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International Journal of Advanced Robotic Systems

Performance Evaluation of RedundantParallel Manipulators AssimilatingMotion/Force TransmissibilityRegular Paper

Fugui Xie, Xin-Jun Liu* and Jinsong Wang

The State Key Laboratory of Tribology & Institute of Manufacturing Engineering,Department of Precision Instruments and Mechanology, Tsinghua University, China*Corresponding author e-mail: [email protected]

Received 03 Sep 2011; Accepted 03 Dec 2011

2011 Xie et al.; licensee InTech. This is an open access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract Performance evaluation is one of the most important issues in the field of parallel kinematic

manipulators (PKMs). As a very important class of PKMs, the redundant PKMs have been studied intensively.

However, the performance evaluation of this type of

PKMs is still unresolved and a challenging endeavor. In this paper, indices that assimilate motion/force

transmissibility are proposed to evaluate the performance

of redundant

PKMs.

To

illustrate

the

application

of

these

indices, three PKMs with different kinds of redundancies

are taken as examples, and performance atlases are plotted based on the definitions of the indices.

Transmissibility comparisons between redundant PKMs

and the corresponding non redundant ones are carried out. To determine the inverse solutions of the PKMs with

kinematic redundancy, an optimization strategy is

presented, and the rationality of this method is demonstrated. The indices introduced here can be

applied to the performance evaluation of redundant

parallel manipulators.

Keywords Performance evaluation, Redundant parallel manipulator, Motion/force transmissibility.

1. Introduction

As an important complementary counterpart of serial manipulators, parallel kinematic manipulators (PKMs) have been studied intensively for more than twenty years

for their advantages of compact structure, high stiffness, lower moving inertia, high load toweight ratio, high

dynamic performance, and high accuracy potential.

Usually, a PKM

provides

a stiff

connection

between

the

payload and the base, and has a complex structure in

terms of its motion and constraints. These characteristics also constitute certain disadvantages, such as the

relatively small workspace, low dexterity and abundant

singularities. Of note is the finding that the singularity makes the limited workspace of PKMs even smaller [1].

Unfortunately, singularity exists for most PKMs. For

example, the classic 3RRR PKM [24], singular loci exist in its workspace and the singularity of this manipulator is

very complex. As is well known, a manipulator works at

singular configurations may create serious problems and

use of such a situation should be avoided. But even so,

the properties such as stiffness and accuracy deteriorate quickly when working at near singular configurations.

113Fugui Xie, Xin-Jun Liu and Jinsong Wang: Performance Evaluation ofRedundant Parallel Manipulators Assimilating Motion/Force Transmissibility

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The most feasible methods for dealing with singularity avoidance and singularity free workspace enlargement

are to use actuation or kinematic redundancy [35]. Redundancy can, in general, improve the ability and

performance of PKMs [2, 6, 7]. Actuation redundancy

means that the mobility of a manipulator is less than the number of its actuated joints, while kinematic

redundancy means that the mobility is greater than the degrees of freedom (DOFs) needed to set an arbitrary pose of the mobile platform [4, 6, 8].

Recently, redundant actuation of robotic manipulators [7,

912] and machine tools [13, 14] has become a field

drawing active research. Actuation redundancy consists of two categories: the activation of passive DOFs in the

joint space, and the introduction of new and active DOFs in the joint space [15]. Actuation redundancy in PKMs

can decrease or eliminate singularity, increase manipulability [15], increase payload and acceleration of the mobile platform [16], further improve the efficiency

and reliability by eliminating actuator singularity or

force unconstrained configurations [4,17], and improve the repeatability by controlling the direction of the

internal forces to reduce the effects of joint backlashes

[18]. Actuation redundancy can improve the transmission properties and yield an optimal load distribution among

actuators by increasing the homogeneity of force

transmission and manipulator stiffness [16]. It can also

optimize the actuator torque and make up the actuator

fault [19,

20].

