A geometrical formulation for the workspace of parallel manipulators

Robotica ◽  
2021 ◽  
pp. 1-11
Author(s):  
Matteo Russo ◽  
Marco Ceccarelli
Keyword(s):  
Parallel Mechanism ◽  
Parallel Manipulator ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Analytical Models ◽  
Kinematic Chains ◽  
Motion Constraints ◽  
Geometrical Formulation ◽  
Geometric Formulation ◽  
Algebraic Formulation

Abstract In study this paper, a geometric formulation is proposed to describe the workspace of parallel manipulators by using a recursive approach as an extension of volume generation for solids of revolution. In this approach, the workspace volume and boundary for each limb of the parallel manipulator is obtained with an algebraic formulation derived from the kinematic chain of the limb and the motion constraints on its joints. Then, the overall workspace of the mechanism can be determined as the intersection of the limb workspaces. The workspace of different kinematic chains is discussed and classified according to its external shape. An algebraic formulation for the inclusion of obstacles in the computation is also proposed. Both analytical models and numerical simulations are reported with their advantages and limitations. An example on a 3-SPR parallel mechanism illustrates the feasibility of the formulation and its efficiency.

Type Synthesis of 3-DOF Spherical Parallel Manipulators Based on Screw Theory1

2004 ◽  
Vol 126 (1) ◽  
pp. 101-108 ◽  
Author(s):  
Xianwen Kong ◽  
Cle´ment M. Gosselin
Keyword(s):  
Parallel Manipulator ◽  
Screw Theory ◽  
General Procedure ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Type Synthesis ◽  
Validity Condition ◽  
First Time ◽  
Spherical Parallel Manipulator ◽  
Selection Of

A spherical parallel manipulator (SPM) refers to a 3-DOF (degree-of-freedom) parallel manipulator generating 3-DOF spherical motion. A method is proposed for the type synthesis of SPMs based on screw theory. The wrench systems of a spherical parallel kinematic chain (SPKC) and its legs are first analyzed. A general procedure is then proposed for the type synthesis of SPMs. The type synthesis of legs for SPKCs, the type synthesis of SPKCs, as well as the selection of inputs of SPMs are dealt with in sequence. An input validity condition of SPMs is proposed. SPKCs with and without inactive joints are synthesized. The number of overconstraints of each SPKC is also given. The phenomenon of dependent joint groups in an SPKC is revealed for the first time.

Direct Position Analysis in Analytical Form of Parallel Manipulators Generating Structures With Topology SR-2PS

2004 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Analytical Form ◽  
Solution Procedure ◽  
Generic Structure ◽  
Kinematic Chains ◽  
Position Analysis ◽  
In Series ◽  
The One ◽  
Wide Family

A wide family of parallel manipulators (PMs) is the one that groups all the PMs with three legs where the legs become kinematic chains constituted of a passive spherical pair (S) in series with either a passive prismatic pair (P) or a passive revolute pair (R) when the actuators are locked. The topologies of the structures generated by these manipulators, when the actuators are locked, are ten. One out of these topologies is the SR-2PS topology (one SR leg and two PS legs). This paper presents an algorithm that determines all the assembly modes of the structures with topology SR-2PS in analytical form. The presented algorithm can be applied without changes to solve, in analytical form, the direct position analysis of any parallel manipulator which generates a SR-2PS structure when the actuators are locked. In particular, the closure equations of a generic structure with topology SR-2PS are written. The eliminant of this system of equations is determined and the solution procedure is presented. Finally, the proposed procedure is applied to a real case. This work demonstrates that the solutions of the direct position analysis of any parallel manipulator which generates a SR-2PS structure when the actuators are locked are at most eight.

