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Computer-aided structural synthesis of 5-DOF parallel mechanisms and the establishment of kinematic structure databases
Mechanism and Machine Theory 83 (2014) 14–30
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Computer-aided structural synthesis of 5-DOF parallel mechanisms and the establishment of kinematic structure databases Huafeng Ding a,b,⁎, Wenao Cao a, Changwang Cai a, Andrés Kecskeméthy c a b c
Hebei Provincial Key Laboratory of Parallel Robot and Mechatronic System, Yanshan University, 066004 Qinhuangdao, China State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, 200240 Shanghai, China Mechanics and Robotics, University of Duisburg-Essen, 47048 Duisburg, Germany
a r t i c l e
i n f o
Article history: Received 8 April 2014 Received in revised form 3 August 2014 Accepted 23 August 2014 Available online xxxx Keywords: Computer-aided Structural synthesis 5-DOF parallel mechanism Kinematic structure database
a b s t r a c t Structural synthesis is one of the most important steps in the conceptual design of mechanisms and robots. The existing synthesis methods for spatial mechanisms are mainly performed by manual enumeration, which is hard to realize the automation and computerization of the conceptual design of mechanisms and robots. This paper proposes a computer-aided method for the synthesis of both symmetrical and asymmetrical 5-DOF parallel mechanisms and the establishment of corresponding kinematic structure databases, which is conductive to kinematic structure design in human–computer interaction. First, a new structural representation is proposed, with which the complete topological information of a parallel mechanism can be represented by a string convenient for computer processing. Then, the computerized synthesis of 5-DOF limbs is performed and the corresponding limb database is established. Third, a computer-aided synthesis method for 5-DOF parallel mechanisms is developed and the corresponding kinematic structure databases are established. Finally, based on the databases, the structural design of 5-DOF parallel mechanisms applied to computed tomography (CT) scanners is conducted, illustrating the effectiveness of the method. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Structural synthesis plays an important role in the conceptual design of mechanisms and robots, and the research on the issue has been a hot topic for over a century and continues to be so [1]. With the development of computer technology, the computerized structural synthesis becomes feasible, which can save human labor and benefit further automation and intelligence of mechanical conceptual design. Dobrjanskyj and Freudenstein [2] made a first attempt to synthesize planar mechanisms automatically. Later, computer-aided structural synthesis of planar mechanisms has been studied by many researchers [3–8], and atlas databases of some planar mechanisms have been established [7,8], with which the designers can conveniently select the feasible kinematic structures for a specified design task in the conceptual design stage. However, computer-aided structural synthesis of spatial mechanisms can be found in little literature. The existing relevant methods are performed by means of manual enumeration. Huang and Li [9] proposed the constraint-synthesis method based on screw theory and enumerated a lot of symmetrical spatial parallel mechanisms. Kong and Gosselin [10], and Fang and Tsai [11] also synthesized many parallel mechanisms based on the principle of constraint synthesis. Gao et al. [12,13] proposed the concept of Gf
⁎ Corresponding author at: Hebei Provincial Key Laboratory of Parallel Robot and Mechatronic System, Yanshan University, 066004 Qinhuangdao, China. E-mail address: [email protected] (H. Ding).
http://dx.doi.org/10.1016/j.mechmachtheory.2014.08.013 0094-114X/© 2014 Elsevier Ltd. All rights reserved.
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set to synthesize parallel mechanisms. In addition, the synthesis methods based on displacement group theory were studied by some scholars [14–16]. With those manual methods, a lot of spatial parallel mechanisms have been synthesized. However, those spatial mechanisms were scattered in literature so that designers can hardly find the desired kinematic structures for a specified design task. Moreover, it is difficult to synthesize systematically asymmetrical parallel mechanisms for their great quantities and varieties. The study of automatic structural synthesis of mechanisms and robots is an effective solution to those problems. 5-DOF parallel mechanisms, which can output three-rotation and two-translation (3R2T) or three-translation and two-rotation (3T2R), have a wide range of applications such as medical devices, welding equipment and machine tools. Piccin et al. [17] proposed a 5-DOF parallel mechanism applied to computed tomography (CT) scanner. Kong and Gosselin [18] synthesized some 5-DOF parallel manipulators based on screw theory. Zhu and Huang [19] enumerated eighteen fully symmetrical 3R2T parallel manipulators with better actuating modes. Motevalli et al. [20] synthesized 3T2R parallel manipulators with prismatic actuators on the base. Li and Huang [15] synthesized 3R2T parallel mechanisms using group theory. Gogu [21] synthesized 3T2R parallel robots via the theory of linear transformations and evolutionary morphology. However, asymmetrical 5-DOF parallel mechanisms with great quantities and varieties have not been synthesized systematically, and computer-aided structural synthesis of 5-DOF mechanisms has not been investigated. In order to realize the computerization of conceptual design of mechanisms and robots, in this paper, we propose a computeraided method to synthesize 5-DOF parallel mechanisms with both symmetrical and asymmetrical structures and establish corresponding kinematic structure databases. The paper is organized as follows. In Section 2, basic concepts of screw theory are given. In Section 3, the structural representation of parallel mechanisms is presented. In Section 4, 5-DOF limbs are synthesized and the corresponding database is established. In Section 5, a computer-aided method for the structural synthesis of 5-DOF parallel mechanisms is proposed. In Section 6, one of the applications is discussed for computed tomography (CT) scanner. 2. Basic concepts In screw theory [22], a unit screw $ is defined by a pair of vectors: $ ¼ ðS; S 0 Þ ¼ ðS; r S þ pS Þ ¼ ða b c; d e f Þ
ð1Þ
where S denotes the unit vector specifying the direction of the screw axis, r denotes the position vector of any point on the screw axis with respect to a reference frame, and p is called the pitch. The unit screw associated with a prismatic pair or a constraint couple can be expressed by (0; S) with p = ∞, and the unit screw associated with a revolute pair or a constraint force is expressed by (S; r × S) with p = 0. For two given screws, $ = (S; S0) and $ r = (Sr; Sr0), their reciprocal product is defined as r
r
$ ∘ $ ¼ S • S0 þ S0 • S
r
ð2Þ
where “°” denotes the reciprocal product. If $ ∘ $ r is equal to zero, $ denotes a twist (motion screw), and $ r denotes a wrench (constraint screw). The geometrical relation of two reciprocal screws can be also expressed as follows [23]. Proposition 1. The wrench of a constraint force is reciprocal to the twist of a revolute joint if and only if their axes are coplanar (parallel or intersecting) with each other; the wrench of a constraint force is reciprocal to the twist of a prismatic joint if and only if their axes are perpendicular to each other; the twist of a prismatic joint is always reciprocal to the wrench of a constraint couple. For a limb, when the types of joints and the geometrical relations between joint axes are known, the constraints contributed by the limb can be easily determined according to Proposition 1. 3. Structure representation of parallel mechanisms Generally, a parallel mechanism consists of a moving platform connected to a fixed platform by several serial limbs, and its kinematic structure is completely determined by the types of kinematic pairs and the invariant geometrical relations between joint axes. Here, a structure representation of parallel mechanisms is proposed as follows. 3.1. Representation of limbs 3.1.1. Types of kinematic pairs Kinematic pairs can be represented by some simple symbols, as shown in Table 1. Composite pairs U, S and C can be regarded as the combinations of two R pairs with orthogonal axes, three R pairs with concurrent axes, and one R pair and one P pair with a collinear axis, respectively. Therefore, a limb can be considered as the one only containing pairs R and P. For convenience, kinematic pairs of a limb are labeled with numbers 1 ~ t from the fixed platform to the moving platform in turn.
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Table 1 Representations of kinematic pairs. Pairs
Prismatic
Revolute
Cylindric
Universal
Spherical
Representations
P
R
C
U
S
3.1.2. Relations between adjacent joint axes The geometrical relations between adjacent joint axes are invariant because they are connected by rigid links. Such relations include six types listed in Table 2. 3.1.3. Sub-chains Three adjacent R or P pairs generating planar motion constitute a 3-DOF planar sub-chain, in which all the R pairs are parallel to each other and all the P pairs are perpendicular to the R pairs. Any two pairs of a 3-DOF planar sub-chain constitute a 2-DOF planar sub-chain, such as RR, RP and PP. Generally, the normal of the motion plane of a planar sub-chain is called the normal of the subchain. Three concurrent R pairs form a 3-DOF spherical sub-chain, and two intersecting R pairs form a 2-DOF spherical sub-chain. A spherical pair and a universal pair can be regarded as a 3-DOF and a 2-DOF spherical sub-chain, respectively. The representations of those planar and spherical sub-chains are shown in Table 3. 3.1.4. Invariant relations between nonadjacent joint axes It is not required to represent the variant relation between two nonadjacent joint axes of a limb in finite continual motion. However, under the condition of some special assemblies, the geometrical relation of two nonadjacent joints in a limb may remain invariant in finite motions. Thus, such relation requires considering, mainly including two cases. One is the geometrical relation of two pairs assembled at two ends of P pairs, for example, the perpendicular relation between the axes of R1 and R2 in Fig. 1(a). The other is the invariant geometrical relation of two pairs connected by a planar sub-chain, for example, the intersecting relation of the axes of R5 and R1 connected by the 3-DOF planar sub-chain consisting of R2, R3 and R4 in Fig. 1(b). 3.1.5. Generation of representation string The representation string of a limb can be generated as follows. First, generate a string following the sequence: the type of pair 1, the relation between the axes of pair 1 and pair 2, the type of pair 2, the relation between the axes of pair 2 and pair 3, …, and up to the type of pair t. Second, check planar or spherical sub-chains in the limb. If so, replace the corresponding parts in the string with the representations of the sub-chains based on Table 3. Third, check invariant nonadjacent geometrical relations in the limb. If there exists such a relation between a pair and a former one, add the labeled number of the former pair and the nonadjacent geometrical relation to the end of the representation of the pair by enclosing them with parenthesis. For example, the representation of the limb shown in Fig. 1(b) is “R + R/R/R + R(1^)”. Finally, add the assembly relations between pair 1 and the fixed platform and between pair t and the moving platform to the beginning and end of the string, respectively. The relation between a pair and a platform (line-plane relation) can be classified into three types: parallel (/), orthogonal (+) and intersecting (^). If a limb is connected to a platform by a P pair of a planar sub-chain, add the relation between the normal of the sub-chain and the platform to the end of the representation of the sub-chain by enclosing it with parenthesis. 3.2. Representation of platforms The representation of a platform consists of the number of limbs and the relations among the axes of the joints connected to the platform. If a limb is connected to a platform by a P pair of a planar sub-chain, the normal of the sub-chain is represented rather than the axis of the P pair. For a parallel mechanism with symmetrical structures, since the joints (planar sub-chains) connected to a platform are assembled symmetrically, the relations among those joint axes (the normals of planar sub-chains) may be parallel, concurrent, forming a polygon and skew, represented by symbols ‘/’, ‘*’, ‘#’ and ‘!’, respectively. Table 2 Representations of geometrical relations between adjacent joint axes. Adjacent joints
Relations Representations
Parallel (/) R/R
Perpendicular (⊥) R⊥R
Orthogonal (+) R+R
Intersecting (^) R^R
Skew (!) R!R
collinear (|) R|R
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Table 3 Representations of sub-chains. Planar sub-chains Types Representations
PRP P + R◇P
Spherical sub-chains
PPR P + P◇R
RPP R + P◇P
RPR R + P◇R
PRR P + R◇R
RRP R/R◇P
PP P△P
PR P△R
RRR R^R*R
RRR R + R*R
3.3. Representation of parallel mechanisms The representation string of a parallel mechanism can be generated as follows: “The representation of the fixed platform — the representation of limb 1 ~ the representation of limb 2 ~ … — the representation of the moving platform”. If some of limbs have the identical representation, for convenience, they can be represented by any one of them. Example 1 The parallel mechanism shown in Fig. 2(a) contains two identical limbs UPR and another limb SPR. It can be represented as “3(1|2!3)-2/R + R + P◇R/~^R^R*R^P△R/-3(1/2⊥3)”. Example 2 The parallel mechanism shown in Fig. 2(b) contains three identical limbs and it can be represented as “3#-/R/R/R⊥R/R/-3*”. 4. Synthesis of 5-DOF limbs and limb database 5-DOF parallel mechanisms can be constructed by only 5-DOF and 6-DOF limbs. Without loss of generality, the following conditions are imposed: (1) Only the 5-DOF limbs which can contribute constraint forces or couples are concerned; (2) 5-DOF limbs including idle DOFs are excluded because they can be replaced by the ones with lower pairs; (3) 5-DOF limbs are regarded as the ones only consisting of pairs R and/or P because composite pairs C, U and S can be expressed as “R|P”, “R + R” and “R^R*R” according to the above representation; (4) Since 6-DOF limbs cannot contribute constraints according to screw theory, only two typical 6-DOF limbs UPS, URS are concerned and the geometrical relations of the two limbs are not required considering. 4.1. Synthesis of 5-R limbs Step 1 Generate all possible combinations of the geometrical relations between adjacent revolute joint axes, shown in Table 4. Since the geometrical relations of each two adjacent R pairs may be +, /, ^, ! and ⊥, the total number of combinations is 54, namely 625. Step 2 Generate the limbs by adding ‘R’ to two sides of each adjacent geometrical relation, shown in Table 5. For example, for a combination “//!^”, the corresponding limb can be obtained as “R/R/R!R^R”. Step 3 Obtain new limbs by replacing those limbs containing three successively intersecting or orthogonal revolute pairs with corresponding spherical sub-chains. For example, the new limb “R/R^R*R⊥R” can be gained from the one “R/R^R^R⊥R”. Step 4 Delete the limbs including idle DOFs. For example, the limb “R/R/R/R!R” should be deleted because it can be replaced by the one “R/R/R!R”. Step 5 Obtain new limbs by adding nonadjacent geometrical relations. For example, the limb “R + R/R/R + R(1^)” can be obtained from the one “R + R/R/R + R”. That is because two R pairs connected a 3-DOF planar sub-chain may remain intersecting. Step 6 Choose the limbs which can provide a constraint force or couple according to Proposition 2.