Force

moment

capabilities

of

redundantly

actuated planar PKMs were investigated by appropriately

adjusting the actuator outputs to their maximal

capabilities, as described in Ref. [21]. Actuation

redundancy is used to eliminate singularity and improve

the orientation ability of Eclipse in Ref. [22].

Actuation redundancy also has its own inherent

drawbacks. It causes challenging internal force problems,

i.e., the force inverse problem no longer has a unique

solution, which will lead to deformation or even material

failure if not tackled appropriately [2, 17].

As a counterpart of actuation redundancy, kinematic redundancy has been widely studied in serial

manipulators, and the focus has been diverted to PKMs of

late. Kinematic redundancy can improve the ability and

performance of a PKM by changing the geometric

parameters of the PKM instead of its basic structure. Due

to the extra DOFs, kinematically redundant PKMs are

inherently capable of more dexterous manipulation [23],

and can not only execute the original output task but also

perform additional tasks such as singularity elimination,

workspace enlargement, dexterity improvement, obstacle

avoidance, force transmission optimization and

unexpected impact compensation [2, 6, 7].

However, the PKMs with kinematic redundancy are more structurally complex in terms of design. This complexity

makes it difficult to guarantee a working process with high precision and high dynamic performance, and it is

also difficult to control because a motion planning

algorithm is needed to choose the most desirable configuration from the infinite possible configurations

that satisfy the constraints of mobile platform [23]. Moreover, in order to achieve real time control performance, this process must be calculated efficiently

[23], but it is very difficult to realize in practice due to the structural complexity, and therefore, sometimes,

actuation redundancy instead of kinematic redundancy is

used to avoid this difficulty in control [24]. It is important to note that the inverse kinematics of a PKM with this

kind of redundancy is no longer unique, which is the main barrier that needs to be removed from the control

process. Moreover, to achieve the mentioned potentials of kinematic redundancy, an appropriate optimization method of determining the inverse kinematics is required.

Aiming at solving this problem, the concept of a single,

discrete, and continuous optimization, which is based on the maximization of the determinant of Jacobian matrix,

was proposed in Ref. [25]. However, the study in Ref. [26]

showed that the dimension of Jacobian matrix does not

conform in a PKM with translational and rotational DOFs.

Therefore, this optimization strategy needs

reconsideration. In this paper, a new optimization

method, based on the locally optimized idea, will be

suggested to

be

used

in

the

generation

of

optimal

inverse

solutions for a PKM with kinematic redundancy.

The two types of redundancies have their own

application fields, and utilizing the advantages while

avoiding the drawbacks is the essential pursuit of

designers.

Optimal kinematic design is always an important and

challenging subject in designing PKMs due to the

closed loop structures, and this problem becomes more

complex with the introduction of redundancy to

non redundant PKMs. In general, there are two issues

involved: performance evaluation and dimension synthesis. Dimension synthesis involves determining the

lengths of links of PKMs, which is a fundamental as well

as challenging process for the design. In the field of

performance evaluation, the popular local conditioning

index (LCI) is unsuitable for application in PKMs with

mixed type of DOFs [26], and it is also defective when

applied to the planar PKMs with only translational DOFs

[27]. A local transmission index (LTI), based on the

transmission angle, was proposed by Wang et al. [27] to

evaluate the motion/force transmissibility, which is

limited in planar non redundant PKMs or non redundant

PKMs with decoupled property. Based on the virtual coefficient of screw theory, Chen et al. [28] proposed a

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generalized transmission index as an index to evaluate the transmissibility in one limb of a non redundant PKM,

and only the output performance was considered. Thereafter, Wang et al. [29] updated the definition of LTI

based on the concept of reciprocal product of screw

theory, which considered both the input and output motion/force transmissibility and evaluated the

motion/force transmissibility of all non redundant PKMs. To the best of our knowledge, the majority of contributions focus only on the non redundant PKMs

[2732] in the domain of optimal kinematic design, the research on the evaluation of motion/force

transmissibility of redundant PKMs, which is an

important and interesting issue to be figured out for their promising potentials and vast application prospects, has

not been reported yet. This paper attempts to make some contributions to this field.

