Comparison of Parallel Kinematic Machines with Three ...

CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 28,aNo. 4,a2015

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DOI: 10.3901/CJME.2015.0128.052, available online at www.springerlink.com; www.cjmenet.com; www.cjme.com.cn

Comparison of Parallel Kinematic Machines with Three Translational Degrees of Freedom and Linear Actuation

PRAUSE Isabel*, CHARAF EDDINE Sami, and CORVES Burkhard

Department of Mechanism Theory and Dynamics of Machines, RWTH Aachen University, Aachen, Germany

Received September 18, 2014; revised January 15, 2015; accepted revised January 28, 2015

Abstract: The development of new robot structures, in particular of parallel kinematic machines(PKM), is widely systematized by

different structure synthesis methods. Recent research increasingly focuses on PKM with less than six degrees of freedom(DOF).

However, an overall comparison and evaluation of these structures is missing. In order to compare symmetrical PKM with three

translational DOF, different evaluation criteria are used. Workspace, maximum actuation forces and velocities, power, actuator stiffness,

accuracy and transmission behavior are taken into account to investigate strengths and weaknesses of the PKMs. A selection scheme

based on possible configurations of translational PKM including different frame configurations is presented. Moreover, an optimization

method based on a genetic algorithm is described to determine the geometric parameters of the selected PKM for an exemplary load

case and a prescribed workspace. The values of the mentioned criteria are determined for all considered PKM with respect to certain

boundary conditions. The distribution and spreading of these values within the prescribed workspace is presented by using box plots for

each criterion. Thereby, the performance characteristics of the different structures can be compared directly. The results show that there

is no “best” PKM. Further inquiries such as dynamic or stiffness analysis are necessary to extend the comparison and to finally select a

PKM.

Keywords: parallel kinematic machines, comparison, benchmark, selection scheme

1 Introduction

The successful implementation of robots and machine tools highly depends on their fundamental kinematics. In contrast to their serial kinematic counterparts, parallel kinematic machines(PKM) come with very good dynamic and stiffness properties, a high positioning accuracy and a good ratio of payload to machine-weight[1–3]. A PKM connects the moving end-effector with at least two limbs (considered as open kinematic chains) to the fixed base. The independent limbs affect the end-effector simultaneously and thus, generate its motion. Depending on the structure and the number of limbs, one to three rotational and/or one to three translational degrees of freedom can be generated[1].

Even if the workspaces of PKM compared to serial kinematic structures are relatively small and the ratio of installation space to workspace tends to be worse, the need for highly dynamic, parallel kinematic machine concepts is significant[1]. While the first machine tools based on PKM, presented at the International Manufacturing Technology Show in Chicago in 1994, consist of a hexapod structure with six degrees of freedom (DOF), not always all six DOF are necessary for modern industrial applications[1]. For

* Corresponding author. E-mail: [email protected] © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2015

example the rotational degree of freedom perpendicular to the end-effector plane in drilling and milling tasks can be omitted[3–4].

Against this background, recent research increasingly focuses on PKM with less than six DOF. These PKM possess, in addition to the reduced number of required components, lower manufacturing and process costs and simpler kinematic models[5–6]. Especially PKM with three translational DOF are able to perform, inter alia, drilling, turning or milling tasks in the context of manufacturing processes[4]. For example, the Orthoglide is primarily used as a machine tool[7–8], whereas Clavel’s Delta robot is used for pick-and-place tasks. There is a variety of other possible kinematic structures, providing purely translational motion of the end-effector with three DOF.

The aim of this paper is to compare symmetrical PKM with three translational DOF using different evaluation criteria in order to point out their strengths and weaknesses. First, the state of the art concerning the comparison of PKM is described(cf. section 2). Then, in section 3, the considered PKM and the selection scheme based on possible configurations of translational PKM are presented. Section 4 focuses on the dimensional synthesis of the selected PKM and includes the description of the optimization method. The analysis of the optimized PKM according to different evaluation criteria is performed in section 5 followed by comparison in section 6.

