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Design of geodesic cable net for space deployable mesh reflectors
Acta Astronautica 119 (2016) 13–21
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Design of geodesic cable net for space deployable mesh reflectors Hanqing Deng, Tuanjie Li n, Zuowei Wang School of Mechano-Electronic Engineering, Xidian University, P.O. Box 188, Xi'an 710071, China
a r t i c l e i n f o
abstract
Article history: Received 17 December 2014 Received in revised form 18 September 2015 Accepted 31 October 2015 Available online 10 November 2015
The main purpose of this paper is to design the geodesic cable net for space deployable mesh reflectors by means of the geodesic differential equation, which is different from the traditional quasi-geodesic cable net. In order to generate the geodesic cable net, two main problems must be addressed. One is to generate the geodesics on the paraboloidal surface, and the other is to form the geodesic cable net according to the prescribed topology. Therefore, through the analysis of geodesic curvature, a general numerical method is presented to calculate the geodesics between arbitrary two points on the paraboloidal surface. Then according to the structural characteristics of mesh reflectors, a dynamic boundary adjustment method is proposed to address the second problem. The key of the method is an optimization model that adjusts the internal geodesics to form the triangular cable net by moving the end-points along the dynamic boundaries. Finally, the feasibility and effectiveness of the proposed design method of geodesic cable net are validated by three numerical examples. & 2015 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Mesh reflectors Geodesics Dynamic boundary adjustment method Geodesic cable net
1. Introduction The demands of mobile communication are increasing greatly since 21st century, especially the coming of 3G, 4G and 5G communication networks. As one of the most important components for high-end communication technologies, space deployable reflectors are grown rapidly [1]. For the disadvantages of small aperture and high weight in the solid surface reflectors, the large-size reflectors applying to space communication are all deployable, including several types [2], such as deployable mesh reflectors, semirigid antennas, and inflatable antennas etc. As the characteristics of lightweight, large aperture and simple structure, deployable mesh reflectors have the majority in-orbit applications [3]. The AstroMesh hoop truss n
Corresponding author. Tel.: þ 86 29 88202470. E-mail address: [email protected] (T. Li).
http://dx.doi.org/10.1016/j.actaastro.2015.10.024 0094-5765/& 2015 IAA. Published by Elsevier Ltd. All rights reserved.
antenna [4] was first applied in the telecommunication satellite Thuraya with the diameter of 12.25 m and the weight of 55 kg. The Engineering Test Satellite VIII [5] has two large deployable reflectors with the dimension of 19 m 17 m. Satellites TerreStar and Skyterra [6] are both unfurlable reflectors with the aperture of 18 m and 22 m, respectively. The deployable mesh reflector consists of cable nets, and a deployable frame structures. And the cable net is thus becoming an indispensable component for deployable mesh reflectors. One important issue is the geometric design which includes the topological configuration and the geometric generation. The main considerations of topology design are the kinematically indeterminate supporting structures and statically indeterminate cable net structures [7]. Several topological configurations have been developed, such as triangular, quadrilateral, and hexagon topological configurations etc. For the given triangular topological configuration, two kinds of cable nets can be generated: triclinic and geodesic cable nets. The triclinic cable net has
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H. Deng et al. / Acta Astronautica 119 (2016) 13–21
been widely applied, which is generated by dividing the inscribed hexagon of circular truss into triangular grid and then mapping onto the paraboloid [8]. And the geodesic cable net is created by the geodesic theory or some geodesic characteristics. If the curvature of each point in a plane curve is equal to zero, the curve is a straight line. Similarly, if the geodesic curvature of each point in a spatial curve is equal to zero, the curve is called a geodesic line. In the intrinsic geometry, the geodesics act as straight lines in a plane. Another definition of geodesics is the shortest path between two points on a curved surface. For this reason, the geodesics are widely used in the geometric design of ship hulls, robot motion planning, terrain navigation and computer 3D modeling [9,10]. On the IUTAM-IASS Symposium, Thomson firstly proposed the concept of “pseudogeodesic” to use in Astromesh hoop truss antennas [11]. Later, in order to generate the triangular cable net with the shortest lengths, the “quasi-geodesic cable net” was used and named “geotensoid” by Tibert [8], in 2002. Then in 2012, Morterolle et al. [12] proposed a numerical formfinding method of geotensoid tension truss for mesh reflectors. They believed that a cable net with uniform tension distribution was the shortest. During this period, some other methods for “geodesic cable net” generation were presented, such as the method proposed by Yang et al. [13,14]. All the existing methods use the concept of “pseudo” or “quasi” geodesic, which has the characteristic of the shortest length. Nevertheless, how to deduce the geodesic cable net from the mathematical point of view is very essential. Based on the geodesic theory in the differential geometry, we propose a general method to design the geodesic cable net for space deployable mesh reflectors. This paper is structured as follows. First, a general numerical method is presented in Section 2 to generate geodesics on the paraboloidal surface. Then a dynamic boundary adjustment method is proposed in Section 3 to form the geodesic cable net with the prescribed topology. And then, three numerical examples are given in Section 4 to illustrate the generation of geodesic cable nets. Finally, some conclusions are summed up in Section 5.
shortest path between two points always exists on the surface. To analyze the shortest path, a parametric surface S is denoted by rðu; vÞ, where u and v are the parametric coordinates, similar to the x, y and z components in the Cartesian coordinates. And a curve C on the surface is denoted by rðtÞ ¼ rðuðtÞ; vðtÞÞ with respect to the parameter t. By differentiating the parametric equation of the curve C, we obtain dr ¼ r u du þ r v dv
ð1Þ
The first fundamental form of the surface S is 2
ds ¼ dr 2 ¼ Edu2 þ2Fdudv þGdv2
ð2Þ
where s is the arc length, E; F and G are called the coefficients of the first fundamental form and given by E ¼ r 2u ¼
∂r ∂r ∂u ∂u
∂r ∂r ∂u ∂v ∂r ∂r G ¼ r 2v ¼ ∂v ∂v F ¼ ru rv ¼
ð3Þ
According to the geodesic curvature equation [16], the geodesic curvature of the curve C is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κg ¼ EG F 2 u″ þ Γ 111 u02 þ 2Γ 112 u0 v0 þ Γ 122 v02 v0 ð4Þ v″ þ Γ 211 u02 þ 2Γ 212 u0 v0 þ Γ 222 v02 u0 with the Christoffel symbols Γ ijk ði; j; k ¼ 1; 2Þ 8 2FF u þ FEv > Γ 111 ¼ GEu2ðEG > F2Þ > > > > GEv FGu 1 > > Γ 12 ¼ 2ðEG F 2 Þ > > > > > Γ 1 ¼ 2GF v GGu þ FGv > < 22 2ðEG F 2 Þ u EF v þ FEu > Γ 211 ¼ 2EF2ðEG > > F2 Þ > > > > Γ 2 ¼ EGu FE2v > > 12 2ðEG F Þ > > > > 2FF v þ FGu > Γ : 222 ¼ EGv2ðEG F2Þ
ð5Þ
where Eu ; Ev ; F u ; F v ; Gu and Gv are partial derivatives of E; F and G with respect to u and v, respectively. 2.2. Geodesics on paraboloid
2. Geodesics on paraboloid of revolution In Cartesian coordinate system, f denotes the focal length, the standard equation of the paraboloid is
2.1. Geodesic curvature For a periodic and aperiodic non-uniform rational Bspline surface (NURBS), Wolter [15] has proved that the
z¼
x2 þy2 4f
Fig. 1. Composition of a hoop truss reflector antenna.
