1. INTRODUCTION

For e�ective utilization of robotic automation in complicated tasks and/or in complicated en-

vironments, it is essential that robot manipulators have more number of inputs than the bare

minimum. This makes redundancy a very useful concept in robotics. A non-redundant manipu-

lator performs well in well-conditioned tasks in structured environments, but it may face serious

di�culties in a general situation due to its inherent mechanical non-linearities and the unstruc-

tured nature of the environment as well. Redundant manipulators o�er additional means for

control to circumvent such di�culties. Detailed analysis of redundancy in robotics can be found

in Nakamura [1] and in references therein.

In serial manipulators, the redundancy is essentially kinematic where the manipulator pos-

sesses more degrees of freedom than the dimension of the task-space. The additional freedoms

available in redundant manipulators can be utilized for various purposes like obstacle avoid-

ance, singularity avoidance, optimization of dexterity or some other performance criteria, satis-

faction of some constraints including joint limits, fault tolerance etc, as discussed by various

authors [2±14]. Actuation redundancy in manipulators with closed kinematic chains has also

been studied (see Ref. [15], for example). However, possibilities of redundancy in parallel manip-

ulators and their e�ective utilization have not been studied extensively until now.

Compared to serial manipulators, parallel manipulators attracted research interest more

recently primarily through the proposal of the aircraft simulator mechanism (now known as the

Stewart platform) by Stewart [16] and the suggestion by Hunt [17] that such parallel-actuated

mechanisms can be used as robot manipulators with advantage over serial manipulators in cer-

tain applications. The study of structural kinematics of parallel manipulators by Hunt [18] and

the mechanics of the Stewart platform by Fichter [19] and others established that parallel manip-

ulators possess certain characteristics in striking contrast to serial ones. Two important obser-

vations about parallel manipulators are of central interest to the present work. First, parallel

manipulators exhibit a deep symmetry and duality against serial manipulators as demonstrated

by Waldron and Hunt [20] in their theory of series±parallel duality which is a consequence of

symmetry between twist and wrench systems and the reciprocity between instantaneous kin-

ematics and statics as propounded by Ball [21]. Secondly, works on parallel manipulator singu-

larities [22±30] show that static singularities (which are completely absent in open-chain serial

727

Example Paper for Anaylsis of Paragraph Structuring

manipulators) of parallel manipulators appear within the workspace with continuous expansethereby rendering the manipulator uncontrollable in some regions of the workspace.

So far as redundancy is concerned, kinematic redundancy is not possible in parallel manipula-tors in the strict sense, because incorporation of more than one actuation in a particular limb ofa parallel manipulator destroys its in-parallel character and it becomes a hybrid manipulator. Afew kinematically redundant hybrid manipulators are also discussed in literature, for examplethe development of 10-d.o.f. hybrid manipulator UPSarm by Cheng [31] and the proposal of the10-d.o.f. manipulator having 3-d.o.f. parallel modules by Mingyang et al. [32]. The 9-d.o.f.manipulator resembling the Stewart platform with an intermediate star connection of actuatorsproposed and studied by Zanganeh and Angeles [33, 34] is also essentially a hybrid manipulatorwith a complicated series±parallel chain in parallel with three in-parallel actuations. Thoughsuch arrangements are useful in avoiding kinematic singularities, avoidance of force singularitiesis quite di�cult by such measures. A direct approach towards avoidance of force singularities inparallel manipulators has to be made through force redundancy, i.e. by the use of additionalsupport(s) in parallel, which has been mentioned as type III redundancy by Lee and Kim [35]and discussed by Collins and Long [36] in the context of twist/wrench decomposition in serialand parallel manipulators. It can be argued that force redundancies are actuation redundanciesin some sense. Nevertheless, to understand the force redundancy completely and to utilize ite�ectively, it should be studied as series±parallel dual concept of kinematic redundancy and pos-sibilities of its use should be explored in that framework.

