what is a parallel robot

8/11/2019 What is a Parallel Robot

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Senior Lecturer Dr. Eng. Sergiu-Dan StanTechnical University of Cluj-NapocaDepartment of Mechanisms, Precision Mechanics and MechatronicsB-dul Muncii, Nr. 103-105, Corp C, Et. III, C304400641 Cluj-Napoca, ROMANIA, EU

Tel. +40-(0)264-401755e-mail. [email protected]://www.east.utcluj.ro/mec/mmfm/Colegi/stan.htmlhttp://www.sergiustan.ro/

What is a Parallel Manipulator?

A parallel manipulator consists of a moving platform and a fixed base, connected by several

linkages (also called legs).

Figure 1. Classic Stewart platform (6 DOF)

The Stewart Platform is characterized by high force-torque capacity, high structural rigidity,and low moving mass.

Figure 2. A typically PKM with 3 DOF designed for milling applications
mailto:[email protected]://www.east.utcluj.ro/mec/mmfm/Colegi/stan.htmlhttp://www.sergiustan.ro/http://www.sergiustan.ro/http://www.east.utcluj.ro/mec/mmfm/Colegi/stan.htmlmailto:[email protected]


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As PKM are used for more difficult tasks, control requirements increase in complexity tomeet these demands. The implementation for PKMs often differs from their serialcounterparts, and the dual relationship between serial and parallel manipulators oftenmeans one technique which is simple to implement on serial manipulators is difficult forPKMs (and vice versa). Because parallel manipulators result in a loss of full constraint atsingular configurations, any control applied to a parallel manipulator must avoid suchconfigurations. The manipulator is usually limited to a subset of the usable workspace sincethe required actuator torques will approach infinity as the manipulator approaches asingular configuration. Thus, some method must be in place to ensure that the manipulatorsavoid those configurations.

Parallel robots have many advantages comparing to the serial robots, such as high flexibility,high stiffness, and high accuracy. To achieve a higher accuracy the static and dynamicbehavior must be better understood. The problems concerning kinematics and dynamics ofparallel robots are as a rule more complicated than those of serial one.

Modeling and simulation of integrated environment for parallel robots

Modeling and simulation are important and essential stages in the engineering design andproblem solving process because it allows to prevent the risks and to lower the costs thatappear with the design, construction and testing stage of a new robot. In the design processof parallel robots different tools can be used to model and simulate a robot. There arehowever different modeling and simulation tools which can be used together with oneanother to help into the development of a model. These different tools can vary according toits functionality and according to its best use for Modeling. For example some tools are best

suitable for first functionality model such as CAD software ( SolidWorks ). SolidWorks ismainly a CAD tool although one can also do some simulation in SolidWorks , it is not the besttool used for simulation. For that reason other simulation tools such as ADAMS can be usedinstead to perform the dynamical simulation of the robot. The same thing, ADAMS is againnot the proper CAD tool so one cannot get detailed geometrical information about the partsand elements of the parallel Robots. ADAMS does not also perform the simulation of thefunctionality of the Robot, so it only shows how a robot would move in reality and one canget real physical information about the Robot and how it would perform dynamically inreality. For simulation and control of a parallel robot other software tool such as MATLABcan be used to simulate the parallel robot.

Figure 3. Integrated Environment for simulation and modeling of parallel robots

ROBOT

MODEL


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With the help of SolidWorks software we can make: 1) the CAD model of the parallel robot,2) detailed modeling of the parallel robot components, 3) choose appropriate material forparts from the material library, 4) assembling parts and components, 5) get geometrical andphysical information about the parts [2].With ADAMS software we can determine: 1) the simulation of the dynamical movement ofthe parallel robot, 2) deal with joints and actuators, 3) simulate the real movement of theparallel robot. With MATLAB software we can elaborate and simulate the control of parallelrobots.Unlike parallel robots, a serial robot is an open-chain structure consisting of several linksconnected in series. The advantages of parallel robots as compared to serial ones are:

higher payload-to-weight ratio since the payload is carried by several links in parallel, higher accuracy due to non-cumulative joint error, higher structural rigidity, since the load is usually carried by several links in parallel

and in some structures in compression-traction mode only,

location of motors at or close to the base, simpler solution of the inverse kinematics equations.In the CAD model note that parallel robots usually are as standard design mechanism, inother words a design model for a parallel robot would only be the effort of designing onechain which is usually repeated symmetrically for the whole robot.