The remainder of this paper is organized as follows. The next section, firstly, clarifies the classification of

redundancy, and, secondly, presents the outcomes of the

motion/force transmissibility analysis within the range of non redundant PKMs. Based on this, the motion/force

transmission indices for redundant PKMs are defined

accordingly. In Section 3, three PKMs with different types of redundancy are taken as examples to illustrate the

application of the related indices, and the performance

comparisons between the redundant PKMs and the

non redundant ones are also carried out. Conclusions are

provided in

the

last

section.

2. Transmission indices of redundant PKMs

2.1 Classification of redundancy

Let us assume that the DOFs of the manipulator are n , i.e.,

the number of the input actuators is n , the mobility of the

mobile platform is m , the number of mutual interference

actuators is k (i.e., any one of the k motors cannot be

moved freely when the rest motors are locked), and the

number of actuation redundancy actuators is r , then, the number of redundant actuators is nm. If 0n m , this

manipulator is redundant. If 0n m and r n m , this redundancy is referred to as actuation redundancy, and this type of redundancy can be classified into two

categories: redundantly actuated and branch redundant, i.e., the activation of passive DOFs in the joint space, and

the introduction of new, active DOFs in the joint space

[15]. If 0n m and 0k , this manipulator is

kinematically redundant. Further, if 0n m and 0 r n m , this redundancy is called a mixed

redundancy, this type of redundancy is rarely used in

practice in view of the difficulty in avoiding the inherent

drawbacks of both actuation and kinematical

redundancies.

2.2 Transmissibility analysis of nonredundant PKMs

Screw theory has been successfully used in the analysis of motion/force transmissibility of the non redundant PKMs

[2830], in the analysis of both kinematics and dynamics

[3335], and in type synthesis [36]. It is also the mathematical foundation for indices calculation in this

article. The basic concepts of screw theory have been introduced in detail in Ref. [28, 29], and can be summarized as following.

The twist and wrench screws in a manipulator can be

expressed by 1 1( ; )t S p r and

2 2( ; )wS f p f r f f , respectively, and the reciprocal

product or virtual coefficient can be expressed as

1 2 ( ) cos sinw t S S p p d , which is the power of

the two screws in the physical meaning, where, and

f represent the scalars of velocity and force, and

f are unit vectors of the two screw axes, 1r and 2r

represent the position vectors, d and are the

distance and angle between the two screw axes,

respectively.

Based on the concept of reciprocal product or virtual

coefficient of screw theory, a generalized transmission index

(GTI) considering only the output transmission performance was defined in [28], which can be expressed

as

1 2

2 21 2 max , ,

( ) cos sin

max ( )w t

w t

p p d w t

p p d S S GTI

S S p p d

(1)

and this expression has been referred to as the power

coefficient in Ref. [29].

In order to evaluate both input and output transmission

performance of a non redundant manipulator, the input

transmission index and output transmission index of the ith

leg based on the concept of power coefficient have been

defined

in

Ref.

[29],

and

can

be

expressed

as

max

Ti Iii

Ti Ii

S S

S S

and

max

Ti Oii

Ti Oi

S S

S S

, respectively.

Consequently, the local transmission index (LTI), which was

suggested to be the evaluation metric of motion/force

transmissibility of non redundant manipulators, was

defined in Ref. [29] as

min ,i i ( 1,2 , ..., )i n , (2)

which

is

independent

of

any

coordinate

frame ,

and

[0,1] .