PRAUSE Isabel, et al: Comparison of Parallel Kinematic Machines with Three Translational Degrees of Freedom and Linear Actuation

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2 State of the Art

MCCLOY[9] compared a planar serial with two planar

parallel kinematic machines: the serial structure RR and the parallel structures RRRPR as well as RRRRR. The analyzed criteria are the size of the workspace, the required performance of the actuators, as well as the stiffness of the structure, assuming all limbs except the drives as ideal rigid. In order to compare the different structures, a global task is defined. A common type of load is selected and the presented criteria are finally combined to a general performance criterion.

GOSSELIN, et al[10], compared different parallel manipulators with six DOF regarding their workspace and kinematic properties. The translational and rotational workspaces as well as the stiffness properties of the structures are used as evaluation criteria. To provide a comparison, the structures are standardized by the definition of a maximum actuator load for each actuator.

A comparison of four parallel manipulators with three translational DOF is performed by TSAI, et al[6]. These include the 3-RUU, the 3-UPU, the 3-PUU(with intersecting rails) and the 3-PUU(with parallel rails) manipulator. A dimensional synthesis is carried out in terms of the maximum, however, well-conditioned workspace. The rigidity of the structures, as well as the mass inertia properties are also considered. To a great extent none of the analyzed structures is better than the others. Only the 3-PUU manipulator with intersecting rails shows a generally good, balanced performance. It becomes clear that a comparison of different structures raises problems: not only that different criteria may be relevant for a certain task, but the parallel manipulator must also be adapted to the respective performance features of the defined task[6]. A similar approach can be found in Ref. [11].

The aim here is the comparison between two PKM with three DOF, of which one has three, the other four limbs. The optimization of the well-conditioned workspace using the global conditioning index is the most important criterion. Furthermore, a comparison of stiffness properties, based on stiffness matrices and derived from the virtual work approach, is carried out.

In Ref. [12] a general set of benchmark criteria is created. The following criteria are included: the ratio of workspace to installation space, the stiffness (determined by measurements), the smallest occurring natural angular frequency (determined by experiments), as well as the maximum possible acceleration of the end-effector, if any actuator has the same motion pattern. In Ref. [12] parallel manipulators from the Multipteron family, i.e. the Tripteron[13] and the Quadrupteron[14], are compared. In case of both structures, forward and inverse kinematics are independent of the link lengths; thus, the motion of the end-effector is not influenced by manufacturing tolerances of the links.

3 Considered Parallel Kinematic Machines

3.1 Selection scheme

In this paper only symmetric PKM with three translational DOF are considered. Furthermore, all limbs have the same kinematic structure. The selection scheme of suitable PKM is based on the constraint-synthesis approach, especially the virtual chain approach introduced by Ref. [15]. This approach uses the screw theory to synthesize new kinematic structures for parallel manipulators. Only simple joints(DOF is 1) such as prismatic(P) and revolute joints (R) are considered. The maximum number of simple joints within one limb is restricted to five. This leads to eight different classes(cf. Table 1) of possible limb configurations for PKM. Within each class several joint alignments are possible, so that the number of configurations can be up to 30 per class.

Table 1. Structure selection based on Ref. [15]

Class NoC

NoC after

applying

criteria

1+2

NoC after

applying

criteria

1–4

Resulting limb

configurations

after applying

criteria 1–5

3P 1 – – –

3R-1P 4 4 1 CRR

2R-2P 6 6 5 RPC, CPR, CRP,

PRC, PCR

1R-3P 4 – – –

5R 5 5 – –

4R-1P 30 9 5 UPU, PUU, CRU,

CUR, RCU

3R-2P 30 – – –

2R-3P 10 – – –

To reduce the number of configurations (NoC)—from 90

in total––the following criteria are applied: (1) Within one limb the number of prismatic joints does not exceed two due to general disadvantages prismatic joints offer, e.g. sticking[16]; (2) No inactive joints within one limb occur to reduce cost by avoiding unnecessary components; (3) First or second joint of each limb is actuated[16]; (4) Linear actuation, and additionally; (5) Only three joints(DOF is more than 1) per limb[16] are allowed. For the last criterion, simple joints are combined: the combination of two revolute joints with perpendicular, intersecting axes is a universal joint(U); the combination of a revolute joint and a prismatic joint with identical motion axes is a cylindrical joint(C).

Table 1 gives an overview of the resulting NoC after applying the mentioned criteria.

