, is a kinematic pair that can be manufactured in different ways. In each limb, three adjacent revolute pairs have parallel axes and the remaining revolute pair has an axis that is not parallel to the axes of the other revolute pairs. The mobility analysis of the manipulators of this new family is addressed by taking into account two different choices for the actuated pairs. One of the results of this analysis is that the geometry of a translational parallel manipulator free from singularities can be defined for a particular choice of the actuated pairs. © 2002 Wiley Periodicals, Inc.     

Simulation and computer-aided kinematic design of three-degree-of-freedom spherical parallel manipulators

Parallel robotic manipulators are complex mechanical systems that lead to involved kinematic and dynamic equations. Hence, the design of such systems is in general not intuitive, and advanced simulation and design tools specialized for this type of architecture are highly desirable. This article discusses the kinematic simulation and computer-aided design of three-degree-of-freedom spherical parallel manipulators with either prismatic or revolute actuators. The kinematic analysis of spherical parallel manipulators is first reviewed. Solutions for the direct and inverse kinematic problems are given, and the expressions for the singularity loci are then introduced. The determination of the workspace of this type of manipulator is also addressed. Finally, a computer package developed specifically for the CAD of spherical parallel manipulators is presented. This package allows the interactive analysis of manipulators of arbitrary architecture including the representation of the workspace, the representation of singularities, and the graphic animation of trajectories specified either by the direct or the inverse kinematic module. It can be used for the design of any spherical parallel three-degree-of-freedom actuated mechanism, which can find many applications in high-performance robotic systems. © 3995 John Wiley & Sons, Inc.     

基于一种新的姿态描述的并行结构机器人工作空间解析解

本文针对并行结构机器人的结构特点建立一种新的姿态描述,给出了相应的定义;并基于这 种描述及Grassmann线几何对6-3并行结构机器人工作空间的奇异问题进行分析,得到 一个简单而有效的非奇异边界,及该区域内的杆长约束条件下保守的工作空间解析解．且与 模型结构无关,具有普适性．因此只要并行结构机器人的模型描述采用了新的方法,那 么它的运动学研究就可以通过这种方法来进行．     

Geometrical conditions for the design of partial or full isotropic hexapods

This paper presents a methodology for the design of PKMs (parallel kinematic machines) with defined isotropy and stiffness. Partial isotropy or full isotropy can be achieved by suitable design choices. The former is useful for five axis applications, while the latter for six axis manipulators. The paper summarizes the concept of full and partial isotropy, and for a wide class of hexapods defines in analytical form the conditions to achieve it exactly. These conditions can be used to design isotropic parallel manipulators. The methodology requires that the six legs have to be divided into two groups (terns). The legs belonging to one tern are mutually identical and are positioned with radial symmetry with respect to the TCP (tool center point). The paper shows that the manipulator structure can be defined in term of 13 design parameters, the value of six of them are chosen in order to achieve the required isotropy and stiffness properties, while the remaining seven parameters can be used to optimize the structure. The design criterion here presented assures that stiffness isotropy, force, and velocity isotropy are all achieved contemporarily. This methodology can be practically applied to a large family of hexapods. © 2005 Wiley Periodicals, Inc.     

Flagged Parallel Manipulators

The conditions for a parallel manipulator to be flagged can be simply expressed in terms of linear dependencies between the coordinates of its leg attachments, both on the base and on the platform. These dependencies permit to describe the manipulator singularities in terms of incidences between two flags (hence, the name ldquoflaggedrdquo). Although these linear dependencies might look, at first glance, too restrictive, in this paper, the family of flagged manipulators is shown to contain large subfamilies of six-legged and three-legged manipulators. The main interest of flagged parallel manipulators is that their singularity loci admit a well-behaved decomposition with a unique topology irrespective of the metrics of each particular design. In this paper, this topology is formally derived and all the cells, in the configuration space of the platform, of dimension 6 (nonsingular) and dimension 5 (singular), together with their adjacencies, are worked out in detail.     