Kinematical representation of parallel robots shows in a simple way the kinematicalstructure of a parallel robot. Mainly parallel robots can be represented with respect to thestructure of their parallel mechanism. A parallel mechanics is a repetition of number ofmechanical chains which are connecting the platform to the base (as previously defined). Forexample it was chosen the Hexaglide parallel robot with six degrees of freedom [3].

Figure 4. Hexaglide parallel robot (6 DOF)

3 DOF parallel robot


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Figure 5. Kinematics scheme of the 3-RRR planar parallel robot

To control the movement of the end-effector of the robot the program uses the inversekinematics, the user will enter the position of the end-effector and the program willcompute the angles of all three actuators. The computation of the angles is made in to steps,in the first step is determine the position of the points A, B, and C, all three points aresituated in the corner of an equilateral triangle, the position of the mass center of thetriangle is known P x and Py , also the angle q between the end-effector and the Ox axe isgiven, L3 represent the distance from P to A.

)cos(3 L P A x x (1)

)sin(3 L P A y y (2)

Knowing the position of point A, B, C of the end-effector the problem is reduced to aninverse kinematics problem of a serial robot with 2 arms, fig. 3

Figure 6. Kinematics scheme of the 2-RR manipulator


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11 )(2 L x x K O B (3)

12 )(2 L y y K O B (4)

2221

223 )()( O BO B y y x x L L K (5)

)],(2tan[2

)],(2tan[2

1323

22

2121

1323

22

2121

K K K K K K a

K K K K K K a

(6)

The angle have two solutions (Eq. 6), this is possible because the mechanism can have thepoint B in the same position in two situations when the elbow is up and when the elbow isdown.

Figure 7. Planar parallel minirobot with 3 DOF

The CAD model of the parallel minirobot with 3 dof is presented in figure 2.

Figure 8. CAD model of the planar parallel minirobot with 3 DOF


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Figure 9. Configuration of the 3-RRR planar parallel minirobotfor x=0.2, y=0.15, =45

Figure 10. Planar 2 DOF mini parallel robot.The working envelope to machine size using variable length struts is dependent on the

following factors:1. The length of the extended and retracted actuator (L min , Lmax );2. Limitations due to the joint angle range.

Fig. 11 clearly illustrates the limiting effect of the joint limits.


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Figure 11. Workspace of the micro parallel robot with variable length struts

Kinematic analysis

Robot kinematics deal with the study of the robot motion as constrained by the geometry ofthe links. Typically, the study of the robot kinematics is divided into two parts, inversekinematics and forward (or direct) kinematics. The inverse kinematics problem involves aknown pose (position and orientation) of the output platform of the robot to a set of input

joint variables that will achieve that pose.

The forward kinematics problem involves the mapping from a known set of input jointvariables to a pose of the moving platform that results from those given inputs. However,the inverse and forward kinematics problems of our parallel robot can be described in closedform.

The kinematics relation between x and q of this 2 DOF mini parallel robot can be expressedsolving the:

f(x, q )=0 (1)

Then the inverse kinematics problem of the parallel robot can be solved by writing thefollowing equations:

(2)

The TCP position can be calculated by using inverted transformation, from (2), thus thedirect kinematics of the robot can be described as:

b

qbq x P

2

22

221

(3)22

1 P P xq y

221 P P y xq

222 )( P P y xbq


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where the values of the x p, y P can be easily determined.

The forward and the inverse kinematics problems were solved under the MATLABenvironment and it contains a user friendly graphical interface. The user can visualize thedifferent solutions and the different geometric parameters of the parallel robot can bemodified to investigate their effect on the kinematics of the robot.

This graphical user interface can be a valuable and effective tool for the workspace analysisand the kinematics of the parallel robots. The designer can enhance the performance of hisdesign using the results given by the presented graphical user interface. The Matlab-basedprogram is written to compute the forward and inverse kinematics of the parallel robot with2 degrees of freedom. It consists of several MATLAB scripts and functions used forworkspace analysis and kinematics of the parallel robot. A friendly user interface wasdeveloped using the MATLAB-GUI (graphical user interface). Several dialog boxes guide theuser through the complete process.

