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Actuation schemes of a spatial 3-PPRR parallel mechanism J-S Zhao, F Chu, Z-J Feng and J S Dai Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 2012 226: 228 originally published online 23 September 2011 DOI: 10.1177/0954406211412683 The online version of this article can be found at: http://pic.sagepub.com/content/226/1/228

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Actuation schemes of a spatial 3-PPRR parallel mechanism J-S Zhao1*, F Chu1, Z-J Feng1, and J S Dai2,3 1 Department of Precision Instruments and Mechanology, Tsinghua University, Beijing, P. R. China 2 Centre for Advanced Mechanisms and Robotics, Tianjin University, Tianjin, P. R. China 3 King’s College London, University of London, London, United Kingdom The manuscript was received on 21 February 2011 and was accepted after revision for publication on 16 May 2011. DOI: 10.1177/0954406211412683

Abstract: This paper presents a systematic method to analyse actuation schemes. With the wide application of complex spatial mechanisms in engineering the number of excess constraints included in a mechanism is increasing. As a result, there are an increasing number of cases in which the number of actuators needed to control an end-effector is larger than the number of degrees of freedom possessed by the end-effector. The mechanism architecture is first presented and then the mobility of the mechanism is analysed in terms of independent parameters in position constraint equations. Finally, the possible and feasible actuation schemes for the endeffector are generated and analysed. Keywords: possible actuation scheme, feasible actuation scheme, actuation of the end-effector, mechanism, Burnside’s lemma

1

INTRODUCTION

Mechanisms to create mobility in systems have been the subject of investigation for over 150 years [1]. Early examples of the use of parallel mechanisms include Gough’s tyre testing machine and Stewart’s motion simulator [2]. Recently, full degree-of-freedom mechanisms have been investigated and applied in systems such as the Eclipse-RP [3] and Eclipse II [4]. Detailed reviews on this area can be found in Gogu [1, 5] and Dai et al. [2]. Traditionally, the mobility of a mechanism is thought of in terms of the smallest number of independent parameters that can be used to define the configuration of the mechanism [6–8]. Gogu [1, 5, 9] proposed a method to analyse the mobility created by using parallel mechanisms that was based on the use linear transformations. The over-constrained spatial parallel mechanism can be used to create high levels of platform stability *Corresponding author: Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, P. R. China. email: [email protected].

against external loads. However, it is often the case that the number of actuators required by this type of approach is greater than the number of degrees of freedom of the end-effector. Thus, there is considerable interest in developing methods to extend the motion control of actuators in order that fewer actuators are required for a specific task. This paper presents the results of an investigation into mobility and actuation schemes for the 3-PPRR spatial parallel mechanism shown in Fig. 1. The particular characteristic of this mechanism is that the end-effector has only a single degree of freedom but requires a minimum of two actuators for its control. The fact that the free motions and constraints of a rigid body are reciprocal in screw theory considerably simplifies the degree-of-freedom analysis. Regarding the reciprocity between the free motions (twists) and the constraints (wrenches) of a rigid body, the conclusions deduced from one can be equally obtained from the other. Therefore, reciprocal screw theory [10–13] has been extensively utilized to analyse the mobility of a mechanism [2, 14–23]. Dai [24] has presented a review on recent developments in the analysis of rigid-body displacements. Hunt [11], Phillips

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Actuation schemes of a spatial 3-PPRR parallel mechanism

Fig. 1

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A 3-PPRR spatial parallel mechanism and the coordinate frame

[12, 13], Phillips and Hunt [25], Waldron [26], Ohwovoriole and Roth [27], Dai and Jones [28–30], Gibson and Hunt [31, 32], Parkin [33], and Rico Martinez and Duffy [34–36] have all made recent contributions to the literature on the application of screw theory. Gibson and Hunt [31, 32], Parkin [33], Rico Martinez and Duffy [34–36], Tsai and Lee [37], Zhang and Xu [38], Fang and Huang [39], Huang and Wang [40], and Zhao et al. [41] have likewise contributed to the literature on the identification of principal screws. Screw theory has been applied to the study of instantaneous kinematics [42–44], instantaneous mobility [45, 46], singularity analysis of mechanisms [47, 48], statics of parallel manipulators [49], and common constraints analysis [2, 11–21, 23, 26]. Zhao et al. [50] proposed a reciprocal-screw-theory-based method to investigate the degree-of-freedom level for an end-effector of a parallel mechanism. This method could be used to study the problem considered in this paper. However, since the mechanism shown in Fig. 1 is highly symmetric it is the current authors’ contention that the problem is amenable to analysis based on the geometric relationships between its kinematic chains; with the mobility of the mechanism being subsequently generated from the use of position constraint equations.

