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Type synthesis of 2T1R-type parallel kinematic mechanisms and the application in manufacturing
Robotics and Computer-Integrated Manufacturing 30 (2014) 1–10
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Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim
Type synthesis of 2T1R-type parallel kinematic mechanisms and the application in manufacturing Fugui Xie a,c,n, Xin-Jun Liu a,b, Zheng You c, Jinsong Wang a a
Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China Beijing Key Lab of Precision/Ultra-precision Manufacturing Equipments and Control, Tsinghua University, Beijing 100084, China c Department of Precision Instruments, Tsinghua University, Beijing 100084, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 6 November 2012 Received in revised form 5 May 2013 Accepted 11 July 2013
This paper investigates the type synthesis of 2T1R-type (T: translational degree of freedom (DoF) and R: rotational DoF) parallel kinematic mechanisms (PKMs). A type synthesis method based on Grassmann Line Geometry and Line-graph Method is introduced. Some basic criterions of Grassmann Line Geometry are briefly summarized, and the Line-graph Method is presented sequentially. In order to uncover the relationship between DoF-line graph and constraint-line graph, the dual rule is brought in and explained in detail. Based on these foundations, the technological process of the type synthesis is given. Thereafter, the type synthesis of 2T1R-type PKMs is carried out and the results are listed. Taken as an application example, a synthesized 3-DoF mechanism is chosen as the parallel module of a five-axis hybrid machine tool, which is capable of five-face machining in one setup. The developed prototype is introduced and applied into the machining of a part with freeform surfaces. The presented type synthesis method is concise and can be used in the type synthesis of other PKMs. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Parallel kinematic mechanism Type synthesis Hybrid machine tool Five-face machining
1. Introduction As the increasing need for the processing of complex parts with freeform surfaces, such as the turbines and marine propellers, multi-axis machine tools with parallel kinematics [1,2] or containing parallel modules [3] are becoming a research hotspot and attracting more and more attentions from both academia and industry. Some such kind of machine tools has been developed in the field. Variax [4] and Hexapod [5] are the first developed and also the most typical machine tools with fully parallel kinematics. They are based on the classical six degrees of freedom (DoFs) fully parallel kinematic mechanism (PKM), i.e., the Gough–Stewart platform [6,7]. Both of them need six-axis computer numerical control (CNC) and have relative small workspace compared to their assembly footprint. Thereafter, a five-axis fully parallel machine tool Metrom [8] is developed based on a 5-DoF PKM. The rotational capability of this machine can achieve 901 [9], which is a very important advantage to the improvement of the machining flexibility. In comparison, the five-axis machine tools containing parallel modules, especially the three-DoF parallel modules [10], are more popular in industry, such as the well-known Tricept machine [11] and Ecospeed series machine centers [12].
n Corresponding author at: Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China. Tel.: +86 10 62781820. E-mail address: [email protected] (F. Xie).
0736-5845/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.rcim.2013.07.002
Tricept machine contains a parallel module with three translational DoFs and a serially added articulated head with two rotational DoFs, its configuration can be represented by P{3T}&S {2R} (P{}: parallel module; S{}: serial module; T: translational DoF; and R: rotational DoF). Similarly, the Exechon [13] and the Verne [14] can also be classified into this category. But the Verne is composed of a three-translational-DoF parallel module and a twoDoF rotary table. In practical application, this type of machine tools has realized the agile machining, but the finished surface is sometimes scratched by the high-speed rotating cutter due to the serially added two-rotational-DoF modules. Ecospeed machine represents another type of machine tools with the configuration of P{1T2R}&S{2T}, in which the Space-5H machine center is included. The parallel module of the Ecospeed machine is called Sprint Z3 [15], which is a 3-PRS [16,17] (P: prismatic joint; R: revolute joint; and S: spherical joint) PKM in principle. The parasitic motions of 3-PRS mechanism have been analyzed in Ref. [18]. Similarly, the Space-5H contains a Hermes tool head [19], which also has complicated parasitic motions. In this category, the 1T2R-type PKMs, i.e., Sprint Z3 and Hermes tool heads, have realized the linked motions of the two rotational DoFs. This characteristic has greatly improved the machining performance and efficiency, but the parasitic motions of these mechanisms have resulted in complicated kinematics in terms of control and are disadvantageous to the further improvement of the machining accuracy. Moreover, the rotational capability is relatively limited.
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In all of the 3T2R-type hybrid machine tools, there is another configuration P{2T1R}&S{1T1R} besides the P{3T}&S{2R} and P {1T2R}&S{2T}, and this configuration may provide a better solution for the five-axis machining application. It is obvious that the two rotational DoFs are respectively assigned into parallel and serial modules in this configuration. Therefore, the problems brought in by the assignment of two rotational DoFs in one module are avoided, and the high rotational capability of the parallel module P{2T1R} can be achieved due to the elimination of the coupling influence of the two rotational DoFs. Serially combined a translational DoF with large traveling capability and a revolute DoF with continuously rotational capability, i.e., the serial module S{1T1R}, the P{2T1R}&S{1T1R}-type machine tools are capable of five-face machining in one setup, which is imperative to modern machine tools characterized by high flexibility and high efficiency. Therefore, the investigation of 2T1R-type PKMs by means of a type synthesis method based on Grassmann Line Geometry will be carried out in this paper, and a machine tool with the configuration of P{2T1R}&S{1T1R} will be introduced. Type synthesis is to bring out the mechanism configuration according to the required DoFs [20,21], and the mechanism configuration is fundamental and decisive to the performance of the developed mechanism and the consequent equipment. It is well known that the creation of a novel mechanism with useful function is a meaningful and also very challenging issue in the domain of PKMs [22]. This issue becomes more complicated for lower mobility PKMs [23], such as the 2T1R-type PKMs, due to the complex constraints in each limb. In general, type synthesis is carried out under the guidance of certain theory, which is the most important and also the most challenging problem to be figured out in the field. For such a reason, a lot of efforts have been contributed to this area, and many effective and novel theories or methods have been proposed. Among them, the methods based on displacement group theory [24,25] of Lie Group Algebra, screw theory [26,27], GF set [28], single-opened-chains units [29] and theory of linear transformation and geometrical analysis [30] are more popular. Based on the presented methods, many interesting mechanisms have been proposed. This has greatly enriched the diversity of the mechanisms. However, these methods are all based on certain complicated mathematic theories. To carry out the type synthesis under the guidance of these methods, designers need to master the relatively complicated procedures and the profound mathematic theories first. This is a barrier to the widespread application of the mentioned methods. In this paper, a concise and visual type synthesis method will be introduced and suggested as the theoretical foundation for the type synthesis of lower mobility PKMs. The remainder of this paper is organized as follows. Section 2 introduces the type synthesis based on Grassmann Line Geometry and Line-graph Method. Some basic criterions and rules are presented, and the technological process is summarized accordingly. Thereafter, the type synthesis of 2T1R-type PKMs based on the introduced method is carried out. Section 3 discusses the application of 2T1R-type PKMs in manufacturing. As an example, a hybrid machine tool capable of five-face milling in one setup is introduced. Section 4 concludes the paper.
