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Force redundancy in parallel manipulators: theoretical and practical issues
PII:
Mech. Mach. Theory Vol. 33, No. 6, pp. 727±742, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-114X/98 $19.00 + 0.00 S0094-114X(97)00094-3
FORCE REDUNDANCY IN PARALLEL MANIPULATORS: THEORETICAL AND PRACTICAL ISSUES BHASKAR DASGUPTA{ and T. S. MRUTHYUNJAYA Department of Mechanical Engineering, Indian Institute of Science, Bangalore-560 012, India (Received 15 November 1996) AbstractÐIn contrast to kinematic redundancy (widely studied in literature) in serial manipulators, the natural choice should be force (static) redundancy in parallel manipulators. In this paper, the force redundancy has been studied as the series±parallel dual concept of kinematic redundancy and its implications in kinematics and dynamics of parallel manipulators are described. In particular, its eective utilization in reduction and elimination of static singularities of parallel manipulators is demonstrated. Numerical studies are also presented for two parallel manipulators to demonstrate the singularity reduction by a single degree of redundancy. # 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
For eective utilization of robotic automation in complicated tasks and/or in complicated environments, 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 manipulator performs well in well-conditioned tasks in structured environments, but it may face serious diculties in a general situation due to its inherent mechanical non-linearities and the unstructured nature of the environment as well. Redundant manipulators oer additional means for control to circumvent such diculties. 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 possesses 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 avoidance, singularity avoidance, optimization of dexterity or some other performance criteria, satisfaction 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 manipulators and their eective 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 certain 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 manipulators possess certain characteristics in striking contrast to serial ones. Two important observations 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 kinematics and statics as propounded by Ball [21]. Secondly, works on parallel manipulator singularities [22±30] show that static singularities (which are completely absent in open-chain serial {Author to whom correspondence should be addressed at: Indian Institute of Technology, Department of Mechanical Engineering, Kanpur 208 016, India. Tel.Ð0091-512-597095; FaxÐ0091-512-597995; EmailÐ[email protected]. 727
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manipulators) of parallel manipulators appear within the workspace with continuous expanse thereby rendering the manipulator uncontrollable in some regions of the workspace. So far as redundancy is concerned, kinematic redundancy is not possible in parallel manipulators in the strict sense, because incorporation of more than one actuation in a particular limb of a parallel manipulator destroys its in-parallel character and it becomes a hybrid manipulator. A few kinematically redundant hybrid manipulators are also discussed in literature, for example the development of 10-d.o.f. hybrid manipulator UPSarm by Cheng [31] and the proposal of the 10-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 actuators proposed and studied by Zanganeh and Angeles [33, 34] is also essentially a hybrid manipulator with a complicated series±parallel chain in parallel with three in-parallel actuations. Though such arrangements are useful in avoiding kinematic singularities, avoidance of force singularities is quite dicult by such measures. A direct approach towards avoidance of force singularities in parallel manipulators has to be made through force redundancy, i.e. by the use of additional support(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 serial and parallel manipulators. It can be argued that force redundancies are actuation redundancies in some sense. Nevertheless, to understand the force redundancy completely and to utilize it eectively, it should be studied as series±parallel dual concept of kinematic redundancy and possibilities of its use should be explored in that framework. In this paper, the force redundancy in parallel manipulators is studied as the dual concept of kinematic redundancy in serial manipulators. In particular, the possibilities of singularity avoidance and elimination through force redundancy has been explored. This aspect is very important for parallel manipulators for which static singularities pose insurmountable barriers among segments of workspace as discussed in Dasgupta and Mruthyunjaya [37], so far as safe and eective operation is concerned. In the next section, the concept of force redundancy is studied and its implications in various problems of kinematics and dynamics are discussed. Section 3 is devoted to an exploration of possibilities of avoidance and complete elimination of force singularities, where fundamental distinctions between the relationships of singularity and redundancy in the cases of serial and parallel manipulators are established. In Section 4, numerical studies of force redundancy in relation to workspace and singularities are presented for two parallel manipulators. Finally, in the last section, conclusions of the present work are summarized and suggestions for future work are enumerated. 2. FORCE REDUNDANCY
Parallel manipulators can have a wide variety of structures with various possibilities of connections and actuations. Out of them, those having straightforward (and unique) inverse kinematics (e.g. the generalized Stewart platform) constitute a subclass (hereafter referred to as ``simple parallel manipulators'') which exhibits the contrast against serial manipulators implied by series±parallel duality in the most prominent manner. The comparison between the two is summarized in Table 1. From Table 1, the duality exhibited by simple parallel manipulators against serial manipulators is clear. Other parallel manipulators also exhibit these characteristics. Besides, each limb of a general (non-simple) parallel manipulators may involve the inverse kinematic complexity of a serial chain, though it will be much simpler and decoupled from other limbs compared with direct kinematics or inverse statics of the entire manipulator. To take advantage of the complete correspondence between the two classes, the analyses have been performed in the present work keeping in view the subclass of simple parallel manipulators. The concepts are nevertheless roughly applicable to all fully parallel manipulators in general. Due to the distinctive features of the two classes, serial manipulators are to be preferred in applications requiring large workspace and manoeuverability, while parallel ones are preferable in applications where precise positioning, high load-carrying capacity and good dynamic performance are of paramount importance. In other words, the primary objective of a serial manip-
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Table 1. Comparison between serial and simple parallel manipulators Serial manipulators
Simple parallel manipulators
Actuators Direct position transformation Inverse position transformation Direct motion transformation
In series Straightforward and unique Complicated and multiple Well-de®ned and unique
Inverse motion transformation
Inverse force transformation
Not well-de®ned; may be non-existent, unique or in®nite Not well-de®ned; may be non-existent, unique or in®nite Well-de®ned and unique
In parallel Complicated and multiple Straightforward and unique Not well-de®ned; may be non-existent, unique or in®nite Well-de®ned and unique
Singularity Natural description Preferred property Preferred application
Kinematic In joint-space Dexterity Gross motion
Direct force transformation
Well-de®ned and unique Not well-de®ned; may be non-existent, unique or in®nite Static In Cartesian space Stiness Precise positioning
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 roles in application. Kinematic singularities in a serial manipulator restrict its motion capabilities and can be remedied (to a great extent) by kinematic redundancy. On the other hand, static or force singularities in a parallel manipulator restrict its load-carrying capabilities. Hence, the concept that arises as the series±parallel dual to the kinematic redundancy of serial manipulators is that of static or force redundancy, which has particular importance regarding the avoidance of static or force singularities, the only singularities in the case of simple parallel manipulators and the more 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) is the two-legged (and 2-d.o.f. also) parallel manipulator shown in Fig. 1(b), Cartesian position p(x, y) of the end-eector de®ning the task-space coordinates in both cases. In the serial manipulator, inverse velocity transformation is unique at a non-singular con®guration and not wellde®ned at a singularity. Similar is the case of inverse force transformation in the parallel manipulator. In the serial manipulator, singularities are found on two circles in the task-space (completely extended and completely folded poses) and are of kinematic nature. In the parallel
Fig. 1. Kinematic redundancy and force redundancy.