Identification and resolution of structural and algorithmic singularity in redundancy control of serial manipulations

A major difficulty that has haunted most researchers in the process of optimal redundancy resolution of robotic manipulators is the instability observed in even very simple numerical simulations. This numerical instability is not related to the structurally singular configurations of the manipulators, and in the literature has been referred to as “algorithmic singularity,” “artificial singularity,” or “unavoidable singularity.” In this work, conditions on both structural and algorithmic singularities are studied based on the Singular Value Decomposition of the Jacobian matrix, and, hence, a singularity-free control algorithm for redundant manipulators is developed and resolved as the Lagrange problem of optimal control. It is shown that many well-known methods for optimal redundant manipulation in the literature, including the Extended Jacobian Technique, most of constraint function-based methods, and most of the previously reported methods on global optimization techniques, are all special cases of the formulation provided here. Further, the necessary conditions of the global optimality for this general formulation are derived in explicit form and the source of “algorithmic singularity” is rigorously identified and resolved. © 2995 John Wiley & Sons, Inc.     

Design of reactionless 3-DOF and 6-DOF parallel manipulators using parallelepiped mechanisms

In this paper, the design of reactionless 3-degree-of-freedom (DOF) and 6-DOF parallel manipulators is presented. At first, the design and dynamic balancing of a novel 3-DOF parallel mechanism referred to as the parallelepiped mechanism are addressed. Two types of actuation schemes of the mechanism are considered, and the two corresponding mechanical structures are designed. The balancing equations are derived by imposing that the center of mass of the mechanism is fixed and that the total angular momentum is constant with respect to a fixed point. Optimization is performed to determine the counterweights and counter-rotations based on the balancing conditions. Numerical examples of reactionless 3-DOF parallelepiped mechanisms are given. The dynamic simulation software ADAMS is used to simulate the motion of the mechanisms and to verify that the mechanisms are reactionless at all times and for arbitrary trajectories. Finally, the 3-DOF parallelepiped mechanisms are used as legs to synthesize reactionless 6-DOF parallel manipulators.     

Orientationability analyses of a special class of the Stewart–Gough parallel manipulators using the unit quaternion representation

This paper addresses the orientation-singularity and orientationability analyses of a special class of the Stewart–Gough parallel manipulators whose moving and base platforms are two similar semi-symmetrical hexagons. Employing a unit quaternion to represent the orientation of the moving platform, an analytical expression representing the singularity locus of this class of parallel manipulators in a six-dimensional Cartesian space is obtained. It shows that for a given orientation, the position-singularity locus is a cubic polynomial expression in the moving platform position parameters, and for a given position, the orientation-singularity locus is an analytical expression but not a polynomial directly with respect to the mobile platform orientation parameters. Further inspection shows that for the special class of parallel manipulators, there must exist a nonsingular orientation void in the orientation space around the orientation origin for each position in the position-workspace. Therefore, a new performance index referred to as orientationability is introduced to describe the orientation capability of the special class of manipulators at a given position. A discretization algorithm is proposed for the computation of the orientationability of the special class of manipulators. Moreover, effects of the design parameters and position parameters on the orientationability are investigated in details. Based on the orientationability performance index, another novel performance index referred to as practical orientationability is presented which represents the practical orientation capability of the manipulator at a given position. The practical orientationability not only can satisfy all the kinematic demands and constraints of such class of manipulators, but also can guarantee that the manipulator is nonsingular.     

Inverse statics analysis of planar parallel manipulators via Grassmann-Cayley algebra

The wrench Jacobian matrix plays an important role in the statics and singularity analysis of planar parallel manipulators (PPMs). The Jacobian matrix can be calculated based on the conventional Plücker coordinate method. However, this method cannot be applied when two links are in parallel. A new approach is proposed for the analysis of the forward and inverse wrench Jacobian matrix using Grassmann-Cayley algebra (GCA). A symbolic formula for the inverse statics analysis is obtained based on the Jacobian. The proposed method can be applied when two links are in parallel. The approach is explained in detail based on a planar 3-RPR PPM example, and the analysis procedure for nine other PPMs is also presented. This novel approach to deriving the statics can be applied to spatial parallel manipulators and redundant cases of PPMs.