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2.3 Definitions of transmission indices for redundant PKMs

2.3.1 Local minimizedtransmission index (LMTI) for actuation redundancy manipulators

For this type of manipulators ( 0r n m ), there exists mutual interference among the k actuators, and it is

difficult to predict force distribution among these mutual interference actuators in advance [2, 17]. Indeed, actuation redundancy has the ability to optimize the internal forces in the device, but this cannot be realized only by kinematic optimum design. Within the scope of

performance evaluation, we cannot evaluate the motion/force transmissibility exactly as we did for non redundant manipulators. Hence, we suggest a local

minimized transmission index (LMTI) to evaluate the motion/force transmissibility of the actuation redundancy

manipulators, which is summarized as follows.

By removing r actuation redundancy actuators from the k mutual interference actuators (i.e., making the r actuation redundancy actuators passive), q non redundant manipulators will be generated, and

r k q C . According to the definition of local transmission

index [29] mentioned above, there is an LTI value for each

manipulator with respect to the designated pose, denoted

by i ( 1, 2, . .., )i q .

From the q non redundant manipulators, we can find a

non redundant manipulator which can transmit motion/force better than others in a given pose of the

mobile platform, and take the LTI value of this manipulator as the local minimized transmission index

(LMTI), which can be expressed as

1 2max , , ..., q , r k q C . (3)

This index reflects the minimum motion/force

transmission performance in a designated pose of the actuation redundancy manipulators. The larger value of

LMTI

indicates

that

the

more

efficient

motion/force

transmission would take place. According to the definition of LTI, LMTI is frame free, and [0,1] . For

the purpose of high speed and high quality of motion/force transmission, the most widely accepted

range for LTI is sin45 1 , or sin40 1 , [27, 30]. Therefore, the corresponding limit for LMTI should be

sin45 or sin40 . When the value of LMTI is

within this range, the good motion/force transmission

takes place for actuation redundancy manipulators.

2.3.2 Local optimaltransmission index (LOTI) for kinematic redundancy manipulators:

There is no mutual interference actuator for kinematic

redundancy manipulators ( 0n m , 0k ), but the

inverse kinematics is not unique [23], the solution set can be denoted by G , and, for any potential positions of the

actuators, the value of the local transmission index , i.e., ,

will be different. There exists a maximal value ( g G ) when the inputs vary in the range of the

potential positions, and take this maximal value as the

local optimaltransmission index (LOTI) to evaluate the motion/force transmission performance for the given pose of a kinematically redundant manipulator. When this

maximal value is taken, the inputs, i.e., the subset g , are referred to as the optimal inverse kinematic solution for the given pose of the mobile platform. There is

g , g G . (4)

This index indicates the best motion/force transmission performance in a given pose of the kinematic redundancy manipulators. Similarly, the larger value of LOTI indicates that the more efficient motion/force transmission would take place, and LOTI is frame free,

[0,1] . To have good motion/force transmission

performance for kinematic redundancy manipulators, the

limit for LOTI should be sin45 or sin40

according to Section 2.3.1.

The optimal inverse kinematic solution, i.e., g , generated

here can realize the best motion/force transmissibility of a PKM with kinematic redundancy in the given pose of its mobile platform. Over a considered workspace, all the

optimal inverse kinematic solutions can be determined, and these solutions can be applied in the process of

control.

3. Examples

In order

to

introduce

the

applications

of

the

proposed

indices and demonstrate their effectiveness in the analysis

of the motion/force transmissibility of redundant PKMs, three PKMs with different kinds of redundancy are taken

as examples and the corresponding atlases of the indices

are plotted. Based on the proposed indices and the LTI, performance comparisons between the redundant PKMs

and their non redundant ones are also presented. Due to space constraints, calculations for related indices are presented in Appendixes in detail, the main procedure

and results are directly presented in the following sections.


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3.1 Actuation redundancy: redundantly actuated

In Ref. [27], we have presented the optimal design of a spatial 3DOF parallel manipulator, as shown in Fig. 1.

When this mechanism was used in a 5axis prototype

called SPKM165 [37] as a parallel module, we found that the accuracy and stiffness along the yaxis are not as good

as that along other directions. To overcome this weakness, a new redundantly actuated PKM, i.e., the joint along the yaxis is actuated, is developed, as shown in Fig. 2.

