A large ring deployable mechanism for space satellite antenna

Aerospace Science and Technology 58 (2016) 498–510

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A large ring deployable mechanism for space satellite antenna Xiaozhi Qi a,b , Hailin Huang a,b,∗ , Bing Li a,b,∗ , Zongquan Deng a,c a b c

State Key Laboratory of Robotics and System (HIT), Harbin, 150001, PR China Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, 518055, PR China School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, 150001, PR China

a r t i c l e

i n f o

Article history: Received 23 June 2016 Received in revised form 31 July 2016 Accepted 13 September 2016 Available online 20 September 2016 Keywords: Deployable mechanism Space antenna Kinematic analysis Driving mode Deployment test

a b s t r a c t With the development of space technology, new demands for satellite communications services, space and earth observations drive the requirement for large aperture space antennas. In order to meet the missions, this paper presents a kind of novel single mobility deployable ring mechanism based on a set of planar six-bar linkages. The ring mechanism has very high deploy/fold ratio, which is suitable for building large scale satellite deployable antenna. The mobile assembly of the ring structure and synchronous movement compatibility conditions are investigated, then the synchronization movement of the six-bar linkage modules can be realized by using close-loop cable and dual slider-crank mechanisms, which ensure the single mobility of the whole ring mechanism. Two types of driving mode for the deployable ring mechanism including cable driven and torsion spring driven are studied, and the high rigidity driving joints are designed. Finally, a ground experimental prototype of a 3.9 meter in diameter is fabricated to show the feasibility of the proposed mechanism, the deployment accuracy are also tested and the results show the good repeatability for the ring deployable mechanism. © 2016 Elsevier Masson SAS. All rights reserved.

1. Introduction Deployable mechanism refers to a kind of mechanism which can be transformed from a compact folded state to an anticipatory deployed form, and can become a complete stable structure with adaptive capacity of supporting loads [1,2]. Due to the good performance at space applications, deployable mechanisms are widely used in building large space structures, such as deployable mast, deployable antenna, and so on, and play a significant role in space missions such as earth observation, telecommunications, scientific researches etc. Because of the high storability and light weight, the deployable antenna with flexible cable net is one of the highly desired antennas in aerospace applications [3,4]. Many efforts, including innovative design concepts, analysis methods and experiments, were contributed to enhance the deployable antennas technology. Large deployable antennas have been built for space missions with different structural schemes, however, most of them can be classified as radial structures (e.g. Lockheed’s WrapRib antenna, Harris’ Rigid-Rib and ESA’s MBB antenna HingedRib), modular structures (e.g. JAXA’s ETSVIII, Tashkent’s KRT10, OKB-MEI’s TKSA-6) and peripheral truss (e.g. Northrop-Grumman’s

*

Corresponding authors at: Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, 518055, PR China. E-mail addresses: [email protected] (X. Qi), [email protected] (H. Huang), [email protected] (B. Li), [email protected] (Z. Deng). http://dx.doi.org/10.1016/j.ast.2016.09.014 1270-9638/© 2016 Elsevier Masson SAS. All rights reserved.

AstroMesh, Harris’ hoop-truss and hoop-column, GTU’s MIR reflector experiment, ESA’s LDA) [5,6]. Among these, the peripheral ring antennas with the advantage of self-synchronization, high thermoelastic stability and deployment reliability is investigated for the past few years. Ever since early 1980s for the Soviet space programs originated at GTU, researches into several pantograph ring concepts and associated technologies have been made [7,8]. In 1999 a 5.5-m peripheral pantograph structure with radial tensed membrane ribs was flown and deployed on the MIR station. Pantographs and derived linkages have been extensively investigated at GTU, TUM and other research groups [9]. On the other hand, double rings structure, such as Cambridge’s Deployable Mesh Reflector, it consists of two peripheral and concentric pantograph rings with different heights radially connected by a third set of pantograph pairs, in spite of the apparent complexity, the overall mechanism has single mobility and good stiffness and accuracy [10]. Chinese researchers have proposed two types of doublering deployable truss concepts which are based on parallelogram mechanism, structural stiffness are verified by prototype experiments [11]. Astro Aerospace Corporation developed one kind of AstroMesh deployable antenna, which consists of two symmetrical parabolic cable nets, one metal mesh reflector surface and one deployable ring mechanism, they have over 15 years of continuous development history [12,13]. JAXA has also developed two truss antennas with the diameter of 19 meters, which works for the satellite communication service with engineering test satellite

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ETS-VIII launched in December 2006. The antenna was supported by 14 hexagonal truss modules with a diameter of 4.9 meters, and each module contains of mesh surface, cable network and deployable structures composed of six basic deployable units [14]. Research team led by L. Datashvili developed the concept of double pantograph based peripheral ring [15], which satisfies the demand of significantly less mass and smaller folded volume while maintaining stability and deployment reliability. Apart from the above mentioned design concern, our research group in HIT has done some efforts about large deployable mechanism, such as synthesis of deployable mechanism [16], mobility analysis [17], optimization design [18], cable net form-finding [19] and so on. This paper introduces our recent work on the novel design concept of large mesh deployable antennas. In order to meet the demands of satellite communications services and earth observation missions, one kind of the ring mechanism is presented, which is composed of multiple deployable modules of six-bar linkage. The ring mechanism has the very good deploy/fold ratio, which can be used to build large scale satellite deployable antenna. The degree of freedom of the mechanism is one, and the deployment process can be controlled by the cables. Since there is no use of gears and other big joints, the mass of the overall mechanism is very light. The rest part of this paper consists of the following five sections. Section 2 presents the design concept of the basic deployable unit, and the geometric modeling of the ring deployable mechanism is studied. The synchronous movement compatibility conditions are presented and the kinematics analysis of ring deployable mechanism is investigated in Section 3. The detailed designs consisted of deployment driving modes and driving joints are developed in Section 4. One ground experimental prototype of a 3.9 meter in diameter is fabricated and the deployment accuracies are tested in Section 5. Finally, Section 6 summarizes the work of this paper and puts forward the suggestions for future work. 2. Proposal of ring deployable mechanism For mesh reflector antenna, it is mainly made up of flexible cable net and support truss. The front net of flexible cable net system which is an approximate parabolic is the working surface of antenna, and the precision of the surface shape directly determines the performance of the antenna. The support truss system is to drive the cable net system smoothly to the working configuration, which could ensure the cable mesh reflector meet high profile accuracy requirements. The precision of cable net reflector is affected by multiple factors, with the rising of the antenna size, the building of flexible cable network to meet high profile accuracy is very difficult, it is desired to design the space truss system which has high rate of collapse, high reliability and the light mass, in order to better support the cable mesh reflector and increase the precision of its surface shape. In this paper, a module with planar six-bar mechanism as the basic of large ring expansion mechanism is proposed; it can be used to build the support structure of large-scale network reflectors. 2.1. Basic deployable unit As shown in Fig. 1, a six-bar linkage consists of two vertical rods and four short rods, all the rods are connected by rotational joints and a single closed loop of the linkage can be formed. The lengths a, b, c , d, e , f of six rods within the mechanism satisfy the following geometrical relationship,

