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Cable-truss hybrid double-layer deployable mechanical network constructed of Bennett linkages and planar symmetric four-bar linkages
Mechanism and Machine Theory 133 (2019) 459–480
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Cable-truss hybrid double-layer deployable mechanical network constructed of Bennett linkages and planar symmetric four-bar linkages Xiaoke Song a, Hongwei Guo b,∗, Sanjun Liu c, Fei Meng a, Qiping Chen d, Rongqiang Liu b, Zongquan Deng b a
School of Mechatronics and Vehicle Engineering, East China Jiaotong University, PR China State Key Laboratory of Robotics and System, Harbin Institute of Technology, PR China c Shanghai Aerospace Equipment Manufacturer Co. Ltd., PR China d Key Laboratory of Conveyance and Equipment Ministry of Education, East China Jiaotong University, PR China b
a r t i c l e
i n f o
Article history: Received 9 August 2018 Revised 26 November 2018 Accepted 4 December 2018
Keywords: Deployable mechanism Double-layer deployable mechanical network Hybrid linkage Bennett linkage
a b s t r a c t This research proposes a single degree of freedom (DOF) cable-truss hybrid double-layer deployable mechanical network (DLDMN) consisting of equilateral Bennett linkages and planar symmetric four-bar linkages. The kinematics of the equilateral Bennett linkage is investigated. Then a single-loop four-bar hybrid linkage is constructed using the equilateral Bennett linkage and planar symmetric four-bar linkage. The original form of the Bennett linkage in the hybrid linkage is replaced by its alternative form and a synchronous mechanism is introduced to obtain a hybrid linkage interlayer vertical pillar with an adjustable height. The interlayer vertical pillar is employed to a single-layer mechanical network consisting of Bennett linkages. Connecting these pillars with cables, a cable-truss hybrid DLDMN that could fit various surfaces is assembled. An offset paraboloid antenna is set up using the DLDMN. A prototype is manufactured to verify the feasibility of the design. The proposed single DOF DLDMN has the advantages of high deployment ratio of spatial single-loop linkages and normative motion of planar linkages. The proposed network can be used to construct a large-scale spaceborne deployable antenna with high stiffness. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Deployable mechanisms are the mechanisms that can be folded to a small size and be deployed into a large predetermined shape. These mechanisms are extensively employed to various scenarios, such as spaceborne deployable antennas, solar panels and civil architectures. Deployment ratio is a key index for evaluating the performance of deployable mechanisms, especially for spaceborne deployable antenna, because the limited volume of rocket payload bay raises a rigorous requirement for deployment ratio. Because the motion of single-loop overconstrained spatial linkage (SOSL) is spatial, it can be deployed to a planar polygon and be folded to a bundle of rods with the length of a single link, leading to its higher deployment ratio than a planar linkage [1]. Thus, SOSL is a good candidate for deployable mechanisms. To date, many scholars,
∗
Corresponding author. E-mail address: [email protected] (H. Guo).
https://doi.org/10.1016/j.mechmachtheory.2018.12.003 0094-114X/© 2018 Elsevier Ltd. All rights reserved.
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such as Chen and You [2–7], Yu [8], Lu [9], Huang [10–12], Song [13] and Qi [14,15], have investigated deployable mechanisms constructed with SOSLs and have obtained substantial achievements. A large-scale spaceborne deployable antenna should have good performance in folding and deploying, also with high stiffness, light weight and simple and reliable drive and control. Most of the current deployable mechanisms constructed with SOSLs are of single layer with low integral stiffness in large scale. If the link stiffness is enhanced to obtain high integral stiffness, the weight will be increased, as the link weight would increase in the meantime. Faced with the increased link weight, these mechanisms are not feasible. Compared with the single-layer deployable mechanical network, the doublelayer deployable mechanical network (DLDMN) can significantly increase the stiffness and fundamental frequency of antenna with limited weight increase. Deployable mechanisms must have simple and reliable drive and control to meet the demands for space mission and thus they usually have only one degree of freedom (DOF). It is difficult to construct the mechanical network constructed with SOSLs as a DLDMN while keeping its single DOF in the meantime, because of the spatial SOSL motion and complex motion path. The methods to construct DLDMN have been widely explored. Pellegrino [16] presented a method to establish a doublelayer parabolic cylinder antenna by composites and proposed a ‘hollow solid’ structural concept that comprised curved surfaces formed from thin sheets. Guan [17] designed a bidirectional deployable parabolic cylinder antenna using the same method as Pellegrino’s. Huang [10] designed a double-layer planar deployable mechanism using Myard linkages, in which the Myard linkages in the upper and lower layers used the common platform to coordinate movements. Ding [18,19] presented a synthesis method of two-layer and two-loop spatial mechanisms with coupling chains and designed a family of such deployable mechanisms. The design of interlayer pillar is the key to construct a single-DOF DLDMN by SOSL. The interlayer pillar protrudes from the single-layer mechanical network, coordinates the motion of the two layers and keeps the single DOF of the entire mechanism. A hybrid linkage constructed with SOSLs and parallel mechanisms can be used in designing the interlayer pillar. Zheng [20] proposed a hybrid linkage incorporating Bennett linkages, Bricard linkages and parallel mechanisms, which can transform the spatial motion of SOSL to normative rectilinear motion. Ma [21] presented a Bennett-spherical 6R metamorphic linkage, whose branch is constructed with Bennett linkage and spherical linkage. Kong and Jin [22] presented a type of multi-mode metamorphic parallel mechanism, which is construct with Bricard linkage and sub-chains, the sub-chains are constituting of parallel revolute joints, the Bricard linkage and sub-chain constitute hybrid joints. In the construction process of the DLDMN, the mobility and kinematic characteristics of the hybrid linkage should be analyzed. Feng [23] analyzed the kinematics of the plane-symmetric Bricard Linkage. López-Custodio [24,25] analyzed the kinematics of the plane-symmetric and line-symmetric Bricard linkages respectively. Dai [26,27] analyzed the mobility of a complex structured ball and metamorphic mechanisms of foldable/erectable kinds based on screw theory. In the current research, a single-layer mechanical network constructed with the alternative form of Bennett linkages is used as the basic layer; the interlayer pillar is designed using the hybrid linkage composed of Bennett linkage and planar symmetric four-bar linkage; and a cable-truss hybrid DLDMN is constructed. Then, the mechanical networks with multiple shapes are obtained through fitting using the DLDMN. Lastly, the cable-truss hybrid DLDMN is employed to construct a paraboloid deployable antenna. A prototype is fabricated to verify the design. The hybrid linkage is used as a basic unit to design the deployable mechanism, thereby formulating a novel method to design a single-DOF cable-truss hybrid DLDMN. The cable-truss hybrid DLDMN has the advantages of single DOF, high-stiffness and light weight, and it can fit lots of surfaces. This research was conducted on the basis of the work in [13]. The symbol definition in this paper was cited from [13]. The readers are invited to read this paper in combination with [13]. The rest of the paper is organised as follows. In Section 2, a hybrid linkage interlayer pillar is constructed with equilateral Bennett linkage and planar symmetric four-bar linkage. Section 3 presents the kinematics analysis of the interlayer pillar. In Section 4, the single-DOF cable-truss hybrid DLDMN is constructed with Bennett linkages and hybrid linkage interlayer pillars. In Section 5, the link areal density of the DLDMN is analysed to evaluate its performance. In Section 6, a paraboloid deployable antenna is designed using the cable-truss hybrid DLDMN, and a prototype is fabricated. Finally, conclusions are elaborated in Section 7. 2. Hybrid linkage constructed with equilateral Bennett linkage and planar symmetric four-bar linkage 2.1. Equilateral Bennett linkage The configuration of Bennett linkage, a well-known four-bar spatial over-constrained linkage, is shown in Fig. 1. More details on Bennett linkage can be found in [28]. Bennett linkage follows Eq. (1).
⎧ a = a3 = a, a2 = a4 = b ⎪ ⎨ 1 α1 = α3 = α , α2 = α4 = β d = 0, ( i = 1, 2, 3, 4 ) ⎪ ⎩ sini α sin β a
=
b
where ai is the link length, α i is the twist angle of the link, di is the joint offset of the linkage, and i = 1, 2, 3, 4.
(1)
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Fig. 1. Configuration of Bennett linkage.
P
D Q Fig. 2. Equilateral Bennett linkage.
Fig. 3. Alternative form of Bennett linkage.
The revolute variables of Bennett linkage follow Eq. (2).
θ1 + θ3 = 2 π , θ2 + θ4 = 2 π tan θ21 tan θ22 =
sin β +2 α
(2)
sin β −2 α
where θ i represents the revolute variable. Eq. (2) indicates that the DOF of Bennett linkage is 1. The object of this research is the equilateral Bennett linkage, its configuration is shown in Fig. 2. This linkage is used as the basic unit to construct a single-layer deployable mechanical network. Equilateral Bennett linkage follows the following conditions:
a=b=l α+β =π
(3)
For the equilateral Bennett linkage in this research, we define θ1 = θ and θ2 = ϕ and then obtain some of its properties from [4,5]. These properties are the basis for the subsequent analysis in Section 3, and they can be found in the Appendix as Eq. (A-1).
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Fig. 4. Compact configuration of the alternative form of Bennett linkage.
Equilateral Bennett linkage is a plane symmetric linkage with intersecting axes of opposite joints, and this feature stays the same during movement. For the equilateral Bennett linkage ABCD in Fig. 2, axes A and C intersect at point P, axes B and D intersect at point Q and plane PAC is perpendicular to plane QBD. 2.2. Alternative form of equilateral Bennett linkage Chen and You transformed equilateral Bennett linkage to improve its deployment performance and obtained the alternative form of Bennett linkage [4,5], as shown in Fig. 3. In the alternative form, the links are not perpendicular to the revolute joints, and they have the following offset lengths relative to the revolute joints: AE = CG = c and BF = DH = d, as shown in Fig. 3. In Fig. 3, the links and the dashed lines represent the alternative and original forms of Bennett linkage, respectively. For the alternative form, it is defined that θ f and ϕ f represent the revolute variables θ and ϕ , respectively, in the folded form, and θ d and ϕ d represent the revolute variables θ and ϕ , respectively, in the deployed form. Some equations of the geometric parameters of the alternative form are presented in the Appendix as Eq. (A-2). These equations will be used in Section 3. Apart from the mathematical model of the alternative form, Chen and You also proposed its actual model using the links with square section. The deployed form of the actual model is a rhombus, while the folded form is a bundle of four rods which are lumped together, formulating good deployment performance for the linkage. Huang proposed a compact configuration of the alternative form of Bennett linkage [10]. The deployed and folded forms of this model are shown in Fig. 4.
