Structural design and model fabrication of cable-rib tensioned deployable parabolic cylindrical antenna

Shunji ZHANG, Baoyan DUAN, Shuxin ZHANG,*, Nan WANG

a Shaanxi Key Laboratory of Space Solar Power Station System, Xi’an 710071, China

b School of Electromechanical Engineering, Xidian University, Xi’an 710071, China

c Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing 100101 China

KEYWORDS Form-finding design;Kinematic analysis;Mechanism design;Parabolic cylindrical antenna;Principle prototype

Abstract Deployable mechanisms with light weight and high storage ratio have received considerable attention for space applications.To meet the requirements of space missions,a parabolic cylindrical deployable antenna based on cable-rib tension structures is proposed and verified by a physical prototype.The parabolic cylindrical antenna adopts simple parallel four-bar mechanisms to construct the basic deployable unit, and the cylindrical direction dimension can be easily extended by modularization,which has obvious advantages in storage ratio and area density.Considering the complexity of the entire antenna structure design,including cable networks and flexible trusses,the form-finding design optimization model of a parabolic cylindrical antenna is established using the force density sensitivity method,and then the kinematics analysis of the deployable mechanism is carried out.Finally, a single-module prototype with a deployable diameter of 4 m × 2 m was designed and fabricated.The results of the ground deployment process test and surface accuracy measurements show that the antenna has good feasibility and practicability.

1.Introduction

With the rapid development of space technology, many scientific applications such as satellite communications, deep space exploration, and electronic reconnaissance have put forward requirements for larger apertures and higher precision space antennas.1–3The large-aperture antenna has higher gain and directivity,which can effectively improve the satellite observation range and reduce the size of ground receiving equipment.Due to the limitations of vehicle volume and carrying capacity,large aperture space antennas must have the characteristics of light weight and high acceptance ratio.4–6In the launch phase,the antennas are stored in the vehicle fairing;after the satellite is in orbit,they can be expanded to the working state under the action of the drive system.

Numerous studies have been conducted on deployable mechanism design and engineering prototypes to improve the stiffness-to-mass ratio and storage efficiency of large deployable antennas.Medzmariashvili et al.7presented a conical truss structure constructed using V-folded bars.No diagonal elements are needed for locking the geometry owing to the trapezoidal section of the antenna, which has the advantages of light weight and high structural stiffness.Qi et al.8developed a large ring deployable mechanism based on planar sixbar linkages.The closed-loop cable and dual slider-crank mechanisms were used to ensure single mobility of the entire mechanism.Under the condition of the same stowed height,the mechanism has a smaller storage volume than that of the AstoMesh antenna.Morterolle et al.9studied a new conceptual design of a supporting truss structure based on scissor mechanisms associated with flexible joints.These flexible joints are used to replace complex articulations while allowing the storage of the energy required for deployment.Structural dynamic simulations and modal tests were performed to verify the high rigidity of the structure.Zhang et al.10proposed two strategies for the inverse design of plane morphing scissor structures with end constraints.Different structural forms can be obtained by adding hinges or telescopic rods without changing the span.Dai et al.11proposed a double-ring deployable antenna truss structure and fabricated a 4.2 m scaled prototype.Compared with single-ring systems with a low mass gain, the double-ring deployable truss can significantly improve the structural stiffness of the antenna.Sun et al.12designed an H-style deployable mechanism for large-space mesh antennas and then developed a double-ring truss structure13to improve the stiffness of the antenna.By adding auxiliary rods on both sides of the slider, the H-style truss structure significantly improves the storage ratio of the antenna, which is suitable for the development of largeaperture and super-aperture space antennas.Cai et al.14investigated the deployment behavior of a scalable planar gossamer space structure based on Miura-ori pattern,introduced a variable Poisson’s ratio model considering the wrinkling effect to improve the existing membrane simulations, and explored the effects of its geometric parameters and initial imperfections on the mechanical properties of the membrane.Semler et al.15proposed a cable-rib tensioned antenna for providing mobile data and voice communication.On this basis, Liu et al.16–18conducted a more detailed structural design and surface accuracy optimization.The cable-rib tensioned antenna adopts multiple sets of auxiliary cables to ensure sufficient structural stiffness instead of the traditional method of increasing the number of bars and hinges.Compared with the above truss structure design method, this antenna has more application advantages in terms of light weight and a high storage rate.

