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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Chinese Journal of Aeronautics, (2019), xxx(xx): xxx–xxx
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method Bo HAN a, Yundou XU a,b, Jiantao YAO a,b, Dong ZHENG a, Luyao GUO a, Yongsheng ZHAO a,b,* a
Parallel Robot and Mechatronic System Laboratory of Hebei Province, Yanshan University, Qinhuangdao 066004, China Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministry of National Education, Yanshan University, Qinhuangdao 066004, China b
Received 2 June 2019; revised 30 June 2019; accepted 5 July 2019
KEYWORDS Constraint-synthesis method; Deployable mechanism; Ring truss antenna; Screw theory; Type synthesis
Abstract Space deployable antenna is the key equipment in realizing the communication and data transmission between the spacecraft and the earth. In order to enrich the configurations of deployable antennas, the type synthesis of deployable mechanisms for ring truss antenna is conducted in this study. First, the principle of the constraint-synthesis method based on screw theory is briefly described, the structure of the ring truss deployable antenna and its folding principle are analyzed, and the ring truss mechanism is divided into upper edges, lower edges and linkages. Then, based on the constraint-synthesis method, the type synthesis of the basic unit edges is carried out, a series of basic unit mechanisms are obtained from combining the basic unit edge mechanisms, and five mechanism units with fewer joints and simple structures are selected. Furthermore, simulation models of the five ring truss deployable mechanisms are built in Solidworks and Matlab software, and the deploying process is verified by the movement simulation. Finally, mechanism characteristics of the five mechanisms are analyzed and discussed, and a prototype is manufactured, verifying the analysis in this paper. This research provides a new way for the type synthesis of spatial deployable mechanisms, and the ring truss deployable mechanisms obtained in this study can be well applied in the field of aerospace. Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction * Corresponding author at: Parallel Robot and Mechatronic System Laboratory of Hebei Province, Yanshan University, Qinhuangdao 066004, China. E-mail address: [email protected] (Y. ZHAO). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
With the rapid advancement of aerospace technology, space missions such as manned spaceflights, earth observation, deep space exploration and space communication missions keep growing in number, and the sizes of space mechanisms in various aerospace projects are becoming increasingly larger to meet the needs of different space missions.1–6 Subject to the limited space in the launcher, deployable mechanisms, which
https://doi.org/10.1016/j.cja.2019.07.015 1000-9361 Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: HAN B et al. Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.07.015
2 can be folded during storage and transport and fully deployed in the working environment, are widely applied to satellite platforms, space stations, space telescopes and other spacecraft. Owing to their favorable application prospects, deployable mechanisms have become one of the research hotspots in the field of aerospace.7–11 One of the important applications of deployable mechanisms in the field of aerospace is the deploying and supporting mechanisms for large-diameter space antennas,12 on which much research in different countries has already been undertaken.13 Cherniavsky et al. used the polygon scaling mechanism to construct a flat-panel deployable antenna mechanism14; Lu et al. proposed a planar deployable antenna mechanism by using the Hoekens unit mechanism with linear output15; references16 and17 each separately put forward a deployable unit mechanism, and the deployable antennas composed of these unit mechanisms were studied. Takamatsu and Onoda proposed a unit mechanism with a spatial diagonal structure to construct a large planar deployable antenna mechanism18; Vu et al. constructed a variety of planar deployable antenna mechanisms based on the pyramidal deployable unit mechanism19; Yang and Zhang proposed a quadrangular pyramid unit based on a seven-bar closed-loop mechanism which can be used to construct a planar deployable mechanism.20 All the above deployable mechanisms for space antennas belong to the planar deployable mechanisms. Compared with planar deployable antennas, mesh deployable antennas, which have curved reflective surfaces formed by a tensioned cable net structure, are more widely applied in large-diameter space antennas. Currently, the largediameter mesh deployable antennas mainly include the rib deployable antenna,21 the truss deployable antenna22–24 and the ring truss deployable antenna.25 The ring truss deployable antenna has a high folding rate and light weight; moreover, its weight does not increase in proportion to the diameter increase. These advantages make it the ideal form for largediameter space antennas. Escrig proposed the Pactruss double-ring truss deployable mechanism in 1985,26 and the United States launched the AstroMesh deployable antenna in 2000,27 which is a single-ring truss deployable antenna composed of a plurality of planar diagonal stretching units. Datashvili et al. constructed a ring truss deployable antenna mechanism based on the scissors mechanism units.28 Shi et al. constructed a single-ring truss deployable mechanism and a double-ring truss deployable mechanism based on the slider-crank planar deployable mechanism units.29 You and Pellegrino constructed a cable-stiffened ring truss deployable structure based on scissors mechanism units.30 Wang and Kong constructed a class of deployable mechanisms based on Bricard linkages, 8R or 10R single-loop linkages.31 Guan et al. constructed a double-ring truss deployable antenna which has the same structure as that of the Pactruss mechanism proposed by Escrig.32–35 The above studies achieved deployable antenna mechanisms by assembling existing mechanisms or proposing novel deployable unit mechanisms. However, none of them involve the type synthesis of mechanisms for deployable antennas. Deng et al. studied the type synthesis of the single loop deployable mechanisms with revolute joints36; references17,37,38 studied the type synthesis of unit mechanisms for the deployable antenna based on graph theory. The deployable unit mechanisms studied in these references can only be
B. HAN et al. assembled into the planar deployable antenna. Screw theory is an efficient analytical method with obvious advantages in mobility analysis and type synthesis of multiple closed-loop mechanisms.39–42 Reference43 studied the type synthesis of the tetrahedral truss deployable antenna mechanism based on screw theory; however, the tetrahedral truss antenna differs widely from with the ring truss antenna. To date, no research on the type synthesis of deployable mechanisms for ring truss antenna has been reported. From the perspective of the initial principle configuration, this paper studied the type synthesis of deployable mechanisms for ring truss antenna, and obtained five new types of ring truss deployable mechanisms. This study aims to provide a new way for the configuration design of ring truss antennas, illustrating clearly the comprehensive analysis process of the type synthesis of such spatial deployable mechanisms. The whole paper is organized as follows. The principle of the constraint-synthesis method based on screw theory is described in Section 2. Section 3 presents the structure and folding process analysis of the ring truss mechanism. Type synthesis of the basic unit edges and the basic units are respectively presented in Sections 4 and 5. In Section 6, five types of ring truss deployable mechanisms are constructed and the deploying processes are simulated, their characteristics analyzed, and a prototype manufactured. Finally, conclusions are presented in Section 7, wherein the present work is summarized. 2. Principle of the constraint-synthesis method based on screw theory According to screw theory, a screw in space can be expressed by a dual vector: S S $¼ ¼ ð1Þ S0 þ hS r S þ hS where S represents the unit direction vector of the screw axis, r the vector from the coordinate origin to any point on the screw axis, and h the pitch of the screw. $ in Eq. (1) can represent the motion of a rigid body or a force acting on the rigid body. When it represents the motion of a rigid body, it is called a twist, and when it represents the force acting on the rigid body, a wrench. When h = 0 in Eq. (1), the form of the screw is simplified to S S ¼ ð2Þ $¼ rS S0 $ in Eq. (2) is a line vector that can represent either the twist of a revolute joint (R) with S as the axis of the joint or a pure force for with S as the axis of the force. When h ! 1 in Eq. (1), the form of the screw is simplified to T $ ¼ 0T ; ST ð3Þ where 0 in Eq. (3) is a 3 1 null vector, and S is a direction vector. $ in Eq. (3) is an even vector that can represent either the twist of a prismatic joint (P) with its direction parallel to S or a couple whose direction is parallel to S. Other types of joint can be formed by combining the prismatic joint and the revolute joint. For example, a cylindric joint (C) is equivalent to a prismatic joint and a revolute joint,
Please cite this article in press as: HAN B et al. Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.07.015
Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method and a spherical joint (S) is equivalent to three intersecting but not coplanar revolute joints. T A screw, $r ¼ STr ; ST0r , and a set of screws, $1, $2, . . ., $n, are said to be reciprocal if they satisfy the condition: $j $r ¼ Sj S0r þ Sr S0j ¼ 0
j ¼ 1; 2; ; n
ð4Þ
where ‘‘ ” represents the reciprocal product, and $j the jth screw of the screw set. If a set of screws, $1, $2, . . ., $n, represents the joint twists associated with a supporting branch of a mechanism, then $r, the reciprocal to the joint twists, represents the constraint wrench exerted by the supporting branch. The motion of the mechanism is completely determined by the combined effect of all of the constraint wrenches of each branch. The type synthesis process of the constraint-synthesis method can be given as follows. First, the motion requirements for the target mechanism are analyzed, the required twist system obtained, and calculate the constraint wrench system which is reciprocal to the mechanism twist system is calculated. Then the branch constraint wrench systems are distributed according to the mechanism constraint wrench system. Each branch twist system can then be correspondingly obtained from the branch constraint wrench systems. Next, the branch kinematic chains are constructed according to the branch twist systems. Finally, all of the branch kinematic chains are distributed properly to guarantee that the combination of all the branch constraint wrench systems is equal to the desired mechanism constraint wrench system. 3. Structure and folding process analysis of the ring truss The United States launched the AstroMesh deployable antenna in 2000, which is a typical ring truss deployable antenna composed of a plurality of planar diagonal stretching units with a diameter of 12.25 meters. The AstroMesh deployable antenna and its components are shown in Fig. 1. In Fig. 1, we can see the structure of the ring truss deployable antenna. A fully centrosymmetric ring truss makes up the supporting structure of the whole antenna, the nets are attached to the upper and lower edge surfaces of the ring truss mechanism and are connected by tension ties. By relying on the prestress of the ties, the front net can be stretched into a parabolic shape needed during the working condition, and the mesh is connected with the front net to gather and reflect signals.