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Fig. 5. Movement of the alternative form of equilateral Bennett linkage.
Fig. 6. Hybrid linkage constructed with equilateral Bennett linkage and planar symmetric four-bar linkage.
In Fig. 4, the blue dashed lines represent the original form of Bennett linkage. AP BP CP DP represents the Bennett linkage P. For a convenient explanation, the units 1 – 4 and units A and P in Figs. 8, 10 and 11 in [13] are defined as the basic and transition units, respectively. Units A are the transition units between units with an included angle, while units P are the transition units between parallel units. The compact configuration is the same as the transition unit between the basic units of the mechanical network in [13]. The Bennett linkage in Fig. 4 is the Bennett linkage P in Figs. 10 and 12 in [13]. From [13], it can be known that the deployed and folded forms of the transition and basic units are the opposite. Their geometric parameters satisfy the following relation:
1 1
θ f = P θd θd = P θ f
(4)
where i θ j , (i = 1, 2, 3, 4, A, P; j=d, f ) represents θ d and θ f of unit i. The other geometric parameters of transition unit P can be found in the Appendix as Eq. (A-3), which will be used in Section 3. More details on the alternative form of Bennett linkage can be found in [4,5,13]. 2.3. Hybrid linkage constructed with equilateral Bennett linkage and planar symmetric four-bar linkage Fig. 5 shows the movement of the alternative form of equilateral Bennett linkage. The blue dashed lines and the links represent the original and alternative forms of Bennett linkage, respectively. The equilateral Bennett linkage in this research is symmetric about planes APC and BQD, with PQ⊥AC and PQ⊥BD. It is defined that a line m that passes dot P and is perpendicular to plane APC and that a line n that passes dot Q and is perpendicular to plane BQD. Fig. 5 shows that axes A and C have a rotation motion about line m, while axes B and D have a rotation motion about line n. If the links of the Bennett linkage are ignored, then axes A and C and line m can be regarded as a virtual revolute joint, with line m for virtual axis and axes A and C for virtual links. Axes B and D and line n can be regarded as a virtual revolute joint in the same way. Therefore, combining equilateral Bennett linkage with planar symmetric four-bar linkage can construct a hybrid linkage, as shown in Fig. 6. In Fig. 6, the links GH, GF and GK form a hybrid joint, while the links EF, EH and EJ form a hybrid joint too. In the movement of the hybrid linkage, links KS, JS, GK and EJ and Bennett linkage EFGH form a virtual planar symmetric four-bar linkage KSJP, whereas the movement of links KS, JS, GK and EJ does not interfere with that of Bennett linkage EFGH. Thus, the DOF of the hybrid linkage is 1. In Fig. 6, the links JS and KS on plane APC and virtual joint APC form the virtual planar symmetric four-bar linkage KSJP. The detailed configuration of this linkage is shown in Fig. 7.
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Fig. 7. Configuration of the virtual planar symmetric four-bar linkage in the hybrid linkage.
Fig. 8. Hybrid linkage for the transition unit of Bennett linkage mechanical network.
The hybrid linkage in Fig. 7 satisfies the following conditions:
$S $K $J JS = KS KP = JP
$m (5)
where $i denotes the screw of axis i. For a clear explanation, we set KG = JE, KG⊥GP and JE⊥EP. We also set KG = JE and ∠KGP = ∠JEP on the condition that KP = JP and KP and JP are symmetric about plane BQD. In [13], we have researched the networking of Bennett linkage and found that the revolute variables of units 1 and P are equal, i.e. 1 θ1 = P θ1 . Thus, the value of ࢬAPC in Fig. 7 can be obtained from Eq. (A-1). In the right triangle PKG, CG = c, KG is a constant value and PC can be calculated from Eq. (A-1). From the triangle relation, we have:
√ PK = KG2 + PG2 ∠KPG = arctan KG PG
(6)
In Fig. 7, the following condition is satisfied:
1
∠APC + ∠KPG + ∠KPS =π KS = ∠KP = ∠PS ∠KPS KSP PKS ∠KPS + ∠KSP + ∠PKS = π
2
(7)
The distance PS from point S of the planar symmetric four-bar linkage to virtual hinge point P of the Bennett linkage can be obtained from Eq. (7). 2.4. Compact hybrid linkage constructed of the compact alternative form of Bennett linkage and planar symmetric four-bar linkage For the hybrid linkage in Fig. 6, when Bennett linkage EFGH moves, axis $S also moves on line PQ about virtual axis $m . According to [13], when the alternative form of Bennett linkage is used to construct the mechanical network, the transition units are equilateral Bennett linkages. In the current research, the hybrid linkage in Fig. 6 is employed to the transition Bennett units in the mechanical network. Then, a hybrid linkage extruding from the mechanical network is obtained. Fig. 8 shows the result when Bennett linkage P in Fig. 10 in [13] is used as the basic Bennett linkage to construct the hybrid linkage. For the hybrid linkage in Fig. 8, the Bennett linkage is the compact configuration of its alternative form. Accordingly, this linkage is named as a compact hybrid linkage.
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Fig. 9. Interlayer vertical pillar in the DLDMN.
Fig. 10. Schematic of the deployed and folded forms of the interlayer pillar.
2.5. Compact hybrid linkage added with a synchronous mechanism The interlayer pillar of the DLDMN should have a certain height and be perpendicular to the network surface. Hence, a synchronous mechanism is added to the compact hybrid linkage to construct a compact hybrid linkage interlayer vertical pillar with an adjustable height. This pillar is shown in Fig. 9. For the interlayer vertical pillar in Fig. 9, synchronous mechanism RSTM is a eudipleural linkage with two slide crank mechanisms. For the linkage in Fig. 9, RS = TS and RM = TM, and the extension rod is integrated with the guide rod. Besides, slide block M and revolute joint M are also combined. This design guarantees that the extension rod is always perpendicular to the network surface and has a certain height. The value of the length CG is constant, so that the value of PS can be obtained from Section 2.3. The length of the extension rod lv is constant. Thus, the height lH from the vertex of the vertical pillar to peak P of the Bennett linkage can be calculated as follows:
lH = PS + lv
(8)
When the alternative form of Bennett linkage is in deployed and folded forms, the interlayer vertical pillar is also in deployed and folded forms, as presented by Fig. 10(a) and (b). 3. Kinematic analysis of the hybrid linkage interlayer vertical pillar Fig. 11 shows the result when the hybrid linkage interlayer vertical pillar is employed to the single-layer mechanical network constructed with Bennett linkages. With the definition in [13], two basic Bennett linkages are units 1 and 3, the transition Bennett linkage is unit P and the hybrid linkage interlayer vertical pillar is only located at unit P. The revolute variables of units 1, 3 and P are equal and synchronous. When units 1 and 3 are deployed, unit P is folded; when units 1 and 3 are folded, unit P is deployed. Fig. 12 presents the calculation process of the height lH , which is from the vertex of the vertical pillar to the peak P of the Bennett linkage. Firstly, the geometric parameters of transition unit P are derived from those of basic unit 1 in Eq. (A-3), according to the relation between basic unit 1 and transition unit P. The revolute variables of transition unit P are then
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Fig. 11. Mechanical network constructed with Bennett linkages with interlayer vertical pillars.
Fig. 12. Calculation process of the height lH .
derived from those of basic unit 1. Accordingly, the angle ࢬAPC and lengths PA and PC are calculated using Eq. (A-1) in the Appendix. Secondly, the distance PS is calculated based on the selected values of CG , KG . Finally, the height lH is obtained with the addition of the length lv . Therefore, it can be concluded that the geometric parameters of the hybrid linkage are determined by those of the basic units, and that this linkage has limited variation range and effect on the PS. The proportion of PS in lH is also limited. The length lv is a free parameter whose value does not affect the mobility of the interlayer vertical pillar. Accordingly, the expected lH can be obtained by adjusting lv and then the deployable antenna curve is fitted using this property.
4. Cable-truss hybrid double-layer deployable mechanical network based on hybrid linkage interlayer vertical pillar It is known that the alternative form of Bennett linkage can be used to construct a single-layer deployable mechanical network. On this basis, the transition Bennett linkage is used to construct the hybrid linkage interlayer vertical pillar, which can support and connect components between layers. Then, the double-layer deployable mechanical network is constructed.
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Fig. 13. Mechanical network constructed of the alternative form of Bennett linkages.
4.1. Networking of the Bennett linkages The single-layer mechanical network constructed with the alternative form of Bennett linkages includes planar and cylindrical networks. The curvature of the cylindrical surface can be adjusted flexibly, which covers some parts, such as the cylindrical and parabolic cylindrical surfaces. The planar and cylindrical mechanical networks are shown in Fig. 13 where the basic Bennett linkages are connected by transition Bennett linkages [13]. From [13], it can be known that transition Bennett linkages are divided into two types: units P and A. Unit P is used to construct the hybrid linkage interlayer vertical pillar. 4.2. Mechanical network added with hybrid linkage interlayer vertical pillars The analysis in Section 2.3 indicates that the interlayer vertical pillars do not affect the movement of the Bennett linkage. The hybrid linkage on unit P is therefore constructed in the mechanical network. The planar mechanical network in Fig. 13(a) is used as the basic network to construct the interlayer vertical pillars. The resulting mechanical network is shown in Fig. 14. In Fig. 14, numbers 1, 2, …, 9 and Z1, Z2, …, Z9 represent the unit and interlayer vertical pillar numbers, respectively. From the folded form to deployed form, units 1, 2, …, 9 and the interlayer vertical pillars are always parallel. If the basic network is a cylindrical network, then the interlayer vertical pillars also keep the cylinder in the deployed form. Fig. 15 shows the resulting mechanical network with the cylindrical network as the basic network. The comparison between Figs. 14 and 15 indicates that the folded forms of the two mechanical networks are nearly the same, but their deployed forms are different. 4.3. Construction of the double-layer deployable mechanical network The interlayer vertical pillar always coincides with the centre line PQ of the Bennett linkage of the hybrid linkage and is perpendicular to diagonal lines AC and BD. Therefore, these pillars can be used as nodes and be connected with cables to construct a cable structure. The cable structure and single-layer mechanical network form a cable-truss hybrid double-layer deployable mechanical network, as shown in Fig. 16.