In the fields of astronomical observation,space-based early warning and resource detection, space beam scanning conducted by antennas is required to expand the observation range.Compared with the paraboloid antenna that generates pen beams, the paraboloid cylindrical antenna adopts a line feed array for one-dimensional scanning, which can greatly reduce the complexity of the antenna and its auxiliary structures.Parabolic cylindrical antennas have become one of the main development directions of space antennas because of their high gain, strong directivity, and easy multiband transmission.19,20To solve the problem of imprecise measurement of precipitation and its related processes by the first rainfall measurement radar, Rahmat-Samii et al.21proposed an offset membrane parabolic cylindrical antenna.The parabolic cylinder reflector surface was formed by supporting the thin membrane structure with a hinge mechanism, and two linear feed arrays were adopted to improve the accuracy of rainfall retrieval.Soykasap et al.22developed an elastic thin-sheet parabolic cylindrical antenna with a high stiffness-to-mass ratio.A parabolic cylindrical configuration was formed by cutting and splicing isotropic thin-sheet materials, and the stored elastic potential energy was used to deploy the antenna.Xiao et al.23developed an optimizing accuracy method of a parabolic cylindrical antenna based on stiffness analysis.The truss structures were formed by the networking of Bennett linkage mechanisms and had good expansibility, but the area density is higher and the storage ratio is relatively low, which makes it difficult to meet the design requirements of large parabolic cylindrical antennas.Several studies have applied the classical parabolic antenna deployable mechanism to parabolic cylindrical antennas.The basic deployable unit of the obtained parabolic cylindrical antenna is the same as that of the ring parabolic antenna; only the layout mode is changed.Lin et al.24,25presented a high-stiffness parabolic cylindrical antenna based on a modular antenna formed by several groups of basic deployable units assembled along the parabolic and cylindrical directions.The modular and distributed truss structure can ensure high structural stiffness of the antenna,but the mass and storage volume of the antenna will increase accordingly.Sun et al.26designed a peripheral truss parabolic cylindrical antenna based on an AstroMesh unit.The rods on one side of the truss are all in the same plane,which leads to a decrease in the structural performance compared to the ring antenna.After the antenna is folded, a large amount of space remains in the center of the truss, and the overall storage ratio is relatively low.

On the basis of summarizing the existing deployable mechanism, a deployable parabolic cylindrical antenna based on cable-rib tension structures is proposed.The parabolic cylinder antenna adopts parallel four-bar mechanisms to construct the basic deployable unit, and the longitudinal scale of the parabolic cylinder can be conveniently extended through modularization, which is suitable for building large scale satellite deployable antenna.The geometry configuration of parabolic cylindrical is formed with fewer rods, and multiple groups of auxiliary cables are designed to ensure sufficient structural stiffness, which has the advantages of light weight and high storage ratio.The deployable truss only has one degree of freedom, and a two-dimensional synchronous movement can be realized under the action of the driving system.

The remainder of this paper is organized as follows.Section 2 presents the design concept of the proposed parabolic cylindrical antenna, and the performance parameters of the antenna are evaluated and compared.In Section 3, the form-finding design optimization model of the entire parabolic cylindrical antenna is established.In Section 4, the kinematics model of the deployable mechanism is established, and the transient kinematics response of the mechanism endpoints is simulated.In Section 5, a single-module prototype with a deployable aperture of 4 m × 2 m is designed and manufactured, and the corresponding ground deployment process test and surface accuracy measurement are presented.Finally,the work of this study is summarized in Section 6.

2.Innovative design of parabolic cylindrical antenna

2.1.Single deployable module

Generally,the reflector surface of a parabolic cylinder antenna is formed by stretching a parabola.Based on the special form of a parabolic cylinder,we can first design the parabolic direction deployable mechanism and then consider the cylindrical direction deployable mechanism to connect the adjacent parabolic direction deployable mechanisms to obtain the entire deployable truss.Here, the direction of parabola stretching is called cylindrical direction.

As shown in Fig.1, the twofold bar mechanism consists of rods ab and bc connected to a rotating joint.Two groups of twofold bar mechanisms are spliced together,forming a geometry very close to a parabolic shape.The folding and deployment of the antenna in the parabolic direction can be realized by twofold bar mechanisms turning around their respective rotating joints.To ensure the synchronous movement of rods ab and bc, a parallel four-bar mechanism is constructed by adding rods cf and de.Finally, the central rod gh and central slider f are connected to the parallel four-bar mechanism to form the basic deployable unit of the parabolic cylindrical antenna.

The parabolic direction deployable mechanism can be obtained by mirroring and splicing two groups of basic deployable units around the central rod gh, as shown in Fig.2.Because of the sharing of the central rod and slider, the rods of the parabolic direction deployable mechanism can move synchronously.From Fig.1, when the unit is folded, the rods ad, bc and de, cf move parallel to each other and finally coincide on a straight line.To avoid rod interference during the movement process and improve the overall storage rate of the antenna, a parabolic direction deployable mechanism is processed in layers.In Fig.2, rods of the same color are on the same plane, and rods of different colors are on different planes.

To facilitate the processing and manufacture of parabolic cylindrical antennas, the cylindrical direction deployable mechanism uses similar basic units,but the rod size and layering mode are slightly different.As shown in Fig.3, rod cb is extended such that the end is in the same vertical direction as point a,and the cylindrical direction deployable mechanism is obtained by mirroring and splicing two groups of basic deployable units in the vertical direction.

The cylindrical direction deployable mechanism and adjacent parabolic direction deployable mechanism have a common central rod and slider, which are connected to each other to form a single deployable module, as shown in Fig.4.The synchronous deployment or folding of the deployable truss can be better realized with the central slider sliding up and down.The deployable truss is formed by splicing several deployable modules along the cylindrical direction, and several groups of auxiliary cables are designed at the back of the truss to improve its structural stiffness, as shown in Fig.5.The cable net system is composed of a front cable net and adjustable tension ties.The front cable net is suspended on the deployable truss and forms an ideal parabolic cylindrical shape under the action of tension ties.The wire mesh is attached to the back of the front cable net and is used to receive and transmit electromagnetic waves.

2.2.Dimension analysis of deployable mechanism

The parabolic cylindrical deployable truss is composed of parabolic direction deployable mechanisms and cylindrical direction deployable mechanisms.The dimensions of each rod must be calculated based on the characteristic parameters of the parabolic cylindrical antenna.Taking a prime focus parabolic cylindrical reflector as an example, and assuming that the deployable aperture and focal length of the parabolic cylindrical antenna are D×L and F,respectively,the relationship between the size of each rod and the characteristic parameters of the antenna is analyzed.