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manipulator, they are static singularities and are found on the line joining the ®xed pivots b1 and b2. Now, considering only the position (and disregarding the orientation) of the end-eector in the plane, the 3-d.o.f. serial manipulator shown in Fig. 1(c) is a kinematically redundant manipulator having one additional link (and an actuation) in series. For this manipulator, at a given con®guration, the inverse velocity transformation will be a ®bre (an in®nitely many valued function), while the inverse force transformation will be unique. In the same way, a static or force redundancy can be introduced in the parallel manipulator of Fig. 1(b) in the form of an additional leg in parallel getting thereby a statically redundant 2-d.o.f. (3 degree-of-constraint) parallel manipulator shown in Fig. 1(d). For this manipulator, at any given con®guration, the inverse force transformation is a ®bre, while the inverse velocity transformation is unique. This shows the duality of kinematic redundancy of serial manipulators and force redundancy of parallel manipulators. However, it is to be observed here that the redundant serial manipulator of Fig. 1(c) still has an in®nite number of singularities at the completely extended (and possibly completely folded) con®gurations, while the redundant parallel manipulator of Fig. 1(d) is entirely free from singularities as long as the three ®xed pivots b1, b2, b3 are non-collinear. This indicates that though the kinematic singularities of serial manipulators remain the same in number{ even after the introduction of redundancy (which merely redistributes the singular poses in the workspace), force redundancy in parallel manipulators may be able to reduce the singularities and even eliminate them completely. This aspect will be discussed in detail for the general case in Section 3. In order to understand the implications of force redundancy in the kinematics and dynamics of a parallel manipulator, let us consider a simple parallel manipulator (de®ned earlier) with mdimensional 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 : Rm 4Rn
1
which uniquely de®nes Y in terms of X. In the case of a non-simple parallel manipulator, the relationship will not be explicit as Equation (1) and there is a possibility of multiple (but ®nite number 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 solution for an arbitrary choice of Y. However, the value of Y in practice is obtained from sensor data which is expected to be consistent (up to the error in measurement). In such a situation, more equations than unknowns will simplify the direct position kinematics compared to the corresponding non-redundant case. In addition, the additional equation may restrict the number of solutions. The velocity kinematics of a simple parallel manipulator is given by _ @f X_ Y @X
2
where the matrix @f/@X is analytic and de®nes a unique inverse velocity transformation. In the case of non-simple parallel manipulators, the matrix @f/@X will not be analytic because one fac_ (in some tor of it will require an inversion. Singularity of that factor may make some of the y's _ limbs) ill-de®ned without aecting other y's (in other limbs), i.e. various limbs are totally decoupled so far as inverse kinematics is concerned. The direct velocity kinematics problem is a linear 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.
Force redundancy in parallel manipulators
731
sors are expected to be consistent and a pseudo-inverse solution is expected to cancel out measurement errors also to some extent. The static relationship between the actuator forces F and the generalized forces T developed at the end-eector is given by T HF
3
mn
is analytic in the case of simple parallel manipwhere the force transformation matrix H $ R ulators and actuator forces are mapped uniquely to the end-eector forces. In order to exert a desired force T at the end-eector, the actuator forces F can be chosen in in®nite number of ways because of the force redundancy. This redundancy can be resolved by pseudo-inverse or by meeting some other optimization or constraint criteria. In particular, a minimax solution for F, i.e. minimization of the maximum of the actuator forces, may be useful in meeting actuator constraints. It should be mentioned in passing that the jacobian symmetry between inverse kinematics and direct statics of the redundant parallel manipulator (as in any parallel manipulator) is manifest in the relationship. @f HT @X
4
The role of the force redundancy in the static relationship, i.e. Equation (3) is directly carried to the dynamics of the manipulator along with the usual additional terms, i.e. Z T HF MX
5
Introduction of kinematic redundancy to a serial manipulator can be used to enhance its workspace. On the other hand, force redundancy in a parallel manipulator tends to further restrict its workspace, 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 becomes more usable by the reduction of the barriers of singularities, as will be seen in the next two sections. 3. SINGULARITY AVOIDANCE AND ELIMINATION
A non-redundant parallel manipulator is at a force singularity when the force transformation matrix 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 (m ÿ 1)-dimensional manifold in the m-dimensional task-space. As such, it partitions the workspace into two or more segments which are not reachable from one another without encountering a singularity. Similarly, for a statically redundant parallel manipulator, force singularities are characterized by 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
n
Though Cm submatrices of order m m can be formed from H by taking m columns at a time, only n ÿ (m ÿ 1) conditions given by Equation (7) are independent. Thus, singularities are found at the intersection of n ÿ (m ÿ 1) hypersurfaces forming a lower dimensional manifold. The dimension of the singularity manifold (DOSM) is given by DOSM m ÿ n ÿ m 1 m ÿ 1 ÿ n ÿ m m ÿ 1 ÿ DOR m ÿ 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 = (m ÿ 1) or above,
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B. Dasgupta and T. S. Mruthyunjaya
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 relations 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 standpoint, 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-eector 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 incorporate 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 1 ÿ DOR n ÿ 1 ÿ n ÿ m m ÿ 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 (m ÿ 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 redundancy. There is no such restriction, however, on the dimension of the manifold of force singularity, 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.