Fig. 1. Invariant relations between two nonadjacent kinematic joint axes.
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Fig. 2. (a) 3(1|2!3)-2/R + R + P◇R/~^ R^R*R^P△R/-3(1/2⊥3), (b) 3#-/R/R/R⊥R/R/-3*.
Proposition 2. A 5-R limb providing a constraint couple (called couple-limb) must contain a 3-DOF planar sub-chain and a 2-DOF planar sub-chain, and a 5-R limb providing a constraint force (called force-limb) must contain a 3-DOF planar sub-chain or a 3-DOF spherical sub-chain. Proof. Because the constraint couple of a limb must be perpendicular to all the R pairs of the limb according to Proposition 1, a 5-R couple-limb must contain two and only two different axis directions. Thus, five R pairs can be divided into two groups: One constitutes a 3-DOF planar sub-chain and the other constitutes a 2-DOF planar sub-chain. Without loss of generality, 5-R limbs can be classified into the following four cases. Case 1. Containing a 3-DOF planar sub-chain. Generally, such a limb “R/R/R!R!R” is shown in Fig. 3(a). Choose a point A on axis L2 of R4 and make a line L3 parallel to axis L1 of R3 through point A. Assume that point B is the intersecting point between axis L4 of R5 and plane γ constituted by lines L2 and L3. Through point B, make a line L5 parallel to L3. So, line L5 is intersecting with both the axes of R4 and R5 and parallel to the axes of R1, R2 and R3. Therefore, line vector L5 represents the constraint force of the limb according to Proposition 1. Case 2. Only containing a 2-DOF planar sub-chain or without any planar sub-chain. Generally, such a limb “R/R!R!R!R” is shown in Fig. 3(b). Make a line L3 parallel to axis L1 of R2 through point A on L2. Assume that points B and C are respectively the intersecting points of axes L4, L5 and plane γ constituted by lines L2 and L3. Generally, line BC is not parallel to L3. Therefore, there doesn't exist any line vector coplanar with the axes of the five R pairs. Namely, the limb cannot provide a constraint force. Similarly, for a limb without any planar sub-chain, for example, the one “R!R!R!R!R”, cannot provide a constraint force. Case 3. Containing a 3-DOF spherical sub-chain. Generally, such a limb “R^R*R!R!R” is shown in Fig. 3(c). Assume that axis L2 of R5 and plane γ constituted by point A and axis L1 of R4 are intersecting at point B. Then, line AB is intersecting with all the revolute joint axes. Thus line vector AB represents the constraint force of the limb. Case 4. Only containing a 2-DOF spherical sub-chain or without any spherical sub-chain. Generally, such a limb “R^R!R!R!R” is shown in Fig. 3(d). Assume that points B and C are respectively the intersecting points of axes L2, L3 and plane γ constituted by point A and axis L1 of R3. Generally, line BC doesn't pass through point A. Therefore, there doesn't exist any line vector coplanar with every axis of the five R pairs. Namely, the limb cannot provide a constraint force. Similarly, a limb without spherical sub-chains cannot provide a constraint force. Thus, for a 5-R limb containing a 3-DOF planar or spherical sub-chain, there is always a line vector coplanar with all the R pairs of the limb. Otherwise, there is no such a line vector. Thus, a 5-R force-limb must contain a 3-DOF planar or spherical sub-chain.
Table 4 Combinations of adjacent geometrical relations. No. 1 No. 2 No. 3 No. 4 No. 5
//// //^^ //++ //⊥⊥ //!!
No. 6 No. 7 No. 8 No. 9 No. 10
///^ //^+ //+⊥ //⊥! /^//
No. 11 No. 12 No. 13 No. 14 No. 15
///+ //^⊥ //+! //!/ /^/^
No. 16 No. 17 No. 18 No. 19 …
///⊥ //^! //⊥/ //!^ …
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Table 5 Limbs with only adjacent geometrical relations. No. 1 No. 2 No. 3 No. 4 No. 5
R/R/R/R/R R/R/R^R^R R/R/R + R + R R/R/R⊥R⊥R R/R/R!R!R
No. 6 No. 7 No. 8 No. 9 No. 10
R/R/R/R^R R/R/R^R + R R/R/R + R⊥R R/R/R⊥R!R R/R^R/R/R
No. 11 No. 12 No. 13 No. 14 No. 15
R/R/R/R + R R/R/R^R⊥R R/R/R + R!R R/R/R!R/R R/R^R/R^R
No. 16 No. 17 No. 18 No. 19 …
R/R/R/R⊥R R/R/R^R!R R/R/R⊥R/R R/R/R!R^R …
4.2. Synthesis of 5-RP limbs Since a planar chain constituted only by R pairs is kinematically equivalent to the one constituted by pairs R and P, 5-RP limbs can be obtained from the synthesized 5-R limbs by replacing some R pairs with corresponding P pairs. For a 5-R limb containing a planar sub-chain, one or two R pairs of the sub-chain can be replaced with P pairs. For a 5-R limb containing a 3-DOF spherical sub-chain, except three R pairs constituting the spherical sub-chain, the remaining two R pairs are kinematically equivalent to two P pairs, since the spherical sub-chain can provide the parallel rotations to the two R pairs by linear combinations so that two corresponding 2-DOF planar sub-chains can be generated. For example, for the limb “R^R*R!R!R” shown in Fig. 4(a), the spherical sub-chain consisting of R1, R2 and R3 can provide the rotation R41 parallel to R4 and the rotation R51 parallel to R5. So an equivalent limb “R^R*R!P!P” can be obtained, shown in Fig. 4(b). The 5-RP limbs derived from 5-R ones with planar sub-chains and spherical sub-chains are shown in Tables 6 and 7, respectively. 4.3. 5-DOF limb database According to Proposition 1, the following characteristics can be obtained for the constraints of 5-DOF limbs: (1) For a 5-DOF force-limb containing a 3-DOF planar sub-chain, the direction of its constraint force is determined, which is the normal of the sub-chain. Except the planar sub-chain, if the remaining two pairs constitute a 2-DOF spherical sub-chain, the position of the constraint force is further determined. Namely, the constraint force passes through the intersecting point of the spherical sub-chain. (2) For a 5-DOF force-limb containing a 3-DOF spherical sub-chain, the position of its constraint force is determined. Namely, the constraint force passes through the intersecting point of the spherical sub-chain. Except the sub-chain, if the remaining two pairs constitute a 2-DOF planar sub-chain, the direction of the constraint force is further determined, which is the normal of the sub-chain. If the remaining two pairs constitute a 2-DOF spherical sub-chain as well, the direction of the constraint force is further determined, which is along the line through the two intersecting points of the two spherical sub-chains. (3) For a 5-DOF couple-limb, the direction of its constraint couple is determined, which is perpendicular to the axes of R pairs. For a 5-DOF limb, the above-mentioned constraint characteristics are invariant (full-cycle) in continual motion, so they are called full-cycle constraint characteristics of the limb, which can be described by a six-dimensional array {A1, B1, C1, D1, E1, F1}. The first three parameters A1, B1 and C1 denote the characteristics of the constraint direction. The last three parameters D1, E1 and F1 denote the characteristics of the constraint position. For a 5-DOF force-limb, parameters A1, B1 and C1 can be determined as: If a planar sub-chain connects to the fixed platform, A1 = 1, B1 = 0 and C1 = 0; If a planar sub-chain connects to the moving platform, C1 = 1, A1 = 0 and B1 = 0; If a planar sub-chain locates
Fig. 3. (a) ‘R/R/R!R!R’ limb, (b)‘R/R!R!R!R’ limb, (c) ‘R^R*R!R!R’ limb, (d) ‘R^R!R!R!R’ limb.