3
The fully deployed ring truss mechanism has a prism shape. The more edges it has, the more similar it is to a ring. As shown in Fig. 2, the ring truss can be simplified to a polygonal prism, and the vertices of the prism are the nodes on the truss. The ring truss can be divided into three parts: upper edges, lower edges and linkages. The upper and lower edges are respectively connected to the front net and the rear net. An upper edge, a lower edge and two linkages make up a planar unit, which is the basic unit of the ring truss deployable mechanism. The ideal folded form of the ring truss deployable antenna mechanism is a straight line in space, and a point in the crosssection view. The cross-sectional shape of the ring truss deployable mechanism is a regular polygon, and each edge in the polygon is the top view of a basic unit. During the folding process, the two vertices on each edge of the polygon move close to each other, the length of each edge shortens, and each edge moves close to the center. The cross-sectional shape of the ring truss mechanism and its folding process are shown in Fig. 3. 4. Motion analysis and type synthesis of the basic unit edge From Figs. 2 and 3, we can see that when the ring truss is folded, the two vertices of each edge on the cross-section will move to a coincident position because the plane on which the basic unit is located is vertical to the cross-section of the ring truss; therefore, the two vertices of each edge on the basic unit will move to a collinear position where the line connected by the two vertices is vertical to the cross section. Further analysis of the movement of the two vertices shows that they can coincide or be located in a straight line when are located in a collinear position. This process can be seen as both nodes moving toward a target axis which can pass through the initial position of either of the two nodes or be located in the middle of the two nodes. The folding processes of the two nodes are shown in Fig. 4. When the target axis passes through the initial position of one node, the other node will move to the target axis; when the target axis is located in the middle of the two nodes, both nodes will move to the target axis. The type synthesis process of the basic unit edge can be described as follows. First, the mechanism synthesis is conducted for the process whereby the node moves to the target axis and then the edge mechanism is obtained. Next, the upper edge and lower edge are connected through the linkage mechanism to form the basic unit, and finally, the basic units are combined to obtain the whole ring truss deployable mechanism. The movement of the node to the target axis can be achieved in two ways: linear movement and curve movement. Next, the type synthesis of the mechanisms for the basic unit edge is carried out based on these two movements. 4.1. Type synthesis of the supporting mechanism for the linear movement of the node
Fig. 1
AstroMesh deployable antenna and its components.
The node moves in a straight-line trajectory to the target axis in the plane which is defined by the node and the target axis, and the posture of the node during its movement is unchanged. Take node A as an example. Assume that the target axis is the z axis and the movement plane of the node is the xOz plane.
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Fig. 2
Fig. 3
Ring truss mechanism.
Cross-sectional shape of the ring truss mechanism and its folding process.
Eq. (5) represents the translational movement along the direction of (a, 0, b), and a, b are real numbers that are not zero at the same time. Solving the reciprocal screw of the twist in Eq. (5) yields the constraint wrenches applied to node A 3T 2 $r1 b 0 a 0 6 6 $r 7 60 1 0 0 6 27 6 6 r7 60 0 0 1 7 $ $r ¼ 6 ¼ 6 6 37 6 6 r7 40 0 0 0 4 $4 5 r $5 0 0 0 0 2
Fig. 4
Folding processes of the two nodes.
The schematic of the relative motion between node A and the target axis is shown in Fig. 5. The kinematic twist of node A is $m ¼ ½ 0 0
0 a 0 b T
ð5Þ
0 0 0 1
3T 0 7 07 7 07 7 7 05
ð6Þ
0 1
The translational movement is a one-dimensional movement which can be achieved by a planar mechanism in the xOz plane. The constraints in a plane include two vertical constraint forces and a constraint couple perpendicular to the plane. For the translational movement of node A, the constraint wrenches in the plane are $r ¼
$r1 $r4
T
¼
b 0 a 0 0 0 0 0 0 1
0 0
T ð7Þ
$r1 in Eq. (7) is a constraint force along the direction of (b, 0, a), and $r4 is a constraint couple along the y axis. If the two constraint wrenches in Eq. (7) are achieved by one branch, then by solving the reciprocal screw of the constraint wrenches in Eq. (7), the kinematic twist can be obtained as $m ¼ ½ 0 0 0
Fig. 5
Relative motion between node A and the target axis.
a 0 b T
ð8Þ
The kinematic twist in Eq. (8) can be achieved by a single P mechanism, as shown in Fig. 6.
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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Fig. 7 Fig. 6
PPPR mechanism.
Single P mechanism.
If the two constraint wrenches in Eq. (7) are achieved by two branches with one branch providing the constraint force $r1 and the other the constraint couple $r4 , then solving the reciprocal screws of the two constraint wrenches obtains the basic solution systems ( $m 0 0 a 0 b T 11 ¼ ½ 0 ð9Þ $m 1 0 0 0 0 T 12 ¼ ½ 0 (
5
0 $m 41 ¼ ½ 0
0 c 0 d T
$m 1 42 ¼ ½ 0
0 d 0 c T
ð10Þ
where c, d are real numbers that are not zero at the same time. m $m 11 represents a P joint along the direction of (a, 0, b), $12 an m R joint in which the axis direction is along the y axis, $41 a P joint along the direction of (c, 0, d), and $m 42 a P joint along the direction of (d, 0, c). On the basis of Eq. (9), we can construct the RP and PR branches, and on the basis of Eq. (10), the PP branch. Combining the branches obtains the PPRP mechanism and the PPPR mechanisms, and the latter is shown in Fig. 7. Since the basic unit is a planar mechanism, the planar unconstrained mechanisms can be used as redundant branches to construct the mechanisms. The planar unconstrained mech-
Fig. 8
anisms which can achieve three degrees of freedom (DOFs) in a plane are the RRR, RRP, RPR, PRR, RPP, PRP and PPR mechanisms.17 When the two constraint wrenches in Eq. (7) are achieved by a single P mechanism, through adding the seven planar unconstrained redundant branches, we can obtain the P + RRR, P + RRP, P + RPR, P + PRR, P + RPP, P + PRP and P + PPR mechanisms, which are shown in Fig. 8. The two constraint wrenches in Eq. (7) can also be achieved by several branches whereby the constraint wrench system provided by each branch is a subset of Eq. (7), and the mechanism can also contain several planar unconstrained branches. In this way, a wide variety of mechanisms can be obtained. However, these mechanisms are complicated. Since the mechanisms used in the aerospace require simple and reliable configurations, the analysis and type synthesis of these complicated mechanisms are not studied here. 4.2. Type synthesis of the supporting mechanism for the curve movement of the node The curve movement can be described as a node swinging to approach the target axis, and this process can be achieved by a pendulum movement. Take node A as an example. Assume that the target axis is the z axis, the movement plane of the node is the xOz plane defined by the node and the target axis,
Seven mechanisms composed of a single P and planar unconstrained mechanisms.
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the coordinate of the movement trajectory center to which node A swings is R (a, 0, b), and the axis which the node A swings around is along the y axis. The schematic of the relative motion between node A and the target axis is shown in Fig. 9. The kinematic twist of node A is $m ¼ ½ 0 1
0 b 0 a T
ð11Þ
This movement is also a planar movement. By solving the reciprocal screw of the twist in Eq. (11), the constraint wrenches applied to node A in the plane can be obtained as r T $1 1 0 0 0 b 0 T ð12Þ ¼ $r ¼ $r2 0 0 1 0 a 0 $r2
$r1
where is a constraint force along the direction of (c, 0, b), is a constraint force passing through point (a, 0, d) and along the direction of the z axis, and c and d are arbitrary real numbers. If the two constraint wrenches in Eq. (12) are achieved by one branch, then by solving the reciprocal screw of the constraint wrenches in Eq. (12), the kinematic twist can be obtained as $m ¼ ½ 0 1
0 b 0 a T
ð13Þ
The kinematic twist in Eq. (13) can be achieved by a single R mechanism, which is shown in Fig. 10. If the two constraint wrenches in Eq. (12) are achieved by two branches with one branch providing the constraint force $r1 and the other the constraint couple $r2 , then solving the reciprocal screws of the two constraint wrenches can obtain the basic solution systems ( $m 0 0 0 0 1 T 11 ¼ ½ 0 ð14Þ m $12 ¼ ½ 0 1 0 b 0 0 T
Fig. 9
Relative motion between node A and the target axis.
Fig. 11
(
PRRP mechanism.
