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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.
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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
PKM; (b) The LTI atlas of the 5R.
(a)
(b)
Figure 9. Actuator B3 is removed: (a) kinematic scheme of 5R
PKM; (b) The LTI atlas of the 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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