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Fig. 4. (a) A limb including a spherical pair, (b) its kinematically equivalent limb.
between the two platforms, B1 = 1, A1 = 0 and C1 = 0. Parameters D1, E1 and F1 can be determined as: If a spherical sub-chain without orthogonal relation connects to the fixed platform, D1 = 1, E1 = 0 and F1 = 0, otherwise, D1 = 2, E1 = 0 and F1 = 0; If a spherical sub-chain without orthogonal relation connects to the moving platform, D1 = 0, E1 = 0 and F1 = 1, otherwise, D1 = 0, E1 = 0 and F1 = 2. For a 5-DOF couple-limb, both parameters A1 and C1 are 1 and other parameters B1, D1, E1 and F1 are always 0. 5-DOF limb database can be established based on the synthesized limbs and their full-cycle constraint characteristics, shown in Fig. 5. 5. Synthesis of 5-DOF parallel mechanisms and kinematic structure databases In a parallel mechanism, the constraints imposed on the moving platform by the limbs and the geometrical relations between such constraints constitute the constraint mode of the mechanism. From the view of linear algebra, those constraints span a linear space, called the platform constraint space. The constraints provided by a limb also span a space, called the limb constraint space. The relations between the platform and limb constraint spaces are fP g ¼ fL1 g∨fL2 g∨fL3 g∨⋯∨fLn g
ð3Þ
where {P} denotes the platform constraint space, {Li} the constraint space of i-th limb, and ∨ the union operation. From Eq. (3), each limb constraint space is a subspace of the platform constraint space. For 5-DOF parallel mechanisms, the feasible constraint modes can be obtained according to the following steps: Step 1 Specify the desired output motions, namely, 3T2R or 3R2T. Step 2 Determine the independent constraint imposed on the moving platform according to the reciprocal equation, and represent the platform constraint space spanned by the independent constraint. Step 3 Enumerate all subspaces of the platform constraint space, each of which is just a possible limb constraint space of the limb for constructing the desired parallel mechanisms. Further, determine the possible constraint provided by a limb. Step 4 Enumerate all the feasible constraint modes according to the number of limbs providing constraint. When a feasible constraint mode is given, the desired parallel mechanisms corresponding to the mode can be synthesized as follows. (1) Identify the constraint characteristics of the limbs for constructing the parallel mechanisms corresponding to the mode and the geometrical relations between the directions and positions of the constraints of different limbs. (2) Choose the feasible limbs from the 5-DOF limb database according to the constraint characteristics. (3) Generate all combinations of the chosen limbs. Table 6 5-RP limbs with a planar sub-chain. No. 1 No. 2 No. 3 No. 4
P + P◇R^R + R P + P◇R + R + R R!P + R◇P!R R!R|P + P◇R
No. 5 No. 6 No. 7 No. 8
R + P◇R^R + R R + P◇R + R + R R!R + P◇P!R R!R⊥P + P◇R
No. 9 No. 10 No. 11 No. 12
P◇R/R^R + R P◇R/R + R + R R + R + P◇P!R R + R⊥P + P◇R
No. 13 No. 14 No. 15 …
R + P◇P!R + R R + P◇P|R + R R!R/R◇P!R …
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Table 7 5-RP limbs with a spherical sub-chain. No. 1 No. 2 No. 3 No. 4
R^R*R/R + P R^R*R/R/P P!R + R^R*R P|R + R^R*R
No. 5 No. 6 No. 7 No. 8
R^R*R + R + P R^R*R + R/P P!R^R^R*R P|R^R^R*R
No. 9 No. 10 No. 11 No. 12
R^R*R!R + P R^R*R!R/P P!R!R^R*R P/R/R^R*R
No. 13 No. 14 No. 15 …
R^R*R^R + P R^R*R^R/P P!R⊥R^R*R …
(4) Establish assembly conditions between the joints (or sub-chains) of different limbs on platforms for each limb combination, according to the relations between the constraints of different limbs. (5) Traverse all assembly relations between single limb and platforms. (6) Store the synthesized parallel mechanisms in corresponding kinematic structure databases. 5.1. Three-rotation and two-translation (3R2T) parallel mechanisms The moving platform is imposed by an independent constraint force, and the platform constraint space can be represented as r fP g ¼ k $ F ¼ fkð a
b c; 0
0
0 Þðk ∈RÞg:
ð4Þ
There are two subspaces. One is {P} itself and the other is null space {0}. Thus, the constraint provided by a limb for constructing 3R2T parallel mechanisms is r
$l ¼
r
A ¼ $ F ¼ ð a b c; 0 B¼0
0
0Þ:
ð5Þ
Fig. 5. 5-DOF limb database.
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Table 8 Constraint modes of 3R2T parallel mechanisms with five limbs. Mode 1: 5A
Mode 2: 4A1B
Mode 3: 3A2B
Mode 4: 2A3B
Mode 5: 1A4B
For 3R2T parallel mechanisms containing five limbs, according to the number of limbs for contributing the constraint force, five feasible constraint modes can be obtained, shown in Table 8. For 3R2T parallel mechanisms including four limbs, according to the number of the limbs contributing the constraint force, four feasible constraint modes can be obtained, shown in Table 9. Likewise, for 3R2T parallel mechanisms including three and two limbs, there are five feasible constraint modes, shown in Table 10. Take mode 1 as an example to show the synthesis process. Five constraint forces (5A) from different limbs have the identical direction and pass through a common point. One possibility is that 3R2T mechanisms are constructed by the 5-DOF limbs with the constraint directions determined by the planar sub-chains connected to the fixed platform and constraint positions determined by the spherical sub-chains connected to the moving platform. The full-cycle constraint characteristic of such limbs is {1 0 0 0 0 1}, called 3R2T-type-1 limbs, for example, the limb “R/R/R!R^R”; The other possibility is that 3R2T mechanisms are constructed by the 5-DOF limbs with the constraint directions determined by the planar sub-chains connected to the moving platform and constraint positions determined by the spherical sub-chains connected to the fixed platform. The constraint characteristic of such limbs is {0 0 1 1 0 0}, called 3R2T-type-2 limbs, for example, the limb “R^R*R!R/R”; Another possibility is that 3R2T mechanisms are constructed by the 5-DOF limbs with two spherical sub-chains connected to the fixed and moving platform, respectively. The constraint characteristic of such limbs is {0 0 0 1 0 1}, called 3R2T-type-3 limbs, for example, the limb “R^R*R!R^R”. Take 3R2T-type-1 limbs for example. Choose such limbs from the 5-DOF limb database, shown in Table 11. Generate the combinations of any five 3R2T-type-1 limbs, shown in Table 12.
Table 9 Constraint modes of 3R2T parallel mechanisms with four limbs. Mode 6: 4A
Mode 7: 3A1B
Mode 8: 2A2B
Mode 9: 1A3B
Table 10 Constraint modes of 3R2T parallel mechanisms with three and two limbs. Mode 10: 3A
Mode 11: 2A1B
Mode 12: 1A2B
Mode 13: 2A
Mode 14:1A1B
B
Table 11 3R2T-type-1 limbs. No. 1 No. 2 No. 3 No. 4
R/R/R + R^R R/R!R^R*R R△P|R^R*R P△R!R^R*R
No. 5 No. 6 No. 7 No. 8
R/R/R⊥R^R R△P/R^R*R P△P/R^R*R P + R◇R^R^R
No. 9 No. 10 No. 11 No. 12
R/R/R!R^R R△P + R^R*R P△R/R^R*R P + P◇R^R^R
No. 13 No. 14 No. 15 …
R/R^R^R*R R△P^R^R*R P△R + R^R*R …
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Table 12 Combinations of any five 3R2T-type-1 limbs. No. 1 No. 2 No. 3 No. 4
R/R/R!R^R~R/R^R^R*R~R/R + R^R*R~R/R⊥R^R*R~R/R!R^R*R R/R/R^R^R~R/R/R!R^R~R/R^R^R*R~R/R + R^R*R~R/R!R^R*R R/R/R⊥R^R~R/R^R^R*R~R/R + R^R*R~R/R⊥ R^R*R~R/R!R^R*R P + R◇R(+)^R^R~P + P◇R(+)^R^R~P + P◇R(+) + R^R~P + P◇R(+)!R^R~P + R◇P(+)!R^R
No. 5 No. 6 No. 7 …
R/R/R!R^R~R/R/R!R^R~R/R/R!R^R~R/R/R!R^R~R/R/R!R^R R/R/R⊥R^R~R/R/R!R^R~R/R^R^R*R~R/R + R^R*R~R/R!R^R*R R/R!R^R*R~R/R!R^R*R~R/R/R!R^R~R/R/R^R^R~R/R/R + R^R …
Fig. 6. The kinematic structure database of 3R2T parallel mechanisms.
Fig. 7. The interface of 3R2T parallel mechanisms.
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Table 13 Constraint modes of 3T2R parallel mechanisms with five limbs. Mode 1: 5D
Mode 2: 4D1E
Mode 3: 3D2E
Mode 4: 2D3E
Mode 5: 1D4E
Table 14 Constraint modes of 3T2R parallel mechanisms with four limbs. Mode 6: 4D
Mode 7: 3D1E
Mode 8: 2D2E
Mode 9: 1D3E
For each combination of 3R2T-type-1 limbs, its assembly satisfies: The normals of all planar sub-chains connected to the fixed platform are parallel to each other to obtain the identical constraint direction. The intersecting points of all spherical sub-chains connected to the moving platform are coincident to obtain the identical constraint position. For single 3R2T-type-1 limb, the normal of its planar sub-chain may be parallel (/) to, orthogonal (+) to or intersecting (^) with the fixed platform. The intersecting points of its spherical sub-chain may be on or outside the moving platform, so the pair connected to the moving platform may be parallel (/) to, or intersecting (^) with the moving platform. The parallel geometrical relation between the normals of the planar sub-chains connected to the fixed platform is invariant in continual motion. The coincident geometrical relation between the spherical sub-chains connected to the moving platform is also invariant in continual motion. Thus, the constraint forces provided by different limbs remain collinear and the synthesized mechanisms have full-cycle DOFs. Likewise, 3R2T parallel mechanisms corresponding to other feasible constraint modes can be also synthesized. The synthesized parallel mechanisms can be stored in a kinematic structure database with some kinematic characteristics, such as DOFs, output motions, the number of constraint limbs, the number of identical limbs, and the number of concurrent axes, shown in Fig. 6. A human–computer interactive interface has been developed, shown in Fig .7. 5.2. Three-translation and two-rotation (3T2R) parallel mechanisms The moving platform is imposed by one independent constraint couple, and the platform constraint space can be represented as r fP g ¼ k $ C ¼ fkð 0 0 0; a
b
c Þðk ∈ R Þg:
ð6Þ
The subspaces are {P} itself and null space {0}. Thus, the constraint provided by a limb for constructing 3T2R parallel mechanisms is r
$l ¼
r
D ¼ $C ¼ ð 0 E¼0
0
0; a
b
cÞ:
ð7Þ
Table 15 Constraint modes of 3T2R parallel mechanisms with three and two limbs. Mode 10: 3D
Mode 11: 2D1E
Mode 12: 1D2E
Mode 13:2D
Mode 14: 1D1E
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Table 16 3T2R-type limbs. No. 1 No. 2 No. 3 No. 4
R/R/R^R/R R/R⊥R/R/R P + P◇R + R/R P + P◇R!R/R
No. 5 No. 6 No. 7 No. 8
R/R/R + R/R R/R!R/R/R R + P◇R + R/R R + P◇R!R/R
No. 9 No. 10 No. 11 No. 12
R/R/R⊥R/R P + R◇R^R/R R/R◇P|R/R R/R^R + P◇R
No. 13 No. 14 No. 15 …
R/R/R!R/R P + P◇R^R/R P + R◇P|R/R …
Table 17 Combinations of any five 3T2R-type limbs. No. 1 No. 2 No. 3 No. 4
R/R/R!R/R~R/R/R + R/R~R/R^R/R/R~R/R + R/R/R~R/R!R/R/R R/R/R!R/R~R/R/R!R/R~R/R/R!R/R~R/R/R!R/R ~R/R/R!R/R R/R/R + R/R~R/R/R⊥R/R~R/R + R/R/R~R/R⊥R/R/R~R/R/R!R/R P + R◇R(/) + R/R~P + P◇R(/) + R/R~R/R◇P/R/R~R/R|P + R◇R~R/R⊥R + P◇R
No. 5 No. 6
R/R/R!R/R~R/R/R + R/R~R/R⊥R/R/R~R/R + R/R/R~R/R!R/R/R R/R/R^R/R~R/R/R⊥R/R~R/R/R!R/R~R/R/R + R/R~R/R⊥R/R/R
No. 7 …
R/R/R + R/R~R/R^R/R/R~R/R⊥R/R/R~R/R + R/R/R~R/R!R/R/R …
For 3T2R parallel mechanisms with five limbs, the number of the limbs provided the constraint couple may be five, four, three, two and one, so there are five feasible constraint modes, shown in Table 13. Likewise, for 3T2R parallel mechanisms with four limbs, four feasible constraint modes can be obtained, shown in Table 14. For 3T2R parallel mechanisms with three and two limbs, five feasible constraint modes can be obtained, shown in Table 15. Take mode 1 as an example. Five constraint couples (5D) have identical direction. Correspondingly, 3T2R parallel mechanisms can be constructed by 5-DOF couple-limbs whose constraint characteristics are {1 1 0 0 0 0}, called 3T2R-type limbs. Choose all 3T2R-type limbs from the limb database, shown in Table 16. Generate all combinations of any five 3T2R-type limbs, shown in Table 17. For each combination, its assembly satisfies: The R pairs nearest to the fixed platform in different limbs are parallel to each other; The R pairs nearest to the moving platform in different limbs are also parallel to each other. Both the parallel geometrical relations between the R pairs nearest to the fixed platform in different limbs and between the R pairs nearest to the moving platform in different limbs are invariant in continual motion. Thus, the constraint couples provided by different limbs remain parallel and the synthesized mechanisms have full-cycle DOFs.