Theoretically, the accuracy and stiffness along the yaxis should be better because the motion/force transmissibility

along the yaxis has been improved by the introduction of

actuation redundancy. To verify this, we will use the suggested index to evaluate the performance of this

redundant PKM, and compare this result with that of the non redundant one.

(a)

(b)

Figure 1. A spatial 3DOF PKM: (a) CAD model; (b) kinematic

scheme.

Here, the dimensional parameters of the two PKMs are specified as

180 mmi i i B P l ( 1,2,3i ), 1 2 112 mm P P l ,

1 2 266 mmT T d , 3 1 87 mm PP r and

1 3 2 3 133 2 mmT T T T .

Parameter 2r represents the distance from point P to

line 3 3 B T , then 2 133 mmr .

For this mechanism, k=3, r=1, n=4, and m=3, then 13 3

r k q C C . By removing actuators B 1 , B 2 , B 3 ,

respectively, three non redundant PKMs can be generated.

For a given pose of the mobile platform, the LTI values

can be calculated and represented by 1 2, and 3 ,

respectively. The corresponding derivation process is

presented in detail in Appendix A. Then, the LMTI of the

redundant PKM can be generated as

1 2 3max , , . (5)

When the PKM moves along the zaxis, i.e., three inputs

are driven in synchronization with each other, the motion/force transmissibility remains unchanged.

Therefore, z=0 is assumed. With this information, the LMTI distribution atlas of the redundant PKM and the LTI distribution atlas of the non redundant one, with

respect to the practical workspace defined by

50 mm, 50 mm y and 25 , 90 , are

generated as shown in Figs. 3 and 4, respectively. Due to the decoupling property of the manipulators, the distribution loci of the related indices are parallel.

(a)

(b) Figure 2. The redundant 3DOF PKM: (a) CAD model; (b)

kinematic scheme.

Comparing Figs. 3 and 4, it is evident that the

transmission performance along the yaxis has been improved dramatically by the introduction of redundant

actuation, and this result is expected. In order to evaluate the extent of improvement of the transmissibility,

LMTI LTI100

LTI is used to express it numerically,

and the distribution of in the identical workspace is


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also plotted as shown in Fig. 5, from which one may see

that in the area defined by 50 mm, 50 mm y and

10 , 50 the improvement of transmissibility is prominent.

Figure 3. The LMTI atlas of the redundant PKM.

Figure 4. The LTI atlas of the non redundant PKM.

Figure 5. The atlas of in the workspace.

By this comparison, it can be concluded that the

redundantly actuated redundancy contributes greatly to

the motion/force transmissibility of a manipulator.

Note that, the value of LMTI represents the minimum motion/force transmission performance, and the actual

transmissibility of the redundant PKM may be better than

that depicted in Fig. 3.

3.2 Actuation redundancy: branchredundant

In this section, an extensively studied redundant 3RRR PKM is taken as an example, as shown in Fig. 6, which has been developed by the introduction of an additional

identical limb to the classic 5R PKM.

Figure 6. A redundant 3RRR PKM.

The parameters of this manipulator used here are

1 2 2 3 1 3 200 mm B B B B B B , 100 mmi i B C

( 1, 2,3i ) and 100 mmiC P ( 1,2,3i ).

For this manipulator, k=3, r=1, n=3, and m=2, then 13 3

r k q C C . Removing the actuators B 1 , B 2 , and B 3 ,

respectively, three non redundant PKMs (5R) can be

generated as shown in Figs. 7(a), 8(a) and 9(a). For a given

position of point P in the workspace, the LTI values of the

three non redundant PKMs can be calculated and

represented by 1 , 2 and 3 . The corresponding

derivation process can be found in Appendix B. Then, the

LMTI value of the redundant PKM can be generated as

1 2 3max , , . (6)

The corresponding atlases of 1 , 2 and 3 have

been presented in Figs. 7(b), 8(b) and 9(b), respectively,

and the LMTI atlas of the redundant 3RRR PKM is

shown in Fig. 10.

