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General method of structural synthesis of parallel mechanisms
archives of civil and mechanical engineering 16 (2016) 256–268
Available online at www.sciencedirect.com
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Original Research Article
General method of structural synthesis of parallel mechanisms J. Bałchanowski * Department of Biomedical Engineering, Mechatronics and Theory of Mechanisms, Faculty of Mechanical Engineering, Wrocław University of Technology, ul. Łukasiewicza 7/9, 50-371 Wrocław, Poland
article info
abstract
Article history:
This paper presents a method of the structural synthesis of spatial or planar parallel
Received 28 July 2015
mechanisms. The method exploits the combined strengths of the Baranov method and
Accepted 23 November 2015
the intermediate chain method. It makes possible to create complete sets of parallel
Available online
mechanism solutions. Relations for the number and structure of branches are provided. The successive steps in the procedure for generating a structural form of a closed branch, the
Keywords:
opening of the branch, the way of constructing a parallel mechanism from branches with
Structural synthesis
negative and zero degree of freedom (DoF) and the connection of the drives separated from
Intermediate chain method
the branches are described.
Parallel mechanisms
The proposed method of structural synthesis was applied to generate a complete set of branch solutions, written using numerals representing the number of links in a branch and the number of kinematic pairs of proper class, and to demonstrate the possibilities of creating mechanisms with a different number of branches. The end result was a complete set of possible spatial parallel mechanism solutions for the required mobility. # 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
1.
Introduction
Parallel mechanisms with a multibranch (also called multileg or multilimb) connection between the platform (driven link) and the base are most often used in manufacturing machines [8,26,32,39]. The working tool (a milling cutter, an abrasive disc, etc.) is equipped with its own drive and is usually mounted on a platform. The guidance of the tool (the platform) by a parallel mechanism increases system stiffness whereby considerable working loads can be carried, high tool position repeatability is ensured and the system overall
dimensions and mass can be reduced. Parallel mechanisms are also widely used as positioning devices, motion simulators and assembly manipulators [4,26,27,32,39]. The drawbacks of parallel mechanisms are their relatively small work zone and the possible occurrence of singular positions [10,11,14,26,32]. The design of parallel mechanisms' structures is generally based on the designer's experience, his/her intuition, accidental ideas or the adaptation of well-known and well-tried solutions. As a result, the same structures tend to be repeated, with the particular designs differing in mainly their mobility – degree of freedom (DoF), geometry (the mutual location of the
* Tel.: +48 72 320 2710. E-mail address: [email protected] http://dx.doi.org/10.1016/j.acme.2015.11.002 1644-9665/# 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
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kinematic pairs, the shape of the members, etc.) and the type of drives used [18,26,31,32]. The aim of this research was to develop a general method of the structural synthesis of spatial parallel mechanisms and apply it to create a complete set of feasible and technically viable structural designs of such mechanisms.
2. Method of structural synthesis of parallel mechanisms Parallel mechanisms are characterized by a multibranch connection between the platform and the base or between the moving links and the base [5,8,26,32,36,39]. If all the branches are identical, the mechanisms are referred to as symmetric, otherwise as asymmetric. Many works, based on different approaches, are devoted to the research on the structure of parallel mechanisms. Early research on parallel mechanisms concentrated on synthesis of the six degrees of freedom Stewart-Gough-type platform with six branches. In the last 25 years the type synthesis of parallel mechanisms with fewer than six degrees of freedom have attracted the researchers' attention [18,32]. In 2002 Merlet in [33] has stated: ‘‘Synthesis of parallel robot is an open field . . . and, in my opinion, one of the main issues for the development of parallel robots in practice’’ indicating the direction of further research on designing parallel mechanisms. Kong in 2003 in [28] and Gogu in 2008 in the book [18] made a review of the applied methods of the parallel mechanisms' structural synthesis. The approaches to the synthesis can be divided into couple main groups: the approach based on the screw theory, the approach based on the displacement group theory, the approaches based on kinematics and mobility criterion. Gogu added to the existing synthesis methods an approach based on the theory of linear transformations. The general approach to structural synthesis of parallel mechanisms, generating a specified motion pattern, based on the screw theory were proposed by Kong in [28] and by Gosselin in [29]. Equally numerous works on the structure of parallel systems deal with methods based on the screw algebra, e.g. papers by Hunt [24], Tsai [16], Carricato [13]. Methods based on the displacement group theory applied to the synthesis of parallel mechanisms are discussed by
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Herve [21–23], Li et al. [30], Karouia [25]. Generally, these methods are the most appropriate for the structural synthesis of parallel mechanisms with a predetermined motion pattern. Methods based on the theory of linear transformations and evolutionary morphology presented by Gogu [17,18] enable obtaining, in a systematic way, the structural solutions of the parallel mechanism with two to six degrees of freedom. Structural synthesis methods are often based on analysis of the degrees of freedom (mobility criterion) of the whole mechanism or its particular branches and have been discussed in the last 15 years inter alia by Bałchanowski [3–5,9], Romaniak [36,37], Alizade [1] and Rasim [38]. Methods of synthesis based on the graph theory are described among others by Ding in [15]. A complex method based on the Baranov method [12] and the intermediate chain method [34,35] for the structural synthesis of parallel mechanisms is developed in this paper. The basics of this method have been developed by the author and presented in [6] and [7]. An important problem not addressed by the latter two methods is the determination of the number of branches connecting the platform with the drivers and the base. Exploiting the strengths of the two methods the new method of the structural synthesis of parallel mechanisms provides an answer to the question concerning the number of branches and their structure and thereby makes the structural synthesis of parallel mechanisms possible. The proposed method is more universal and remedies the deficiencies and inconveniences of the previous methods.
2.1.
Determination of number of branches
The number of branches is interrelated not only with mobility (DoF) Mt of the mechanism being designed, but also with the type and number nc of drivers fixed in the base, number f1c of actuated pairs and number ns of variable-length links (linear actuators) which may occur in the intermediate chain connecting the platform with the base and even with a driver. Using the intermediate chain method [34,35] one can find that each mechanism consists of base 0, platform p with DoF Mp, a driver or drivers nc with DoF Mc and chain U of intermediate links (Fig. 1) with DoF MU. Under this assumption the mobility of a mechanism can be expressed as follows: Mt ¼ Mc þ Mp þ MU
(1)
Fig. 1 – Assumptions for intermediate chain method – division of mechanism into platform p, base 0 and drivers-at-base nc: (a) empty chain U, (b) chain comprising actuated pairs f1c and variable-length link ns.