The rest of this paper is organized as follows. Section 2 presents a geometrical method to analyse the degree-of-freedom level of the end-effector and the mobility of the mechanism. Section 3 reports the possible actuation schemes that can be obtained using Burnside’s lemma and the statics of the mechanism under various actuation schemes.

2 ARCHITECTURE AND MOBILITY OF THE MECHANISM The primary characteristics of the mechanism shown in Fig. 1 are that the end-effector, E1 E2 E3 , is connected to the fixed base, B1 B2 B3 , through three PPRR kinematic chains, and the two prismatic pairs, Bi , Ci ði ¼ 1, 2, 3Þ, in their chain are orthogonal to each other. These three PPRR kinematic chains, Bi Ci Di Ei ði ¼ 1, 2, 3Þ, are identical in topology [30]. In what follows, the degree-of-freedom level of the endeffector, E1 E2 E3 , will be investigated by exploiting the geometric characteristics of the mechanism; and then the position of the end-effector relative to the fixed base will be established in a single Cartesian coordinate frame, after which the mobility of the mechanism will be discussed from the viewpoint of the Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science


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J-S Zhao, F Chu, Z-J Feng, and J S Dai

number of independent parameters of the position equations. Before developing the first topic, further observations on the structural geometry of the mechanism are made in the hope that it might reveal relationships that could be used to solve this problem. As shown in Fig. 2, in each PPRR kinematic chain of the end-effector, the links are always in the same plane. For instance, kinematic chain B1 C1 D1 E1 is always located in the 1 -plane, B2 C2 D2 E2 is always in the 2 -plane, and B3 C3 D3 E3 is always in the3 plane. Obviously, each PPRR series kinematic chain only allows its distal connector to take three independent planar motions within the plane in which the kinematic chain is located: two translations and one rotation about the normal of the plane. In other words, no rotations about the direction parallel to the plane are possible. Consequently, the end-effector, E1 E2 E3 , being constrained by three such kinematic chains only has one translation along the

Fig. 2

common intersecting line, OB OE , of the three planes, 1 , 2 , and 3 . Therefore, it is explicit that the end-effector of the mechanism only has one translational degree of freedom along the common intersecting line of the three planes of its kinematic chains. The common intersecting line is also the centralized line of the mechanism, and therefore, the mechanism shown in Fig. 1 is also called the centralized moving parallel mechanism. It should be noted that only two PPRR series kinematic chains are necessary to make the end-effector E1 E2 E3 trace the straight line OB OE . The third, and redundant, kinematic chain is used mainly to increase the rigidness and stability of the end-effector. However, this can lead to an increase in the number of actuators. As previously mentioned the end-effector E1 E2 E3 can only move up and down the centralized line OB OE . Therefore, one can assume that the z-axis is along this centralized line, the x-axis is along the sliding direction of the prismatic pair B1 , and the

Geometry constraints for the parallel mechanism (a) different painting schemes for the coloured two-striped flags and (b) actuation schemes for the two chains



origin of the coordinate frame is at OB which is the crossing point of the three fixed guide lines. Therefore, the y-axis is determined in accordance with the right-hand rule. The Cartesian coordinate frame is shown in Fig. 1. Consequently, the position constraints can be obtained for each kinematic chain when the displacement of the end-effector in the z-direction is assumed to be h. It is also assumed that the position parameter of the prismatic pair Bi ði ¼ 1, 2, 3Þ relative to the centre of the fixed base, OB , is ai , the position parameter of the prismatic pair Ci relative to pair Bi is bi , the length of the link Di Ei is li , the rotational angle relative to the horizontal line is Di , and the radius of the circumscribed circle of the end-effector is r. Then, position constraint equations for the mechanism can be established. The constraint equations for the second kinematic chain located in the 2 -plane can be expressed as a2 r ¼ l2 cos D2 ð1Þ h b2 ¼ l2 sin D2 The constraint equations for the other two kinematic chains have exactly the same form. Therefore, the position constraint equations for the mechanism system are 8 a1 r ¼ l1 cos D1 > > > > h b1 ¼ l1 sin D1 > > < a2 r ¼ l2 cos D2 ð2Þ h b2 ¼ l2 sin D2 > > > > a r ¼ l3 cos D3 > > : 3 h b3 ¼ l3 sin D3 where r is a known structure parameter, and h, ai , bi , Di ði ¼ 1, 2, 3Þ, are unknown variables. From equation (2), it is easy to see that the mechanism system has ten variables but only six constraint equations. Therefore, there are four independent variables in the system. The mobility of a mechanism has been defined as ‘the number of independent coordinates required to define its position’ [8]. Therefore, the mobility of the mechanism shown in Fig. 1 based on this definition should be four. Thus, the mechanism can be completely controlled if one adds four actuators to the mechanism. Is this always true in reality? Section 3 will discuss this question in detail by applying combinatorial mathematics and Burnside’s lemma. 3