Figure 12. Graphical User Interface (GUI) for solving inverse kinematics of the 2 DOF planarparallel robot in MATLAB environment.

The user can modify the geometry of the 2 DOF parallel robot. The program visualizes thecorresponding kinematics results with the new inputs.


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Figure 13. The GUI for calculus of workspace for the planar 2 DOF mini parallel robot.

Case I:Conditions:

bqq min2min1 , bq max1 , bq max2 a) for y>0

Figure 14. The workspace of the planar 2 DOF mini parallel robot is shown as the shadingregion.

b) for y , there exist two regions of the workspace


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Case II:Conditions:

bqq min2min1 , bq max1 , bq max2 a) for y>0


b) for y , there exist two regions of the workspace


Case III:Conditions:

bqq min2min1 , bq max1 , bq max2


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Case IV:Conditions:

bqq min2min1 , bq max1 , bq max2


Case V:Conditions: bqq min2min1 , min2max1 qbq , min1max2 qbq


Case VI:Conditions: bqq min2min1 , min2max1 qbq , min1max2

qbq


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Case VII:Conditions: bq min1 , bq max1 , bq min2 , bq max2 ,

bqq min2min1 , bqq max2max1


Description of the 2 DOF parallel structure

A planar parallel robot is formed when two or more planar kinematic chains act together ona common rigid platform. For simplicity, the origin of the fixed base frame {B} is located atbase joint A with its x-axis towards base joint B, and the origin of the moving frame {M} islocated in TCP, as shown in Fig. 15. The position of the moving frame {M} in the base frame{B} isx=(xP, yP)

T and the actuated joint variables are represented by q=(q1, q 2)T.


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Figure 15. Kinematic scheme of the 2 DOF parallel robot

The kinematics relation between x and q of this 2 DOF mini parallel robot can be

expressed solving the:

f(x, q )=0 (1)

Then the inverse kinematics problem of the parallel robot can be solved by writing

the following equations:

2211 P p yl xq ;

2222 P p yl xq (2)

The designer can enhance the performance of his design using the results given by thepresented graphical user interface. The Delphi-based program is written to compute theforward and inverse kinematics of the parallel robot with 2 degrees of freedom.


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Figure 15 . Assembly of the robot, CAD model of 2 DOF parallel mini robot

A general planar micro parallel robot with two degrees of freedom activated by prismatic joints is shown in Fig. 1.

Fig. 16. Variable length struts micro parallel robot


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Fig. 2. The general kinematic scheme of a 2 DOF parallel robot with translation actuators

Fig. 3. Assembly of the robot, CAD model of 2 DOF parallel mini robot

Fig. 5. Workspace limits of Biglide

The mechanism PRRRP realizes a wide workspace as well as high-speed. Analysis,visualization of workspace is an important aspect of performance analysis. A numerical

algorithm to generate reachable workspace of parallel manipulators is introduced.


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This section presents the methodology to determine the workspace of the 2 DOF miniparallel robot. It consists of several MATLAB scripts and functions used for workspaceanalysis and kinematics of the parallel robot.

A friendly user interface was developed using the MATLAB-GUI (graphical user interface).Several dialog boxes guide the user through the complete process.

Fig. 6. The GUI for calculus of workspace for the planar 2 DOF mini parallel robot.

A fundamental characteristic that must be taken into account in the dimensional design of

robot manipulators is the area of their workspace.It is crucial to calculate the workspace and its boundaries with perfect precision, because

they influence the dimensional design, the manipulators positioning in the workenvironment, and its dexterity to execute tasks.

a) Workspace for the planar 2 DOF mini parallel robot, case mmqq 100max2max1


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b) Workspace for the planar 2 DOF mini parallel robot, case mmqq 200max2max1

c) Workspace for the planar 2 DOF mini parallel robot, case mmqq 400max2max1

Fig. 7. Different regions of workspace for different lengths of stroke of actuators a), b) and c)

Factors that limit the workspace of a given parallel manipulator include actuator limits, leginterference, and singular configurations.


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Fig. 8. Two different areas of workspace in two possible configurations

C. Singularity analysis

Because singularity leads to a loss of the controllability and degradation of the naturalstiffness of manipulators, the analysis of parallel manipulators has drawn considerableattention.