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Fig. 14. Planar mechanical network added with interlayer vertical pillars.
Fig. 15. Cylindrical mechanical network added with interlayer vertical pillars.
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Fig. 16. Cable-truss hybrid double-layer deployable mechanical network.
Fig. 17. Cable-truss hybrid double-layer mechanical network to fit paraboloid surface (cable layer fitting, planar truss layer).
The planar mechanical network in Fig. 16 can be used as a planar deployable mechanism. The red dashed lines represent the cables. All the interlayer vertical pillars have the same height. The interconnected cables and the links of the Bennett linkages form the cable and truss layers, respectively, which further form the cable-truss hybrid double-layer deployable mechanical network. The cable is flexible and can bend arbitrarily. Specifically, the folded form of the double-layer deployable mechanical network in Fig. 16 is similar to the form in Fig. 14(c). The process of constructing the hybrid linkage reveals that the movements of hybrid linkages belonging to different units are independent of each other. Thus, for interlayer vertical pillars Z1, Z2, …, Z9, the heights of the extension rods do not affect the movement of the mechanical network. Accordingly, the heights of the extension rods lv in Fig. 16 can be adjusted to fit the cable layer on curved surfaces. As a result, the DLDMNs with different curved surfaces can be constructed. Using this method, a deployable mechanical network with paraboloid surface is constructed, as shown in Fig. 17. In this network, the cable layer is used to fit the surface. The centre of the truss layer in Fig. 17, which is a planar mechanical network, coincides with the rotation centre axis of the paraboloid surface. The heights of the pillars are determined by the pointcuts of the paraboloid surface and the pillars. The heights of the pillars in Figs. 16 and 17 can be adjusted to fit other types of surface using the fitting method similar to that in Fig. 17. From the construction process of compact hybrid linkage, it can be known that this linkage is only related to one direction of the transition units and has no relation with the transition units in the other direction. Thus, the truss layer can use the mechanical network with a cylindrical surface. When the cylindrical surface truss layer is adopted with curvature as its free parameter, the truss layer can be used to fit the deployable mechanism with a cylindrical surface. In this case, the cable structure is used to improve the stiffness of the DLDMNs. Alternately, the cable layer can be used to fit the surface of deployable mechanisms. In the first case, the surface types are limited to the types of the truss layer constructed with Bennett linkages, such as cylindrical and parabolic cylindrical surfaces and the planar surface in Fig. 17. In the second case, the surface types are not limited because of the arbitrary adjustment of the cable layer. Figs. 18 and 19 show the DLDMNs which use the truss layer to fit the surface. The truss layer in Fig. 18 fits the cylindrical surface, whereas the truss layer in Fig. 19 fits the parabolic cylindrical surface. Fig. 20 shows the resulting DLDMN when the cylindrical surface is used as the truss layer and the cable layer is used to fit the surface. The surface fitted in this network is a paraboloid surface. Compared with the planar truss layer, the cylindrical truss surface is able to reduce the heights of the interlayer vertical pillars in some situations, which can increase the deployment ratio of the deployable mechanisms. All the DLDMNs shown in Figs. 16–20 are presented in deployed forms. The folded forms of these DLDMNs are similar to those in Fig. 14(c).
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Fig. 18. Cable-truss hybrid DLDMN to fit cylindrical surface (truss layer fitting).
5. Link areal density analysis of the cable-truss hybrid DLDMN The link plays an important role in the deployable framework antenna because it directly affects the folding volume, weight, deploying method and accuracy. The cable-truss hybrid DLDMN proposed in this research can be deployed to a large cover area with few links, which reduces the weight and folding volume of the antenna. Fig. 21 shows the basic units of the common deployable framework antenna at present: units of the cable-truss hybrid DLDMN in this research, units of the tetrahedron deployable antenna and units of the hexagonal prism deployable antenna. Comparison is conducted in the total link length and cover area in the deployed form of the planar mechanical network constructed with the three types of units. The results reveal the excellent performance of the cable-truss hybrid DLDMN. Because the cable takes up very few volume and weight compared with links, the cable layer in the cable-truss hybrid DLDMN can be ignored. For a convenient comparison, it is defined that the link lengths and the interlayer heights in planar part of the unit are equal and that the mechanical networks use basic units to fit planar surface. In the unit of cable-truss hybrid DLDMN, we define ω = 45◦ , that is, AC⊥BD. The link lengths are given by:
OA = OB = OC = OD = L1 OM = L2
(9)
According to Eq. (9), the cover area of the unit of cable-truss hybrid DLDMN is 2L21 . For the unit of the tetrahedron deployable antenna, it is defined that triangle ABC is an equilateral triangle. The link lengths are given by:
AB = BC = CA = L1 OM = L2
(10)
The units of the tetrahedron deployable antenna are distributed symmetrically in the two layers. As a result, the cover √ area of the unit is calculated to be 23 L21 . For the unit of the hexagonal prism deployable antenna, we define that
MA = MB = MC = MD = ME = MF = L1 MN = L2
(11) √
Therefore, the cover area of the unit for the hexagonal prism deployable antenna is 3 2 3 L21 . Fig. 22 presents the planar mechanical networks constructed with the three types of units. For a convenient comparison, it is assumed that the three types of mechanical networks have nearly same cover area. The cover area of the unit indicates
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Fig. 19. Cable-truss hybrid DLDMN to fit parabolic surface (truss layer fitting). Table 1 Properties of units of deployable framework antenna. Type
Total link length
Total link number
Cover area
Cable-truss hybrid DLDMN Tetrahedron deployable antenna Hexagonal prism deployable antenna
80L1 + 20L2 252L1 + 45 3L21 + 9L22 180L1 + 61L2 + 90 L21 + L22
100 387 331
40L21 38.97L21 38.97L21
that the cable-truss DLDMN, hexagonal prism deployable antenna and tetrahedron deployable antenna have 20, 15 and 45 units, respectively. In Fig. 22, the red dashed line and the blue heavy line represent the cable and the basic unit in the mechanical network, respectively. Table 1 displays the total link length, total link number and cover area for the three types of planar mechanical network. As shown by Table 1, when the cover area for the three types of DLDMN are nearly equal, the total link number and total link length of the cable-truss DLDMN are less than those of the two other deployable antennas. This feature demonstrates that the cable-truss hybrid DLDMN is an excellent solution for the deployable antenna. For the convenience of comparison, the link length in unit cover area is defined, that is, total link length/total cover area, as the link areal density. Low link areal density indicates large link sparsity, high utilisation efficiency, low antenna weight and small folding volume. The link lengths in these types of antenna are L1 = 1 m and L2 = 0.3 m considering of the size of the spaceborne deployable antenna at present. After the introduction of the specific parameters into the link, Table 2 is formulated. From Table 2, it can be observed that the link areal density of the cable-truss hybrid DLDMN is only 28.7% of that of the tetrahedron deployable antenna and is only 24.7% of that of the hexagonal prism deployable antenna. The link is sparse for the cable-truss hybrid DLDMN because parts of the links are replaced by cables, and the link utilisation efficiency is increased using the hinge combination method with special design.
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Fig. 20. Cable-truss hybrid DLDMN to fit paraboloid surface (cable layer fitting, cylindrical truss layer).
Fig. 21. Units of common deployable framework antennas.
Fig. 22. Three types of deployable planar framework antenna.
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Fig. 23. Process to fit the surface of deployable antenna by the cable-truss hybrid DLDMN.
Table 2 Properties of units of deployable framework antenna (specific parameters). Type
Total link length (m)
Cover area (m2 )
Link areal density
Cable-truss hybrid DLDMN Tetrahedron deployable antenna Hexagonal prism deployable antenna
86 292.26 339.84
40 38.97 38.97
2.15 7.5 8.72
6. Deployable paraboloid antenna implemented with cable-truss hybrid DLDMN 6.1. Process of fitting surface for the DLDMN Before designing a deployable mechanism, it is required to determine the surface size and accuracy of the deployable mechanism based on its requirements. Then, the stiffness, fundamental frequency, deployed and folded sizes of the deployable mechanism should be determined according to the connected satellite main part and the working environment. When these parameters are determined, the deployable mechanism can be designed.
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Fig. 24. Schematic of the offset feed paraboloid antenna.
Fig. 25. Cable-truss hybrid DLDMN to fit offset feed paraboloid antenna.
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Fig. 26. Interlayer pillars in the mechanical network.
Fig. 27. Schematic of the cable-truss hybrid DLDMN.