An underlined joint indicates an actuated joint. Due to the fact that for three DOF three actuators are necessary and that the PKM should consist of three limbs (symmetrical assembly), every limb has one actuated joint.

The proposed selection scheme leads to eleven structures

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for translational parallel manipulators with three DOF which are further analyzed and extensively compared in the following sections.

3.2 Description of the selected PKM

As already mentioned, the selected PKM consist of three limbs connecting the fixed base(frame) with the moving platform(end-effector). Each limb has up to two links (lengths l1, l2) and up to three joints (j1, j2, j3) with one DOF or two DOFs. To achieve a symmetrical assembly, different frame configurations are possible(cf. Fig. 1).

Fig. 1 Frame configurations

The frame configurations shown in Fig. 1 are represented by the following symbols. Δ—Triangular configuration: The actuation axes are all parallel to one plane, but not

parallel to each other. ┴—Orthogonal configuration: The actuation axes are perpendicular to each other. ||—Parallel configuration: The actuation axes are parallel to each other. *—Star-shaped configuration: The actuation axes intersect in one common point. It is clear that the orthogonal configuration is a special

version of the star-shaped configuration. Furthermore, the orientation of the motion axes of the joints within a limb is shown in Table 2. The index after the joint variable describes the joint alignment within one limb. An identical index means that the joint axes are oriented parallel to one another. P┴ denotes that the motion axis of the prismatic joint is perpendicular to the following joint axis. If the axes’ orientation is not maintained, the purely translational motion of the end-effector cannot be provided.

Table 2. Axes’ orientation and frame configurations

TypeAxes’

orientation

Frame configuration Refs

Δ ┴ || *

CRR C1R1R1 –1) –1) [13, 17] RPC R1P┴C1 –2) –3) [17] CPR C1P┴R1 –2) [17] CRP C1R1P┴ –4) –2) [17] PRC P┴R1C1 –4) [17–19] PCR P┴C1R1 [17, 20]

UPU U12P2U21 –5) –3) –3) [17] U12P┴U21 –3) –3) [17, 21]

PUU P┴U12U21 [17, 21, 22] CRU C1R2U21 –6) [17] CUR C1U12R2 –6) [17, 23] RCU R1C2U21 –2) –2) [17]

1) All the axes of the R joints are not parallel to a plane[15]. 2) The axes of the R joints are not all parallel[15]. 3) First joint is no prismatic joint. 4) Excluded due to design criteria, where cylindrical joint axes

at the end-effector would intersect. 5) Excluded due to design criteria, where second U joint axis

is parallel to P joint axis. 6) Three lines each perpendicular to all the axes of the R joints

within a leg are not parallel to a plane[15]. To demonstrate the procedure of comparison, only PKM

with triangular and/or star-shaped frame configurations are selected. In case of triangular frame configurations the prismatic joint is actuated, whereas in case of star-shaped frame configurations the actuated joints are the first joints of each limb, because the prismatic/cylindrical joint axes form the frame. Due to design criteria (possibly colliding cylindrical joint axes) the PKM R1C2U21 is omitted. In summary, the following eleven structures are compared (cf. Table 3).

4 Dimensional Synthesis

For the dimensional synthesis, the selected PKM have to be described according to their kinematic and kinetostatic properties. For that purpose, the inverse kinematics and the Jacobian matrix are derived analytically. These calculations are based on the references listed in Table 2, but they are not reported for the sake of conciseness.

With regard to the dimensional synthesis and the illustration of the analysis methods, we introduce two reference frames(cf. Fig. 2): R0(O, x, y, z) describes the reference frame attached to the fixed base and RP(OP, xP, yP, zP) designates the reference frame attached to the end-effector. All axes of RP are parallel to those belonging to R because the end-effector is supposed to perform only translational motions.

The three limbs are symmetrically assembled for both, the triangular and the star-shaped configuration. This leads to the following geometric parameters (α, β):

120 ,= (1)

120 .= (2)


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Table 3. Considered structures for further analysis

Axes’ orientation

Frame configuration

Δ *

C1R1R1 –

R1P┴C1 –

C1P┴R1 resp. C1P┴R1

C1R1P┴ –

P┴R1C1 –

P┴C1R1 –

U12P2U21 – –

U12P┴U21 –

P┴U12U21 –

C1R2U21 –

C1U12R2 –

(R1C2U21) –

Fig. 2. Reference frames and geometric parameter Additionally, for the star-shaped configuration, the angle

ψ is introduced. It designates the inclination of the actuation axes with respect to the xy-plane.