For a cylindrical reflector with given parameters, the equation in the global coordinate system can be expressed as

where hgrepresents the distance between the center of the cable net and deployable truss.

Ignoring the joint size, the relationship between the dimensions of the rod in the parabolic direction, and the characteristic parameters of the antenna can be obtained from Fig.6.

where α represents the angle between rod ab and the horizontal direction, and lijrepresents the length of rod ij.

Fig.1 Schematic diagram of basic deployable unit.

Fig.2 Parabolic direction deployable mechanism.

Fig.3 Cylindrical direction deployable mechanism.

Fig.4 Deployment process for single module.

Fig.5 Components of parabolic cylindrical antenna.

The truss structures of the cable-rib tensioned parabolic cylindrical antenna are shown in Fig.7, whose stowed configuration is approximately a cuboid shape.The deployable height of the antenna is the sum of the length of rod gh and the projection length of rod ab in the vertical direction, and its deployable height and volume can be expressed respectively as

The stowed dimension of parabolic cylindrical antenna is directly related to the number of deployable modules.To ensure that the height of the cylindrical deployable mechanism does not exceed that of the parabolic deployable mechanism,the minimum number Nminof modules along the cylindrical direction is constrained, whereas the stowed height is the length of the longest rod ad.

where function ceil() denotes rounding up.

As adopting a layer structure for the deployable truss, the parabola direction and the cylindrical direction deployable mechanism may interfere after being completely folded.The joint length at the central bar and the central slider of the parabolic deployable mechanism should be larger than the projection length of the cylindrical deployable mechanism in the parabola direction.Assuming that the outer diameter of the rod is d, the stowed dimension in the parabolic direction can be regarded as the sum of the outer diameters of 9 rods.Considering the gaps at the joints after the antenna is folded, the stowed dimension of a single deployable module in the cylindrical direction can be regarded as the sum of the outer diameters of the 4 rods.If the number of deployable modules is n,the stowed volume of the parabolic cylindrical antenna can be expressed as

Fig.6 Relationship between dimensions of rod in parabola direction and characteristic parameters.

Fig.7 Deployable truss of cable-rib tensioned parabolic cylindrical antenna.

2.3.Deployment driven design

As the deployable mechanism is a complex system comprising several sets of motion pairs and internal rods,it is complicated,and therefore, difficult to analyze directly.Thus, a basic deployable unit is selected to obtain relevant information about the parabolic cylinder antenna, as shown in Fig.8.Assuming that the central rod is a fixed member, the deployable mechanism consists of 5 moving parts, 12 rotating pairs,and 1 sliding pair.

The motion of a spatial mechanism can be represented by a screw.The direction of the screw denotes the axis direction of

Fig.8 Reference coordinate system of the basic deployable unit.the motion pair,and the line distance of the screw indicates the position of the motion pair.The reference coordinate system of the basic deployable unit is established as shown in Fig.8.The direction of the Z-axis is consistent with that of the rotation pair at the point g, the Y-axis is along the straight line where the rod gh is located, and the X-axis is determined by the right-hand criterion.The degree of freedom of the mechanism is calculated by the reciprocal screw theory.

Let lgb=lab=l1,lcf=l2.According to the geometric constraints provided by Eq.(2), the screw coordinates of the 7 kinematic pairs can be expressed as

The deployable truss is formed by the array assembly of a plurality of basic deployable units in the parabolic direction and the cylindrical direction.The movement of the central slider can control the motion of each connected basic deployable unit, so the degree of freedom of the deployable truss is the same as that of the basic deployable unit, both of which are 1.Meanwhile, the cylindrical direction deployable mechanism and the adjacent parabolic direction deployable mechanism share a central rod, a slider and a moving pair.The up and down sliding of the central slider can drive the twodimensional synchronous deployment of the cylindrical direction deployable mechanism and the parabolic direction deployable mechanism.

When the basic deployable unit is completely folded, the deployment angle θ of the mechanism is 0.The four moving screws S2, S3, S4and S5of the parallel four-bar mechanism are located in the same plane.The four-bar mechanism is in the singular position of degree of freedom,and the mechanism with single mobility has two degrees of freedom instantly.The torsion spring must be installed on the rotating joint to provide the driving force needed for the initial deployment.In the later stage of deployment, the cable net is gradually tightened and the driving force of the antenna increases sharply,so the combination of a torsion spring and driving cable is more reasonable.

The combined drive mode of the torsion spring and driving cable is adopted,as shown in Fig.9.To make the driving cable less exposed to the outside of the rods,the original central rod and slider are replaced with the upper and lower sleeves without changing the degree of freedom of the deployable mechanism.The driving cable runs through the deployable mechanism in the cylindrical direction, starting at the lower end of the lower sleeve, bypassing the pulley of the upper sleeve upward, and then passing through the lower end of the lower sleeve.When the length of the driving cable is shortened, it drives the upper sleeve to slide down and realizes the synchronous deployment of the entire parabolic cylindrical antenna.

2.4.Antenna performance evaluation and comparison

To better evaluate the performance parameters of the parabolic cylindrical antenna, the cable-rib tensioned parabolic cylindrical antenna is compared with a representative modular truss parabolic cylindrical antenna24,25and a peripheral truss parabolic cylindrical antenna26and some performance parameters are investigated.