ð6Þ
H. Deng et al. / Acta Astronautica 119 (2016) 13–21
15
Fig. 2. Cable net consisted of geodesics. (a) Cable net topology and (b) cable net consisted of geodesics.
Eq. (6) can be expressed as the parameter equation ! u2 rðu; vÞ ¼ u cos ðvÞ; u sin ðvÞ; ð7Þ 4f The first fundamental form of Eq. (7) is obtained ! u2 2 ds ¼ 1 þ 2 du2 þ u2 dv2 4f
ð8Þ
in which, the relevant parameters are E ¼ 1þ
Γ 111 ¼
u2 4f
2
;
G ¼ u2 2
u 2
4f þu2
Γ 211 ¼ 0;
F ¼ 0;
;
Γ 112 ¼ 0;
1 Γ 212 ¼ ; u
Γ 222 ¼ 0
Γ 122 ¼
4f u 2
4f þu2 ð9Þ Fig. 3. Geodesic boundaries and dynamic boundary nodes.
Substituting Eq. (9) into Eq. (4), the geodesic curvature of the paraboloid is obtained sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" ! # 2 u4 u u02 4f u v02 0 2 κ g ¼ u2 þ 2 u″ þ 2 2 v v″ þ u0 v0 u0 u 4f 4f þ u2 4f þ u2
general geodesic equations of the paraboloid 8 2 02 < u″ þ u2 u02 4f 2 u v ¼ 0 2 2 4f þ u
4f þ u
: v″ þ 2u0 v0 ¼ 0
ð11Þ
u
ð10Þ According to the definition of geodesic, the geodesics are curves with zero geodesic curvature, i.e. κ g 0. The curves on the paraboloid can be divided into the following four situations. 0
0
1) When u ¼ 0; v ¼ 0, the κg is equal to zero, which indicates the curve is just a point. 2) When u0 ¼ 0; v0 a0, u can be denoted by a constant u ¼ c0 , and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4f c0 v03 c0 2 κg ¼ 2 1þ 2 4f þ c0 2 4f If and only if c0 ¼ 0, the κ g is equal to zero, which indicates the curve is also a point. 3) When u0 a 0; v0 ¼ 0, κ g 0 is obtained by substituting them into Eq. (10). So the curves are the geodesics, as well as the generatrices of the paraboloid. 4) When u0 a0; v0 a0, in order to obtain the geodesics, the coefficients of Eq. (10) must be zeros, yielding the
2.3. Solution of geodesic equation By combining Eq. (11), we have 2 v″=v0 ¼ u0 u
ð12Þ
Integrating both sides of Eq. (12), it yields 0
v ¼
c1 u2
where c1 is a constant. Substituting Eq. (13) into Eq. (8), we obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c1 u2 þ 4f dv ¼ du 2f u u2 c1 2
ð13Þ
ð14Þ
Then integrating both sides of Eq. (14), the expression of v we can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 c1 u2 þ4f v¼ duþ c2 ð15Þ 2f u u2 c1 2 where c2 is a constant. c1 and c2 can be solved with the
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H. Deng et al. / Acta Astronautica 119 (2016) 13–21
Fig. 4. Geodesic boundaries of geodesic cable net.
boundary conditions of the end-points AðuA ; vA Þ and BðuB ; vB Þ. Because the integration expression of Eq. (15) is too complicated to be solved directly, the relaxation method [16] is adopted to solve the second order ordinary differential equation, Eq. (11). Eq. (11) is transformed into the first order ordinary differential equations 8 du > > ¼p > > ds > > > > > dv > > > < ds ¼ q 2 ð16Þ dp u p2 4f u q2 > > ¼ 2 þ 2 > > > ds 4f þ u2 4f þ u2 > > > > > dq 2 > > : ¼ pq ds u
which can be rewritten as follows. Fk ¼ ðF 1k ; F 2k ; F 3k ; F 4k ÞT ¼ ¼ 0;
Yk Yk 1 1 ðG þ Gk 1 Þ sk sk 1 2 k
k ¼ 2; 3; U U U ; m
ð21Þ 0
Given a starting vector Y , Eq. (21) can be solved by the quadratically convergent Newton's method with the following iteration scheme 8 Fk ¼ Jaco ΔY k > < ð22Þ > :Y ¼ Y þ ΔY kþ1 k k where Jaco is the Jacobian matrix of F with respect to Y.
3. Geodesic cable net
Eq. (16) can be rewritten as 8 dy > > > < ds ¼ g > yðAÞ ¼ αA > > : yðBÞ ¼ α B
3.1. Composition of mesh reflectors ð17Þ
with y ¼ ðu; v; p; qÞT g¼
!T 2 p; q; 2 þ 2 ; pq u 4f þ u2 4f þ u2 u p2
2
4f u q2
ð18Þ
A deployed mesh reflector is conceived with the concept of tension truss, which is a light and inherently structure that can be precisely and repeatedly deployed regardless of environment. As illustrated in Fig. 1, the hoop truss reflector antenna is consisted of five parts: supporting truss, front net, tension ties, back net (or rear net) and the RF reflective mesh fixed to the front net. 3.2. Difficulty in geodesic cable net design
and the boundary conditions αA ¼ ðuA ; vA ; pA ; qA ÞT αB ¼ ðuB ; vB ; pB ; qB ÞT
ð19Þ
where s A ½A; B, points A and B are two end-points of the geodesic. The geodesic between points A and B, as an arc length parameterized curve with parameter s, is discretized by the finite differences with m points, satisfying A ¼ s1 o s2 o :: : o sm ¼ B. Then Eq. (17) is approximated by the trapezoidal rule Y k Y k 1 1 ¼ ðGk þ Gk 1 Þ; 2 sk sk 1
k ¼ 2; 3; U U U; m
ð20Þ
where the boundary conditions are Y 1 ¼ αA ; Y m ¼ αB . According to the boundary conditions, Eq. (20) has ðm 1Þ 4 unknown variables and ðm 1Þ 4 equations,
According to the numerical method in Section 2, it is easy to get the geodesic between arbitrary two points on the paraboloidal surface. Nevertheless, there is still a difficulty to generate the geodesic cable net, as shown in Fig. 2. It can be seen from Fig. 2 that the cable net consisted of geodesics is not coincident with its topological configuration, in which all cables form the prescribed triangular cable net. However, in the cable net consisted of geodesics with equally spaced nodes along the circumference, a set of three-band geodesic intersections appear in the positions of “topological node”, as shown in the enlarged portion in Fig. 2(b). This area is called as the “irregular region”. These three-band geodesics do not intersect at unique points, and thus result in some polygons generated around the positions. Here, the unique point existing in the prescribed topology is defined as “topological node”.