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Fig. 2 – Division of chain U* into branches: (a) empty chain U* and drives separated from it, (b) chain U* divided into ngc branches.
method, one can notice that branches can be characterized by zero DoF (Mgj = 0) provided that a actuated pair is in the branch, a variable-length link is added to the branch, or the branch itself is a variable-length link. In formulas (4) and (5) there are ng branches with DoF not equal to zero. The number of branches with zero DoF (ng0) in the mechanism is determined by the number of actuated pairs ( p1c) and the number of variable-length links (ns) excluded from chain U (Fig. 2b). A static analysis of a zero branch shows that such a branch will never carry any external load if it does not contain a actuated pair or a variable-length link in its structure [5–7]. Exploiting the above observation, the number of zero branches (ng0) can be expressed by the relation:
According to the assumptions, unknown chain U can be empty (Fig. 1a) or contain a specific number f1c of actuated pairs (always class I translational or rotational pairs – in the article it is assumed that the pair's class number is described using Roman numerals and indicates the number of degrees of connectivity (freedom) of the pair) and a specific number nS of variable-length links s (Fig. 1b) being drivers. If variablelength links occur in chain U, the size of the task of searching for an intermediate chain increases whereby the task becomes more labour-intensive. In order simplify the task, variable-length links s and actuated pairs will be excluded from chain U (Fig. 2a). Variable-length links s (Fig. 2a) are characterized by degree of freedom Ms = 0 which can be decomposed into real mobility Mrs = 1 and local mobility MrL = 1. Variable-length links always make the system more rigid. Movement can take place only as a result of a change in the length of the links. Taking into account the number of variable-length links in chain U (Fig. 2), the DoF of empty chain U* (Fig. 2a), taking into consideration (1), can be expressed as follows:
Considering the number of nonzero branches (ng), conditions (4) and (5) and the allowable number ng0 of zero branches (6), the total number of branches (ngc) in the parallel mechanism must satisfy the following condition:
MU ¼ Mt þ nS Mc Mp
2ngc ng þ f 1c þ nS
(2)
The theoretical mobility Mt of a rational mechanism must satisfy [6] the following relation: Mt ¼ nc þ f 1c
(3)
In order to design parallel mechanisms with required mobility Mt (3) chain U* must be decomposed into branches gj (Fig. 2b), the DoF of which must satisfy the following trivial condition: (a) for symmetric branches: MU ¼ ng Mgj
(4)
(b) for asymmetric branches: MU ¼
ng X Mgj
ng0 ¼ f 1c þ nS
(7)
Relation (7) defines the range of change in the total number of branches (ngc) occurring in designed rational parallel mechanisms in which drivers forming class I pairs with the base, and actuated pairs and variable-length links included in the chain or in the branches connecting the platform with the drivers or the base can serve as a drive.
2.2.
Determination of the branches topology
Knowing number ngc of branches (7) and their mobility (DoF) Mgj (4) and (5) one can begin to determine the structure of the branches themselves using the Artobolevski–Dobrovolski equation [2,6,18,34,35]:
(5)
j¼1
(6)
Mgj ¼ 6kj
5 X ð6iÞf i
(8)
i¼1
where Mgj – the DoF of the jth branch, ng – the number of branches in a parallel mechanism with DoF Mgj 6¼ 0. Relations (4) and (5) are used to calculate the number (ng) of non-zero branches. Considering the properties of the Baranov
where kj – the number of links in the jth branch, i – the degree of connectivity (freedom) the class of a kinematic pair, fi – the number of pairs with the degree of connectivity i.
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Fig. 3 – Illustration of branch closure.
By transforming relation (8) one gets a formula describing an open branch structure: 6kj Mgj ¼
5 X ð6iÞf i
(9)
i¼1
By solving relation (9) one acquires only initial information about the structure of an open branch, concerning the number of links and the number and class of pairs or half-pairs (nodes) included in the branch. This information is too scant to
explicitly define the form of the branch. In order to obtain more information the open branch (Fig. 3a) was closed by introducing an additional link z into the branch (Fig. 3b) [6,7,9,34]. For the closed branch the following conditions must be satisfied [34,35]: kj þ 1 ¼ n2 þ n3 þ þ nw X 2 f i ¼ 2n2 þ 3n3 þ þ wnw
(10)
where nw – the number of links with w nodes (half-pairs).
Fig. 4 – Illustration of connection of branches to platform p and the base 0: (a) links with negative DoF, (b) links with zero DoF, (c) variable-length links, and (d) interconnections between links.
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Table 1 – Branch structures described with numbers. 2
Mgj kj f1 f2 f3
1 1 0 1
2 2 1 0
2 1 0 3
3 4 0 0
2 1 2
1 1 3 1
4 3 2 1
2 4 0
4 0 2
2 2 0 1
1 2 0
3 3 1 0
Table 2 – Nodality of branch links. kj + 1
2
P fi n2 n3
2 2 0
3
4
3 3 0
4 1 2
4 4 0
5 5 2 2
5 5 0
6 3 2
Table 3 – Number of links and pairs in branch. kj
1
2
2
P fi
2
3
4
3 4
4 5
5
6
ðlÞ
2 0 3
1 2 2
0 4 5 0 0
3 1 2
2 3 1
1 5 0
2 1 1 1
3 3 0 1
2 2 0
1 1 3
4 4 1 0
3 0 3
1 4 1
2 2 2
Relation (10) supplies more information about the structure of closed branches, concerning the nodality of the links in the branch and their number. Then connections between the links with specific nodality w need to be determined. The properties of connection matrix A are used for this purpose. For each closed structure, matrix A, defining the connections between the links, assumes the following form [34]: 2 3 a12 . . . a1kj þ1 a11 6 a21 a22 . . . a2kjþ1 7 7 (11) A¼6 4 ... akl ... ... 5 akjþ1 1 akjþ1 2 . . . akjþ1 kjþ1
Table 4 – Adjacency matrices for closed branches (Nw – name of mechanism link, where: w – nodality of link, l – denotes the number of consecutive links with nodality w).