2. Type synthesis of 2T1R-type PKMs based on Grassmann Line Geometry and Line-graph Method 2.1. Theoretical foundations: Grassmann Line Geometry and Line-graph Method Grassmann Line Geometry is a systematic theory in mathematics, and can be used to investigate the geometrical features of spatial line-clusters. Under this frame, the linear dependence of line vectors
in a line-cluster can be identified by comparing their number and dimension. This makes Grassmann Line Geometry very suitable to be used in the relevant research in mechanism theory, especially the DoFs and constraints of a mechanism. For such a reason, Grassmann Line Geometry has been extensively applied to PKMs' singularity analysis [31,32] in the field, and its practicability is well affirmed. In the present paper, Grassmann Line Geometry will be used to deal with the type synthesis of 2T1R-type PKMs. 2.1.1. Basic criterions of Grassmann Line Geometry Using Grassmann Line Geometry to carry out the type synthesis of PKMs, the dimensions of the corresponding line-clusters should be identified first. This will be very helpful to investigate the DoFs and constraints of a mechanism, and lay the foundation for the type synthesis. The following criterions can be used as the theory basis for this process: (1) There are at most three independent lines or only two independent parallel lines in a plane; (2) in all of the spatially parallel lines, there are only three independent parallel lines; (3) in all of the coplanar and concurrent lines, there are only two independent lines; (4) among all of the concurrent lines in space, there are only three independent lines; (5) for two sets concurrent lines (or one set concurrent lines and one set parallel lines) in two different planes and the intersections lie in the intersecting line of the two planes, there are only three independent lines; and (6) in two or more parallel planes or planes that intersect at one line, there are at most five independent lines. Under the guidance of these criterions, the dimensions of lineclusters can be identified and the line-clusters can be classified into different categories according to the identified dimensions. Some typical line-clusters under different dimensions are presented in Table 1. 2.1.2. Line-graph Method Based on Grassmann Line Geometry introduced in Section 2.1.1 and the line-clusters listed in Table 1, a mechanism's DoFs and constraints can be expressed by the corresponding line-clusters with assigned physical meanings. In this paper, a line-cluster with specific physical meaning, i.e., DoFs or constraints, is called line graph. In general, a line graph is composed of linear vectors or couples. Imposed the corresponding physical meanings, four basic elements with definite meanings can be generated as shown in Table 2. In Table 2, red color represents constraint; blue color represents DoF; line without arrows represents vector; and line with double arrows represents couple. A line graph constituted by these elements can be used to express an n-dimensional ðn≤6Þ freedomspace or constraint-space. The manner using freedom- or constraint-space line graphs to express the DoFs or constraints of a mechanism is referred as Line-graph Method. By using this method, the DoFs or constraints of a mechanism can be visually investigated. Moreover, the line graphs have clear and definite physical meaning. Therefore, this method is intuitive and concise. Under this expression, the qualitative analysis of the DoFs and constraints of a mechanism can be carried out in an intuitive and visual way. This paper mainly investigates the type synthesis of 2T1R-type PKMs. To illustrate the application of the Line-graph Method, some typical three-dimensional line graphs which can be used to express the 3-DoF motions are presented in Table 3, and the
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Table 1 Typical line-clusters classified by different dimensions. Dimension
Typical line-clusters
1 2
Planar concurrent
Parallel and coplanar
Non-coplanar
3
Spatial concurrent
Coplanar
Concurrent in two different planes
Non-parallel and non-intersecting
Coplanar or passing through one point
Regulus of lines
4
Concurrent with one line
Concurrent with two lines
5
Intersect with one line
Non-singular line-cluster
Table 2 Basic elements in a line graph and the corresponding meanings. Basic element
Mathematic meaning
Physical meaning
Vector
Rotational DoF
Vector
Constraint force
Couple
Translational DoF
Couple
Constraint couple
corresponding physical meanings are also included in this table. If the color of the line graphs in Table 3 is changed into red, the corresponding line graphs can be used to express threedimensional force- or couple-constraints according to the definitions given in Table 2.
2.2. Dual rules and technological process of the type synthesis Based on Grassmann Line Geometry and Line-graph Method, the DoFs and constraints of a mechanism can be visually represented by line graphs. Obviously, a line graph contains a lot of lines and couples, and the number of independent lines or couples in a line graph, i.e., the dimension of the line graph, can be identified according to the criterions listed in Section 2.1.1. In order to carry out type synthesis, the relationship between the DoF-line graphs and the constraint-line graphs should be uncovered. Actually, there exists a close relationship between the two kinds of line graphs. This paper will bring in dual principle of DoFs and constraints to deal with this problem, and the technological process of the type synthesis will be consequently presented.