H. Deng et al. / Acta Astronautica 119 (2016) 13–21
17
3.3. Dynamic boundary adjustment method
Fig. 5. Geodesic generation between two points.
Fig. 6. Iteration process of Newton's method. Table 1 Geometric parameters. Parameters Aperture (m) Focal length (m)
Subdivision number
Value
9
20
10
Discrete number
In order to form the triangular cable net inside the hexagon, the geodesic boundaries of the cable net must be firstly determined. In the design phase, a cable net is always selected because of the shorter total length, the larger efficient area (the area of inscribed hexagon) and the higher surface accuracy. According to these requirements, some types of geodesic boundaries are available, as shown in Fig. 4. The first type is a pure geodesic boundary consisting of a geodesic between two vertexes of the inscribed hexagon. The second type is a “straight line” boundary, which is a curve on the paraboloidal surface that is mapped by the straight line between the two vertexes. The third type is a curve with the sag-to-span ratio [17]. After determining the geodesic boundaries of the cable net, the dynamic boundary nodes can be obtained and the internal geodesics can be calculated with the end-points. On account of the difficulty in the geodesic cable net design, a mathematical model for generating the triangular cable net is established to improve the intersection situation of internal geodesics. It is assumed that the irregular region exists in each position of topological node in the internal geodesic cable net (a three-band intersection is regarded as an infinitesimal region with zero area). The intersection situation of internal geodesics of the dynamical adjustment process can be then measured by the areas of irregular regions. The positions of dynamic nodes in the geodesic boundaries are regarded as the design variables, the minimum areas of irregular regions are regarded as the objective function, and some structural characteristics are regarded as the constraints. The dynamical boundary adjustment method is thus descried as an optimization model, that is: Find
M ki
ðk ¼ 1; 2; ⋯; 6;
1000
min Such a situation highlights the difficulty in geodesic cable net design: how to form a triangular cable net which is coincident with its prescribed topology. In this paper, the concept of “dynamic boundary node” is proposed to adjust the internal geodesics, as shown in Fig. 3. The dynamic boundary nodes are the end-points of internal geodesics, which offer a way to improve the intersection situation of internal geodesics. Because the nodes on the truss are usually related to the manufacture, they must be the uniform nodes. So the dynamic boundary nodes can only be selected inside the cable net. Such a dynamic boundary that offers the end-points of internal geodesics is called as the “geodesic boundary” here. Before a cable net is generated, only the six vertexes of the inscribed hexagon are fixed points. Two feasible kinds of geodesic boundaries can be determined by the fixed points. One is the diagonal lines of hexagon, and the other is the six sides of hexagon that is adopted in this paper. Therefore, the geodesic cable net can be divided into three parts: internal geodesic cables, geodesic boundaries and the boundary cables connected to the supporting truss.
z¼
N um X
sΔj
i ¼ 1; 2; ⋯; N 1Þ
ðj ¼ 1; 2; ⋯; NumÞ
j¼1
s:t: with
1 o Mki oM sΔj o ϵ sΔj ¼ GetArea P m;n ; P n;q ; P q;m P m;n ¼ GetPoint ðC m ; C n Þ
ð23Þ
where M ki is the position of the end-point in the geodesic boundary. M is the number of the points in the geodesic boundary. k is the serial number of the six vertexes. And N is the subdivision number of one side of the inscribed hexagon. sΔj is the area of the irregular region in the jth topological node. Num is the total number of topological nodes. ε is the allowance error. GetArea P m;n ; P n;q ; P q;m denotes a function to get the area of the irregular region consisting of the three points P m;n ; P n;q and P q;m . And GetPoint ðC m ; C n Þ denotes a function to get the intersection point from two discrete curves C m and C n . A detailed description of GetArea P m;n ; P n;q ; P q;m and GetPoint ðC m ; C n Þ are given in the Appendix. The dynamic boundary adjustment method is described in detail as follows. First, according to the structural parameters
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H. Deng et al. / Acta Astronautica 119 (2016) 13–21
Fig. 7. Five cable nets. (a) 1st geodesic cable net; (b) 2nd geodesic cable net; (c) 3rd geodesic cable net; (d) triclinic cable net; (e) geotensoid cable net.
Table 2 Results of the cable nets. Item
1st geodesic cable net
2nd geodesic cable net
3rd geodesic cable net
Triclinic cable net
Geotensoid cable net
Lengths (m) Efficient area (m2)
998.66 268.96
996.17 259.81
992.83 241.11
998.04 259.81
993.81 241.67
H. Deng et al. / Acta Astronautica 119 (2016) 13–21
of the reflector, such as the aperture, focus length, cable net topological configuration and the subdivision number etc., the appropriate geodesic boundaries for the geodesic cable net are selected. Second, some points of the six geodesic boundaries are selected to act as the dynamic boundary nodes. Finally, by moving the dynamic boundary nodes along the geodesic boundaries, the intersection situation of internal geodesics is adjusted to form the triangular cable net.