In some configurations, the robot cannot be fully controlled. Most parallel robots sufferfrom the presence of singular configurations in their workspace that limit the machineperformances.

In the case of parallel manipulators and closed-loop mechanisms, singularity analysis ismuch more difficult since such mechanisms contain unactuated joints and joints with morethan one degree of freedom.

In general, closed-form solutions for singular curves/surfaces for parallel manipulators ofarbitrary architecture requires elimination of unwanted variables from several nonlineartranscendental equations, and this is quite difficult.

Based on the forward and inverse Jacobian matrices, three kinds of singularities of parallelmanipulators can be obtained [10]. Let q denote the actuated joint variables, and let x

describe the location of the moving platform. The kinematic constraints imposed by thelimbs are expressed as f(x,q) = 0. Differentiating with respect to time, a relation between theinput joint rates and the end-effector output velocity is obtained as:

q J x J q x (7)

wheredx f

J x and dq f

J q . Hence, the overall jacobian matrix J, can be written as:

x J q (8)where xq J J J

1 .


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Singular configurations should be avoided. The singular configurations (also calledsingularities) of a parallel robot may appear inside the workspace or at its boundaries.

The first kind of singularity

The first kind of singularity occurs when the following condition is satisfied:0,)det(J x 0)det(J q (9)

This kind of singularity corresponds to the limit of the workspace.

The second kind of singularity

The second kind of singularity occurs when we have following:0,)det(J x 0)det(J q (10)

The physical interpretation of this kind of singularity is that even if all of the input velocitiesare zero, there are still be instantaneous motion of the end-effector. In this configuration,the manipulator loses stiffness and becomes uncontrollable. This kind of singularity islocated inside the workspace of the manipulator. Such a singularity is very difficult to locateonly by analyzing and expanding the equation 0)det(J q . A numerical method is thus agood selection for solving this problem.

The third kind of singularity

The third kind of singularity occurs when both:

0)det(J)det(J xq (11)

This kind of singularity corresponds to the first and second type of singularity occurringsimultaneously. This singularity is both configuration and architecture dependent.

Parallel singularities are particularly undesirable because they cause the followingproblems:

a high increase of forces in joints and links, that may damage the structure, a decrease of the mechanism stiffness that can lead to uncontrolled motions of the tool

though actuated joints are locked.


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Fig. 9. Singular configuration for the planar 2 DOF mini parallel robot



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Fig. 13 Transmission quality index for PRRRP mini parallel robot


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Fig. 3 CAD model of the parallel robot

The forward and the inverse kinematics problems were solved under the MATLAB

environment and it contains a user friendly graphical interface.

Fig. 4. Graphical User Interface for solving IKP

Fig. 5. Robot configuration for X P=25 mm Y P=60 mm


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Fig. 6. Robot configuration for X P=35 mm Y P=60 mm

Fig. 2. Six degrees-of-freedom micro parallel robot


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Fig. 3. Workspace views of the HEXAPOD micro parallel robot with six degrees-of-freedom

Fig. 4. Workspace views of the HEXAPOD micro parallel robot with six degrees-of-freedom

The possible workspace of the robot is of a great importance for optimization of theparallel robots. Without the ability to solve the workspace is impossible to state that therobot can fulfill any work task. The general analysis of the workspace consists in workspacedetermination using the described discretization method.


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Fig. 5. Graphical User Interface for determining the shape of the HEXAPOD micro parallelrobot with six degrees-of-freedom

The workspace is the volume in the space case where the tool centre point (TCP) can becontrolled and moved continuously and unobstructed. The workspace is limited by singularconfigurations . At singularity poses it is not possible to establish definite relations betweeninput and output coordinates. Such poses must be avoided by the control.

Fig. 6. Top view of the workspace of the HEXAPOD micro parallel robot with six degrees-of-freedom

The robotics literature contains various indices of performance [19], [20], such as theworkspace index W .