In this paper, the force redundancy in parallel manipulators is studied as the dual concept ofkinematic redundancy in serial manipulators. In particular, the possibilities of singularity avoid-ance and elimination through force redundancy has been explored. This aspect is very importantfor parallel manipulators for which static singularities pose insurmountable barriers among seg-ments of workspace as discussed in Dasgupta and Mruthyunjaya [37], so far as safe and e�ectiveoperation is concerned.

In the next section, the concept of force redundancy is studied and its implications in variousproblems of kinematics and dynamics are discussed. Section 3 is devoted to an exploration ofpossibilities of avoidance and complete elimination of force singularities, where fundamental dis-tinctions between the relationships of singularity and redundancy in the cases of serial and par-allel manipulators are established. In Section 4, numerical studies of force redundancy inrelation to workspace and singularities are presented for two parallel manipulators. Finally, inthe last section, conclusions of the present work are summarized and suggestions for futurework are enumerated.

2. FORCE REDUNDANCY

Parallel manipulators can have a wide variety of structures with various possibilities of con-nections and actuations. Out of them, those having straightforward (and unique) inverse kin-ematics (e.g. the generalized Stewart platform) constitute a subclass (hereafter referred to as``simple parallel manipulators'') which exhibits the contrast against serial manipulators impliedby series±parallel duality in the most prominent manner. The comparison between the two issummarized in Table 1.

From Table 1, the duality exhibited by simple parallel manipulators against serial manipula-tors is clear. Other parallel manipulators also exhibit these characteristics. Besides, each limb ofa general (non-simple) parallel manipulators may involve the inverse kinematic complexity of aserial chain, though it will be much simpler and decoupled from other limbs compared withdirect kinematics or inverse statics of the entire manipulator. To take advantage of the completecorrespondence between the two classes, the analyses have been performed in the present workkeeping in view the subclass of simple parallel manipulators. The concepts are neverthelessroughly applicable to all fully parallel manipulators in general.

Due to the distinctive features of the two classes, serial manipulators are to be preferred inapplications requiring large workspace and manoeuverability, while parallel ones are preferablein applications where precise positioning, high load-carrying capacity and good dynamic per-formance are of paramount importance. In other words, the primary objective of a serial manip-

ulator is to move an object, while that of a parallel one is to support a load. In the light of this,we can view singularities in both the classes of manipulators as hampering their respective rolesin application. Kinematic singularities in a serial manipulator restrict its motion capabilities andcan be remedied (to a great extent) by kinematic redundancy. On the other hand, static or forcesingularities in a parallel manipulator restrict its load-carrying capabilities. Hence, the conceptthat arises as the series±parallel dual to the kinematic redundancy of serial manipulators is thatof static or force redundancy, which has particular importance regarding the avoidance of staticor force singularities, the only singularities in the case of simple parallel manipulators and themore challenging ones in the case of other parallel manipulators.

For example, the series±parallel dual of the 2-d.o.f. serial manipulators shown in Fig. 1(a) isthe two-legged (and 2-d.o.f. also) parallel manipulator shown in Fig. 1(b), Cartesian positionp(x, y) of the end-e�ector de®ning the task-space coordinates in both cases. In the serial manip-ulator, inverse velocity transformation is unique at a non-singular con®guration and not well-de®ned at a singularity. Similar is the case of inverse force transformation in the parallel manip-ulator. In the serial manipulator, singularities are found on two circles in the task-space (com-pletely extended and completely folded poses) and are of kinematic nature. In the parallel

Table 1. Comparison between serial and simple parallel manipulators

Serial manipulators Simple parallel manipulators

Actuators In series In parallelDirect position transformation Straightforward and unique Complicated and multipleInverse position transformation Complicated and multiple Straightforward and uniqueDirect motion transformation Well-de®ned and unique Not well-de®ned; may be non-existent,

unique or in®niteInverse motion transformation Not well-de®ned; may be non-existent,

unique or in®niteWell-de®ned and unique

Direct force transformation Not well-de®ned; may be non-existent,unique or in®nite

Well-de®ned and unique

Inverse force transformation Well-de®ned and unique Not well-de®ned; may be non-existent,unique or in®nite

Singularity Kinematic StaticNatural description In joint-space In Cartesian spacePreferred property Dexterity Sti�nessPreferred application Gross motion Precise positioning

Fig. 1. Kinematic redundancy and force redundancy.

manipulator, they are static singularities and are found on the line joining the ®xed pivots b1and b2.