4. Numerical simulations 4.1. Generation of single geodesic This simulation is to calculate the geodesic between two points (10, 0, 2.5) and (0, 10, 2.5) on the paraboloidal surface with the focus length of 10 m. As shown in Fig. 5, the curve mapped by the straight line between the two points is chosen as the initial curve. The coordinates of 1000 discrete points of the initial curve are taken as the initial iteration values of Newton's method. The calculated geodesic between two points is shown in Fig. 5 and the iteration process is shown in Fig. 6. In the numerical simulation, the generated geodesic is 0.0330 m shorter than the initial curve. In Fig. 6, the iterative error is denoted as the 2-norm of F, and the iteration process is convergence to the acceptable error of 7.6393 10 9. 4.2. Generation of geodesic cable net In this simulation, cable nets of the mesh reflector with AM1 configuration proposed by AstroMesh [7] are taken as examples. The geometric parameters are listed in Table 1. The geodesic cable nets with three types of geodesic boundaries, the triclinic and geotensoid cable nets are designed, respectively. The projection views from the aperture plane of them are shown in Fig. 7. Fig. 7(d) is the triclinic cable net generated by dividing the inscribed hexagon into the triangular grid and then mapping onto the paraboloid. Fig. 7(e) is the geotensoid cable net Table 3 Geometrical and material parameters. Item
Diameter (m)
Elastic modulus (GPa)
Poisson ratio
Cables Truss
0.002 0.02
20 200
0.3 0.3
19
designed by the equal tension method [12]. The analysis result of these cable nets are listed in Table 2. The “Subdivision number” is the number of cables in the radius. It can be noted from Fig. 7 and Table 2 that three types of geodesic cable nets are different with each other because of different geodesic boundaries. Compared with the triclinic cable net, the first type enlarges the efficient area with the bulgy pure geodesic boundary, but the total length increases a little. The second type has the same “straight line” geodesic boundaries with the triclinic cable net. So it has the same efficient area but a shorter total length. And due to the concave geodesic boundaries with the sag-to-span ratio of 0.1, the third type has a smaller efficient area but the shortest total length. Among these cable nets, the performance of the 3rd geodesic cable net is close to the geotensoid cable net. 4.3. Form-finding analysis The form-finding analysis of the reflectors shown in Fig. 7 is performed in this simulation. The geometrical and material parameters of the cables and truss are listed in Table 3. The pretension distributions of the three geodesic cable nets and the triclinic cable net are designed by the plane projection method [18]. The plane projection method is based on the projection principle. The equilibrium tension of the projected plane cable net structure is firstly obtained by a force equilibrium method. The obtained cable tensions of the plane cable net structure are then projected to the spatial cable net for obtaining the spatial cable tensions. The pretension of the geotensoid cable net is calculated by the equal tension method [12]. The equal tension method is based on a force density strategy coupled with geometrical constraints and the uniform tension is achieved by iterating force density coefficients. For a topological configuration shown in Fig. 2, the pretension of the front net in the geotensoid cable net equals each other. But due to the special boundary cables, the pretensions of the geotensoid cable net are not all the same for the AM1 configuration. The back net of each reflector is the same as the front net in size, as well as the pretension in value. The detailed form-finding results are listed in Table 4. It is noted from Table 4 that the performances of the 3rd geodesic cable net are better than the geotensoid cable net, including the ratios of maximum to minimum tensions of front net and tension ties, and the surface root mean square (RMS) error. Although the tension uniformity of 1st geodesic cable net is slightly worse than that of other two geodesic cable nets, it has a higher surface accuracy. On the contrary, the 3rd geodesic cable net has a better tension uniformity
Table 4 Form-finding results. Item
1st geodesic cable net 2nd geodesic cable net 3rd geodesic cable net Triclinic cable net Geotensoid cable net
Tensions of front net
Tensions of tension ties
Mean (N)
Maximum/Minimum
Mean (N)
Maximum/Minimum
10 10 10 10 10
3.2311 2.8434 2.0941 2.8656 2.2811
1.6174 1.6105 1.5887 1.5301 1.4906
1.6206 1.6031 1.4856 1.7216 1.7997
RMS (mm)
0.4076 0.4194 0.4357 0.4300 0.4499
20
H. Deng et al. / Acta Astronautica 119 (2016) 13–21
and a worse surface accuracy. And the 2nd geodesic cable net has the compromise performances. But it is better than the triclinic cable net in both tension uniformity and surface accuracy.
5. Conclusions
formula 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 > > > a ¼ ðP1x P2x Þ þðP1y P2y Þ þðP1z P2z Þ > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < b ¼ ðP2x P3x Þ2 þ ðP2y P3y Þ2 þ ðP2z P3z Þ2 > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > : c ¼ ðP3x P1x Þ2 þðP3y P1y Þ2 þ ðP3z P1z Þ2 ρ ¼ ða þb þ cÞ=2
In order to generate the geodesic cable nets, two main problems are solved in this paper. The geodesic design method and the geodesic cable net generation method are proposed. The main results and conclusions are summarized as follows. (1) By deriving the general geodesic equations of the paraboloidal surface and applying the relaxation method, we propose the general numerical method for geodesic generation, which helps to calculate the geodesics between arbitrary two points on the paraboloidal surface. This method is very efficient and convenient, because of no needs to solve the second order ordinary differential equation. (2) The dynamic boundary adjustment method is presented to make the three-band geodesics intersect in the topological nodes, which can form the geodesic cable net with the prescribed topology, shortest total length and higher surface accuracy. (3) Three types of geodesic cable net have different performances in the aspects of the geometric design and form-finding analysis because of their different boundaries. The 1st geodesic cable net has the larger efficient area. The 2nd geodesic cable net has shorter total length and the larger efficient area. And the 3rd geodesic cable net has the shortest total length.
GetAreaðP1; P2; P3Þ ¼
ðA:1Þ
ðA:2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρðρ aÞðρ bÞðρ cÞ
ðA:3Þ
Appendix B: Intersecting point of two discrete curves For two discrete curves on the paraboloidal surface, we propose a numerical method for calculating the intersecting point. The method is denoted by GetPoint ðC1 ; C2 Þ and the detailed steps are as follows. 1) Projecting the discrete curves onto the x–y plane in the coordinate system, we obtain two planar curves: C1 ¼ fX1 ; Y 1 g and C2 ¼ fX2 ; Y 2 g. 2) As shown in Fig. B.1(a), we can obtain the intersecting region R as follows.
R ¼ fX; Y g ¼ x A X; yA Y xm1 r x rxm2 ; ym1 ry r ym2 ðB:1Þ where
xm1 ¼ max minfX1g; minfX2g
xm2 ¼ minfmaxfX1g; maxfX2gg
ym1 ¼ max minfY1g; minfY2g ym2 ¼ minfmaxfY1g; maxfY2gg And then we get the x-coordinates in the region R
Xp ¼ ðfXg \ fX1 gÞ [ ðfXg \ fX2 gÞ
ðB:2Þ
Acknowledgment This project is supported by National Natural Science Foundation of China (Grant no. 51375360).
Appendix A: Calculation of the triangle area For three spatial points Piðx; y; zÞ; i ¼ 1; 2; 3, the corresponding triangle area can be calculated by Heron's
3) As shown in Fig. B.1(b), by the linear interpolation, we
redraw the two curves according to the above Xp as:
C1p ¼ Xp ; Y 1p and C2p ¼ Xp ; Y 2p . 4) If there is a coincident point between two curves, it is the intersecting point, and then goes to the step 7). If not, go to the
next step.
5) fΔY g ¼ Y 1p Y 2p . If Δyk Δyk þ 1 o 0, we choose four points: p1 xk ; y1k , p2 xk þ 1 ; y1k þ 1 , p3 xk ; y2k and p4 xk þ 1 ; y2k þ 1 , as shown in Fig. B.1(c), where k is the point number.
Fig. B.1. Intersection of two discrete curves. (a) intersecting region R; (b) new curves; (c) intersecting point.