2.2.1. Dual rule–Blanding rule Different from the traditional serial kinematic mechanism (SKM), the DoFs of the PKMs cannot be generated by a simple superposition of the DoFs of each limb. Therefore, the DoF-line graph cannot be directly decomposed into each limb. In fact, the end-effector of a PKM is constrained by multi-limbs, and the constraints of the end-effector can be generated by a superposition of the constraints of each limb. So, the constraint-line graph instead of the DoF-line graph is used and decomposed to generate the kinematics of each limb in this paper. The DoFs and constraints of a mechanism are a pair of conceptions describing the property of a mechanism from two different aspects. Therefore, they can be mutually transformed. Blanding proposed a basic rule that can reflect the mutually transformable relationship between the DoF-line graphs and constraint-line graphs of a mechanism. This rule is called dual rule or Blanding rule and can be summarized as follows. Provided that a line graph ℜ contains n independent (nonredundant) lines, then the corresponding dual graph ℜ′ contains (6-n) independent (non-redundant) lines and each line in line graph ℜ intersects with all lines in the dual graph ℜ′. According to the Blanding rule, the dual constraint-line graph ℜ′ can be uniquely identified when the DoF-line graph ℜ is given, and vice versa. Taking the physical meaning (DoFs and constraints) of the line graphs presented in Table 2 into account, a generalized Blanding rule [20] can be summarized as (1) The axes of rotational DOFs of a mechanism intersect with the lines of all constraint forces and are orthogonal to the axes of all constraint couples. (2) The axes of translational DOFs of a mechanism are orthogonal to the lines of all constraint forces and are arbitrary to the axes of all constraint couple. This generalized Blanding rule is very helpful to realize the mutual transformation between the DoF-line graphs and the
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Table 3 Some three-dimensional freedom spaces expressed by Line-graph Method. Line graph
Physical meaning
2T1R (Parallel)
2T1R (Vertical)
1T2R (Vertical)
1T2R (Parallel)
3T
3R
constraint-line graphs. On this basis, the investigation of DoFs and constraints of a mechanism can be carried out in a concise and visual way. All of these have laid the theoretical foundation for the type synthesis to be carried out in this paper. 2.2.2. Technological process of the type synthesis Based on the foundations established in the previous sections, a type synthesis method based on Grassmann Line Geometry and Line-graph Method will be introduced in this section, and the corresponding technological process of this method will be presented as follows. (1) Portray the DoF-line graph ℜDoFs according to the required DoFs of the mechanism to be designed. (2) Derive the corresponding constraint-line graph ℜ′CST from the DoF-line graph ℜDoFs using the dual rule presented in Section 2.2.1. (3) Generate the subspaces of the derived constraint-line graph ℜ′CST according to the equivalent relationship of line graphs, and the dimension of each subspace is same as that of the constraint-line graph ℜ′CST . (4) Select a subspace and decompose it, and configure the constraint-line graphs for each limb. In general, the number of limbs is not less than the DoFs of the mechanism to be designed. (5) Derive the corresponding DoF-line graphs for each limb from the constraint-line graphs generated in last step using the dual rule. (6) Configure the kinematic joints for each limb according to the corresponding DoF-line graphs, and generate the topological configurations for all limbs. (7) Constitute the mechanism configuration using the generated limbs and check the continuity of mechanism's motions. (8) If the synthesized mechanism does not have continuous motions, the synthesis stage should back to Step 6; if the derived mechanism has continuous motions, the type synthesis based on this subspace is finished, and another subspace should be selected and the type synthesis stage should start from Step 4.
N
Y
Fig. 1. Flow chart of the type synthesis based on Grassmann Line Geometry and Line-graph Method.
The whole technological process of the type synthesis can be summarized in a flow chart as shown in Fig. 1. 2.3. Type synthesis of 2T1R-type PKMs For the 2T1R-type PKMs, the rotational axis of the revolute DoF can be parallel or vertical to the plane composed of the translational axes (see Table 3). Therefore, the type synthesis of PKMs with 2T1R should be discussed separately. This paper focuses on the parallel situation, and the results for the vertical situation will be presented directly. The configurations of three chains and an active prismatic or revolute joint in each chain are used in the type synthesis of this paper. For the parallel situation, the DoF-line graph can be generated as shown in Fig. 2(a). According to the Blanding rule, the dual
Fig. 2. Line graphs: (a) DoF-line graph and (b) constraint-line graph.
constraint-line graph (denoted by Α) can be derived as shown in Fig. 2(b). This atlas represents two-dimensional planar couples and one-dimensional force with the axis parallel to the plane. According to the flow chart presented in Fig. 1, the atlas presented in Fig. 2(b) should be decomposed to generate the constraint-line graph of each limb. The subspaces of the constraint-line graph Α can be one-dimensional (a force or a couple), two-dimensional (a force and a couple or two couples) and threedimensional (a force and two couples). All of the subspaces can be
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DoF-line graph of the third limb is composed of three-dimensional translations and one-dimensional rotation, such a DoF-line graph can be realized by a PRC (C: cylindrical joint) kinematic chain. The three limbs can constitute a spatial 3-DoF (2T1R) PKM, which has continuous motions and nonsingular workspace. For case II, the type synthesis process and the corresponding result are presented in Table 5. The first limb realizes one-dimensional force constraint, and the second and third limbs realize one-dimensional couple constraint individually. According to the Blanding rule, the DoFline graph of the first limb is composed of two-dimensional translations and three-dimensional rotations, such a DoF-line graph can be realized by a PRS kinematic chain; the DoF-line graphs of the second and third limbs are composed of three-dimensional translations and two-dimensional rotations, such DoF-line graphs can be realized by PR (Pa)RR (Pa: parallelogram linkage) kinematic chains. The three limbs can also constitute a spatial 3-DoF (2T1R) PKM, which has continuous motions. When the stated constraint-line graph is decomposed in a different way, a different type synthesis result will be derived. In this paper, we just present some typical PKMs generated by the type synthesis method introduced here, and the results are shown in Table 6. For the vertical situation (rotational axis is vertical to the translational plane), some typical PKMs can also be derived by the similar type synthesis process, the corresponding constraintline graphs and PKMs are listed in Table 6.
taken as the constraint-line graph for each limb on the premise that Α ¼ Αsub1 ∪Αsub2 ∪Αsub3 (Αsubi represents the constraint-line graph of the ith limb; ∪ represents the union operation). Following this methodology, all possible results can be obtained by different permutations and combinations of the subspaces. For the space limitation of the presented paper, only two typical cases are discussed here and presented in Tables 4 and 5. Case I: only two-dimensional subspaces of the three limbs are considered, the result listed in Table 4 can be generated. Where, Αsub1 and Αsub2 represent one-dimensional force and onedimensional couple constraints; Αsub3 represents two-dimensional couple constraints. Case II: only one-dimensional subspaces of the three limbs are considered, the result listed in Table 5 can be generated. Where, Αsub1 represents one-dimensional force constraint; Αsub2 represents one-dimensional couple constraint; Αsub3 represents onedimensional couple constraint. For case I, the type synthesis process and the corresponding result are presented in Table 4. There are one-dimensional force constraint and one-dimensional couple constraint for the first two limbs and two-dimensional couple constraints for the third limb. According to the Blanding rules, the DoF-line graphs of the first two limbs are the same and composed of two-dimensional translations and two-dimensional rotations, such a DoF-line graph can be realized by a PRU (U: universal joint) kinematic chain; the Table 4 One result of the type synthesis based on constraint-line graph presented in Fig. 2(b). Configuration Constraint-line graph for each limb
DoF-line graph for each limb
Model and kinematic chain
One-dimensional force constraint and one-dimensional couple constraint
Two-dimensional translations and two-dimensional rotations
PRU
One-dimensional force constraint and one-dimensional couple constraint
Two-dimensional translations and two-dimensional rotations
PRU
Two-dimensional couple constraints
Three-dimensional translations and one-dimensional rotation
PRC
1th limb
2th limb
3th limb
Mechanism
Two translational DoFs and one rotational DoF (
: Prismatic joint;
: Revolute joint;
: Universal joint;
: Cylindrical joint)
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Table 5 Another result of the type synthesis based on constraint-line graph presented in Fig. 2(b). Configuration
Constraint-line graph for each limb
DoF-line graph for each limb
Model and kinematic chain
One-dimensional force constraint
Two-dimensional translations and three-dimensional rotations
PRS
One-dimensional couple constraint
Three-dimensional translations and two-dimensional rotations
PR(Pa)RR
One-dimensional couple constraint
Three-dimensional translations and two-dimensional rotations
PR(Pa)RR
1th limb
2th limb
3th limb
Mechanism
Two translational DoFs and one rotational DoF (
: Prismatic joint;
Note that, neither the decomposition of the constraint-line graph nor the constitution of kinematic joints is unique in the type synthesis process. Therefore, the results presented in Table 6 are not exhaustive and just some typical PKMs generated under some typical situations. From the type synthesis process of 2T1R-type PKMs, it can be concluded that the type synthesis method used in this paper is concise and intuitive. The type synthesis process can be visually presented due to the use of Grassmann Line Geometry and the Line-graph Method. Integrating the concepts of DoFs and constraints in a mechanism, each step of the type synthesis process has clear and definite physical meaning. Therefore, this method should have well prospective in practical applications, and can be used in the type synthesis of other PKMs with lower mobility.