A parabolic cylindrical antenna with a deployable aperture of 4 m × 8 m and a focal length of 1.5 m is considered as an example, and the corresponding finite element models of the three antennas are established.Based on the approximate fundamental frequency of the deployable state, the mass, deployable, and stowed dimensions of the three types of parabolic cylindrical antennas are comprehensively compared.For convenience and simplicity, each antenna truss element has only one cross-section.The detailed performance parameters of the parabolic cylindrical antenna are presented in Table 1.

The deployable fundamental frequencies of the three parabolic cylindrical antennas are approximately 0.315 Hz.The modular parabolic cylindrical antenna has a large number of members and high structural stiffness, the inner and outer diameters of the truss rods are smaller, the peripheral truss parabolic cylindrical antenna adopts a square peripheral truss,and the rods on one side of the deployable truss are all in the same plane,which leads to a decrease in the structural stiffness compared with the ring truss.Larger inner and outer diameters of the rods are required for the peripheral truss parabolic cylindrical antenna,while for the cable-rib tensioned parabolic cylindrical antennas with the help of auxiliary cables,the inner and outer diameters of the truss rods are between those of the three antennas.

Fig.9 Arrangement mode of drive cable.

Table 1 Performance comparison of three parabolic cylindrical antennas.

Compared with the modular antenna and the peripheral truss antenna, the mass and volume storage ratios of the proposed parabolic cylindrical antenna are significantly improved.It can be seen from Table 1 that the stowed dimension of the proposed antenna in the parabolic direction is not affected by the number of modules but is only related to the outer diameter of the bar,while the stowed dimension in the cylindrical direction is also increased by 33.5% and 48.05%, respectively, compared with the other antennas.For large-aperture parabolic cylindrical deployable antennas, the proposed cable-rib tensioned cylindrical antenna has greater application prospects.

3.Form-finding design of parabolic cylindrical antenna

3.1.Cable net configuration design

For the rotating paraboloid antenna, the cable-mesh reflector is composed of many triangular patches, while the parabolic cylindrical antenna has different geometric characteristics in the parabolic direction and the cylindrical direction, so the rectangular patch is used to approach the surface shape of the parabolic cylindrical reflector.The cable net system of the parabolic cylindrical antenna is shown in Fig.10, which is mainly composed of the front cable net and the tension ties.The front cable net can also be divided into parabola direction cables and cylindrical direction cables.Parabolic direction cables are arranged above the parabolic deployable mechanism and forms a parabolic shape under the action of the tension ties.Cylindrical direction cables connect the adjacent direction parabolic cables to form rectangular grids to carry the wire mesh.

In the gravity-free environment of space, the cable element is in a straight line due to the action of its own tension.Therefore, there are inevitable principle errors when multi-segment straight cables are used to approach the surface shape of parabolic cylinder.The reflector surface of parabolic cylinder antenna can be regarded as formed by the parabolic stretching along its cylindrical direction.To reduce the principle errors of approaching the ideal parabolic cylinder,the cable elements in the parabolic direction should best approach the ideal parabolic shape.The problem of optimal approximation between a single cable element and an ideal parabola is discussed as below.

Denote an ideal parabola with focal length F by p1, whose parabola equation is shown in Eq.(1).Take any segment of cable element in parabolic direction as shown in Fig.11, its two end nodes are A （x0,y0) and B （x1,y1) respectively, and the projection length in the x-axis direction is L.It can be obtained from the position relationship between nodes A and B and the parabola equation.

where h1and h2represent the offset distance between nodes A,B and the ideal paraboloid p1in the z-direction.With the change of coordinate parameters x0, y0, projection length L and offset distance h1, h2, line segment AB can represent the cable element corresponding to any parabolic arc segment.

Fig.10 Cable net system of parabolic cylindrical antenna.

Fig.11 Arbitrary cable element AB in parabola direction.

The straight line equation where the segment AB is located can be obtained from the spatial coordinates of nodes A and B.

Eq.(14)represents the z-direction variance between the line segment AB and the ideal parabola p1.It can be found that the principle error is only related to the projection length L of the cable element, and the offset distance h1and h2,but has nothing to do with the specific position coordinates of the node.Considering that the projection length L is given, there must be a minimum variance in the z-direction between the line segment AB and the ideal parabola p1.By calculating the extreme value of Eq.(14), the relationship between the projection length L and the offset distance h1and h2can be obtained.

The ideal parabola p1is translated -L2/24F along the z-axis to get the parabola p2.From the coordinate equations of nodes A and B, it is known that when reaching the best approximation to the ideal parabola, the nodes at both ends of the cable element should be located on the parabola p2,which is coaxial with the parabola p1, and has equal focal length.By substituting Eqs.(16) and (17) into Eq.(14), the minimum root mean square error δrmsin z-direction can be obtained.

From the above derivation process, it can be known that when the cable segment with equal projection length is used to approach the ideal parabola p1, the nodes at both ends of all cable segments should be located on the parabola p2with equal focal length and coaxial with the parabola p1.The root mean square error in the z-direction between the cable-mesh reflector and the ideal parabolic cylinder can be minimized at the same time.