A Novel Parallel Mechanism With 3-RRUR Legs

2007 ◽  
Author(s):  
Yanwen Li ◽  
Yueyue Zhang ◽  
Lumin Wang ◽  
Zhen Huang
Keyword(s):  
Machine Tools ◽  
Parallel Mechanism ◽  
Parallel Manipulator ◽  
Degrees Of Freedom ◽  
Parallel Manipulators ◽  
Forward Kinematics ◽  
Symmetrical Structure ◽  
Forward Kinematic ◽  
Accumulated Error

This paper investigates a novel 4-DOF 3-RRUR parallel manipulator, the number and the characteristics of its degrees of freedom are determined firstly, the rational input plan and the invert and forward kinematic solutions are carried out then. The corresponding numeral example of the forward kinematics is given. This type of parallel manipulators has a symmetrical structure, less accumulated error, and can be used to construct virtual-axis machine tools. The analysis in this paper will play an important role in promoting the application of such manipulators.

A 2-legged XY parallel flexure motion stage with minimised parasitic rotation

2014 ◽  
Vol 228 (17) ◽  
pp. 3156-3169 ◽  
Author(s):  
Guangbo Hao
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Design Approach ◽  
Analytical Models ◽  
Geometrical Parameters ◽  
Element Analysis ◽  
Motion Range ◽  
Cross Axis

XY compliant parallel manipulators (aka XY parallel flexure motion stages) have been used as diverse applications such as atomic force microscope scanners due to their proved advantages such as eliminated backlash, reduced friction, reduced number of parts and monolithic configuration. This paper presents an innovative stiffness centre based approach to design a decoupled 2-legged XY compliant parallel manipulator in order to better minimise the inherent parasitic rotation and have a more compact configuration. This innovative design approach makes all of the stiffness centres, associated with the passive prismatic (P) modules, overlap at a point that all of the applied input forces can go through. A monolithic compact and decoupled XY compliant parallel manipulator with minimised parasitic rotation is then proposed using the proposed design approach based on a 2-PP kinematically decoupled translational parallel manipulator. Its load–displacement and motion range equations are derived, and geometrical parameters are determined for a specified motion range. Finite element analysis comparisons are also implemented to verify the analytical models with analysis of the performance characteristics including primary stiffness, cross-axis coupling, parasitic rotation, input and output motion difference and actuator nonisolation effect. Compared with the existing XY compliant parallel manipulators obtained using 4-legged mirror-symmetric constraint arrangement, the proposed XY compliant parallel manipulators based on stiffness centre approach mainly benefits from fewer legs resulting in reduced size, simpler modelling as well as smaller lost motion. Compared with existing 2-legged designs with the conventional arrangement, the present design has smaller parasitic rotation, which has been proved from the finite element analysis results.

The differential calculus of screws: Theory, geometrical interpretation, and applications

2009 ◽  
Vol 223 (6) ◽  
pp. 1449-1468 ◽  
Author(s):  
J J Cervantes-Sánchez ◽  
J M Rico-Martínez ◽  
G González-Montiel ◽  
E J González-Galván
Keyword(s):  
Differential Calculus ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Higher Order ◽  
Abstract Concepts ◽  
Serial Manipulators ◽  
Kinematic Chains ◽  
Finite Displacements ◽  
Typical Shape ◽  
Derivatives Of

This article presents a novel and original formula for the higher-order time derivatives, and also for the partial derivatives of screws, which are successively computed in terms of Lie products, thus leading to the automation of the differentiation process. Through the process and, due to the pure geometric nature of the derivation approach, an enlightening physical interpretation of several screw derivatives is accomplished. Important applications for the proposed formula include higher-order kinematic analysis of open and closed kinematic chains and also the kinematic synthesis of serial and parallel manipulators. More specifically, the existence of a natural relationship is shown between the differential calculus of screws and the Lie subalgebras associated with the expected finite displacements of the end effector of an open kinematic chain. In this regard, a simple and comprehensible methodology is obtained, which considerably reduces the abstraction level frequently required when one resorts to more abstract concepts, such as Lie groups or Lie subalgebras; thus keeping the required mathematical background to the extent that is strictly necessary for kinematic purposes. Furthermore, by following the approach proposed in this article, the elements of Lie subalgebra arise in a natural way — due to the corresponding changes in screws through time — and they also have the typical shape of the so-called ordered Lie products that characterize those screws that are compatible with the feasible joint displacements of an arbitrary serial manipulator. Finally, several application examples — involving typical, serial manipulators — are presented in order to prove the feasibility and validity of the proposed method.