POSSIBLE ACTUATION SCHEME ANALYSIS BASED ON CYCLIC GROUPS

For the mechanism shown in Fig. 1, the total number of kinematic pairs is 12, any one of which can be assigned an actuator. Therefore, the number of

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possible actuation schemes can be immediately calculated using the combinatorial formula 4 N ¼ C12 ¼

12 11 10 9 ¼ 495 4321

ð3Þ

However, among these 495 possible actuation schemes, there are some schemes that are identical when the symmetry of the mechanism structure is taken into account. For the sake of convenience in this paper, each actuation scheme is represented by the symbols of the actuated pairs in the mechanism. For example, actuation scheme B1 B2 B3 C1 represents that the kinematic pairs B1 , B2 , B3 , and C1 are each assigned an actuator. This actuation scheme can also be denoted by B1 C1 B2 B3 which is effectively identical to B1 B2 C2 B3 and B1 B2 B3 C3 in what occurs in the movement of the end-effector. That is to say, actuation schemes B1 C1 B2 B3 , B1 B2 C2 B3 , and B1 B2 B3 C3 are three different schemes in equation (3) but should be grouped into one actuation scheme because they are identical in topology when symmetry is taken into account. Consequently, the number of different actuation schemes needs to be obtained. According to the position constraint equation set (2), it is explicit that the mobility of the mechanism is four. Therefore, amongst the three kinematic chains of the end-effector there must be one kinematic chain in which there are two kinematic pairs at least that will be selected as the actuators. As a result, the actuation schemes for this three-chain mechanism can be roughly divided into three groups: (a) the maximum number of actuators for each kinematic chain is 2; (b) the maximum number of actuators for each kinematic chain is 3; (c) the maximum number of actuators for each kinematic chain is 4. The first group can be split into two further types. One type is that after the two actuators are assigned to one kinematic chain, the other two actuators are evenly allocated to the other two kinematic chains; and the other type is that after the two actuators are assigned to one kinematic chain, the other two actuators are simultaneously allocated to any one of the other two kinematic chains. Calculating the number of actuation schemes for the first type can be divided into two steps: the first one is to find the number of possible means to select the two that are to be the actuators from the four pairs in the kinematic chain, and the second step is to find the different choices in selecting one pair from the other two kinematic chains. For the first step, the Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science


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number of possible selections can be easily expressed by the combinatorial formula N11 ¼ C42 ¼

43 ¼6 21

For the second step, the group actions on a finite set should be first recalled for the sake of simplicity. Assume to be a target set and G to be a cyclic group acting on . For every a 2 , there is n o

a ¼ g ða Þg 2G where a is called the orbit of under the action of G, and a is a representative element of the orbit. According to Burnside’s lemma [51], the number of orbits of a finite set under the operations of finite group G is N12 ¼

1 X 1 þ2 þþm n jGm j g 2G

Fix g1 ¼ 41 þ2 ¼ 41þ1 ¼ 16

ð5Þ

Fix g2 ¼ 4 ¼ 41 ¼ 4

ð6Þ

and substituting equations (5) and (6) into equation (4) yields

1 X Fix g jG j g 2G

where jG j denotes the order of group G, Fix g represents the number of a 2 fixed by g , the summation operates on every element of the group, and Fix g ¼ a a 2 , g ða Þ ¼ a . For each a that satisfies the relationship g ða Þ ¼ a, the actuation schemes corresponding to the same cycle of g must be the same. The number of cycles that g contains is 1 þ 2 þ þ m ; thus the number of possible selections of the actuation schemes that satisfy the criterion of g ða Þ ¼ a is n 1 n2 nn ¼ n 1 þ2 þþn where n is the total number of actuators that can be assigned for a kinematic chain. Therefore, the number of different schemes can be obtained when actuation 1 þ2 þþn is substituted into Burnside’s Fix g ¼ n lemma [51] N12 ¼

