A new methodology for optimal kinematic design of parallel mechanisms

Transcription

1 Mechanism and Machine Theory 42 (2007) A new methodology for optimal kinematic design of parallel mechanisms Xin-Jun Liu *, Jinsong Wang Institute of Manufacturing Engineering, Department of Precision Instruments, Tsinghua University, Beijing , China Received 7 November 2005; received in revised form 2 June 2006; accepted 12 August 2006 Available online 2 October 2006 Abstract This paper addresses the general issue of optimal kinematic design of parallel mechanisms. Optimal design is one of the most challenging issues in the field. To solve the design problem ideally, the difficulties that one should solve can be summarized as: (a) reducing the number of design parameters; (b) specifying the bounds of each parameter reasonably; (c) defining a parameter design space, in which the optimal kinematic design can be implemented logically; (d) providing all possible optimal results. This paper proposes an optimal kinematic design methodology, which is referred to as Performance-Chart based Design Methodology (PCbDM), for parallel mechanisms with fewer than five linear parameters. Some steps in this methodology are also helpful for the objective-function based optimal design. The results of this paper will be very useful in developing the computer-aided design system for parallel mechanisms. The proposed design methodology can be also applied in serial and parallel robots or any other mechanisms. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Parallel mechanism; Optimal design; Performance criteria; Performance chart 1. Introduction In the past two decades, parallel mechanisms have attracted more and more researchers attention in terms of industrial applications, especially in the field of machine tools, due to their relative advantages, e.g., high stiffness, high payload capability, low moving inertia, and so on. For such reasons, more and more parallel mechanisms with specified number and type of degree of freedom (DoF) have been proposed. However, optimal kinematic design is always one of the most challenging issues in the field. Abbreviations: PFNM, parameter-finiteness normalization method; DoF, degree of freedom; GE, greater than or equal to; R, revolute joint; P, prismatic joint; S, spherical joint; U, universal joint; CL, characteristic length; NL, natural length; PDS, parameter design space; GCMIW, good-condition maximal inscribed workspace; SM(s), similarity mechanism(s); BSM, basic similarity mechanism; LCI, local conditioning index; GCI, global conditioning index; GCW, good-condition workspace; MIC, maximal inscribed circle; MIW, maximal inscribed workspace; PCbDM, performance-chart based design methodology. * Corresponding author. Tel.: ; fax: address: XinJunLiu@mail.tsinghua.edu.cn (X.-J. Liu) X/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi: /j.mechmachtheory 转载

2 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) For optimal kinematic design of parallel mechanisms, most methods first develop an objective function and then reach a result using numerical methods with an algorithm [1 3]. These methodologies have common disadvantages, i.e., the objective function is highly non-linear and the process is iterative and time consuming. They may provide an optimal result; however, the user cannot know how optimal the result is. In the field of machine design, one of the most reasonable designs is that achieved from comparison. This method needs the use of performance charts, which are widely used in the classical design and most design manuals. Using performance charts, the antagonism of multiple criteria can be identified globally. A mechanism usually has several design parameters, and each of them, especially link lengths, can have any value between zero and infinity. In order to present the performance chart in a finite space, it is necessary to normalize the design parameters involved. The most used normalization approach is that dividing all parameters by one of them [4,5]. However, this kind of normalization cannot make sure that the normalized parameters are finite. Thus, the developed design space cannot be used to plot performance charts. This paper introduces a significant normalization technique, which is referred to as the parameter-finiteness normalization method (PFNM). This normalization makes sure that not only are the normalized parameters finite, but the sum of all parameters is constant as well. The analysis in this paper will show that the normalization factor does not change the similarity of mechanisms performances. This is actually the most important point in kinematic design. When it is applied to mechanisms that have linear parameters, the method has the advantages of (1) reducing parameter number, (2) studying a type of mechanism completely, (3) remaining performance similarity of mechanisms, and (4) implementing the optimal kinematic design in a limited parameter space. This is very useful for the representation of a performance chart and for performance comparison of different mechanisms. Using this normalization technique, the parameter design space, which can be used to analyze performance and to implement the optimal design of a mechanism, will be developed. Based on the normalization method and the performance similarity of mechanisms, a novel optimal design methodology will be proposed accordingly Terminology We shall use the term a type of mechanism to mean a mechanism with a specified kinematic structure, and a kind of mechanism to indicate the type of mechanism where each parameter is assigned with a specified value. For example, the 5R 2-DoF symmetric parallel mechanism is a type of mechanism. The mechanism has three linear characteristic parameters L 1, L 2 and L 3. Such a mechanism with L 1 = 15 mm, L 2 =12mm and L 3 = 13 mm is referred to as a kind of mechanism. If each parameter is assigned a specified value, the set of these n parameters is defined as a combination. Every combination corresponds to a kind of mechanism Notation n the characteristic parameter number of a mechanism L i the linear parameter with dimension l i the normalized parameter of L i C n =(L 1,L 2,...,L n ) the combination of a dimensional mechanism with n parameters c n =(l 1,l 2,...,l n ) the combination of a non-dimensional (normalized) mechanism with n parameters P ¼½C n 1 ; Cn 2 ; Cn 3 ;...Š the dimensional mechanism space p ¼½c n 1 ; cn 2 ; cn 3 ;...Š the non-dimensional (normalized) mechanism space 2. Related problems existing in the design of parallel mechanisms In order to perform a specified task, the parameters of a parallel mechanism usually should be optimized. As each of the parameters can have any value between zero and infinity and performance criteria are usually antagonistic, the optimal design is one of the most challenging issues in the field. Suppose that a mechanism has n characteristic linear parameters, each of which is denoted as L i (1 6 i 6 n). These n parameters are included in the Jacobian matrix J of the mechanism. For such a reason, the Jacobian matrix is closely related