$m 0 0 1 21 ¼ ½ 0
0 0 T
$m 1 0 0 22 ¼ ½ 0
0 a T
ð15Þ
$m 11 in Eq. (14) represents a P joint along the direction of the z axis, $m 12 an R joint for which the axis passes the point (0, 0, b) and along the direction of the y axis, $m 21 a P joint along the direction of the x axis, and $m 22 represents an R joint for which the axis passes the point (a, 0, 0) and along the direction of the y axis. By combining the joints, we can obtain the PRRP mechanism, as shown in Fig. 11. When the two constraint wrenches in Eq. (12) are achieved by a single R mechanism, we can also obtain new mechanisms through adding the planar unconstrained redundant branches. These new mechanisms are the R + RRR, R + RRP, R + RPR, R + PRR, R + RPP, R + PRP and R + PPR mechanisms, as shown in Fig. 12. The two constraint wrenches in Eq. (12) can also be achieved by several branches. The constraint wrench system provided by each branch is a subset of Eq. (12), and the mechanism can contain several planar unconstrained branches. The analysis and type synthesis of this mechanism are not studied here. From Figs. 9 to 12, we can see that the posture of the node is changed in the moving process. Due to the fact that the mesh is connected with the nodes in the ring truss deployable antenna, the posture change of the node can potentially cause the mesh and the nodes to wind together and become stuck, leading to failure in deploying the antenna. To solve this problem, we can add an R joint at the connection location of the node and the rod. This R joint has no influence on the movement of the node; instead, it only changes its posture. In this way, the single R mechanism is changed to an RR mechanism, as shown in Fig. 13. The RR mechanism has two DOFs: one of the R joints can achieve the deploying and folding movements, and the other can change the posture of the node. After the addition of this second R joint to the seven mechanisms shown in Fig. 12, seven additional mechanisms can be obtained as shown in Fig. 14. 4.3. Analysis and optimization of the basic unit edge
Fig. 10
Single R mechanism.
The mechanisms synthesized in Sections 4.1 and 4.2 can achieve the movement of the node. Sixteen different mechanisms were obtained in total, and their configurations are listed in Table 1. The ideal folded form of the ring truss deployable antenna mechanism is a straight line. Thus, the ring truss mechanism requires a high folding ratio in the circumferential direction.
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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Fig. 12
Fig. 13
Seven mechanisms composed of a single R and planar unconstrained mechanisms.
RR mechanism.
However, the basic unit edge is on the circumferential direction, and a P joint will occupy more space than an R joint. The maximum folding rate of the P joint is 2; therefore the mechanisms in Table 1 need to be screened, thereby obtaining the RR and RRRR mechanisms.
Fig. 14
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The RR and RRRR mechanisms in Table 1 can achieve the movement of the node. When the target axis passes through the initial position of a node, the other node will move to the target axis, and the supporting mechanism of the node can be either the RR or RRRR mechanism. When the target axis is located in the middle of the two nodes, both nodes will move to the target axis, and the supporting mechanism of each node can also be either the RR or RRRR mechanism. Consequently, we can obtain the RRR mechanism, the RRRR mechanism and the RRRRRRR mechanism, i.e., the 2R, 3R, 4R, 5R and 7R mechanisms, as shown in Fig. 15. 5. Type synthesis and analysis of the basic unit The ring truss deployable mechanism is formed by a plurality of basic units. The mechanisms for the basic unit edges are
Seven mechanisms composed by double R joints and planar unconstraint mechanisms.
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B. HAN et al. struct the basic unit. From the basic unit edge mechanisms shown in Fig. 15 and the RRP planar unconstrained mechanism, five deployable mechanisms can be obtained, as shown in Fig. 17.
Sixteen mechanisms.
Table 1 DOFs
Number of components
Number of joints
Configurations
1
2 4
1 4
3 5
2 5
P PRRR, PRRP, PRPR, PPRR, PRPP, PPRP, PPPR, RRRR, RRRP, RRPR, RRPP RR RRPRR, RRPRP, RRPPR
2
shown in Fig. 15, and these mechanisms can be used to construct the basic units. From Fig. 2, we can see that an upper edge and a lower edge can be connected to construct a basic unit. There are two ways to achieve the connection of the two edges: one is through a connecting rod, and the other is the upper edge and the lower edge sharing a joint or a rod. 5.1. Upper and lower edges connected by a rod When the upper and the lower edges are connected by a connecting rod, the mechanism of the upper edge and the lower edge can be one of the mechanisms shown in Fig. 15, and the basic unit mechanisms shown in Fig. 16 can be obtained. The basic unit mechanism is a planar one. The upper edge or lower edge can be one of the mechanisms in Fig. 15, and the other edge mechanism can be replaced by one of the seven planar unconstrained mechanisms. Since the ring truss mechanism requires a high folding rate in the circumferential direction, the P joint in the planar unconstrained mechanisms can only lie along the direction perpendicular to the cross section of the ring truss. The suitable planar unconstrained mechanisms after screening are the RRR and RRP mechanisms. The RRR mechanism is the same as the RRR basic unit edge mechanism shown in Fig. 15, so it can also be removed, thus leaving only the RRP planar unconstrained mechanism to con-
Fig. 15
5.2. Upper and lower edges connected by a shared joint or rod The upper edge mechanism and the lower edge mechanism can be one of the mechanisms shown in Fig. 15. When the upper and lower edges share a joint or a rod to achieve the connection, we can only obtain the scissors 5R mechanism, which is a combination of two 3R mechanisms, as shown in Fig. 18. Since the scissors 5R mechanism has only one DOF, and the four joints at the ends of the rods are located at the vertices of the basic unit, we can also add a planar unconstrained mechanism to the scissors 5R mechanism to form new mechanisms. Similarly, removing the planar unconstrained mechanisms with P joint, keeping the RRR mechanism, and adding an additional RRR mechanism to the scissors 5R mechanism, we can obtain the scissors 6R and scissors 7R mechanisms shown in Fig. 19. In this way, we have completed the type synthesis of the basic unit mechanisms. Twenty-three mechanisms are obtained as shown in Figs. 16–19, and all of these mechanisms can be used to construct a ring truss deployable mechanism. 6. Construction and analysis of the ring truss deployable mechanism 6.1. Basic unit mechanism selection and combination analysis Each mechanism unit in the Section 5 can be used as a basic unit mechanism to form the ring truss deployable mechanism. Considering the structure complexity and the number of joint clearances, we choose five units with simple structure and a small number of joints to construct the ring truss deployable mechanism. The basic units selected are shown in Fig. 20.
Five mechanisms for the basic unit edge.
Please cite this article in press as: HAN B et al. Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.07.015
Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Fig. 16
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Fifteen mechanisms for the basic unit.
The basic unit mechanisms in Fig. 20 are used to construct the ring truss deployable mechanisms. The combined mechanisms of the 4R mechanisms, 6R mechanisms, the scissors 7R mechanisms and their half folded states are shown in Fig. 21. When the 2R + RRP mechanism is used to construct the ring truss deployable mechanism, the P joint in the mechanism will not work if each mechanism unit is arranged in the same position, thus the mechanism is the same as the 4R mechanism unit. To render the P joint a role in the mechanism, the mechanism units should be arranged upside down in the opposite direction, and since the P joint needs to move on the vertical rod, the length of the vertical rod needs to be longer than
the length of the cross rod. The combination mechanism and its half folded state are shown in Fig. 22. Similar to the above analysis, when the 3R + RRP is used mechanism to construct the ring truss deployable mechanism, due to the limitations of the mechanism unit structure, the mechanism units need to be arranged symmetrically, and the length of the vertical rod needs to be longer than the length of the cross rod. The combination mechanism and its half folded state are shown in Fig. 23. The basic unit mechanisms in Fig. 20 and the combination mechanisms in Figs. 21–23 are the initial principle configurations of the unit mechanisms in the ring truss deployable antenna, which only show the mechanism configurations with-
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Fig. 17
Fig. 18
Five deployable mechanisms for the basic unit.
Scissors 5R mechanism.
out involving the specific structural design. The mechanisms shown in Figs. 20(a) and 21(a) are now commonly used in ring truss deployable antennas, the AstroMesh antenna being one of them. In structural design, a telescopic diagonal support mechanism is usually added to this mechanism, and the 4R mechanism will turn into a 4R1P mechanism, as shown in Fig. 24. The P joint in the 4R1P mechanism shown in Fig. 24 is usually a rope or a hollow telescopic rod which is running through a rope, and can be used to drive the mechanism. When a plurality of mechanism units are combined to a ring truss mechanism, the whole mechanism usually has multiple DOFs. To reduce the DOFs of the whole mechanism, synchronous joints are often added at the nodes shared by two mechanism units,
Fig. 19
and a typical synchronous joint is shown in Fig. 25. The diagonal support mechanism and the synchronous joint can also be used in the 6R, 2R + RRP and 3R + RRP mechanisms shown in Fig. 20. The scissors 7R mechanism has the diagonals and only one DOF, thus it does not need the diagonal support mechanism and synchronous joints. Although the diagonal support mechanism and the synchronous joints can be added to the initial principle mechanisms shown in Figs. 21–23 during the structural design, they cannot change the topologies of the initial principle mechanisms and have no effect on the deployable and foldable properties of these mechanisms. Since this paper focuses on the analysis and synthesis of deployable mechanisms for ring truss antennas from the perspective of mechanism principles and deployable properties, the structure design is not analyzed in detail here. 6.2. Deploying process simulation and verification of the constructed ring truss deployable mechanisms Based on the five selected basic unit mechanisms and their combinations shown in Figs. 20–23, five ring truss deployable mechanisms can be constructed. To verify the correctness of the above theoretical analysis, the simulation models of the five ring truss deployable mechanisms are built in Solidworks software and Matlab software to simulate the deploying process.
Scissors 6R and 7R mechanism.
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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Fig. 20
Fig. 21
11
Selected five basic units.
Three combinations of the selected basic unit mechanisms.
Fig. 22
Combination of the 2R + RRP mechanisms.
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Fig. 23
Fig. 24
4R1P mechanism.
Fig. 25
Synchronous joint.
Combination of the 3R + RRP mechanism.