Force redundancy in parallel manipulators
733
4. NUMERICAL STUDIES
In order to visualize the eect of force redundancy on singularities and workspaces of parallel manipulators, two parallel manipulators, namely the 3-d.o.f. planar parallel manipulator and the 6-d.o.f. generalized Stewart platform are studied. In each case, comparisons are made between a non-redundant manipulator and a redundant manipulator with a single degree of redundancy, 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 redundant manipulator with four legs shown in Fig. 2(a) and (b) are compared for their workspaces and singularities. Denoting the i-th base point by bi, i-th platform point (in platform frame) by pi, position of platform reference point by t = [x y]T and the platform orientation by y, we have the i-th leg vector as Si t Rpi ÿ bi where
cos y R sin y
ÿsin y cos y
The i-th leg length is given by Li kSi k
Fig. 2. 3-d.o.f. planar parallel manipulators.
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B. Dasgupta and T. S. Mruthyunjaya
and the unit vector along the leg as si Si =Li The actuator forces F F1
F3 T
F2
in the non-redundant case and F F1
F2
F3
F 4 T
in the redundant case are related to the force R and moment M at the platform taken together as T Rx
Ry
MT
by the Equation (3), where the force transformation matrix H is given by s2 s3 s1 HNR b1 s1 b2 s2 b3 s3 for the non-redundant case and by HRed
s2 b2 s 2
s1 b1 s 1
s3 b3 s 3
s4 b3 s 4
for the redundant case. In each of the two cases, a suciently large rectangular area (to enclose the workspace) in the x ÿ y plane is scanned for various orientations y of the platform and the above computations are 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 being within the limits, is taken as a point on the workspace boundary. At each point, the condition number (ratio of greatest to least singular values) of H is evaluated and whenever it is found to be above a prescribed threshold, the point is taken as a singular point. Finally, the workspace boundary and singularities are plotted together. The description of the manipulators used for the numerical study are as follows. (All dimensions are in metres.) Base points: 0:0 0:2 0:4 0:2 b1 b2 b3 b4 0:0 0:0 0:1 0:2 Platform points (in platform frame): p1
p2
p3
p4
ÿ0:15 0:0
ÿ0:1 ÿ0:1
0:15 0:0
ÿ0:1 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 manipulator and all four for the redundant one. The rectangular area scanned is between xlow= ÿ 1.2 to 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 intervals of 308 of orientation. Similarly, plots of workspace boundary and singularities with orientations at the interval of 308 are shown superimposed in Figs 4 and 5 for the non-redundant and redundant cases, respectively. The plots in the Fig. 4 show that singularities for various orientations are found on quadratic curves, a fact that can be veri®ed analytically through a few elementary column transformations on the matrix H followed by the expansion of its determinant. In ad-
Force redundancy in parallel manipulators
735
Fig. 3. Workspace of non-redundant PPM.
dition, it shows how the singularities partition the workspace into dierent segments and obstruct 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 succeeds in ®nding singular points at other orientations also, it is guaranteed that the continuous singularity barriers of Fig. 4 are removed and the complete workspace can be used eectively, 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 platform shown in Fig. 6 and its redundant version with one additional leg are next compared for their workspaces and singularities. The analysis is similar to the previous example except that here the 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 zT and yx
yy
y z T
Mx
My
The wrench at the platform is T RT
MT T Rx
Ry
Rz
Mz T
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B. Dasgupta and T. S. Mruthyunjaya
Fig. 4. Workspace and singularities of non-redundant PPM.
which is related to the actuator forces F F1
F2
F3
F4
F6 T
F5
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 s2 s3 s4 s5 s6 s1 HNR b1 s1 b2 s2 b3 s3 b4 s4 b5 s5 b6 s6 in the non-redundant case and by s2 s1 HRed b1 s 1 b2 s 2
s3 b3 s 3
s4 b4 s 4
s5 b5 s 5
s6 b6 s6
s7 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, {Here, for the workspace determination, the restrictions of joint limits of the spherical joints have not been considered.
Force redundancy in parallel manipulators
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Fig. 5. Workspace and singularities of redundant PPM.
results are presented for a single orientation only with dierent plots showing sections at dierent values of the z-coordinate. Description of the manipulators is given below. (All dimensions are in metres.) Base points: 2 3 0:6 0:1 ÿ0:3 ÿ0:3 0:20 0:5 0:5 0:3 ÿ0:4 ÿ0:30 ÿ0:2 0:0 5 b1 b2 b3 b4 b5 b6 b7 4 0:2 0:5 0:0 0:1 0:0 0:0 ÿ0:05 0:0 0:0 Platform points (in platform frame): p1
p2
p3
p4
p5
p6
2
0:3 p7 4 0:0 0:1
0:3 0:2 0:0
0:0 0:3 0:0
ÿ0:2 ÿ0:15 0:1 ÿ0:20 ÿ0:1 ÿ0:05
0:15 ÿ0:15 ÿ0:05
3 0:0 ÿ0:2 5 0:0
Leg-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 platform and all seven for the redundant one. The lower and upper limits for the region for scanning 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-
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Fig. 6. The Stewart platform.
Fig. 7. Workspace of Stewart platform.
Force redundancy in parallel manipulators
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Fig. 8. Workspace and singularities of Stewart platform.
tation selected is Y yx
yy
yz T 0:3
ÿ 0:2
0:2T
Plots of boundaries of workspace sections at 12 dierent values of the z-coordinate at intervals of 0.2 m are shown in Fig. 7. Superposition of plots of singularities and boundaries of workspace sections 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. In fact, they are quartic curves, because the singularity surface of the Stewart platform for a given orientation of the platform is a quartic surface as can be shown with some elementary column transformations on the matrix H. The obstruction and partitioning of the workspace due to singularities here are found to be more severe than the previous example. Plots in Fig. 9 show a drastic reduction of singularity (shown by crosses in this ®gure) by a single degree of redundancy which virtually frees the entire workspace for eective use, of course at the cost of reducing the total 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 as an illustration. Singularities of the non-redundant manipulator partition this section of the workspace into six segments, as seen in Fig. 10(a). From any one of them, the adjacent two segments and the opposite one are de®nitely unreachable through a singularity-free path. To the other two segments, a path, if possible, might require circuitous detours on the way in the zdirection and/or in the orientation variables. By the force redundancy, all these singularities van-
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Fig. 9. Workspace and singularities of redundant Stewart platform.