3 1212 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) to the parameters. The kinematics, workspace, singularity, dynamics and other performances are also closely dependent on them. The optimal kinematic design of a mechanism is actually the determination of these parameters with respect to prescribed specifications. It is noteworthy that, in this paper, the input parameters are not defined as the design parameters due to the fact that they will eventually not be included in the Jacobian matrix. This will be explained in detail in Section 5.1. As it is one of the most important and challenging issues in the field of parallel mechanisms, optimal design has attracted more and more researchers attention [1 6]. Many methods have been proposed for the design of specified mechanisms. The most commonly used method is the objective-function based optimal design. The methodology is first to establish an objective function with specified constraints and then search the result utilizing an optimization algorithm. This method is time-consuming and it is hard to reach the globally optimum target due to difficulties such as the non-finiteness of each parameter, the antagonism of multiple criteria and the assignment of its initial value. Another method can be to use the performance chart (atlas), which is widely used in the classical design and most design manuals. A performance chart can show the relationship between a performance index and associated design parameters in a limited space, globally and visually. Then, performance charts can show how antagonistic the involved criteria are. For such reasons, this method can be an ideal one. Compared with the result achieved from the objective-function based method, the optimum result is fuzzy. But the method is more flexible due to the fact that the optimal design method provides not a unique solution to a design problem. That means the designer can adjust the optimum result appropriately according to his design conditions. The most important for the performance-chart based optimal design is the presentation of performance charts. As it is well known, each parameter of a mechanism can have any value between zero and infinity. This is actually the biggest problem in the design method since we cannot illustrate a chart in an infinite space. Thus, the parameter infinity is the most troublesome problem in the kinematic design of a parallel mechanism if performance charts are used. The problems arising can then be summarized as: (a) how to reduce the design parameter number; (b) how to specify the bound of each parameter reasonably; (c) how to define a parameter design space, in which the optimal kinematic design can be implemented logically and (d) if there is such a design space, how to deal with the relationship between mechanisms with infinite and finite parameters. 3. Proposed solution To solve the problems mentioned in Section 2, one should first find a technique to define logically the limit of each linear parameter and, at the same time, retain the performance similarity of the mechanism. The parameter normalization technique is one solution to these problems. The key problem in the parameter normalization is the selection of a normalization factor. In order to analyze the performance of all fourbar mechanisms, a kind of normalization method was proposed by Yang [7]. To handle the problem of non-homogeneous physical units in Jacobian matrix, the concepts of characteristic length (CL) and natural length (NL) were introduced [8 10]. The normalization methods were, in fact, established on the factor that is the average of related linear parameters. The CL and NL were used to normalize the dimensional elements in Jacobian matrix, but could not make the elements finite due to the fact that they were not the average of all dimensional elements. Actually, the finiteness, which is very important for our problem, is not necessary for such a kind of normalization. Anyway, the idea, especially that of Yang of generating normalization factor is helpful for us. Here, we extend the idea to a general case. Suppose that there are n characteristic parameters in a mechanism, which are denoted as L i (i =1,2,...,n). Let D ¼ Xn i¼1 L i =d ð1þ where d can be any positive number. Here, the parameter D is defined as the normalization factor of the mechanism. One can obtain n non-dimensional parameters l i by means of l i ¼ L i =D ð2þ