H. Deng et al. / Acta Astronautica 119 (2016) 13–21
6) According to the four points, we obtain the intersecting point by the diagonal equations. x y
! ¼
y1k þ 1 y1k
xk xk þ 1
y2k þ 1 y2k
xk xk þ 1
!1
! y1k þ 1 y1k xk þ xk xk þ 1 y1k y2k þ 1 y2k xk þ xk xk þ 1 y2k
ðB:3Þ 7) Projecting the planar intersecting point back to the paraboloidal surface with Eq. (6), we get the finial intersecting point as follows. GetPoint ðC1 ; C2 Þ ¼ x; y; x2 þ y2 =4f ðB:4Þ
References [1] W. Imbriale, S. Gao, L. Boccia, Space Antenna Handbook, John Wiley & Sons, New York, 2012. [2] T.J. Li, Deployment analysis and control of deployable space antenna, Aerosp. Sci. Technol. 18 (1) (2012) 42–47. [3] S. Ozawa, K. Shintate, A. Tsujihata, 30 m class lightweight large deployable reflector, in: Proceedings of the 5th IEEE European Conference on Antennas and Propagation (EUCAP), Rome, Italy, 2011, pp. 3354–3358. [4] M. Mobrem, S. Kuehn, C. Spier, E. Slimko, Design and performance of astromesh reflector onboard soil moisture active passive spacecraft, in: Proceedings of the IEEE Aerospace Conference, Big Sky, MT, USA, 2012. [5] A. Meguro, K. Shintate, M. Usui, A. Tsujihata, In-orbit deployment characteristics of large deployable antenna reflector onboard Engineering Test Satellite VIII, Acta Astronaut. 65 (9) (2009) 1306–1316. [6] S. Dennis, T. Ann, S. Rodney, M. James, Design, integration, and deployment of the TerreStar 18-meter reflector, in: Proceedings of the 28th AIAA International Communications Satellite Systems Conference, Anaheim, California, USA, 2010, AIAA 2010-8855.
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[7] Z.W. Wang, T.J. Li, X.F. Ma, Method for generating statically determinate cable net topology configurations of deployable mesh antennas, J. Struct. Eng. 141 (7) (2015) 04014182. [8] A.G. Tibert, Deployable Tensegrity Structures for Space Applications (Ph.D. dissertation), Department of Mechanics, Royal Institute of Technology, Sweden, 2002. [9] J. Sánchez-Reyes, R. Dorado, Constrained design of polynomial surfaces from geodesic curves, Comput.-Aided Des. 40 (1) (2008) 49–55. [10] P. Bose, A. Maheshwari, S. Chang, S. Wuhrer, A survey of geodesic paths on 3D surfaces, Comput. Geom.: Theory Appl. 44 (9) (2011) 486–498. [11] M.W. Thomson, The Astromesh deployable reflector, in: Proceedings of IEEE Antennas and Propagation Society International Symposium, 3, 1999, pp. 1516–1519. [12] S. Morterolle, B. Maurin, J. Quirant, C. Dupuy, Numerical formfinding of geotensoid tension truss for mesh reflector, Acta Astronaut. 76 (2012) 154–163. [13] B. Yang, H. Shi, M. Thomson, H. Fang, Optimal design of initial surface profile of deployable mesh reflectors via static modeling and quadratic programming, in: Proceedings of the 50th AIAA/ASME/ ASCE/AHS/ ASC Structural Dynamics and Material Conference, Palm Springs, California, 2009, AIAA 2009-2173. [14] H. Shi, B. Yang, H. Fang, Offset-feed surface mesh generation for design of space deployable mesh reflectors, in: Proceedings of the 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Boston, Massachusetts, 2013, AIAA 20131526. [15] F.E. Wolter, Interior Metric, Shortest Paths and Loops in Riemannian Manifolds with not Necessarily Smooth Boundary (Master's thesis), Free University of Berlin, Berlin, 1979. [16] N.M. Patrikalakis, T. Maekawa, Shape Interrogation for Computer Aided Design and Manufacturing, Springer, New York, 2002. [17] A.G. Tibert, S. Pellegrino, Furlable reflector concept for small satellites, in: Proceedings of the 42nd AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics, and Material Conference, Seattle, WA, 2001, AIAA 2001-1261. [18] H.Q. Deng, T.J. Li, Z.W. Wang, X.F. Ma, Pretension design of space mesh reflector antennas based on projection principle, J. Aerosp. Eng. 28 (6) (2015) 04014182. 10.1061/(ASCE)AS. 19435525.0000483.
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Design of geodesic cable net for space deployable mesh reflectors Hanqing Deng, Tuanjie Li n, Zuowei Wang School of Mechano-Electronic Engineering, Xidian University, P.O. Box 188, Xi'an 710071, China
a r t i c l e i n f o
abstract
Article history: Received 17 December 2014 Received in revised form 18 September 2015 Accepted 31 October 2015 Available online 10 November 2015
The main purpose of this paper is to design the geodesic cable net for space deployable mesh reflectors by means of the geodesic differential equation, which is different from the traditional quasi-geodesic cable net. In order to generate the geodesic cable net, two main problems must be addressed. One is to generate the geodesics on the paraboloidal surface, and the other is to form the geodesic cable net according to the prescribed topology. Therefore, through the analysis of geodesic curvature, a general numerical method is presented to calculate the geodesics between arbitrary two points on the paraboloidal surface. Then according to the structural characteristics of mesh reflectors, a dynamic boundary adjustment method is proposed to address the second problem. The key of the method is an optimization model that adjusts the internal geodesics to form the triangular cable net by moving the end-points along the dynamic boundaries. Finally, the feasibility and effectiveness of the proposed design method of geodesic cable net are validated by three numerical examples. & 2015 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Mesh reflectors Geodesics Dynamic boundary adjustment method Geodesic cable net
1. Introduction The demands of mobile communication are increasing greatly since 21st century, especially the coming of 3G, 4G and 5G communication networks. As one of the most important components for high-end communication technologies, space deployable reflectors are grown rapidly [1]. For the disadvantages of small aperture and high weight in the solid surface reflectors, the large-size reflectors applying to space communication are all deployable, including several types [2], such as deployable mesh reflectors, semirigid antennas, and inflatable antennas etc. As the characteristics of lightweight, large aperture and simple structure, deployable mesh reflectors have the majority in-orbit applications [3]. The AstroMesh hoop truss n
Corresponding author. Tel.: þ 86 29 88202470. E-mail address: [email protected] (T. Li).
http://dx.doi.org/10.1016/j.actaastro.2015.10.024 0094-5765/& 2015 IAA. Published by Elsevier Ltd. All rights reserved.