The number of the ring truss deployable mechanism units is 12, and the length of the unit and the height of the ring truss after full deployment is 400 mm and 600 mm, respectively. The five ring truss deployable mechanisms have the same profile, as shown in Fig. 26. Although the profiles of the five ring truss deployable mechanisms are the same after full deployment, their structural forms are different in the fully folded state and the deploying process due to different mechanism configurations. The deploying process simulations of the five ring truss deployable mechanisms carried out in Solidworks software are shown in Fig. 27. Numerical simulations of the deploying processes are conducted in Matlab software, and the deploying processes are shown in Fig. 28. The five ring truss deployable mechanisms shown in Figs. 27 and 28 complete the deploying process with the desired movement during the simulation. Through these figures, the correctness of the synthesis process of the ring truss deployable antenna mechanism based on the constraint-synthesis method can be verified. 6.3. Mechanism characteristics analysis The ring truss deployable mechanisms shown in Figs. 27 and 28 have simple structures, and therefore can all be used to
Fig. 26
Profile of the fully deployed ring truss mechanism.
the support mechanism for ring truss antennas. From these two figures, we can see that they have different characteristics, which are listed as follows. (1) When the whole mechanism folds, the height of the ring truss deployable mechanism constructed by 4R mechanisms is higher than the others. Consequently, it will occupy a larger space in storage and transport. (2) Relative to the other mechanisms shown in Figs. 27 and 28, the mechanism constructed by 4R mechanisms has fewer joints. Therefore, the joint clearance is smaller than the other mechanisms, rendering it a higher motion accuracy and relatively higher structural stiffness than the other mechanisms. (3) The ring truss deployable mechanisms constructed by 4R, 6R and scissors 7R mechanisms are all constructed by rods and R joints. In the space environment, the R joint has higher movement flexibility than the P joint. (4) The ring truss deployable mechanism constructed by scissors 7R mechanisms has one DOF; therefore, during the deploying process, it only needs one input drive.
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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Fig. 27
13
Deploying processes in Solidworks software.
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Fig. 28
Deployment processes in Matlab software.
However, all the other mechanisms have multiple DOFs, and therefore need synchronous joints added to the nodes to ensure the synchronism of the deployable movement. (5) Relative to the other mechanisms shown in Figs. 27 and 28, the ring truss deployable mechanism constructed by scissors 7R mechanisms has triangular support struc-
tures when fully deployed, rendering it a higher structural stiffness than the other mechanisms which need an additional diagonal support mechanism when applied in engineering. The above analyses describe the characteristics of the five ring truss deployable mechanisms shown in Figs. 27 and 28.
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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Fig. 29
15
Prototype of the ring truss deployable mechanism constructed by scissors 7R mechanisms.
In practical engineering applications, we can weigh the advantages and disadvantages of these mechanisms according to specific situations, and choose the most suitable mechanism. From the perspective of fewer DOFs and flexibility movements, we chose the one with scissors 7R mechanisms to manufacture a prototype to verify the previous analyses, as shown in Fig. 29. Fig. 29 shows the fully folded state, half folded state and the fully deployed state of the ring truss deployable mechanism constructed by scissors 7R mechanisms, and the previous analysis can be verified. In future work we will carry out performance analysis and evaluation for the ring truss deployable mechanisms obtained in this paper, and comprehensively consider the engineering practice requirements when conducting the structural design and optimization for these ring truss deployable mechanisms. 7. Conclusions (1) Based on the constraint-synthesis method, the type synthesis of deployable mechanisms for ring truss antennas was conducted in this study, providing good reference for the design of other deployable mechanisms. (2) The structure and the folding principle of the ring truss deployable mechanism were analyzed and the mechanisms for the basic unit edges were obtained through two types of node movements. A plurality of unit mechanisms were obtained and five unit mechanisms with simple structure and a small number of joints were selected. (3) Five ring truss deployable mechanisms were constructed based on the selected unit mechanisms, and the deploying features of these mechanisms were verified by the simulation of the deploying processes. (4) Different characteristics of the five ring truss deployable mechanisms were analyzed, and a prototype based on the scissors 7R mechanisms was manufactured, whose deploying process verified the previous analysis in this paper. (5) The mechanisms obtained in this paper are the initial principle configurations, and the structural design and optimization analyses will be carried out in further research.
Acknowledgements This study was co-supported by the National Natural Science Foundation of China (No. 51675458), the Key Project of Natural Science Foundation of Hebei Province of China (No. E201720335) and the Youth Top Talent Project of Hebei Province Higher Education of China (No. BJ2017060). References 1. Li TJ, Jiang J, Deng HQ, Lin ZC, Wang ZW. Form-finding methods for deployable mesh reflector antennas. Chin J Aeronaut 2013;26(5):1276–82. 2. Fowler PW, Guest SD, Schulze B. Mobility of a class of perforated polyhedral. Int. J. Solids Struct. 2016;85–86:105–13. 3. Qi XZ, Huang HL, Miao ZH, Li B, Deng ZQ. Design and mobility analysis of large deployable mechanisms based on planesymmetric bricard linkage. J Mech Des, Trans ASME 2017;139(2) 022302. 4. Nie R, He BY, Zhang LH, Fang YG. Deployment analysis for space cable net structures with varying topologies and parameters. Aerosp Sci Technol 2017;68:1–10. 5. Chen Y, Feng J, Sun QZ. Lower-order symmetric mechanism modes and bifurcation behavior of deployable bar structures with cyclic symmetry. Int. J. Solids Struct. 2018;139–140:1–14. 6. Cao WA, Yang DH, Ding HF. Topological structural design of umbrella-shaped deployable mechanisms based on new spatial closed-loop linkage units. J Mech Des, Trans ASME 2017;140(6) 062302. 7. Xing ZG, Zheng GT. Deploying process modeling and attitude control of a satellite with a large deployable antenna. Chin J Aeronaut 2014;27(2):299–312. 8. Ding XL, Li X, Xu K, Yang QL, Pu HL. Study on the behavior of solar array deployment with root hinge drive assembly. Chin J Aeronaut 2012;25(2):276–84. 9. Roovers K, Temmerman ND. Deployable scissor grids consisting of translational units. Int J Solids Struct 2017;121:45–61. 10. Yang GG, Duan BY, Du JL, Zhang YQ. Shape pre-adjustment of deployable mesh antennas considering space thermal loads. Proc Inst Mech Eng Part G J Aerosp Eng 2018;232(1):143–55. 11. Tanaka H, Shibutani Y, Lzumi S, Sakai S. Planar mobility modes of 8-bar-jointed structures with a single degree of freedom. Int J Solids Struct 2012;49(13):1712–22. 12. Liu RW, Guo HW, Liu RQ, Wang HX, Tang DW, Song XK. Shape accuracy optimization for cable-rib tension deployable antenna structure with tensioned cables. Acta Astronaut 2017;140:66–77.
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16 13. Han B, Xu YD, Yao JT, Zheng D, Zhang S, Zhao YS. Kinematic characteristics analysis and assembly application of a spatial symmetric 7R mechanism. Acta Aeronautica et Astronautica Sinica 2019;40(4):422536 [Chinese]. 14. Cherniavsky AG, Gulyayev VI, Gaidaichuk VV, Fedoseev AI. Large deployable space antennas based on usage of polygonal pantograph. J Aerosp Eng 2005;18(3):139–45. 15. Lu SN, Zlatanov D, Ding XL, Molfino R. A new family of deployable mechanisms based on the Hoekens linkage. Mech Mach Theory 2014;73(2):130–53. 16. Chu ZR, Deng ZQ, Qi XZ, Li B. Modeling and analysis of a large deployable antenna structure. Acta Astronaut 2014;95(1):51–60. 17. Wang Y, Liu RQ, Yang H, Cong Q, Guo HW. Design and deployment analysis of modular deployable structure for large antennas. J Spacecr Rockets 2015;52(4):1101–11. 18. Takamatsu KA, Onoda J. New deployable truss concepts for large antenna structures or solar concentrators. J Spacecr Rockets 2015;28(3):330–8. 19. Vu KK, Richard Liew JYR, Anandasivam K. Deployable tensionstrut structures: from concept to implementation. J Constr Steel Res 2006;62(3):195–209. 20. Yang Y, Zhang WX. Kinematic investigation and assembly application of a spatial symmetric 6R mechanism. Acta Aeronautica et Astronautica Sinica 2014;35(12):3459–69 [Chinese]. 21. Takano T, Miura K, Natori M, Hanayama E. Deployable antenna with 10-m maximum diameter for space use. IEEE Trans Antennas Propag 2004;52(1):2–11. 22. Xu Y, Guan FL. Structure–electronic synthesis design of deployable truss antenna. Aerosp Sci Technol 2013;26(1):259–67. 23. Guan FL, Shou JJ, Hou GY, Zhang JJ. Static analysis of synchronism deployable antenna. J Zhejiang Univ: Sci 2006;7 (8):1365–71. 24. Xu YD, Guo JW, Guo LY, Liu WL, Yao JT, Zhao YS. Design and analysis of a truss deployable antenna mechanism based on a 3UU-3URU unit. Chin J Aeronaut 2020;33. https://doi.org/ 10.1016/j.cja.2018.12.008. 25. Dai L, Guan FL, Guest JK. Structural optimization and model fabrication of a double-ring deployable antenna truss. Acta Astronaut 2014;94(2):843–51. 26. Escrig F. Expandable space structures. Space Struct 1985;1 (2):79–91. 27. Meguro A, Tsujihata A, Tsujihata N, Homma M. Technology status of the 13 m aperture deployment antenna reflectors for engineering test satellite VIII. Acta Astronaut 2000;47(2):147–52. 28. Datashvili L, Endler S, Wei B, et al. Study of mechanical architectures of large deployable space antenna apertures: From design to tests. CEAS Space J 2013;5(3–4):169–84.