ish except at a localized spot (that too inside the void of the workspace) and the complete section of the workspace is connected, as seen in Fig. 10(b). The slight loss in the area of the section is insigni®cant in comparison. 5. CONCLUSIONS
The concept of force redundancy has been analysed in the context of parallel manipulators as the series-parallel dual concept of kinematic redundancy of serial manipulators. It has been shown that force redundancy can be used to reduce or even eliminate force singularities from parallel manipulator, while kinematic singularity can be at most redistributed and avoided by kinematic redundancy in serial manipulators. Thus, force singularities can be of eective use in tackling the problem of singularities and to increase the stiness of parallel manipulators, though costing some overhead and restricting the workspace to some extent. Numerical studies on two parallel manipulators show that addition of a single degree of redundancy can reduce singularities drastically and improve the quality of the workspace to a great extent, in comparison to which the slight reduction in the workspace volume is insigni®cant. The scope of future works includes development of strategies for redundancy resolution for various applications. Minimization of actuator forces and satisfaction of actuator constraints are possibly the most apparent objectives that can be met with the force redundancy. In addition, kinematic synthesis of redundant parallel manipulators also will be of interest so far as optimal placement of redundant limb(s) is concerned. The optimization of the non-redundant
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Fig. 10. Sample section: workspace and singularities.
manipulator geometry also should be considered to examine the relative advantages and costs of incorporation of redundancy. REFERENCES 1. Nakamura, Y., Advanced Robotics: Redundancy and Optimization, Addison±Welsey, Reading, MA, 1991.
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2. Whitney, D. E., Transactions of the ASME Journal of Dynamic Systems, Measurement and Control, 1972, 94, 303± 309. 3. Liegeois, A., IEEE Transaction of Systems, Man and Cybernetics, 1977, 7(12), 842±868. 4. Maciejewski, A. A. and Klein, C. A., International Journal of Robotics Research, 1985, 4(3), 109±117. 5. Klein, C. A. and Blaho, B. E., International Journal of Robotics Research, 1987, 6(2), 72±83. 6. Hollerbach, J. M. and Suh, K. C., IEEE Journal of Robotics and Automation, 1987, 3, 308±316. 7. Nakamura, Y. and Hanafusa, H., International Journal of Robotics Research, 1987, 6(1), 32±42. 8. Baker, D. R. and Wampler, C. W. II, International Journal of Robotics Research, 1988, 7(2), 3±21. 9. Martin, D. P., Baillieul, J. and Hollerbach, J. M., IEEE Transactions on Robotics and Automation, 1989, 5(4), 529± 533. 10. Podhorodeski, R. P., Goldenberg, A. A. and Fenton, R. G., Transactions of the ASME Journal of Mechanical Design, 1993, 115, 530±539. 11. Lewis, C. L. and Maciejewsky, A. A., Computers and Electrical Engineering, 1994, 20(3), 273±288. 12. Wang, J. T., Transactions of the ASME Journal of Mechanical Design, 1995, 117, 454±459. 13. Kircanski, M., International Journal of Robotics Research, 1995, 14(4), 382±398. 14. Roberts, R. G. and Maciejewski, A. A., IEEE Transactions on Robotics and Automation, 1996, 12(4), 543±552. 15. Nakamura, Y. and Ghodoussi, M., IEEE Transactions on Robotics and Automation, 1989, 5(3), 294±302. 16. Stewart, D., Proceedings of the Institute of Mechanical Engineers, Part I, 1965±1966, 180(15), 371±386. 17. Hunt, K. H., in Kinematic Geometry of Mechanisms. Clarendon Press, Oxford, 1978. 18. Hunt, K. H., Transactions of the ASME Journal of Mechanisms Transmission and Automation in Design, 1983, 105, 705±712. 19. Fichter, E. F., International Journal of Robotics Research, 1986, 5(2), 157±182. 20. Waldron, K. J. and Hunt, K. H., International Journal of Robotics Research, 1991, 10(5), 473±480. 21. Ball, R. S., A Treatise on the Theory of Screws, Cambridge University Press, Cambridge, MA, 1900. 22. Merlet, J.-P., International Journal of Robotics Research, 1989, 8(5), 45±56. 23. Gosselin, C. and Angeles, J., IEEE Transactions on Robotics and Automation, 1990, 6(3), 281±290. 24. Sefrioui, J. and Gosselin, C. M., Robotics and Autonomous Systems, 1992, 10, 209±224. 25. Gosselin, C. M. and Sefrioui, J., Robotics, Spatial Motion and Mechanical Systems, DEV 45, 1992, ASME, 329±335. 26. Liu, K., Lewis, F., Lebret, G. and Taylor, D., Journal of Intelligent Robotic Systems, 1993, 8, 287±308. 27. Sefrioui, J. and Gosselin, C. M., Mechanism and Machine Theory, 1995, 30(4), 533±551. 28. Mohammadi Daniali, H. R., Zsombor-Murray, P. J. and Angeles, J., Mechanism and Machine Theory, 1995, 30(5), 665±678. 29. Collins, C. L. and Long, G. L., IEEE Transactions on Robotics and Automation, 1995, 11(5), 661±669. 30. Takeda, Y. and Funabashi, H., JSME International Journal Series C, 1996, 39(1), 85±93. 31. Cheng, H. H., Transactions of the ASME Journal of Dynamic Systems, Measurement and Control, 1994, 116, 687± 701. 32. Mingyang, Z., Tomg, G., Ge, C., Qunming, L. and Dalong, T., Proceedings of the IEEE International Conference on Robotics and Automation, 1995, 221±226. 33. Zanganeh, K. E. and Angeles, J., Proceedings of the IEEE International Conference on Robotics and Automation, 1994, 3043±3048. 34. Zanganeh, K. E. and Angeles, J., Proceedings of the IEEE International Conference on Robotics and Automation, 1994, 3049±3054. 35. Lee, S. and Kim, S., Proceedings of the IEEE International Conference on Robotics and Automation, 1994, 77±82. 36. Collins, C. L. and Long, G. L., Proceedings of the IEEE International Conference on Robotics and Automation, 1995, 526±531. 37. Dasgupta, B. and Mruthyunjaya, T. S., Mechanism and Machine Theory, 1998, 33(6), 711±725.