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Fig. 7. Lateral view of the workspace of the HEXAPOD micro parallel robot with six degrees-of-freedom

C. Performance evaluation

Beside workspace which is an important design criterion, transmission quality index isanother important criterion. The transmission quality index couples velocity and force

transmission properties of a parallel robot, i.e. power features [21]. Its definition runs:

1

2

J J

E T (3)

where E is the unity matrix. T is between 0


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Fig. 8. Transmission quality index for HEXAPOD micro parallel robot with six degrees-of-freedom (3D view)

Fig. 9. Transmission quality index for HEXAPOD micro parallel robot with six degrees-of-freedom (top view)


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As it can be seen, the micro parallel robot presents better performances in the middle of

its workspace, as presented in Fig. 6-7.

2. 3-RRR PARALLEL ROBOT AND WORKSPACE ANALYSIS

The planar micro 3 DOF parallel robot is shown in Fig.1. This structure is also known as 3-RRR robot. Since mobility of this micro parallel robot is three, three actuators are required tocontrol this robot.

Fig. 1. 3-DOF micro parallel robot of type 3-RRR

The workspace of a robot is defined as the set of all endeffector configurations which canbe reached by some choice of joint coordinates. As the reachable locations of an end-effector are dependent on its orientation, a complete representation of the workspaceshould be embedded in a 6-dimensional workspace for which there is no possible graphicalillustration; only subsets of the workspace may therefore be represented.

The knowledge of the workspace of a 3 DOF micro parallel robot is very important inplanning a dexterous manipulation task. The workspace is one of the most importantkinematic properties of robots, even by practical viewpoint because of its impact on robotdesign. In this section, the workspace of the proposed robot will be discussed systematically.Here, we propose an approach to compute and visualize the workspace of a 3 DOF microparallel robot. Micro parallel robots are good candidates for microminiaturization into amicrodevice.

Case I.

Workspace can be determined by using discretization method. There were identified severalcases of workspace.


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(1)

The main disadvantage of parallel robots is their small workspace in comparison to serialarms of similar size. Despite the advantages of parallel manipulators there are certaindisadvantages to be encountered such as complicated kinematics and dynamics, manysingular configurations, and poor workspace availability.

It is very important to analyze the area and the shape of workspace for parameters givenrobot in the context of industrial application.

Fig. 2. Workspace of the 3-RRR parallel micro robot


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Fig. 3. Workspace of the 3-RRR parallel micro robot

Case II. , ,

(2)

Singular configurations were identified as it is presented in the following figures.

Fig. 4. Singular configuration, =0


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Fig. 5. Singular configuration,

Fig. 6. Singular configurations,

Fig. 2. CAD model of the 3-DOF UPU micro parallel robot


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Fig. 3. The GUI for calculus of workspace for the 3 DOF UPU micro parallel robot.

Resolution of the latter is very important for any manipulator design. If we restrictourselves to a 3-DOF parallel robot (a UPU parallel robot for example) we will find that thelink lengths limit the workspace. On the other hand, there will necessarily be a designlimitation of some sort on the link lengths. Also one must have a compact design which iscapable of full manoeuvring. The workspace is primarily limited by the boundary ofsolvability of inverse kinematics. Then the workspace is limited by the reachable extent ofdrives and joints, occurrence of singularities and by the link and platform collisions.Analysis,visualization of workspace is an important aspect of performance analysis. A numericalalgorithm to generate reachable workspace of parallel manipulators is introduced. Thissection presents the methodology to determine the workspace of the 3 DOF micro parallelrobot. It consists of several MATLAB scripts and functions used for workspace analysis andkinematics of the parallel robot. A friendly user interface was developed using the MATLAB-GUI (graphical user interface). Several dialog boxes guide the user through the completeprocess. In the followings are presented the workspace of the UPU parallel robot fordifferent values of diameter of the circle for the lower platform, d .


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Fig. 4. Workspace of the 3-DOF UPU parallel robot, d=100mm

Fig. 5. Workspace of the 3-DOF UPU micro parallel robot, d=110mm


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This graphical user interface (Fig. 2) can be a valuable and effective tool for theworkspace analysis and the kinematics of the parallel robots. The designer can enhance theperformance of his design using the results given by the presented graphical user interface.