⎧ ⎨a=d b=c=e= f ⎩ b ≤ a /2

(1)

499

Fig. 1. Schematic diagram of six-bar linkage.

From Fig. 1, when the unit is folded, the four short rods BC , C D and A F , E F are parallel with each other. Then, two vertical rods A B, D E on both sides move towards left and right respectively, the joints C , F of short connecting rod move up and down respectively. When short connecting rods rotate to horizontal position and are locked, the unit could achieve maximum deployable configuration, similar to a rectangle. The two adjacent elements can be connected by sharing vertical rod together, in order to achieve kinematic synchronicity, two double-crank slider mechanisms are installed in both ends of the vertical supporting rods as shown in Fig. 2. Due to sharing the vertical supporting rod, the slider and the prismatic pair, the short rods of the adjacent units can realize the synchronous deployment or folding. Multiple basic units can be assembled through the same method and a ring expansion mechanism can be built. 2.2. Geometric modeling of ring deployable mechanism Due to the fact that the paraboloid can focus a lot of parallel signals on one point, so the parabolic form or part of approximate paraboloid fitted with spherical surface is used as the working surface of the satellite antenna, and the satellite signal receiver is located at the focus of the paraboloid. According to the different positions of the rotary axis of the paraboloid and the antenna working surface, the satellite antenna can be divided into two types, prime focus antenna and offset antenna, as shown in Fig. 3. The prime focus antenna has circular contour, and the offset antenna has elliptic contour. Generally, the multiple units are designed with the same geometric parameters to construct the ring deployable mechanism, the supporting truss of prime focus antenna’s is round, and the basic assembly principle adopts regular polygon to approximate a circle. The design parameters are simple, and the angles between the adjacent units are equal. As shown in Fig. 4, a regular polygon is inscribed in a circle, the number of edges is n, α is the angle between the adjacent edges, β is the exterior angle of polygon, R is the radius of antenna, b is the length of the short connecting rod on the surrounding truss antenna. Then, the angle between the adjacent units can be gained as follow,

α = 180 − β = 180◦ − 360◦ /n

(2)

As the antenna reflector cable network is compos of approximate triangles, the number n of the units in the circular truss is the integer times of 6. By Eq. (2), we can get the angles between the adjacent units in the circular truss corresponding to different number n of edges. When the number n of edges is determined, the angles of the connection joints between the adjacent units can be gained. According to Fig. 4, the diameter D of antenna can be obtained,

D = 2R =

2b sin(β/2)

=

2b sin(180◦ /n)

(3)

From Eq. (3), it can be seen that the deployment diameter of the ring mechanism is directly proportional to the length b of the short connecting rod, and the deployment diameter will increase with the increase of the number n of units.

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Fig. 2. Deployment process of two units.

Fig. 5. Angle between chords inscribed inside of elliptic arc.

Fig. 3. Two types of antenna surface.

Fig. 4. Regular polygon inscribed in the circle.

For the large offset antenna, the outer contour of its supporting truss is ellipse, so the angles between each side in the polygon inscribed inside of the ellipse are not equal, the calculation is relatively complex. We take the dodecagon inscribed inside of the ellipse as an example to explain the calculation process. Due to the elliptical symmetry, a quarter of elliptic arc can be chosen to analyze, as shown in Fig. 5. Firstly, the coordinates of each node which satisfy standard equations of elliptic arc must be determined, and the chord lengths are known equal, then, we can get the following equations,

⎧ ⎨ | A0 A1| = | A1 A2| = | A2 A3| 2 2 ⎩ x i + y i = 1 ( i = 0 ∼ 3) 2 2

a

(4)

b

Eq. (4) could obtain the only one solution, the coordinates of each node can be obtained. Then the different angles between the chords shown in Fig. 5 can be solved by the following equations,

⎧ y1 ⎪ α0 = arctan ⎪ ⎪ a − x1 ⎪ ⎪ ⎪ ⎪ ⎪ | A A |2 − | A 0 A 1 |2 − | A 1 A 2 |2 ⎪ ⎪ α1 = arccos 0 2 ⎨ 2| A 0 A 1 || A 1 A 2 | ⎪ α = arctan x2 ⎪ 3 ⎪ ⎪ b − y2 ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎩ α2 = arccos | A 1 A 3 | − | A 1 A 2 | − | A 2 A 3 | 2| A 1 A 2 || A 2 A 3 |

(5)

Fig. 6. Ring mechanisms for different number n of units.