In this research, the cable-truss hybrid DLDMN is employed to implement the deployable antenna and the cable layer to fit the surface. The accuracy of the cable layer is determined by the nodes constructed with the interlayer vertical pillars. Thus, it is required to firstly determine the node number based on the surface size and accuracy of the antenna. When the node number is determined, the number of the basic unit of the DLDMN can be determined. Then, the side length, angle of the basic unit and the connection relation between the basic units can be determined by the deployed surface. On these bases and the folded volume, the section size of the link in the unit can finally be determined. Deployable mechanisms are commonly symmetric because symmetric structure can decrease the design difficulty and increase the uniformity of the unit parameters. For the symmetric deployable mechanism, the truss layer with a symmetric structure is chosen and the middle pillars are used to fit the centre symmetric curve. The height of the middle pillars is firstly designated and then the height of the other pillars is determined depending on the intersections of the deployed surface and the pillars of the mechanical network. For the target surface, surface equation fsurface (x, y, z ) = 0 is used. When the relationship between the position of DLDMN and that of the target surface is determined, the line equation f pillar (x, y, z ) = 0 of the pillars can be obtained by the geometric relation. The intersections of the two equations are the intersection point coordinates of the pillars and the target surface.
fsurface (x, y, z ) = 0 fpillar (x, y, z ) = 0
(12)
On the basis of the coordinates of intersection point and point S in the pillar, the length lv of the extension rod in the pillar can be obtained. Now, the geometric parameters of the DLDMN are determined. Then the folded volume and weight are calculated and compared with the working requirements. If the requirements are met, then the pillars are connected with cables, the stiffness and fundamental frequency are calculated and compared with the requirements. If the requirements are not met, then the geometric parameters are modified, followed by the restart of the design and calculation. The cycle is repeated until all the requirements are met. The detailed process is shown in Fig. 23. 6.2. Design of the double-layer deployable paraboloid antenna A paraboloid antenna is designed following the process in Section 6.1. 6.2.1. Design of the truss layer and the interlayer vertical pillars Paraboloid antenna has two types: positive and offset feed antennas. For positive feed antenna, the feed support equipment blocks the electromagnetic wave. Thus, offset feed antenna is chosen as our target, which is shown in Fig. 24. The reflector of the antenna in Fig. 24 is obtained when parabolic segment AB rotates around the X axis. The cable-truss hybrid DLDMN is used to fit this reflector. To achieve a simple design and a high deployment ratio, the truss layer is selected as a cylindrical surface, and cable layer is used to fit the paraboloid surface. The detailed fitting method is shown in Fig. 25. The curve AB in Fig. 25(c) is the generating line of the target paraboloid surface, and it is located at the centre of the paraboloid surface. Straight line A B is tangent with curve AB at its midpoint; straight line CD, the symmetrical centre line
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Fig. 28. 3D model of the interlayer pillars with carbon fibre bar.
of the truss layer, is parallel to A B . Given that the equation of the parabola is z2 = 11040x, the focal length of the parabola is f = 2760mm, the coordinate of point A is ( −324.3 0 586.4 ) and that of point B is ( 210.8 0 2555.2 ). Considering of the deployment ratio and the convenience of the design and manufacture, the geometric parameters of the Bennett linkage in the truss layer are set as L = 450 mm, ω = 55◦ , f = 51 mm and λ = 50◦ and the included angle as ∠1 = 10◦ . As the mechanical network has 20 units, it also has 20 interlayer vertical pillars. These pillars are shown in Fig. 26. In Fig. 26, column Y = 0 represents the column with less number, and the heights of the pillars are symmetric about column Y = 0. According to the parameters of the paraboloid reflector and the mechanical network, the length lv of the extension rods in Fig. 25 can be obtained. The calculation results are displayed in Table 3. The geometric parameters are bilateral symmetric about column Y = 0. Thus, only the parameters of one side are given. On the basis of the aforementioned parameters, a DLDMN is designed. The deployed and folded forms of this network are shown in Fig. 27. The link in the DLDMN in this research supports and connects the revolute joints and resists external forces. The carbon fibre bar has the advantages of low weight and good mechanical properties, leading to its extensive application to aerospace equipment. To decrease the weight of the mechanical network, two connectors are connected through the carbon fibre bar. For the hybrid linkage in the mechanical network, which is similar to that in Fig. 9, the planar four-bar linkage interferes with the Bennett linkage of the truss layer. The hybrid linkage is designed to avoid interference, as shown in Fig. 28.
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Table 3 Length of extension rods of interlayer pillars (mm). Y Z
0
1
2
0 1 2 3
308.5 229.3 229 300.8
287.7 207.8 206.8 278
223.9 141.9 138.8 208.2
Fig. 29. Basic unit of the cable-truss hybrid DLDMN.
Fig. 30. DLDMN with added cables.
The extension rod is also substituted with the carbon fibre bar to decrease the weight. Fig. 29 presents the resulting interlayer vertical pillar and the unit of the truss layer. 6.2.2. Design of the cable layer Apart from the ability to fit the special surface, the mechanical network should have sufficient stiffness and strength to resist external forces and disturbances. For the DLDMN in Fig. 27, the cable layer fits the antenna reflector with a paraboloid surface, and it also forms a double-layer rigid structure with the truss layer and the interlayer vertical pillars. To increase the stiffness of the rigid structure, several methods can be adopted, such as increasing the stiffness of the truss layer and interlayer vertical pillars, increasing the tension force in the cable of the cable layer and optimising the configuration of the mechanical network. The first method increases the stiffness of the links, which will inevitably increase the weight of the mechanical network. The second method has oversized tension force that requires high stiffness of the truss layer and interlayer pillar, which will also increase the weight. The cable is lighter than the link, and it can be seen as the link under tensile force in the deployed form. Thus, it is feasible to increase the stiffness without increasing the weight by increasing the number of the cables and changing the configuration of the mechanical network. The truss and cable layers in Fig. 27 are connected through the interlayer vertical pillars, and the structure is not a closed envelope. The structure with closed envelope has high stiffness. Accordingly, side and slant cables are added to the structure in Fig. 27. The resulting structure is shown in Fig. 30. In the figure, the blue long-dotted dashed line represents
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Joint of the wirerope
a) Folded form
b) Intermediate form
c) Deployed form
Fig. 31. Deployed and folded forms of the basic unit for cable-truss hybrid DLDMN.
Fig. 32. Deployed and folded forms of the cable-truss hybrid DLDMN.
the side cable, the blue double-dotted dashed line represents the slant cable and the red dashed line represents the original cable. The above indicates that the DLDMN forms a close envelope. The connections between the layers also increase, which increases the stiffness of the mechanical network. 6.2.3. Prototype A prototype of the DLDMN is designed and manufactured to verify the scheme for its application to large-size spaceborne antenna. The deployed and folded forms of the basic unit and the mechanical network are shown in Figs. 31 and 32, respec-
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LMS test system computer sensors Force hammer Multi-channel vibration test and analysis system
Fig. 33. Free mode test system of the deployable antenna.
Table 4 The first three order natural frequencies obtained from the test for the deployable antenna. Configuration Frequency(Hz) Order
Truss layer
Truss layer + cable layer
1 2 3
10.055 17.336 27.747
16.715 37.789 57.816
tively. Thin wirerope is used as the cable, and the wirerope is layout at the top and bottom of the interlayer vertical pillars, as shown in Fig. 31(b). The wirerope passes through the holes of the component and is fixed by chucks. For the comparison of stiffness, the prototype of the single-layer configuration for the DLDMN is also presented, as shown in Fig. 32(C). The free mode of the single-layer configuration and double-layer configuration of the mechanical network are tested using LMS test system. The test system is shown in Fig. 33, and the results are shown in Table 4. The natural frequencies in free mode represent the stiffness of the mechanical network. As can be seen from Table 4, the first three order natural frequencies of the double-layer configuration is much higher than that of the single-layer configuration, meaning that the stiffness of the double-layer configuration is also much higher than that of the single-layer configuration.
7. Conclusion This research proposes a single-DOF cable-truss hybrid DLDMN. The kinematics of equilateral Bennett linkage is analysed, and virtual joints are found in its movement. Then, equilateral Bennett linkage and planar symmetric four-bar linkage are combined to construct a single-loop four-bar hybrid linkage. The original form of the Bennett linkage in the hybrid linkage is replaced by its alternative form and a synchronous mechanism is introduced to obtain a hybrid linkage of compact form, through which rectilinear motion can be achieved in the certain direction of the vertical pillar. The hybrid linkage interlayer vertical pillar is employed to the single-layer Bennett linkage mechanical network to obtain a cable-truss hybrid DLDMN. The height of the interlayer vertical pillar has no relation with the mobility of the mechanical network. Thus, the mechanical network is used to fit various surfaces, such as planar, cylindrical and paraboloid surfaces. The mechanical networks not only have single DOF but also have high stiffness. Using the cable-truss hybrid DLDMN, an offset feed paraboloid antenna is designed and a prototype is fabricated to verity the feasibility of the proposed cable-truss hybrid DLDMN. The proposed double-layer deployable antenna with high stiffness can fit all kinds of spaceborne antenna including those with planar and those with curved surfaces. This research provides a reference for the design of other spaceborne antennas.
Acknowledgment This work was financially supported by the Joint Funds of the National Natural Science Foundation of China (Grant Nos. U1637207 and U1613201), in part by the National Natural Science Foundation of China (Grant Nos. 51575119, 51675114, 51675180 and 51565011), in part by the Natural Science Foundation for Distinguished Young Scholars of Jiangxi Province (Grant No. 20171BCB23059) and the Key Research Program of Jiangxi Province (Grant No. 20171BBE50039).