Comparing the atlases shown in Figs. 7(b), 8(b) and 9(b) with that in Fig. 10, it can be deduced that the

branch redundant redundancy evidently contributes to

the improvement of the motion/force transmission

performance. The atlas in Fig. 10 illustrates that the

distribution of the transmission performance index is

more uniform in the workspace, and the isotropy of the

manipulator has also been improved. Note that, the

atlases presented in Figs. 7(b), 8(b) and 9(b) are actually 120 rotational symmetry, this is consistent to the

configurations of the mechanisms presented in Figs. 7(a),

8(a) and 9(a).


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3.3 Kinematic redundancy

By introducing an active slider P to the classic 5R PKM, a kinematically redundant manipulator can be generated as

shown in Fig. 11, the non dimensional parameters of the

manipulator are 1 0.85r , 2 1.6r and 3 0.55r .

For this manipulator, k=0, i.e., there is no mutual

interference actuators. However, the inverse kinematics of

this manipulator is no longer unique, i.e., for a given

position of the point C , numerous solutions of the inputs

subject to the constraints of the mechanism can be derived,

and among all the potential solutions, there exists a

solution g , which can make the LTI of the PKM achieve its

maximum value , then the LOTI can be expressed as

, (7)

and g is the optimal inverse kinematic solution for the

given position of point C. The corresponding derivation

process has been elaborated in Appendix C.

(a)

(b)

Figure 7. Actuator B1 is removed: (a) kinematic scheme of 5R

PKM; (b) The LTI atlas of the 5R.

The atlas of LOTI and singular loci is presented in Fig. 12. In order to depict the effect of the kinematic redundancy, the corresponding performance of a non redundant 5R PKM, as shown in Fig. 13, is evaluated, and the atlas of

singular loci and LTI is shown in Fig. 14.

Comparing the singular loci shown in Figs. 12 and 14, it can

be concluded that the singularity free workspace has been dramatically enlarged due to the introduction of kinematic redundancy. The workspace with good transmission

performance of the redundant PKM is obviously larger than that of the non redundant PKM. Therefore, kinematic

redundancy contributes greatly to the singularity avoidance, singularity free workspace enlargement and motion/force

transmission performance improvement.

(a)

(b) Figure 8. Actuator B2 is removed: (a) kinematic scheme of 5R


(a)

(b)

Figure 9. Actuator B3 is removed: (a) kinematic scheme of 5R


Generally, the

inverse

kinematic

solution

is

unique

for

a

specific working mode of a parallel manipulator. However, in the practical application of the kinematically redundant


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PKM shown in Fig. 11, the inverse kinematic solution is no longer unique. An appropriate optimization strategy is

necessary to determine the inverse solutions. Based on the optimization of the motion/force transmission

performance, the optimal inverse kinematic solutions

generated here are qualified for this task. In order to verify the rationality of this method, a rectangular

workspace, as shown in Fig. 15, is considered, the positions of slider P are generated and plotted in Fig. 16.

Figure 10. The LMTI atlas of the redundant 3RRR PKM.

Figure 11. A manipulator with kinematic redundancy.

Figure 12. The atlas of LOTI and singular loci of the kinematically

redundant manipulator.

The slider P moves smoothly over the rectangular workspace of the mobile platform; that is, no jumps or transient impacts occur during the locating process of the actuator. Therefore, this optimization method can be used to determine inverse kinematic solutions in the control

process, and these optimal kinematic inverse solutions can maximize the motion/force transmissibility of kinematically redundant PKMs.

Figure 13. The non redundant 5R PKM.

Figure 14. The atlas of singular loci and LTI of the nonredundant 5R.