4.1 Boundary conditions

Both the reachable and the prescribed workspace of the PKM are obtained by discretization of a sufficiently large space. For every point of this space, the actuator positions and internal joint parameters are calculated by means of the inverse kinematic model(InvKin). Several boundary conditions(BCi) are applied to assure that the calculated values are feasible.

First, it is checked if the joint parameters are real for the desired position p of the end-effector:

( )( )( )

1, if img 0,

0, if img 0.1

InvKinBC

InvKin

ìï =ï=íï ¹ïîp (3)

Then, the calculated actuator positions q(p) are compared

to the admissible maximum and minimum displacements (qmax, qmin) (given by the stroke length, cf. section 4.3):

( )( )( ) ( )( )( )( ) ( )( )

max min2

max min

1, if max min ,

0, if max V min .

q qBC

q q

ìïïï=íï > <ïïî

q p q pp

q p q p

≤ ≥

(4) If the two boundary conditions BC1 and BC2 are fulfilled,

the considered point p is a valid point for the PKM and hence, a point of its reachable workspace. The definition of additional boundary conditions is required with regard to the optimization and the prescribed workspace. Since the actuator velocities generally increase when minimizing the actuator forces, in particular, the maximum of the actuator velocities q (p) should not exceed a certain limit ( maxq ,

minq ). The minimum actuator velocity is equal to 0. The


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maximum velocity for linear drives is set to 3.5 m/s,

( )( )( )( )( )

max3

max

1, if max ,

0, if max .

qBC

q

ìïïï=íï >ïïî

q pp

q p

≤

(5)

Furthermore, the radius of the end-effector rB is smaller

than the radius of the frame rA(cf. Fig. 2). This condition (Eq. 6) can only be applied for a triangular frame configuration (in this case: no radius of the frame):

( )

4

1, if ,

0, else.B Ar r

BCì <ïï=íïïî

p (6)

At last, the sum of all link lengths( )ilå and the

displacements of the prismatic/cylindrical joints (qS) not attached to the frame should not exceed the value of 1 m:

( )( )( )( )

5

1, if max abs 1,

0, else.

i SlBC

ìï + <åïï=íïïïî

q pp (7)

For sure, there are other approaches to determine

boundary conditions. For example, Ref. [24] analyzes the singularity positions of the PKM. Further work could include those approaches.

4.2 Optimization method

The aim is to minimize the actuator forces for a given external load at the end-effector within a prescribed workspace. The prescribed workspace Wpre is a cuboid with side lengths(xS, yS) of 0.2 m and a height zS of 0.1 m (cf. Fig. 3). The coordinates of its midpoint MS are shifted with zMS only along the positive z-axis of the reference frame R0 attached to the fixed base. In this case zMS is a variable parameter and hence the first optimization parameter.

Fig. 3. Prescribed workspace Wpre

Additionally, the following load case is assumed for

further calculations:

( )T2 2 10 N,EE = -F ( )T m

1 1 1 .sEE =v (8)

The actuator forces are considered separately for each

limb for the given payload. Thereby, the objective function is written as follows:

( ) ( )( ),max ,min min max ,q q i EEf f=Optpar F (9)

with Optpar as vector of all parameters to be optimized.

Due to complexity reasons and according to Eq. (10), only three points in each Cartesian direction of the pre-scribed workspace are considered during optimization. Hence, in total 27 points have to fulfill the mentioned boundary conditions (cf. gray points in Fig. 3):

( )

( )

( )

Tvec

Tvec

Tvec

0.5 0.5 ,

0.5 0.5 ,

0.5 0.5 .

MS S MS MS S

MS S MS MS S

MS S MS MS S

x x x x x

y y y y y

z z z z z

= + -

= + -

= + -

x

y

z

(10)

The minimization problem is solved by using the genetic

algorithm of the MATLAB Optimization Toolbox[25]. Based on the evolution, there are randomly generated solution candidates for the mentioned minimization problem above. The objective function value improves and ideally achieves the optimum by changing the solution candidates according to different rules[26]. The initialization of the algorithm randomly creates a population(here size of 250) describing the amount of solution candidates(individuals). Within this population, individuals are further changed by varying the individual parameters(mutation) and evaluated by the fitness function(fitness). As Eq. (9) describes a minimization problem, individuals with lesser fitness are selected(selection) and used as starting values for new generations[27]. The maximum number of iterations is set to 1000.