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Appendix Properties of the equilateral Bennett linkage:
⎧ θ tan 2 tan ϕ2 = cos1 α ⎪ ⎪ ⎪ ⎪ ⎪ cos ϕ ⎪ cos ∠APC = 2 1+ −1 ⎪ 1−cos θ ⎪ ⎪ ⎨ 1+cos θ
cos ∠BQD = 2 1−cos ϕ − 1
⎪ ⎪ (1+cos ϕ )(1−cos θ ) ⎪ PA = PC = l ⎪ ⎪ −2(cos ϕ +cos θ ) ⎪ ⎪ ⎪ ⎪ (1+cos θ )(1−cos ϕ ) ⎩ QB = QD = l
(A-1)
−2(cos ϕ +cos θ )
Geometry parameters for the alternative form of Bennett linkage:
⎧ (1+cos ϕ f )(1−cos θ f ) ⎪ c = l ⎪ −2(cos ϕ f +cos θ f ) ⎪ ⎨ (1−cos ϕ f )(1+cos θ f ) d=l −2(cos ϕ f +cos θ f ) ⎪ ⎪ ⎪ ⎩ θ θ 2 −tan α tan θ2d tan 2f = sec2 θ2d sec2 2f + tan2 α
(A-2)
Geometry parameters of the transition unit P:
⎧P 1 ⎪ ⎨ α = 2π − α 1 P θ1 = θ1 ⎪ ⎩l = L f sin1 α −kFU l1 f P
(A-3)
d1 +kP l1
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Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Cable-truss hybrid double-layer deployable mechanical network constructed of Bennett linkages and planar symmetric four-bar linkages Xiaoke Song a, Hongwei Guo b,∗, Sanjun Liu c, Fei Meng a, Qiping Chen d, Rongqiang Liu b, Zongquan Deng b a
School of Mechatronics and Vehicle Engineering, East China Jiaotong University, PR China State Key Laboratory of Robotics and System, Harbin Institute of Technology, PR China c Shanghai Aerospace Equipment Manufacturer Co. Ltd., PR China d Key Laboratory of Conveyance and Equipment Ministry of Education, East China Jiaotong University, PR China b
a r t i c l e
i n f o
Article history: Received 9 August 2018 Revised 26 November 2018 Accepted 4 December 2018
Keywords: Deployable mechanism Double-layer deployable mechanical network Hybrid linkage Bennett linkage
a b s t r a c t This research proposes a single degree of freedom (DOF) cable-truss hybrid double-layer deployable mechanical network (DLDMN) consisting of equilateral Bennett linkages and planar symmetric four-bar linkages. The kinematics of the equilateral Bennett linkage is investigated. Then a single-loop four-bar hybrid linkage is constructed using the equilateral Bennett linkage and planar symmetric four-bar linkage. The original form of the Bennett linkage in the hybrid linkage is replaced by its alternative form and a synchronous mechanism is introduced to obtain a hybrid linkage interlayer vertical pillar with an adjustable height. The interlayer vertical pillar is employed to a single-layer mechanical network consisting of Bennett linkages. Connecting these pillars with cables, a cable-truss hybrid DLDMN that could fit various surfaces is assembled. An offset paraboloid antenna is set up using the DLDMN. A prototype is manufactured to verify the feasibility of the design. The proposed single DOF DLDMN has the advantages of high deployment ratio of spatial single-loop linkages and normative motion of planar linkages. The proposed network can be used to construct a large-scale spaceborne deployable antenna with high stiffness. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Deployable mechanisms are the mechanisms that can be folded to a small size and be deployed into a large predetermined shape. These mechanisms are extensively employed to various scenarios, such as spaceborne deployable antennas, solar panels and civil architectures. Deployment ratio is a key index for evaluating the performance of deployable mechanisms, especially for spaceborne deployable antenna, because the limited volume of rocket payload bay raises a rigorous requirement for deployment ratio. Because the motion of single-loop overconstrained spatial linkage (SOSL) is spatial, it can be deployed to a planar polygon and be folded to a bundle of rods with the length of a single link, leading to its higher deployment ratio than a planar linkage [1]. Thus, SOSL is a good candidate for deployable mechanisms. To date, many scholars,
∗
Corresponding author. E-mail address: [email protected] (H. Guo).
https://doi.org/10.1016/j.mechmachtheory.2018.12.003 0094-114X/© 2018 Elsevier Ltd. All rights reserved.
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such as Chen and You [2–7], Yu [8], Lu [9], Huang [10–12], Song [13] and Qi [14,15], have investigated deployable mechanisms constructed with SOSLs and have obtained substantial achievements. A large-scale spaceborne deployable antenna should have good performance in folding and deploying, also with high stiffness, light weight and simple and reliable drive and control. Most of the current deployable mechanisms constructed with SOSLs are of single layer with low integral stiffness in large scale. If the link stiffness is enhanced to obtain high integral stiffness, the weight will be increased, as the link weight would increase in the meantime. Faced with the increased link weight, these mechanisms are not feasible. Compared with the single-layer deployable mechanical network, the doublelayer deployable mechanical network (DLDMN) can significantly increase the stiffness and fundamental frequency of antenna with limited weight increase. Deployable mechanisms must have simple and reliable drive and control to meet the demands for space mission and thus they usually have only one degree of freedom (DOF). It is difficult to construct the mechanical network constructed with SOSLs as a DLDMN while keeping its single DOF in the meantime, because of the spatial SOSL motion and complex motion path. The methods to construct DLDMN have been widely explored. Pellegrino [16] presented a method to establish a doublelayer parabolic cylinder antenna by composites and proposed a ‘hollow solid’ structural concept that comprised curved surfaces formed from thin sheets. Guan [17] designed a bidirectional deployable parabolic cylinder antenna using the same method as Pellegrino’s. Huang [10] designed a double-layer planar deployable mechanism using Myard linkages, in which the Myard linkages in the upper and lower layers used the common platform to coordinate movements. Ding [18,19] presented a synthesis method of two-layer and two-loop spatial mechanisms with coupling chains and designed a family of such deployable mechanisms. The design of interlayer pillar is the key to construct a single-DOF DLDMN by SOSL. The interlayer pillar protrudes from the single-layer mechanical network, coordinates the motion of the two layers and keeps the single DOF of the entire mechanism. A hybrid linkage constructed with SOSLs and parallel mechanisms can be used in designing the interlayer pillar. Zheng [20] proposed a hybrid linkage incorporating Bennett linkages, Bricard linkages and parallel mechanisms, which can transform the spatial motion of SOSL to normative rectilinear motion. Ma [21] presented a Bennett-spherical 6R metamorphic linkage, whose branch is constructed with Bennett linkage and spherical linkage. Kong and Jin [22] presented a type of multi-mode metamorphic parallel mechanism, which is construct with Bricard linkage and sub-chains, the sub-chains are constituting of parallel revolute joints, the Bricard linkage and sub-chain constitute hybrid joints. In the construction process of the DLDMN, the mobility and kinematic characteristics of the hybrid linkage should be analyzed. Feng [23] analyzed the kinematics of the plane-symmetric Bricard Linkage. López-Custodio [24,25] analyzed the kinematics of the plane-symmetric and line-symmetric Bricard linkages respectively. Dai [26,27] analyzed the mobility of a complex structured ball and metamorphic mechanisms of foldable/erectable kinds based on screw theory. In the current research, a single-layer mechanical network constructed with the alternative form of Bennett linkages is used as the basic layer; the interlayer pillar is designed using the hybrid linkage composed of Bennett linkage and planar symmetric four-bar linkage; and a cable-truss hybrid DLDMN is constructed. Then, the mechanical networks with multiple shapes are obtained through fitting using the DLDMN. Lastly, the cable-truss hybrid DLDMN is employed to construct a paraboloid deployable antenna. A prototype is fabricated to verify the design. The hybrid linkage is used as a basic unit to design the deployable mechanism, thereby formulating a novel method to design a single-DOF cable-truss hybrid DLDMN. The cable-truss hybrid DLDMN has the advantages of single DOF, high-stiffness and light weight, and it can fit lots of surfaces. This research was conducted on the basis of the work in [13]. The symbol definition in this paper was cited from [13]. The readers are invited to read this paper in combination with [13]. The rest of the paper is organised as follows. In Section 2, a hybrid linkage interlayer pillar is constructed with equilateral Bennett linkage and planar symmetric four-bar linkage. Section 3 presents the kinematics analysis of the interlayer pillar. In Section 4, the single-DOF cable-truss hybrid DLDMN is constructed with Bennett linkages and hybrid linkage interlayer pillars. In Section 5, the link areal density of the DLDMN is analysed to evaluate its performance. In Section 6, a paraboloid deployable antenna is designed using the cable-truss hybrid DLDMN, and a prototype is fabricated. Finally, conclusions are elaborated in Section 7. 2. Hybrid linkage constructed with equilateral Bennett linkage and planar symmetric four-bar linkage 2.1. Equilateral Bennett linkage The configuration of Bennett linkage, a well-known four-bar spatial over-constrained linkage, is shown in Fig. 1. More details on Bennett linkage can be found in [28]. Bennett linkage follows Eq. (1).
⎧ a = a3 = a, a2 = a4 = b ⎪ ⎨ 1 α1 = α3 = α , α2 = α4 = β d = 0, ( i = 1, 2, 3, 4 ) ⎪ ⎩ sini α sin β a
=
b
where ai is the link length, α i is the twist angle of the link, di is the joint offset of the linkage, and i = 1, 2, 3, 4.
(1)
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Fig. 1. Configuration of Bennett linkage.
P
D Q Fig. 2. Equilateral Bennett linkage.
Fig. 3. Alternative form of Bennett linkage.
The revolute variables of Bennett linkage follow Eq. (2).
θ1 + θ3 = 2 π , θ2 + θ4 = 2 π tan θ21 tan θ22 =
sin β +2 α
(2)
sin β −2 α
where θ i represents the revolute variable. Eq. (2) indicates that the DOF of Bennett linkage is 1. The object of this research is the equilateral Bennett linkage, its configuration is shown in Fig. 2. This linkage is used as the basic unit to construct a single-layer deployable mechanical network. Equilateral Bennett linkage follows the following conditions:
a=b=l α+β =π
(3)
For the equilateral Bennett linkage in this research, we define θ1 = θ and θ2 = ϕ and then obtain some of its properties from [4,5]. These properties are the basis for the subsequent analysis in Section 3, and they can be found in the Appendix as Eq. (A-1).
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Fig. 4. Compact configuration of the alternative form of Bennett linkage.
Equilateral Bennett linkage is a plane symmetric linkage with intersecting axes of opposite joints, and this feature stays the same during movement. For the equilateral Bennett linkage ABCD in Fig. 2, axes A and C intersect at point P, axes B and D intersect at point Q and plane PAC is perpendicular to plane QBD. 2.2. Alternative form of equilateral Bennett linkage Chen and You transformed equilateral Bennett linkage to improve its deployment performance and obtained the alternative form of Bennett linkage [4,5], as shown in Fig. 3. In the alternative form, the links are not perpendicular to the revolute joints, and they have the following offset lengths relative to the revolute joints: AE = CG = c and BF = DH = d, as shown in Fig. 3. In Fig. 3, the links and the dashed lines represent the alternative and original forms of Bennett linkage, respectively. For the alternative form, it is defined that θ f and ϕ f represent the revolute variables θ and ϕ , respectively, in the folded form, and θ d and ϕ d represent the revolute variables θ and ϕ , respectively, in the deployed form. Some equations of the geometric parameters of the alternative form are presented in the Appendix as Eq. (A-2). These equations will be used in Section 3. Apart from the mathematical model of the alternative form, Chen and You also proposed its actual model using the links with square section. The deployed form of the actual model is a rhombus, while the folded form is a bundle of four rods which are lumped together, formulating good deployment performance for the linkage. Huang proposed a compact configuration of the alternative form of Bennett linkage [10]. The deployed and folded forms of this model are shown in Fig. 4.