3.2.Auxiliary cable design

In the parabolic cylindrical antenna, the cable net system is connected to the deployable truss by tension ties.The cable net acting on the deployable truss is mainly reflected in the tension ties.After the configuration and form-finding design of the cable net is completed, the cable force of the tension ties is extracted and applied to the deployable truss.The deformation of the antenna under the action of the cable net is shown in Fig.12, which has been magnified 20 times.The black virtual line represents the truss structure before deformation,while the colored line indicates the deformed truss structure.

It can be found from Fig.12 that due to the small number of rods in the cylindrical direction, the main deformation of the parabolic cylindrical antenna comprises symmetrical warping about the constraint nodes,and the whole truss structure is in the shape of ‘‘a(chǎn)rch bridge”along the cylindrical direction.As the modules far away from the constrained nodes follow the previous modules to produce rigid displacement,the deformation of the parabolic cylindrical antenna will be more serious with the increase of the antenna cylindrical direction length.The deformation of trusses with different antenna lengths under the action of cable net is shown in Table 2.It can be found that when the length of the antenna is 12 m,the maximum deformation and the root mean square of the truss node deformation are 91.4 mm and 40.3 mm respectively.The overall deformation is very obvious, and it is difficult to meet the accuracy requirements of reflector surface through form-finding design.

Fig.12 Cloud diagram of antenna deformation under the action of cable net.

Table 2 Truss node deformation under action of cable net.

Considering the advantages of high stiffness-mass ratio and simple structure of cable-rib tensioned structures, several groups of auxiliary cables are designed in the back direction of weak antenna stiffness to reduce the influence of antenna deformation.According to the deformation of the antenna and the structure form of the deployable truss, three types of auxiliary cables are designed as shown in Fig.13.Considering that the main deformation of the antenna occurs in the cylindrical direction, one group of auxiliary cables are arranged along the parabolic direction, and the other two groups are arranged along the cylindrical direction.

3.3.Optimization model of form-finding design

The truss stiffness of the parabolic cylindrical antenna is relatively small.When the cable net is suspended on the truss, the truss nodes are easy to produce large deformation under the action of cable net.Therefore, a reasonable form-finding design is particularly important for this kind of antenna.The truss,cable net,and auxiliary cables can be designed as a coupled whole by establishing a system equilibrium equation for the entire parabolic cylindrical antenna27.

For the cable-network structures of parabolic cylindrical antenna, the node equilibrium equation in the equilibrium state can be expressed as

Fig.13 Distribution of auxiliary cables.

Ignoring the influence of connection joints on form-finding design, a flexible truss model is established by using 6-degreeof-freedom Euler-Bernoulli beam elements to consider the axial strain and bending deformation of truss members.Suppose Ngrepresents the number of truss nodes connected to the cable net, and Nbrepresents the set of truss constraint nodes.When the truss nodes are subjected to cable net tension,its structural stiffness equation can be expressed as

For the parabolic cylindrical antenna, the force of the flexible truss on the cable net and the force of the cable net on the flexible truss are a pair of interaction forces with equal magnitude and opposite direction.Thus,there is an equilibrium relationship as shown in Eq.(26).

Substitute Eqs.(20) and (24) into Eq.(26), the coordinates of cable network boundary nodes and free nodes in the balanced state can be obtained

When the topological relationship between the cable net and the flexible truss is determined,any set of force density values can be used to determine the node positions and tension distribution of the cable network in the equilibrium state.However, the parabolic cylindrical antennas have specific requirements for reflector accuracy and tension distribution according to the design requirements in practical application,28so it is necessary to further establish a form-finding design optimization model to find the best parabolic cylindrical antenna configuration that meets the mission requirements.

An optimization model of the form-finding design of a parabolic cylindrical antenna is established using the method introduced in our previous work.29The increment of force density is taken as the design variable,the node RMS error of the reflector surface is taken as the optimization objective,and the maximum and minimum values of cable tension are used to obtain a more uniform tension distribution;the maximum deformation of the truss nodes is constrained to reduce the antenna deformation.The optimization model of the form-finding design of a parabolic cylindrical antenna can be expressed as

3.4.Numerical case

To better illustrate the form-finding process of a parabolic cylindrical antenna, a prime focus parabolic cylindrical antenna composed of four deployable modules is discussed and analyzed.The aperture of the antenna is 4 m × 8 m,the focal length is 1.5 m, and the cable net is divided into 15 segments in the direction of the parabola.In the process of form-finding, the maximum and minimum of cable tension are set to 60 N and 2 N respectively, and the maximum node deformation of deployable truss is set to 0.01 m.Meantime,the finite element model of the whole parabolic cylindrical antenna is established to illustrate the effectiveness of the optimization model of form-finding design.The cable element is modeled using Link10, and the truss element is modeled using Beam4.The detailed element parameter information is presented in Table 3.The form-finding design results are substituted into ANSYS to solve the real equilibrium state, and the final configurations of parabolic cylindrical antennas obtained by the two methods are compared.

After six iterations,the reflector node accuracy was reduced from 66.81 mm to 5.34×10-3mm.The force density sensitivity matrix provides gradient information for the form-finding design, and the iterations are greatly reduced.The formfinding design results are substituted into ANSYS, and the design results of the two methods are summarized in Table 4 for comparative analysis.The tension distribution cloud diagram of parabolic cylindrical antenna after form-finding is shown in Fig.14.It can be found that the minimum cable tension is 2.10 N and the maximum cable tension is 60.19 N,which are consistent with the design requirements.The tension of the auxiliary cable is mostly above 55 N, which indicates that the designed auxiliary cables play an important role in reducing antenna deformation.