3. Typical application in manufacturing It is well known that the five-axis machining capability is indispensable for modern machine tools. In Fig. 3(a), a typical fiveaxis milling application is presented. Both the five-axis positioning capability and the rotational flexibility should be considered. To process parts with freeform surfaces, the high rotational capability is necessary and the capability of five-face machining in one setup
: Revolute joint;
: Spherical joint)
[see Fig. 3(b)] is required in some working conditions. This requires the machine tools can realize the vertical–horizontal transformation. Therefore, to design a machine tool capable of five-face machining is meaningful. In the field of processing parts with freeform surfaces, the fiveaxis machine tool with hybrid architecture is a good solution. It is well known that the hybrid machine tool can combine the benefits of both PKMs and SKMs while avoiding the drawbacks of either. On this basis, the hybrid structure is helpful to enlarge the workspace and allow a flexible axis arrangement on the premise of moderate complexity and reasonable technological risks. Therefore, a hybrid machine tool capable of five-face machining will be proposed in the following sections. For hybrid machine tools, the restriction of the limited rotational capability of the parallel module is the main barrier to the achievement of the desired performance and should be broken through first. In the type synthesis of 2T1R-type PKMs, a spatial PKM (the CAD model is shown in Fig. 4(a)) is generated in Table 4. This mechanism is composed of three limbs, the first two limbs are PRU kinematic chains and the third limb is a PRC kinematic chain. The kinematic scheme is presented in Fig. 4(b). From which we can see that, the third limb is equivalent to a slider-crank mechanism in kinematics. In Ref. [33], the rotational capability of slider-crank
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Table 6 The results of the type synthesis of 2T1R-type PKMs. Atlases
Typical constraint-line graphs and the corresponding PKMs
Rotational axis is parallel to the translational plane
Rotational axis is vertical to the translational plane
Note: Atlas with red lines represents constraint-line graph and atlas with blue lines represents DoF-line graph.
Fig. 3. Five-axis machining: (a) a typical application and (b) five-face machining in one setup.
mechanism has been investigated, and the results are briefly summarized here and presented in Fig. 5. Obviously, the rotational capability of the slider-crank mechanism is more than 901. Therefore, the PKM presented in Fig. 4 can also achieve the same rotational capability. That is to say that this spatial PKM can realize the vertical–horizontal transformation. This is a great advantage to the application in manufacturing. To achieve the required five-face machining capability in one setup, a serial module with 1T1R-DoF should be combined to this
parallel mechanism. In order to realize three-dimensional translations, the translational DoF of the serial module should be perpendicular to the translational plane of the parallel module. Taking advantage of the vertical–horizontal transformation capability of the parallel module, a rotational DoF with continuous rotation capability should be used and its axis should be perpendicular to that of the rotational DoF of the parallel module. With such a configuration, the part to be processed can rotate continuously and the tool can orientate from vertical to horizontal direction, i.e., the
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Fig. 4. A 2T1R-type PKM with high rotational capability: (a) CAD model and (b) kinematic scheme.
Fig. 5. A slider-crank mechanism: (a) rotational capability and (b) transmission performance.
Fig. 6. The 3T2R-DoF hybrid mechanism: (a) CAD model and (b) kinematic scheme.
five-face machining in one setup can be achieved. The CAD model and kinematic scheme are presented in Fig. 6(a) and (b), respectively.
Based on the five-axis hybrid mechanism presented in Fig. 6, a prototype has been developed. The detailed development process has been discussed in Ref. [34], and will not be presented here for
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Fig. 7. The developed prototype: (a) exterior appearance and (b) horizontal working condition.
Fig. 8. Part processed with the prototype: (a) three-axis cavity milling and (b) five-axis curved-contour milling.
the space limitation of this paper. The exterior appearance of the prototype is shown in Fig. 7(a), and the horizontal working configuration is shown in Fig. 7(b). The developed machine tool is applied to process a part with freeform surfaces. The part presented in Fig. 8(a) is processed with three-axis cavity milling, and that presented in Fig. 8(b) is processed with five-axis curved-contour milling.
with high rotational capability is used to develop a hybrid machine tool with five-face machining capability. From the type synthesis process presented in this paper, it can be concluded that the type synthesis method introduced here is concise and intuitive and can be used in the type synthesis of other PKMs.
Acknowledgments 4. Conclusion The type synthesis of 2T1R-type PKMs was investigated in this paper. A type synthesis method based on Grassmann Line Geometry and Line-graph Method was introduced. In order to identify the dimension of a line graph, some criterions of Grassmann Line Geometry were summarized. Taking the concepts of DoF and constraint into consideration, the Line-graph Method, which can visually represent the DoFs and constraints in a mechanism by means of the corresponding line graphs, was presented. On this basis, the Blanding rule and the generalized Blanding rule were brought in to uncover the dual relationship between DoF-line graph and constraint-line graph. Thereafter, the technological process of the type synthesis method was given, and the type synthesis of 2T1Rtype PKMs was carried out sequentially. Some typical results were listed in this paper, and a 3-DoF spatial PKM was taken as the parallel module of a hybrid machine tool capable of five-face machining in one setup. The developed machine tool was introduced and applied to the machining of a part with freeform surfaces. The main contribution of this paper is to propose a type synthesis method based on Grassmann Line Geometry and Linegraph Method. On this basis, some new 2T1R-type PKMs are derived by using the proposed method and a 3-DoF mechanism
This work was supported in part by the National Natural Science Foundation of China under Grant 51135008 and 51021064, and by the China Postdoctoral Science Foundation under Grant no. 2012M520256.