The 2 degree of freedom Parallel Kinematic Machine

The Parallel Kinematic Machine considered in this paper is the 2-dof planar parallelmechanism shown in Figure 1. The PKM consists of a five-bar mechanism connected to abase by two rotary actuators, which control the two output degrees of freedom of the end-effector. The actuators are joined to the base and platform by means of revolute jointsidentified by the letters A D. It will be assumed that AO=OC=AC/2. The coordinates of pointP, the end-effector point, are ( x P , y P). In more general terms, the actuator joint angles are the

input variables, i.e. v=[q1 , q 2]T

2

. The global coordinates of the working point P form theoutput coordinates, i.e. u=[ x P , y P]

T

2. The 2-dof PKM may be used only for positioning P in


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the x-y plane. It is evident that this manipulator thus has 2-dof. Thus, the generalizedcoordinates for this kind of PKM are therefore given by:

q=[uT, vT]T=[ x P , y P, q1 , q 2]T

4 (1)

In general, factors imposed by the physical construction of the planar parallelmanipulator, which limit the workspace, may be related to the input variables or acombination of input, output and intermediate variables. An example of former type for theplanar parallel manipulator are joint angles limits, and of the latter, limits on the angulardisplacement of the revolute joints connecting the legs to the ground and to the platform.These limiting factors are described by means of inequality constraints and may,respectively, take the general forms:

vmin v vmax (2)

gmin g(u , v) gmax (3)

The above general definitions are necessary in order to facilitate the mathematicaldescription of kinematics and workspaces of the 2-dof planar PKM. In this study mixedconstraints, represented by (3), are not taken into consideration.

Figure 1. Parallel Kinematic Machine with 2 degrees of freedom


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3. The kinematics and condition number of the manipulatorKinematic analysis of five-bar mechanism is needed before carrying out derivations for themathematical model. It is considered the five-bar mechanism with revolute joints as inFigure 1. It is known the length of the links as well as the fixed joints coordinates. The five-bar mechanism is symmetric toward Oy-axis, thus l a = l d = l respectively l b = l c = L. Actuatorsare placed in A and C . Attaching to each link a vector, on the OABPO respectively OCDPO, wecan write successively the relations:

.DPCDOCOP;BPABOAOP (4)Based on the above relations, the coordinates of the point P have the following forms:

.sinsinsinsin

;coscos2

coscos2

4231

4231

q Lql q Lql y

q Lql d

q Lql d

x

P

P (5)

3.1 Direct Kinemtics Problem AnalysisIn this part, kinematics of a planar POM articulated with revolute type joints has beenformulated to solve direct kinematics problem, where the position, velocity and accelerationof the PKM end effector are required for a given set of joint position, velocity andacceleration.

The Direct Kinematic Problem (DKP) of PKM is an important research direction ofmechanics, which is also the most basic task of mechanic movement analysis and the basesuch as mechanism velocity, mechanism acceleration, force analysis, error analysis,workspace analysis, dynamical analysis and mechanical integration. For this kind of PKMsolving DKP is easy. Coordinates of point P in the case when values of joint angles are known

1q and 2q are obtained from relations:

.)(y ,2 4 P2

D B

D B P P y y x x x AC BC D D x (6)

where:

1B1B2D2D

DB2

DBDDBDBD2

DB2

DB

DBD22

DP2D

2D

2DB

2BP

2DP

2B

2B

2D

2D

qsinly;qcosl2d

x;qsinly;qcosl2d

x

)xx(A2)yy(x2)xx)(yy(y2D;)xx()yy(C

A)yy(y2A)Lyx()yy(B);LLyxyx(21

A

(7)

The speed of the point P is obtained differentiating the relations (1). Thus results:

3

1B

y

xA JV

VJ

where .)sin(0

0)sin( ;sincos

sincos

42

31

44

33

qq Ll

qq Ll J q Lq L

q Lq L J B A

(8)

or .3

1

J

V

V

y

x (9)

where

.)sin(cos)sin(cos

)sin(sin)sin(sin

)sin( 423314

423314

34

1

qqqqq

qqqqqq

qq L

J J J A B (10)

And J represents the Jacobian matrix.Acceleration of the point P are obtained by differentiating of relation (9), as it yealds:


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.3

1

3

1

J

dt

d J

A

A

y

x (11)

Figure 2. The two forward kinematic models: (a) the up-configuration and (b) the down-configuration

3.2 Inverse Kinematics Problem AnalysisBased on the inverse kinematics analysis are determined the motion lows of the actuatorlinks function of the kinematics parameters of point P.