Because of symmetry, the whole angles between adjacent units within elliptic contour can be obtained. According to the different angles, the different connection joints can be designed. Fig. 6 demonstrates the apertures of multiple ring mechanisms for different number n of units. The up and down ends of the multiple vertical supporting rods in the ring mechanism can be connected to the flexible cable net structure, and they are used to support the cable mesh reflector, as shown in Fig. 7. In the same store height of the case, the ring mechanism built by the units presented above has larger diameter; similarly, compared with the existing mechanism, building the antenna with the same deployable area by these units can reduce the number of units effectively. 2.3. Evaluation indexes In order to better evaluate the performance of the mechanism in storage volume and deployment area, three evaluation indexes of deployable mechanism are presented, as shown in Fig. 8:

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Fig. 9. Schematic diagram of AstroMesh antenna truss.

Fig. 7. Deployable mesh reflector antenna.

Fig. 8. Evaluation parameters of ring deployable mechanism.

Height-Ratio, the ratio between furl height h and antenna diameter D, namely HR = h/ D; Diameter-Ratio, the ratio between furl diameter d and antenna diameter D, namely DR = d/ D; Volume-Ratio, the ratio between furl volume v and the largest deployment envelope volume V , namely VR = (d2 h)/( D 2 H ). The coverage area of ring mechanism deployed is similar to round, the antenna diameter D can be calculated by Eq. (3), and the furl height h and deployment height H are consistent, which is about two times of the length b of the short connecting rod, namely, h = 2b. Suppose that when furled, the rods are circularly arranged in parallel, the diameter of the rods is d , each unit contains an average of five rods, so the maximum envelope diameter d can be approximately calculated as follows,

d=

5d

(6)

sin(180◦ /n)

The mechanism’s Height-Ratio, Diameter-Ratio and VolumeRatio can be calculated separately,

⎧ ⎪ ⎪ HR = ⎪ ⎪ ⎪ ⎪ ⎨ DR = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ VR =

h D d

= sin(180◦ /n) =

5d

D 2b d2 h 25d 2 D2 H

=

(7)

4b2

From Eqs. (7), it can be seen that the mechanism’s HeightRatio decreases with the increase of side number n, and DiameterRatio and Volume-Ratio have constant values, don’t change with the change of side number n. Nowadays, the successful case as ring truss antenna in the world is AstroMesh antenna of America, whose basic unit is the parallelogram mechanism, as shown in Fig. 9. By using the geometrical parameters shown in Fig. 9, the three evaluation indexes of AstroMesh antenna can be calculated, respectively,

⎧ (a + b1 ) sin(180◦ /n) ⎪ ⎪ ⎪ HRAstromesh = ⎪ b1 ⎪ ⎪ ⎪ ⎨ 4d DRAstromesh = b1 ⎪ ⎪ ⎪  2 (a + b ) ⎪ ⎪ 16d 1 ⎪ ⎪ VRAstromesh = ⎩ 2 ab1

(8)

Fig. 10. Evaluation indices with side number n.

Under the condition of assuming that the furled height is determinate, the three evaluation indexes can be compared between the mechanisms presented this paper and AstroMesh antenna mechanism, as shown in Fig. 10. It can be seen that the three indexes of mechanism presented this paper are better than that of AstroMesh antenna. When the antenna store height is certain value, the mechanism can be used to build antenna with larger diameter and smaller store volume. The proposed ring mechanism is composed of multiple deployable modules of six-bar linkage, which is very simple at the aspect of structure. By using close-loop cable and dual slider-crank mechanisms, the mechanism could realize deployment synchronization. AstroMesh antenna mechanism contains synchronization movement by using multiple conical gear pairs, which increase the system weight and manufacturing difficulty. Our proposed mechanism is simpler than AstroMesh antenna. Due to parallelogram mechanism, AstroMesh antenna, is accomplished by changing the lengths of ring leading to the increase in the transport package length. This feature considerably limits its application area, particularly when the reflector antenna size is large. However, the transport package length of our proposed mechanism has no change during the deployment process. Therefore, it has very good deploy-fold ratio, which can be used to build large scale satellite deployable antenna. 3. Kinematics analysis 3.1. Geometry compatibility conditions of synchronous movement For the ring deployable mechanism composed of multiple closed loops, the mechanism contains multiple internal bars, kinematic pairs and flexible cable units, thus the basic unit module from the deployable mechanism will be studied in detail to solve the kinematic problem. As shown in Fig. 11, the unit module consists of two vertical rods, four chord rods, four crank connecting rods and four sliders. Fixing the vertical rod in the left side, then, the unit contains 13 moving parts and 18 motion pairs, and the degree of freedom about the unit module mechanism can be obtained as follows:

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Fig. 13. Synchronization cable system.

Therefore, the module only has one form of motion and the sliders from two vertical rods must be synchronized. Because the adjacent modules share two sliders and one vertical rod, all sliders must be synchronized. 3.2. Kinematic model

Fig. 11. Schematic diagram of unit module.

F = 3n − 2P L = 3 × 13 − 2 × 18 = 3

(9)

From Eq. (9) we know that the degree of freedom for the unit module is 3, that means there exist three independent joint variables of four parameters as shown in Fig. 11. Therefore, its movement process is not determinate, and could appear a variety of motion configurations, which is difficult to guarantee the motions of two vertical rods horizontally. In this paper, three joint variables α1 , α2 , α3 are selected to analyze the four configurations in the deployment process of mechanism, as shown in Fig. 12, where the crank connecting rods and the sliders are omitted. We can find that only when it meets α1 = α2 = α3 , the mechanism could achieve expected kinematic configuration. So two constraints must be introduced to the mechanism, and the degree of freedom can become 1. The translational motion between two vertical rods can be achieved. It must ensure that the angles between the chord rods and the vertical rod are equal, namely, the geometry compatibility conditions ensuring the mechanism deployed stably are given as follows:



α1 = α2 α3 = α4

(10)

In order to satisfy the geometry compatibility conditions, it must ensure that the two sliders on the vertical rod move to the opposite direction at the same time, and their speeds must be equal. In this paper, the synchronous movement of the two sliders is achieved by using the synchronization cable system, as shown in Fig. 13, a closed loop cable is passed through pulleys at the ends of the vertical rod A B, and is fixed in the two sliders C and D, respectively. When the slider C moves to the left, it drives the cable C B–B D move to counterclockwise direction, and drives the slider D to move to the right at the same time, obviously, the displacement variations of two sliders are equal, namely C =  D . Similarly, when the slider D moves to the right, it drives the cable D A– AC move, the slider C will move to the left at the same time. Through using a closed loop cable and two fixed pulleys, the synchronous movement of sliders on the vertical rod can be achieved, at the same time, the geometry compatibility conditions are satisfied. There will exist only one independent joint variable of four parameters α1 , α2 , α3 , α4 , then its degree of freedom becomes one.