4 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) Therefore, X n i¼1 l i ¼ d ð3þ which not only reduces the parameter number from n to n 1 but yields the bound of each normalized parameter l i as well, i.e., l n ¼ d Xn 1 i¼1 l i ð4þ and 0 6 l i 6 d ð5þ It is noteworthy that, in some cases, as parameters l i (i =1,2,...,n) should be used to set up the mechanism and make it work, there will be other conditions on these parameters. Eqs. (3) and (5) together with these conditions actually define an (n 1)-dimensional finite space. Such a space is defined as the parameter design space (PDS). From Eqs. (3) and (5), one can see that the limit of the PDS depends on the value of d. For convenience of prescribing the bound of every normalized parameter and expressing the PDS in a limited space, although the parameter d can be any positive number, we usually assign it with an integer, typically the number 1 or n. Itis noteworthy that the selection of parameter d just determines the size of the PDS but will not affect the shape of the PDS and the final result. If d =1,D is the sum of all characteristic parameters. When d = n, D is the average of all parameters. Whatever the parameter d is, using Eqs. (1) (5), a mechanism can be changed from one with dimension to one with no dimension. What is most important is that this technique also changes an n- dimensional problem to an (n 1)-dimensional one and, at the same time, defines the limits of each of the normalized parameters. This is one of the most important contributions to mechanisms. For such a reason, the method is referred to as the parameter-finiteness normalization method (PFNM). We can see that the size of the mechanism with Dl i is similar to that of the mechanism with l i. With different D, there will be different kinds of mechanisms. Here, the mechanisms with Dl i (with different D) are defined as the similarity mechanisms (SMs), and the mechanism with l i is defined as the basic similarity mechanism (BSM). All of the SMs have a same ratio between the characteristic parameters Dl i. The ratio is also the same as that between l i. It is noteworthy that the selection of the parameter d does not affect the ratio. For example, considering a mechanism with L 1 = 6 mm and L 2 = 4 mm, if d = 1, there is l 1 /l 2 = 0.6/0.4 = 1.5. If d = 5, there is l 1 /l 2 = 3/2 = 1.5. Therefore, the selection of the parameter d will not affect the application of PFNM in the analysis and design of mechanisms and the results. The characteristic parameters L i = Dl i of a mechanism make up an n-dimensional space. As each of the parameters can have any value between zero and infinity, this space is infinite. Therefore, the dimensional mechanism space P ¼½C n 1 ; Cn 2 ; Cn 3 ;...Š is an infinite space and the combination Cn =(L 1,L 2,...,L n ) consists in such an infinite space. Since all parameter l i are limited, the normalized mechanism space p ¼ ½c n 1 ; cn 2 ; cn 3 ;...Š, which is actually the PDS, is finite and the combination cn =(l 1,l 2,...,l n ) is one element in this finite space. Therefore, the PFNM establishes the relation between the elements in a finite space and an infinite space. Every combination in the infinite space P can find its unique counterpart in the finite space p. But, one element in the finite space corresponds to an infinite number of elements in the infinite space. The defined SMs are in the space P, while the BSM exists in the space p, i.e., the PDS. The parameter number affects the difficulty of kinematic design of the mechanism. Here, based on the PFNM, we give some examples with different parameter numbers. In the following examples, d = n = 1 The case n = 1 means that there is only one characteristic parameter L 1 in the mechanism. For example, the PRRRP 2-DoF parallel mechanism and the RPRPR 2-DoF parallel mechanism as shown in Fig. 1(a) and (b),