Classification of 6-SPS Parallel Manipulators According to Their Components

2000 ◽  
Author(s):  
Xianwen Kong ◽  
Clément M. Gosselin
Keyword(s):  
Parallel Manipulator ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Rigid Bodies ◽  
Type Synthesis ◽  
Forward Displacement ◽  
Quadratic Equations ◽  
Straight Lines ◽  
Forward Displacement Analysis

Abstract The complexity of the forward displacement analysis (FDA) of 6-SPS parallel manipulators1 varies to a great extent with the change of their geometric parameters. This paper presents a classification of the 6-SPS parallel manipulators according to their components. At first, we give the components for the 6-SPS parallel manipulator. A component refers to a part of the 6-SPS kinematic chain in which the number of actuators is equivalent to the degree of freedom. In addition to the commonly used rigid bodies, points and (straight) lines are also taken as elements of the components. Type synthesis of the 6-SPS parallel manipulators is then performed. The influence of the types of components on the maximal numbers of configurations and the degrees of the characteristic polynomials of the 6-SPS parallel manipulators is then revealed. The number of redundant sensors needed to reduce the FDA of 6-SPS parallel manipulators to the solution of several univariate quadratic equations in sequence based on the component method is also presented.

A novel class of generalized parallel manipulators with high rotational capability

2020 ◽  
Vol 234 (23) ◽  
pp. 4599-4619
Author(s):  
Chunxu Tian ◽  
Dan Zhang ◽  
Jian Liu
Keyword(s):  
Parallel Manipulator ◽  
Degrees Of Freedom ◽  
Structural Characteristics ◽  
Synthesis Method ◽  
Parallel Manipulators ◽  
Rigid Bodies ◽  
Kinematic Chains ◽  
Moving Platforms ◽  
Moving Platform ◽  
Limb Structure

A conventional parallel manipulator is characterized by connecting one moving platform with two or more serial kinematic limbs. Since each limb is independently supporting one moving platform, the moving platform must be a rigid body with several kinematic pairs fixed on it. However, for generalized parallel manipulators with articulated moving platforms, the moving platforms are not limited to rigid bodies but including serial kinematic chains or internal kinematic joints. The introduction of articulated moving platforms allows for improving the kinematic performance of generalized parallel manipulators, especially for rotational capability. On account of the structural characteristics of the moving platforms, it also poses a significant challenge in the construction of the structures of manipulators. This research raises a new method for the type synthesis of generalized parallel manipulators with novel articulated moving platforms. The proposed method introduces a striking shortcut for the limb structure analysis of mechanisms with high rotational capability. In this paper, a class of generalized parallel manipulator with different degrees of freedom from 3 to 6 are constructed by using the constraint synthesis method, and several examples are provided to demonstrate the feasibility of the advocated method. At last, the 3T3R generalized parallel manipulator is taken as an example to analyze the inverse kinematics, and the evaluation of the workspace is conducted to verify the rotational capacity.

Forward Problem Singularities of Manipulators Which Become PS-2RS or 2PS-RS Structures When the Actuators Are Locked

2003 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Rigid Body ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Geometric Interpretation ◽  
Forward Problem ◽  
End Effector ◽  
Kinematic Chains ◽  
Position And Orientation ◽  
Manipulator Design