Two constraints are introduced in one basic unit through using synchronous cable system, which make the sliders in each vertical rod move along the opposite direction synchronously. Then, the mechanism has only one translational degree of freedom. Because the slider-crank mechanisms within the unit have influence of guaranteeing synchronization movement between the adjacent units, for the purpose of the analysis of function relationship between input and output parameters of the mechanism, the slidercrank mechanisms can be simplified as the constraint conditions into the six-bar mechanism. Then, such analysis is relatively simple, and it does not have any effect on the analysis results. Firstly, the global coordinate system O − X Y Z is set up as shown in Fig. 14(a), Z direction is along the vertical rod upward, Y direction is along the diameter direction of ring mechanism and pointed to the center O , X direction is determined by the right-hand rule. Each vertical rod is labeled counter-clockwise from 0 to 11, at the same time, the local coordinate system i − xyz is established, y direction of each system is pointed to the center O , all the z axes are upward, x axes are determined by the right-hand rule. Only there are translation and rotation around z axis between adjacent coordinate systems, the rotational angle is the exterior angle β of the adjacent units. Fig. 14(b) is the local coordinate system in a single module, according to the geometry relation in the figure, we can obtain the coordinates of joints A, B, C , D, E, F in the local coordinate system,

  ⎧ A= 0 0 0 ⎪ ⎪   ⎪ ⎪ ⎪ B= 0 0 a ⎪ ⎪ ⎪ ⎨ C =  b sin(α ) cos(β/2) b sin(α ) sin(β/2) a − b cos(α )    (11) ⎪ D = 2b sin(α ) cos(β/2) 2b sin(α ) sin(β/2) a ⎪ ⎪ ⎪ ⎪ E =  2b sin(α ) cos(β/2) 2b sin(α ) sin(β/2) 0  ⎪ ⎪ ⎪   ⎩ F = b sin(α ) cos(β/2) b sin(α ) sin(β/2) b cos(α ) The homogeneous transformation matrix between the adjacent coordinate systems is as follows,

⎡

cos(β) ⎢ sin(β) T=⎢ ⎣ 0 0

⎤ − sin(β) 0 2b sin(α ) cos(β/2) cos(β) 0 2b sin(α ) sin(β/2) ⎥ ⎥ ⎦ 0 1 0 0

0

Fig. 12. Four configurations of mechanism with different geometric conditions.

1

(12)

X. Qi et al. / Aerospace Science and Technology 58 (2016) 498–510

503

Fig. 14. Coordinate system of the ring mechanism.

Fig. 16. Movement trajectory of ring deployable mechanism. Fig. 15. Contour trajectory of the ring mechanism.

By using Eq. (12), all the transformation matrices between the local coordinate system and global coordinate system can be obtained, for example, the homogeneous transformation matrix Ti from the i-th coordinate system to the global coordinate system is, i

  

Ti = T . . . T = Ti

(13)

The joint space coordinates in each unit can be transformed to the global coordinate system, the relation between the input and output parameters in the ring mechanism can be quickly solved. 3.3. Simulation example Take the ring mechanism with twelve units as an example, the kinematics analysis is performed. The length a of vertical rod is 1000 mm, the length b of short connecting rod is 500 mm, the input parameter of deployment process satisfies α = ωt, where ω = π /180 rad/s. So the mechanism can be fully deployed in 90 s, the input angle α changes 1◦ per second. The motion trajectory of the vertical rods and the outer contour variation can be obtained, as shown in Fig. 15, it can be seen that the ring mechanism completely symmetrically deploy, and all the vertical rods move along certain fixed straight lines. According to this rule, the plane orbits of the suspension experiment device can be determined. Fig. 16 is the space movement trajectory of the ring mechanism, from which we can find that the deployment diameter is close to 4 m.

Fig. 17. CAD model of ring deployable mechanism composed of twelve basic units.

Fig. 17 is the CAD model of ring deployable mechanism composed of twelve basic units, the mechanism can deploy from the furled configuration to a ring structure. Because of the physical deviation of the rods and the joints, its deployment diameter is

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Fig. 18. Arrangements of cable-driven mode.