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Fig. 5 – Graphic forms of closed branches forming parallel mechanisms.
Each row and column of the connection matrix are assigned to a specific link and directly indicate the number of connections which the link forms with the other links in the system. Matrix elements akl describing the number of connections (kinematic pairs) between links k and l must satisfy the following conditions [34]: akl ¼ alk akl ¼ 0 for kj þ1 X akl ¼ w
k¼l
characterized by negative mobility (Mgj < 0), it can be connected in any way (e.g. to a platform p and a driver or the base 0) (Fig. 4a). In the case of a zero branch (Mgj = 0), prior to connection an actuated pair or a variable-length link s must
(12)
k¼1
By solving the equations of conditions (10) and (11) one can explicitly determine the form of a closed branch or an intermediated chain.
2.3.
Creation of basic diagrams of mechanism
Having a numerical description of the structure of branches one can present them in a graphic form in order to check the DoF of the contours [19,20] and to determine closing link z (Fig. 3b). By opening a branch one obtains free half-pairs needed to connect the branch to a platform and the base or to a platform, the base and a driver. When connecting branches to the initial links (a platform, the base and drivers) one should pay attention to the mobility of the branches. If a branch is
Fig. 6 – Illustration of effect of kinematic pair classes on branch mobility.
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be added to the branch and then the latter must be connected in the same way as a branch with negative DoF (Fig. 4b). Also the connection of a zero branch in the form of a variable-length link is arbitrary (Fig. 4c). Moreover, if the link closing the branch is three- or more nodal, the branches can be connected together (Fig. 4d). The correct connection of all the branches and then the incorporation of the other actuated pairs and variable-length
links into the branches ends the structural synthesis stage which results in a set of all theoretically possible solutions [6,7,9].
3.
Structural synthesis of parallel mechanisms
The structural synthesis stage is one of the first major stages in the process of design. The adopted assumptions are crucial
Fig. 7 – Open branches with mobility Mgj = S1.
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Table 5 – Possible numbers of branches used to construct mechanism with Mt = 3. ngc
6
5
Mgj = 2 Mgj = 1 Mgj = 0
0 3 3
1 1 3
4 1 1 2
3 0 3 1
1 1 1
2 0 3 0
1 1 0
since too complex or technically unfeasible solutions, containing a large number of links and pairs of higher class IV and V (which can be the source of undesirable local mobilities), must be eliminated right at the beginning. Considering the above, it
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was assumed that such pairs cannot occur in a branch and the number of class III pairs was limited to three. This comes down to the following conditions: f 4 ¼ f 5 ¼ 0; f 3 3:
(13)
In order to ensure the possibility of mounting a (rotational or translational) drive in each of the branches (this requires at least one pair of class I in the branch) the following inequality must be satisfied: f 1 > 0:
Fig. 8 – Basic structures of symmetric mechanisms with mobility Mt = 3 (part 1).
(14)
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Moreover, it was assumed that the number of links in the jth branch kj (without taking into account variable-length links) should be within the interval: 1kj 4:
(15)
The above assumptions (13)–(16) were made only for the needs of this paper in order to limit the number of solutions and they do not stem from the theoretical basis of the method. If the branches satisfy relation (16), it is possible to create parallel mechanisms with mobility in the range:
A parallel mechanism the number of branches (ngc) of which is within the interval defined by relation (7) can be constructed from branches characterized by different DoF. In order to limit the number of solutions it was decided to use branches whose DoF satisfied the following condition:
2Mt 6
2Mgj 0
6kj Mgj ¼ 5f 1 þ 4f 2 þ 3f 3
(16)
(17)
By transforming relation (9) and taking into account (12) one gets the following formula describing the structure of the sought branch:
Fig. 9 – Basic structures of symmetric mechanisms with mobility Mt = 3 (part 2).
(18)
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There are three independent variables in relation (18). Since they belong to the set of natural numbers, including zero, the number of solutions is finite under the assumptions (13)–(16). The structures (denoted with numerals) obtained from relation (18) are shown in Table 1. By solving relation (10) for the data contained in Table 1 the results shown in Table 2, describing the nodality of the links in a branch, were obtained. Using the results shown in Tables 1–2, the numbers of branch links and the proper numbers of kinematic pairs were presented in Table 3.
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The numerical results contained in Tables 1–3 are presented in the form of connection matrix A in Table 4 and in the graphic form in Fig. 5. Among the solutions there are four simple single-contour solutions consisting of solely two-node links, and five two-contour solutions containing two- and three-node links.
3.1.
Basic structures of branches
On the basis of the matrix contained in Table 4 the graphic forms of the closed branches are presented in Fig. 5. The distinguished
Fig. 10 – Basic structures of symmetric mechanisms with mobility Mt = 3 (part 3).
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Fig. 11 – Basic structures of symmetric mechanisms with mobility Mt = 3 (part 4).
link is a closed link z which should be disconnected in order to prepare the branch (by opening it) for connection to a platform and the base. Fig. 5 shows all the forms of branches which enable the construction of symmetric and asymmetric parallel mechanisms with mobility (2 Mt 6). Having removed the closing link one obtains the graphic form of the branch together with symbols of the pairs, but without inserted classed of the pairs. Therefore after opening the branch one should substitute the determined classes of the pairs (Table 1) for the symbols of the pairs, whereby a branch with the required mobility is obtained. Fig. 6 shows an exemplary open branch with three links and P f j ¼ 4 , taken from Fig. 5, in which four pairs kj ¼ 3; appropriate classes of pairs were substituted (using Table 1) for the symbols of pairs. In this way branches characterized by different DoF were obtained. Doing the same with the other branches (Fig. 6) one obtains a set of all the branches needed to construct parallel mechanisms. Taking the above observations into account, the closed branches were opened (Fig. 5) and after arranging the kinematic pairs basic diagrams of branches for the particular DoF were obtained. An analysis of the solutions shows that the solutions for branches with DoF Mgj = 2 (64 solutions [9]) and for branches with DoF Mgj = 1 (57 solutions) are the most numerous, whereas the set of branches with Mgj = 0 comprises 30 solutions [9]. Moreover, most of the solutions are solutions obtained using two-contour closed branches. The selected set of structural diagrams of the branches with DoF Mgj = 1 is shown in Fig. 7.