Fig. 8. The kinematic structure database of 3T2R parallel mechanisms.
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H. Ding et al. / Mechanism and Machine Theory 83 (2014) 14–30
Fig. 9. The interface of 3T2R parallel mechanisms.
Likewise, 3T2R parallel mechanisms corresponding to other feasible constraint modes can be also synthesized. The synthesized parallel mechanisms are also stored in a kinematic structure database with some kinematic characteristics, shown in Fig. 8. Fig. 9 presents a corresponding human–computer interactive interface.
6. Applications A 5-DOF parallel mechanism for CT scanner is proposed in literature [17], which consists of two identical 5R limbs and another limb equivalent to SRU, shown in Fig. 10(a). Although the desired output motion of the mechanism is three-translation and two-rotation (3T2R), in fact, the output motion of the mechanism is changeable and it can be 3T2R or 3R2T.
Fig. 10. Two configurations of the 5-DOF parallel mechanism.
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Fig. 11. A new candidate mechanism.
For limb 1 of the mechanism, R11 is parallel to R15. R12, R13 and R14 are parallel to each other and perpendicular to R11 and R15. R11 and R21 are collinear. R15 and R25 are also collinear. Limb 3 is a 6-DOF limb. The representation of the mechanism is “3(1|2!3)-2/R⊥R/R/ R⊥R(1/)/~SRU-3(1|2!3)”.
Fig. 12. Another new candidate mechanism.
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H. Ding et al. / Mechanism and Machine Theory 83 (2014) 14–30
Fig. 13. Two new typical candidate mechanisms for CT scanner.
A reference frame can be set up on the fixed platform, in which the origin locates at one point in the fixed platform, Z-axis is perpendicular to the fixed platform upward and Y-axis is parallel to R11. For the initial configuration shown in Fig. 10(a), the twist system of limb 1 is 0
$ 11 B $ 12 B $1 ¼ B B $ 13 @$ 14 $ 15
¼ ½0; ¼ ½1; ¼ ½1; ¼ ½1; ¼ ½0;
1; 0; 0; 0; 1;
0; 0; 0; 0; 0;
ðl11 ; ðl12 ; ðl13 ; ðl14 ; ðl15 ;
m11 ; m12 ; m13 ; m14 ; m15 ;
n11 Þ ð0; n12 Þ ð1; n13 Þ ð1; n14 Þ ð1; n15 Þ ð0;
1; 0; 0; 0; 1;
0Þ 0Þ 0Þ 0Þ 0Þ
1 C C C: C A
ð8Þ
According to Eq. (3), the wrench system of limb 1 is r
$1 ¼ ð0; 0; 0; 0; 0; 1Þ
ð9Þ
which denotes a constraint couple along Z-axis. Likewise, the wrench system of limb 2 is also r
r
$ 2 ¼ $ 1 ¼ ð0; 0; 0; 0; 0; 1Þ:
ð10Þ
Thus, the moving platform is imposed by an independent constraint couple along Z-axis and it can output three translation and two-rotation (3T2R) about X-axis and Y-axis. After the moving platform rotates about X-axis, the configuration becomes the one as shown in Fig. 10(b). In this condition, the twist system of limb 1 has changed to 0
$ 11 B$ B 12 B $1 ¼ B B $ 13 B @ $ 14 $ 15
¼ 0; ¼ 1; ¼ 1; ¼ 1; ¼ 0;
0 1; 0; l11 ; 0 0; 0; l 12 ; 0 0; 0; l 13 ; 0 0; 0; l 14 ;
1 0 0 m11 ; n11 ð0; 1; 0Þ 0 0 C m12 ; n12 ð1; 0; 0Þ C C 0 0 C: m13 ; n13 ð1; 0; 0Þ C C 0 0 m14 ; n14 ð1; 0; 0Þ A 0 0 0 cosðβ15 Þ; cosðγ15 Þ; l 11 ; m 11 ; n11 ð0; cosðβ15 Þ; cosðγ15 ÞÞ
ð11Þ
The wrench system of limb 1 is 0 r 0 0 $ 1 ¼ 1; 0; 0; l 11 ; m11 ; n11 ð1; 0; 0Þ which denotes a constraint force parallel to X-axis through the intersecting point of R11 and R15.
ð12Þ
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Likewise, the wrench system of limb 2 is also 0 r 0 0 r $ 2 ¼ $ 1 ¼ 1; 0; 0; l 11 ; m11 ; n11 ð1; 0; 0Þ :
ð13Þ
The moving platform is imposed by an independent constraint force and it can output three-rotation and two-translation (3R2T). Therefore, the mechanism may output three-translation and two-rotation or three-rotation and two-translation in finite continual motion, but three-rotation and two-translation are not the desired output motion. So the mechanism is not necessarily a good one applied to CT scanner. According some kinematic characteristics, such as the number of constraint limbs, the types of kinematic pairs, the desired output motion and the number of identical limbs, some new candidate mechanisms which can always output three-translation and tworotation in finite continual motion can be selected from the established kinematic structure database of 3T2R parallel mechanisms, such as “3(1/2!3)-2/R/R/R⊥R/R/~SRU-3(1/2!3)”, “3(1/2!3)-2/R/R⊥R/R/R/~ SRU-3(1/2!3)”, “3(1/2!3)-2/R/R + R/R/R/~SRU-3 (1/2! 3)” and “3(1/2!3)-2/R/R/R + R/R/~SRU-3(1/2!3)”. Two typical candidate mechanisms “3(1/2!3)-2/R/R/R⊥R/R/~SRU-3(1/2!3)” and “3(1/2!3)-2/R/R⊥R/R/R/~ SRU-3(1/2!3)” from the kinematic structure database are shown in Figs. 11 and 12, respectively. For the mechanism “3(1/2!3)-2/R/R/R⊥R/R/~SRU-3(1/2!3)”, shown in Fig. 13(a), R11, R12 and R13 are parallel to each other. R13 is perpendicular to R14. R14 and R15 are parallel to each other. Limb 2 has the same geometry as limb 1. R11 and R21 are parallel to each other. R15 and R25 are also parallel to each other. Limb 3 is a 6-DOF limb SRU. For the mechanism “3(1/2!3)-2/R/R⊥R/R/R/~SRU-3(1/2!3)”, shown in Fig. 13(b), R11 and R12 are parallel to each other. R12 is perpendicular to R13. R13, R14 and R15 are parallel to each other. Limb 2 has the same geometry as limb 1. R11 and R21 are parallel to each other. R15 and R25 are also parallel to each other. Limb 3 is a 6-DOF limb SRU. Comparing with the existing mechanism for CT scanner, the above two mechanisms not only can always output three-translation and two-rotation in finite continual motion, but also can have simpler assembly requirements. Maybe, the two mechanisms are more suitable for CT scanner. 7. Conclusions In this paper, a computer-aided method for the structure synthesis of 5-DOF parallel mechanisms is proposed, which can be used to systematically synthesize 5-DOF parallel mechanisms with asymmetrical and symmetrical structures. The structural characteristics of 5-DOF limbs which can provide a constraint force or couple are presented. Based on the proposed method, the process of structural synthesis is performed by a computer program, and the corresponding kinematic structure databases have been established. The human–computer interactive interface has been developed for designers. With these features, the kinematic structure design can be performed in a human–computer interactive manner. The relevant study of this paper is expected to benefit the automation and computerization of the mechanical creative design. Acknowledgments The authors are grateful to the project (Nos. 51275437, 51422509) supported by NSFC, the Hebei Nature Science Foundation (Nos. E2014203117, E2012203154), the Research Fund for the Doctoral Program of Higher Education (No. 20131333110018), the Fok YingTong Education Foundation (No. 141049), and the open project of State Key Laboratory of Mechanical System and Vibration (No. MSV2014-15). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
T.S. Mruthyunjaya, Kinematic structure of mechanisms revisited, Mech. Mach. Theory 38 (2003) 279–320. L. Dobrjanskyj, F. Freudenstein, Some applications of graph theory to the structural analysis of mechanisms, ASME J. Eng. Ind. 89B (1967). T.S. Mruthyunjaya, M.R. Raghavan, Computer aided analysis of the structure of kinematic chains, Mech. Mach. Theory 19 (1984) 357–368. Y.-X. Wang, H.-S. Yan, Computerized rules-based regeneration method for conceptual design of mechanisms, Mech. Mach. Theory 37 (2002) 833–849. H. Ding, Z. Huang, D. Mu, Computer-aided structure decomposition theory of kinematic chains and its applications, Mech. Mach. Theory 43 (2008) 1596–1609. H. Ding, P. Huang, B. Zi, A. Kecskemethy, Automatic synthesis of kinematic structures of mechanisms and robots especially for those with complex structures, Appl. Math. Model. 36 (2012) 6122–6131. H. Ding, F. Hou, A. Kecskeméthy, Z. Huang, Synthesis of the whole family of planar 1-DOF kinematic chains and creation of their atlas database, Mech. Mach. Theory 47 (2012) 1–15. H. Ding, W. Cao, A. Kecskemethy, Z. Huang, Complete atlas database of 2-DOF kinematic chains and creative design of mechanisms, J. Mech. Des. Trans. ASME 134 (2012) 0310061–03100610. Z. Huang, Q. Li, Type synthesis of symmetrical lowermobility parallel mechanisms using the constraint synthesis method, Int. J. Robot. Res. 22 (2003) 59–79. X. Kong, C.M. Gosselin, Type synthesis of parallel mechanisms, Springer, Berlin, 2007. Y. Fang, L.W. Tsai, Structure synthesis of a class of 4-DOF and 5-DOF parallel manipulators with identical limb structures, Int. J. Robot. Res. 21 (2002) 799–810. F. Gao, W. Li, X. Zhao, Z. Jin, H. Zhao, New kinematic structures for 2-, 3-, 4-, and 5-DOF parallel manipulator designs, Mech. Mach. Theory 37 (2002) 1395–1411. F. Gao, J. Yang, Q.J. Ge, Type synthesis of parallel mechanisms having the second class Gf sets and two dimensional rotations, J. Mech. Robot. 3 (2011) 011003. P. Fanghella, C. Galletti, Metric relations and displacement groups in mechanism and robot kinematics, J. Mech. Des. Trans. ASME 117 (1995) 470–478. Q. Li, Z. Huang, J.M. Herve, Type synthesis of 3R2T 5-DOF parallel mechanisms using the lie group of displacements, IEEE Trans. Robot. Autom. 20 (2004) 173–180. J.M. Herve, Lie group of rigid body displacements, a fundamental tool for mechanism design, Mech. Mach. Theory 34 (1999) 719–730. O. Piccin, B. Bayle, B. Maurin, M. de Mathelin, Kinematic modeling of a 5-DOF parallel mechanism for semi-spherical workspace, Mech. Mach. Theory 44 (2009) 1485–1496. X. Kong, C.M. Gosselin, Type synthesis of 5-DOF parallel manipulators based on screw theory, J. Robot. Syst. 22 (2005) 535–547.