5 1214 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) Fig. 1. Mechanisms with one characteristic parameter. respectively, are such mechanisms. The normalization factor is L 1 itself. From Eq. (2), there is l 1 = 1, which indicates that the PDS is a point n = 2 For this case, there are two characteristic parameters L 1 and L 2, e.g., the PRRRP 2-DoF parallel mechanism actuated vertically by linear actuators and the 3-RPR parallel mechanism shown in Fig. 2(a) and (b), respectively. The Star-like robot [11] shown in Fig. 2(c), the 3-RPS parallel mechanism [12] and the linear Delta robot [13] also have two characteristic parameters. The parameters for the Star-like robot are the length of each leg and the radius of the moving platform. They are the length of each leg and the margin between the radius of the base and that of the moving platform for the Delta robot. The normalization factor of these mechanisms is D = L 1 + L 2. As there is l 1 + l 2 = 1 for the normalized parameters, the PDS is actually a closed line section. Particularly, there is the parameter condition l 1 P l 2, i.e., l 1 P 0.5 for the mechanism shown in Fig. 2(a). The PDS of the mechanism is a closed interval l 1 2 [0.5,1] n = 3 The parallel mechanisms, such as the 5R planar 2-DoF parallel mechanism (shown in Fig. 3(a)), Delta robot [14] and the 3-PRS parallel mechanism (Fig. 3(b)), have three parameters. Other 3-DoF translational parallel mechanisms with revolute actuators, for instance Tsai s mechanism [15], are also such mechanisms. Like the 3-PRS mechanism, most non-translational parallel mechanisms actuated vertically by linear actuators, for instance the 6-PUS 6-DoF parallel mechanism and the HALF manipulator [16] shown in Fig. 3(c) and (d), respectively, have three characteristic parameters as well. For these mechanisms, there is l 1 + l 2 + l 3 = 1. The normalized parameters should be specified as 0<l 1,l 2,l 3 < 1 and l 2 P jl 3 l 1 j. An additional condition l 1 > l 3 should be given to the 3-PRS mechanism, the 6-PUS mechanism and the HALF manipulator. In any case, the PDS for any one of these mechanisms is a closed planar space. For example, the PDS for the 3-PRS parallel mechanism is shown as the isosceles triangle ABC in Fig n = 4 Most parallel mechanisms with revolute actuators, such as the 6-RRRS mechanism [12], the Hexa mechanism [17], the 3-RRR parallel mechanism shown in Fig. 5(a), the CaPaMan mechanism [18] shown in Fig. 2. Mechanisms with two characteristic parameters.

6 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) Fig. 3. Mechanisms with three characteristic parameters. Fig. 4. PDS of the 3-PRS parallel mechanism shown in Fig. 3(b). Fig. 5. Mechanisms with four characteristic parameters.

7 1216 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) Fig. 6. PDS of the 3-RRR parallel mechanism shown in Fig. 5(a). Fig. 5(b), the 321-HEXA mechanism [19], the TURIN mechanism [20] and the HALF manipulator with revolute actuators [21] have four characteristic parameters. When n = 4, there is l 1 + l 2 + l 3 + l 4 = 1, which defines a unit cube. As there always exist other parameter conditions, the PDS of such a mechanism is usually a three-dimensional polyhedron. For example, to make sure that the 3-RRR parallel mechanism shown in Fig. 5(a) can be assembled, the parameter conditions l 2 + l 3 + l 4 P l 1 (l ) and l 2 + l 3 + l 1 P l 4 (l ) must be satisfied. Therefore, the PDS of the mechanism is actually the polyhedron ABCDEFG shown in Fig n P 5 Most fully parallel mechanisms with k DoFs consist of k legs, each of which is made up of three joints and two moving links. Here, for parallel mechanisms, such as Delta robot, Tsai s mechanism, Star-like manipulator, CaPaMan, HALF manipulator, and many others, which contain simple mechanisms in their legs, the simple mechanism is thought of as the combination of one characteristic link and one joint. In addition, parallel mechanisms are usually fully symmetric in kinematic structure. That means the parameters of one leg are the same as those of the others in number and length. Therefore, the characteristic parameter number of a fully parallel mechanism usually cannot exceed 4. The case n P 5 arises if a parallel mechanism is not symmetric. For example, a 5R 2-DoF dissymmetric parallel mechanism has five characteristic parameters. The parameter number of a planar 3-RRR dissymmetric parallel mechanism can be up to 12. No matter how many parameters there are, using the PFNM, the number of parameters can be reduced from n to n 1, and the n 1 normalized parameters are limited. But, the Fig. 7. A linear Delta robot.