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Contents lists available at SciVerse ScienceDirect
Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim
Type synthesis of 2T1R-type parallel kinematic mechanisms and the application in manufacturing Fugui Xie a,c,n, Xin-Jun Liu a,b, Zheng You c, Jinsong Wang a a
Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China Beijing Key Lab of Precision/Ultra-precision Manufacturing Equipments and Control, Tsinghua University, Beijing 100084, China c Department of Precision Instruments, Tsinghua University, Beijing 100084, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 6 November 2012 Received in revised form 5 May 2013 Accepted 11 July 2013
This paper investigates the type synthesis of 2T1R-type (T: translational degree of freedom (DoF) and R: rotational DoF) parallel kinematic mechanisms (PKMs). A type synthesis method based on Grassmann Line Geometry and Line-graph Method is introduced. Some basic criterions of Grassmann Line Geometry are briefly summarized, and the Line-graph Method is presented sequentially. In order to uncover the relationship between DoF-line graph and constraint-line graph, the dual rule is brought in and explained in detail. Based on these foundations, the technological process of the type synthesis is given. Thereafter, the type synthesis of 2T1R-type PKMs is carried out and the results are listed. Taken as an application example, a synthesized 3-DoF mechanism is chosen as the parallel module of a five-axis hybrid machine tool, which is capable of five-face machining in one setup. The developed prototype is introduced and applied into the machining of a part with freeform surfaces. The presented type synthesis method is concise and can be used in the type synthesis of other PKMs. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Parallel kinematic mechanism Type synthesis Hybrid machine tool Five-face machining
1. Introduction As the increasing need for the processing of complex parts with freeform surfaces, such as the turbines and marine propellers, multi-axis machine tools with parallel kinematics [1,2] or containing parallel modules [3] are becoming a research hotspot and attracting more and more attentions from both academia and industry. Some such kind of machine tools has been developed in the field. Variax [4] and Hexapod [5] are the first developed and also the most typical machine tools with fully parallel kinematics. They are based on the classical six degrees of freedom (DoFs) fully parallel kinematic mechanism (PKM), i.e., the Gough–Stewart platform [6,7]. Both of them need six-axis computer numerical control (CNC) and have relative small workspace compared to their assembly footprint. Thereafter, a five-axis fully parallel machine tool Metrom [8] is developed based on a 5-DoF PKM. The rotational capability of this machine can achieve 901 [9], which is a very important advantage to the improvement of the machining flexibility. In comparison, the five-axis machine tools containing parallel modules, especially the three-DoF parallel modules [10], are more popular in industry, such as the well-known Tricept machine [11] and Ecospeed series machine centers [12].
n Corresponding author at: Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China. Tel.: +86 10 62781820. E-mail address: [email protected] (F. Xie).
0736-5845/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.rcim.2013.07.002
Tricept machine contains a parallel module with three translational DoFs and a serially added articulated head with two rotational DoFs, its configuration can be represented by P{3T}&S {2R} (P{}: parallel module; S{}: serial module; T: translational DoF; and R: rotational DoF). Similarly, the Exechon [13] and the Verne [14] can also be classified into this category. But the Verne is composed of a three-translational-DoF parallel module and a twoDoF rotary table. In practical application, this type of machine tools has realized the agile machining, but the finished surface is sometimes scratched by the high-speed rotating cutter due to the serially added two-rotational-DoF modules. Ecospeed machine represents another type of machine tools with the configuration of P{1T2R}&S{2T}, in which the Space-5H machine center is included. The parallel module of the Ecospeed machine is called Sprint Z3 [15], which is a 3-PRS [16,17] (P: prismatic joint; R: revolute joint; and S: spherical joint) PKM in principle. The parasitic motions of 3-PRS mechanism have been analyzed in Ref. [18]. Similarly, the Space-5H contains a Hermes tool head [19], which also has complicated parasitic motions. In this category, the 1T2R-type PKMs, i.e., Sprint Z3 and Hermes tool heads, have realized the linked motions of the two rotational DoFs. This characteristic has greatly improved the machining performance and efficiency, but the parasitic motions of these mechanisms have resulted in complicated kinematics in terms of control and are disadvantageous to the further improvement of the machining accuracy. Moreover, the rotational capability is relatively limited.
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In all of the 3T2R-type hybrid machine tools, there is another configuration P{2T1R}&S{1T1R} besides the P{3T}&S{2R} and P {1T2R}&S{2T}, and this configuration may provide a better solution for the five-axis machining application. It is obvious that the two rotational DoFs are respectively assigned into parallel and serial modules in this configuration. Therefore, the problems brought in by the assignment of two rotational DoFs in one module are avoided, and the high rotational capability of the parallel module P{2T1R} can be achieved due to the elimination of the coupling influence of the two rotational DoFs. Serially combined a translational DoF with large traveling capability and a revolute DoF with continuously rotational capability, i.e., the serial module S{1T1R}, the P{2T1R}&S{1T1R}-type machine tools are capable of five-face machining in one setup, which is imperative to modern machine tools characterized by high flexibility and high efficiency. Therefore, the investigation of 2T1R-type PKMs by means of a type synthesis method based on Grassmann Line Geometry will be carried out in this paper, and a machine tool with the configuration of P{2T1R}&S{1T1R} will be introduced. Type synthesis is to bring out the mechanism configuration according to the required DoFs [20,21], and the mechanism configuration is fundamental and decisive to the performance of the developed mechanism and the consequent equipment. It is well known that the creation of a novel mechanism with useful function is a meaningful and also very challenging issue in the domain of PKMs [22]. This issue becomes more complicated for lower mobility PKMs [23], such as the 2T1R-type PKMs, due to the complex constraints in each limb. In general, type synthesis is carried out under the guidance of certain theory, which is the most important and also the most challenging problem to be figured out in the field. For such a reason, a lot of efforts have been contributed to this area, and many effective and novel theories or methods have been proposed. Among them, the methods based on displacement group theory [24,25] of Lie Group Algebra, screw theory [26,27], GF set [28], single-opened-chains units [29] and theory of linear transformation and geometrical analysis [30] are more popular. Based on the presented methods, many interesting mechanisms have been proposed. This has greatly enriched the diversity of the mechanisms. However, these methods are all based on certain complicated mathematic theories. To carry out the type synthesis under the guidance of these methods, designers need to master the relatively complicated procedures and the profound mathematic theories first. This is a barrier to the widespread application of the mentioned methods. In this paper, a concise and visual type synthesis method will be introduced and suggested as the theoretical foundation for the type synthesis of lower mobility PKMs. The remainder of this paper is organized as follows. Section 2 introduces the type synthesis based on Grassmann Line Geometry and Line-graph Method. Some basic criterions and rules are presented, and the technological process is summarized accordingly. Thereafter, the type synthesis of 2T1R-type PKMs based on the introduced method is carried out. Section 3 discusses the application of 2T1R-type PKMs in manufacturing. As an example, a hybrid machine tool capable of five-face milling in one setup is introduced. Section 4 concludes the paper.