B. HAN et al. 29. Shi C, Guo HW, Liu RQ, Deng ZQ. Configuration optimization and structure design of the double-layer hoop deployable antenna mechanism. J Astronautics 2016;37(7):869–78 [Chinese]. 30. You Z, Pellegrion S. Cable-stiffened pantographic deployable structures. Part 2: mesh reflector. AIAA J 1997;35(8):1348–55. 31. Wang JY, Kong XW. Deployable mechanisms constructed by connecting orthogonal Bricard linkages, 8R or 10R single-loop linkages using S joints. Mech Mach Theory 2018;120:178–91. 32. Guan FL, Dai L, Xia MM. Pretension optimization and verification test of double-ring deployable cable net antenna based on improved PSO. Aerosp Sci Technol 2014;32(1):19–25. 33. Xu Y, Guan FL, Xu X, Wang HJ, Zheng Y. Development of a novel double-ring deployable mesh antenna. Int J Antennas Propag 2012;2012 375463. 34. Xu Y, Guan FL, Zheng Y, Zhao ML. Kinematic analysis of the deployable truss structures for space applications. J Aerosp Technolog Manage 2012;4(4):453–62. 35. Xu Y, Guan FL, Chen JJ, Zheng Y. Structural design and static analysis of a double-ring deployable truss for mesh antennas. Acta Astronaut 2012;81(2):545–54. 36. Deng ZQ, Huang HL, Li B, Liu RQ. Synthesis of deployable/foldable single loop mechanisms with revolute joints. J Mech Robot 2011;3(3) 031006. 37. Warnaar DB, Chew M. Kinematic synthesis of deployablefoldable truss structures using graph theory, part 1: graph generation. J Mech Des, Trans ASME 1995;117(1):112–6. 38. Wang Y, Deng ZQ, Liu RQ, Yang H, Guo HW. Topology structure synthesis and analysis of spatial pyramid deployable truss structures for satellite SAR antenna. Chin J Mech Eng Engl Ed 2014;27(4):683–92. 39. Huang Z, Zhao YS, Zhao TS. Advanced Spatial Mechanism. 2nd ed. Beijing: Higher Education Press; 2014. p. 339–78. 40. Kong XW, Gosselin CM. Type synthesis of 4-DOF SP-equivalent parallel manipulators: a virtual chain approach. Mech Mach Theory 2006;41(11):1306–19. 41. Xu YD, Yao JT, Zhao YS. Type synthesis of spatial mechanisms for forging manipulators. Proc Inst Mech Eng Part C J Mech Eng Sci 2012;226(9):2320–30. 42. Zhao XY, Zhao TS, Wang C, Tian X, Chen YH. Type synthesis and analysis of parallel mechanisms with sub-closed-loops. Mech Mach Theory 2018;120:140–65. 43. Xu YD, Chen LL, Liu WL, Yao JT, Zhu JL, Zhao YS. Type synthesis of the deployable mechanisms for the truss antenna using the method of adding constraint chains. J. Mech. Robot 2018;10(4) 041002.
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method Bo HAN a, Yundou XU a,b, Jiantao YAO a,b, Dong ZHENG a, Luyao GUO a, Yongsheng ZHAO a,b,* a
Parallel Robot and Mechatronic System Laboratory of Hebei Province, Yanshan University, Qinhuangdao 066004, China Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministry of National Education, Yanshan University, Qinhuangdao 066004, China b
Received 2 June 2019; revised 30 June 2019; accepted 5 July 2019
KEYWORDS Constraint-synthesis method; Deployable mechanism; Ring truss antenna; Screw theory; Type synthesis
Abstract Space deployable antenna is the key equipment in realizing the communication and data transmission between the spacecraft and the earth. In order to enrich the configurations of deployable antennas, the type synthesis of deployable mechanisms for ring truss antenna is conducted in this study. First, the principle of the constraint-synthesis method based on screw theory is briefly described, the structure of the ring truss deployable antenna and its folding principle are analyzed, and the ring truss mechanism is divided into upper edges, lower edges and linkages. Then, based on the constraint-synthesis method, the type synthesis of the basic unit edges is carried out, a series of basic unit mechanisms are obtained from combining the basic unit edge mechanisms, and five mechanism units with fewer joints and simple structures are selected. Furthermore, simulation models of the five ring truss deployable mechanisms are built in Solidworks and Matlab software, and the deploying process is verified by the movement simulation. Finally, mechanism characteristics of the five mechanisms are analyzed and discussed, and a prototype is manufactured, verifying the analysis in this paper. This research provides a new way for the type synthesis of spatial deployable mechanisms, and the ring truss deployable mechanisms obtained in this study can be well applied in the field of aerospace. Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction * Corresponding author at: Parallel Robot and Mechatronic System Laboratory of Hebei Province, Yanshan University, Qinhuangdao 066004, China. E-mail address: [email protected] (Y. ZHAO). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
With the rapid advancement of aerospace technology, space missions such as manned spaceflights, earth observation, deep space exploration and space communication missions keep growing in number, and the sizes of space mechanisms in various aerospace projects are becoming increasingly larger to meet the needs of different space missions.1–6 Subject to the limited space in the launcher, deployable mechanisms, which
https://doi.org/10.1016/j.cja.2019.07.015 1000-9361 Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: HAN B et al. Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.07.015
2 can be folded during storage and transport and fully deployed in the working environment, are widely applied to satellite platforms, space stations, space telescopes and other spacecraft. Owing to their favorable application prospects, deployable mechanisms have become one of the research hotspots in the field of aerospace.7–11 One of the important applications of deployable mechanisms in the field of aerospace is the deploying and supporting mechanisms for large-diameter space antennas,12 on which much research in different countries has already been undertaken.13 Cherniavsky et al. used the polygon scaling mechanism to construct a flat-panel deployable antenna mechanism14; Lu et al. proposed a planar deployable antenna mechanism by using the Hoekens unit mechanism with linear output15; references16 and17 each separately put forward a deployable unit mechanism, and the deployable antennas composed of these unit mechanisms were studied. Takamatsu and Onoda proposed a unit mechanism with a spatial diagonal structure to construct a large planar deployable antenna mechanism18; Vu et al. constructed a variety of planar deployable antenna mechanisms based on the pyramidal deployable unit mechanism19; Yang and Zhang proposed a quadrangular pyramid unit based on a seven-bar closed-loop mechanism which can be used to construct a planar deployable mechanism.20 All the above deployable mechanisms for space antennas belong to the planar deployable mechanisms. Compared with planar deployable antennas, mesh deployable antennas, which have curved reflective surfaces formed by a tensioned cable net structure, are more widely applied in large-diameter space antennas. Currently, the largediameter mesh deployable antennas mainly include the rib deployable antenna,21 the truss deployable antenna22–24 and the ring truss deployable antenna.25 The ring truss deployable antenna has a high folding rate and light weight; moreover, its weight does not increase in proportion to the diameter increase. These advantages make it the ideal form for largediameter space antennas. Escrig proposed the Pactruss double-ring truss deployable mechanism in 1985,26 and the United States launched the AstroMesh deployable antenna in 2000,27 which is a single-ring truss deployable antenna composed of a plurality of planar diagonal stretching units. Datashvili et al. constructed a ring truss deployable antenna mechanism based on the scissors mechanism units.28 Shi et al. constructed a single-ring truss deployable mechanism and a double-ring truss deployable mechanism based on the slider-crank planar deployable mechanism units.29 You and Pellegrino constructed a cable-stiffened ring truss deployable structure based on scissors mechanism units.30 Wang and Kong constructed a class of deployable mechanisms based on Bricard linkages, 8R or 10R single-loop linkages.31 Guan et al. constructed a double-ring truss deployable antenna which has the same structure as that of the Pactruss mechanism proposed by Escrig.32–35 The above studies achieved deployable antenna mechanisms by assembling existing mechanisms or proposing novel deployable unit mechanisms. However, none of them involve the type synthesis of mechanisms for deployable antennas. Deng et al. studied the type synthesis of the single loop deployable mechanisms with revolute joints36; references17,37,38 studied the type synthesis of unit mechanisms for the deployable antenna based on graph theory. The deployable unit mechanisms studied in these references can only be
B. HAN et al. assembled into the planar deployable antenna. Screw theory is an efficient analytical method with obvious advantages in mobility analysis and type synthesis of multiple closed-loop mechanisms.39–42 Reference43 studied the type synthesis of the tetrahedral truss deployable antenna mechanism based on screw theory; however, the tetrahedral truss antenna differs widely from with the ring truss antenna. To date, no research on the type synthesis of deployable mechanisms for ring truss antenna has been reported. From the perspective of the initial principle configuration, this paper studied the type synthesis of deployable mechanisms for ring truss antenna, and obtained five new types of ring truss deployable mechanisms. This study aims to provide a new way for the configuration design of ring truss antennas, illustrating clearly the comprehensive analysis process of the type synthesis of such spatial deployable mechanisms. The whole paper is organized as follows. The principle of the constraint-synthesis method based on screw theory is described in Section 2. Section 3 presents the structure and folding process analysis of the ring truss mechanism. Type synthesis of the basic unit edges and the basic units are respectively presented in Sections 4 and 5. In Section 6, five types of ring truss deployable mechanisms are constructed and the deploying processes are simulated, their characteristics analyzed, and a prototype manufactured. Finally, conclusions are presented in Section 7, wherein the present work is summarized. 2. Principle of the constraint-synthesis method based on screw theory According to screw theory, a screw in space can be expressed by a dual vector: S S $¼ ¼ ð1Þ S0 þ hS r S þ hS where S represents the unit direction vector of the screw axis, r the vector from the coordinate origin to any point on the screw axis, and h the pitch of the screw. $ in Eq. (1) can represent the motion of a rigid body or a force acting on the rigid body. When it represents the motion of a rigid body, it is called a twist, and when it represents the force acting on the rigid body, a wrench. When h = 0 in Eq. (1), the form of the screw is simplified to S S ¼ ð2Þ $¼ rS S0 $ in Eq. (2) is a line vector that can represent either the twist of a revolute joint (R) with S as the axis of the joint or a pure force for with S as the axis of the force. When h ! 1 in Eq. (1), the form of the screw is simplified to T $ ¼ 0T ; ST ð3Þ where 0 in Eq. (3) is a 3 1 null vector, and S is a direction vector. $ in Eq. (3) is an even vector that can represent either the twist of a prismatic joint (P) with its direction parallel to S or a couple whose direction is parallel to S. Other types of joint can be formed by combining the prismatic joint and the revolute joint. For example, a cylindric joint (C) is equivalent to a prismatic joint and a revolute joint,