The node deformation cloud of parabolic cylindrical antenna after form-finding is shown in Fig.15.It can be found that the maximum deformation of the truss node is 10.54 mm,which is only 5.4% error from the required maximum deformation constraint.The node deformation of the reflector surface mainly occurs in the edge position, while most of the internal node deformation is less than 1 mm.The accuracy of the reflector surface is 0.32 mm by Ansys analysis, which meets the actual engineering requirements.

Table 3 Geometric material parameters of the element.

Table 4 Detailed form-finding design results of parabolic cylindrical antenna.

Fig.15 Node deformation of parabolic cylindrical antenna.

4.Kinematic and dynamic analysis

The kinematics analysis of parabolic cylindrical antenna is based on the geometric and topological relationship of the deployable mechanism, and the transient kinematics response during the deployment process is studied by planning the deployable angular velocity of the mechanism.This analysis mainly includes kinematics modeling,velocity and acceleration analysis and so on.The cable-rib tensioned parabolic cylindrical antenna is formed by the splicing of several deployable modules, including a number of motion closed loops, internal parts, and motion pairs, and its kinematics must be simplified during the modeling process.Ignoring the influence of the geometric length of each joint, the joint is simplified as a point,and the rod is simplified as a line segment for kinematic analysis.

4.1.Kinematic model

For a deployable truss composed of n modules, the global coordinate system is established at the middle module of the truss, as shown in Fig.16.The origin of the global coordinate system O0coincides with the upper end of the central rod,and the coordinate axis Z0is vertically upward along the direction of the center rod.Axis Y0is in the cylindrical direction and is perpendicular to axis Z0.The coordinate axis, X0, is determined using the right-hand criterion.The coordinate system O0- X0Y0Z0is not only the global coordinate system but also the local coordinate system of the middle module.Each module is numbered from the middle to both ends along the cylindrical direction, where the module number in the same direction as Y0increases gradually and decreases gradually in the opposite direction.Subsequently, the local coordinate system Oi- XiYiZiof each module is established according to the marked number.The coordinate origin of each local coordinate system coincides with the upper endpoint of the central rod of the deployable module.In the deployment process,there is only a translation relationship between each local coordinate system and the global coordinate system;therefore,the directions of the three axes are the same as those of the global coordinate system.

Considering the symmetry and similarity of the deployable mechanism, only one side of the parabolic deployable mechanism is analyzed and other node positions can be obtained by geometric relations.The local coordinate system established in the ith deployable module is shown in Fig.17.Assuming that the dimensions of the rods meet the requirements lgibi=laibi=l1, lcifi=l2, The endpoint coordinates of the rods in the local coordinate system can be expressed as

Fig.16 Global coordinate system of deployable truss.

Fig.17 Local coordinate system of a single deployable module.

where leis the length of the rod jk;C indicates the shape function matrix,and×represents the distance between node q and node j.

The motion of point q in the global coordinate system O0- X0Y0Z0can be regarded as the sum of the plane motion of point q in the local coordinate system Oi- XiYiZi, and the translation motion of the local coordinate system Oi- XiYiZiin the global coordinate system O0- X0Y0Z0.Therefore, the coordinate rqof point q in the global coordinate system can be expressed as follows:

4.2.Numerical case

To reduce the deployment impact and vibration and ensure the smooth deployment process of the parabolic cylindrical antenna, a cubic polynomial function is used to plan the deployable angular velocity ˙θ of the mechanism with time t as the independent variable.The cubic polynomial function is defined as follows:

where t0and t1are the start and end times, and h0and h1are the function values corresponding to the start and end times,respectively.

Using the deployable strategy comprising of the first acceleration, then uniform speed, and finally deceleration, the planned angular velocity function of the mechanism is as follows:

where T is the total deployable time.

The angular displacement and angular acceleration functions of the mechanism can be obtained through integral and differential calculations on Eq.(36), respectively.The angular displacement function of the mechanism is given by Eq.(37).The boundary conditions on the generalized coordinate θ are considered:when the antenna is stowed(t=0),the deployable angle θ of the mechanism is 0; after full deployment (t = T),the angle θ changes to π/2.Thus, the angular velocity ˙θsof the uniform stage can be obtained, that is, ˙θs=5π/8T.

As long as the total deployable time T is determined, the corresponding parabolic cylindrical antenna motion-planning parameters can be obtained from Eqs.(36) and (37).In this paper, the kinematics analysis of deployable truss is carried out by taking a prime focus parabolic cylindrical antenna with a deployable aperture of 4 m×8 m and a focal length of 1.5 m as an example.Assuming that the total time T is 250 s, the angular velocity, ˙θs=0.0079 rad/s, of the uniform stage can be obtained.The angular displacement, angular velocity, and angular acceleration function curves of the mechanism during the deployment process are shown in Fig.18.It can be found that the truss accelerates in the first 50 s,decelerates in the last 50 s,and moves at a uniform speed in the middle 150 s.In the whole deployment process, the velocity changes slowly, the acceleration is small and there is no sudden change.The motion trajectories of the endpoints are shown in Fig.19.The deployable truss expands symmetrically to both sides,and each endpoint moves along a fixed straight line.

Fig.18 Variation curves of angular displacement, angular velocity and angular acceleration.

Fig.20 Displacement curves of the endpoints.