2. Type synthesis of 2T1R-type PKMs based on Grassmann Line Geometry and Line-graph Method 2.1. Theoretical foundations: Grassmann Line Geometry and Line-graph Method Grassmann Line Geometry is a systematic theory in mathematics, and can be used to investigate the geometrical features of spatial line-clusters. Under this frame, the linear dependence of line vectors
in a line-cluster can be identified by comparing their number and dimension. This makes Grassmann Line Geometry very suitable to be used in the relevant research in mechanism theory, especially the DoFs and constraints of a mechanism. For such a reason, Grassmann Line Geometry has been extensively applied to PKMs' singularity analysis [31,32] in the field, and its practicability is well affirmed. In the present paper, Grassmann Line Geometry will be used to deal with the type synthesis of 2T1R-type PKMs. 2.1.1. Basic criterions of Grassmann Line Geometry Using Grassmann Line Geometry to carry out the type synthesis of PKMs, the dimensions of the corresponding line-clusters should be identified first. This will be very helpful to investigate the DoFs and constraints of a mechanism, and lay the foundation for the type synthesis. The following criterions can be used as the theory basis for this process: (1) There are at most three independent lines or only two independent parallel lines in a plane; (2) in all of the spatially parallel lines, there are only three independent parallel lines; (3) in all of the coplanar and concurrent lines, there are only two independent lines; (4) among all of the concurrent lines in space, there are only three independent lines; (5) for two sets concurrent lines (or one set concurrent lines and one set parallel lines) in two different planes and the intersections lie in the intersecting line of the two planes, there are only three independent lines; and (6) in two or more parallel planes or planes that intersect at one line, there are at most five independent lines. Under the guidance of these criterions, the dimensions of lineclusters can be identified and the line-clusters can be classified into different categories according to the identified dimensions. Some typical line-clusters under different dimensions are presented in Table 1. 2.1.2. Line-graph Method Based on Grassmann Line Geometry introduced in Section 2.1.1 and the line-clusters listed in Table 1, a mechanism's DoFs and constraints can be expressed by the corresponding line-clusters with assigned physical meanings. In this paper, a line-cluster with specific physical meaning, i.e., DoFs or constraints, is called line graph. In general, a line graph is composed of linear vectors or couples. Imposed the corresponding physical meanings, four basic elements with definite meanings can be generated as shown in Table 2. In Table 2, red color represents constraint; blue color represents DoF; line without arrows represents vector; and line with double arrows represents couple. A line graph constituted by these elements can be used to express an n-dimensional ðn≤6Þ freedomspace or constraint-space. The manner using freedom- or constraint-space line graphs to express the DoFs or constraints of a mechanism is referred as Line-graph Method. By using this method, the DoFs or constraints of a mechanism can be visually investigated. Moreover, the line graphs have clear and definite physical meaning. Therefore, this method is intuitive and concise. Under this expression, the qualitative analysis of the DoFs and constraints of a mechanism can be carried out in an intuitive and visual way. This paper mainly investigates the type synthesis of 2T1R-type PKMs. To illustrate the application of the Line-graph Method, some typical three-dimensional line graphs which can be used to express the 3-DoF motions are presented in Table 3, and the
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Table 1 Typical line-clusters classified by different dimensions. Dimension
Typical line-clusters
1 2
Planar concurrent
Parallel and coplanar
Non-coplanar
3
Spatial concurrent
Coplanar
Concurrent in two different planes
Non-parallel and non-intersecting
Coplanar or passing through one point
Regulus of lines
4
Concurrent with one line
Concurrent with two lines
5
Intersect with one line
Non-singular line-cluster
Table 2 Basic elements in a line graph and the corresponding meanings. Basic element
Mathematic meaning
Physical meaning
Vector
Rotational DoF
Vector
Constraint force
Couple
Translational DoF
Couple
Constraint couple
corresponding physical meanings are also included in this table. If the color of the line graphs in Table 3 is changed into red, the corresponding line graphs can be used to express threedimensional force- or couple-constraints according to the definitions given in Table 2.
2.2. Dual rules and technological process of the type synthesis Based on Grassmann Line Geometry and Line-graph Method, the DoFs and constraints of a mechanism can be visually represented by line graphs. Obviously, a line graph contains a lot of lines and couples, and the number of independent lines or couples in a line graph, i.e., the dimension of the line graph, can be identified according to the criterions listed in Section 2.1.1. In order to carry out type synthesis, the relationship between the DoF-line graphs and the constraint-line graphs should be uncovered. Actually, there exists a close relationship between the two kinds of line graphs. This paper will bring in dual principle of DoFs and constraints to deal with this problem, and the technological process of the type synthesis will be consequently presented.