3.2.
Considering the possibility of constructing mechanisms with mobility Mt = 3, with a different possible number of branch combinations (Table 5) and branch forms (e.g. there are 57 branches with Mgj = 1, Fig. 7), it becomes apparent that the number of solutions is very large. Exemplary solutions of symmetric parallel mechanisms without local mobility (Mt = 3), each consisting of three branches (ngc = 3) with mobility Mgj = 1 (Table 5), are presented in Figs. 8–11 (114 solutions). Using the proposed method of the structural synthesis of parallel mechanisms one can create a very numerous set of parallel mechanisms, containing all the solutions which satisfy the requirements set down in this paper. The set contains the solutions actually used in practice (e.g. DELTA, IRB340, FANUC M-1iA and Tsai 3-upu manipulators) [8,26,32,39], but the majority are unknown, but applicable solutions. Using the set of solutions one will be able to design an optimal (according to the adopted criteria) parallel mechanism with a chosen number of branches, which will satisfy the originality criterion (among other criteria).
3.3.
Kinematic schemes of parallel mechanisms
In the next step, using the basic structural schemes of parallel mechanisms one can begin to construct kinematic diagrams.
Basic diagrams of parallel mechanisms
In the next step of the structural synthesis of parallel mechanisms all the open branches are connected, in all the possible ways, to the platform p and the base 0 as shown in Fig. 4. This labour-intensive task leads to the creation of a complete set of parallel mechanism structures. Limiting the analysis to parallel mechanisms with real mobility M = Mt = 3 (without local mobility), the possible numbers of branches used to construct both symmetric and asymmetric basic diagrams of the mechanisms are presented in Table 5.
Fig. 12 – Used physical forms of kinematic pairs.
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Fig. 13 – Kinematic schemes of selected parallel mechanism.
The procedure for creating kinematic schemes consists in substituting the physical forms of kinematic pairs for their symbols. From among the many theoretical forms of lower class I, II and III pairs only the forms schematically shown in Fig. 12 were used in this study. As regards pairs of class I, only rotational and translational pairs were permitted to be used. Cylindrical pairs and universal (Cardan) joints were permitted for class II pairs while spherical joints were permitted for class III pairs. By substituting the physical forms of kinematic pairs (Fig. 12) for their symbols in the mechanisms shown in Figs. 8–11 one can create a complete set of kinematic diagrams of the considered parallel mechanisms. Since the set of solutions is very numerous it was decided to illustrate the procedure for structure no. 6 selected from Fig. 8, for which the created kinematic diagrams are collected in Fig. 13. The obtained 14 kinematic diagrams of symmetric parallel mechanisms with mobility (M = Mt = 3) are the result of the application of the proposed method of structural synthesis to only one basic parallel mechanism diagram (Fig. 8). In order to obtain a complete set the above actions should be repeated for each of the structural mechanism solutions contained in Figs. 8–11.
4.
Conclusion
The developed method is based on the properties of the Baranov method and those of the intermediate chain method. By
exploiting the strengths of the latter two methods a new method, which enables one to create complete sets of symmetric and asymmetric parallel mechanisms with the number of branches described by relation (6), with mobility in the range 2 Mt 6, has been developed. The advantage of this method is its ability to create all the theoretically possible solutions with separate branches or with interconnected branches. Already the initial results obtained using this method show that the generated set of solutions comprises the existing atypical solutions arrived at through reasoning and intuition. Moreover, the method can be easily extended, by changing branch mobility Mgj in formula (8) to cover branches with local mobility. As shown in the work the received collection of the new structures of parallel mechanisms, even when placed in the work limits (13–15), are quite numerous. The large number of solutions makes it difficult to select the proper mechanism for the intended use. Therefore, in the next research stage it is necessary to develop the criteria for the selection of the useful or optimal structures. This way the number of the selected mechanisms will be reduced to the most useful for the technical applications. The proposed method of the structural synthesis of spatial parallel mechanisms, could also be easily changed, into a form suitable for planar mechanisms. The presented results of the structural synthesis should contribute to the better understanding and wider use of this interesting group of mechanisms.
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[20] A. Gronowicz, W. Twaróg, Mobility as a criterion in selection of the kinematic systems in structural synthesis, Systems: Journal of Transdisciplinary Systems Science 8 (2003) 153–160. [21] J.M. Hervé, Design of parallel manipulators via the displacement group, in: Proc. of the 9th World Congress on the Theory of Machines and Mechanisms, Milano, Italy, (1995) 2079–2082. [22] J. Hervé, The Lie group of rigid body displacements, a fundamental tool for mechanism design, Mechanism and Machine Theory 34 (1999) 719–730. [23] J. Hervé, New translational parallel manipulators with extensible parallelogram, in: Proc of the 11th World Congress in Mechanism and Machine Science, vol. 4, China Machine Press, 2004 1599–1603. [24] K.H. Hunt, Structural kinematics of in-parallel-actuated robot arms, Transactions ASME Journal of Mechanical Design 105 (1983) 705–712. [25] M. Karouia, J. Hervé, New parallel wrists: special limbs with motion dependency, in: Lenarčič, Galletti (Eds.), On Advances in Robot Kinematics, Kluwer Academic Publishers, 2004 371–380. [26] J. Knapczyk, A. Morecki, The Basics of Robotics, WNT, Warszawa, 1999 (in Polish). [27] J. Knapczyk, G. Tora, Motion planning of a novel 2-dof parallel manipulator applied as driving simulator of the wheel loader, International Journal of Applied Mechanics and Engineering 19 (3) (2014) 599–608. [28] X. Kong, Type synthesis and kinematics of general and analytic parallel mechanisms, (Ph.D. thesis), Faculte des Sciences et de Genie, Universite Laval, Quebec, 2003. [29] X. Kong, C.M. Gosselin, Type synthesis of 3-dof translational parallel manipulators based on screw theory, Transactions ASME Journal of Mechanical Design 126 (2004) 83–92. [30] Q. Li, Z. Huang, J. Hervé, Type synthesis of 3R2T 5-DOF parallel mechanisms using the Lie group of displacements, IEEE Transactions of Robotics and Automation 20 (2) (2004) 173–180. [31] E.-Ch. Lovasz, S. Grigorescu, D. Mărgineanu, C. Gruescu, C. Pop, V. Ciupe, I. Maniu, Geared linkages with linear actuation used as kinematic chains of a planar parallel manipulator, mechanisms, transmissions and applications, Mechanisms and Machine Science 31 (2015) 21–31. [32] J.-P. Merlet, Parallel Manipulators, Kluwer Academic Publishers, London, 2000. [33] J.-P. Merlet, Still a long way to go on the road for parallel mechanisms, in: Keynote speech in ASME 2002 DETC Conference, Montreal, Canada, 2002 http://www-sop.inria.fr/ coprin/equipe/merlet/asme/asme2002.html. [34] S. Miller, Kinematic Systems. Basics of Design, PWN, Warszawa, 1988 (in Polish). [35] S. Miller, A. Gronowicz, E. Adamczyk, Some remarks on finding a solution for the structure when designing a mechanism, Mechanism and Machine Theory 16 (6) (1981) 645–651. [36] K. Romaniak, Structural synthesis of parallel mechanisms (in Polish), Modelowanie Inżynierskie 11 (42) (2011) 359–367. [37] K. Romaniak, Generalized methods of kinematic chains structural synthesis, International Journal of Applied Mechanics and Engineering 15 (3) (2010) 821–829. [38] A. Rasim, C. Bayram, E. Gezgin, Structural synthesis of serial platform manipulators, Mechanism and Machine Theory 42 (2007) 580–599. [39] L.-W. Tsai, Robot Analysis, the Mechanics and Parallel Manipulators, John Wiley & Sons Inc., New York, 1999.