Now, considering only the position (and disregarding the orientation) of the end-e�ector inthe plane, the 3-d.o.f. serial manipulator shown in Fig. 1(c) is a kinematically redundant manip-ulator having one additional link (and an actuation) in series. For this manipulator, at a givencon®guration, the inverse velocity transformation will be a ®bre (an in®nitely many valued func-tion), while the inverse force transformation will be unique. In the same way, a static or forceredundancy can be introduced in the parallel manipulator of Fig. 1(b) in the form of an ad-ditional leg in parallel getting thereby a statically redundant 2-d.o.f. (3 degree-of-constraint) par-allel manipulator shown in Fig. 1(d). For this manipulator, at any given con®guration, theinverse force transformation is a ®bre, while the inverse velocity transformation is unique. Thisshows the duality of kinematic redundancy of serial manipulators and force redundancy of par-allel manipulators. However, it is to be observed here that the redundant serial manipulator ofFig. 1(c) still has an in®nite number of singularities at the completely extended (and possiblycompletely folded) con®gurations, while the redundant parallel manipulator of Fig. 1(d) isentirely free from singularities as long as the three ®xed pivots b1, b2, b3 are non-collinear. Thisindicates that though the kinematic singularities of serial manipulators remain the same in num-ber{ even after the introduction of redundancy (which merely redistributes the singular poses inthe workspace), force redundancy in parallel manipulators may be able to reduce the singular-ities and even eliminate them completely. This aspect will be discussed in detail for the generalcase in Section 3.

In order to understand the implications of force redundancy in the kinematics and dynamicsof a parallel manipulator, let us consider a simple parallel manipulator (de®ned earlier) with m-dimensional task-space and n actuations in parallel (n>m).

The relationship between the input coordinates

Y � �y1 y2 . . . yn�T

and output coordinates

X � �x1 x2 . . . xm�T

will be in the form of an explicit inverse kinematic relationship as

Y � f�X�; f : Rm4Rn �1�which uniquely de®nes Y in terms of X. In the case of a non-simple parallel manipulator, the re-lationship will not be explicit as Equation (1) and there is a possibility of multiple (but ®nitenumber of) solutions of inverse position kinematics. The direct position kinematics problem,which involves the solution of Equation (1) for X if over-speci®ed and there may be no solutionfor an arbitrary choice of Y. However, the value of Y in practice is obtained from sensor datawhich is expected to be consistent (up to the error in measurement). In such a situation, moreequations than unknowns will simplify the direct position kinematics compared to the corre-sponding non-redundant case. In addition, the additional equation may restrict the number ofsolutions.

The velocity kinematics of a simple parallel manipulator is given by

_Y � @f

@X_X �2�

where the matrix @f/@X is analytic and de®nes a unique inverse velocity transformation. In thecase of non-simple parallel manipulators, the matrix @f/@X will not be analytic because one fac-tor of it will require an inversion. Singularity of that factor may make some of the _y's (in somelimbs) ill-de®ned without a�ecting other _y's (in other limbs), i.e. various limbs are totallydecoupled so far as inverse kinematics is concerned. The direct velocity kinematics problem is alinear system of a larger number of equations than unknowns. Again, practical data from sen-

{Number of singularities =1m ÿ 1 where m is the task-space dimension.

sors are expected to be consistent and a pseudo-inverse solution is expected to cancel outmeasurement errors also to some extent.