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[19] S.-J. Zhu, Z. Huang, Eighteen fully symmetrical 5-Dof 3R2T parallel manipulators with better actuating modes, Int. J. Adv. Manuf. Technol. 34 (2007) 406–412. [20] B. Motevalli, H. Zohoor, S. Sohrabpour, Structural synthesis of 5 Dofs 3t2r parallel manipulators with prismatic actuators on the base, Robot. Auton. Syst. 58 (2010) 307–321. [21] G. Gogu, Structural Synthesis of Maximally Regular T3r2-Type Parallel Robots Via Theory of Linear Transformations and Evolutionary Morphology, Cambridge University Press, 40 West 20th Street, New York, NY 10011-4211, United States, 2009. 79–101. [22] K.H. Hunt, Kinematic Geometry of Mechanisms, Oxford University Press, Oxford, 1978. [23] Z. Huang, Q. Li, H. Ding, Theory of Parallel Mechanisms, Springer, Dordrecht, 2012.
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Computer-aided structural synthesis of 5-DOF parallel mechanisms and the establishment of kinematic structure databases Huafeng Ding a,b,⁎, Wenao Cao a, Changwang Cai a, Andrés Kecskeméthy c a b c
Hebei Provincial Key Laboratory of Parallel Robot and Mechatronic System, Yanshan University, 066004 Qinhuangdao, China State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, 200240 Shanghai, China Mechanics and Robotics, University of Duisburg-Essen, 47048 Duisburg, Germany
a r t i c l e
i n f o
Article history: Received 8 April 2014 Received in revised form 3 August 2014 Accepted 23 August 2014 Available online xxxx Keywords: Computer-aided Structural synthesis 5-DOF parallel mechanism Kinematic structure database
a b s t r a c t Structural synthesis is one of the most important steps in the conceptual design of mechanisms and robots. The existing synthesis methods for spatial mechanisms are mainly performed by manual enumeration, which is hard to realize the automation and computerization of the conceptual design of mechanisms and robots. This paper proposes a computer-aided method for the synthesis of both symmetrical and asymmetrical 5-DOF parallel mechanisms and the establishment of corresponding kinematic structure databases, which is conductive to kinematic structure design in human–computer interaction. First, a new structural representation is proposed, with which the complete topological information of a parallel mechanism can be represented by a string convenient for computer processing. Then, the computerized synthesis of 5-DOF limbs is performed and the corresponding limb database is established. Third, a computer-aided synthesis method for 5-DOF parallel mechanisms is developed and the corresponding kinematic structure databases are established. Finally, based on the databases, the structural design of 5-DOF parallel mechanisms applied to computed tomography (CT) scanners is conducted, illustrating the effectiveness of the method. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Structural synthesis plays an important role in the conceptual design of mechanisms and robots, and the research on the issue has been a hot topic for over a century and continues to be so [1]. With the development of computer technology, the computerized structural synthesis becomes feasible, which can save human labor and benefit further automation and intelligence of mechanical conceptual design. Dobrjanskyj and Freudenstein [2] made a first attempt to synthesize planar mechanisms automatically. Later, computer-aided structural synthesis of planar mechanisms has been studied by many researchers [3–8], and atlas databases of some planar mechanisms have been established [7,8], with which the designers can conveniently select the feasible kinematic structures for a specified design task in the conceptual design stage. However, computer-aided structural synthesis of spatial mechanisms can be found in little literature. The existing relevant methods are performed by means of manual enumeration. Huang and Li [9] proposed the constraint-synthesis method based on screw theory and enumerated a lot of symmetrical spatial parallel mechanisms. Kong and Gosselin [10], and Fang and Tsai [11] also synthesized many parallel mechanisms based on the principle of constraint synthesis. Gao et al. [12,13] proposed the concept of Gf
⁎ Corresponding author at: Hebei Provincial Key Laboratory of Parallel Robot and Mechatronic System, Yanshan University, 066004 Qinhuangdao, China. E-mail address: [email protected] (H. Ding).
http://dx.doi.org/10.1016/j.mechmachtheory.2014.08.013 0094-114X/© 2014 Elsevier Ltd. All rights reserved.
H. Ding et al. / Mechanism and Machine Theory 83 (2014) 14–30
15
set to synthesize parallel mechanisms. In addition, the synthesis methods based on displacement group theory were studied by some scholars [14–16]. With those manual methods, a lot of spatial parallel mechanisms have been synthesized. However, those spatial mechanisms were scattered in literature so that designers can hardly find the desired kinematic structures for a specified design task. Moreover, it is difficult to synthesize systematically asymmetrical parallel mechanisms for their great quantities and varieties. The study of automatic structural synthesis of mechanisms and robots is an effective solution to those problems. 5-DOF parallel mechanisms, which can output three-rotation and two-translation (3R2T) or three-translation and two-rotation (3T2R), have a wide range of applications such as medical devices, welding equipment and machine tools. Piccin et al. [17] proposed a 5-DOF parallel mechanism applied to computed tomography (CT) scanner. Kong and Gosselin [18] synthesized some 5-DOF parallel manipulators based on screw theory. Zhu and Huang [19] enumerated eighteen fully symmetrical 3R2T parallel manipulators with better actuating modes. Motevalli et al. [20] synthesized 3T2R parallel manipulators with prismatic actuators on the base. Li and Huang [15] synthesized 3R2T parallel mechanisms using group theory. Gogu [21] synthesized 3T2R parallel robots via the theory of linear transformations and evolutionary morphology. However, asymmetrical 5-DOF parallel mechanisms with great quantities and varieties have not been synthesized systematically, and computer-aided structural synthesis of 5-DOF mechanisms has not been investigated. In order to realize the computerization of conceptual design of mechanisms and robots, in this paper, we propose a computeraided method to synthesize 5-DOF parallel mechanisms with both symmetrical and asymmetrical structures and establish corresponding kinematic structure databases. The paper is organized as follows. In Section 2, basic concepts of screw theory are given. In Section 3, the structural representation of parallel mechanisms is presented. In Section 4, 5-DOF limbs are synthesized and the corresponding database is established. In Section 5, a computer-aided method for the structural synthesis of 5-DOF parallel mechanisms is proposed. In Section 6, one of the applications is discussed for computed tomography (CT) scanner. 2. Basic concepts In screw theory [22], a unit screw $ is defined by a pair of vectors: $ ¼ ðS; S 0 Þ ¼ ðS; r S þ pS Þ ¼ ða b c; d e f Þ
ð1Þ
where S denotes the unit vector specifying the direction of the screw axis, r denotes the position vector of any point on the screw axis with respect to a reference frame, and p is called the pitch. The unit screw associated with a prismatic pair or a constraint couple can be expressed by (0; S) with p = ∞, and the unit screw associated with a revolute pair or a constraint force is expressed by (S; r × S) with p = 0. For two given screws, $ = (S; S0) and $ r = (Sr; Sr0), their reciprocal product is defined as r
r
$ ∘ $ ¼ S • S0 þ S0 • S
r
ð2Þ
where “°” denotes the reciprocal product. If $ ∘ $ r is equal to zero, $ denotes a twist (motion screw), and $ r denotes a wrench (constraint screw). The geometrical relation of two reciprocal screws can be also expressed as follows [23]. Proposition 1. The wrench of a constraint force is reciprocal to the twist of a revolute joint if and only if their axes are coplanar (parallel or intersecting) with each other; the wrench of a constraint force is reciprocal to the twist of a prismatic joint if and only if their axes are perpendicular to each other; the twist of a prismatic joint is always reciprocal to the wrench of a constraint couple. For a limb, when the types of joints and the geometrical relations between joint axes are known, the constraints contributed by the limb can be easily determined according to Proposition 1. 3. Structure representation of parallel mechanisms Generally, a parallel mechanism consists of a moving platform connected to a fixed platform by several serial limbs, and its kinematic structure is completely determined by the types of kinematic pairs and the invariant geometrical relations between joint axes. Here, a structure representation of parallel mechanisms is proposed as follows. 3.1. Representation of limbs 3.1.1. Types of kinematic pairs Kinematic pairs can be represented by some simple symbols, as shown in Table 1. Composite pairs U, S and C can be regarded as the combinations of two R pairs with orthogonal axes, three R pairs with concurrent axes, and one R pair and one P pair with a collinear axis, respectively. Therefore, a limb can be considered as the one only containing pairs R and P. For convenience, kinematic pairs of a limb are labeled with numbers 1 ~ t from the fixed platform to the moving platform in turn.