antenna [4] was first applied in the telecommunication satellite Thuraya with the diameter of 12.25 m and the weight of 55 kg. The Engineering Test Satellite VIII [5] has two large deployable reflectors with the dimension of 19 m 17 m. Satellites TerreStar and Skyterra [6] are both unfurlable reflectors with the aperture of 18 m and 22 m, respectively. The deployable mesh reflector consists of cable nets, and a deployable frame structures. And the cable net is thus becoming an indispensable component for deployable mesh reflectors. One important issue is the geometric design which includes the topological configuration and the geometric generation. The main considerations of topology design are the kinematically indeterminate supporting structures and statically indeterminate cable net structures [7]. Several topological configurations have been developed, such as triangular, quadrilateral, and hexagon topological configurations etc. For the given triangular topological configuration, two kinds of cable nets can be generated: triclinic and geodesic cable nets. The triclinic cable net has
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H. Deng et al. / Acta Astronautica 119 (2016) 13–21
been widely applied, which is generated by dividing the inscribed hexagon of circular truss into triangular grid and then mapping onto the paraboloid [8]. And the geodesic cable net is created by the geodesic theory or some geodesic characteristics. If the curvature of each point in a plane curve is equal to zero, the curve is a straight line. Similarly, if the geodesic curvature of each point in a spatial curve is equal to zero, the curve is called a geodesic line. In the intrinsic geometry, the geodesics act as straight lines in a plane. Another definition of geodesics is the shortest path between two points on a curved surface. For this reason, the geodesics are widely used in the geometric design of ship hulls, robot motion planning, terrain navigation and computer 3D modeling [9,10]. On the IUTAM-IASS Symposium, Thomson firstly proposed the concept of “pseudogeodesic” to use in Astromesh hoop truss antennas [11]. Later, in order to generate the triangular cable net with the shortest lengths, the “quasi-geodesic cable net” was used and named “geotensoid” by Tibert [8], in 2002. Then in 2012, Morterolle et al. [12] proposed a numerical formfinding method of geotensoid tension truss for mesh reflectors. They believed that a cable net with uniform tension distribution was the shortest. During this period, some other methods for “geodesic cable net” generation were presented, such as the method proposed by Yang et al. [13,14]. All the existing methods use the concept of “pseudo” or “quasi” geodesic, which has the characteristic of the shortest length. Nevertheless, how to deduce the geodesic cable net from the mathematical point of view is very essential. Based on the geodesic theory in the differential geometry, we propose a general method to design the geodesic cable net for space deployable mesh reflectors. This paper is structured as follows. First, a general numerical method is presented in Section 2 to generate geodesics on the paraboloidal surface. Then a dynamic boundary adjustment method is proposed in Section 3 to form the geodesic cable net with the prescribed topology. And then, three numerical examples are given in Section 4 to illustrate the generation of geodesic cable nets. Finally, some conclusions are summed up in Section 5.
shortest path between two points always exists on the surface. To analyze the shortest path, a parametric surface S is denoted by rðu; vÞ, where u and v are the parametric coordinates, similar to the x, y and z components in the Cartesian coordinates. And a curve C on the surface is denoted by rðtÞ ¼ rðuðtÞ; vðtÞÞ with respect to the parameter t. By differentiating the parametric equation of the curve C, we obtain dr ¼ r u du þ r v dv
ð1Þ
The first fundamental form of the surface S is 2
ds ¼ dr 2 ¼ Edu2 þ2Fdudv þGdv2
ð2Þ
where s is the arc length, E; F and G are called the coefficients of the first fundamental form and given by E ¼ r 2u ¼
∂r ∂r ∂u ∂u
∂r ∂r ∂u ∂v ∂r ∂r G ¼ r 2v ¼ ∂v ∂v F ¼ ru rv ¼
ð3Þ
According to the geodesic curvature equation [16], the geodesic curvature of the curve C is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κg ¼ EG F 2 u″ þ Γ 111 u02 þ 2Γ 112 u0 v0 þ Γ 122 v02 v0 ð4Þ v″ þ Γ 211 u02 þ 2Γ 212 u0 v0 þ Γ 222 v02 u0 with the Christoffel symbols Γ ijk ði; j; k ¼ 1; 2Þ 8 2FF u þ FEv > Γ 111 ¼ GEu2ðEG > F2Þ > > > > GEv FGu 1 > > Γ 12 ¼ 2ðEG F 2 Þ > > > > > Γ 1 ¼ 2GF v GGu þ FGv > < 22 2ðEG F 2 Þ u EF v þ FEu > Γ 211 ¼ 2EF2ðEG > > F2 Þ > > > > Γ 2 ¼ EGu FE2v > > 12 2ðEG F Þ > > > > 2FF v þ FGu > Γ : 222 ¼ EGv2ðEG F2Þ
ð5Þ
where Eu ; Ev ; F u ; F v ; Gu and Gv are partial derivatives of E; F and G with respect to u and v, respectively. 2.2. Geodesics on paraboloid
2. Geodesics on paraboloid of revolution In Cartesian coordinate system, f denotes the focal length, the standard equation of the paraboloid is
2.1. Geodesic curvature For a periodic and aperiodic non-uniform rational Bspline surface (NURBS), Wolter [15] has proved that the
z¼
x2 þy2 4f
Fig. 1. Composition of a hoop truss reflector antenna.
ð6Þ
H. Deng et al. / Acta Astronautica 119 (2016) 13–21
15
Fig. 2. Cable net consisted of geodesics. (a) Cable net topology and (b) cable net consisted of geodesics.
Eq. (6) can be expressed as the parameter equation ! u2 rðu; vÞ ¼ u cos ðvÞ; u sin ðvÞ; ð7Þ 4f The first fundamental form of Eq. (7) is obtained ! u2 2 ds ¼ 1 þ 2 du2 þ u2 dv2 4f
ð8Þ
in which, the relevant parameters are E ¼ 1þ
Γ 111 ¼
u2 4f
2
;
G ¼ u2 2
u 2
4f þu2
Γ 211 ¼ 0;
F ¼ 0;
;
Γ 112 ¼ 0;
1 Γ 212 ¼ ; u
Γ 222 ¼ 0
Γ 122 ¼
4f u 2
4f þu2 ð9Þ Fig. 3. Geodesic boundaries and dynamic boundary nodes.
Substituting Eq. (9) into Eq. (4), the geodesic curvature of the paraboloid is obtained sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" ! # 2 u4 u u02 4f u v02 0 2 κ g ¼ u2 þ 2 u″ þ 2 2 v v″ þ u0 v0 u0 u 4f 4f þ u2 4f þ u2
general geodesic equations of the paraboloid 8 2 02 < u″ þ u2 u02 4f 2 u v ¼ 0 2 2 4f þ u
4f þ u
: v″ þ 2u0 v0 ¼ 0
ð11Þ
u
ð10Þ According to the definition of geodesic, the geodesics are curves with zero geodesic curvature, i.e. κ g 0. The curves on the paraboloid can be divided into the following four situations. 0
0
1) When u ¼ 0; v ¼ 0, the κg is equal to zero, which indicates the curve is just a point. 2) When u0 ¼ 0; v0 a0, u can be denoted by a constant u ¼ c0 , and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4f c0 v03 c0 2 κg ¼ 2 1þ 2 4f þ c0 2 4f If and only if c0 ¼ 0, the κ g is equal to zero, which indicates the curve is also a point. 3) When u0 a 0; v0 ¼ 0, κ g 0 is obtained by substituting them into Eq. (10). So the curves are the geodesics, as well as the generatrices of the paraboloid. 4) When u0 a0; v0 a0, in order to obtain the geodesics, the coefficients of Eq. (10) must be zeros, yielding the
2.3. Solution of geodesic equation By combining Eq. (11), we have 2 v″=v0 ¼ u0 u
ð12Þ
Integrating both sides of Eq. (12), it yields 0
v ¼
c1 u2
where c1 is a constant. Substituting Eq. (13) into Eq. (8), we obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c1 u2 þ 4f dv ¼ du 2f u u2 c1 2
ð13Þ
ð14Þ
Then integrating both sides of Eq. (14), the expression of v we can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 c1 u2 þ4f v¼ duþ c2 ð15Þ 2f u u2 c1 2 where c2 is a constant. c1 and c2 can be solved with the
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H. Deng et al. / Acta Astronautica 119 (2016) 13–21
Fig. 4. Geodesic boundaries of geodesic cable net.