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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method and a spherical joint (S) is equivalent to three intersecting but not coplanar revolute joints. T A screw, $r ¼ STr ; ST0r , and a set of screws, $1, $2, . . ., $n, are said to be reciprocal if they satisfy the condition: $j $r ¼ Sj S0r þ Sr S0j ¼ 0
j ¼ 1; 2; ; n
ð4Þ
where ‘‘ ” represents the reciprocal product, and $j the jth screw of the screw set. If a set of screws, $1, $2, . . ., $n, represents the joint twists associated with a supporting branch of a mechanism, then $r, the reciprocal to the joint twists, represents the constraint wrench exerted by the supporting branch. The motion of the mechanism is completely determined by the combined effect of all of the constraint wrenches of each branch. The type synthesis process of the constraint-synthesis method can be given as follows. First, the motion requirements for the target mechanism are analyzed, the required twist system obtained, and calculate the constraint wrench system which is reciprocal to the mechanism twist system is calculated. Then the branch constraint wrench systems are distributed according to the mechanism constraint wrench system. Each branch twist system can then be correspondingly obtained from the branch constraint wrench systems. Next, the branch kinematic chains are constructed according to the branch twist systems. Finally, all of the branch kinematic chains are distributed properly to guarantee that the combination of all the branch constraint wrench systems is equal to the desired mechanism constraint wrench system. 3. Structure and folding process analysis of the ring truss The United States launched the AstroMesh deployable antenna in 2000, which is a typical ring truss deployable antenna composed of a plurality of planar diagonal stretching units with a diameter of 12.25 meters. The AstroMesh deployable antenna and its components are shown in Fig. 1. In Fig. 1, we can see the structure of the ring truss deployable antenna. A fully centrosymmetric ring truss makes up the supporting structure of the whole antenna, the nets are attached to the upper and lower edge surfaces of the ring truss mechanism and are connected by tension ties. By relying on the prestress of the ties, the front net can be stretched into a parabolic shape needed during the working condition, and the mesh is connected with the front net to gather and reflect signals.
3
The fully deployed ring truss mechanism has a prism shape. The more edges it has, the more similar it is to a ring. As shown in Fig. 2, the ring truss can be simplified to a polygonal prism, and the vertices of the prism are the nodes on the truss. The ring truss can be divided into three parts: upper edges, lower edges and linkages. The upper and lower edges are respectively connected to the front net and the rear net. An upper edge, a lower edge and two linkages make up a planar unit, which is the basic unit of the ring truss deployable mechanism. The ideal folded form of the ring truss deployable antenna mechanism is a straight line in space, and a point in the crosssection view. The cross-sectional shape of the ring truss deployable mechanism is a regular polygon, and each edge in the polygon is the top view of a basic unit. During the folding process, the two vertices on each edge of the polygon move close to each other, the length of each edge shortens, and each edge moves close to the center. The cross-sectional shape of the ring truss mechanism and its folding process are shown in Fig. 3. 4. Motion analysis and type synthesis of the basic unit edge From Figs. 2 and 3, we can see that when the ring truss is folded, the two vertices of each edge on the cross-section will move to a coincident position because the plane on which the basic unit is located is vertical to the cross-section of the ring truss; therefore, the two vertices of each edge on the basic unit will move to a collinear position where the line connected by the two vertices is vertical to the cross section. Further analysis of the movement of the two vertices shows that they can coincide or be located in a straight line when are located in a collinear position. This process can be seen as both nodes moving toward a target axis which can pass through the initial position of either of the two nodes or be located in the middle of the two nodes. The folding processes of the two nodes are shown in Fig. 4. When the target axis passes through the initial position of one node, the other node will move to the target axis; when the target axis is located in the middle of the two nodes, both nodes will move to the target axis. The type synthesis process of the basic unit edge can be described as follows. First, the mechanism synthesis is conducted for the process whereby the node moves to the target axis and then the edge mechanism is obtained. Next, the upper edge and lower edge are connected through the linkage mechanism to form the basic unit, and finally, the basic units are combined to obtain the whole ring truss deployable mechanism. The movement of the node to the target axis can be achieved in two ways: linear movement and curve movement. Next, the type synthesis of the mechanisms for the basic unit edge is carried out based on these two movements. 4.1. Type synthesis of the supporting mechanism for the linear movement of the node
Fig. 1
AstroMesh deployable antenna and its components.
The node moves in a straight-line trajectory to the target axis in the plane which is defined by the node and the target axis, and the posture of the node during its movement is unchanged. Take node A as an example. Assume that the target axis is the z axis and the movement plane of the node is the xOz plane.
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Fig. 2
Fig. 3
Ring truss mechanism.
Cross-sectional shape of the ring truss mechanism and its folding process.
Eq. (5) represents the translational movement along the direction of (a, 0, b), and a, b are real numbers that are not zero at the same time. Solving the reciprocal screw of the twist in Eq. (5) yields the constraint wrenches applied to node A 3T 2 $r1 b 0 a 0 6 6 $r 7 60 1 0 0 6 27 6 6 r7 60 0 0 1 7 $ $r ¼ 6 ¼ 6 6 37 6 6 r7 40 0 0 0 4 $4 5 r $5 0 0 0 0 2
Fig. 4
Folding processes of the two nodes.
The schematic of the relative motion between node A and the target axis is shown in Fig. 5. The kinematic twist of node A is $m ¼ ½ 0 0
0 a 0 b T
ð5Þ
0 0 0 1
3T 0 7 07 7 07 7 7 05
ð6Þ
0 1
The translational movement is a one-dimensional movement which can be achieved by a planar mechanism in the xOz plane. The constraints in a plane include two vertical constraint forces and a constraint couple perpendicular to the plane. For the translational movement of node A, the constraint wrenches in the plane are $r ¼
$r1 $r4
T
¼
b 0 a 0 0 0 0 0 0 1
0 0
T ð7Þ
$r1 in Eq. (7) is a constraint force along the direction of (b, 0, a), and $r4 is a constraint couple along the y axis. If the two constraint wrenches in Eq. (7) are achieved by one branch, then by solving the reciprocal screw of the constraint wrenches in Eq. (7), the kinematic twist can be obtained as $m ¼ ½ 0 0 0
Fig. 5
Relative motion between node A and the target axis.
a 0 b T
ð8Þ
The kinematic twist in Eq. (8) can be achieved by a single P mechanism, as shown in Fig. 6.
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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Fig. 7 Fig. 6
PPPR mechanism.
Single P mechanism.
If the two constraint wrenches in Eq. (7) are achieved by two branches with one branch providing the constraint force $r1 and the other the constraint couple $r4 , then solving the reciprocal screws of the two constraint wrenches obtains the basic solution systems ( $m 0 0 a 0 b T 11 ¼ ½ 0 ð9Þ $m 1 0 0 0 0 T 12 ¼ ½ 0 (
5
0 $m 41 ¼ ½ 0
0 c 0 d T
$m 1 42 ¼ ½ 0
0 d 0 c T
ð10Þ
where c, d are real numbers that are not zero at the same time. m $m 11 represents a P joint along the direction of (a, 0, b), $12 an m R joint in which the axis direction is along the y axis, $41 a P joint along the direction of (c, 0, d), and $m 42 a P joint along the direction of (d, 0, c). On the basis of Eq. (9), we can construct the RP and PR branches, and on the basis of Eq. (10), the PP branch. Combining the branches obtains the PPRP mechanism and the PPPR mechanisms, and the latter is shown in Fig. 7. Since the basic unit is a planar mechanism, the planar unconstrained mechanisms can be used as redundant branches to construct the mechanisms. The planar unconstrained mech-
Fig. 8
anisms which can achieve three degrees of freedom (DOFs) in a plane are the RRR, RRP, RPR, PRR, RPP, PRP and PPR mechanisms.17 When the two constraint wrenches in Eq. (7) are achieved by a single P mechanism, through adding the seven planar unconstrained redundant branches, we can obtain the P + RRR, P + RRP, P + RPR, P + PRR, P + RPP, P + PRP and P + PPR mechanisms, which are shown in Fig. 8. The two constraint wrenches in Eq. (7) can also be achieved by several branches whereby the constraint wrench system provided by each branch is a subset of Eq. (7), and the mechanism can also contain several planar unconstrained branches. In this way, a wide variety of mechanisms can be obtained. However, these mechanisms are complicated. Since the mechanisms used in the aerospace require simple and reliable configurations, the analysis and type synthesis of these complicated mechanisms are not studied here. 4.2. Type synthesis of the supporting mechanism for the curve movement of the node The curve movement can be described as a node swinging to approach the target axis, and this process can be achieved by a pendulum movement. Take node A as an example. Assume that the target axis is the z axis, the movement plane of the node is the xOz plane defined by the node and the target axis,
Seven mechanisms composed of a single P and planar unconstrained mechanisms.