The values of joint angles iq , (i = 14 ) knowing the coordinates x P , y P of point P, may becomputed with the following relations:

;M N

arctgP

)P NMarctgq;

AC

)AC(BBarctg2q 2

2222

3

222i

1

1-or 1= i

,EF

)EF( b barctg2q;

ef )ef (BB

arctg2q222

4

222i

2

(12)

.sinsin;coscos; ;2 ;2

;2

;2

2;2

;2 ;2

2;2

;

2

2 ;

2

;2 ;

2

2

212122

2222

2222

2222

2222

ql ql d ql ql P L N L M

Ll yd

x F d

x L E Ll yd

xC

l y Bd

xl A Ll yd

x f

d xl e Ll y

d xc L yb

d x La

P P P P P

P P P P

P P P P P

From Eq. (15), one can see that there are four solutions for the inverse kinematicsproblem of the 2-dof PKM. These four inverse kinematics models correspond to four types ofworking modes (see Fig. 3).

-8 -6 -4 -2 0 2 4 6 8-5

0

5

10

15

y

xO

-8 -6 -4 -2 0 2 4 6 8-5

0

5

10

15

y

xO


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a) b) c) d)

Figure 3. The four inverse kinematics models: (a)+ model; (b) + model; (b) model;(d)++ model.

4. Singularities analysis of the planar 2-dof PKMIn the followings, vector v is used to denote the actuated joint coordinates of themanipulator, representing the vector of kinematic input. Moreover, vector u denotes theCartesian coordinates of the manipulator gripper, representing the kinematic output. Thevelocity equations of the PKM can be rewritten as:

0vBuA (13)

Where T qq 21,v , T P P y x ,u , and where A and B are square matrices of dimension 2,called Jacobian matrices, with 2 the number of degrees of freedom of the PKM. Referring toEq. (13), (Gosselin and Angeles, 1990), have defined three types of singularities which occurin parallel kinematics machines.

(I) The first type of singularity occurs when det( B)=0. These configurations correspond toa set of points defining the outer and internal boundaries of the workspace of the PKM.

(II) The second type of singularity occurs when det( A)=0. This kind of singularitycorresponds to a set of points within the workspace of the PKM.

(III) The third kind of singularity when the positioning equations degenerate. This kind ofsingularity is also referred to as an architecture singularity (Stan, 2003). This occurs when thefive points ABCDP are collinear.

a) b)

-8 -6 -4 -2 0 2 4 6 8-5

0

5

10

15

y

xO

-8 -6 -4 -2 0 2 4 6 8-5

0

5

10

15

y

xO

-8 -6 -4 -2 0 2 4 6 8-5

0

5

10

15

y

xO

-8 -6 -4 -2 0 2 4 6 8-5

0

5

10

15

y

xO

-5 0 5 10 15-5

0

5

10

15

y

xO

-5 0 5 10 15-5

0

5

10

15

y

xO


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c) d)

Figure 4. Some configurations of singularities: (a) the configuration when l b and l c arecompletely extended (b) both legs are completely extended; (c) the second leg is completely

extended and (d) the first leg is completely extended.

In this paper, it will be used to analyze the second type of singularity of the 2-dof PKMintroduced above in order to find the singular configuration with this type of PKM. For the

first type of singularity, the singular configurations can be obtained by computing theboundary of the workspace of the PKM. Furthermore, it is assumed that the third type ofsingularity is avoided by a proper choice of the kinematic parameters.

For this PKM, we can use the angular velocities of links l c and l b as the output vector.Matrix A is then written as:

)sin()(ins

)cos()cos(

3b4c

3b4c

ql ql

ql ql A (14)

From Eq. (14), one then obtains:

).sin()det( 34bc qql l A (15)From Eq. (15), it is clear that when ....,,2,1,0n,n 34 qq then .)det( 0 A In

other words if the two links l c and l b are along the same line, the PKM is in a configurationwhich corresponds to be second type of singularity.

5. Optimal design of the planar 2-dof PKMThe performance index chosen corresponds to the workspace of the PKM. Workspace isdefined as the region that the output point P can reach if q1 and q2 changes from 2 withoutthe consideration of interference between links and the singularities. There were identifiedfive types of workspace shapes for the 2-dof PKM. Each workspace is symmetric about the x and y axes. Workspace was determined using a program made in MATLAB.