Mech. Mach. Theory Vol. 33, No. 6, pp. 727±742, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-114X/98 $19.00 + 0.00 S0094-114X(97)00094-3
FORCE REDUNDANCY IN PARALLEL MANIPULATORS: THEORETICAL AND PRACTICAL ISSUES BHASKAR DASGUPTA{ and T. S. MRUTHYUNJAYA Department of Mechanical Engineering, Indian Institute of Science, Bangalore-560 012, India (Received 15 November 1996) AbstractÐIn contrast to kinematic redundancy (widely studied in literature) in serial manipulators, the natural choice should be force (static) redundancy in parallel manipulators. In this paper, the force redundancy has been studied as the series±parallel dual concept of kinematic redundancy and its implications in kinematics and dynamics of parallel manipulators are described. In particular, its eective utilization in reduction and elimination of static singularities of parallel manipulators is demonstrated. Numerical studies are also presented for two parallel manipulators to demonstrate the singularity reduction by a single degree of redundancy. # 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
For eective utilization of robotic automation in complicated tasks and/or in complicated environments, 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 manipulator performs well in well-conditioned tasks in structured environments, but it may face serious diculties in a general situation due to its inherent mechanical non-linearities and the unstructured nature of the environment as well. Redundant manipulators oer additional means for control to circumvent such diculties. 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 possesses 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 avoidance, singularity avoidance, optimization of dexterity or some other performance criteria, satisfaction 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 manipulators and their eective 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 certain 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 manipulators possess certain characteristics in striking contrast to serial ones. Two important observations 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 kinematics and statics as propounded by Ball [21]. Secondly, works on parallel manipulator singularities [22±30] show that static singularities (which are completely absent in open-chain serial {Author to whom correspondence should be addressed at: Indian Institute of Technology, Department of Mechanical Engineering, Kanpur 208 016, India. Tel.Ð0091-512-597095; FaxÐ0091-512-597995; EmailÐ[email protected]. 727
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manipulators) of parallel manipulators appear within the workspace with continuous expanse thereby rendering the manipulator uncontrollable in some regions of the workspace. So far as redundancy is concerned, kinematic redundancy is not possible in parallel manipulators in the strict sense, because incorporation of more than one actuation in a particular limb of a parallel manipulator destroys its in-parallel character and it becomes a hybrid manipulator. A few kinematically redundant hybrid manipulators are also discussed in literature, for example the development of 10-d.o.f. hybrid manipulator UPSarm by Cheng [31] and the proposal of the 10-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 actuators proposed and studied by Zanganeh and Angeles [33, 34] is also essentially a hybrid manipulator with a complicated series±parallel chain in parallel with three in-parallel actuations. Though such arrangements are useful in avoiding kinematic singularities, avoidance of force singularities is quite dicult by such measures. A direct approach towards avoidance of force singularities in parallel manipulators has to be made through force redundancy, i.e. by the use of additional support(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 serial and parallel manipulators. It can be argued that force redundancies are actuation redundancies in some sense. Nevertheless, to understand the force redundancy completely and to utilize it eectively, it should be studied as series±parallel dual concept of kinematic redundancy and possibilities of its use should be explored in that framework. In this paper, the force redundancy in parallel manipulators is studied as the dual concept of kinematic redundancy in serial manipulators. In particular, the possibilities of singularity avoidance and elimination through force redundancy has been explored. This aspect is very important for parallel manipulators for which static singularities pose insurmountable barriers among segments of workspace as discussed in Dasgupta and Mruthyunjaya [37], so far as safe and eective operation is concerned. In the next section, the concept of force redundancy is studied and its implications in various problems of kinematics and dynamics are discussed. Section 3 is devoted to an exploration of possibilities of avoidance and complete elimination of force singularities, where fundamental distinctions between the relationships of singularity and redundancy in the cases of serial and parallel manipulators are established. In Section 4, numerical studies of force redundancy in relation to workspace and singularities are presented for two parallel manipulators. Finally, in the last section, conclusions of the present work are summarized and suggestions for future work are enumerated. 2. FORCE REDUNDANCY
Parallel manipulators can have a wide variety of structures with various possibilities of connections and actuations. Out of them, those having straightforward (and unique) inverse kinematics (e.g. the generalized Stewart platform) constitute a subclass (hereafter referred to as ``simple parallel manipulators'') which exhibits the contrast against serial manipulators implied by series±parallel duality in the most prominent manner. The comparison between the two is summarized in Table 1. From Table 1, the duality exhibited by simple parallel manipulators against serial manipulators is clear. Other parallel manipulators also exhibit these characteristics. Besides, each limb of a general (non-simple) parallel manipulators may involve the inverse kinematic complexity of a serial chain, though it will be much simpler and decoupled from other limbs compared with direct kinematics or inverse statics of the entire manipulator. To take advantage of the complete correspondence between the two classes, the analyses have been performed in the present work keeping in view the subclass of simple parallel manipulators. The concepts are nevertheless roughly applicable to all fully parallel manipulators in general. Due to the distinctive features of the two classes, serial manipulators are to be preferred in applications requiring large workspace and manoeuverability, while parallel ones are preferable in applications where precise positioning, high load-carrying capacity and good dynamic performance are of paramount importance. In other words, the primary objective of a serial manip-
Force redundancy in parallel manipulators
729
Table 1. Comparison between serial and simple parallel manipulators Serial manipulators
Simple parallel manipulators
Actuators Direct position transformation Inverse position transformation Direct motion transformation
In series Straightforward and unique Complicated and multiple Well-de®ned and unique
Inverse motion transformation
Inverse force transformation
Not well-de®ned; may be non-existent, unique or in®nite Not well-de®ned; may be non-existent, unique or in®nite Well-de®ned and unique
In parallel Complicated and multiple Straightforward and unique Not well-de®ned; may be non-existent, unique or in®nite Well-de®ned and unique
Singularity Natural description Preferred property Preferred application
Kinematic In joint-space Dexterity Gross motion
Direct force transformation
Well-de®ned and unique Not well-de®ned; may be non-existent, unique or in®nite Static In Cartesian space Stiness Precise positioning
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 roles in application. Kinematic singularities in a serial manipulator restrict its motion capabilities and can be remedied (to a great extent) by kinematic redundancy. On the other hand, static or force singularities in a parallel manipulator restrict its load-carrying capabilities. Hence, the concept that arises as the series±parallel dual to the kinematic redundancy of serial manipulators is that of static or force redundancy, which has particular importance regarding the avoidance of static or force singularities, the only singularities in the case of simple parallel manipulators and the more 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) is the two-legged (and 2-d.o.f. also) parallel manipulator shown in Fig. 1(b), Cartesian position p(x, y) of the end-eector de®ning the task-space coordinates in both cases. In the serial manipulator, inverse velocity transformation is unique at a non-singular con®guration and not wellde®ned at a singularity. Similar is the case of inverse force transformation in the parallel manipulator. In the serial manipulator, singularities are found on two circles in the task-space (completely extended and completely folded poses) and are of kinematic nature. In the parallel
Fig. 1. Kinematic redundancy and force redundancy.