8 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) PDS cannot be expressed in a three-dimensional space due to the fact that it exists in a space with the dimensions GE Relationship between a BSM and its SMs As it is presented in previous sections, the PFNM cannot only reduce the characteristic parameter number but specify the bounds of each normalized parameter logically as well. The normalization makes it possible to establish a design space in a limited space. If the technique can be used in parallel mechanisms, especially mechanisms with fewer than five parameters, the analysis and optimal design issues can be implemented completely and perfectly. To this end, the inherent relationship between a BSM and its corresponding SMs should be made clear. As shown in Eq. (2), a BSM and any one of its SMs have the relationship L i = Dl i in size. The planar workspace of a SM should be D 2 -time that of its BSM. The factor will be D 3 for spatial workspaces. Since the normalization factor D cannot change the inherent characteristics of the Jacobian matrix, a BSM and any one of its SMs will be also similar in performance. We give here two examples to illustrate the relationships in details. The first example is the linear Delta robot as shown in Fig. 7. The inverse kinematics, Jacobian matrix and workspace analysis were discussed in [13]. Here, we recall the problems in brief. For a prescribed position (x,y,z) of the point O 0, the inputs of points B k (k = 1,2,3) for the working mode shown in Fig. 7 can be obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z k ¼ L 2 2 ðx x kþ 2 ðy y k Þ 2 þ z ð6þ where x k = L 1 cos/ k, y k = L 1 sin / k, L 1 = R r and / k =2(k 1)p/3. The Jacobian matrix can be written as 2 3 x L 1 cos / 1 y L 1 sin / 1 1 q 1 q 1 x L 1 cos / J ¼ 2 y L 1 sin / 2 1 ð7þ q 2 q x L 1 cos / 3 y L 1 sin / q 3 q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 in which q k ¼ L 2 2 ðx L 1 cos / k Þ 2 ðy L 1 sin / k Þ 2 ðk ¼ 1; 2; 3Þ. The local conditioning index (LCI) l is defined as the reciprocal of the condition number of the Jacobian matrix, i.e., l ¼ 1=kJkkJ 1 k where kæk denotes the Euclidean norm of the matrix. The index is usually used to evaluate the control accuracy, dexterity and isotropy of a mechanism [22 24]. This number is to be kept as large as possible. If the number can be unity, the matrix is an isotropic one, and the mechanism is in an isotropic configuration. The index l also indicates the distance from the singularity. Theoretical workspace contains the singular poses and their neighborhoods, where the mechanism will be out of control. Thus, the task workspace is usually the region that some poses are excluded from the theoretical workspace with respect to a specified LCI. In order to evaluate the global behavior of a mechanism in a workspace, the global conditioning index (GCI) was defined by Gosselin and Angeles [24] as Z. Z g ¼ 1=jdW dw ð9þ W W in which W is the workspace. The GCI is actually the average of LCI over a workspace. It is noteworthy that, as the average cannot describe the deviation between the maximum and minimum values of LCI, the GCI itself cannot give a full-scaled description of the overall global kinematic performance. One solution to this problem can be that based on the assumption that the minimum LCI values are the same but the task workspaces are different. The minimum LCI can be specified with respect to the design specification. The set of poses in which the LCI values are greater than or equal to (GE) the minimum LCI is defined as the good-con- ð8þ

9 1218 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) dition workspace (GCW). The GCI is then the average of LCI over the GCW. Therefore, if such a GCI value is bigger, we can conclude that the mechanism has a better kinematic performance in its workspace. In this paper, the GCW is used in Eq. (9). As workspace, LCI and GCI are the criteria in almost all designs, these indices are used here to compare a mechanism with its normalized mechanism. Eq. (7) shows that the Jacobian matrix is independent of z. That means, in every z-section of the workspace, the performance will be constant. To analyze the performance, we can just be concerned with one workspace section in the O xy-plane. The Jacobian matrix also includes no input parameters z k. Then, if the input is disregarded, the maximal region that the mechanism can reach in the O xy-plane is determined by the intersection of three circles C k, which can be written as C k : ðx x k Þ 2 þðy y k Þ 2 ¼ L 2 2 ; k ¼ 1; 2; 3 ð10þ The intersecting region is defined as the maximal workspace in the O xy-plane, which is denoted as W xy. In the maximal workspace, the GCW of the mechanism is irregular in shape. The GCW can be characterized by its maximal inscribed circle (MIC) [13]. The region circled by the MIC is referred to as the good-condition maximal inscribed workspace (GCMIW), which is denoted as W GCMIW. Replacing the parameters L i in Eq. (10) by Dl i,it is not hard to conclude that the maximal workspace of a SM is D 2 -time that of its BSM. From Eq. (7), one can see that the Jacobian matrix of a SM of the mechanism at point (x,y) is same to that of the BSM at point (x/ D,y/D). As a result, the corresponding LCI values are the same. Therefore, the GCMIWs of a SM and its BSM also have the D 2 -time relation. The GCI values in their corresponding GCMIWs will be finally the same as well. For example, if the specified LCI is 0.3, the GCMIWs of a SM with parameters L 1 = 8 mm and L 2 =12mm and its BSM with l 1 = 0.4 and l 2 = 0.6 (D = 20 mm) are mm 2 and ( /20 2 ), respectively. The GCI values over their GCMIWs are both , and the distributions of LCI for them are shown in Fig. 8(a) and (b), respectively. One can see that the distributions are similar to each other. The first example shows a comparison of positional workspaces. The second concerns the orientational workspace of a mechanism. The angle is defined as the figure formed by two lines diverging from a common point or by two planes diverging from a common line. Thus, the angle is usually measured by the ratio of two linear parameters. For such a reason, the normalization factor D cannot lead to any change in the orientational capability of a mechanism. For example, Fig. 9 shows the maximal orientational capability that the HALF manipulator with linear actuators [16] can reach at one point. The capability at point O 0 can be calculated by n max ¼ 180 p cos 1 L 1 L 3 þ L 2 þ cos 1 L 1 L 2 L 3 ð11þ Fig. 8. Distribution of the LCI in the GCW of the linear Delta mechanism: (a) for the SM; (b) for the BSM.