-5 0 5 10 15-5

0

5

10

15

y

xO

-5 0 5 10 15-5

0

5

10

15

y

xO


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Figure 5. Workspace of the 2-dof Parallel Kinematic Machine

Three DOF parallel robots

The parallel robots are with 3 degrees-of-freedom parallel manipulator comprising a fixedbase platform and a payload platform, linked together by three independent, identical, openkinematic chains (Fig. 1-2).

The TRIGLIDE parallel robot consists of a spatial parallel structure with three translationaldegrees of freedom and is driven by three linear actuators. The platform is connected witheach drive by two links forming a parallelogram and allowing only translational movementsof the platform and keeping the platform parallel to the base plane. An additional rotationalaxis can be mounted on the working platform to adjust the orientation of the end-effector.

The three drives of the structure are arranged in the base plane at intervals of 120 degreesstar-shaped. Thus the structure has a workspace which is nearly round or triangle-shaped.

Fig. 1. TRIGLIDE parallel robot with 3 DOF.

The DELTA linear parallel robot with 3 DOF is shown in Fig. 2 and the geometric parametersare presented in Fig. 3, where the moving platform is connected to the base platform bythree identical serial chains.

Each of the three chains contains one spatial parallelogram. The parallelogram is actuallythe vertices of which are four spherical joints


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Fig. 2. DELTA parallel robot with linear actuators.

Mathematical model

To analyze the kinematics model of the parallel robots, two relative coordinate frames areassigned, as shown in Fig. 3. A static Cartesian coordinate frame XYZ is fixed at the center ofthe base while a mobile Cartesian coordinate frame X PYPZP is assigned to the center of themobile platform. A i, (i = 1, 2, 3) and B i, (i = 1, 2, 3) are the joints that are located at thecenter of the base and the platform passive joints respectively. A middle link L 2 is installedbetween the mobile and fixed platform.


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where i is computed as .120)1( ii Its obtained also

i Ar from Eq. (3):

0

sin

cos

ii

ii

A q

q

r i

(3)

where i is computes as .120)1( ii From Eq. (3) yields f i :

|| (4)

(5)From Eq. (4) we obtain Eq. (5) and by reformulating Eq. (5) is obtained:

(6)and

(7)by substitution from Eq. (7) we obtain the inverse kinematics problem of the TRIGLIDEparallel robot from Fig. 1 by solving the following equations:

(8)The forward and the inverse kinematics problems were solved under the MATLAB

environment and it contains a user friendly graphical interface.For DELTA linear the closed-form solutions for both the inverse and forward

kinematics have been developed in [14]. Here, for convenience, we recall the inversekinematics briefly.


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a) 3 DOF DELTA linear parallel robot

b) Fixed platform

c) Mobile platform

Fig. 5. Schematic diagram of mobile and fixed platform for DELTA linear parallel robot.

Let R and r be the radii of the base and the platform passing though joints Pi and Bi (i = 1,2, 3), respectively.

(9) 000

23

23

0

22

321 R R

R R R

P P P


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(10)

(11)

After computing the position of joints Pi and Bi the inverse kinematics of the DELTAparallel robot with linear actuators can be solved by writing following equations:

( ) (12)

Eq. (12) represents the inverse kinematics problem of the DELTA linear parallel robot.

Workspace evaluationIn this section, the workspace of the proposed robots will be discussed systematically. It is

very important to analyze the area and the shape of workspace for parameters given robotin the context of industrial application. The workspace is primarily limited by the boundaryof solvability of inverse kinematics. Then the workspace is limited by the reachable extent ofdrives and joints, occurrence of singularities and by the link and platform collisions. Theparallel robots TRIGLIDE and DELTA linear realize a wide workspace. Analysis, visualization ofworkspace is an important aspect of performance analysis. A numerical algorithm togenerate reachable workspace of parallel manipulators is introduced. Other design specificfactors such as the end-effector size, drive volumes have been neglected for simplification.

000

23

23

0

22

321 r r

r r r

B B B

111

000

000

321 uuu


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Fig. 6. The GUI for calculus of workspace for the TRIGLIDE 3 DOF parallel robot.

Fig. 7. The GUI for calculus of workspace for the DELTA linear 3 DOF parallel robot

what is a parallel robot

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