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the coordinate of the movement trajectory center to which node A swings is R (a, 0, b), and the axis which the node A swings around is along the y axis. The schematic of the relative motion between node A and the target axis is shown in Fig. 9. The kinematic twist of node A is $m ¼ ½ 0 1
0 b 0 a T
ð11Þ
This movement is also a planar movement. By solving the reciprocal screw of the twist in Eq. (11), the constraint wrenches applied to node A in the plane can be obtained as r T $1 1 0 0 0 b 0 T ð12Þ ¼ $r ¼ $r2 0 0 1 0 a 0 $r2
$r1
where is a constraint force along the direction of (c, 0, b), is a constraint force passing through point (a, 0, d) and along the direction of the z axis, and c and d are arbitrary real numbers. If the two constraint wrenches in Eq. (12) are achieved by one branch, then by solving the reciprocal screw of the constraint wrenches in Eq. (12), the kinematic twist can be obtained as $m ¼ ½ 0 1
0 b 0 a T
ð13Þ
The kinematic twist in Eq. (13) can be achieved by a single R mechanism, which is shown in Fig. 10. If the two constraint wrenches in Eq. (12) are achieved by two branches with one branch providing the constraint force $r1 and the other the constraint couple $r2 , then solving the reciprocal screws of the two constraint wrenches can obtain the basic solution systems ( $m 0 0 0 0 1 T 11 ¼ ½ 0 ð14Þ m $12 ¼ ½ 0 1 0 b 0 0 T
Fig. 9
Relative motion between node A and the target axis.
Fig. 11
(
PRRP mechanism.
$m 0 0 1 21 ¼ ½ 0
0 0 T
$m 1 0 0 22 ¼ ½ 0
0 a T
ð15Þ
$m 11 in Eq. (14) represents a P joint along the direction of the z axis, $m 12 an R joint for which the axis passes the point (0, 0, b) and along the direction of the y axis, $m 21 a P joint along the direction of the x axis, and $m 22 represents an R joint for which the axis passes the point (a, 0, 0) and along the direction of the y axis. By combining the joints, we can obtain the PRRP mechanism, as shown in Fig. 11. When the two constraint wrenches in Eq. (12) are achieved by a single R mechanism, we can also obtain new mechanisms through adding the planar unconstrained redundant branches. These new mechanisms are the R + RRR, R + RRP, R + RPR, R + PRR, R + RPP, R + PRP and R + PPR mechanisms, as shown in Fig. 12. The two constraint wrenches in Eq. (12) can also be achieved by several branches. The constraint wrench system provided by each branch is a subset of Eq. (12), and the mechanism can contain several planar unconstrained branches. The analysis and type synthesis of this mechanism are not studied here. From Figs. 9 to 12, we can see that the posture of the node is changed in the moving process. Due to the fact that the mesh is connected with the nodes in the ring truss deployable antenna, the posture change of the node can potentially cause the mesh and the nodes to wind together and become stuck, leading to failure in deploying the antenna. To solve this problem, we can add an R joint at the connection location of the node and the rod. This R joint has no influence on the movement of the node; instead, it only changes its posture. In this way, the single R mechanism is changed to an RR mechanism, as shown in Fig. 13. The RR mechanism has two DOFs: one of the R joints can achieve the deploying and folding movements, and the other can change the posture of the node. After the addition of this second R joint to the seven mechanisms shown in Fig. 12, seven additional mechanisms can be obtained as shown in Fig. 14. 4.3. Analysis and optimization of the basic unit edge
Fig. 10
Single R mechanism.
The mechanisms synthesized in Sections 4.1 and 4.2 can achieve the movement of the node. Sixteen different mechanisms were obtained in total, and their configurations are listed in Table 1. The ideal folded form of the ring truss deployable antenna mechanism is a straight line. Thus, the ring truss mechanism requires a high folding ratio in the circumferential direction.
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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Fig. 12
Fig. 13
Seven mechanisms composed of a single R and planar unconstrained mechanisms.
RR mechanism.
However, the basic unit edge is on the circumferential direction, and a P joint will occupy more space than an R joint. The maximum folding rate of the P joint is 2; therefore the mechanisms in Table 1 need to be screened, thereby obtaining the RR and RRRR mechanisms.
Fig. 14
7
The RR and RRRR mechanisms in Table 1 can achieve the movement of the node. When the target axis passes through the initial position of a node, the other node will move to the target axis, and the supporting mechanism of the node can be either the RR or RRRR mechanism. When the target axis is located in the middle of the two nodes, both nodes will move to the target axis, and the supporting mechanism of each node can also be either the RR or RRRR mechanism. Consequently, we can obtain the RRR mechanism, the RRRR mechanism and the RRRRRRR mechanism, i.e., the 2R, 3R, 4R, 5R and 7R mechanisms, as shown in Fig. 15. 5. Type synthesis and analysis of the basic unit The ring truss deployable mechanism is formed by a plurality of basic units. The mechanisms for the basic unit edges are
Seven mechanisms composed by double R joints and planar unconstraint mechanisms.
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B. HAN et al. struct the basic unit. From the basic unit edge mechanisms shown in Fig. 15 and the RRP planar unconstrained mechanism, five deployable mechanisms can be obtained, as shown in Fig. 17.
Sixteen mechanisms.
Table 1 DOFs
Number of components
Number of joints
Configurations
1
2 4
1 4
3 5
2 5
P PRRR, PRRP, PRPR, PPRR, PRPP, PPRP, PPPR, RRRR, RRRP, RRPR, RRPP RR RRPRR, RRPRP, RRPPR
2
shown in Fig. 15, and these mechanisms can be used to construct the basic units. From Fig. 2, we can see that an upper edge and a lower edge can be connected to construct a basic unit. There are two ways to achieve the connection of the two edges: one is through a connecting rod, and the other is the upper edge and the lower edge sharing a joint or a rod. 5.1. Upper and lower edges connected by a rod When the upper and the lower edges are connected by a connecting rod, the mechanism of the upper edge and the lower edge can be one of the mechanisms shown in Fig. 15, and the basic unit mechanisms shown in Fig. 16 can be obtained. The basic unit mechanism is a planar one. The upper edge or lower edge can be one of the mechanisms in Fig. 15, and the other edge mechanism can be replaced by one of the seven planar unconstrained mechanisms. Since the ring truss mechanism requires a high folding rate in the circumferential direction, the P joint in the planar unconstrained mechanisms can only lie along the direction perpendicular to the cross section of the ring truss. The suitable planar unconstrained mechanisms after screening are the RRR and RRP mechanisms. The RRR mechanism is the same as the RRR basic unit edge mechanism shown in Fig. 15, so it can also be removed, thus leaving only the RRP planar unconstrained mechanism to con-
Fig. 15
5.2. Upper and lower edges connected by a shared joint or rod The upper edge mechanism and the lower edge mechanism can be one of the mechanisms shown in Fig. 15. When the upper and lower edges share a joint or a rod to achieve the connection, we can only obtain the scissors 5R mechanism, which is a combination of two 3R mechanisms, as shown in Fig. 18. Since the scissors 5R mechanism has only one DOF, and the four joints at the ends of the rods are located at the vertices of the basic unit, we can also add a planar unconstrained mechanism to the scissors 5R mechanism to form new mechanisms. Similarly, removing the planar unconstrained mechanisms with P joint, keeping the RRR mechanism, and adding an additional RRR mechanism to the scissors 5R mechanism, we can obtain the scissors 6R and scissors 7R mechanisms shown in Fig. 19. In this way, we have completed the type synthesis of the basic unit mechanisms. Twenty-three mechanisms are obtained as shown in Figs. 16–19, and all of these mechanisms can be used to construct a ring truss deployable mechanism. 6. Construction and analysis of the ring truss deployable mechanism 6.1. Basic unit mechanism selection and combination analysis Each mechanism unit in the Section 5 can be used as a basic unit mechanism to form the ring truss deployable mechanism. Considering the structure complexity and the number of joint clearances, we choose five units with simple structure and a small number of joints to construct the ring truss deployable mechanism. The basic units selected are shown in Fig. 20.
Five mechanisms for the basic unit edge.
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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Fig. 16
9
Fifteen mechanisms for the basic unit.
The basic unit mechanisms in Fig. 20 are used to construct the ring truss deployable mechanisms. The combined mechanisms of the 4R mechanisms, 6R mechanisms, the scissors 7R mechanisms and their half folded states are shown in Fig. 21. When the 2R + RRP mechanism is used to construct the ring truss deployable mechanism, the P joint in the mechanism will not work if each mechanism unit is arranged in the same position, thus the mechanism is the same as the 4R mechanism unit. To render the P joint a role in the mechanism, the mechanism units should be arranged upside down in the opposite direction, and since the P joint needs to move on the vertical rod, the length of the vertical rod needs to be longer than
the length of the cross rod. The combination mechanism and its half folded state are shown in Fig. 22. Similar to the above analysis, when the 3R + RRP is used mechanism to construct the ring truss deployable mechanism, due to the limitations of the mechanism unit structure, the mechanism units need to be arranged symmetrically, and the length of the vertical rod needs to be longer than the length of the cross rod. The combination mechanism and its half folded state are shown in Fig. 23. The basic unit mechanisms in Fig. 20 and the combination mechanisms in Figs. 21–23 are the initial principle configurations of the unit mechanisms in the ring truss deployable antenna, which only show the mechanism configurations with-
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Fig. 17
Fig. 18
Five deployable mechanisms for the basic unit.