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Table 1 Representations of kinematic pairs. Pairs
Prismatic
Revolute
Cylindric
Universal
Spherical
Representations
P
R
C
U
S
3.1.2. Relations between adjacent joint axes The geometrical relations between adjacent joint axes are invariant because they are connected by rigid links. Such relations include six types listed in Table 2. 3.1.3. Sub-chains Three adjacent R or P pairs generating planar motion constitute a 3-DOF planar sub-chain, in which all the R pairs are parallel to each other and all the P pairs are perpendicular to the R pairs. Any two pairs of a 3-DOF planar sub-chain constitute a 2-DOF planar sub-chain, such as RR, RP and PP. Generally, the normal of the motion plane of a planar sub-chain is called the normal of the subchain. Three concurrent R pairs form a 3-DOF spherical sub-chain, and two intersecting R pairs form a 2-DOF spherical sub-chain. A spherical pair and a universal pair can be regarded as a 3-DOF and a 2-DOF spherical sub-chain, respectively. The representations of those planar and spherical sub-chains are shown in Table 3. 3.1.4. Invariant relations between nonadjacent joint axes It is not required to represent the variant relation between two nonadjacent joint axes of a limb in finite continual motion. However, under the condition of some special assemblies, the geometrical relation of two nonadjacent joints in a limb may remain invariant in finite motions. Thus, such relation requires considering, mainly including two cases. One is the geometrical relation of two pairs assembled at two ends of P pairs, for example, the perpendicular relation between the axes of R1 and R2 in Fig. 1(a). The other is the invariant geometrical relation of two pairs connected by a planar sub-chain, for example, the intersecting relation of the axes of R5 and R1 connected by the 3-DOF planar sub-chain consisting of R2, R3 and R4 in Fig. 1(b). 3.1.5. Generation of representation string The representation string of a limb can be generated as follows. First, generate a string following the sequence: the type of pair 1, the relation between the axes of pair 1 and pair 2, the type of pair 2, the relation between the axes of pair 2 and pair 3, …, and up to the type of pair t. Second, check planar or spherical sub-chains in the limb. If so, replace the corresponding parts in the string with the representations of the sub-chains based on Table 3. Third, check invariant nonadjacent geometrical relations in the limb. If there exists such a relation between a pair and a former one, add the labeled number of the former pair and the nonadjacent geometrical relation to the end of the representation of the pair by enclosing them with parenthesis. For example, the representation of the limb shown in Fig. 1(b) is “R + R/R/R + R(1^)”. Finally, add the assembly relations between pair 1 and the fixed platform and between pair t and the moving platform to the beginning and end of the string, respectively. The relation between a pair and a platform (line-plane relation) can be classified into three types: parallel (/), orthogonal (+) and intersecting (^). If a limb is connected to a platform by a P pair of a planar sub-chain, add the relation between the normal of the sub-chain and the platform to the end of the representation of the sub-chain by enclosing it with parenthesis. 3.2. Representation of platforms The representation of a platform consists of the number of limbs and the relations among the axes of the joints connected to the platform. If a limb is connected to a platform by a P pair of a planar sub-chain, the normal of the sub-chain is represented rather than the axis of the P pair. For a parallel mechanism with symmetrical structures, since the joints (planar sub-chains) connected to a platform are assembled symmetrically, the relations among those joint axes (the normals of planar sub-chains) may be parallel, concurrent, forming a polygon and skew, represented by symbols ‘/’, ‘*’, ‘#’ and ‘!’, respectively. Table 2 Representations of geometrical relations between adjacent joint axes. Adjacent joints
Relations Representations
Parallel (/) R/R
Perpendicular (⊥) R⊥R
Orthogonal (+) R+R
Intersecting (^) R^R
Skew (!) R!R
collinear (|) R|R
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Table 3 Representations of sub-chains. Planar sub-chains Types Representations
PRP P + R◇P
Spherical sub-chains
PPR P + P◇R
RPP R + P◇P
RPR R + P◇R
PRR P + R◇R
RRP R/R◇P
PP P△P
PR P△R
RRR R^R*R
RRR R + R*R
3.3. Representation of parallel mechanisms The representation string of a parallel mechanism can be generated as follows: “The representation of the fixed platform — the representation of limb 1 ~ the representation of limb 2 ~ … — the representation of the moving platform”. If some of limbs have the identical representation, for convenience, they can be represented by any one of them. Example 1 The parallel mechanism shown in Fig. 2(a) contains two identical limbs UPR and another limb SPR. It can be represented as “3(1|2!3)-2/R + R + P◇R/~^R^R*R^P△R/-3(1/2⊥3)”. Example 2 The parallel mechanism shown in Fig. 2(b) contains three identical limbs and it can be represented as “3#-/R/R/R⊥R/R/-3*”. 4. Synthesis of 5-DOF limbs and limb database 5-DOF parallel mechanisms can be constructed by only 5-DOF and 6-DOF limbs. Without loss of generality, the following conditions are imposed: (1) Only the 5-DOF limbs which can contribute constraint forces or couples are concerned; (2) 5-DOF limbs including idle DOFs are excluded because they can be replaced by the ones with lower pairs; (3) 5-DOF limbs are regarded as the ones only consisting of pairs R and/or P because composite pairs C, U and S can be expressed as “R|P”, “R + R” and “R^R*R” according to the above representation; (4) Since 6-DOF limbs cannot contribute constraints according to screw theory, only two typical 6-DOF limbs UPS, URS are concerned and the geometrical relations of the two limbs are not required considering. 4.1. Synthesis of 5-R limbs Step 1 Generate all possible combinations of the geometrical relations between adjacent revolute joint axes, shown in Table 4. Since the geometrical relations of each two adjacent R pairs may be +, /, ^, ! and ⊥, the total number of combinations is 54, namely 625. Step 2 Generate the limbs by adding ‘R’ to two sides of each adjacent geometrical relation, shown in Table 5. For example, for a combination “//!^”, the corresponding limb can be obtained as “R/R/R!R^R”. Step 3 Obtain new limbs by replacing those limbs containing three successively intersecting or orthogonal revolute pairs with corresponding spherical sub-chains. For example, the new limb “R/R^R*R⊥R” can be gained from the one “R/R^R^R⊥R”. Step 4 Delete the limbs including idle DOFs. For example, the limb “R/R/R/R!R” should be deleted because it can be replaced by the one “R/R/R!R”. Step 5 Obtain new limbs by adding nonadjacent geometrical relations. For example, the limb “R + R/R/R + R(1^)” can be obtained from the one “R + R/R/R + R”. That is because two R pairs connected a 3-DOF planar sub-chain may remain intersecting. Step 6 Choose the limbs which can provide a constraint force or couple according to Proposition 2.
Fig. 1. Invariant relations between two nonadjacent kinematic joint axes.
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Fig. 2. (a) 3(1|2!3)-2/R + R + P◇R/~^ R^R*R^P△R/-3(1/2⊥3), (b) 3#-/R/R/R⊥R/R/-3*.
Proposition 2. A 5-R limb providing a constraint couple (called couple-limb) must contain a 3-DOF planar sub-chain and a 2-DOF planar sub-chain, and a 5-R limb providing a constraint force (called force-limb) must contain a 3-DOF planar sub-chain or a 3-DOF spherical sub-chain. Proof. Because the constraint couple of a limb must be perpendicular to all the R pairs of the limb according to Proposition 1, a 5-R couple-limb must contain two and only two different axis directions. Thus, five R pairs can be divided into two groups: One constitutes a 3-DOF planar sub-chain and the other constitutes a 2-DOF planar sub-chain. Without loss of generality, 5-R limbs can be classified into the following four cases. Case 1. Containing a 3-DOF planar sub-chain. Generally, such a limb “R/R/R!R!R” is shown in Fig. 3(a). Choose a point A on axis L2 of R4 and make a line L3 parallel to axis L1 of R3 through point A. Assume that point B is the intersecting point between axis L4 of R5 and plane γ constituted by lines L2 and L3. Through point B, make a line L5 parallel to L3. So, line L5 is intersecting with both the axes of R4 and R5 and parallel to the axes of R1, R2 and R3. Therefore, line vector L5 represents the constraint force of the limb according to Proposition 1. Case 2. Only containing a 2-DOF planar sub-chain or without any planar sub-chain. Generally, such a limb “R/R!R!R!R” is shown in Fig. 3(b). Make a line L3 parallel to axis L1 of R2 through point A on L2. Assume that points B and C are respectively the intersecting points of axes L4, L5 and plane γ constituted by lines L2 and L3. Generally, line BC is not parallel to L3. Therefore, there doesn't exist any line vector coplanar with the axes of the five R pairs. Namely, the limb cannot provide a constraint force. Similarly, for a limb without any planar sub-chain, for example, the one “R!R!R!R!R”, cannot provide a constraint force. Case 3. Containing a 3-DOF spherical sub-chain. Generally, such a limb “R^R*R!R!R” is shown in Fig. 3(c). Assume that axis L2 of R5 and plane γ constituted by point A and axis L1 of R4 are intersecting at point B. Then, line AB is intersecting with all the revolute joint axes. Thus line vector AB represents the constraint force of the limb. Case 4. Only containing a 2-DOF spherical sub-chain or without any spherical sub-chain. Generally, such a limb “R^R!R!R!R” is shown in Fig. 3(d). Assume that points B and C are respectively the intersecting points of axes L2, L3 and plane γ constituted by point A and axis L1 of R3. Generally, line BC doesn't pass through point A. Therefore, there doesn't exist any line vector coplanar with every axis of the five R pairs. Namely, the limb cannot provide a constraint force. Similarly, a limb without spherical sub-chains cannot provide a constraint force. Thus, for a 5-R limb containing a 3-DOF planar or spherical sub-chain, there is always a line vector coplanar with all the R pairs of the limb. Otherwise, there is no such a line vector. Thus, a 5-R force-limb must contain a 3-DOF planar or spherical sub-chain.
Table 4 Combinations of adjacent geometrical relations. No. 1 No. 2 No. 3 No. 4 No. 5
//// //^^ //++ //⊥⊥ //!!
No. 6 No. 7 No. 8 No. 9 No. 10
///^ //^+ //+⊥ //⊥! /^//
No. 11 No. 12 No. 13 No. 14 No. 15
///+ //^⊥ //+! //!/ /^/^
No. 16 No. 17 No. 18 No. 19 …
///⊥ //^! //⊥/ //!^ …
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Table 5 Limbs with only adjacent geometrical relations. No. 1 No. 2 No. 3 No. 4 No. 5
R/R/R/R/R R/R/R^R^R R/R/R + R + R R/R/R⊥R⊥R R/R/R!R!R
No. 6 No. 7 No. 8 No. 9 No. 10
R/R/R/R^R R/R/R^R + R R/R/R + R⊥R R/R/R⊥R!R R/R^R/R/R
No. 11 No. 12 No. 13 No. 14 No. 15
R/R/R/R + R R/R/R^R⊥R R/R/R + R!R R/R/R!R/R R/R^R/R^R
No. 16 No. 17 No. 18 No. 19 …
R/R/R/R⊥R R/R/R^R!R R/R/R⊥R/R R/R/R!R^R …
4.2. Synthesis of 5-RP limbs Since a planar chain constituted only by R pairs is kinematically equivalent to the one constituted by pairs R and P, 5-RP limbs can be obtained from the synthesized 5-R limbs by replacing some R pairs with corresponding P pairs. For a 5-R limb containing a planar sub-chain, one or two R pairs of the sub-chain can be replaced with P pairs. For a 5-R limb containing a 3-DOF spherical sub-chain, except three R pairs constituting the spherical sub-chain, the remaining two R pairs are kinematically equivalent to two P pairs, since the spherical sub-chain can provide the parallel rotations to the two R pairs by linear combinations so that two corresponding 2-DOF planar sub-chains can be generated. For example, for the limb “R^R*R!R!R” shown in Fig. 4(a), the spherical sub-chain consisting of R1, R2 and R3 can provide the rotation R41 parallel to R4 and the rotation R51 parallel to R5. So an equivalent limb “R^R*R!P!P” can be obtained, shown in Fig. 4(b). The 5-RP limbs derived from 5-R ones with planar sub-chains and spherical sub-chains are shown in Tables 6 and 7, respectively. 4.3. 5-DOF limb database According to Proposition 1, the following characteristics can be obtained for the constraints of 5-DOF limbs: (1) For a 5-DOF force-limb containing a 3-DOF planar sub-chain, the direction of its constraint force is determined, which is the normal of the sub-chain. Except the planar sub-chain, if the remaining two pairs constitute a 2-DOF spherical sub-chain, the position of the constraint force is further determined. Namely, the constraint force passes through the intersecting point of the spherical sub-chain. (2) For a 5-DOF force-limb containing a 3-DOF spherical sub-chain, the position of its constraint force is determined. Namely, the constraint force passes through the intersecting point of the spherical sub-chain. Except the sub-chain, if the remaining two pairs constitute a 2-DOF planar sub-chain, the direction of the constraint force is further determined, which is the normal of the sub-chain. If the remaining two pairs constitute a 2-DOF spherical sub-chain as well, the direction of the constraint force is further determined, which is along the line through the two intersecting points of the two spherical sub-chains. (3) For a 5-DOF couple-limb, the direction of its constraint couple is determined, which is perpendicular to the axes of R pairs. For a 5-DOF limb, the above-mentioned constraint characteristics are invariant (full-cycle) in continual motion, so they are called full-cycle constraint characteristics of the limb, which can be described by a six-dimensional array {A1, B1, C1, D1, E1, F1}. The first three parameters A1, B1 and C1 denote the characteristics of the constraint direction. The last three parameters D1, E1 and F1 denote the characteristics of the constraint position. For a 5-DOF force-limb, parameters A1, B1 and C1 can be determined as: If a planar sub-chain connects to the fixed platform, A1 = 1, B1 = 0 and C1 = 0; If a planar sub-chain connects to the moving platform, C1 = 1, A1 = 0 and B1 = 0; If a planar sub-chain locates
Fig. 3. (a) ‘R/R/R!R!R’ limb, (b)‘R/R!R!R!R’ limb, (c) ‘R^R*R!R!R’ limb, (d) ‘R^R!R!R!R’ limb.