2.2.1. Dual rule–Blanding rule Different from the traditional serial kinematic mechanism (SKM), the DoFs of the PKMs cannot be generated by a simple superposition of the DoFs of each limb. Therefore, the DoF-line graph cannot be directly decomposed into each limb. In fact, the end-effector of a PKM is constrained by multi-limbs, and the constraints of the end-effector can be generated by a superposition of the constraints of each limb. So, the constraint-line graph instead of the DoF-line graph is used and decomposed to generate the kinematics of each limb in this paper. The DoFs and constraints of a mechanism are a pair of conceptions describing the property of a mechanism from two different aspects. Therefore, they can be mutually transformed. Blanding proposed a basic rule that can reflect the mutually transformable relationship between the DoF-line graphs and constraint-line graphs of a mechanism. This rule is called dual rule or Blanding rule and can be summarized as follows. Provided that a line graph ℜ contains n independent (nonredundant) lines, then the corresponding dual graph ℜ′ contains (6-n) independent (non-redundant) lines and each line in line graph ℜ intersects with all lines in the dual graph ℜ′. According to the Blanding rule, the dual constraint-line graph ℜ′ can be uniquely identified when the DoF-line graph ℜ is given, and vice versa. Taking the physical meaning (DoFs and constraints) of the line graphs presented in Table 2 into account, a generalized Blanding rule [20] can be summarized as (1) The axes of rotational DOFs of a mechanism intersect with the lines of all constraint forces and are orthogonal to the axes of all constraint couples. (2) The axes of translational DOFs of a mechanism are orthogonal to the lines of all constraint forces and are arbitrary to the axes of all constraint couple. This generalized Blanding rule is very helpful to realize the mutual transformation between the DoF-line graphs and the
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Table 3 Some three-dimensional freedom spaces expressed by Line-graph Method. Line graph
Physical meaning
2T1R (Parallel)
2T1R (Vertical)
1T2R (Vertical)
1T2R (Parallel)
3T
3R
constraint-line graphs. On this basis, the investigation of DoFs and constraints of a mechanism can be carried out in a concise and visual way. All of these have laid the theoretical foundation for the type synthesis to be carried out in this paper. 2.2.2. Technological process of the type synthesis Based on the foundations established in the previous sections, a type synthesis method based on Grassmann Line Geometry and Line-graph Method will be introduced in this section, and the corresponding technological process of this method will be presented as follows. (1) Portray the DoF-line graph ℜDoFs according to the required DoFs of the mechanism to be designed. (2) Derive the corresponding constraint-line graph ℜ′CST from the DoF-line graph ℜDoFs using the dual rule presented in Section 2.2.1. (3) Generate the subspaces of the derived constraint-line graph ℜ′CST according to the equivalent relationship of line graphs, and the dimension of each subspace is same as that of the constraint-line graph ℜ′CST . (4) Select a subspace and decompose it, and configure the constraint-line graphs for each limb. In general, the number of limbs is not less than the DoFs of the mechanism to be designed. (5) Derive the corresponding DoF-line graphs for each limb from the constraint-line graphs generated in last step using the dual rule. (6) Configure the kinematic joints for each limb according to the corresponding DoF-line graphs, and generate the topological configurations for all limbs. (7) Constitute the mechanism configuration using the generated limbs and check the continuity of mechanism's motions. (8) If the synthesized mechanism does not have continuous motions, the synthesis stage should back to Step 6; if the derived mechanism has continuous motions, the type synthesis based on this subspace is finished, and another subspace should be selected and the type synthesis stage should start from Step 4.
N
Y
Fig. 1. Flow chart of the type synthesis based on Grassmann Line Geometry and Line-graph Method.
The whole technological process of the type synthesis can be summarized in a flow chart as shown in Fig. 1. 2.3. Type synthesis of 2T1R-type PKMs For the 2T1R-type PKMs, the rotational axis of the revolute DoF can be parallel or vertical to the plane composed of the translational axes (see Table 3). Therefore, the type synthesis of PKMs with 2T1R should be discussed separately. This paper focuses on the parallel situation, and the results for the vertical situation will be presented directly. The configurations of three chains and an active prismatic or revolute joint in each chain are used in the type synthesis of this paper. For the parallel situation, the DoF-line graph can be generated as shown in Fig. 2(a). According to the Blanding rule, the dual
Fig. 2. Line graphs: (a) DoF-line graph and (b) constraint-line graph.
constraint-line graph (denoted by Α) can be derived as shown in Fig. 2(b). This atlas represents two-dimensional planar couples and one-dimensional force with the axis parallel to the plane. According to the flow chart presented in Fig. 1, the atlas presented in Fig. 2(b) should be decomposed to generate the constraint-line graph of each limb. The subspaces of the constraint-line graph Α can be one-dimensional (a force or a couple), two-dimensional (a force and a couple or two couples) and threedimensional (a force and two couples). All of the subspaces can be
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DoF-line graph of the third limb is composed of three-dimensional translations and one-dimensional rotation, such a DoF-line graph can be realized by a PRC (C: cylindrical joint) kinematic chain. The three limbs can constitute a spatial 3-DoF (2T1R) PKM, which has continuous motions and nonsingular workspace. For case II, the type synthesis process and the corresponding result are presented in Table 5. The first limb realizes one-dimensional force constraint, and the second and third limbs realize one-dimensional couple constraint individually. According to the Blanding rule, the DoFline graph of the first limb is composed of two-dimensional translations and three-dimensional rotations, such a DoF-line graph can be realized by a PRS kinematic chain; the DoF-line graphs of the second and third limbs are composed of three-dimensional translations and two-dimensional rotations, such DoF-line graphs can be realized by PR (Pa)RR (Pa: parallelogram linkage) kinematic chains. The three limbs can also constitute a spatial 3-DoF (2T1R) PKM, which has continuous motions. When the stated constraint-line graph is decomposed in a different way, a different type synthesis result will be derived. In this paper, we just present some typical PKMs generated by the type synthesis method introduced here, and the results are shown in Table 6. For the vertical situation (rotational axis is vertical to the translational plane), some typical PKMs can also be derived by the similar type synthesis process, the corresponding constraintline graphs and PKMs are listed in Table 6.
taken as the constraint-line graph for each limb on the premise that Α ¼ Αsub1 ∪Αsub2 ∪Αsub3 (Αsubi represents the constraint-line graph of the ith limb; ∪ represents the union operation). Following this methodology, all possible results can be obtained by different permutations and combinations of the subspaces. For the space limitation of the presented paper, only two typical cases are discussed here and presented in Tables 4 and 5. Case I: only two-dimensional subspaces of the three limbs are considered, the result listed in Table 4 can be generated. Where, Αsub1 and Αsub2 represent one-dimensional force and onedimensional couple constraints; Αsub3 represents two-dimensional couple constraints. Case II: only one-dimensional subspaces of the three limbs are considered, the result listed in Table 5 can be generated. Where, Αsub1 represents one-dimensional force constraint; Αsub2 represents one-dimensional couple constraint; Αsub3 represents onedimensional couple constraint. For case I, the type synthesis process and the corresponding result are presented in Table 4. There are one-dimensional force constraint and one-dimensional couple constraint for the first two limbs and two-dimensional couple constraints for the third limb. According to the Blanding rules, the DoF-line graphs of the first two limbs are the same and composed of two-dimensional translations and two-dimensional rotations, such a DoF-line graph can be realized by a PRU (U: universal joint) kinematic chain; the Table 4 One result of the type synthesis based on constraint-line graph presented in Fig. 2(b). Configuration Constraint-line graph for each limb
DoF-line graph for each limb
Model and kinematic chain
One-dimensional force constraint and one-dimensional couple constraint
Two-dimensional translations and two-dimensional rotations
PRU
One-dimensional force constraint and one-dimensional couple constraint
Two-dimensional translations and two-dimensional rotations
PRU
Two-dimensional couple constraints
Three-dimensional translations and one-dimensional rotation
PRC
1th limb
2th limb
3th limb
Mechanism
Two translational DoFs and one rotational DoF (
: Prismatic joint;
: Revolute joint;
: Universal joint;
: Cylindrical joint)
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Table 5 Another result of the type synthesis based on constraint-line graph presented in Fig. 2(b). Configuration
Constraint-line graph for each limb
DoF-line graph for each limb
Model and kinematic chain
One-dimensional force constraint
Two-dimensional translations and three-dimensional rotations
PRS
One-dimensional couple constraint
Three-dimensional translations and two-dimensional rotations
PR(Pa)RR
One-dimensional couple constraint
Three-dimensional translations and two-dimensional rotations
PR(Pa)RR
1th limb
2th limb
3th limb
Mechanism
Two translational DoFs and one rotational DoF (
: Prismatic joint;
Note that, neither the decomposition of the constraint-line graph nor the constitution of kinematic joints is unique in the type synthesis process. Therefore, the results presented in Table 6 are not exhaustive and just some typical PKMs generated under some typical situations. From the type synthesis process of 2T1R-type PKMs, it can be concluded that the type synthesis method used in this paper is concise and intuitive. The type synthesis process can be visually presented due to the use of Grassmann Line Geometry and the Line-graph Method. Integrating the concepts of DoFs and constraints in a mechanism, each step of the type synthesis process has clear and definite physical meaning. Therefore, this method should have well prospective in practical applications, and can be used in the type synthesis of other PKMs with lower mobility.