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Original Research Article
General method of structural synthesis of parallel mechanisms J. Bałchanowski * Department of Biomedical Engineering, Mechatronics and Theory of Mechanisms, Faculty of Mechanical Engineering, Wrocław University of Technology, ul. Łukasiewicza 7/9, 50-371 Wrocław, Poland
article info
abstract
Article history:
This paper presents a method of the structural synthesis of spatial or planar parallel
Received 28 July 2015
mechanisms. The method exploits the combined strengths of the Baranov method and
Accepted 23 November 2015
the intermediate chain method. It makes possible to create complete sets of parallel
Available online
mechanism solutions. Relations for the number and structure of branches are provided. The successive steps in the procedure for generating a structural form of a closed branch, the
Keywords:
opening of the branch, the way of constructing a parallel mechanism from branches with
Structural synthesis
negative and zero degree of freedom (DoF) and the connection of the drives separated from
Intermediate chain method
the branches are described.
Parallel mechanisms
The proposed method of structural synthesis was applied to generate a complete set of branch solutions, written using numerals representing the number of links in a branch and the number of kinematic pairs of proper class, and to demonstrate the possibilities of creating mechanisms with a different number of branches. The end result was a complete set of possible spatial parallel mechanism solutions for the required mobility. # 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
1.
Introduction
Parallel mechanisms with a multibranch (also called multileg or multilimb) connection between the platform (driven link) and the base are most often used in manufacturing machines [8,26,32,39]. The working tool (a milling cutter, an abrasive disc, etc.) is equipped with its own drive and is usually mounted on a platform. The guidance of the tool (the platform) by a parallel mechanism increases system stiffness whereby considerable working loads can be carried, high tool position repeatability is ensured and the system overall
dimensions and mass can be reduced. Parallel mechanisms are also widely used as positioning devices, motion simulators and assembly manipulators [4,26,27,32,39]. The drawbacks of parallel mechanisms are their relatively small work zone and the possible occurrence of singular positions [10,11,14,26,32]. The design of parallel mechanisms' structures is generally based on the designer's experience, his/her intuition, accidental ideas or the adaptation of well-known and well-tried solutions. As a result, the same structures tend to be repeated, with the particular designs differing in mainly their mobility – degree of freedom (DoF), geometry (the mutual location of the
* Tel.: +48 72 320 2710. E-mail address: [email protected] http://dx.doi.org/10.1016/j.acme.2015.11.002 1644-9665/# 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
archives of civil and mechanical engineering 16 (2016) 256–268
kinematic pairs, the shape of the members, etc.) and the type of drives used [18,26,31,32]. The aim of this research was to develop a general method of the structural synthesis of spatial parallel mechanisms and apply it to create a complete set of feasible and technically viable structural designs of such mechanisms.
2. Method of structural synthesis of parallel mechanisms Parallel mechanisms are characterized by a multibranch connection between the platform and the base or between the moving links and the base [5,8,26,32,36,39]. If all the branches are identical, the mechanisms are referred to as symmetric, otherwise as asymmetric. Many works, based on different approaches, are devoted to the research on the structure of parallel mechanisms. Early research on parallel mechanisms concentrated on synthesis of the six degrees of freedom Stewart-Gough-type platform with six branches. In the last 25 years the type synthesis of parallel mechanisms with fewer than six degrees of freedom have attracted the researchers' attention [18,32]. In 2002 Merlet in [33] has stated: ‘‘Synthesis of parallel robot is an open field . . . and, in my opinion, one of the main issues for the development of parallel robots in practice’’ indicating the direction of further research on designing parallel mechanisms. Kong in 2003 in [28] and Gogu in 2008 in the book [18] made a review of the applied methods of the parallel mechanisms' structural synthesis. The approaches to the synthesis can be divided into couple main groups: the approach based on the screw theory, the approach based on the displacement group theory, the approaches based on kinematics and mobility criterion. Gogu added to the existing synthesis methods an approach based on the theory of linear transformations. The general approach to structural synthesis of parallel mechanisms, generating a specified motion pattern, based on the screw theory were proposed by Kong in [28] and by Gosselin in [29]. Equally numerous works on the structure of parallel systems deal with methods based on the screw algebra, e.g. papers by Hunt [24], Tsai [16], Carricato [13]. Methods based on the displacement group theory applied to the synthesis of parallel mechanisms are discussed by
257
Herve [21–23], Li et al. [30], Karouia [25]. Generally, these methods are the most appropriate for the structural synthesis of parallel mechanisms with a predetermined motion pattern. Methods based on the theory of linear transformations and evolutionary morphology presented by Gogu [17,18] enable obtaining, in a systematic way, the structural solutions of the parallel mechanism with two to six degrees of freedom. Structural synthesis methods are often based on analysis of the degrees of freedom (mobility criterion) of the whole mechanism or its particular branches and have been discussed in the last 15 years inter alia by Bałchanowski [3–5,9], Romaniak [36,37], Alizade [1] and Rasim [38]. Methods of synthesis based on the graph theory are described among others by Ding in [15]. A complex method based on the Baranov method [12] and the intermediate chain method [34,35] for the structural synthesis of parallel mechanisms is developed in this paper. The basics of this method have been developed by the author and presented in [6] and [7]. An important problem not addressed by the latter two methods is the determination of the number of branches connecting the platform with the drivers and the base. Exploiting the strengths of the two methods the new method of the structural synthesis of parallel mechanisms provides an answer to the question concerning the number of branches and their structure and thereby makes the structural synthesis of parallel mechanisms possible. The proposed method is more universal and remedies the deficiencies and inconveniences of the previous methods.