4040 mm, the furled height is 1020 mm, the furled diameter is 420 mm, then HR is 0.252, and DR is 0.104, VR is 0.018, which are very close to the theoretical values. So we can conclude that bigger deployment area can be constructed by increasing the unit numbers. 4. Drive design of deployable ring structure Ring deployable mechanism is composed of multiple internal rods, multiple kinematic pairs and multiple closed loops, which requires multi-point drive at the same time in the deployment process, otherwise, it is difficult to meet the requirements of synchronous movement and steady expansion. Because the deployable mechanism works in the space environment, which limits the quantity and the mass of the motor drive system, also it requires drive system as simple and light as possible. To drive the deployable mechanism is relatively important issue, when an antenna support structure scheme was determined. The drive design of the antenna mainly contains deployment driving mode and driving joint design. 4.1. Deployment driving mode In this paper, two kinds of driving mode suitable for space ring deployable mechanism are presented, cable-driven mode and torsion spring driven mode. No matter which kind of driving mode, it must ensure that each unit of deployable mechanism is deployed synchronously. As shown above, the adjacent units keep the synchronicity through the double crank-slider mechanisms, and for a single module, it is realized by arranging slider synchronous cable system on the vertical support rod. (1) Cable driven mode. Cable-driven mode is that the mechanism deployment only rely on the cable driven by the motor, no other energy is needed. The cable drive system includes motor, cables, fixed pulleys at the ends of vertical support rods, movable pulleys consolidated with the slider, and displacement compensation spring, etc. As shown in Fig. 18, the cable-driven system includes two drive cables, one end of the cable is fastened to connect with vertical support rod by connecting with the compensation spring, another end is connected to the hub installed on the motor. The cables, which are driven by the motor, move along the direction of arrows as shown in Fig. 18, and make two sliders move along the vertical support rod so that both ends can move up and down, then the short rods are smoothly deployed through the slider-crank mechanism. The double crank-slider mechanisms and the slider synchronous cable system guarantee the movement synchronicity of the ring deployable mechanism. In order to avoid the cable appears slack phenomenon, the compression springs, which provide some driving resistance, are installed in the position between slider and the vertical rod ends. The displacement

Fig. 19. Arrangements of torsion spring driven mode.

compensation spring is arranged at the end of cable in case of the cable tension increases suddenly at the moment when the mechanism is fully expanded, which play a role of overload protection for the motor. After the mechanism is fully expanded, the joints are locked, at the same time, the motor imposes certain tension on the cable, then the cable is also locked. When the deployed mechanism is locked, the cable could keep certain tension, and the displacement compensation spring can avoid the cable relaxation as the time changes. The motor output torque is transformed to multiple driving forces, which overcome internal frictions of kinematic pairs and provides prestressing force for cable net structure. Cable-driven mode has high stability and reliability, and the deployment speed is easy to control; the disadvantages are that with the increase of deployment angle, the motor should produce more and more driving torque to overcome the resistance, therefore, more external energy is needed. (2) Torsion spring driven mode. Torsion spring driven mode is that the mechanism deployment replies on the elastic potential energy of the built-in torsion spring mounted in the joints, at the same time, the development rate is controlled through cable which is released by the drive motor. The torsion spring driving system includes motors, cables, fixed pulleys, torsion springs installed in rotary joints and displacement compensation spring, etc. As shown in Fig. 19, the constant torque springs are respectively installed in the rotational joints A, B, C , D, and both ends of the spring are respectively fixed on the rods connected with the rotational joints. Two cables are interlaced arranged as shown in the figure, and one end of the cable is fastened to connect with vertical support rod by connecting with the compensation spring, another end is connected to the hub installed on the motor. When the mechanism is in the folded state, the torsion springs in the compression state have the largest output torque. When the furled mechanism is unlocked, the short connecting rods are deployed under the action of the constant torque springs, at the same time, the motors begin to release the cable which runs along the direction of arrows as shown in Fig. 19 to control deployment rate, in order to avoid the collision phenomenon caused by the impact of the large torsion springs. When fully expanded, the rotational joints are locked, and the cables are tensioned by motor reversal to increase the truss’s internal tension and enhance the overall stiffness of the truss. Torsion spring driven mode depends entirely on the energy stored in torsion springs, which deploy the mechanism. So the design requirements of the torsion spring are very high, which request that the torsion provided by the spring is always greater than the friction resistance of the kinematic pairs in the whole deployment process, and the angular displacement output is greater than 180◦ to make the mechanism fully expanded. Meanwhile, the output torsion of the spring is not too large, on the one hand, the mass of spring is increased, on the other hand, the impact to

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Fig. 22. Equivalence principle diagram of torsion springs.

lease cable is longer by torsion spring driven mode, but the motor drive torque is small and does less work. To sum up, each drive modes has its merits. 4.2. Driving joint Fig. 20. Cable arrangements scheme of two kinds of drive modes.

Fig. 21. Tendencies of cable length with change of deployment angle.

the motor and the vibration phenomenon could appear. Reasonable design of torsion spring can reduce the energy needed and less demanding for the motor. For above two kinds of driven modes, the variation tendencies of the cable length with the change of deployment angle are different. The variation rule of cable length is mainly studied for a single module. Fig. 20(a) is the cable arrangement scheme of cable-driven mode, and Fig. 20(b) is the cable arrangement scheme of the torsion spring driven mode. Two driven modes both adopt two cables, the cable length in cable-driven mode is L 1 , the cable length in torsion spring driven mode is L 2 . By the geometric relationships in Fig. 20, their expressions can be obtained as follows:





L 1 = 2b + 2c cos α + 2 d2 − c 2 sin2 α

√

L 2 = 2 a2 + b2 − 2ab cos α

(14)

where, a refers to the distance between two fixed pulleys of vertical support rod; b refers to the length of the short connecting rod; c stands for the length of the crank; d stands for the length of crank connecting rods; α represents the deployment angle. Let a = 1000 mm, b = 500 mm, c = 100 mm, d = 150 mm, we can obtain the variation tendencies of cable length with the change of deployment angle under two kinds of drive modes, as shown in Fig. 21. It can be seen that the cable length L 1 is slowly decreased with the change of the angle α from 0◦ to 90◦ , which illustrates that the cable is tightening; however, the cable length L 2 is fast increased, which illustrates that the cable is released. For the same geometry parameters of the mechanism, the range of cable length variation in driving mode 1 is much smaller than driving mode 2. The reason is that the slide block movement range is small by cable-driven mode, but the motor drive torque is bigger. The re-