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Fig. 4. (a) A limb including a spherical pair, (b) its kinematically equivalent limb.
between the two platforms, B1 = 1, A1 = 0 and C1 = 0. Parameters D1, E1 and F1 can be determined as: If a spherical sub-chain without orthogonal relation connects to the fixed platform, D1 = 1, E1 = 0 and F1 = 0, otherwise, D1 = 2, E1 = 0 and F1 = 0; If a spherical sub-chain without orthogonal relation connects to the moving platform, D1 = 0, E1 = 0 and F1 = 1, otherwise, D1 = 0, E1 = 0 and F1 = 2. For a 5-DOF couple-limb, both parameters A1 and C1 are 1 and other parameters B1, D1, E1 and F1 are always 0. 5-DOF limb database can be established based on the synthesized limbs and their full-cycle constraint characteristics, shown in Fig. 5. 5. Synthesis of 5-DOF parallel mechanisms and kinematic structure databases In a parallel mechanism, the constraints imposed on the moving platform by the limbs and the geometrical relations between such constraints constitute the constraint mode of the mechanism. From the view of linear algebra, those constraints span a linear space, called the platform constraint space. The constraints provided by a limb also span a space, called the limb constraint space. The relations between the platform and limb constraint spaces are fP g ¼ fL1 g∨fL2 g∨fL3 g∨⋯∨fLn g
ð3Þ
where {P} denotes the platform constraint space, {Li} the constraint space of i-th limb, and ∨ the union operation. From Eq. (3), each limb constraint space is a subspace of the platform constraint space. For 5-DOF parallel mechanisms, the feasible constraint modes can be obtained according to the following steps: Step 1 Specify the desired output motions, namely, 3T2R or 3R2T. Step 2 Determine the independent constraint imposed on the moving platform according to the reciprocal equation, and represent the platform constraint space spanned by the independent constraint. Step 3 Enumerate all subspaces of the platform constraint space, each of which is just a possible limb constraint space of the limb for constructing the desired parallel mechanisms. Further, determine the possible constraint provided by a limb. Step 4 Enumerate all the feasible constraint modes according to the number of limbs providing constraint. When a feasible constraint mode is given, the desired parallel mechanisms corresponding to the mode can be synthesized as follows. (1) Identify the constraint characteristics of the limbs for constructing the parallel mechanisms corresponding to the mode and the geometrical relations between the directions and positions of the constraints of different limbs. (2) Choose the feasible limbs from the 5-DOF limb database according to the constraint characteristics. (3) Generate all combinations of the chosen limbs. Table 6 5-RP limbs with a planar sub-chain. No. 1 No. 2 No. 3 No. 4
P + P◇R^R + R P + P◇R + R + R R!P + R◇P!R R!R|P + P◇R
No. 5 No. 6 No. 7 No. 8
R + P◇R^R + R R + P◇R + R + R R!R + P◇P!R R!R⊥P + P◇R
No. 9 No. 10 No. 11 No. 12
P◇R/R^R + R P◇R/R + R + R R + R + P◇P!R R + R⊥P + P◇R
No. 13 No. 14 No. 15 …
R + P◇P!R + R R + P◇P|R + R R!R/R◇P!R …
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Table 7 5-RP limbs with a spherical sub-chain. No. 1 No. 2 No. 3 No. 4
R^R*R/R + P R^R*R/R/P P!R + R^R*R P|R + R^R*R
No. 5 No. 6 No. 7 No. 8
R^R*R + R + P R^R*R + R/P P!R^R^R*R P|R^R^R*R
No. 9 No. 10 No. 11 No. 12
R^R*R!R + P R^R*R!R/P P!R!R^R*R P/R/R^R*R
No. 13 No. 14 No. 15 …
R^R*R^R + P R^R*R^R/P P!R⊥R^R*R …
(4) Establish assembly conditions between the joints (or sub-chains) of different limbs on platforms for each limb combination, according to the relations between the constraints of different limbs. (5) Traverse all assembly relations between single limb and platforms. (6) Store the synthesized parallel mechanisms in corresponding kinematic structure databases. 5.1. Three-rotation and two-translation (3R2T) parallel mechanisms The moving platform is imposed by an independent constraint force, and the platform constraint space can be represented as r fP g ¼ k $ F ¼ fkð a
b c; 0
0
0 Þðk ∈RÞg:
ð4Þ
There are two subspaces. One is {P} itself and the other is null space {0}. Thus, the constraint provided by a limb for constructing 3R2T parallel mechanisms is r
$l ¼
r
A ¼ $ F ¼ ð a b c; 0 B¼0
0
0Þ:
ð5Þ
Fig. 5. 5-DOF limb database.
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Table 8 Constraint modes of 3R2T parallel mechanisms with five limbs. Mode 1: 5A
Mode 2: 4A1B
Mode 3: 3A2B
Mode 4: 2A3B
Mode 5: 1A4B
For 3R2T parallel mechanisms containing five limbs, according to the number of limbs for contributing the constraint force, five feasible constraint modes can be obtained, shown in Table 8. For 3R2T parallel mechanisms including four limbs, according to the number of the limbs contributing the constraint force, four feasible constraint modes can be obtained, shown in Table 9. Likewise, for 3R2T parallel mechanisms including three and two limbs, there are five feasible constraint modes, shown in Table 10. Take mode 1 as an example to show the synthesis process. Five constraint forces (5A) from different limbs have the identical direction and pass through a common point. One possibility is that 3R2T mechanisms are constructed by the 5-DOF limbs with the constraint directions determined by the planar sub-chains connected to the fixed platform and constraint positions determined by the spherical sub-chains connected to the moving platform. The full-cycle constraint characteristic of such limbs is {1 0 0 0 0 1}, called 3R2T-type-1 limbs, for example, the limb “R/R/R!R^R”; The other possibility is that 3R2T mechanisms are constructed by the 5-DOF limbs with the constraint directions determined by the planar sub-chains connected to the moving platform and constraint positions determined by the spherical sub-chains connected to the fixed platform. The constraint characteristic of such limbs is {0 0 1 1 0 0}, called 3R2T-type-2 limbs, for example, the limb “R^R*R!R/R”; Another possibility is that 3R2T mechanisms are constructed by the 5-DOF limbs with two spherical sub-chains connected to the fixed and moving platform, respectively. The constraint characteristic of such limbs is {0 0 0 1 0 1}, called 3R2T-type-3 limbs, for example, the limb “R^R*R!R^R”. Take 3R2T-type-1 limbs for example. Choose such limbs from the 5-DOF limb database, shown in Table 11. Generate the combinations of any five 3R2T-type-1 limbs, shown in Table 12.
Table 9 Constraint modes of 3R2T parallel mechanisms with four limbs. Mode 6: 4A
Mode 7: 3A1B
Mode 8: 2A2B
Mode 9: 1A3B
Table 10 Constraint modes of 3R2T parallel mechanisms with three and two limbs. Mode 10: 3A
Mode 11: 2A1B
Mode 12: 1A2B
Mode 13: 2A
Mode 14:1A1B
B
Table 11 3R2T-type-1 limbs. No. 1 No. 2 No. 3 No. 4
R/R/R + R^R R/R!R^R*R R△P|R^R*R P△R!R^R*R
No. 5 No. 6 No. 7 No. 8
R/R/R⊥R^R R△P/R^R*R P△P/R^R*R P + R◇R^R^R
No. 9 No. 10 No. 11 No. 12
R/R/R!R^R R△P + R^R*R P△R/R^R*R P + P◇R^R^R
No. 13 No. 14 No. 15 …
R/R^R^R*R R△P^R^R*R P△R + R^R*R …
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Table 12 Combinations of any five 3R2T-type-1 limbs. No. 1 No. 2 No. 3 No. 4
R/R/R!R^R~R/R^R^R*R~R/R + R^R*R~R/R⊥R^R*R~R/R!R^R*R R/R/R^R^R~R/R/R!R^R~R/R^R^R*R~R/R + R^R*R~R/R!R^R*R R/R/R⊥R^R~R/R^R^R*R~R/R + R^R*R~R/R⊥ R^R*R~R/R!R^R*R P + R◇R(+)^R^R~P + P◇R(+)^R^R~P + P◇R(+) + R^R~P + P◇R(+)!R^R~P + R◇P(+)!R^R
No. 5 No. 6 No. 7 …
R/R/R!R^R~R/R/R!R^R~R/R/R!R^R~R/R/R!R^R~R/R/R!R^R R/R/R⊥R^R~R/R/R!R^R~R/R^R^R*R~R/R + R^R*R~R/R!R^R*R R/R!R^R*R~R/R!R^R*R~R/R/R!R^R~R/R/R^R^R~R/R/R + R^R …
Fig. 6. The kinematic structure database of 3R2T parallel mechanisms.
Fig. 7. The interface of 3R2T parallel mechanisms.