3. Typical application in manufacturing It is well known that the five-axis machining capability is indispensable for modern machine tools. In Fig. 3(a), a typical fiveaxis milling application is presented. Both the five-axis positioning capability and the rotational flexibility should be considered. To process parts with freeform surfaces, the high rotational capability is necessary and the capability of five-face machining in one setup
: Revolute joint;
: Spherical joint)
[see Fig. 3(b)] is required in some working conditions. This requires the machine tools can realize the vertical–horizontal transformation. Therefore, to design a machine tool capable of five-face machining is meaningful. In the field of processing parts with freeform surfaces, the fiveaxis machine tool with hybrid architecture is a good solution. It is well known that the hybrid machine tool can combine the benefits of both PKMs and SKMs while avoiding the drawbacks of either. On this basis, the hybrid structure is helpful to enlarge the workspace and allow a flexible axis arrangement on the premise of moderate complexity and reasonable technological risks. Therefore, a hybrid machine tool capable of five-face machining will be proposed in the following sections. For hybrid machine tools, the restriction of the limited rotational capability of the parallel module is the main barrier to the achievement of the desired performance and should be broken through first. In the type synthesis of 2T1R-type PKMs, a spatial PKM (the CAD model is shown in Fig. 4(a)) is generated in Table 4. This mechanism is composed of three limbs, the first two limbs are PRU kinematic chains and the third limb is a PRC kinematic chain. The kinematic scheme is presented in Fig. 4(b). From which we can see that, the third limb is equivalent to a slider-crank mechanism in kinematics. In Ref. [33], the rotational capability of slider-crank
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Table 6 The results of the type synthesis of 2T1R-type PKMs. Atlases
Typical constraint-line graphs and the corresponding PKMs
Rotational axis is parallel to the translational plane
Rotational axis is vertical to the translational plane
Note: Atlas with red lines represents constraint-line graph and atlas with blue lines represents DoF-line graph.
Fig. 3. Five-axis machining: (a) a typical application and (b) five-face machining in one setup.
mechanism has been investigated, and the results are briefly summarized here and presented in Fig. 5. Obviously, the rotational capability of the slider-crank mechanism is more than 901. Therefore, the PKM presented in Fig. 4 can also achieve the same rotational capability. That is to say that this spatial PKM can realize the vertical–horizontal transformation. This is a great advantage to the application in manufacturing. To achieve the required five-face machining capability in one setup, a serial module with 1T1R-DoF should be combined to this
parallel mechanism. In order to realize three-dimensional translations, the translational DoF of the serial module should be perpendicular to the translational plane of the parallel module. Taking advantage of the vertical–horizontal transformation capability of the parallel module, a rotational DoF with continuous rotation capability should be used and its axis should be perpendicular to that of the rotational DoF of the parallel module. With such a configuration, the part to be processed can rotate continuously and the tool can orientate from vertical to horizontal direction, i.e., the
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Fig. 4. A 2T1R-type PKM with high rotational capability: (a) CAD model and (b) kinematic scheme.
Fig. 5. A slider-crank mechanism: (a) rotational capability and (b) transmission performance.
Fig. 6. The 3T2R-DoF hybrid mechanism: (a) CAD model and (b) kinematic scheme.
five-face machining in one setup can be achieved. The CAD model and kinematic scheme are presented in Fig. 6(a) and (b), respectively.
Based on the five-axis hybrid mechanism presented in Fig. 6, a prototype has been developed. The detailed development process has been discussed in Ref. [34], and will not be presented here for
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Fig. 7. The developed prototype: (a) exterior appearance and (b) horizontal working condition.
Fig. 8. Part processed with the prototype: (a) three-axis cavity milling and (b) five-axis curved-contour milling.
the space limitation of this paper. The exterior appearance of the prototype is shown in Fig. 7(a), and the horizontal working configuration is shown in Fig. 7(b). The developed machine tool is applied to process a part with freeform surfaces. The part presented in Fig. 8(a) is processed with three-axis cavity milling, and that presented in Fig. 8(b) is processed with five-axis curved-contour milling.
with high rotational capability is used to develop a hybrid machine tool with five-face machining capability. From the type synthesis process presented in this paper, it can be concluded that the type synthesis method introduced here is concise and intuitive and can be used in the type synthesis of other PKMs.
Acknowledgments 4. Conclusion The type synthesis of 2T1R-type PKMs was investigated in this paper. A type synthesis method based on Grassmann Line Geometry and Line-graph Method was introduced. In order to identify the dimension of a line graph, some criterions of Grassmann Line Geometry were summarized. Taking the concepts of DoF and constraint into consideration, the Line-graph Method, which can visually represent the DoFs and constraints in a mechanism by means of the corresponding line graphs, was presented. On this basis, the Blanding rule and the generalized Blanding rule were brought in to uncover the dual relationship between DoF-line graph and constraint-line graph. Thereafter, the technological process of the type synthesis method was given, and the type synthesis of 2T1Rtype PKMs was carried out sequentially. Some typical results were listed in this paper, and a 3-DoF spatial PKM was taken as the parallel module of a hybrid machine tool capable of five-face machining in one setup. The developed machine tool was introduced and applied to the machining of a part with freeform surfaces. The main contribution of this paper is to propose a type synthesis method based on Grassmann Line Geometry and Linegraph Method. On this basis, some new 2T1R-type PKMs are derived by using the proposed method and a 3-DoF mechanism
This work was supported in part by the National Natural Science Foundation of China under Grant 51135008 and 51021064, and by the China Postdoctoral Science Foundation under Grant no. 2012M520256.
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