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manipulator, they are static singularities and are found on the line joining the ®xed pivots b1 and b2. Now, considering only the position (and disregarding the orientation) of the end-eector in the plane, the 3-d.o.f. serial manipulator shown in Fig. 1(c) is a kinematically redundant manipulator having one additional link (and an actuation) in series. For this manipulator, at a given con®guration, the inverse velocity transformation will be a ®bre (an in®nitely many valued function), while the inverse force transformation will be unique. In the same way, a static or force redundancy can be introduced in the parallel manipulator of Fig. 1(b) in the form of an additional leg in parallel getting thereby a statically redundant 2-d.o.f. (3 degree-of-constraint) parallel manipulator shown in Fig. 1(d). For this manipulator, at any given con®guration, the inverse force transformation is a ®bre, while the inverse velocity transformation is unique. This shows the duality of kinematic redundancy of serial manipulators and force redundancy of parallel manipulators. However, it is to be observed here that the redundant serial manipulator of Fig. 1(c) still has an in®nite number of singularities at the completely extended (and possibly completely folded) con®gurations, while the redundant parallel manipulator of Fig. 1(d) is entirely free from singularities as long as the three ®xed pivots b1, b2, b3 are non-collinear. This indicates that though the kinematic singularities of serial manipulators remain the same in number{ even after the introduction of redundancy (which merely redistributes the singular poses in the workspace), force redundancy in parallel manipulators may be able to reduce the singularities and even eliminate them completely. This aspect will be discussed in detail for the general case in Section 3. In order to understand the implications of force redundancy in the kinematics and dynamics of a parallel manipulator, let us consider a simple parallel manipulator (de®ned earlier) with mdimensional 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 : Rm 4Rn
1
which uniquely de®nes Y in terms of X. In the case of a non-simple parallel manipulator, the relationship will not be explicit as Equation (1) and there is a possibility of multiple (but ®nite number 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 solution for an arbitrary choice of Y. However, the value of Y in practice is obtained from sensor data which is expected to be consistent (up to the error in measurement). In such a situation, more equations than unknowns will simplify the direct position kinematics compared to the corresponding non-redundant case. In addition, the additional equation may restrict the number of solutions. The velocity kinematics of a simple parallel manipulator is given by _ @f X_ Y @X
2
where the matrix @f/@X is analytic and de®nes a unique inverse velocity transformation. In the case of non-simple parallel manipulators, the matrix @f/@X will not be analytic because one fac_ (in some tor of it will require an inversion. Singularity of that factor may make some of the y's _ limbs) ill-de®ned without aecting other y's (in other limbs), i.e. various limbs are totally decoupled so far as inverse kinematics is concerned. The direct velocity kinematics problem is a linear 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.
Force redundancy in parallel manipulators
731
sors are expected to be consistent and a pseudo-inverse solution is expected to cancel out measurement errors also to some extent. The static relationship between the actuator forces F and the generalized forces T developed at the end-eector is given by T HF
3
mn
is analytic in the case of simple parallel manipwhere the force transformation matrix H $ R ulators and actuator forces are mapped uniquely to the end-eector forces. In order to exert a desired force T at the end-eector, the actuator forces F can be chosen in in®nite number of ways because of the force redundancy. This redundancy can be resolved by pseudo-inverse or by meeting some other optimization or constraint criteria. In particular, a minimax solution for F, i.e. minimization of the maximum of the actuator forces, may be useful in meeting actuator constraints. It should be mentioned in passing that the jacobian symmetry between inverse kinematics and direct statics of the redundant parallel manipulator (as in any parallel manipulator) is manifest in the relationship. @f HT @X
4
The role of the force redundancy in the static relationship, i.e. Equation (3) is directly carried to the dynamics of the manipulator along with the usual additional terms, i.e. Z T HF MX
5
Introduction of kinematic redundancy to a serial manipulator can be used to enhance its workspace. On the other hand, force redundancy in a parallel manipulator tends to further restrict its workspace, 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 becomes more usable by the reduction of the barriers of singularities, as will be seen in the next two sections. 3. SINGULARITY AVOIDANCE AND ELIMINATION
A non-redundant parallel manipulator is at a force singularity when the force transformation matrix 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 (m ÿ 1)-dimensional manifold in the m-dimensional task-space. As such, it partitions the workspace into two or more segments which are not reachable from one another without encountering a singularity. Similarly, for a statically redundant parallel manipulator, force singularities are characterized by 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
n
Though Cm submatrices of order m m can be formed from H by taking m columns at a time, only n ÿ (m ÿ 1) conditions given by Equation (7) are independent. Thus, singularities are found at the intersection of n ÿ (m ÿ 1) hypersurfaces forming a lower dimensional manifold. The dimension of the singularity manifold (DOSM) is given by DOSM m ÿ n ÿ m 1 m ÿ 1 ÿ n ÿ m m ÿ 1 ÿ DOR m ÿ 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 = (m ÿ 1) or above,
732
B. Dasgupta and T. S. Mruthyunjaya
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 relations 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 standpoint, 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-eector 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 incorporate 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 1 ÿ DOR n ÿ 1 ÿ n ÿ m m ÿ 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 (m ÿ 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 redundancy. There is no such restriction, however, on the dimension of the manifold of force singularity, 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.