Fig.20 shows the displacement curves of the mechanism endpoints in the global coordinate system.Due to the symmetry of the deployable truss, it is only necessary to analyze the motion of the endpoints 1–3.It can be found that the deployment process of the mechanism is smooth as a whole, there is no sudden change, and the velocity at the end of the deployment process is very slow.The displacement function of the endpoint calculates the first derivative and the second derivative of time, respectively, and the variation curves of velocity and acceleration with time can be obtained, as shown in Fig.21 and Fig.22.It can be found that the farther the endpoint is from the constraint nodes, the greater the variation trend of velocity and acceleration.Therefore, more attention should be paid to the changes of the velocity and acceleration of the distal rods during the deployment control of the mechanism,so as to avoid unreasonable motion planning and structural impact oscillation.Fig.23 shows the deployable state of the truss at different angles, which shows that the mechanism can be developed smoothly according to the planned motion parameters.

Fig.19 Trajectories of the endpoints.

Fig.21 Velocity curves of the endpoints.

Fig.22 Acceleration curves of the endpoints.

5.Prototype and experiments

5.1.Single module prototype

To verify the feasibility and practicability of the proposed cable-rib tension parabolic cylindrical antenna, a prototype of a parabolic cylindrical antenna is designed and manufactured.Considering the limitations of economic cost and placement space, the prototype is reduced from four deployable modules to a single deployable module, and the antenna deployable aperture is reduced from 4 m × 8 m to 4 m × 2 m.

Fig.23 Deployable state of the truss at different angles.

A single-module prototype of the parabolic cylindrical antenna is illustrated in Fig.24.The deployable truss is a prototype composed of two parabolic deployable mechanisms,one cylindrical deployable mechanism, and two end auxiliary rods.The deployable aperture of the parabolic cylindrical antenna is 4 m×2 m,and the deployable height is 1.7 m.After the antenna is completely stowed, the length of the antenna is 0.28 m in the parabolic direction, 0.205 m in the cylindrical direction, and 1.38 m in height.The ratio of deployable volume to stowed volume is 171.69.To reduce the overall weight of the antenna, the rods of the antenna use carbon fiber materials, and the joints are made of aluminum alloy materials.After the measurement,the total weight of the parabolic cylinder principle prototype is 10.82 kg.

Meanwhile, to conduct the surface accuracy measurement and deployment experiments of the antenna on the ground, a ground support device of the principle prototype is designed,as shown in Fig.25.The support device mainly includes a detachable frame,linear slide rail,and driving motor.The bottom of the central rod is connected to a slide rail using a linear bearing.When the antenna is deployed,the linear bearing is in rolling contact with the connected rail through an internal steel ball, which can effectively reduce the influence of the sliding rail friction resistance on the deployment process.To facilitate the surface accuracy measurement and deployment experiments, a principle prototype with a linear slide rail is installed on the detachable frame.

5.2.Deployment function test

The functional test of ground parabolic cylindrical antenna is carried out to verify the feasibility of the mechanism scheme.The driving cable runs through the cylindrical direction deployable mechanism and is connected with the driving motor on the support device.The output displacement length and pulse frequency of the motor are set by the planned kinematic parameters, so as to control the tightening speed of the driving cable.

In the initial stage of the antenna deployment, torsion springs at the joints store a large amount of elastic potential energy.The deployable truss expands rapidly under the action of the torsion springs, which requires the motor to properly improve the speed of tightening the driving cable.When the antenna is deployed to a certain angle, it requires a steady tightening speed of the motor to maintain a steady movement.When the antenna is about to be unfolded in place, the cable net is gradually tightened, then a larger driving cable tension is needed, the speed of the motor tightening the driving cable should be slowed down.Therefore, a deployment strategy of the first acceleration, then uniform speed, and then deceleration is adopted to set the motor parameters.The deployment process of the parabolic cylindrical prototype is shown in Fig.26.The test results show that the deployable mechanism can be successfully driven by combined drive mode of the torsion spring and driving cable.

5.3.Surface accuracy measurement

The surface accuracy of the reflector is an important index for measuring and evaluating the performance of the antenna,which directly affects its electrical performance.Owing to the inevitable introduction of errors in the antenna processing and assembly,there is a difference between the physical model and the design model, resulting in poor antenna surface accuracy.30Therefore, it is necessary to measure and adjust the parabolic cylindrical antennas.

Fig.25 Ground support device for principle prototype.

The parabolic cylindrical antenna is a flexible structure;therefore, the measurement of surface accuracy should be non-contact to avoid external interference caused by contact and collision.In addition, owing to the high accuracy of the cable-mesh reflector itself, the measurement method adopted should also have sufficiently high measurement accuracy.The V-STARS photogrammetry system provided by the American GSI Company is used to measure the prototype,as shown in Fig.27.The V-STARS photogrammetry system is mainly composed of a measuring camera, a marking point,a coding point, and a measuring ruler.In the range of 10 m,the measurement accuracy can reach 0.08 mm, which fully meets the surface accuracy measurement requirements of the prototype.

Fig.26 Prototype of parabolic cylinder antenna.

Fig.27 V-STARS photogrammetry system.

Fig.28 Measurement environment of principle prototype.