The fitness function contains the boundary conditions described in section 4.1. Let gh be the function value of a tested boundary condition h. The boundary conditions are distinguished between inequalities with an upper bound ub:

,h hg ub< (11)

and respectively with a lower bound lb:

.h hg lb> (12)

If an individual fulfills a boundary condition, its fitness

remains constant. If an individual does not comply with a constraint, it is not eliminated, but punished instead by adding a penalty value. This value indicates how to adjust the following individuals and thus affects the efficiency of the algorithm. The penalty values consist of a constant


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value C1(>107) and a linear term. The linear term is the difference between the test value gh and the upper or lower bound(ubh, lbh) respectively and it is multiplied by a constant C2(>107). Thus, early convergences of the algorithm can be avoided. If a boundary condition is violated by an upper bound, the fitness is

( ) 2 1,•h hfitness fitness g ub C C= + - + (13)

and in case of a lower bound respectively:

( ) 2 1.•h hfitness fitness lb g C C= + - + (14)

However, if all tested boundary conditions are complied,

the resulting fitness corresponds to the computed objective function value(cf. Eq. (9)). 4.3 Variable parameters

The variable parameters are the parameters to be optimized. In Table these parameters are summarized, depending on the PKM. rA is the radius of the frame, rB is the radius of the end-effector respectively(cf. Fig. 2). zMS is the distance between the origin of the base reference frame and the center of the workspace(MS) according to Fig. 3. Depending on the structure, there are one or two links within one limb(l1, l2). Finally, for structures with a star-shaped configuration, the pitch Ψ of the prismatic/cylindrical joint axes is an optimization parameter. Because the inverse kinematic problem and the Jacobian matrix calculation is the same for CPR*/CRP*, PRC*/PCR* and RPCΔ/CPRΔ respectively, finally only eight PKM are optimized.

Table 4. Optimization parameters

PKM zMS/m rA/m rB/m l1/m l2/m Ψ/(°)

CRR*

0.12 ×

0.11

0.29

0.25

30

CPR*/CRP*

0.54 ×

0.1 × ×

17

PRC*/PCR*

0.28 ×

0.26

0.39 ×

5

PUU*

0.32 ×

0.23

0.45 ×

5

CRU*

0.42 ×

0.42

0.1

0.42

25

CUR*

0.92 ×

0.15

0.49

0.49

30

RPCΔ/CPRΔ

0.59

0.5

0.16 × × ×

UPUΔ

0.45

0.5

0.12 × × ×

To find meaningful solutions and to be able to compare

the PKM, the optimization parameters have to be restricted before the optimization starts. For both, the fixed base and the end-effector, a radius of 0.1 m for the lower bound and of 0.5 m for the upper bound is set. These bounds are also

applied for the link lengths li. The angle Ψ can vary between 5° and 30°. Besides, the stroke length for the prismatic/cylindrical joints is restricted to 0.4 m. If the prismatic joint is the first joint of the limb the length varies between 0 and 0.4 m, otherwise between 0.4 m and 0.8 m. The final optimized parameters for all PKM can be found in Table 4 as well.

4.4 Optimized workspace

Fig. 4 shows the prescribed workspace within the resulting, reachable workspaces of the optimized structures by solving the inverse kinematic problem. Due to the symmetrical assembly, the reachable workspaces are symmetrical for all structures as well.

Fig. 4. Usable (blue) and prescribed (red) workspace

of different PKM

It can be assumed that for PKM with greater zMS, the

installation space is larger(e.g. CPR*, CUR*). Furthermore, the ratio of prescribed to reachable workspace differs. For example, for PRC* or RPCΔ the volume of the workspace is


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perfectly utilized, whereas the prescribed workspace for the CUR* or the UPUΔ is only a small part of their reachable workspace.

5 Kinetostatic Analysis

The analysis of the optimized PKM is based on different criteria described in the following paragraphs with regard to the prescribed workspace Wpre(cf. Fig. 3). The load case applied is the same as for the optimization in section 4.2(cf. Eq. (8)).