represents the kinematic pairs, Bi , Ci , Di , Ei ði ¼ 1, 2Þ shown in Fig. 1. Now, the problem is transformed into the following question: how many striped flags are there having two equal stripes each of which can be coloured red, blue, green, or yellow? The results correspond to the actuation schemes for the two kinematic chains. According to equation (4), the number of different coloured two-striped flagsis the number of orbits of the set ¼ r b g y under the operations of group G 2 . As n ¼ 4, g1 ¼ ð1Þð2Þ, g2 ¼ ð12Þ, one can find that

ð4Þ

m

where g is expressed in 11 22 mm -format, Gm is a cyclic group, and the summation is operated on every permutation of Gm . Now, if the end-effector, E1 E2 E3 , shown in Fig. 1 is thought of as an equilateral triangle and has three identical kinematic chains in its topology, Bi Ci Di Ei ði ¼ 1, 2, 3Þ, the symmetries for the two kinematic chains that the other two actuators will be assigned to can be expressed by a cyclic group G2 ¼ g1 g2 where g1 ¼ ð1Þð2Þ and g1 indicates the identical cycle, g2 ¼ ð12Þ indicates a mirror operation. When every kinematic pair of each chain is considered to be different and to be encoded as numbers to indicate their differences, the possible actuators in each kinematic chain forms a finite set,

¼ 1 2 3 4 , the element number of which

N12 ¼

1 1 X Fix g ¼ 42 þ 4 ¼ 10 jG2 j g 2G 2

ð7Þ

The different striped flags are shown in Fig. 3(a) and (b) and the corresponding actuation schemes are illustrated in Fig. 3(b). The arrows indicate the locations of the actuators. Therefore, the total number of actuation schemes for the first type in the first group can be expressed as N1 ¼ N11 N12 ¼ 6 10 ¼ 60 For the second type of the first group, the symmetries for the two kinematic chains can also be expressed by a cyclic group G2 ¼ ð1Þð2Þ ð12Þ . In this type, there are two kinematic chains each one of which has two pairs that can be selected as the actuators. Therefore, there are C42 ¼ ð4 3Þ ð2 1Þ ¼ 6 selections for each chain. As a result, there will be two chains; each one of which contains two actuators in this type. Consequently, according to Burnside’s lemma [51], the different number of possible actuation schemes in essence is the number of orbits of a six-element set under the actions of group G. Therefore, the number of actuation schemes will be N13 ¼

1 1 X Fix g ¼ 1 62 þ 1 6 ¼ 21 jG2 j g 2G 2

when the symmetries of the two kinematic chains are considered to be identical in topology. Therefore, the total number of actuation schemes for the first group can be obtained as N1 ¼ N11 N12 þ N13 ¼ 6 10 þ 21 ¼ 81


ð8Þ


Fig. 3

233

Actuation schemes for the two kinematic chains of the mechanism


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For the second group, the number of possible actuation schemes can be directly calculated using the combinatorial formula N2 ¼ C 34 C41 ¼ 4 4 ¼ 16

ð9Þ

Similarly, the number of possible actuation schemes for the third group can also be directly computed using the combinatorial formula N3 ¼ C 44 ¼ 1

ð10Þ

The number of different actuation schemes obtained by considering the symmetries in the mechanism architecture can be obtained from equations (8), (9), and (10) N ¼ N1 þ N2 þ N3 ¼ 81 þ 16 þ 1 ¼ 98

ð11Þ

Equation (11) indicates that the number of different possible actuation schemes is 98. In other words, there are 98 different actuation schemes for the mechanism shown in Fig. 1 that could be used in engineering applications. That is to say, any one of the 98 different actuation schemes can be selected as the final design. Are all the possible actuation schemes feasible in reality? This is a new problem that has not previously been addressed in the literature. It should be highlighted that this phenomenon results from the over-constrained nature of the mechanism. 4