The static relationship between the actuator forces F and the generalized forces T developedat the end-e�ector is given by

T � HF �3�where the force transformation matrix H $ Rm� n is analytic in the case of simple parallel manip-ulators and actuator forces are mapped uniquely to the end-e�ector forces. In order to exert adesired force T at the end-e�ector, the actuator forces F can be chosen in in®nite number ofways because of the force redundancy. This redundancy can be resolved by pseudo-inverse orby meeting some other optimization or constraint criteria. In particular, a minimax solution forF, i.e. minimization of the maximum of the actuator forces, may be useful in meeting actuatorconstraints.

It should be mentioned in passing that the jacobian symmetry between inverse kinematics anddirect statics of the redundant parallel manipulator (as in any parallel manipulator) is manifestin the relationship.

@f

@X� HT �4�

The role of the force redundancy in the static relationship, i.e. Equation (3) is directly carried tothe dynamics of the manipulator along with the usual additional terms, i.e.

M�X� ZZZ� T � HF �5�Introduction of kinematic redundancy to a serial manipulator can be used to enhance its work-space. On the other hand, force redundancy in a parallel manipulator tends to further restrict itsworkspace, as in this case workspace consists of poses allowed by all the constraints in parallel.However, such reduction is overcompensated by the fact that the available workspace becomesmore usable by the reduction of the barriers of singularities, as will be seen in the next two sec-tions.

3. SINGULARITY AVOIDANCE AND ELIMINATION

A non-redundant parallel manipulator is at a force singularity when the force transformationmatrix H is singular (i.e. rank-de®cient). By the condition of vanishing of the determinant of H,we get the singularity hypersurface which is an (mÿ 1)-dimensional manifold in the m-dimen-sional task-space. As such, it partitions the workspace into two or more segments which are notreachable from one another without encountering a singularity.

Similarly, for a statically redundant parallel manipulator, force singularities are characterizedby the rank-de®ciency of m� n matrix H, i.e.

rank �H� < m �6�or,

det�h� � 0 8h 2 fm�m submatrices of Hg �7�Though nCm submatrices of order m�m can be formed from H by taking m columns at a time,only nÿ (mÿ 1) conditions given by Equation (7) are independent. Thus, singularities are foundat the intersection of nÿ (mÿ 1) hypersurfaces forming a lower dimensional manifold. Thedimension of the singularity manifold (DOSM) is given by

DOSM � mÿ �nÿm� 1� � �mÿ 1� ÿ �nÿm�� �mÿ 1� ÿDOR � �mÿ 1� �8�

where DOR is the degree of redundancy.Thus, it can be seen that force redundancy can be used to reduce the force singularities to a

lower dimensional manifold in the task-space. In particular, making DOR = (mÿ 1) or above,

the singularities in the workspace can be reduced in a ®nite number of con®gurations or even

completely eliminated, provided that all the limbs are independent such that none of the re-

lations given by the Equation (7) becomes an identity.

Each degree of redundancy, however, is associated with a reduction in the workspace volume

due to both the explicit constraints and possible mechanical interference of the limbs.

Additional links and actuator add to the bulk of the entire manipulator also and add to the

inertia of the system. Hence, addition of redundant limbs should be properly examined in view

of these costs and the advantage gained by the singularity reduction. From a practical stand-

point, reduction of the dimension of the singularity manifold by just one through a single degree

of redundancy is good enough, because a manifold of lower dimension can always be bypassed

and the reduced singularities can always be avoided in planning a path from one point to

another in the task-space. As a matter of fact, the term singularity avoidance should be properly

interpreted in the context of parallel manipulators. In the case of serial manipulators, it is often

used to mean the ®nding of a non-singular con®guration for a given point in the task-space.

For parallel manipulators, this makes little sense. Hence, in the context of parallel manipulators,

singularity avoidance should always be interpreted as path-planning with end-e�ector poses

away from singularities.