2.1.
Determination of number of branches
The number of branches is interrelated not only with mobility (DoF) Mt of the mechanism being designed, but also with the type and number nc of drivers fixed in the base, number f1c of actuated pairs and number ns of variable-length links (linear actuators) which may occur in the intermediate chain connecting the platform with the base and even with a driver. Using the intermediate chain method [34,35] one can find that each mechanism consists of base 0, platform p with DoF Mp, a driver or drivers nc with DoF Mc and chain U of intermediate links (Fig. 1) with DoF MU. Under this assumption the mobility of a mechanism can be expressed as follows: Mt ¼ Mc þ Mp þ MU
(1)
Fig. 1 – Assumptions for intermediate chain method – division of mechanism into platform p, base 0 and drivers-at-base nc: (a) empty chain U, (b) chain comprising actuated pairs f1c and variable-length link ns.
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Fig. 2 – Division of chain U* into branches: (a) empty chain U* and drives separated from it, (b) chain U* divided into ngc branches.
method, one can notice that branches can be characterized by zero DoF (Mgj = 0) provided that a actuated pair is in the branch, a variable-length link is added to the branch, or the branch itself is a variable-length link. In formulas (4) and (5) there are ng branches with DoF not equal to zero. The number of branches with zero DoF (ng0) in the mechanism is determined by the number of actuated pairs ( p1c) and the number of variable-length links (ns) excluded from chain U (Fig. 2b). A static analysis of a zero branch shows that such a branch will never carry any external load if it does not contain a actuated pair or a variable-length link in its structure [5–7]. Exploiting the above observation, the number of zero branches (ng0) can be expressed by the relation:
According to the assumptions, unknown chain U can be empty (Fig. 1a) or contain a specific number f1c of actuated pairs (always class I translational or rotational pairs – in the article it is assumed that the pair's class number is described using Roman numerals and indicates the number of degrees of connectivity (freedom) of the pair) and a specific number nS of variable-length links s (Fig. 1b) being drivers. If variablelength links occur in chain U, the size of the task of searching for an intermediate chain increases whereby the task becomes more labour-intensive. In order simplify the task, variable-length links s and actuated pairs will be excluded from chain U (Fig. 2a). Variable-length links s (Fig. 2a) are characterized by degree of freedom Ms = 0 which can be decomposed into real mobility Mrs = 1 and local mobility MrL = 1. Variable-length links always make the system more rigid. Movement can take place only as a result of a change in the length of the links. Taking into account the number of variable-length links in chain U (Fig. 2), the DoF of empty chain U* (Fig. 2a), taking into consideration (1), can be expressed as follows:
Considering the number of nonzero branches (ng), conditions (4) and (5) and the allowable number ng0 of zero branches (6), the total number of branches (ngc) in the parallel mechanism must satisfy the following condition:
MU ¼ Mt þ nS Mc Mp
2ngc ng þ f 1c þ nS
(2)
The theoretical mobility Mt of a rational mechanism must satisfy [6] the following relation: Mt ¼ nc þ f 1c
(3)
In order to design parallel mechanisms with required mobility Mt (3) chain U* must be decomposed into branches gj (Fig. 2b), the DoF of which must satisfy the following trivial condition: (a) for symmetric branches: MU ¼ ng Mgj
(4)
(b) for asymmetric branches: MU ¼
ng X Mgj
ng0 ¼ f 1c þ nS
(7)
Relation (7) defines the range of change in the total number of branches (ngc) occurring in designed rational parallel mechanisms in which drivers forming class I pairs with the base, and actuated pairs and variable-length links included in the chain or in the branches connecting the platform with the drivers or the base can serve as a drive.
2.2.
Determination of the branches topology
Knowing number ngc of branches (7) and their mobility (DoF) Mgj (4) and (5) one can begin to determine the structure of the branches themselves using the Artobolevski–Dobrovolski equation [2,6,18,34,35]:
(5)
j¼1
(6)
Mgj ¼ 6kj
5 X ð6iÞf i
(8)
i¼1
where Mgj – the DoF of the jth branch, ng – the number of branches in a parallel mechanism with DoF Mgj 6¼ 0. Relations (4) and (5) are used to calculate the number (ng) of non-zero branches. Considering the properties of the Baranov
where kj – the number of links in the jth branch, i – the degree of connectivity (freedom) the class of a kinematic pair, fi – the number of pairs with the degree of connectivity i.
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Fig. 3 – Illustration of branch closure.
By transforming relation (8) one gets a formula describing an open branch structure: 6kj Mgj ¼
5 X ð6iÞf i
(9)
i¼1
By solving relation (9) one acquires only initial information about the structure of an open branch, concerning the number of links and the number and class of pairs or half-pairs (nodes) included in the branch. This information is too scant to
explicitly define the form of the branch. In order to obtain more information the open branch (Fig. 3a) was closed by introducing an additional link z into the branch (Fig. 3b) [6,7,9,34]. For the closed branch the following conditions must be satisfied [34,35]: kj þ 1 ¼ n2 þ n3 þ þ nw X 2 f i ¼ 2n2 þ 3n3 þ þ wnw
(10)
where nw – the number of links with w nodes (half-pairs).