Force redundancy in parallel manipulators
733
4. NUMERICAL STUDIES
In order to visualize the eect of force redundancy on singularities and workspaces of parallel manipulators, two parallel manipulators, namely the 3-d.o.f. planar parallel manipulator and the 6-d.o.f. generalized Stewart platform are studied. In each case, comparisons are made between a non-redundant manipulator and a redundant manipulator with a single degree of redundancy, 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 redundant manipulator with four legs shown in Fig. 2(a) and (b) are compared for their workspaces and singularities. Denoting the i-th base point by bi, i-th platform point (in platform frame) by pi, position of platform reference point by t = [x y]T and the platform orientation by y, we have the i-th leg vector as Si t Rpi ÿ bi where
cos y R sin y
ÿsin y cos y
The i-th leg length is given by Li kSi k
Fig. 2. 3-d.o.f. planar parallel manipulators.
734
B. Dasgupta and T. S. Mruthyunjaya
and the unit vector along the leg as si Si =Li The actuator forces F F1
F3 T
F2
in the non-redundant case and F F1
F2
F3
F 4 T
in the redundant case are related to the force R and moment M at the platform taken together as T Rx
Ry
MT
by the Equation (3), where the force transformation matrix H is given by s2 s3 s1 HNR b1 s1 b2 s2 b3 s3 for the non-redundant case and by HRed
s2 b2 s 2
s1 b1 s 1
s3 b3 s 3
s4 b3 s 4
for the redundant case. In each of the two cases, a suciently large rectangular area (to enclose the workspace) in the x ÿ y plane is scanned for various orientations y of the platform and the above computations are 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 being within the limits, is taken as a point on the workspace boundary. At each point, the condition number (ratio of greatest to least singular values) of H is evaluated and whenever it is found to be above a prescribed threshold, the point is taken as a singular point. Finally, the workspace boundary and singularities are plotted together. The description of the manipulators used for the numerical study are as follows. (All dimensions are in metres.) Base points: 0:0 0:2 0:4 0:2 b1 b2 b3 b4 0:0 0:0 0:1 0:2 Platform points (in platform frame): p1
p2
p3
p4
ÿ0:15 0:0
ÿ0:1 ÿ0:1
0:15 0:0
ÿ0:1 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 manipulator and all four for the redundant one. The rectangular area scanned is between xlow= ÿ 1.2 to 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 intervals of 308 of orientation. Similarly, plots of workspace boundary and singularities with orientations at the interval of 308 are shown superimposed in Figs 4 and 5 for the non-redundant and redundant cases, respectively. The plots in the Fig. 4 show that singularities for various orientations are found on quadratic curves, a fact that can be veri®ed analytically through a few elementary column transformations on the matrix H followed by the expansion of its determinant. In ad-
Force redundancy in parallel manipulators
735
Fig. 3. Workspace of non-redundant PPM.
dition, it shows how the singularities partition the workspace into dierent segments and obstruct 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 succeeds in ®nding singular points at other orientations also, it is guaranteed that the continuous singularity barriers of Fig. 4 are removed and the complete workspace can be used eectively, 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 platform shown in Fig. 6 and its redundant version with one additional leg are next compared for their workspaces and singularities. The analysis is similar to the previous example except that here the 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 zT and yx
yy
y z T
Mx
My
The wrench at the platform is T RT
MT T Rx
Ry
Rz
Mz T
736
B. Dasgupta and T. S. Mruthyunjaya
Fig. 4. Workspace and singularities of non-redundant PPM.
which is related to the actuator forces F F1
F2
F3
F4
F6 T
F5
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 s2 s3 s4 s5 s6 s1 HNR b1 s1 b2 s2 b3 s3 b4 s4 b5 s5 b6 s6 in the non-redundant case and by s2 s1 HRed b1 s 1 b2 s 2
s3 b3 s 3
s4 b4 s 4
s5 b5 s 5
s6 b6 s6
s7 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, {Here, for the workspace determination, the restrictions of joint limits of the spherical joints have not been considered.
Force redundancy in parallel manipulators
737
Fig. 5. Workspace and singularities of redundant PPM.
results are presented for a single orientation only with dierent plots showing sections at dierent values of the z-coordinate. Description of the manipulators is given below. (All dimensions are in metres.) Base points: 2 3 0:6 0:1 ÿ0:3 ÿ0:3 0:20 0:5 0:5 0:3 ÿ0:4 ÿ0:30 ÿ0:2 0:0 5 b1 b2 b3 b4 b5 b6 b7 4 0:2 0:5 0:0 0:1 0:0 0:0 ÿ0:05 0:0 0:0 Platform points (in platform frame): p1
p2
p3
p4
p5
p6
2
0:3 p7 4 0:0 0:1
0:3 0:2 0:0
0:0 0:3 0:0
ÿ0:2 ÿ0:15 0:1 ÿ0:20 ÿ0:1 ÿ0:05
0:15 ÿ0:15 ÿ0:05
3 0:0 ÿ0:2 5 0:0
Leg-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 platform and all seven for the redundant one. The lower and upper limits for the region for scanning 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-
738
B. Dasgupta and T. S. Mruthyunjaya
Fig. 6. The Stewart platform.
Fig. 7. Workspace of Stewart platform.
Force redundancy in parallel manipulators
739
Fig. 8. Workspace and singularities of Stewart platform.
tation selected is Y yx
yy
yz T 0:3
ÿ 0:2
0:2T
Plots of boundaries of workspace sections at 12 dierent values of the z-coordinate at intervals of 0.2 m are shown in Fig. 7. Superposition of plots of singularities and boundaries of workspace sections 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. In fact, they are quartic curves, because the singularity surface of the Stewart platform for a given orientation of the platform is a quartic surface as can be shown with some elementary column transformations on the matrix H. The obstruction and partitioning of the workspace due to singularities here are found to be more severe than the previous example. Plots in Fig. 9 show a drastic reduction of singularity (shown by crosses in this ®gure) by a single degree of redundancy which virtually frees the entire workspace for eective use, of course at the cost of reducing the total 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 as an illustration. Singularities of the non-redundant manipulator partition this section of the workspace into six segments, as seen in Fig. 10(a). From any one of them, the adjacent two segments and the opposite one are de®nitely unreachable through a singularity-free path. To the other two segments, a path, if possible, might require circuitous detours on the way in the zdirection and/or in the orientation variables. By the force redundancy, all these singularities van-
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Fig. 9. Workspace and singularities of redundant Stewart platform.