The possibility of reduction and elimination of force singularities of parallel manipulators is

in sharp contrast with serial manipulators for which the dimension of the singularity manifold

remains unaltered even after the introduction of redundancy. Unfortunately, this distinction has

been overlooked by some previous researchers and as assertion like ``redundancy can only move

the original singularities from some points to some other points, it cannot remove or even

reduce the singular positions'' has been made in the context of the Stewart platform (see Liu et

al. [26]) under the wrong premise that redundancy in parallel manipulators must have the same

nature and behaviour as in serial manipulators. To safeguard against such pitfalls and to incor-

porate concepts from the class of serial manipulators into that of parallel manipulators, it is

necessary to bear in mind that what we have between the two classes is a duality, and not just a

similarity. Keeping this in view, the above-mentioned distinction can be explained easily.

The force singularities in a statically redundant parallel manipulator can be reduced in the

task-space as shown above owing to the fact that the con®guration of the manipulator can be

speci®ed by the m task-space coordinates only{ and the matrix H can be expressed in terms of

these coordinates. In comparison, the speci®cation of con®guration of a kinematically redundant

serial manipulator requires the n joint-space coordinates which enter the jacobian matrix and

subsequently the condition for the kinematic singularity also, namely

rank �J� < m �9�Hence, the only expectation that can be had is the dimension of the singularity manifold

(DOSM) in the joint-space being given by

DOSM � dim �Joint Space� ÿ 1ÿDOR

� nÿ 1ÿ �nÿm� � mÿ 1

as indeed is the fact!

Looking from the aspect of task-space also, we arrive at the same conclusion. It is well

known that workspace boundary of a manipulator appears at kinematic singularities (except at

segments dictated by joint limits) and the boundary of an m-dimensional workspace has to be

an (mÿ 1)-dimensional manifold, i.e. a hypersurface. Therefore, it is quite natural that the

dimension of the manifold of kinematic singularity cannot be reduced by measures like redun-

dancy. There is no such restriction, however, on the dimension of the manifold of force singular-

ity, and hence it can be reduced.

The next section shows numerical case-studies of singularity reduction in the workspaces of

two parallel manipulators.

{Apart from the branches of the inverse kinematics, in the case of a non-simple parallel manipulator.

4. NUMERICAL STUDIES

In order to visualize the e�ect of force redundancy on singularities and workspaces of parallelmanipulators, two parallel manipulators, namely the 3-d.o.f. planar parallel manipulator andthe 6-d.o.f. generalized Stewart platform are studied. In each case, comparisons are madebetween a non-redundant manipulator and a redundant manipulator with a single degree ofredundancy, i.e. with one additional limb.

4.1. The 3-d.o.f. planar parallel manipulator

The 3 d.o.f. planar parallel manipulator (PPM) with three legs and the corresponding redun-dant manipulator with four legs shown in Fig. 2(a) and (b) are compared for their workspacesand singularities. Denoting the i-th base point by bi, i-th platform point (in platform frame) bypi, position of platform reference point by t = [x y]T and the platform orientation by y, we havethe i-th leg vector as

Si � t�Rpi ÿ bi

where

R � cos y ÿsin ysin y cos y

� �The i-th leg length is given by

Li � kSik

Fig. 2. 3-d.o.f. planar parallel manipulators.

and the unit vector along the leg as

si � Si=Li

The actuator forces

F � �F1 F2 F3�T

in the non-redundant case and

F � �F1 F2 F3 F4�T

in the redundant case are related to the force R and moment M at the platform taken togetheras

T � �Rx Ry M�T

by the Equation (3), where the force transformation matrix H is given by

HNR � s1 s2 s3b1 � s1 b2 � s2 b3 � s3

� �for the non-redundant case and by

HRed � s1 s2 s3 s4b1 � s1 b2 � s2 b3 � s3 b3 � s4

� �for the redundant case.

In each of the two cases, a su�ciently large rectangular area (to enclose the workspace) in thexÿ y plane is scanned for various orientations y of the platform and the above computationsare performed. Any point, for which some of the leg-lengths are found equal (numerically,close) to either the lower limit (Lmin) or the upper limit (Lmax) with the other leg lengths beingwithin the limits, is taken as a point on the workspace boundary. At each point, the conditionnumber (ratio of greatest to least singular values) of H is evaluated and whenever it is found tobe above a prescribed threshold, the point is taken as a singular point. Finally, the workspaceboundary and singularities are plotted together.