Fig. 4 – Illustration of connection of branches to platform p and the base 0: (a) links with negative DoF, (b) links with zero DoF, (c) variable-length links, and (d) interconnections between links.
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Table 1 – Branch structures described with numbers. 2
Mgj kj f1 f2 f3
1 1 0 1
2 2 1 0
2 1 0 3
3 4 0 0
2 1 2
1 1 3 1
4 3 2 1
2 4 0
4 0 2
2 2 0 1
1 2 0
3 3 1 0
Table 2 – Nodality of branch links. kj + 1
2
P fi n2 n3
2 2 0
3
4
3 3 0
4 1 2
4 4 0
5 5 2 2
5 5 0
6 3 2
Table 3 – Number of links and pairs in branch. kj
1
2
2
P fi
2
3
4
3 4
4 5
5
6
ðlÞ
2 0 3
1 2 2
0 4 5 0 0
3 1 2
2 3 1
1 5 0
2 1 1 1
3 3 0 1
2 2 0
1 1 3
4 4 1 0
3 0 3
1 4 1
2 2 2
Relation (10) supplies more information about the structure of closed branches, concerning the nodality of the links in the branch and their number. Then connections between the links with specific nodality w need to be determined. The properties of connection matrix A are used for this purpose. For each closed structure, matrix A, defining the connections between the links, assumes the following form [34]: 2 3 a12 . . . a1kj þ1 a11 6 a21 a22 . . . a2kjþ1 7 7 (11) A¼6 4 ... akl ... ... 5 akjþ1 1 akjþ1 2 . . . akjþ1 kjþ1
Table 4 – Adjacency matrices for closed branches (Nw – name of mechanism link, where: w – nodality of link, l – denotes the number of consecutive links with nodality w).
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Fig. 5 – Graphic forms of closed branches forming parallel mechanisms.
Each row and column of the connection matrix are assigned to a specific link and directly indicate the number of connections which the link forms with the other links in the system. Matrix elements akl describing the number of connections (kinematic pairs) between links k and l must satisfy the following conditions [34]: akl ¼ alk akl ¼ 0 for kj þ1 X akl ¼ w
k¼l
characterized by negative mobility (Mgj < 0), it can be connected in any way (e.g. to a platform p and a driver or the base 0) (Fig. 4a). In the case of a zero branch (Mgj = 0), prior to connection an actuated pair or a variable-length link s must
(12)
k¼1
By solving the equations of conditions (10) and (11) one can explicitly determine the form of a closed branch or an intermediated chain.
2.3.
Creation of basic diagrams of mechanism
Having a numerical description of the structure of branches one can present them in a graphic form in order to check the DoF of the contours [19,20] and to determine closing link z (Fig. 3b). By opening a branch one obtains free half-pairs needed to connect the branch to a platform and the base or to a platform, the base and a driver. When connecting branches to the initial links (a platform, the base and drivers) one should pay attention to the mobility of the branches. If a branch is
Fig. 6 – Illustration of effect of kinematic pair classes on branch mobility.
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be added to the branch and then the latter must be connected in the same way as a branch with negative DoF (Fig. 4b). Also the connection of a zero branch in the form of a variable-length link is arbitrary (Fig. 4c). Moreover, if the link closing the branch is three- or more nodal, the branches can be connected together (Fig. 4d). The correct connection of all the branches and then the incorporation of the other actuated pairs and variable-length
links into the branches ends the structural synthesis stage which results in a set of all theoretically possible solutions [6,7,9].
3.
Structural synthesis of parallel mechanisms
The structural synthesis stage is one of the first major stages in the process of design. The adopted assumptions are crucial
Fig. 7 – Open branches with mobility Mgj = S1.
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Table 5 – Possible numbers of branches used to construct mechanism with Mt = 3. ngc
6
5
Mgj = 2 Mgj = 1 Mgj = 0
0 3 3
1 1 3
4 1 1 2
3 0 3 1
1 1 1
2 0 3 0
1 1 0
since too complex or technically unfeasible solutions, containing a large number of links and pairs of higher class IV and V (which can be the source of undesirable local mobilities), must be eliminated right at the beginning. Considering the above, it
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was assumed that such pairs cannot occur in a branch and the number of class III pairs was limited to three. This comes down to the following conditions: f 4 ¼ f 5 ¼ 0; f 3 3:
(13)
In order to ensure the possibility of mounting a (rotational or translational) drive in each of the branches (this requires at least one pair of class I in the branch) the following inequality must be satisfied: f 1 > 0:
Fig. 8 – Basic structures of symmetric mechanisms with mobility Mt = 3 (part 1).
(14)
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Moreover, it was assumed that the number of links in the jth branch kj (without taking into account variable-length links) should be within the interval: 1kj 4:
(15)
The above assumptions (13)–(16) were made only for the needs of this paper in order to limit the number of solutions and they do not stem from the theoretical basis of the method. If the branches satisfy relation (16), it is possible to create parallel mechanisms with mobility in the range:
A parallel mechanism the number of branches (ngc) of which is within the interval defined by relation (7) can be constructed from branches characterized by different DoF. In order to limit the number of solutions it was decided to use branches whose DoF satisfied the following condition:
2Mt 6
2Mgj 0
6kj Mgj ¼ 5f 1 þ 4f 2 þ 3f 3
(16)
(17)
By transforming relation (9) and taking into account (12) one gets the following formula describing the structure of the sought branch:
Fig. 9 – Basic structures of symmetric mechanisms with mobility Mt = 3 (part 2).
(18)
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There are three independent variables in relation (18). Since they belong to the set of natural numbers, including zero, the number of solutions is finite under the assumptions (13)–(16). The structures (denoted with numerals) obtained from relation (18) are shown in Table 1. By solving relation (10) for the data contained in Table 1 the results shown in Table 2, describing the nodality of the links in a branch, were obtained. Using the results shown in Tables 1–2, the numbers of branch links and the proper numbers of kinematic pairs were presented in Table 3.