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Fig. 5. Movement of the alternative form of equilateral Bennett linkage.
Fig. 6. Hybrid linkage constructed with equilateral Bennett linkage and planar symmetric four-bar linkage.
In Fig. 4, the blue dashed lines represent the original form of Bennett linkage. AP BP CP DP represents the Bennett linkage P. For a convenient explanation, the units 1 – 4 and units A and P in Figs. 8, 10 and 11 in [13] are defined as the basic and transition units, respectively. Units A are the transition units between units with an included angle, while units P are the transition units between parallel units. The compact configuration is the same as the transition unit between the basic units of the mechanical network in [13]. The Bennett linkage in Fig. 4 is the Bennett linkage P in Figs. 10 and 12 in [13]. From [13], it can be known that the deployed and folded forms of the transition and basic units are the opposite. Their geometric parameters satisfy the following relation:
1 1
θ f = P θd θd = P θ f
(4)
where i θ j , (i = 1, 2, 3, 4, A, P; j=d, f ) represents θ d and θ f of unit i. The other geometric parameters of transition unit P can be found in the Appendix as Eq. (A-3), which will be used in Section 3. More details on the alternative form of Bennett linkage can be found in [4,5,13]. 2.3. Hybrid linkage constructed with equilateral Bennett linkage and planar symmetric four-bar linkage Fig. 5 shows the movement of the alternative form of equilateral Bennett linkage. The blue dashed lines and the links represent the original and alternative forms of Bennett linkage, respectively. The equilateral Bennett linkage in this research is symmetric about planes APC and BQD, with PQ⊥AC and PQ⊥BD. It is defined that a line m that passes dot P and is perpendicular to plane APC and that a line n that passes dot Q and is perpendicular to plane BQD. Fig. 5 shows that axes A and C have a rotation motion about line m, while axes B and D have a rotation motion about line n. If the links of the Bennett linkage are ignored, then axes A and C and line m can be regarded as a virtual revolute joint, with line m for virtual axis and axes A and C for virtual links. Axes B and D and line n can be regarded as a virtual revolute joint in the same way. Therefore, combining equilateral Bennett linkage with planar symmetric four-bar linkage can construct a hybrid linkage, as shown in Fig. 6. In Fig. 6, the links GH, GF and GK form a hybrid joint, while the links EF, EH and EJ form a hybrid joint too. In the movement of the hybrid linkage, links KS, JS, GK and EJ and Bennett linkage EFGH form a virtual planar symmetric four-bar linkage KSJP, whereas the movement of links KS, JS, GK and EJ does not interfere with that of Bennett linkage EFGH. Thus, the DOF of the hybrid linkage is 1. In Fig. 6, the links JS and KS on plane APC and virtual joint APC form the virtual planar symmetric four-bar linkage KSJP. The detailed configuration of this linkage is shown in Fig. 7.
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Fig. 7. Configuration of the virtual planar symmetric four-bar linkage in the hybrid linkage.
Fig. 8. Hybrid linkage for the transition unit of Bennett linkage mechanical network.
The hybrid linkage in Fig. 7 satisfies the following conditions:
$S $K $J JS = KS KP = JP
$m (5)
where $i denotes the screw of axis i. For a clear explanation, we set KG = JE, KG⊥GP and JE⊥EP. We also set KG = JE and ∠KGP = ∠JEP on the condition that KP = JP and KP and JP are symmetric about plane BQD. In [13], we have researched the networking of Bennett linkage and found that the revolute variables of units 1 and P are equal, i.e. 1 θ1 = P θ1 . Thus, the value of ࢬAPC in Fig. 7 can be obtained from Eq. (A-1). In the right triangle PKG, CG = c, KG is a constant value and PC can be calculated from Eq. (A-1). From the triangle relation, we have:
√ PK = KG2 + PG2 ∠KPG = arctan KG PG
(6)
In Fig. 7, the following condition is satisfied:
1
∠APC + ∠KPG + ∠KPS =π KS = ∠KP = ∠PS ∠KPS KSP PKS ∠KPS + ∠KSP + ∠PKS = π
2
(7)
The distance PS from point S of the planar symmetric four-bar linkage to virtual hinge point P of the Bennett linkage can be obtained from Eq. (7). 2.4. Compact hybrid linkage constructed of the compact alternative form of Bennett linkage and planar symmetric four-bar linkage For the hybrid linkage in Fig. 6, when Bennett linkage EFGH moves, axis $S also moves on line PQ about virtual axis $m . According to [13], when the alternative form of Bennett linkage is used to construct the mechanical network, the transition units are equilateral Bennett linkages. In the current research, the hybrid linkage in Fig. 6 is employed to the transition Bennett units in the mechanical network. Then, a hybrid linkage extruding from the mechanical network is obtained. Fig. 8 shows the result when Bennett linkage P in Fig. 10 in [13] is used as the basic Bennett linkage to construct the hybrid linkage. For the hybrid linkage in Fig. 8, the Bennett linkage is the compact configuration of its alternative form. Accordingly, this linkage is named as a compact hybrid linkage.
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Fig. 9. Interlayer vertical pillar in the DLDMN.
Fig. 10. Schematic of the deployed and folded forms of the interlayer pillar.
2.5. Compact hybrid linkage added with a synchronous mechanism The interlayer pillar of the DLDMN should have a certain height and be perpendicular to the network surface. Hence, a synchronous mechanism is added to the compact hybrid linkage to construct a compact hybrid linkage interlayer vertical pillar with an adjustable height. This pillar is shown in Fig. 9. For the interlayer vertical pillar in Fig. 9, synchronous mechanism RSTM is a eudipleural linkage with two slide crank mechanisms. For the linkage in Fig. 9, RS = TS and RM = TM, and the extension rod is integrated with the guide rod. Besides, slide block M and revolute joint M are also combined. This design guarantees that the extension rod is always perpendicular to the network surface and has a certain height. The value of the length CG is constant, so that the value of PS can be obtained from Section 2.3. The length of the extension rod lv is constant. Thus, the height lH from the vertex of the vertical pillar to peak P of the Bennett linkage can be calculated as follows:
lH = PS + lv
(8)
When the alternative form of Bennett linkage is in deployed and folded forms, the interlayer vertical pillar is also in deployed and folded forms, as presented by Fig. 10(a) and (b). 3. Kinematic analysis of the hybrid linkage interlayer vertical pillar Fig. 11 shows the result when the hybrid linkage interlayer vertical pillar is employed to the single-layer mechanical network constructed with Bennett linkages. With the definition in [13], two basic Bennett linkages are units 1 and 3, the transition Bennett linkage is unit P and the hybrid linkage interlayer vertical pillar is only located at unit P. The revolute variables of units 1, 3 and P are equal and synchronous. When units 1 and 3 are deployed, unit P is folded; when units 1 and 3 are folded, unit P is deployed. Fig. 12 presents the calculation process of the height lH , which is from the vertex of the vertical pillar to the peak P of the Bennett linkage. Firstly, the geometric parameters of transition unit P are derived from those of basic unit 1 in Eq. (A-3), according to the relation between basic unit 1 and transition unit P. The revolute variables of transition unit P are then
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Fig. 11. Mechanical network constructed with Bennett linkages with interlayer vertical pillars.
Fig. 12. Calculation process of the height lH .
derived from those of basic unit 1. Accordingly, the angle ࢬAPC and lengths PA and PC are calculated using Eq. (A-1) in the Appendix. Secondly, the distance PS is calculated based on the selected values of CG , KG . Finally, the height lH is obtained with the addition of the length lv . Therefore, it can be concluded that the geometric parameters of the hybrid linkage are determined by those of the basic units, and that this linkage has limited variation range and effect on the PS. The proportion of PS in lH is also limited. The length lv is a free parameter whose value does not affect the mobility of the interlayer vertical pillar. Accordingly, the expected lH can be obtained by adjusting lv and then the deployable antenna curve is fitted using this property.
4. Cable-truss hybrid double-layer deployable mechanical network based on hybrid linkage interlayer vertical pillar It is known that the alternative form of Bennett linkage can be used to construct a single-layer deployable mechanical network. On this basis, the transition Bennett linkage is used to construct the hybrid linkage interlayer vertical pillar, which can support and connect components between layers. Then, the double-layer deployable mechanical network is constructed.
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Fig. 13. Mechanical network constructed of the alternative form of Bennett linkages.
4.1. Networking of the Bennett linkages The single-layer mechanical network constructed with the alternative form of Bennett linkages includes planar and cylindrical networks. The curvature of the cylindrical surface can be adjusted flexibly, which covers some parts, such as the cylindrical and parabolic cylindrical surfaces. The planar and cylindrical mechanical networks are shown in Fig. 13 where the basic Bennett linkages are connected by transition Bennett linkages [13]. From [13], it can be known that transition Bennett linkages are divided into two types: units P and A. Unit P is used to construct the hybrid linkage interlayer vertical pillar. 4.2. Mechanical network added with hybrid linkage interlayer vertical pillars The analysis in Section 2.3 indicates that the interlayer vertical pillars do not affect the movement of the Bennett linkage. The hybrid linkage on unit P is therefore constructed in the mechanical network. The planar mechanical network in Fig. 13(a) is used as the basic network to construct the interlayer vertical pillars. The resulting mechanical network is shown in Fig. 14. In Fig. 14, numbers 1, 2, …, 9 and Z1, Z2, …, Z9 represent the unit and interlayer vertical pillar numbers, respectively. From the folded form to deployed form, units 1, 2, …, 9 and the interlayer vertical pillars are always parallel. If the basic network is a cylindrical network, then the interlayer vertical pillars also keep the cylinder in the deployed form. Fig. 15 shows the resulting mechanical network with the cylindrical network as the basic network. The comparison between Figs. 14 and 15 indicates that the folded forms of the two mechanical networks are nearly the same, but their deployed forms are different. 4.3. Construction of the double-layer deployable mechanical network The interlayer vertical pillar always coincides with the centre line PQ of the Bennett linkage of the hybrid linkage and is perpendicular to diagonal lines AC and BD. Therefore, these pillars can be used as nodes and be connected with cables to construct a cable structure. The cable structure and single-layer mechanical network form a cable-truss hybrid double-layer deployable mechanical network, as shown in Fig. 16.