10 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) Fig. 9. The orientational capability of HALF manipulator at one point. from which one can see that replacing L i with L i /D (i.e., l i ) will not change the value n max. That means, no matter what the normalization factor D is, the orientational workspaces of a BSM and its SMs are the same. Therefore, a SM and its BSM are similar not only in size but also in performance. This fact indicates that, if we know the performance of a BSM, the performance of all of its SMs can be realized accordingly. Although we just checked the workspace, LCI and GCI, a BSM and its SMs are also similar in other performances that are defined with respect to the Jacobian matrix. We can then reach a result that, if a BSM is optimal, any one of its SMs will be optimal. This relationship is very important for developing a methodology for optimal kinematic design. 5. Development of a new optimal kinematic design methodology 5.1. A general design procedure As it is pointed out in Section 3, every SM in the mechanism space P can find its unique BSM in the PDS and one BSM in the PDS corresponds infinite SMs (with different factor D) in the space P. The analysis results in Section 4 show that the SMs and BSM are similar not only in size but also in performance. No matter how many parameters there are, using the PFNM, the number of parameters can be reduced from n to n 1, and what is more, the n 1 normalized parameters are limited. Whatever kind of design methodology the designer uses, this actually provides some useful information on the parameters. Therefore, the PFNM can be applied in the kinematic design of a mechanism. For the classical objective-function based optimal design, the method solves the problem of parameter limitation. This can really reduce the cost of this optimal method. The detailed description about this will not discussed in this paper. Here we would like to mention that the concept of performance chart has been introduced to analyze the performance of parallel and serial mechanisms with only revolute actuators in Refs. [14,25 28]. However, these articles did not touch the topic of giving a general method to reach an optimal result or optimal region with respect to desired performance(s) in practice. In addition, parallel mechanisms with prismatic actuators were not investigated yet. This paper addresses the issue of optimal kinematic design for parallel mechanisms with both revolute and prismatic actuators systematically, including identification of design parameters, parameter normalization, performance chart illustration, and, most importantly, the finding of all possible optimal solutions. As the PDS (or the PDS section) of a mechanism with fewer than five parameters can be expressed in a twodimensional space, the relationship between a performance index and the parameters (or all possible BSMs) can be illustrated graphically. Therefore, such a mechanism can always be studied completely in a finite space. This fact motivates us to propose a novel design methodology, which is referred to as the Performance-Chart