The workspace is discretized with a step size of 0.01 m. Consequently, the performance of each structure is analyzed at 4851 points within the prescribed workspace. It should be mentioned that at each considered point only the maximum value of the current criteria—except for the local dexterity index the minimum—is taken into account.

5.1 Actuator velocity

The linear velocity of the end-effector p = ( , ,x yp p zp )T is related to the actuator velocity sq = 1 2 3( , ,s s s )T. In

this context the relation is defined as follows:

,s =q Jp (15)

where J is considered as Jacobian matrix.

5.2 Actuator force Also the force transmission depends on the Jacobian

matrix. When an external load FEE is applied at the end-effector, the necessary actuator forces Fq can be calculated by the following equation[28]:

1

2

3

• .q

Tq q EE

q

F

F

F

-

æ ö÷ç ÷ç ÷ç ÷= =ç ÷ç ÷ç ÷÷ç ÷çè ø

F J F (16)

5.3 Power

Based on Eqs. (15) and (16), the required power of the actuators can be expressed as

• .q q s=P F q (17)

5.4 Actuator stiffness

With the compliance matrix Cq of the actuators the displacement of the end-effector Δp due to an applied external force FEE can be calculated. All links and joints are treated as ideal rigid[29]. The compliance cqi (diagonal entries of the compliance matrix) of the prismatic actuators is assumed to be 0.4×10–8 m/N[30].

1 • • • .T

q EE - -=p J C J F (18)

5.5 Positioning error/Accuracy

The positioning error δp measures the maximum

deviation of the position of the end-effector from the target position in relation to the inaccuracies of the actuators δq. Thus, it is an evalutation of the positioning accuracy[29]:

( )1ma .•x -=p J q (19)

The inaccuracies of the actuators δq depend on the

direction of the inaccuracy. As all of the analyzed PKM come with three actuators, eight possible combinations of directions are considered. The positioning error δq of the actuator is 0.052×10–3 m[31].

1

2

3

0 0

0 0

0 0

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 .

1 1 1 1 1 1 1 1

•

q

q

q

æ ö÷ç ÷ç ÷ç ÷=ç ÷ç ÷ç ÷÷çè øæ ö- - - - ÷ç ÷ç ÷ç ÷- - - -ç ÷ç ÷ç ÷÷ç - - - -è ø

q

(20)

δqi is the positioning error of the actuator of limb i. δq is

a 3´8-matrix with a 3´1-vector in each column. This vector describes the displacement of the end-effector in x, y and z direction for the respective direction of motion.

5.6 Local dexterity index

The local dexterity index(LDI) ν is the reciprocal of the condition number κ and makes a statement on the transmission behavior and the occurrence of singular and isotropic configurations[32–33]:

1

1 1.

•v

-= =

J J (21)

The values of the LDI vary between 0(singular

configuration) and 1(isotropic configuration). It should be mentioned that ν depends on the position of the end-effector and thus, is a local property. In a singular configuration the motion behavior of the PKM cannot be predicted due to missing or too many degrees of freedom. Also the force transmission according to desired specifications is not possible[33].

6 Comparison

Based on the kinetostatic analysis described in section 5.2, in this section a comparison of the PKM for the assumed load case(cf. Eq. (8)) is carried out. The results are displayed graphically using box plots. Thus, a reliable statement can be made on the absolute values with respect to each criterion and their distribution within the prescribed workspace. The box plots show the median as a red line, the two quartiles as a blue rectangle, and the two extremes as a horizontal black line(whiskers). If the whisker exceeds


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1.5 times the corresponding interquartile range, the values are marked by a red cross as an outlier.

6.1 Actuator force

Fig. 5 shows the actuator forces within the prescribed workspace for the different PKM. In case of CRR* and CPR*(CRP*) the Jacobian matrix does not depend on the link length and moving prismatic axes. Therefore, the force transmission remains constant over the whole workspace. PRC* and PUU* have the smallest actuation forces, whereas CUR* has, in spite of the outliers, a relatively homogeneous force distribution within the prescribed workspace. CRU*, RPCΔ and UPUΔ show similar properties.