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The numerical results contained in Tables 1–3 are presented in the form of connection matrix A in Table 4 and in the graphic form in Fig. 5. Among the solutions there are four simple single-contour solutions consisting of solely two-node links, and five two-contour solutions containing two- and three-node links.
3.1.
Basic structures of branches
On the basis of the matrix contained in Table 4 the graphic forms of the closed branches are presented in Fig. 5. The distinguished
Fig. 10 – Basic structures of symmetric mechanisms with mobility Mt = 3 (part 3).
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Fig. 11 – Basic structures of symmetric mechanisms with mobility Mt = 3 (part 4).
link is a closed link z which should be disconnected in order to prepare the branch (by opening it) for connection to a platform and the base. Fig. 5 shows all the forms of branches which enable the construction of symmetric and asymmetric parallel mechanisms with mobility (2 Mt 6). Having removed the closing link one obtains the graphic form of the branch together with symbols of the pairs, but without inserted classed of the pairs. Therefore after opening the branch one should substitute the determined classes of the pairs (Table 1) for the symbols of the pairs, whereby a branch with the required mobility is obtained. Fig. 6 shows an exemplary open branch with three links and P f j ¼ 4 , taken from Fig. 5, in which four pairs kj ¼ 3; appropriate classes of pairs were substituted (using Table 1) for the symbols of pairs. In this way branches characterized by different DoF were obtained. Doing the same with the other branches (Fig. 6) one obtains a set of all the branches needed to construct parallel mechanisms. Taking the above observations into account, the closed branches were opened (Fig. 5) and after arranging the kinematic pairs basic diagrams of branches for the particular DoF were obtained. An analysis of the solutions shows that the solutions for branches with DoF Mgj = 2 (64 solutions [9]) and for branches with DoF Mgj = 1 (57 solutions) are the most numerous, whereas the set of branches with Mgj = 0 comprises 30 solutions [9]. Moreover, most of the solutions are solutions obtained using two-contour closed branches. The selected set of structural diagrams of the branches with DoF Mgj = 1 is shown in Fig. 7.
3.2.
Considering the possibility of constructing mechanisms with mobility Mt = 3, with a different possible number of branch combinations (Table 5) and branch forms (e.g. there are 57 branches with Mgj = 1, Fig. 7), it becomes apparent that the number of solutions is very large. Exemplary solutions of symmetric parallel mechanisms without local mobility (Mt = 3), each consisting of three branches (ngc = 3) with mobility Mgj = 1 (Table 5), are presented in Figs. 8–11 (114 solutions). Using the proposed method of the structural synthesis of parallel mechanisms one can create a very numerous set of parallel mechanisms, containing all the solutions which satisfy the requirements set down in this paper. The set contains the solutions actually used in practice (e.g. DELTA, IRB340, FANUC M-1iA and Tsai 3-upu manipulators) [8,26,32,39], but the majority are unknown, but applicable solutions. Using the set of solutions one will be able to design an optimal (according to the adopted criteria) parallel mechanism with a chosen number of branches, which will satisfy the originality criterion (among other criteria).
3.3.
Kinematic schemes of parallel mechanisms
In the next step, using the basic structural schemes of parallel mechanisms one can begin to construct kinematic diagrams.
Basic diagrams of parallel mechanisms
In the next step of the structural synthesis of parallel mechanisms all the open branches are connected, in all the possible ways, to the platform p and the base 0 as shown in Fig. 4. This labour-intensive task leads to the creation of a complete set of parallel mechanism structures. Limiting the analysis to parallel mechanisms with real mobility M = Mt = 3 (without local mobility), the possible numbers of branches used to construct both symmetric and asymmetric basic diagrams of the mechanisms are presented in Table 5.
Fig. 12 – Used physical forms of kinematic pairs.
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Fig. 13 – Kinematic schemes of selected parallel mechanism.
The procedure for creating kinematic schemes consists in substituting the physical forms of kinematic pairs for their symbols. From among the many theoretical forms of lower class I, II and III pairs only the forms schematically shown in Fig. 12 were used in this study. As regards pairs of class I, only rotational and translational pairs were permitted to be used. Cylindrical pairs and universal (Cardan) joints were permitted for class II pairs while spherical joints were permitted for class III pairs. By substituting the physical forms of kinematic pairs (Fig. 12) for their symbols in the mechanisms shown in Figs. 8–11 one can create a complete set of kinematic diagrams of the considered parallel mechanisms. Since the set of solutions is very numerous it was decided to illustrate the procedure for structure no. 6 selected from Fig. 8, for which the created kinematic diagrams are collected in Fig. 13. The obtained 14 kinematic diagrams of symmetric parallel mechanisms with mobility (M = Mt = 3) are the result of the application of the proposed method of structural synthesis to only one basic parallel mechanism diagram (Fig. 8). In order to obtain a complete set the above actions should be repeated for each of the structural mechanism solutions contained in Figs. 8–11.
4.
Conclusion
The developed method is based on the properties of the Baranov method and those of the intermediate chain method. By
exploiting the strengths of the latter two methods a new method, which enables one to create complete sets of symmetric and asymmetric parallel mechanisms with the number of branches described by relation (6), with mobility in the range 2 Mt 6, has been developed. The advantage of this method is its ability to create all the theoretically possible solutions with separate branches or with interconnected branches. Already the initial results obtained using this method show that the generated set of solutions comprises the existing atypical solutions arrived at through reasoning and intuition. Moreover, the method can be easily extended, by changing branch mobility Mgj in formula (8) to cover branches with local mobility. As shown in the work the received collection of the new structures of parallel mechanisms, even when placed in the work limits (13–15), are quite numerous. The large number of solutions makes it difficult to select the proper mechanism for the intended use. Therefore, in the next research stage it is necessary to develop the criteria for the selection of the useful or optimal structures. This way the number of the selected mechanisms will be reduced to the most useful for the technical applications. The proposed method of the structural synthesis of spatial parallel mechanisms, could also be easily changed, into a form suitable for planar mechanisms. The presented results of the structural synthesis should contribute to the better understanding and wider use of this interesting group of mechanisms.
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archives of civil and mechanical engineering 16 (2016) 256–268
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