ANALYSIS OF THE FEASIBLE ACTUATION SCHEMES

This section will investigate the possible actuation schemes of the mechanism from the aspect of statics. As previously discussed the end-effector only has one translational degree of freedom along the z-direction as shown in Fig. 1. That is to say that the end-effector only has the ability to output power for loads in the zdirection. Therefore, the load of the end-effector in the z-direction can be denoted by a variable, Fz . Now, the statics equations of the mechanism can be established for each of the 98 actuation schemes. For example, assume that the mechanism is actuated under the B1 C1 B2 B3 actuation scheme. Therefore, the four actuators are assigned to the kinematic pairs B1 , C1 , B2 , and B3 , respectively. Assume that the loads in the z-direction exerted on the terminals of the three kinematic chains from the end-effector are F1 , F2 , and F3 , respectively, then the following equation must hold F1 þ F2 þ F3 ¼ Fz

ð12Þ

Suppose that the angle from the xoy-plane to link Di Ei ði ¼ 1, 2, 3Þ is denoted by i . The following

equations can be written for the first kinematic chain shown in Fig. 4 FC1 F1 ¼ 0 ð13Þ FB1 F1 cot 1 ¼ 0 where cot 1 ¼ xE1 xD1 zE1 zD1 , FB1 represents the horizontal force from joint B1 , and FC1 represents the vertical force from joint C1 . Similar arguments apply to the second kinematic chain, but because there is no actuator at C2 , FC2 ¼ 0, requiring that FB2 ¼ 0 holds simultaneously. The third kinematic chain is equivalent to the second chain and thus FC3 ¼ 0 and FB3 ¼ 0 also holds. Hence, equation (12) and equation (13) yield the inverse static requirement that 8 xE xD1 > FB1 ¼ 1 Fz > > zE1 zD1 > > > < FC1 ¼ Fz ð14Þ > > > FB2 ¼ 0 > > > : FB3 ¼ 0 The theoretical result represented by equation (14) indicates that only the actuators B1 and C1 bear the load exerted by the end-effector, while the other two actuators B2 and B3 do not bear any load in the zdirection. Could these two actuators, B2 and B3 , be removed from the system? Before answering this question, a further analysis on the geometric characteristics should be performed. As the end-effector, E1 E2 E3 , only has one translational degree of freedom along the z-direction, the mechanism structure shown in Fig. 1 can be equivalently substituted by the one shown in Fig. 5. In any of the 1 , 2 , and 3 planes shown in Fig. 2, the kinematic chain oB Bi Ci Di Ei oE ði ¼ 1, 2, 3Þ forms a planar PPRRP five-link mechanism. Obviously, all the links in the five-link mechanism, oB B1 C1 D1 E1 oE , have determined motions when two actuators are assigned to prismatic pairs B1 and C1 , individually. Therefore, the end-effector, E1 E2 E3 , has a determined position along the z-direction. As a result, the relative motions of the links in the five-link mechanism, oB B2 C2 D2 E2 oE , will be determined when a third actuator is also assigned to any one of the pairs C2 , D2 , and E2 . This is also the case for the five-link mechanism, oB B3 C3 D3 E3 oE , when a fourth actuator is allotted to the prismatic pair, C3 . Obviously, these two actuators, C2 (or D2 or E2 ) and C3 (or D3 or E3 ), are simply utilized to keep the configurations of the planar five-link mechanisms, oB B2 C2 D2 E2 oE and oB B3 C3 D3 E3 oE , and they cannot output any useful power that can be used to propel the end-effector. Therefore, these two actuators can be removed in theory. Thus, the B1 C1 B2 B3 actuation



Fig. 4

235

The statics of the mechanism under the B1 C1 B2 B3 actuation scheme

scheme is equivalent to the B1 C1 actuation scheme from the viewpoint of propelling the end-effector. The only differences between these two schemes are that the relative motions of the links in the subchains, B2 C2 D2 E2 and B3 C3 D3 E3 , are determined in the former scheme but not determined in the later one. Alternatively, in order to keep the links of the other kinematic chains, B2 C2 D2 E2 and B3 C3 D3 E3 , to have a determined motion, one can lock any single joint in each chain. However, these two seemingly redundant kinematic chains, B2 C2 D2 E2 and B3 C3 D3 E3 , do contribute to the enhancement of the torsional stiffness and linear stiffness and the stability of the end-effector. This is one of the positive features of the overconstrained mechanism. With the advent of spatial parallel mechanisms, the primary concern of designers has been the degree-offreedom level of the end-effector and its actuation. In this regard, it is necessary to investigate the actuation

