A Study on Workspace of Planar 2-DOF Redundant Drive Parallel Mechanism

CHAI Baoming, WANG Yuandong

College of Mechanical and Electrical Engineering, Hebei University of Engineering, Handan 056038, China

AStudyonWorkspaceofPlanar2-DOFRedundantDriveParallelMechanism

CHAI Baoming, WANG Yuandong*

CollegeofMechanicalandElectricalEngineering,HebeiUniversityofEngineering,Handan056038,China

Theproblemsofresearchonworkspaceofplanarparallelmechanismwereproposedinthispaper.Thentakinga2-DOFparallelrobotastheresearchobject,thispaperpaintedthe2Dmapofworkspacebasedondimensionlessparametersandsolvedtheworkspaceofparallelrobot.Thefunctionalrelationshipsbetweenstructureparametersandworkspaceshapewerebuilt.Afterwardsthe2Dmapofworkspacewasanalysedandthelawofworkspaceinfluencedbystructuresizeparameters,whichprovidedsignificantreferencefordesigningmechanismandplanningtrajectory,wasobtained.Inaddition,thefurtheranalysisofworkspaceofparallelrobotwasalsogreatlysimplified.

parallelrobot,redundantdrive，workspace,structureparameters，functionalrelationship

1.Introduction

Workspace is the area where robot end-effector works and is the important index to measure its performances[1]. The study on workspace includes both defining and analysing workspace. Workspace could be divided into complete workspace, reachable workspace, dextrous workspace and so on. Complete workspace refers to all of the positions and form sets of end-effector where different joint rotating angles can reach. Reachable workspace refers to the position and space sets where end-effector can reach at a certain location and pose. Dextrous workspace refers to the position and space sets where end-effector can reach at any location and pose. Workspace analysis relates to mechanism geometry parameter, structure restraint and driving force range. And there are two methods, numerical method or analytic method, to evaluate the position and form where the end-effector can realize.

This paper focuses on planar 2-DOF redundancy drive parallel robot. Cervantes-Sanchez[2] and Fallahi[3] respectively got the robot workspace boundary by employing implicit function theorem and a common solution between two quadratic equations. Kock and Lijie Zhang[4] have done some research on the workspace of planar 2-DOF redundant drive parallel robot. Nevertheless, the complete mathematical relationships corresponding to structure parameters and workspace are not involved in their studies. This paper systematically studied the workspace of planar 2-DOF redundant drive parallel robot and got the corresponding workspace map according to the built functional relationship between structure parameters and workspace.

2.Relationships between structure parameters and workspace

In order to make the performance of parallel robot symmetrical and simplify the design parameters, this paper chose the parallel robot with symmetrical structure.

The structural model of planar 2-DOF redundant drive parallel robot is presented in Fig.1. Here, point C on end-effector can be connected to steady platformA1A2A3with three series branches.A1,A2andA3are the installation positions of driving motor.

Fig.1 Structural model of parallel robot

To begin with, the size parameters of each robot member should be non-dimensioned in order to systematically study the workspace of robot. And the non-dimensionalization parametric equations can be expressed as follows:

(1)

In whichdrepresents the installation distance from a drive motor to another in the above Eq.(1).

The installation conditions are known to all that the sum of single branch length must be more than the distance from the vertex of equilateral triangle, which is made up of the three vertexes of the platform, to the center. That is:

(2)

The workspace shape of parallel robot can be divided into the following four situations to discuss.

1) When it is satisfied that |l1-l2|=0 oru=λ.

② If (l1+l2)=dor (u+λ)=1, which presents the point (0. 5, 0. 5), the corresponding workspace shapes are shown as the shaded area in fig.b of Fig.2. PointsA1,A2andA3are distributed at the border on the shaded area.

③ If (l1+l2)>dor (u+λ)>1, the workspace shapes corresponding to all of the points of the shaded area are shown as fig.c in Fig.2, pointsA1,A2andA3are distributed in the shaded area.

(xc-A1x)2+(yc-A1y)2=(l1+l2)2

(xc-A1x)2+(yc-A1y)2=(l1-l2)2

(xc-A2x)2+(yc-A2y)2=(l1+l2)2

(xc-A2x)2+(yc-A2y)2=(l1-l2)2

(xc-A3x)2+(yc-A3y)2=(l1+l2)2

(xc-A3x)2+(yc-A3y)2=(l1-l2)2

③ If (l1+l2)>d+|l1-l2|, that is (μ>λ>0.5)∪(λ>μ>0.5), which represents the two areas are respectively surrounded by these functional curves ofu=0.5,u=λandu-λ=-0.5, and these functional curves ofu=λ=0.5 andu-λ=0.5. The workspace shapes corresponding to all of the points in the shaded area are shown as fig.f in Fig.2. The shaded areas are different from the ones in fig.d and fig.e in following respects. PointsA1,A2andA3are arrayed in the shaded area, centres are at pointA1,A2,A3,|l1-l2| is the radius of the three circular sections which cannot be reached.