The description of the manipulators used for the numerical study are as follows. (All dimen-sions are in metres.)

Base points:

�b1 b2 b3 b4� � 0:0 0:2 0:4 0:20:0 0:0 0:1 0:2

� �Platform points (in platform frame):

�p1 p2 p3 p4� � ÿ0:15 ÿ0:1 0:15 ÿ0:10:0 ÿ0:1 0:0 0:15

� �Leg-length limits:

Lmin � 0:2 and Lmax � 1:0 for all legs

The ®rst three of the base points and platform points are taken for the non-redundant manipu-lator and all four for the redundant one. The rectangular area scanned is between xlow=ÿ 1.2to xup=1.5 and ylow=ÿ1.2 to yup=1.2. Steps of 0.005 are taken in both x and y coordinates.Points with condition numbers above 5000 are considered as singular.

Plots of workspace boundary for the non-redundant case are shown in Fig. 3 at the intervalsof 308 of orientation. Similarly, plots of workspace boundary and singularities with orientationsat the interval of 308 are shown superimposed in Figs 4 and 5 for the non-redundant and redun-dant cases, respectively. The plots in the Fig. 4 show that singularities for various orientationsare found on quadratic curves, a fact that can be veri®ed analytically through a few elementarycolumn transformations on the matrix H followed by the expansion of its determinant. In ad-

dition, it shows how the singularities partition the workspace into di�erent segments andobstruct manoeuverability. The plots in Fig. 5 show rare singularities (shown by crosses in this®gure) at isolated points corresponding to some of the orientations. Even if a ®ner scanning suc-ceeds in ®nding singular points at other orientations also, it is guaranteed that the continuoussingularity barriers of Fig. 4 are removed and the complete workspace can be used e�ectively,albeit at the cost of a slight reduction of workspace due to the fourth leg length constraints.

4.2. The generalized Stewart platform

The six-degree-of-freedom parallel manipulator known as the generalized Stewart platformshown in Fig. 6 and its redundant version with one additional leg are next compared for theirworkspaces and singularities. The analysis is similar to the previous example except that herethe vectors are three-dimensional and the rotation matrix is given in terms of the role±pitch±yaw angles as

R � RPY�yz; yy; yx� � Rot�z; yz�Rot�y; yy�Rot�x; yx�where the position and orientation are de®ned by

t � �x y z�T and � �yx yy yz�T

The wrench at the platform is

T � �RT MT�T � �Rx Ry Rz Mx My Mz�T

Fig. 3. Workspace of non-redundant PPM.

which is related to the actuator forces

F � �F1 F2 F3 F4 F5 F6�T

for the non-redundant case and

F � �F1 F2 F3 F4 F5 F6 F7�T

for the redundant case through the Equation (3). The force transformation matrix H is given by

HNR � s1 s2 s3 s4 s5 s6b1 � s1 b2 � s2 b3 � s3 b4 � s4 b5 � s5 b6 � s6

� �in the non-redundant case and by

HRed � s1 s2 s3 s4 s5 s6 s7b1 � s1 b2 � s2 b3 � s3 b4 � s4 b5 � s5 b6 � s6 b7 � s7

� �in the redundant one.

A similar procedure as in the previous example is followed to ®nd the workspace boundary{and singular points. However, since the complete task-space in this case is six-dimensional,

Fig. 4. Workspace and singularities of non-redundant PPM.

{Here, for the workspace determination, the restrictions of joint limits of the spherical joints have not been considered.

results are presented for a single orientation only with di�erent plots showing sections at di�er-ent values of the z-coordinate.