In order to reduce the energy consumption of the motor, the elastic elements are installed in the space mechanism to drive the mechanism motion with the help of their stored elastic potential energy. The built-in spring is easy to install with light mass, which, on the one hand, reduces the number of the motor, on the other hand, improves the joint stiffness. In order to realize design optimization, the springs are not set in all revolute joints, and the finite springs are selected to achieve maximum driving effect. Firstly, the position of driving joint must be determined, and then according to the requirements of task, the geometric parameters of spring can be designed. For the mechanism mentioned in this paper, there mainly exist six rotational joints in a single module, as shown in Fig. 22, due to the symmetry, which can be classified only two kinds of joints, the first kind is four rotational joints A, B, D and E, which are connected to the vertical support rods, the second kind is two rotational joints C and F . Assuming that the vertical rod A B is fixed, then the vertical rod D E only occurs translational motion. The output torque effectiveness of the springs located in two types of joints can be equivalent to a force F on the vertical rod D E horizontally, so if the equivalent force F of spring torque τ in different positions can be solved, the optimal effect of the joint can be determined. According to the principle of virtual work, the equivalent force F 1 of the first kind of joint torque to the rod D E meets,

F 1 δ X = τ δα

(15)

It can be obtained as

F1 =

τ δα δX

=

τ δα τ = δ(2b sin α ) 2b cos α

(16)

Similarly, we can deduce that the equivalent force F 2 of the second kind of joint torque to the rod DE meets,

F 2 δ X = τ δ(2α )

(17)

It can be obtained as

F2 =

τ δ(2α ) δX

=

2τ δ α τ = δ(2b sin α ) b cos α

(18)

By comparing Eq. (16) with Eq. (18), it can be known that F 2 = 2F 1 , for the same torsion spring, the equivalent force produced by four torsion springs installed in joints A, B, D and E is equivalent with that produced by two torsion springs installed in joints C and F . But the second category of the joints only need two torsion springs, which decrease the number of spring, and the overall mass is relatively reduced, therefore the torsion springs are chosen to install to the rotational joint C and F . To determine the torque of torsion spring, the mission requirements of the mechanism must be sought. When fully expanded, the mechanism needs to provide enough tension T for cable network system back to the center of the ring truss. As shown in

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Fig. 23. Diagram of driving tension.

Fig. 23, the equivalent force F 2 produced by a single module is in the plane of the unit and along the outside direction, the angle between the adjacent modules is θ , then tensioning force T can be generated by the geometric relationship obtained by Fig. 23,

T = 2F 2 cos(θ/2)

(19)

Substituting Eq. (18) into Eq. (19), it can be deduced that,

T=

2τ cos(θ/2) b cos α

(20)

By Eq. (20), it can be known that when the angle is close to 90◦ , the tensioning force torsion T produced by spring theoretically is close to infinity, which illustrates that the mechanism fully expanded with self-locked function. Therefore, the smaller torsion spring can be chosen to realize greater tension. The tensioning force produced by spring could make the cable net tensioning, completely. By Eq. (20), the torque τ of torsion spring can be determined as follows:

τ=

T b cos α 2 cos(θ/2)

(21)

According to Eq. (21), the total torque of the drive torsion spring can be calculated, then according to the layout position and quantity, the geometrical design parameters of a single torsion spring can be reasonable determined. The torque of the general torsion spring decreases as the deployment angle becomes large, which is difficult to meet the demand range of the deployment angle from 0◦ to 180◦ , therefore this article adopts the constant torque spring as the driving element. As shown in Fig. 24(a), constant torque spring is made by spiraling the sheet metal, only its outer end can be connected with other components. The torsion angle is very large, which can completely meet that the spring outputs the approximate constant torque with the scope of deployment angle from 0◦ to 180◦ . As shown in Fig. 24(b), the constant force spring is installed to the storage drum, which is in the free state, and the outer end is

fastened to the big torque drum by screw. In order to improve the torque, the reverse installation of constant torque spring can be applied as shown in Fig. 24(b). When the storage drum is in position A, more than half of the big torque drum surface is covered by the spring, which is in the stretch to the maximum configuration. Assuming that big torque drum is static and the distance O 1 O 2 between two drum centers is kept constant, the storage drum will turn in the counter-clockwise direction of arrows in Fig. 24(b) around the torque drum center O 1 under the action of spring torque. At the same time, the torsion spring occurs counter-clockwise rotation around the storage drum center O 2 , which tightens the spring. The storage drum transfers by intermediate position B to position C , the rotation range is 180◦ , and the constant torque spring has a torque acting on the driving joints which keep a certain rigidity. If the big torque drum is fixed to one rod and the storage drum is fixed to another rod, the rotation from 0◦ to 180◦ will appear between the two rods. The output torque of spring can be amplified or reduced by adjusting diameters of the two drums; the magnification is the ratio R /r of the two drums radius. In order to make the driving moment of the spring balance to the rod, generally, the symmetrical arrangement of two driving springs in both sides of the joint is adopted. So for a module of ring mechanism there exist four constant torque springs, the requirements of the driving moment can be meted through adjusting the geometric parameters of the spring. Fig. 25 is the 3D CAD model of a driving joint, the storage drum is installed to one rod and is able to free rotate, the axis of big torque drum which follows another rod rotation is collinear with the joint rotation. The threaded hole is made in the contour surface of the torque drum, which can be connected with one end of the spring. Two rods can rotate 180◦ by relying on the spring torque, the mechanical limit stops rotation when the rod turns to the position of 180◦ , at the same time, the driving joint has built-in torque and high stiffness, the contrarotation of the rod cannot appear. 5. Prototype and experiments In order to verify the feasibility of the ring deployable mechanism which is presented above, a ring deployable antenna prototype which is composed of 12 modules is built, as shown in Fig. 26. The deployable antenna aperture is 3.92 m, the furled diameter is 0.32 m, the height is 1.23 m, the ratio of deployed diameter and furled diameter is 12.25, the overall mass of the antenna is 11.5 kg. For reducing the overall mass of the antenna, each joint is made of aluminum alloy, the rod is made of carbon fiber tube. Torsion spring driven mode is adopted, the deployment rate is controlled

Fig. 24. Application of constant torque spring.

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Fig. 25. CAD model of driving joint.