schemes after obtaining the extent of the degree-offreedom level of an end-effector. In a similar process to the one previously discussed, the other 97 actuation schemes were also comprehensively investigated. The possible actuation schemes of class 1 are listed in Table 1 in appendix 2, all the possible actuation schemes of class 2 are listed in Table 2 in appendix 2, and the possible actuation scheme of class 3 is listed in Table 3 in appendix 2. All the possible actuation schemes for the mechanism shown in Fig. 1 are listed in Tables 1 to 3. Among these 98 actuation schemes, there are 30 kinds that are feasible if one requires every link of the mechanism be controlled. As a matter of fact, only two actuators are effectively utilized to output power to the end-effector even in these 30 completely feasible actuation schemes. They can be analysed using the technique proposed in the previous section. The Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science


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Fig. 5

The equivalent mechanism

other two actuators in these 30 feasible actuation schemes are only used to keep the configurations of the other two kinematic chains. Among the remaining actuation schemes: (a) nine schemes could be used to control the endeffector but not to control all the links of the mechanism; (b) 19 schemes could be used to control the endeffector with one or more actuators but not to control all links of the mechanism; (c) 40 schemes can neither control the end-effector nor control the links of the mechanism. The design must be considered to be a failure when any one of latter 40 actuation schemes is selected for the mechanism. From this analysis it is clear that the actuations of a mechanism are affected by the utilized actuation scheme. Muller [52] investigated the internal preload control of redundantly actuated parallel manipulators which is a good demonstration of the problems

discussed in this paper. The presented method for the analysis of possible and feasible actuation schemes will find great use in the future in the design of mobility mechanisms. 5 CONCLUSIONS This paper presented a systematic process to decide the possible and feasible actuation schemes for a mechanism. First, a 3-PPRR mechanism architecture was presented, and then, the mobility of the mechanism was analysed from the aspect of the number of independent parameters in position constraint equations. Based on the mobility analysis, the number of possible actuation schemes was calculated using a combinatorial formula and Burnside’s lemma. A statics analysis of each actuation scheme was used to obtain the characteristics of every possible actuation scheme. The representative simple example shows that the mobility calculation of a mechanism can only provide the number of actuators that are



needed to control all the links of the mechanism and cannot guarantee that any given actuation scheme is available to control the mechanism. This method can be used to synthesize and optimize the actuation schemes of future complex spatial mechanisms and should allow the creation of a lot of possible actuation schemes. ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China [Grant number 50805083], a Foundation for the Author of National Excellent Doctoral Dissertation of China [Grant number 200741] and Specialized Research Fund for the Doctoral Program of Higher Education of China [Grant number 200800031004]. The authors gratefully acknowledge these support agencies. In addition, the authors would also greatly acknowledge the anonymous reviewers for their invaluable suggestions in enhancing our paper. ß Authors 2011 REFERENCES 1 Gogu, G. Mobility of mechanisms: a critical review. Mech. Mach. Theory, 2005, 40(9), 1068–1097. 2 Dai, J. S., Huang, Z., and Lipkin, H. Mobility of overconstrained parallel mechanisms. Trans. ASME, J. Mech. Des., 2006, 128(1), 220–229. 3 Kim, J., Cho, K. S., Hwang, J. C., Iurascu, C. C., and Park, F. C. Eclipse-RP: a new RP machine based on repeated deposition and machining. Proc. IMechE, Part K: J. Multi-body Dyn., 2002, 216(1), 13–20. 4 Kim, J., Hwang, J. C., Kim, J. S., Iurascu, C. C., Park, F. C., and Cho, Y. M. Eclipse II: a new parallel mechanism enabling continuous 360-degree spinning plus three-axis translational motions. IEEE Trans. Robot. Autom., 2002, 18(3), 367–373. 5 Gogu, G. Structural synthesis of parallel robots: part 1-methodology 2008 (Springer). 6 Shigley, J. E. and Uicher, J. J. Theory of machines and mechanisms, 1980 (McGraw-Hill, New York). 7 Waldron, K. J. and Kinzel, G. L. Kinematics, dynamics, and design of machinery, 1999, pp. 1–42 (John Wiley & Sons, New York). 8 Norton, R. L. Design of machinery—an introduction to the synthesis and analysis of mechanisms and machines, 2001 (McGraw-Hill, New York, USA). 9 Gogu, G. Mobility and spatiality of parallel robots revisited via theory of linear transformations. Eur. J. Mech. A, Solids, 2005, 24(4), 690–711. 10 Ball, R. S. A treatise on the theory of screws, 1998 (Cambridge University Press, Cambridge, UK). 11 Hunt, K. H. Kinematic geometry of mechanisms, 1978 (Oxford University Press, Oxford, UK).