boundary conditions of the end-points AðuA ; vA Þ and BðuB ; vB Þ. Because the integration expression of Eq. (15) is too complicated to be solved directly, the relaxation method [16] is adopted to solve the second order ordinary differential equation, Eq. (11). Eq. (11) is transformed into the first order ordinary differential equations 8 du > > ¼p > > ds > > > > > dv > > > < ds ¼ q 2 ð16Þ dp u p2 4f u q2 > > ¼ 2 þ 2 > > > ds 4f þ u2 4f þ u2 > > > > > dq 2 > > : ¼ pq ds u
which can be rewritten as follows. Fk ¼ ðF 1k ; F 2k ; F 3k ; F 4k ÞT ¼ ¼ 0;
Yk Yk 1 1 ðG þ Gk 1 Þ sk sk 1 2 k
k ¼ 2; 3; U U U ; m
ð21Þ 0
Given a starting vector Y , Eq. (21) can be solved by the quadratically convergent Newton's method with the following iteration scheme 8 Fk ¼ Jaco ΔY k > < ð22Þ > :Y ¼ Y þ ΔY kþ1 k k where Jaco is the Jacobian matrix of F with respect to Y.
3. Geodesic cable net
Eq. (16) can be rewritten as 8 dy > > > < ds ¼ g > yðAÞ ¼ αA > > : yðBÞ ¼ α B
3.1. Composition of mesh reflectors ð17Þ
with y ¼ ðu; v; p; qÞT g¼
!T 2 p; q; 2 þ 2 ; pq u 4f þ u2 4f þ u2 u p2
2
4f u q2
ð18Þ
A deployed mesh reflector is conceived with the concept of tension truss, which is a light and inherently structure that can be precisely and repeatedly deployed regardless of environment. As illustrated in Fig. 1, the hoop truss reflector antenna is consisted of five parts: supporting truss, front net, tension ties, back net (or rear net) and the RF reflective mesh fixed to the front net. 3.2. Difficulty in geodesic cable net design
and the boundary conditions αA ¼ ðuA ; vA ; pA ; qA ÞT αB ¼ ðuB ; vB ; pB ; qB ÞT
ð19Þ
where s A ½A; B, points A and B are two end-points of the geodesic. The geodesic between points A and B, as an arc length parameterized curve with parameter s, is discretized by the finite differences with m points, satisfying A ¼ s1 o s2 o :: : o sm ¼ B. Then Eq. (17) is approximated by the trapezoidal rule Y k Y k 1 1 ¼ ðGk þ Gk 1 Þ; 2 sk sk 1
k ¼ 2; 3; U U U; m
ð20Þ
where the boundary conditions are Y 1 ¼ αA ; Y m ¼ αB . According to the boundary conditions, Eq. (20) has ðm 1Þ 4 unknown variables and ðm 1Þ 4 equations,
According to the numerical method in Section 2, it is easy to get the geodesic between arbitrary two points on the paraboloidal surface. Nevertheless, there is still a difficulty to generate the geodesic cable net, as shown in Fig. 2. It can be seen from Fig. 2 that the cable net consisted of geodesics is not coincident with its topological configuration, in which all cables form the prescribed triangular cable net. However, in the cable net consisted of geodesics with equally spaced nodes along the circumference, a set of three-band geodesic intersections appear in the positions of “topological node”, as shown in the enlarged portion in Fig. 2(b). This area is called as the “irregular region”. These three-band geodesics do not intersect at unique points, and thus result in some polygons generated around the positions. Here, the unique point existing in the prescribed topology is defined as “topological node”.
H. Deng et al. / Acta Astronautica 119 (2016) 13–21
17
3.3. Dynamic boundary adjustment method
Fig. 5. Geodesic generation between two points.
Fig. 6. Iteration process of Newton's method. Table 1 Geometric parameters. Parameters Aperture (m) Focal length (m)
Subdivision number
Value
9
20
10
Discrete number
In order to form the triangular cable net inside the hexagon, the geodesic boundaries of the cable net must be firstly determined. In the design phase, a cable net is always selected because of the shorter total length, the larger efficient area (the area of inscribed hexagon) and the higher surface accuracy. According to these requirements, some types of geodesic boundaries are available, as shown in Fig. 4. The first type is a pure geodesic boundary consisting of a geodesic between two vertexes of the inscribed hexagon. The second type is a “straight line” boundary, which is a curve on the paraboloidal surface that is mapped by the straight line between the two vertexes. The third type is a curve with the sag-to-span ratio [17]. After determining the geodesic boundaries of the cable net, the dynamic boundary nodes can be obtained and the internal geodesics can be calculated with the end-points. On account of the difficulty in the geodesic cable net design, a mathematical model for generating the triangular cable net is established to improve the intersection situation of internal geodesics. It is assumed that the irregular region exists in each position of topological node in the internal geodesic cable net (a three-band intersection is regarded as an infinitesimal region with zero area). The intersection situation of internal geodesics of the dynamical adjustment process can be then measured by the areas of irregular regions. The positions of dynamic nodes in the geodesic boundaries are regarded as the design variables, the minimum areas of irregular regions are regarded as the objective function, and some structural characteristics are regarded as the constraints. The dynamical boundary adjustment method is thus descried as an optimization model, that is: Find
M ki
ðk ¼ 1; 2; ⋯; 6;
1000
min Such a situation highlights the difficulty in geodesic cable net design: how to form a triangular cable net which is coincident with its prescribed topology. In this paper, the concept of “dynamic boundary node” is proposed to adjust the internal geodesics, as shown in Fig. 3. The dynamic boundary nodes are the end-points of internal geodesics, which offer a way to improve the intersection situation of internal geodesics. Because the nodes on the truss are usually related to the manufacture, they must be the uniform nodes. So the dynamic boundary nodes can only be selected inside the cable net. Such a dynamic boundary that offers the end-points of internal geodesics is called as the “geodesic boundary” here. Before a cable net is generated, only the six vertexes of the inscribed hexagon are fixed points. Two feasible kinds of geodesic boundaries can be determined by the fixed points. One is the diagonal lines of hexagon, and the other is the six sides of hexagon that is adopted in this paper. Therefore, the geodesic cable net can be divided into three parts: internal geodesic cables, geodesic boundaries and the boundary cables connected to the supporting truss.