11 1220 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) based Design Methodology (PCbDM) in this paper, for such mechanisms. The methodology can be described as following: S1. Identify all linear characteristic parameters L 1,L 2,...,L n from the Jacobian matrix. S2. Normalize the involved design parameters using the PFNM proposed in this paper and establish the (n 1)-dimensional parameter design space (PDS). S3. Illustrate relationships between involved performance indices, especially the good-condition workspace (GCW) and the GCI over the GCW, and the normalized parameters graphically in the PDS. S4. Utilize performance charts to identify the optimum region meeting kinematic performance constraints. S5. Select a candidate, i.e., BSM (l 1,l 2,...,l n ), from the optimum region. S6. Determine the GCW of the BSM. S7. Calculate the normalization factor by comparing the GCW and the desired task workspace. S8. Determine the design parameters using the factor. The corresponding SM can then be achieved. S9. Calculate the input parameters of the SM with respect to the task workspace. S10. Check the involved performances and input parameters (if commercially available actuators are used) of the SM. If all performances are available and the input parameters are subject to the actuators, all parameters can be finally determined. If not, return to S5 and perform S5 S10 again. It is noteworthy that this design methodology is based on the performance chart. The optimum region in S4 is the intersection between several charts. Since every designer has different design conditions and the conditions cannot be predicted previously, the optimum region will not be unique. For the same reasons, the designer can adjust the BSM in S5 due to the fact that the optimum region provides a non-unique solution to a design problem. Therefore, human interactions will inevitably be used in the design process. Since we cannot consider all design conditions in an integrated equation, compared with the mathematical optimal result or that generated by any automatic way, only the design result in application sense is indeed optimal. Therefore, human interactions are necessary in a practical design. The proposed design methodology has such an advantage that human interaction can be involved. The proposed design process can be described schematically in Fig. 10. Notice that, in the proposed design methodology, the input parameter is not the optimized parameter. It is no doubt that input parameters determine the workspace. For such a reason, in most developed optimal design methods, the input parameters are usually involved in the design process. This actually increases the parameter number. In the optimal design, the workspace is definitely not the only criterion. The local conditioning index must be used to structure a GCW, which is part of the theoretical workspace. Only such a workspace can be accepted in practical. Normally, the input parameters will not be contained in the final Jacobian matrix (see Eq. (7)), and the theoretical workspace can be constrained by singularities. Thus, there is no need to think of the input parameters as the design parameters. Therefore, in our proposed design method, only the parameters that are included in the Jacobian matrix are defined as the characteristic parameters, which are the design parameters. According to this definition, the input parameters are not the characteristic parameters. Undoubtedly, the input parameters should be given finally. This can be done with respect to the prescribed task workspace after the characteristic parameters are optimized and determined. Note that the theoretical workspace of a mechanism with variable-length rods (for example, the mechanisms shown in Fig. 1(b) and Fig. 2(b)) is an infinite region. Maybe, the GCW of such a mechanism will be infinite. Experimentally, considering the stiffness, the maximum length q max must be less than three-time the minimum length q min of a variable-length rod. Then, for such a mechanism, the minimum length q min should be first determined with the nearest pose where the LCI is equal to the specified LCI. The GCW can be achieved by letting q max =3q min. In the proposed design methodology PCbDM, the steps S6 S10 can be applicable in the objective-function based optimal design after a BSM is achieved using the algorithm introduced in [1 3] or any others. It is noteworthy that, for some parallel mechanisms, the angular parameter is usually included in the Jacobian matrix. For such a case, the angular parameter is also a design parameter. As the angular parameter cannot be normalized, the proposed design procedure did not mention the design of such a parameter. But, it does not mean that the design method cannot handle the optimal design of a parallel mechanism with angular

12 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) Fig. 10. Flowchart describing the proposed design method PCbDM. parameters. From the design procedure, we can see that performance charts are very important for the implement of the optimal design. For the parallel mechanisms with angular parameters, we can use the method introduced in [28] to illustrate the performance charts. Therefore, the design methodology can be also applied to the design of those mechanisms. Additionally, since it is possible to illustrate the performance chart of a serial robot after normalizing the design parameters [25], the design methodology proposed here can be applied to serial robots as well. All in all, as long as the performance charts of a mechanism can be presented and performances of the original mechanism and its normalized mechanism are similar with each other, one can use the proposed methodology to design optimally the mechanism Application example As an application example of the proposed PCbDM, here we consider the optimal kinematic design of the linear Delta robot shown in Fig. 7. For the optimal kinematic design, suppose that the desired task workspace of the robot is a cylinder with /20 mm 15 mm, which indicates that the radius of the task workspace is R task = 10 mm and the height of the workspace is H = 15 mm. According to the PCbDM developed in Section 5.1, the process to reach the objective can be as following:

13 1222 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) S1: Identification of design parameters. The design parameters of the mechanism are L 1 = R r and L 2, which can be identified from the Jacobian matrix in Eq. (7). S2: Development of the PDS. Letting d = 1, Eqs. (1) (3) lead to l 1 + l 2 =1.Asl 2 P l 1, there is l Thus, the PDS of the mechanism is a closed interval l 2 2 [0.5,1]. S3: Illustration of relationships between involved performance indices and the normalized parameters graphically. Numerical calculation shows that, if the LCI is specified as 0.3, there is the GCMIW for the normalized mechanisms when l 2 2 [0.5060, ]. The performance charts of the GCMIW W GCMIW and the GCI g over the GCMIW are shown in Fig. 11(a) and (b), respectively. S4: Use of performance charts to identify the optimum region meeting kinematic performance constraints. The performance charts show that the GCMIW W GCMIW reaches its maximum when l 2 = and the GCI is maximum when l 2 = Within the interval l 2 2 [0.5505, ], the two performances are antagonistic. That means it is impossible to find a l 2 solution, i.e., a BSM whose GCMIW and GCI are both best. This is also a universal problem in the optimal design with multi-criteria. In order to identify a solution that both GCMIW and GCI are better, the two performances would both be sacrificed. For example, if the GCMIW is specified as W GCMIW P 0.1 and GCI g P 0.45, we can identify an intersecting region X GCMIW-GCI =[l 2 j l ] (see Fig. 12). Such a region can be one optimal design region if both GCMIW and GCI performances are considered. S5: Selection of a candidate from the optimal region. The region X GCMIW-GCI gives the possible link lengths about the geometric parameters l 2 and l 1 (l 1 =1 l 2 ) of the normalized mechanism. For example, the BSM with l 2 = and l 1 = can be one candidate for the optimal design. S6: Determination of the GCMIW of the BSM. The GCMIW of the selected BSM is W GCMIW = Additionally, its GCI is g = Fig. 11. Performance charts of the linear Delta robot when LCI is specified as 0.3: (a) the GCMIW; (b) the GCI over the GCMIW. Fig. 12. One optimum region with desired GCMIW and GCI.