Fig. 26. Prototype of deployable antenna.

by two releasing cables, which run through ring supporting mechanism. The pretension level of the front and rear cable net ranges from 10N to 20N, the pretension level of the longitudinal tension cable is little, which ranges from 5N to 10N. For mesh reflector antenna, it is mainly made up of flexible cable net and support mechanism. When the antenna is in a compact folded state, the flexible cable net is convolved in the centre of ring mechanism and is fastened by some elastic belts. The flexible cable net system is connected with the ends of vertical rods. When the antenna need being deployed, the support mechanism is to drive the cable net system smoothly to the working configuration, which could ensure the cable mesh reflector is an approximate parabolic. In the space, all parts of the antenna are floated, and the mechanism and the cable net system can be not interfered with each other. As it is difficult to simulate microgravity environment on the ground, this paper employs the way of installing universal wheel on the deployable mechanism to weaken the impact on gravity, owing to that each vertical rod only occurs relative translation motion during the development process of antenna mechanism. The universal wheel is mounted to the bottom of vertical support rod, due to that the rolling friction coefficient between the universal wheel and the ground is small, which can attenuate the influence of ground friction force caused by gravity. 5.1. Deployment functional verification test Deployment function test of deployable antenna is used to verify the feasibility of mechanism scheme. The test system consists of two parts, antenna prototype and motion control system. Motion control system mainly ensures the role of antenna’s steady deployment; its functions include the control of motor’s forward/reversal rotation and the rotational speed to achieve antenna’s unfolding and folding smoothly. Through adjusting the parameters of motors the motor output speed can be controlled. When the mechanism in the folded state, the output moment of the joint torsion spring is in maximum, which requires the cables

quick released to avoid the damage to mechanism from torsion spring at the unlocked instant. When the mechanism is unfolding in a certain extent, it needs to maintain steady movement with the stable releasing speed of the cable. When the mechanism is imminent deployed in place, at this time, the cable network is tensioning, then the tension of the releasing cable reduces, the releasing speed of the cable should be slow down in order to avoid cable slack. Thereby the releasing speed of the cable should be uniformly accelerative motion at first, and then uniform motion in the middle, uniformly retarded motion at last until the speed is at zero. After setting the motor parameters, the mechanism is unlocked and expands slowly on the slick ground; the deployment process is shown in Fig. 27. The results show that the mechanism can be driven by torsion springs to deploy successfully. 5.2. Precision test Because the deployable antenna reflector cable network is supported by surrounding truss, the location accuracy of the deployable mechanism indirectly determines the accuracy of the antenna reflecting surface. If the position of the key point on the mechanism has bigger deviation, it is hard to guarantee the antenna work in demanded precision, it is necessary to make the positioning precision test of the mechanism. 5.2.1. Test principle and scheme This experiment mainly measures the positioning accuracy of the mechanism motion, which needs to measure three-dimensional coordinates of each key point. At present, the equipment which has the ability of three coordinate measuring mainly includes 3D coordinate-measure-machine, total-station, laser tracker and photogrammetric system. The laser tracking measurement system has high precision, high efficiency, fast installation and user-friendly control, it is able to realize space moving target tracking and realtime 3D coordinate measurement. Therefore, the laser tracker is used to achieve 3D coordinate measurement of key points in this paper.

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Fig. 27. Deployment process of antenna mechanism.

Fig. 28. Labeling of 12 key points.

The cable net work surface of mesh reflector antenna is supported by the top end of the vertical supporting rods of the mechanism, so the space three-dimensional coordinates of the top key points of 12 vertical supporting rods are measured in the experiment when the mechanism is fully deployed. As shown in Fig. 29, the ring deployable mechanism can form a dodecagon after it is fully expanded, according to the clockwise direction, label the 12 key points from 1 to 12, the vertical support rod with the key point 1 is in a stationary state. Ten times of fold-and-deploy experiments were carried out to measure 3D coordinates of 12 key points. In each experiment, when the mechanism is fully deployed and the cable net is tensioning, the test data can be obtained by changing the position of the laser target in turn. Due to the external factors, such as motor vibration, the antenna mechanism could appear small overall movement or deflection in different times of experiments, so it is needed to find a way to eliminate the influence. As shown in Fig. 28, assuming that the laser tracker is stationary, all its measurement data is relative to the coordinate system O − xyz. The measurement data obtained in one process of experiment can be transformed into the coordinate system 1 − xyz from the coordinate system O − xyz, which could eliminate the measurement error caused by the fixed support rod movements. The measurement data after coordinate transformation can be used for the subsequent data processing and analysis. Fig. 29 is the experimental picture, and the laser target ball is installed to the mechanism through one custom-made connecting base. 5.2.2. Error analysis of the key points The error distribution of each module is studied by analyzing the side length error of dodecagon formed by the ring deployable mechanism. Firstly, the coordinates of the key points can be obtained by calculating the arithmetic mean of 10 groups of

Fig. 29. Test picture. Table 1 Errors of the distance between adjacent key points. Adjacent key points

Absolute errors (mm)

Relative errors (%)

Adjacent key points

Absolute errors (mm)

Relative errors (%)

1–2 2–3 3–4 4–5 5–6 6–7

0.407 1.329 0.419 1.199 0.416 0.736

0.041 0.132 0.042 0.119 0.041 0.073

7–8 8–9 9–10 10–11 11–12 12–1

3.301 0.416 0.868 0.973 0.898 0.555

0.329 0.041 0.086 0.097 0.089 0.055

measured data after coordinate transformation. Then the distance between adjacent key points is calculated to compare with the theoretical value in turns. The theoretical distance between adjacent points is 1004.64 mm when the mechanism is without pretightening cable network. The absolute errors and relative errors between experimental value and theoretical value are obtained, as shown in Table 1. It can be seen from Table 1, the absolute error of the distance between the adjacent points 7–8 is maximum, which is 3.301 mm and may be caused by incompletely deployment of corresponding module, but its relative error is only 0.329%. Overall, the absolute errors of the modules are in millimeter and the relative errors are very small, which show that the prototype has high machining and assembly accuracy. Some of the values are bigger, such as 2–3, 4–5 and 7–8, which may be due to the large pre-tightening cable net. The appropriate adjustments of related torsion spring and cable network prestress should be done to reduce the errors.