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Table 13 Constraint modes of 3T2R parallel mechanisms with five limbs. Mode 1: 5D
Mode 2: 4D1E
Mode 3: 3D2E
Mode 4: 2D3E
Mode 5: 1D4E
Table 14 Constraint modes of 3T2R parallel mechanisms with four limbs. Mode 6: 4D
Mode 7: 3D1E
Mode 8: 2D2E
Mode 9: 1D3E
For each combination of 3R2T-type-1 limbs, its assembly satisfies: The normals of all planar sub-chains connected to the fixed platform are parallel to each other to obtain the identical constraint direction. The intersecting points of all spherical sub-chains connected to the moving platform are coincident to obtain the identical constraint position. For single 3R2T-type-1 limb, the normal of its planar sub-chain may be parallel (/) to, orthogonal (+) to or intersecting (^) with the fixed platform. The intersecting points of its spherical sub-chain may be on or outside the moving platform, so the pair connected to the moving platform may be parallel (/) to, or intersecting (^) with the moving platform. The parallel geometrical relation between the normals of the planar sub-chains connected to the fixed platform is invariant in continual motion. The coincident geometrical relation between the spherical sub-chains connected to the moving platform is also invariant in continual motion. Thus, the constraint forces provided by different limbs remain collinear and the synthesized mechanisms have full-cycle DOFs. Likewise, 3R2T parallel mechanisms corresponding to other feasible constraint modes can be also synthesized. The synthesized parallel mechanisms can be stored in a kinematic structure database with some kinematic characteristics, such as DOFs, output motions, the number of constraint limbs, the number of identical limbs, and the number of concurrent axes, shown in Fig. 6. A human–computer interactive interface has been developed, shown in Fig .7. 5.2. Three-translation and two-rotation (3T2R) parallel mechanisms The moving platform is imposed by one independent constraint couple, and the platform constraint space can be represented as r fP g ¼ k $ C ¼ fkð 0 0 0; a
b
c Þðk ∈ R Þg:
ð6Þ
The subspaces are {P} itself and null space {0}. Thus, the constraint provided by a limb for constructing 3T2R parallel mechanisms is r
$l ¼
r
D ¼ $C ¼ ð 0 E¼0
0
0; a
b
cÞ:
ð7Þ
Table 15 Constraint modes of 3T2R parallel mechanisms with three and two limbs. Mode 10: 3D
Mode 11: 2D1E
Mode 12: 1D2E
Mode 13:2D
Mode 14: 1D1E
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Table 16 3T2R-type limbs. No. 1 No. 2 No. 3 No. 4
R/R/R^R/R R/R⊥R/R/R P + P◇R + R/R P + P◇R!R/R
No. 5 No. 6 No. 7 No. 8
R/R/R + R/R R/R!R/R/R R + P◇R + R/R R + P◇R!R/R
No. 9 No. 10 No. 11 No. 12
R/R/R⊥R/R P + R◇R^R/R R/R◇P|R/R R/R^R + P◇R
No. 13 No. 14 No. 15 …
R/R/R!R/R P + P◇R^R/R P + R◇P|R/R …
Table 17 Combinations of any five 3T2R-type limbs. No. 1 No. 2 No. 3 No. 4
R/R/R!R/R~R/R/R + R/R~R/R^R/R/R~R/R + R/R/R~R/R!R/R/R R/R/R!R/R~R/R/R!R/R~R/R/R!R/R~R/R/R!R/R ~R/R/R!R/R R/R/R + R/R~R/R/R⊥R/R~R/R + R/R/R~R/R⊥R/R/R~R/R/R!R/R P + R◇R(/) + R/R~P + P◇R(/) + R/R~R/R◇P/R/R~R/R|P + R◇R~R/R⊥R + P◇R
No. 5 No. 6
R/R/R!R/R~R/R/R + R/R~R/R⊥R/R/R~R/R + R/R/R~R/R!R/R/R R/R/R^R/R~R/R/R⊥R/R~R/R/R!R/R~R/R/R + R/R~R/R⊥R/R/R
No. 7 …
R/R/R + R/R~R/R^R/R/R~R/R⊥R/R/R~R/R + R/R/R~R/R!R/R/R …
For 3T2R parallel mechanisms with five limbs, the number of the limbs provided the constraint couple may be five, four, three, two and one, so there are five feasible constraint modes, shown in Table 13. Likewise, for 3T2R parallel mechanisms with four limbs, four feasible constraint modes can be obtained, shown in Table 14. For 3T2R parallel mechanisms with three and two limbs, five feasible constraint modes can be obtained, shown in Table 15. Take mode 1 as an example. Five constraint couples (5D) have identical direction. Correspondingly, 3T2R parallel mechanisms can be constructed by 5-DOF couple-limbs whose constraint characteristics are {1 1 0 0 0 0}, called 3T2R-type limbs. Choose all 3T2R-type limbs from the limb database, shown in Table 16. Generate all combinations of any five 3T2R-type limbs, shown in Table 17. For each combination, its assembly satisfies: The R pairs nearest to the fixed platform in different limbs are parallel to each other; The R pairs nearest to the moving platform in different limbs are also parallel to each other. Both the parallel geometrical relations between the R pairs nearest to the fixed platform in different limbs and between the R pairs nearest to the moving platform in different limbs are invariant in continual motion. Thus, the constraint couples provided by different limbs remain parallel and the synthesized mechanisms have full-cycle DOFs.
Fig. 8. The kinematic structure database of 3T2R parallel mechanisms.
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Fig. 9. The interface of 3T2R parallel mechanisms.
Likewise, 3T2R parallel mechanisms corresponding to other feasible constraint modes can be also synthesized. The synthesized parallel mechanisms are also stored in a kinematic structure database with some kinematic characteristics, shown in Fig. 8. Fig. 9 presents a corresponding human–computer interactive interface.
6. Applications A 5-DOF parallel mechanism for CT scanner is proposed in literature [17], which consists of two identical 5R limbs and another limb equivalent to SRU, shown in Fig. 10(a). Although the desired output motion of the mechanism is three-translation and two-rotation (3T2R), in fact, the output motion of the mechanism is changeable and it can be 3T2R or 3R2T.
Fig. 10. Two configurations of the 5-DOF parallel mechanism.
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Fig. 11. A new candidate mechanism.
For limb 1 of the mechanism, R11 is parallel to R15. R12, R13 and R14 are parallel to each other and perpendicular to R11 and R15. R11 and R21 are collinear. R15 and R25 are also collinear. Limb 3 is a 6-DOF limb. The representation of the mechanism is “3(1|2!3)-2/R⊥R/R/ R⊥R(1/)/~SRU-3(1|2!3)”.
Fig. 12. Another new candidate mechanism.
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Fig. 13. Two new typical candidate mechanisms for CT scanner.
A reference frame can be set up on the fixed platform, in which the origin locates at one point in the fixed platform, Z-axis is perpendicular to the fixed platform upward and Y-axis is parallel to R11. For the initial configuration shown in Fig. 10(a), the twist system of limb 1 is 0
$ 11 B $ 12 B $1 ¼ B B $ 13 @$ 14 $ 15
¼ ½0; ¼ ½1; ¼ ½1; ¼ ½1; ¼ ½0;
1; 0; 0; 0; 1;
0; 0; 0; 0; 0;
ðl11 ; ðl12 ; ðl13 ; ðl14 ; ðl15 ;
m11 ; m12 ; m13 ; m14 ; m15 ;
n11 Þ ð0; n12 Þ ð1; n13 Þ ð1; n14 Þ ð1; n15 Þ ð0;
1; 0; 0; 0; 1;
0Þ 0Þ 0Þ 0Þ 0Þ
1 C C C: C A
ð8Þ
According to Eq. (3), the wrench system of limb 1 is r
$1 ¼ ð0; 0; 0; 0; 0; 1Þ
ð9Þ
which denotes a constraint couple along Z-axis. Likewise, the wrench system of limb 2 is also r
r
$ 2 ¼ $ 1 ¼ ð0; 0; 0; 0; 0; 1Þ:
ð10Þ
Thus, the moving platform is imposed by an independent constraint couple along Z-axis and it can output three translation and two-rotation (3T2R) about X-axis and Y-axis. After the moving platform rotates about X-axis, the configuration becomes the one as shown in Fig. 10(b). In this condition, the twist system of limb 1 has changed to 0
$ 11 B$ B 12 B $1 ¼ B B $ 13 B @ $ 14 $ 15
¼ 0; ¼ 1; ¼ 1; ¼ 1; ¼ 0;
0 1; 0; l11 ; 0 0; 0; l 12 ; 0 0; 0; l 13 ; 0 0; 0; l 14 ;
1 0 0 m11 ; n11 ð0; 1; 0Þ 0 0 C m12 ; n12 ð1; 0; 0Þ C C 0 0 C: m13 ; n13 ð1; 0; 0Þ C C 0 0 m14 ; n14 ð1; 0; 0Þ A 0 0 0 cosðβ15 Þ; cosðγ15 Þ; l 11 ; m 11 ; n11 ð0; cosðβ15 Þ; cosðγ15 ÞÞ
ð11Þ
The wrench system of limb 1 is 0 r 0 0 $ 1 ¼ 1; 0; 0; l 11 ; m11 ; n11 ð1; 0; 0Þ which denotes a constraint force parallel to X-axis through the intersecting point of R11 and R15.
ð12Þ
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Likewise, the wrench system of limb 2 is also 0 r 0 0 r $ 2 ¼ $ 1 ¼ 1; 0; 0; l 11 ; m11 ; n11 ð1; 0; 0Þ :
ð13Þ
The moving platform is imposed by an independent constraint force and it can output three-rotation and two-translation (3R2T). Therefore, the mechanism may output three-translation and two-rotation or three-rotation and two-translation in finite continual motion, but three-rotation and two-translation are not the desired output motion. So the mechanism is not necessarily a good one applied to CT scanner. According some kinematic characteristics, such as the number of constraint limbs, the types of kinematic pairs, the desired output motion and the number of identical limbs, some new candidate mechanisms which can always output three-translation and tworotation in finite continual motion can be selected from the established kinematic structure database of 3T2R parallel mechanisms, such as “3(1/2!3)-2/R/R/R⊥R/R/~SRU-3(1/2!3)”, “3(1/2!3)-2/R/R⊥R/R/R/~ SRU-3(1/2!3)”, “3(1/2!3)-2/R/R + R/R/R/~SRU-3 (1/2! 3)” and “3(1/2!3)-2/R/R/R + R/R/~SRU-3(1/2!3)”. Two typical candidate mechanisms “3(1/2!3)-2/R/R/R⊥R/R/~SRU-3(1/2!3)” and “3(1/2!3)-2/R/R⊥R/R/R/~ SRU-3(1/2!3)” from the kinematic structure database are shown in Figs. 11 and 12, respectively. For the mechanism “3(1/2!3)-2/R/R/R⊥R/R/~SRU-3(1/2!3)”, shown in Fig. 13(a), R11, R12 and R13 are parallel to each other. R13 is perpendicular to R14. R14 and R15 are parallel to each other. Limb 2 has the same geometry as limb 1. R11 and R21 are parallel to each other. R15 and R25 are also parallel to each other. Limb 3 is a 6-DOF limb SRU. For the mechanism “3(1/2!3)-2/R/R⊥R/R/R/~SRU-3(1/2!3)”, shown in Fig. 13(b), R11 and R12 are parallel to each other. R12 is perpendicular to R13. R13, R14 and R15 are parallel to each other. Limb 2 has the same geometry as limb 1. R11 and R21 are parallel to each other. R15 and R25 are also parallel to each other. Limb 3 is a 6-DOF limb SRU. Comparing with the existing mechanism for CT scanner, the above two mechanisms not only can always output three-translation and two-rotation in finite continual motion, but also can have simpler assembly requirements. Maybe, the two mechanisms are more suitable for CT scanner. 7. Conclusions In this paper, a computer-aided method for the structure synthesis of 5-DOF parallel mechanisms is proposed, which can be used to systematically synthesize 5-DOF parallel mechanisms with asymmetrical and symmetrical structures. The structural characteristics of 5-DOF limbs which can provide a constraint force or couple are presented. Based on the proposed method, the process of structural synthesis is performed by a computer program, and the corresponding kinematic structure databases have been established. The human–computer interactive interface has been developed for designers. With these features, the kinematic structure design can be performed in a human–computer interactive manner. The relevant study of this paper is expected to benefit the automation and computerization of the mechanical creative design. Acknowledgments The authors are grateful to the project (Nos. 51275437, 51422509) supported by NSFC, the Hebei Nature Science Foundation (Nos. E2014203117, E2012203154), the Research Fund for the Doctoral Program of Higher Education (No. 20131333110018), the Fok YingTong Education Foundation (No. 141049), and the open project of State Key Laboratory of Mechanical System and Vibration (No. MSV2014-15). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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