ish except at a localized spot (that too inside the void of the workspace) and the complete section of the workspace is connected, as seen in Fig. 10(b). The slight loss in the area of the section is insigni®cant in comparison. 5. CONCLUSIONS
The concept of force redundancy has been analysed in the context of parallel manipulators as the series-parallel dual concept of kinematic redundancy of serial manipulators. It has been shown that force redundancy can be used to reduce or even eliminate force singularities from parallel manipulator, while kinematic singularity can be at most redistributed and avoided by kinematic redundancy in serial manipulators. Thus, force singularities can be of eective use in tackling the problem of singularities and to increase the stiness of parallel manipulators, though costing some overhead and restricting the workspace to some extent. Numerical studies on two parallel manipulators show that addition of a single degree of redundancy can reduce singularities drastically and improve the quality of the workspace to a great extent, in comparison to which the slight reduction in the workspace volume is insigni®cant. The scope of future works includes development of strategies for redundancy resolution for various applications. Minimization of actuator forces and satisfaction of actuator constraints are possibly the most apparent objectives that can be met with the force redundancy. In addition, kinematic synthesis of redundant parallel manipulators also will be of interest so far as optimal placement of redundant limb(s) is concerned. The optimization of the non-redundant
Force redundancy in parallel manipulators
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Fig. 10. Sample section: workspace and singularities.
manipulator geometry also should be considered to examine the relative advantages and costs of incorporation of redundancy. REFERENCES 1. Nakamura, Y., Advanced Robotics: Redundancy and Optimization, Addison±Welsey, Reading, MA, 1991.
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2. Whitney, D. E., Transactions of the ASME Journal of Dynamic Systems, Measurement and Control, 1972, 94, 303± 309. 3. Liegeois, A., IEEE Transaction of Systems, Man and Cybernetics, 1977, 7(12), 842±868. 4. Maciejewski, A. A. and Klein, C. A., International Journal of Robotics Research, 1985, 4(3), 109±117. 5. Klein, C. A. and Blaho, B. E., International Journal of Robotics Research, 1987, 6(2), 72±83. 6. Hollerbach, J. M. and Suh, K. C., IEEE Journal of Robotics and Automation, 1987, 3, 308±316. 7. Nakamura, Y. and Hanafusa, H., International Journal of Robotics Research, 1987, 6(1), 32±42. 8. Baker, D. R. and Wampler, C. W. II, International Journal of Robotics Research, 1988, 7(2), 3±21. 9. Martin, D. P., Baillieul, J. and Hollerbach, J. M., IEEE Transactions on Robotics and Automation, 1989, 5(4), 529± 533. 10. Podhorodeski, R. P., Goldenberg, A. A. and Fenton, R. G., Transactions of the ASME Journal of Mechanical Design, 1993, 115, 530±539. 11. Lewis, C. L. and Maciejewsky, A. A., Computers and Electrical Engineering, 1994, 20(3), 273±288. 12. Wang, J. T., Transactions of the ASME Journal of Mechanical Design, 1995, 117, 454±459. 13. Kircanski, M., International Journal of Robotics Research, 1995, 14(4), 382±398. 14. Roberts, R. G. and Maciejewski, A. A., IEEE Transactions on Robotics and Automation, 1996, 12(4), 543±552. 15. Nakamura, Y. and Ghodoussi, M., IEEE Transactions on Robotics and Automation, 1989, 5(3), 294±302. 16. Stewart, D., Proceedings of the Institute of Mechanical Engineers, Part I, 1965±1966, 180(15), 371±386. 17. Hunt, K. H., in Kinematic Geometry of Mechanisms. Clarendon Press, Oxford, 1978. 18. Hunt, K. H., Transactions of the ASME Journal of Mechanisms Transmission and Automation in Design, 1983, 105, 705±712. 19. Fichter, E. F., International Journal of Robotics Research, 1986, 5(2), 157±182. 20. Waldron, K. J. and Hunt, K. H., International Journal of Robotics Research, 1991, 10(5), 473±480. 21. Ball, R. S., A Treatise on the Theory of Screws, Cambridge University Press, Cambridge, MA, 1900. 22. Merlet, J.-P., International Journal of Robotics Research, 1989, 8(5), 45±56. 23. Gosselin, C. and Angeles, J., IEEE Transactions on Robotics and Automation, 1990, 6(3), 281±290. 24. Sefrioui, J. and Gosselin, C. M., Robotics and Autonomous Systems, 1992, 10, 209±224. 25. Gosselin, C. M. and Sefrioui, J., Robotics, Spatial Motion and Mechanical Systems, DEV 45, 1992, ASME, 329±335. 26. Liu, K., Lewis, F., Lebret, G. and Taylor, D., Journal of Intelligent Robotic Systems, 1993, 8, 287±308. 27. Sefrioui, J. and Gosselin, C. M., Mechanism and Machine Theory, 1995, 30(4), 533±551. 28. Mohammadi Daniali, H. R., Zsombor-Murray, P. J. and Angeles, J., Mechanism and Machine Theory, 1995, 30(5), 665±678. 29. Collins, C. L. and Long, G. L., IEEE Transactions on Robotics and Automation, 1995, 11(5), 661±669. 30. Takeda, Y. and Funabashi, H., JSME International Journal Series C, 1996, 39(1), 85±93. 31. Cheng, H. H., Transactions of the ASME Journal of Dynamic Systems, Measurement and Control, 1994, 116, 687± 701. 32. Mingyang, Z., Tomg, G., Ge, C., Qunming, L. and Dalong, T., Proceedings of the IEEE International Conference on Robotics and Automation, 1995, 221±226. 33. Zanganeh, K. E. and Angeles, J., Proceedings of the IEEE International Conference on Robotics and Automation, 1994, 3043±3048. 34. Zanganeh, K. E. and Angeles, J., Proceedings of the IEEE International Conference on Robotics and Automation, 1994, 3049±3054. 35. Lee, S. and Kim, S., Proceedings of the IEEE International Conference on Robotics and Automation, 1994, 77±82. 36. Collins, C. L. and Long, G. L., Proceedings of the IEEE International Conference on Robotics and Automation, 1995, 526±531. 37. Dasgupta, B. and Mruthyunjaya, T. S., Mechanism and Machine Theory, 1998, 33(6), 711±725.