14 X.-J. Liu, J. Wang / Mechanism and Machine Theory 42 (2007) S7: Calculation of the normalization factor by comparing the GCW and the desired task workspace. Once the GCMIW of the selected BSM is achieved, the normalization factor D can be calculated with respect to the z-section of the desired task workspace. For the example, there is D ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pr 2 task =W GCMIW q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p10 2 =0:1190 ¼ 51:3809 mm. S8: Determination of the parameters using the factor. The parameters of the SM can be determined using Eq. (2), i.e., L 2 = Dl 2 = mm and L 1 = Dl 1 = mm. S9: Calculation of the input parameters. Putting the bottom section of the task workspace in the O xyplane when z = 0, the maximum and minimum input parameters of each actuator can be written as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z max ¼ L 2 2 ðl 1 R task Þ 2 þ H and z min ¼ L 2 2 ðl 1 þ R task Þ 2. Then, there are z max = mm and z min = mm for the obtained SM. S10: Checkout of the involved performances and input parameters. For the linear Delta robot, the GCI values of the BSM and SM are the same as that of the other. If the specified LCI is suitable for the SM, the GCI should not be checked here. In addition, in the case that a commercial actuator is used, the input parameter should be checked as well. If it is not Ok, turn to S5 and repeat S6 S10. Maybe, the special case that no BSM in the optimum region can be suitable occurs. If so, turn to S4 to identify another optimum region and perform S5 S10 again. If every item is suitable, the optimal kinematic design can be finished. 6. Conclusions This paper first introduces one solution to the parameter infinity problem existing in the design issue of parallel mechanisms. By using the parameter-finiteness normalization method (PFNM) presented here, any n-parameter combination, which exists in an n-dimensional infinite space, of a type of mechanism can be represented by a normalized mechanism. The PFNM not only reduces the parameter number from n to n 1 but defines the bound of each parameter logically as well. Therefore, the normalized mechanism, which is referred to as the basic similarity mechanism (BSM) in this paper, consists in an (n 1)-dimensional finite space that is defined as the parameter design space (PDS). A kind of mechanism, whose parameters are D-time those of the BSM, is accordingly defined as the similarity mechanism (SM). The analysis shows that the BSM and all of its SMs (with different normalization factor D) are similar in both size and performance. Accordingly, this paper proposes a novel design methodology, i.e., the Performance-Chart based Design Methodology (PCbDM), for mechanisms with fewer than five linear parameters. Compared with other design methods, the proposed methodology has some advantages as follows: (a) one performance criterion corresponds to one chart, which can graphically and globally show the relationship between the criterion and design parameters; (b) for such a reason in (a), the fact that some performance criteria are antagonistic is no longer a problem in the design; (c) the optimal design can consider multi-objective functions or multi-criteria, and also guarantees the optimal result; (d) the design method provides all possible optimal solutions to a design problem and finally, (e) as the optimal design can be carried out by using performance charts, this methodology shall be said to be acceptable in practice. The results of the paper will be very useful in developing computer-aided design systems for parallel mechanisms. The proposed methodology can be also applied to the optimal kinematic design of serial and parallel robots and other mechanisms. Acknowledgement This work is supported by the National Natural Science Foundation of China (No ), and partly by Tsinghua Basic Research Foundation and the Beijing Nova Program. References [1] M. Stock, K. Miller, Optimal kinematic design of spatial parallel manipulators: application to linear Delta robot, J. Mech. Des. 125 (2004)

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