Before photogrammetry of the prototype, it is necessary to arrange the measurement environment reasonably.First, the reflective marking points are uniformly placed on the reflector surface; next, the measuring ruler is placed smoothly around the antenna, the relative height between the ruler and the marking points should not be too large, the coding points are arranged uniformly and smoothly around the entire structure, and the distance between the coding points should be controlled at approximately 40–60 cm.The measurement environment for the parabolic cylindrical prototype is illustrated in Fig.28.The total number of marking points on the reflector surface is 152, and the measuring ruler is placed at the 30 cm below the mesh reflector.The surrounding shooting method is used to shoot the reflection surface in the indoor windless environment,and the strong light should be avoided to ensure the clarity of the photo marking points.

After photogrammetry is completed, all pictures are imported into the V-STARS image processing software.The 3D coordinates of all marking points on the reflector are obtained by image superposition processing, and a 3D model is constructed.The distribution of the marking points on the reflector of the parabolic cylindrical prototype is shown in Fig.29, where the straight yellow line indicates the position of the measuring ruler.

After the reflector surface node measurement and data processing, the best-fitting parabolic cylinder equation can be determined by the least squares method,so that the root mean square error of the measured reflector node relative to the bestfitting parabolic cylinder is minimal.The general form of the best-fit parabolic cylinder can be expressed as

Fig.29 Distribution of reflector marking points.

where a′, b′, c′, d′, e′, and f′are fitting coefficients.The x- and y-coordinate values of the landmarks are substituted into the best parabolic cylindrical equation to solve the ideal coordinate values z-.Taking Δz= z--z as the deviation between the point and ideal parabolic cylinder, the surface accuracy of the prototype antenna can be expressed as

where δrmsrepresents the root mean square error of the reflector nodes, and n represents the number of reflector nodes measured.

After the error distribution of the reflector is obtained, the adjustment amount of the tension ties is obtained using an adjustment algorithm,31and the parabolic cylinder prototype is adjusted.The variation curve of the node error during the adjustment process is shown in Fig.30.The initial accuracy of the reflector surface is 4.98 mm.Because the influence of joint hinge and wire mesh is not considered in the process of form-finding design, as well as the existence of manufacturing errors, there is a certain error between the design results and the form-finding design results.After 8 rounds of measurement and adjustment,the surface accuracy of the reflector relative to the best fitting paraboloid is reduced to 0.91 mm.This shows that the designed parabolic cylindrical antenna has adjustability and can satisfy the design requirements of high surface accuracy.

Fig.30 Surface accuracy variation curve of reflector during adjustment.

Fig.31 Surface accuracy variation curve of reflector with multiple repetitions of deployment.

Repeated deployment accuracy is to observe the change of reflector surface accuracy through repeated folding and deployment experiments, in order to verify whether the antenna can be accurately deployed to the working position when running in orbit.The surface accuracy variation curve of reflector with multiple repetitions of deployment is shown in Fig.31.The first measurement is the adjusted cable net,and the surface accuracy is high.In the process of repeated deployment, due to motor vibration, cable net pretension,rod flexibility and other factors,the accuracy is lower than that of the first time, but the change trend is relatively stable.The surface accuracy is all within 1.2 mm, which belongs to the experimental acceptable range.The experimental results show that the parabolic cylindrical antenna has high accuracy in deployment, and can meet the stability requirements of the antenna in orbit.

6.Conclusions

To meet the design requirements of high stiffness,high storage ratio, and light weight of large-aperture parabolic cylindrical antennas, a cable-rib tensioned deployable parabolic cylindrical antenna is proposed.The design scheme has the advantages of high storage ratio, low quality and easy modular splicing,which is suitable for building large scale space parabolic cylindrical antenna.The geometric modeling process of the deployment mechanism is systematically studied, and the corresponding relationship between the rod dimension and the geometric characteristic parameters of the antenna is established.The synchronous deployment of the multi-module parabolic cylindrical antenna can be realized by the combined drive of torsion spring and driving cable.The form-finding design optimization model of the parabolic cylindrical antenna is established by using the force density sensitivity method,and the deployment process of the parabolic cylindrical antenna is simulated.A single module ground prototype with a deployable aperture of 4 m × 2 m was designed and fabricated, the dimension of the prototype was 0.28 m × 0.205 m after full stowing, the ratio of the deployable envelope volume to the stowed envelope volume was 171.69, and the total weight of antenna was 10.82 kg.The deployment test results show that the mechanism can be driven by a combination of a torsion spring and driving cable to deploy successfully, and has good deployment performance.After 8 rounds of photogrammetry and profile adjustment, the surface accuracy of the parabolic cylindrical antenna was reduced from the initial 6.98 mm to 0.91 mm.The results indicate that the designed parabolic cylindrical antenna has good adjustability, and can satisfy the high accuracy design requirements.

Suggestions for future work are as follows:(A)to minimize the influence of gravity,a complete gravity elimination scheme needs to be further designed.(B)due to the limitations of funds and sites, only the working performance of the single module was tested.The cooperative deployment experiment of multimodule parabolic cylindrical antenna should be done to verify and improve the prototype.(C) the parabolic cylindrical antenna can be successfully driven by combined drive mode of the torsion spring and driving cable.The dynamic model of deployable mechanism considering cable net, joint frictions and flexible rods needs further investigation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos.51705388 and 51675398) and the Youth Talent Fund of Science and Technology Association of Shaanxi University of China.Thank the staff of the Research Institute of Mechatronics,Xi’dian University,China,for their assistance in the completion of this paper.We would also like to express our gratitude to the Aerospace information Research Institute,Chinese Academy of Sciences for its financial support.