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APPENDIX Notation ai ði ¼ 1, 2, 3Þ

bi ði ¼ 1, 2, 3Þ

Bi Ci Di Ei ði ¼ 1, 2, 3Þ

C Fi ði ¼ 1, 2, 3Þ

Fz FB1 Fix g g G jG j h li N r Di ði ¼ 1, 2, 3Þ i ði ¼ 1, 2, Þ

a


position parameter of the prismatic pair Bi ði ¼ 1, 2, 3Þ relative to the geometry centre of the fixed base, OB position parameter of the prismatic pair Ci relative to pair Bi an actuation scheme in which each joint of Bi , Ci , Di , and Ei is assigned one actuator combinatorial operation loads in z-direction exerted on the terminal of the ith kinematic chain by the endeffector external load in z-direction the horizontal force of joint B1 number of a 2 fixed by g element of the cyclic group cyclic group the order of group G displacement of the endeffector in the z-direction length of the link Di Ei number of actuation schemes radius of the circumscribed circle of the end-effector the rotational angle of joint Di relative to the horizontal line number of cycles contained by g target set orbit of under the action of G


239

APPENDIX 2 Table 1.

Actuation schemes

Actuated kinematic pairs Bi

Ci

Bj

Cj

Ck Dk Ek Dk Ek Ek Bj

Dj Bi

Ej Di

Cj Dj Bi

Ei

Ck Dk Ek Dk Ek Ej Bj

Cj Dj Ci

Di

Ej Bj

Cj Dj Ci

Ei

Cj Dj Di

Ej Ei

Cj Dj Ej

Ej Bj

Ck Dk Ek Dk Ek Ek Bj

Ck Dk Ek Dk Ek Ek

The possible actuation schemes of class 1

Bk Ck Dk Ek Bi Ci Cj Ck Bi Ci Cj Dk Bi Ci Cj Ek Bi Ci Dj Dk Bi Ci Dj Ek Bi Ci Ej Ek Bk Ck Dk Ek Bi Di Cj Ck Bi Di Cj Dk Bi Di Cj Ek Bi Di Dj Dk Bi Di Dj Ek Ek Bk Ck Dk Ek Ck Dk Ek Dk Ek Ek Bk Ck Dk Ek Ck Dk Ek Dk Ek Ek Bk Ck Dk Ek Ci Ei Cj Ck Ci Ei Cj Dk Ci Ei Cj Ek Ci Ei Dj Dk Ci Ei Dj Ek Ci Ei Ej Ek Bk Ck Dk Ek Di Ei Cj Ck Di Ei Cj Dk Di Ei Cj Ek Di E i Dj Dk Di E i Dj E k Di Ei Ej Ek

Feasibility to control End-effector

Links

Notes

Sequence

Bi Ci Bj Bk Bi Ci Bj Ck B i C i B j Dk Bi Ci Bj Ek

Feasible

Feasible

All links are controlled

Bi Di Bj Bk Bi Di Bj Ck Bi Di Bj Dk Bi Di Bj Ek

Unfeasible

Unfeasible

All links are not controlled

Unfeasible

Unfeasible


Feasible

Feasible

All links are controlled

Unfeasible

Unfeasible


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Bi Di Ej Ek Bi Ei Bj Bk Bi Ei Bj Ck Bi Ei Bj Dk Bi Ei Bj Ek Bi Ei Cj Ck Bi Ei Cj Dk Bi Ei Cj Ek Bi Ei Dj Dk Bi Ei Dj Ek Bi Ei Ej Ek Ci Di Bj Bk Ci Di Bj Ck Ci Di Bj Dk Ci Di Bj Ek Ci Di Cj Ck Ci Di Cj Dk Ci Di Cj Ek Ci Di Dj Dk Ci Di Dj Ek Ci Di Ej Ek Ci Ei Bj Bk Ci Ei Bj Ck C i E i B j Dk Ci Ei Bj Ek

Di Ei Bj Bk Di Ei Bj Ck Di Ei Bj Dk Di Ei Bj Ek

(continued)