Fig. 5. Comparison of max. actuator forces within Wpre

6.2 Actuator velocity

Fig. 6 shows the actuator velocities within the prescribed workspace for the different PKM. Because the calculation of actuator velocities is reciprocal to the calculation of actuator forces, the PKM with high actuation forces require lower actuation velocities. Again, CRU*, RPCΔ and UPUΔ show similar properties with a homogeneous velocity transmission.

Fig. 6 Comparison of max. actuator velocities within Wpre

6.3 Power Fig. 7 shows the power distribution over the prescribed

workspace for all PKM. The power depends on the results shown in Fig. 4 and Fig. 5. Hence, CRU*, RPCΔ and UPUΔ require the lowest actuation power.

6.4 Actuator stiffness The maximal displacement of the end-effector due to the

compliance of the actuators is presented in the box plot of Fig. 8. In this case, RPCΔ and UPUΔ with the triangular frame configuration are better than CRU* because of fewer outliers. However, in spite of the greater interquartile range, the maximum end-effector displacements of PRC* and PUU* are lower.

Fig. 7 Comparison of max. actuator power within Wpre

Fig. 8 Comparison of max. end-effector displacements

within Wpre

6.5 Positioning error/Accuracy

The maximum positioning errors for the PKM are displayed in Fig. 9. Again, PRC* and PUU* show the best performance. Looking at the triangular frame configuration, UPUΔ is less prone to positioning errors of the actuators. CRR* and CRU* are in the same range.

Fig. 9 Comparison of max. positioning error within Wpre

6.6 Local dexterity index

Finally, the last box plot in Fig. 10 shows the LDI. It should be pointed out that the overall behavior of CRR*


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with constant Jacobian matrix over the whole workspace is the best. UPUΔ has a higher LDI than RPCΔ. CUR* shows the poorest local dexterity index. This behavior has already been reflected in the previous box plots(cf. outliers).

Fig. 10 Comparison of the min. local dexterity index within Wpre

7 Conclusions

(1) In recent years, a variety of new parallel kinematic machines has been synthesized. But only a few of them were analyzed or even implemented by industry. An overall comparison is missing as well.

(2) Symmetrical parallel kinematic machines(PKM) with three translational DOF are compared by using different evaluation criteria in order to point out their strengths and weaknesses.

(3) The considered PKM and the selection scheme based on possible configurations of translational PKM are presented.

(4) Two frame configurations, the triangular and the star-shaped configuration are analyzed in more detail.

(5) An optimization method based on the genetic algorithm is described to allow the identification of the geometric parameters, e.g. the links length. The optimization is performed for an exemplary load case.

(6) The following criteria are considered to compare the different PKM: actuator forces, actuator velocities, actuator stiffness, accuracy and the local dexterity index.

(7) The results show that there is no “best” PKM for the applied load case.

(8) To allow a benchmark of the PKM for a final selection the criteria should be weighted.

(9) Future research should include the analysis of other PKM, e.g. with rotational actuators or orthogonal frame configuration, according to the described procedure as well as the extension of the calculation methods on dynamic and structural stiffness properties.

(10) The development of structure-specific indicators should also be provided to allow a general comparison of different PKM.

The analysis in this paper is limited to PKM with linear actuation. PKM with rotational actuation can be analyzed in the same way.

References

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Biographical notes PRAUSE Isabel, born in 1984, is currently a PhD candidate at Department of Mechanism Theory and Dynamics of Machines, RWTH Aachen University, Germany. She received her diploma degree in mechanical engineering in 2010 from RWTH Aachen University, Germany. Her research interests include synthesis and analysis of parallel mechanism, mechanism theory, engineering design and development. Tel: +49-241-8095586; E-mail: [email protected] CHARAF EDDINE Sami, born in 1987, is currently a PhD candidate at Department of Mechanism Theory and Dynamics of Machines, RWTH Aachen University, Germany. He received his master degree in 2013 from RWTH Aachen University, Germany. His research interests include dynamics of machines and robotics. E-mail: [email protected]

CORVES Burkhard, born in 1960, is currently a professor and a PhD candidate supervisor at Department of Mechanism Theory and Dynamics of Machines, Aachen University, Germany. His main research interests include mechanism theory, kinematics, dynamics of machines, robotics and mechatronics. E-mail: [email protected]

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