Scissors 5R mechanism.
out involving the specific structural design. The mechanisms shown in Figs. 20(a) and 21(a) are now commonly used in ring truss deployable antennas, the AstroMesh antenna being one of them. In structural design, a telescopic diagonal support mechanism is usually added to this mechanism, and the 4R mechanism will turn into a 4R1P mechanism, as shown in Fig. 24. The P joint in the 4R1P mechanism shown in Fig. 24 is usually a rope or a hollow telescopic rod which is running through a rope, and can be used to drive the mechanism. When a plurality of mechanism units are combined to a ring truss mechanism, the whole mechanism usually has multiple DOFs. To reduce the DOFs of the whole mechanism, synchronous joints are often added at the nodes shared by two mechanism units,
Fig. 19
and a typical synchronous joint is shown in Fig. 25. The diagonal support mechanism and the synchronous joint can also be used in the 6R, 2R + RRP and 3R + RRP mechanisms shown in Fig. 20. The scissors 7R mechanism has the diagonals and only one DOF, thus it does not need the diagonal support mechanism and synchronous joints. Although the diagonal support mechanism and the synchronous joints can be added to the initial principle mechanisms shown in Figs. 21–23 during the structural design, they cannot change the topologies of the initial principle mechanisms and have no effect on the deployable and foldable properties of these mechanisms. Since this paper focuses on the analysis and synthesis of deployable mechanisms for ring truss antennas from the perspective of mechanism principles and deployable properties, the structure design is not analyzed in detail here. 6.2. Deploying process simulation and verification of the constructed ring truss deployable mechanisms Based on the five selected basic unit mechanisms and their combinations shown in Figs. 20–23, five ring truss deployable mechanisms can be constructed. To verify the correctness of the above theoretical analysis, the simulation models of the five ring truss deployable mechanisms are built in Solidworks software and Matlab software to simulate the deploying process.
Scissors 6R and 7R mechanism.
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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Fig. 20
Fig. 21
11
Selected five basic units.
Three combinations of the selected basic unit mechanisms.
Fig. 22
Combination of the 2R + RRP mechanisms.
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Fig. 23
Fig. 24
4R1P mechanism.
Fig. 25
Synchronous joint.
Combination of the 3R + RRP mechanism.
The number of the ring truss deployable mechanism units is 12, and the length of the unit and the height of the ring truss after full deployment is 400 mm and 600 mm, respectively. The five ring truss deployable mechanisms have the same profile, as shown in Fig. 26. Although the profiles of the five ring truss deployable mechanisms are the same after full deployment, their structural forms are different in the fully folded state and the deploying process due to different mechanism configurations. The deploying process simulations of the five ring truss deployable mechanisms carried out in Solidworks software are shown in Fig. 27. Numerical simulations of the deploying processes are conducted in Matlab software, and the deploying processes are shown in Fig. 28. The five ring truss deployable mechanisms shown in Figs. 27 and 28 complete the deploying process with the desired movement during the simulation. Through these figures, the correctness of the synthesis process of the ring truss deployable antenna mechanism based on the constraint-synthesis method can be verified. 6.3. Mechanism characteristics analysis The ring truss deployable mechanisms shown in Figs. 27 and 28 have simple structures, and therefore can all be used to
Fig. 26
Profile of the fully deployed ring truss mechanism.
the support mechanism for ring truss antennas. From these two figures, we can see that they have different characteristics, which are listed as follows. (1) When the whole mechanism folds, the height of the ring truss deployable mechanism constructed by 4R mechanisms is higher than the others. Consequently, it will occupy a larger space in storage and transport. (2) Relative to the other mechanisms shown in Figs. 27 and 28, the mechanism constructed by 4R mechanisms has fewer joints. Therefore, the joint clearance is smaller than the other mechanisms, rendering it a higher motion accuracy and relatively higher structural stiffness than the other mechanisms. (3) The ring truss deployable mechanisms constructed by 4R, 6R and scissors 7R mechanisms are all constructed by rods and R joints. In the space environment, the R joint has higher movement flexibility than the P joint. (4) The ring truss deployable mechanism constructed by scissors 7R mechanisms has one DOF; therefore, during the deploying process, it only needs one input drive.
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Type synthesis of deployable mechanisms for ring truss antenna based on constraint-synthesis method
Fig. 27
13
Deploying processes in Solidworks software.
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Fig. 28
Deployment processes in Matlab software.
However, all the other mechanisms have multiple DOFs, and therefore need synchronous joints added to the nodes to ensure the synchronism of the deployable movement. (5) Relative to the other mechanisms shown in Figs. 27 and 28, the ring truss deployable mechanism constructed by scissors 7R mechanisms has triangular support struc-
tures when fully deployed, rendering it a higher structural stiffness than the other mechanisms which need an additional diagonal support mechanism when applied in engineering. The above analyses describe the characteristics of the five ring truss deployable mechanisms shown in Figs. 27 and 28.
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Fig. 29
15
Prototype of the ring truss deployable mechanism constructed by scissors 7R mechanisms.
In practical engineering applications, we can weigh the advantages and disadvantages of these mechanisms according to specific situations, and choose the most suitable mechanism. From the perspective of fewer DOFs and flexibility movements, we chose the one with scissors 7R mechanisms to manufacture a prototype to verify the previous analyses, as shown in Fig. 29. Fig. 29 shows the fully folded state, half folded state and the fully deployed state of the ring truss deployable mechanism constructed by scissors 7R mechanisms, and the previous analysis can be verified. In future work we will carry out performance analysis and evaluation for the ring truss deployable mechanisms obtained in this paper, and comprehensively consider the engineering practice requirements when conducting the structural design and optimization for these ring truss deployable mechanisms. 7. Conclusions (1) Based on the constraint-synthesis method, the type synthesis of deployable mechanisms for ring truss antennas was conducted in this study, providing good reference for the design of other deployable mechanisms. (2) The structure and the folding principle of the ring truss deployable mechanism were analyzed and the mechanisms for the basic unit edges were obtained through two types of node movements. A plurality of unit mechanisms were obtained and five unit mechanisms with simple structure and a small number of joints were selected. (3) Five ring truss deployable mechanisms were constructed based on the selected unit mechanisms, and the deploying features of these mechanisms were verified by the simulation of the deploying processes. (4) Different characteristics of the five ring truss deployable mechanisms were analyzed, and a prototype based on the scissors 7R mechanisms was manufactured, whose deploying process verified the previous analysis in this paper. (5) The mechanisms obtained in this paper are the initial principle configurations, and the structural design and optimization analyses will be carried out in further research.
Acknowledgements This study was co-supported by the National Natural Science Foundation of China (No. 51675458), the Key Project of Natural Science Foundation of Hebei Province of China (No. E201720335) and the Youth Top Talent Project of Hebei Province Higher Education of China (No. BJ2017060). References 1. Li TJ, Jiang J, Deng HQ, Lin ZC, Wang ZW. Form-finding methods for deployable mesh reflector antennas. Chin J Aeronaut 2013;26(5):1276–82. 2. Fowler PW, Guest SD, Schulze B. Mobility of a class of perforated polyhedral. Int. J. Solids Struct. 2016;85–86:105–13. 3. Qi XZ, Huang HL, Miao ZH, Li B, Deng ZQ. Design and mobility analysis of large deployable mechanisms based on planesymmetric bricard linkage. J Mech Des, Trans ASME 2017;139(2) 022302. 4. Nie R, He BY, Zhang LH, Fang YG. Deployment analysis for space cable net structures with varying topologies and parameters. Aerosp Sci Technol 2017;68:1–10. 5. Chen Y, Feng J, Sun QZ. Lower-order symmetric mechanism modes and bifurcation behavior of deployable bar structures with cyclic symmetry. Int. J. Solids Struct. 2018;139–140:1–14. 6. Cao WA, Yang DH, Ding HF. Topological structural design of umbrella-shaped deployable mechanisms based on new spatial closed-loop linkage units. J Mech Des, Trans ASME 2017;140(6) 062302. 7. Xing ZG, Zheng GT. Deploying process modeling and attitude control of a satellite with a large deployable antenna. Chin J Aeronaut 2014;27(2):299–312. 8. Ding XL, Li X, Xu K, Yang QL, Pu HL. Study on the behavior of solar array deployment with root hinge drive assembly. Chin J Aeronaut 2012;25(2):276–84. 9. Roovers K, Temmerman ND. Deployable scissor grids consisting of translational units. Int J Solids Struct 2017;121:45–61. 10. Yang GG, Duan BY, Du JL, Zhang YQ. Shape pre-adjustment of deployable mesh antennas considering space thermal loads. Proc Inst Mech Eng Part G J Aerosp Eng 2018;232(1):143–55. 11. Tanaka H, Shibutani Y, Lzumi S, Sakai S. Planar mobility modes of 8-bar-jointed structures with a single degree of freedom. Int J Solids Struct 2012;49(13):1712–22. 12. Liu RW, Guo HW, Liu RQ, Wang HX, Tang DW, Song XK. Shape accuracy optimization for cable-rib tension deployable antenna structure with tensioned cables. Acta Astronaut 2017;140:66–77.
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