(xc-A1x)2+(yc-A1y)2=(l1-l2)2

(xc-A2x)2+(yc-A2y)2=(l1-l2)2

(xc-A3x)2+(yc-A3y)2=(l1-l2)2.

③ If (l1+l2)>d+|l1-l2|, that is (μ>λ>0.5)∪(λ>μ>0.5), the workspace shapes corresponding to all of points in the shaded area are shown as fig.nin Fig.2. The shaded area shape is a ring.

Fig.2 Workspace map of parallel robot

3.Results analysis

Both shaded areas centered onA1,A2andA3, with exradius ofl1+l2and inradius of |l1-l2|, in fig.m and fig.n are made up of the intersects of three rings.

The shaded area in fig.n is also a ring, which is made up of these functional curves:

(xc-A1x)2+(yc-A1y)2≤(l1+l2)2

(xc-A2x)2+(yc-A2y)2≤(l1+l2)2

(xc-A3x)2+(yc-A3y)2≤(l1+l2)2

PointsA1,A2andA3are surrounded by the shaded area.

The shaded areas in fig.m are these three areas distributed out of the triangleA1A2A3. Area 1 is enveloped by these functional curves of

(xc-A1x)2+(yc-A1y)2=(l1+l2)2

(xc-A2x)2+(yc-A2y)2=(l1-l2)2

(xc-A3x)2+(yc-A3y)2=(l1-l2)2

Area 2 is enveloped by these functional curves of

(xc-A1x)2+(yc-A1y)2=(l1-l2)2

(xc-A2x)2+(yc-A2y)2=(l1+l2)2

(xc-A3x)2+(yc-A3y)2=(l1-l2)2

Area 3 is enveloped by these functional curves of

(xc-A1x)2+(yc-A1y)2=(l1-l2)2

(xc-A2x)2+(yc-A2y)2=(l1-l2)2

(xc-A3x)2+(yc-A3y)2=(l1+l2)2

4.Conclusions

This paper systematically deduced the functional relationship between structure parameters and workspace of planar 2-DOF redundant drive parallel robot through non-dimensionalizing each robot member size parameter. The effective design space of robot was identified based on non-dimensionalized parametersμandλ, and the effective design space was divided into several sub-domain through taking advantage of functional relationships. The one-to-one relationships between effective design space and workspace shape of robot were established. The 2D map of workspace was painted and analysed. Finally, this paper concluded the law of workspace influenced by structure size parameters, which provided significant reference for designing mechanism and planning trajectory.

[1] Huang Zhen.Mechanism Theory and Control of Parallel Robot[M].Beijing.China Machine Press,1996:12-29.

[2] Cervantes-Ssnehez J J,Rendon-Sanchez J G.A Simplified Approach for Obtaining The Workspace of A Class of 2-dof Planar Parallel Manipulators[J].Mechanism and Machine Theory,1999,34(7):1057-1073.

[3] Fallahi B.A Study of The Workspace of Five-bar Closed Loop Manipulator[J].Mechanism and Machine Theory,1994,29(5):759-765.

[4] Kock S,Sehumaeher W.A Parallelx-yManipulator with Actuation Redundancy for High-speed and Active-stiffness Applications[C]//Proeeedings of the 1998 IEEE International Conference on Robotics and Automation.Leuven,Beigium:[s.n.],1998:2295-2300.

平面2自由度冗余驅(qū)動并聯(lián)機構(gòu)的工作空間

柴保明,王遠東*

河北工程大學(xué) 機電學(xué)院，河北 邯鄲 056038

提出研究平面并聯(lián)機構(gòu)的工作空間問題，并以2-DOF并聯(lián)機器人為研究對象，通過建立平面2自由度冗余驅(qū)動并聯(lián)機器人的結(jié)構(gòu)參數(shù)與工作空間形狀的函數(shù)關(guān)系，繪出了兩者基于無量綱參數(shù)的工作空間二維圖譜，即求出了該平面并聯(lián)機器人的工作空間。對工作空間圖譜進行分析，得到了結(jié)構(gòu)尺寸參數(shù)對工作空間的影響規(guī)律，為該并聯(lián)機器人的機構(gòu)設(shè)計和軌跡規(guī)劃提供了重要參考，簡化了分析并聯(lián)機器人工作空間的難度。

并聯(lián)機器人；工作空間；冗余驅(qū)動；結(jié)構(gòu)參數(shù)；函數(shù)關(guān)系

TP24

2012-10-26

*WANG Yuandong. E-mail:916015379@qq.com

10.3969/j.issn.1001-3881.2013.06.009