The instantaneous forward problem (IFP) singularities of a parallel manipulator (PM) must be determined during the manipulator design and avoided during the manipulator operation, because they are configurations where the end-effector pose (position and orientation) cannot be controlled by acting on the actuators any longer, and the internal loads of some links become infinite. When the actuators are locked, PMs become structures consisting of one rigid body (platform) connected to another rigid body (base) by means of a number of kinematic chains (limbs). The geometries (singular geometries) of these structures where the platform can perform infinitesimal motion correspond to the IFP singularities of the PMs the structures derive from. This paper studies the singular geometries both of the PS-2RS structure and of the 2PS-RS structure. In particular, the singularity conditions of the two structures will be determined. Moreover, the geometric interpretation of their singularity conditions will be provided. Finally, the use of the obtained results in the design of parallel manipulators which become either PS-2RS or 2PS-RS structures, when the actuators are locked, will be illustrated.

Compositional Submanifolds of Prismatic–Universal–Prismatic and Skewed Prismatic–Revolute– Prismatic Kinematic Chains and Their Derived Parallel Mechanisms

2018 ◽  
Vol 10 (3) ◽  
Author(s):  
Xinsheng Zhang ◽  
Pablo López-Custodio ◽  
Jian S. Dai
Keyword(s):  
Parallel Mechanism ◽  
Kinematic Chain ◽  
Planar Motion ◽  
Helical Motion ◽  
Parallel Mechanisms ◽  
Motion Group ◽  
Revolute Joint ◽  
Kinematic Chains ◽  
Prismatic Joint ◽  
Joint Axis

The kinematic chains that generate the planar motion group in which the prismatic-joint direction is always perpendicular to the revolute-joint axis have shown their effectiveness in type synthesis and mechanism analysis in parallel mechanisms. This paper extends the standard prismatic–revolute–prismatic (PRP) kinematic chain generating the planar motion group to a relatively generic case, in which one of the prismatic joint-directions is not necessarily perpendicular to the revolute-joint axis, leading to the discovery of a pseudo-helical motion with a variable pitch in a kinematic chain. The displacement of such a PRP chain generates a submanifold of the Schoenflies motion subgroup. This paper investigates for the first time this type of motion that is the variable-pitched pseudo-planar motion described by the above submanifold. Following the extraction of a helical motion from this skewed PRP kinematic chain, this paper investigates the bifurcated motion in a 3-prismatic–universal–prismatic (PUP) parallel mechanism by changing the active geometrical constraint in its configuration space. The method used in this contribution simplifies the analysis of such a parallel mechanism without resorting to an in-depth geometrical analysis and screw theory. Further, a parallel platform which can generate this skewed PRP type of motion is presented. An experimental test setup is based on a three-dimensional (3D) printed prototype of the 3-PUP parallel mechanism to detect the variable-pitched translation of the helical motion.

Dexterous Workspace of n-PRRR Planar Parallel Manipulators

2012 ◽  
Vol 4 (3) ◽  
Author(s):  
André Gallant ◽  
Roger Boudreau ◽  
Marise Gallant
Keyword(s):  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Divergence Theorem ◽  
End Effector ◽  
Kinematic Chains ◽  
Prismatic Joint ◽  
Planar Surfaces ◽  
Revolute Joints ◽  
Gauss Divergence Theorem ◽  
Planar Parallel Manipulators

In this work, a method is presented to geometrically determine the dexterous workspace boundary of kinematically redundant n-PRRR (n-PRRR indicates that the manipulator consists of n serial kinematic chains that connect the base to the end-effector. Each chain is composed of two actuated (therefore underlined) joints and two passive revolute joints. P indicates a prismatic joint while R indicates a revolute joint.) planar parallel manipulators. The dexterous workspace of each nonredundant RRR kinematic chain is first determined using a four-bar mechanism analogy. The effect of the prismatic actuator is then considered to yield the workspace of each PRRR kinematic chain. The intersection of the dexterous workspaces of all the kinematic chains is then obtained to determine the dexterous workspace of the planar n-PRRR manipulator. The Gauss divergence theorem applied to planar surfaces is implemented to compute the total dexterous workspace area. Finally, two examples are shown to demonstrate applications of the method.

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