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Fig. 30. Deployment repeatability in x, y , z directions.

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Fig. 32. Deployment repeatability of diagonal.

lowest. It can be discovered that the closer to the y direction, the diagonal has the higher precision, on the contrary, the accuracy close to x axis is relative low. At three aspects of three directions, absolute space position and diagonal, the deployment repeatability of the ring deployable mechanism are analyzed, overall repeatability is less than 1 mm, therefore, the precision is in the millimeter level, which is very high and can satisfy completely the requirements of antenna reflector. In the process of tests, motor vibration, cable network and link flexibility are able to influence the accuracy measurement, so some of the larger error occurs, which is within the allowed range of the experiments. Fig. 31. Absolute space error distribution of each key point.

5.2.3. Analysis of deployment repeatability The analysis of deployment repeatability is an analysis method of checking if the antenna can be unfolded into the work configuration in multiple times. Meanwhile, the deployment repeatability is an important evaluation index of the deployable mechanism, the higher deployment repeatability, the higher deployment reliability. We can calculate the deployment accuracies of each point in the process by multiple tests, analyze the testing data of each point, and verify that the ring support truss has high repeatability and high positioning precision. Because the key point 1 is fixed as coordinate system origin, the eleven key points started from key point 2 should be studied. Taking the arithmetic values of 10 times measuring results as reference values, then the errors between testing values and reference values can be calculated. Fig. 30 is the deployment repeatability of 11 key points 2–12 in x, y , z directions. From Fig. 30, it can be seen that it has very high accuracy in the z direction, which distribute within the range of ±0.05 mm, this is mainly because that the bending deformation of the vertical support rod is very small, even can be ignored, when the mechanism is unfolded or folded for many times on the level ground. Relative to the z direction, the deployment repeatability in x and y directions are poorer, which is ±0.6 mm, but considering the random experimental factors, there are 90% of the experimental data distribution within the range of ±0.4 mm, the repeatability is relative high for the 3.9-meter aperture antenna. Fig. 31 is the absolute error distribution between testing values and reference values for each key point. It can be found that the biggest error is 0.7 mm, 87% of the test data are distributed under 0.4 mm. According to measuring data, the distance between diagonal key points can be calculated, taking the arithmetic values of 10 times measuring results as reference values, repeatability about the distance between diagonal key points can be obtained, as shown in Fig. 32, from which it can be seen that the overall repeatability is ±0.7 mm. According to the analysis of different diagonal, the diagonal 1–7 has the highest precision and the diagonal 4–10 is the

6. Conclusion One kind of the deployable ring mechanism which is composed of multiple deployable modules of planar six-bar linkages, is presented and studied. The analysis results of the ring mechanism have demonstrated the feasibility for building large-scale satellite deployable antenna. Detailed geometric modeling methods for prime focus antenna and offset antenna are introduced. The compatibility conditions of the synchronous movement are studied, and the deployment synchronization of the ring mechanism has been realized by using closed loop cables and dual slidercrank mechanisms, which ensure the single degree of freedom for the ring mechanism. The kinematic analysis and simulation of the mechanism are carried out. Two types of driving modes including cable driven and torsion spring driven are introduced, and the high rigidity driving joint has been designed. A ground experiment prototype of 3.9 meter in diameter is fabricated, the furled diameter is 0.32 meter, the height is 1.23 meter, the ratio of deployed diameter and furled diameter is 12.25, and the overall mass of the antenna is 11.5 kg. The deployment test results show that the mechanism can be driven by torsion springs to deploy successfully, which demonstrate good deployment performance for the proposed deployable mechanism. Ten times of fold-and-deploy experiments are carried out to test the deployment repeatability; the results indicate that the mechanism has satisfactory deployment repeatability for the requirements of antenna reflector. Some future work is suggested as follows: (1) to reduce the influence of the gravity as much as possible, the suspension device will be used for the test. (2) In the deployment process, both joint frictions and the flexibility of the rods will affect the mechanism movement, the influence rule should be studied. (3) The dynamics modeling of the ring mechanism with the flexible cable network still need to be investigated. (4) The optical or parabolic performance test should be done to verify and improve the prototype. Conflict of interest statement No conflict of interest.

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Acknowledgements This work was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51521003), in part by the National Natural Science Foundation of China (Grant No. 51505097), in part by State Key Laboratory of Robotics and System (HIT) (Grant No. SKLRS201508B) and in part by the Shenzhen Research Funds (Grant Nos. JCYJ20160427183553203 and JCYJ20150529141408781). References [1] C.J. Gantes, Deployable Structures: Analysis and Design, WIT Press, Boston, 2001. [2] A. Hanaor, R. Levy, Evaluation of deployable structures for space enclosures, Int. J. Space Struct. 16 (2001) 211–229. [3] L. Puig, A. Barton, N. Rando, A review on large deployable structures for astrophysics missions, Acta Astronaut. 67 (2010) 12–26. [4] J. Santiago-Prowald, H. Baier, Advances in deployable structures and surfaces for large apertures in space, CEAS Space J. 5 (2013) 89–115. [5] L. Datashvili, Review and evaluation of the existing designs/technologies for space large deployable apertures, in: International Scientific Conference on Advanced Lightweight Structures and Reflector Antennas, Tbilisi, Georgia, 2009. [6] A. Roederer, Historical overview of the development of space antennas, chap. 7 in: Space Antenna Handbook, Wiley, New York, 2012. [7] E. Medzmariashvili, Transformable Systems, Academy of Science USSR, Georgia, 1990.

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