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Fig. 14. Planar mechanical network added with interlayer vertical pillars.
Fig. 15. Cylindrical mechanical network added with interlayer vertical pillars.
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Fig. 16. Cable-truss hybrid double-layer deployable mechanical network.
Fig. 17. Cable-truss hybrid double-layer mechanical network to fit paraboloid surface (cable layer fitting, planar truss layer).
The planar mechanical network in Fig. 16 can be used as a planar deployable mechanism. The red dashed lines represent the cables. All the interlayer vertical pillars have the same height. The interconnected cables and the links of the Bennett linkages form the cable and truss layers, respectively, which further form the cable-truss hybrid double-layer deployable mechanical network. The cable is flexible and can bend arbitrarily. Specifically, the folded form of the double-layer deployable mechanical network in Fig. 16 is similar to the form in Fig. 14(c). The process of constructing the hybrid linkage reveals that the movements of hybrid linkages belonging to different units are independent of each other. Thus, for interlayer vertical pillars Z1, Z2, …, Z9, the heights of the extension rods do not affect the movement of the mechanical network. Accordingly, the heights of the extension rods lv in Fig. 16 can be adjusted to fit the cable layer on curved surfaces. As a result, the DLDMNs with different curved surfaces can be constructed. Using this method, a deployable mechanical network with paraboloid surface is constructed, as shown in Fig. 17. In this network, the cable layer is used to fit the surface. The centre of the truss layer in Fig. 17, which is a planar mechanical network, coincides with the rotation centre axis of the paraboloid surface. The heights of the pillars are determined by the pointcuts of the paraboloid surface and the pillars. The heights of the pillars in Figs. 16 and 17 can be adjusted to fit other types of surface using the fitting method similar to that in Fig. 17. From the construction process of compact hybrid linkage, it can be known that this linkage is only related to one direction of the transition units and has no relation with the transition units in the other direction. Thus, the truss layer can use the mechanical network with a cylindrical surface. When the cylindrical surface truss layer is adopted with curvature as its free parameter, the truss layer can be used to fit the deployable mechanism with a cylindrical surface. In this case, the cable structure is used to improve the stiffness of the DLDMNs. Alternately, the cable layer can be used to fit the surface of deployable mechanisms. In the first case, the surface types are limited to the types of the truss layer constructed with Bennett linkages, such as cylindrical and parabolic cylindrical surfaces and the planar surface in Fig. 17. In the second case, the surface types are not limited because of the arbitrary adjustment of the cable layer. Figs. 18 and 19 show the DLDMNs which use the truss layer to fit the surface. The truss layer in Fig. 18 fits the cylindrical surface, whereas the truss layer in Fig. 19 fits the parabolic cylindrical surface. Fig. 20 shows the resulting DLDMN when the cylindrical surface is used as the truss layer and the cable layer is used to fit the surface. The surface fitted in this network is a paraboloid surface. Compared with the planar truss layer, the cylindrical truss surface is able to reduce the heights of the interlayer vertical pillars in some situations, which can increase the deployment ratio of the deployable mechanisms. All the DLDMNs shown in Figs. 16–20 are presented in deployed forms. The folded forms of these DLDMNs are similar to those in Fig. 14(c).
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Fig. 18. Cable-truss hybrid DLDMN to fit cylindrical surface (truss layer fitting).
5. Link areal density analysis of the cable-truss hybrid DLDMN The link plays an important role in the deployable framework antenna because it directly affects the folding volume, weight, deploying method and accuracy. The cable-truss hybrid DLDMN proposed in this research can be deployed to a large cover area with few links, which reduces the weight and folding volume of the antenna. Fig. 21 shows the basic units of the common deployable framework antenna at present: units of the cable-truss hybrid DLDMN in this research, units of the tetrahedron deployable antenna and units of the hexagonal prism deployable antenna. Comparison is conducted in the total link length and cover area in the deployed form of the planar mechanical network constructed with the three types of units. The results reveal the excellent performance of the cable-truss hybrid DLDMN. Because the cable takes up very few volume and weight compared with links, the cable layer in the cable-truss hybrid DLDMN can be ignored. For a convenient comparison, it is defined that the link lengths and the interlayer heights in planar part of the unit are equal and that the mechanical networks use basic units to fit planar surface. In the unit of cable-truss hybrid DLDMN, we define ω = 45◦ , that is, AC⊥BD. The link lengths are given by:
OA = OB = OC = OD = L1 OM = L2
(9)
According to Eq. (9), the cover area of the unit of cable-truss hybrid DLDMN is 2L21 . For the unit of the tetrahedron deployable antenna, it is defined that triangle ABC is an equilateral triangle. The link lengths are given by:
AB = BC = CA = L1 OM = L2
(10)
The units of the tetrahedron deployable antenna are distributed symmetrically in the two layers. As a result, the cover √ area of the unit is calculated to be 23 L21 . For the unit of the hexagonal prism deployable antenna, we define that
MA = MB = MC = MD = ME = MF = L1 MN = L2
(11) √
Therefore, the cover area of the unit for the hexagonal prism deployable antenna is 3 2 3 L21 . Fig. 22 presents the planar mechanical networks constructed with the three types of units. For a convenient comparison, it is assumed that the three types of mechanical networks have nearly same cover area. The cover area of the unit indicates
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Fig. 19. Cable-truss hybrid DLDMN to fit parabolic surface (truss layer fitting). Table 1 Properties of units of deployable framework antenna. Type
Total link length
Total link number
Cover area
Cable-truss hybrid DLDMN Tetrahedron deployable antenna Hexagonal prism deployable antenna
80L1 + 20L2 252L1 + 45 3L21 + 9L22 180L1 + 61L2 + 90 L21 + L22
100 387 331
40L21 38.97L21 38.97L21
that the cable-truss DLDMN, hexagonal prism deployable antenna and tetrahedron deployable antenna have 20, 15 and 45 units, respectively. In Fig. 22, the red dashed line and the blue heavy line represent the cable and the basic unit in the mechanical network, respectively. Table 1 displays the total link length, total link number and cover area for the three types of planar mechanical network. As shown by Table 1, when the cover area for the three types of DLDMN are nearly equal, the total link number and total link length of the cable-truss DLDMN are less than those of the two other deployable antennas. This feature demonstrates that the cable-truss hybrid DLDMN is an excellent solution for the deployable antenna. For the convenience of comparison, the link length in unit cover area is defined, that is, total link length/total cover area, as the link areal density. Low link areal density indicates large link sparsity, high utilisation efficiency, low antenna weight and small folding volume. The link lengths in these types of antenna are L1 = 1 m and L2 = 0.3 m considering of the size of the spaceborne deployable antenna at present. After the introduction of the specific parameters into the link, Table 2 is formulated. From Table 2, it can be observed that the link areal density of the cable-truss hybrid DLDMN is only 28.7% of that of the tetrahedron deployable antenna and is only 24.7% of that of the hexagonal prism deployable antenna. The link is sparse for the cable-truss hybrid DLDMN because parts of the links are replaced by cables, and the link utilisation efficiency is increased using the hinge combination method with special design.
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Fig. 20. Cable-truss hybrid DLDMN to fit paraboloid surface (cable layer fitting, cylindrical truss layer).
Fig. 21. Units of common deployable framework antennas.
Fig. 22. Three types of deployable planar framework antenna.
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Fig. 23. Process to fit the surface of deployable antenna by the cable-truss hybrid DLDMN.
Table 2 Properties of units of deployable framework antenna (specific parameters). Type
Total link length (m)
Cover area (m2 )
Link areal density
Cable-truss hybrid DLDMN Tetrahedron deployable antenna Hexagonal prism deployable antenna
86 292.26 339.84
40 38.97 38.97
2.15 7.5 8.72
6. Deployable paraboloid antenna implemented with cable-truss hybrid DLDMN 6.1. Process of fitting surface for the DLDMN Before designing a deployable mechanism, it is required to determine the surface size and accuracy of the deployable mechanism based on its requirements. Then, the stiffness, fundamental frequency, deployed and folded sizes of the deployable mechanism should be determined according to the connected satellite main part and the working environment. When these parameters are determined, the deployable mechanism can be designed.
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Fig. 24. Schematic of the offset feed paraboloid antenna.
Fig. 25. Cable-truss hybrid DLDMN to fit offset feed paraboloid antenna.
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Fig. 26. Interlayer pillars in the mechanical network.
Fig. 27. Schematic of the cable-truss hybrid DLDMN.