4. Conclusions

This article presents the classification of redundancy and

briefly expounds the outcomes of the motion/force transmissibility analysis of nonredundant PKMs. Based on

this, the local minimized transmission index (LMTI) and local

optimaltransmission index (LOTI) are defined as the

evaluation metrics of motion/force transmissibility for PKMs with actuation redundancy and that with kinematic redundancy, respectively. Aiming at determining and optimizing inverse solutions of a PKM with kinematic

redundancy for the control process, the optimal inverse

kinematic solution , which can realize the best motion/force

transmissibility for this type of PKMs, is proposed. To

illustrate the application of the indices, three PKMs with different types of redundancy are taken as examples, and the

corresponding performance atlases based on the indices are presented. The comparisons, between the redundant and the corresponding non redundant PKMs, with respect to the

motion/force transmissibility are carried out. The optimal kinematic design method of redundant PKMs based on the proposed indices will be presented in our future work.

Figure 15. A selected workspace.


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1 2 1 3 1 3 1 2 23 2 2

1 2 1 3

cos , 0, sin ; 0, cos sin , 0

( cos ) ( sin )T

r r r z z r z r r r z S

r r r z z

. (20)

When the actuator B 1 is removed, a new non redundant

PKM is generated. Then,

max

Ti Iii

Ti Ii

S S

S S

( 2, 3i ), (21)

1max

Ti Oii

Ti Oi

S S

S S

( 2, 3i ), (22)

1max

Tr Or r

Tr Or

S S

S S

.

(23)

Where, 2OS and 3OS can be determined by

0

0

0

Oi Cmn

Oi Tj

Oi Tr

S S

S S

S S

( , , , 2, 3i j m n , i j ), (24)

and Or S can be determined by

0

0Or Cmn

Or Ti

S S

S S

( , , 2, 3i m n ). (25)

Then, the LTI of this non redundant PKM can be

expressed as

1 2 3 21 31 1min , , , , r . (26)

Similarly, when the actuator B 2 and B 3 are removed,

respectively, the LTIs of the two non redundant PKMs can be expressed as

2 1 3 12 32 2min , , , , r (27)

and

3 1 2 3 1 2 3min , , , , ,r r r . (28)

Appendix B: Indices derivation of the redundant 3RRR

PKM

The corresponding angular parameters used in this part

are shown in Fig. 17, and values of these parameters can be easily derived when the position of point P is given.

Later, when the actuator B 1 is removed, the LTI of the non redundant manipulator can be derived as

1 2 3 23min sin , sin , sin . (29)

Figure 17. The redundant 3RRR PKM.

Similarly, the other two indices, when actuators B 2 and B 3

are removed respectively, can be generated as

2 1 3 13min sin , sin , sin , (30)

3 1 2 12min sin , sin , sin . (31)

Appendix C: Indices derivation of the kinematically

redundant 5R PKM

The corresponding angular parameters used in this section are shown in Fig. 18.

Figure 18. The kinematically redundant 5R PKM.

When the position of point C is given, the values of these parameters can be easily derived and positions of slider P that satisfy the constraints of the manipulator can be

denoted by G . When the slider P is located at an arbitrary position Pi ( Pi G ), the transmission performance of

this manipulator can be expressed as

1 2 3 0min sin , sin , sin , cos Pi (32)

In the set G , there exists a position Pj ( Pj G ) which

makes the transmission performance of this manipulator

achieve its optimal level, and this maximum value can be

expressed as

max g Pi Pi G

, (33)

Then, the optimal kinematic inverse solution is given by g Pj .


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6. References

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Mechatronics , vol. 19, pp. 761766, 2009.

[2] S.H. Cha, T.A. Lasky, S. A. Velinsky, Kinematically redundant variations of the 3RRR

mechanism and local optimization based singularity avoidance, Mechanics based Design of Structures and

Machines , vol. 35, pp. 1538, 2007.

[3] S.H. Cha, T.A. Lasky, S.A. Velinsky, Determination of the kinematically redundant active prismatic joint

variable ranges of a planar parallel mechanism for

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