240


Table 1. Continued Actuation schemes

Actuated kinematic pairs Bi

Ci

Bj

Di

Ci

Ci

Di

Ei

Di

Ei

Ei

Links

Notes

Sequence

Feasible but overactuated Feasible

Unfeasible

There is one redundant actuator in the actuation scheme while the chain Bk Ck Dk Ek is not controlled There is one redundant actuator in the kinematic chain Bj Cj Dj Ej while the chain Bk Ck Dk Ek is not controlled All links are not controlled. There is one redundant actuator in the actuation scheme while the chain Bk Ck Dk Ek is not controlled All links are not controlled All links are not controlled There is one redundant actuator in the actuation scheme while the chain Bk Ck Dk Ek is not controlled All links are not controlled There is one redundant actuator in the actuation scheme while the chain Bk Ck Dk Ek is not controlled There is one redundant actuator in the actuation scheme while the chain Bk Ck Dk Ek is not controlled There is one redundant actuator in the actuation scheme while the chain Bk Ck Dk Ek is not controlled All links are not controlled.

61

Bi Ci Bj Cj

Dj Ej Dj Ej

B i C i B j Dj Bi Ci Bj Ej B i C i C j Dj Bi Ci Cj Ej

Dj

Ej

Bi Ci Dj Ej

Feasible

Unfeasible

Bj

Dj Ej Dj Ej

Bi Di Bj Dj Bi Di Bj Ej Bi Di Cj Dj Bi Di Cj Ej

Unfeasible

Unfeasible

Feasible

Unfeasible

Dj

Ej

Bi Di Dj Ej

Unfeasible

Unfeasible

Bj

Ej

Bi Ei Bj Ej

Unfeasible

Unfeasible

Cj

Dj Ej

Bi Ei Cj Dj Bi Ei Cj Ej

Feasible

Unfeasible

Dj

Ej

Bi Ei Dj Ej

Unfeasible

Unfeasible

Cj

Dj Ej

Ci Di Cj Dj Ci Di Cj Ej

Feasible but overactuated

Unfeasible

Dj

Ej

C i Di Dj E j

Feasible

Unfeasible

Cj

Ej

Ci Ei Cj Ej

Unfeasible

Dj

Ej

C i E i Dj E j

Feasible but overactuated Feasible

Unfeasible

Dj

Ej

Di Ei Dj Ej

Unfeasible

Unfeasible

Cj

Bi

End-effector

Cj

Cj

Bi

Feasibility to control



62 63 64 65 66

67 68 69 70

71 72 73 74

75 76 77

78

79 80

81


Table 2.

Bi

Ci

The possible actuation schemes of class 2 Actuation schemes

Actuated kinematic pairs Di

Bj Cj Dj Ej

Bi Ci Di Bj Bi Ci Di Cj Bi Ci Di Dj Bi Ci Di Ej

Ei

Bj Cj Dj Ej

Bi Ci Ei Bj Bi Ci Ei Cj Bi Ci Ei Dj Bi Ci Ei Ej


Links

Notes

Sequence


Unfeasible

There is one redundant actuator in the kinematic chain Bi Ci Di Ei while the chain Bk Ck Dk Ek is not controlled There is one redundant actuator in the kinematic chain Bi Ci Di Ei while the chain Bk Ck Dk Ek is not controlled There is one redundant actuator in the kinematic chain Bi Ci Di Ei while the chain Bk Ck Dk Ek is not controlled There is one redundant actuator in the kinematic chain Bi Ci Di Ei while the chain Bk Ck Dk Ek is not controlled

82 83 84 85

Ci

Di

Ei

Bj Cj Dj Ej

Ci Di Ei Bj Ci Di Ei Cj Ci Di Ei Dj Ci Di Ei Ej

Feasible but Overactuated

Unfeasible

Bi

Di

Ei

Bj Cj Dj Ej

Bi Di Ei Bj Bi Di Ei Cj Bi Di Ei Dj Bi Di Ei Ej

Unfeasible

Unfeasible

Table 3.

Bi

Ci

Di

Ei

86 87 88 89

90 91 92 93

94 95 96 97

The possible actuation scheme of class 3 Actuation schemes

Actuated kinematic pairs

241

Bi Ci Di Ei


Links

Notes

Sequence


Unfeasible

There are two redundant actuators in the actuation scheme while the kinematic chains Bj Cj Dj Ej and Bk Ck Dk Ek are not controlled

98


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