Description of the manipulators is given below. (All dimensions are in metres.) Base points:

�b1 b2 b3 b4 b5 b6 b7� �0:6 0:1 ÿ0:3 ÿ0:3 0:20 0:5 0:50:2 0:5 0:3 ÿ0:4 ÿ0:30 ÿ0:2 0:00:0 0:1 0:0 0:0 ÿ0:05 0:0 0:0

24 35Platform points (in platform frame):

�p1 p2 p3 p4 p5 p6 p7� �0:3 0:3 0:0 ÿ0:2 ÿ0:15 0:15 0:00:0 0:2 0:3 0:1 ÿ0:20 ÿ0:15 ÿ0:20:1 0:0 0:0 ÿ0:1 ÿ0:05 ÿ0:05 0:0

24 35Leg-length limits:

Lmin � 0:3 and Lmax � 1:5 for all legs

First six of the base points and platform points are taken for the non-redundant Stewart plat-form and all seven for the redundant one. The lower and upper limits for the region for scan-ning is taken as xlow=ylow=ÿ 2.0 and xup=yup=2.0 and a step-size of 0.01 has been used.Values of the z-coordinate from zlow=ÿ 1.4 to zup=1.4 have been examined at steps of 0.2.Points with condition number 20�103 have been considered as singular. The constant orien-

Fig. 5. Workspace and singularities of redundant PPM.

Fig. 6. The Stewart platform.

Fig. 7. Workspace of Stewart platform.

tation selected is

Y � �yx yy yz�T � �0:3 ÿ 0:2 0:2�T

Plots of boundaries of workspace sections at 12 di�erent values of the z-coordinate at intervalsof 0.2 m are shown in Fig. 7. Superposition of plots of singularities and boundaries of workspacesections at those sections are shown in Figs 8 and 9 for the non-redundant and redundant cases,respectively. The plots in Fig. 8 show that the curves of singularities are quite complicated. Infact, they are quartic curves, because the singularity surface of the Stewart platform for a givenorientation of the platform is a quartic surface as can be shown with some elementary columntransformations on the matrix H. The obstruction and partitioning of the workspace due tosingularities here are found to be more severe than the previous example. Plots in Fig. 9 show adrastic reduction of singularity (shown by crosses in this ®gure) by a single degree of redundancywhich virtually frees the entire workspace for e�ective use, of course at the cost of reducing thetotal workspace by a small amount due to the seventh leg length constraints.

The plots corresponding to the section at z = 0 are shown in an enlarged form in Fig. 10 asan illustration. Singularities of the non-redundant manipulator partition this section of theworkspace into six segments, as seen in Fig. 10(a). From any one of them, the adjacent two seg-ments and the opposite one are de®nitely unreachable through a singularity-free path. To theother two segments, a path, if possible, might require circuitous detours on the way in the z-direction and/or in the orientation variables. By the force redundancy, all these singularities van-

Fig. 8. Workspace and singularities of Stewart platform.

ish except at a localized spot (that too inside the void of the workspace) and the complete sec-tion of the workspace is connected, as seen in Fig. 10(b). The slight loss in the area of the sec-tion is insigni®cant in comparison.

5. CONCLUSIONS

The concept of force redundancy has been analysed in the context of parallel manipulators asthe series-parallel dual concept of kinematic redundancy of serial manipulators. It has beenshown that force redundancy can be used to reduce or even eliminate force singularities fromparallel manipulator, while kinematic singularity can be at most redistributed and avoided bykinematic redundancy in serial manipulators. Thus, force singularities can be of e�ective use intackling the problem of singularities and to increase the sti�ness of parallel manipulators,though costing some overhead and restricting the workspace to some extent. Numerical studieson two parallel manipulators show that addition of a single degree of redundancy can reducesingularities drastically and improve the quality of the workspace to a great extent, in compari-son to which the slight reduction in the workspace volume is insigni®cant.

The scope of future works includes development of strategies for redundancy resolution forvarious applications. Minimization of actuator forces and satisfaction of actuator constraintsare possibly the most apparent objectives that can be met with the force redundancy. In ad-dition, kinematic synthesis of redundant parallel manipulators also will be of interest so far asoptimal placement of redundant limb(s) is concerned. The optimization of the non-redundant

Fig. 9. Workspace and singularities of redundant Stewart platform.

manipulator geometry also should be considered to examine the relative advantages and costs ofincorporation of redundancy.

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