In this research, the cable-truss hybrid DLDMN is employed to implement the deployable antenna and the cable layer to fit the surface. The accuracy of the cable layer is determined by the nodes constructed with the interlayer vertical pillars. Thus, it is required to firstly determine the node number based on the surface size and accuracy of the antenna. When the node number is determined, the number of the basic unit of the DLDMN can be determined. Then, the side length, angle of the basic unit and the connection relation between the basic units can be determined by the deployed surface. On these bases and the folded volume, the section size of the link in the unit can finally be determined. Deployable mechanisms are commonly symmetric because symmetric structure can decrease the design difficulty and increase the uniformity of the unit parameters. For the symmetric deployable mechanism, the truss layer with a symmetric structure is chosen and the middle pillars are used to fit the centre symmetric curve. The height of the middle pillars is firstly designated and then the height of the other pillars is determined depending on the intersections of the deployed surface and the pillars of the mechanical network. For the target surface, surface equation fsurface (x, y, z ) = 0 is used. When the relationship between the position of DLDMN and that of the target surface is determined, the line equation f pillar (x, y, z ) = 0 of the pillars can be obtained by the geometric relation. The intersections of the two equations are the intersection point coordinates of the pillars and the target surface.
fsurface (x, y, z ) = 0 fpillar (x, y, z ) = 0
(12)
On the basis of the coordinates of intersection point and point S in the pillar, the length lv of the extension rod in the pillar can be obtained. Now, the geometric parameters of the DLDMN are determined. Then the folded volume and weight are calculated and compared with the working requirements. If the requirements are met, then the pillars are connected with cables, the stiffness and fundamental frequency are calculated and compared with the requirements. If the requirements are not met, then the geometric parameters are modified, followed by the restart of the design and calculation. The cycle is repeated until all the requirements are met. The detailed process is shown in Fig. 23. 6.2. Design of the double-layer deployable paraboloid antenna A paraboloid antenna is designed following the process in Section 6.1. 6.2.1. Design of the truss layer and the interlayer vertical pillars Paraboloid antenna has two types: positive and offset feed antennas. For positive feed antenna, the feed support equipment blocks the electromagnetic wave. Thus, offset feed antenna is chosen as our target, which is shown in Fig. 24. The reflector of the antenna in Fig. 24 is obtained when parabolic segment AB rotates around the X axis. The cable-truss hybrid DLDMN is used to fit this reflector. To achieve a simple design and a high deployment ratio, the truss layer is selected as a cylindrical surface, and cable layer is used to fit the paraboloid surface. The detailed fitting method is shown in Fig. 25. The curve AB in Fig. 25(c) is the generating line of the target paraboloid surface, and it is located at the centre of the paraboloid surface. Straight line A B is tangent with curve AB at its midpoint; straight line CD, the symmetrical centre line
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Fig. 28. 3D model of the interlayer pillars with carbon fibre bar.
of the truss layer, is parallel to A B . Given that the equation of the parabola is z2 = 11040x, the focal length of the parabola is f = 2760mm, the coordinate of point A is ( −324.3 0 586.4 ) and that of point B is ( 210.8 0 2555.2 ). Considering of the deployment ratio and the convenience of the design and manufacture, the geometric parameters of the Bennett linkage in the truss layer are set as L = 450 mm, ω = 55◦ , f = 51 mm and λ = 50◦ and the included angle as ∠1 = 10◦ . As the mechanical network has 20 units, it also has 20 interlayer vertical pillars. These pillars are shown in Fig. 26. In Fig. 26, column Y = 0 represents the column with less number, and the heights of the pillars are symmetric about column Y = 0. According to the parameters of the paraboloid reflector and the mechanical network, the length lv of the extension rods in Fig. 25 can be obtained. The calculation results are displayed in Table 3. The geometric parameters are bilateral symmetric about column Y = 0. Thus, only the parameters of one side are given. On the basis of the aforementioned parameters, a DLDMN is designed. The deployed and folded forms of this network are shown in Fig. 27. The link in the DLDMN in this research supports and connects the revolute joints and resists external forces. The carbon fibre bar has the advantages of low weight and good mechanical properties, leading to its extensive application to aerospace equipment. To decrease the weight of the mechanical network, two connectors are connected through the carbon fibre bar. For the hybrid linkage in the mechanical network, which is similar to that in Fig. 9, the planar four-bar linkage interferes with the Bennett linkage of the truss layer. The hybrid linkage is designed to avoid interference, as shown in Fig. 28.
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Table 3 Length of extension rods of interlayer pillars (mm). Y Z
0
1
2
0 1 2 3
308.5 229.3 229 300.8
287.7 207.8 206.8 278
223.9 141.9 138.8 208.2
Fig. 29. Basic unit of the cable-truss hybrid DLDMN.
Fig. 30. DLDMN with added cables.
The extension rod is also substituted with the carbon fibre bar to decrease the weight. Fig. 29 presents the resulting interlayer vertical pillar and the unit of the truss layer. 6.2.2. Design of the cable layer Apart from the ability to fit the special surface, the mechanical network should have sufficient stiffness and strength to resist external forces and disturbances. For the DLDMN in Fig. 27, the cable layer fits the antenna reflector with a paraboloid surface, and it also forms a double-layer rigid structure with the truss layer and the interlayer vertical pillars. To increase the stiffness of the rigid structure, several methods can be adopted, such as increasing the stiffness of the truss layer and interlayer vertical pillars, increasing the tension force in the cable of the cable layer and optimising the configuration of the mechanical network. The first method increases the stiffness of the links, which will inevitably increase the weight of the mechanical network. The second method has oversized tension force that requires high stiffness of the truss layer and interlayer pillar, which will also increase the weight. The cable is lighter than the link, and it can be seen as the link under tensile force in the deployed form. Thus, it is feasible to increase the stiffness without increasing the weight by increasing the number of the cables and changing the configuration of the mechanical network. The truss and cable layers in Fig. 27 are connected through the interlayer vertical pillars, and the structure is not a closed envelope. The structure with closed envelope has high stiffness. Accordingly, side and slant cables are added to the structure in Fig. 27. The resulting structure is shown in Fig. 30. In the figure, the blue long-dotted dashed line represents
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Joint of the wirerope
a) Folded form
b) Intermediate form
c) Deployed form
Fig. 31. Deployed and folded forms of the basic unit for cable-truss hybrid DLDMN.
Fig. 32. Deployed and folded forms of the cable-truss hybrid DLDMN.
the side cable, the blue double-dotted dashed line represents the slant cable and the red dashed line represents the original cable. The above indicates that the DLDMN forms a close envelope. The connections between the layers also increase, which increases the stiffness of the mechanical network. 6.2.3. Prototype A prototype of the DLDMN is designed and manufactured to verify the scheme for its application to large-size spaceborne antenna. The deployed and folded forms of the basic unit and the mechanical network are shown in Figs. 31 and 32, respec-
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LMS test system computer sensors Force hammer Multi-channel vibration test and analysis system
Fig. 33. Free mode test system of the deployable antenna.
Table 4 The first three order natural frequencies obtained from the test for the deployable antenna. Configuration Frequency(Hz) Order
Truss layer
Truss layer + cable layer
1 2 3
10.055 17.336 27.747
16.715 37.789 57.816
tively. Thin wirerope is used as the cable, and the wirerope is layout at the top and bottom of the interlayer vertical pillars, as shown in Fig. 31(b). The wirerope passes through the holes of the component and is fixed by chucks. For the comparison of stiffness, the prototype of the single-layer configuration for the DLDMN is also presented, as shown in Fig. 32(C). The free mode of the single-layer configuration and double-layer configuration of the mechanical network are tested using LMS test system. The test system is shown in Fig. 33, and the results are shown in Table 4. The natural frequencies in free mode represent the stiffness of the mechanical network. As can be seen from Table 4, the first three order natural frequencies of the double-layer configuration is much higher than that of the single-layer configuration, meaning that the stiffness of the double-layer configuration is also much higher than that of the single-layer configuration.
7. Conclusion This research proposes a single-DOF cable-truss hybrid DLDMN. The kinematics of equilateral Bennett linkage is analysed, and virtual joints are found in its movement. Then, equilateral Bennett linkage and planar symmetric four-bar linkage are combined to construct a single-loop four-bar hybrid linkage. The original form of the Bennett linkage in the hybrid linkage is replaced by its alternative form and a synchronous mechanism is introduced to obtain a hybrid linkage of compact form, through which rectilinear motion can be achieved in the certain direction of the vertical pillar. The hybrid linkage interlayer vertical pillar is employed to the single-layer Bennett linkage mechanical network to obtain a cable-truss hybrid DLDMN. The height of the interlayer vertical pillar has no relation with the mobility of the mechanical network. Thus, the mechanical network is used to fit various surfaces, such as planar, cylindrical and paraboloid surfaces. The mechanical networks not only have single DOF but also have high stiffness. Using the cable-truss hybrid DLDMN, an offset feed paraboloid antenna is designed and a prototype is fabricated to verity the feasibility of the proposed cable-truss hybrid DLDMN. The proposed double-layer deployable antenna with high stiffness can fit all kinds of spaceborne antenna including those with planar and those with curved surfaces. This research provides a reference for the design of other spaceborne antennas.
Acknowledgment This work was financially supported by the Joint Funds of the National Natural Science Foundation of China (Grant Nos. U1637207 and U1613201), in part by the National Natural Science Foundation of China (Grant Nos. 51575119, 51675114, 51675180 and 51565011), in part by the Natural Science Foundation for Distinguished Young Scholars of Jiangxi Province (Grant No. 20171BCB23059) and the Key Research Program of Jiangxi Province (Grant No. 20171BBE50039).
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Appendix Properties of the equilateral Bennett linkage:
⎧ θ tan 2 tan ϕ2 = cos1 α ⎪ ⎪ ⎪ ⎪ ⎪ cos ϕ ⎪ cos ∠APC = 2 1+ −1 ⎪ 1−cos θ ⎪ ⎪ ⎨ 1+cos θ
cos ∠BQD = 2 1−cos ϕ − 1
⎪ ⎪ (1+cos ϕ )(1−cos θ ) ⎪ PA = PC = l ⎪ ⎪ −2(cos ϕ +cos θ ) ⎪ ⎪ ⎪ ⎪ (1+cos θ )(1−cos ϕ ) ⎩ QB = QD = l
(A-1)
−2(cos ϕ +cos θ )
Geometry parameters for the alternative form of Bennett linkage:
⎧ (1+cos ϕ f )(1−cos θ f ) ⎪ c = l ⎪ −2(cos ϕ f +cos θ f ) ⎪ ⎨ (1−cos ϕ f )(1+cos θ f ) d=l −2(cos ϕ f +cos θ f ) ⎪ ⎪ ⎪ ⎩ θ θ 2 −tan α tan θ2d tan 2f = sec2 θ2d sec2 2f + tan2 α
(A-2)
Geometry parameters of the transition unit P:
⎧P 1 ⎪ ⎨ α = 2π − α 1 P θ1 = θ1 ⎪ ⎩l = L f sin1 α −kFU l1 f P
(A-3)
d1 +kP l1
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