     

Forward kinematics of walking machines with pantograph legs based on selections of independent joints

This article studies the forward kinematics of walking machines with pantograph legs. The walking machine is supported by three legs and each leg has 3 degrees of freedom (dof). Because the body has 6 dof motion, only 6 joint variables among the 9 are independent. Thus, there are 84 possible ways to select the 6 independent joint variables. It is shown that the complexity of the forward position solutions is very much dependent on the selection of independent joints. In the category of 3:2:1 (the 3 numbers denote the number of independent joints at each of the supporting legs), all 54 combinations give closed-form solutions. Among the 27 combinations of category 2:2:2, 10 possess closed-form solutions, 16 yield to higher-order polynomials, and 1 gives no solutions. The 3 combinations of category 3:3:0 give no solutions. The results are useful to control and simulation of walking machines and other parallel systems consisting of pantograph subchains such as parallel manipulators, multifingered hands, and multiply coordinated manipulators. © 1992 John Wiley & Sons, Inc.     

The Isotropic Design of Two General Classes of Planar Parallel Manipulators

Manipulator designs with isotropic architectures have a number of advantages, including high servo accuracy and dexterity. Using the isotropy criterion, isotropic designs of two general classes of planar, three-degree-of-freedom, parallel manipulators with three legs are produced. One of these classes comprises 20 manipulators, while the other, 4. As special cases for each class, the complete set of isotropic designs of 2 manipulators is found. The parameter space of isotropic design of these manipulators is a continuum of at least one dimension. © 2995 John Wiley & Sons, Inc.     

Kinematic Analysis of a Macro–Micro Redundantly Actuated Parallel Manipulator

In this paper the kinematic and Jacobian analysis of a macro–micro parallel manipulator is studied in detail. The manipulator architecture is a simplified planar version adopted from the structure of the Large Adaptive Reflector (LAR), the Canadian design of the next generation of giant radio telescopes. This structure is composed of two parallel and redundantly actuated manipulators at the macro and micro level, which both are cable-driven. Inverse and forward kinematic analysis of this structure is presented in this paper. Furthermore, the Jacobian matrices of the manipulator at the macro and micro level are derived, and a thorough singularity and sensitivity analysis of the system is presented. The kinematic and Jacobian analysis of the macro–micro structure is extremely important to optimally design the geometry and characteristics of the LAR structure. The optimal location of the base and moving platform attachment points in both macro and micro manipulators, singularity avoidance of the system in nominal and extreme maneuvers, and geometries that result in high dexterity measures in the design are among the few characteristics that can be further investigated from the results reported in this paper. Furthermore, the availability of the extra degrees of freedom in a macro–micro structure can result in higher dexterity provided that this redundancy is properly utilized. In this paper, this redundancy is used to generate an optimal trajectory for the macro–micro manipulator, in which the Jacobian matrices derived in this analysis are used in a quadratic programming approach to minimize performance indices like minimal micro manipulator motion or singularity avoidance criterion.     

Stiffness and natural frequency of a 3-DOF parallel manipulator with consideration of additional leg candidates

This paper investigates the stiffness and natural frequency of a 3-DOF parallel manipulator with consideration of additional leg candidates. The stiffness model and natural frequency are derived, and then the stiffness and natural frequency of the manipulators are compared. The simulations show that the stiffness and natural frequency of the parallel manipulator with one or two additional legs are higher than those of the manipulator without additional leg. The stiffness performance and natural frequency of the manipulator with one additional leg can only be improved little by adding the second additional leg. It is better to develop this parallel manipulator by adding only one additional leg to construct a symmetrical architecture.     

Type synthesis of 5‐DOF parallel manipulators based on screw theory

With the introduction of virtual chains to represent the motion patterns of 5‐DOF motions, a classification of 5‐DOF PMs (parallel manipulators) is proposed at first. A general method for the type synthesis of 5‐DOF PMs is then proposed based on screw theory and using the concept of virtual chains. The type synthesis of US‐equivalent PMs is presented in detail to show the application of the proposed approach. US‐equivalent PMs are the parallel counterparts of the 5‐DOF US serial manipulators. For a US‐equivalent PM, the moving platform can rotate arbitrarily about a point moving along a spherical surface. The type synthesis of legs for US‐equivalent PKCs (parallel kinematic chains), the type synthesis of US‐equivalent PKCs, as well as the selection of actuated joints of US‐equivalent PMs are dealt with in sequence. US‐equivalent PKCs with and without inactive joints are synthesized. Several US‐equivalent PMs as well as other classes of 5‐DOF PMs with identical type of legs are obtained. © 2005 Wiley Periodicals, Inc.