z¼
N um X
sΔj
i ¼ 1; 2; ⋯; N 1Þ
ðj ¼ 1; 2; ⋯; NumÞ
j¼1
s:t: with
1 o Mki oM sΔj o ϵ sΔj ¼ GetArea P m;n ; P n;q ; P q;m P m;n ¼ GetPoint ðC m ; C n Þ
ð23Þ
where M ki is the position of the end-point in the geodesic boundary. M is the number of the points in the geodesic boundary. k is the serial number of the six vertexes. And N is the subdivision number of one side of the inscribed hexagon. sΔj is the area of the irregular region in the jth topological node. Num is the total number of topological nodes. ε is the allowance error. GetArea P m;n ; P n;q ; P q;m denotes a function to get the area of the irregular region consisting of the three points P m;n ; P n;q and P q;m . And GetPoint ðC m ; C n Þ denotes a function to get the intersection point from two discrete curves C m and C n . A detailed description of GetArea P m;n ; P n;q ; P q;m and GetPoint ðC m ; C n Þ are given in the Appendix. The dynamic boundary adjustment method is described in detail as follows. First, according to the structural parameters
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H. Deng et al. / Acta Astronautica 119 (2016) 13–21
Fig. 7. Five cable nets. (a) 1st geodesic cable net; (b) 2nd geodesic cable net; (c) 3rd geodesic cable net; (d) triclinic cable net; (e) geotensoid cable net.
Table 2 Results of the cable nets. Item
1st geodesic cable net
2nd geodesic cable net
3rd geodesic cable net
Triclinic cable net
Geotensoid cable net
Lengths (m) Efficient area (m2)
998.66 268.96
996.17 259.81
992.83 241.11
998.04 259.81
993.81 241.67
H. Deng et al. / Acta Astronautica 119 (2016) 13–21
of the reflector, such as the aperture, focus length, cable net topological configuration and the subdivision number etc., the appropriate geodesic boundaries for the geodesic cable net are selected. Second, some points of the six geodesic boundaries are selected to act as the dynamic boundary nodes. Finally, by moving the dynamic boundary nodes along the geodesic boundaries, the intersection situation of internal geodesics is adjusted to form the triangular cable net.
4. Numerical simulations 4.1. Generation of single geodesic This simulation is to calculate the geodesic between two points (10, 0, 2.5) and (0, 10, 2.5) on the paraboloidal surface with the focus length of 10 m. As shown in Fig. 5, the curve mapped by the straight line between the two points is chosen as the initial curve. The coordinates of 1000 discrete points of the initial curve are taken as the initial iteration values of Newton's method. The calculated geodesic between two points is shown in Fig. 5 and the iteration process is shown in Fig. 6. In the numerical simulation, the generated geodesic is 0.0330 m shorter than the initial curve. In Fig. 6, the iterative error is denoted as the 2-norm of F, and the iteration process is convergence to the acceptable error of 7.6393 10 9. 4.2. Generation of geodesic cable net In this simulation, cable nets of the mesh reflector with AM1 configuration proposed by AstroMesh [7] are taken as examples. The geometric parameters are listed in Table 1. The geodesic cable nets with three types of geodesic boundaries, the triclinic and geotensoid cable nets are designed, respectively. The projection views from the aperture plane of them are shown in Fig. 7. Fig. 7(d) is the triclinic cable net generated by dividing the inscribed hexagon into the triangular grid and then mapping onto the paraboloid. Fig. 7(e) is the geotensoid cable net Table 3 Geometrical and material parameters. Item
Diameter (m)
Elastic modulus (GPa)
Poisson ratio
Cables Truss
0.002 0.02
20 200
0.3 0.3
19
designed by the equal tension method [12]. The analysis result of these cable nets are listed in Table 2. The “Subdivision number” is the number of cables in the radius. It can be noted from Fig. 7 and Table 2 that three types of geodesic cable nets are different with each other because of different geodesic boundaries. Compared with the triclinic cable net, the first type enlarges the efficient area with the bulgy pure geodesic boundary, but the total length increases a little. The second type has the same “straight line” geodesic boundaries with the triclinic cable net. So it has the same efficient area but a shorter total length. And due to the concave geodesic boundaries with the sag-to-span ratio of 0.1, the third type has a smaller efficient area but the shortest total length. Among these cable nets, the performance of the 3rd geodesic cable net is close to the geotensoid cable net. 4.3. Form-finding analysis The form-finding analysis of the reflectors shown in Fig. 7 is performed in this simulation. The geometrical and material parameters of the cables and truss are listed in Table 3. The pretension distributions of the three geodesic cable nets and the triclinic cable net are designed by the plane projection method [18]. The plane projection method is based on the projection principle. The equilibrium tension of the projected plane cable net structure is firstly obtained by a force equilibrium method. The obtained cable tensions of the plane cable net structure are then projected to the spatial cable net for obtaining the spatial cable tensions. The pretension of the geotensoid cable net is calculated by the equal tension method [12]. The equal tension method is based on a force density strategy coupled with geometrical constraints and the uniform tension is achieved by iterating force density coefficients. For a topological configuration shown in Fig. 2, the pretension of the front net in the geotensoid cable net equals each other. But due to the special boundary cables, the pretensions of the geotensoid cable net are not all the same for the AM1 configuration. The back net of each reflector is the same as the front net in size, as well as the pretension in value. The detailed form-finding results are listed in Table 4. It is noted from Table 4 that the performances of the 3rd geodesic cable net are better than the geotensoid cable net, including the ratios of maximum to minimum tensions of front net and tension ties, and the surface root mean square (RMS) error. Although the tension uniformity of 1st geodesic cable net is slightly worse than that of other two geodesic cable nets, it has a higher surface accuracy. On the contrary, the 3rd geodesic cable net has a better tension uniformity
Table 4 Form-finding results. Item
1st geodesic cable net 2nd geodesic cable net 3rd geodesic cable net Triclinic cable net Geotensoid cable net
Tensions of front net
Tensions of tension ties
Mean (N)
Maximum/Minimum
Mean (N)
Maximum/Minimum
10 10 10 10 10
3.2311 2.8434 2.0941 2.8656 2.2811
1.6174 1.6105 1.5887 1.5301 1.4906
1.6206 1.6031 1.4856 1.7216 1.7997
RMS (mm)
0.4076 0.4194 0.4357 0.4300 0.4499
20
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and a worse surface accuracy. And the 2nd geodesic cable net has the compromise performances. But it is better than the triclinic cable net in both tension uniformity and surface accuracy.
5. Conclusions
formula 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 > > > a ¼ ðP1x P2x Þ þðP1y P2y Þ þðP1z P2z Þ > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < b ¼ ðP2x P3x Þ2 þ ðP2y P3y Þ2 þ ðP2z P3z Þ2 > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > : c ¼ ðP3x P1x Þ2 þðP3y P1y Þ2 þ ðP3z P1z Þ2 ρ ¼ ða þb þ cÞ=2
In order to generate the geodesic cable nets, two main problems are solved in this paper. The geodesic design method and the geodesic cable net generation method are proposed. The main results and conclusions are summarized as follows. (1) By deriving the general geodesic equations of the paraboloidal surface and applying the relaxation method, we propose the general numerical method for geodesic generation, which helps to calculate the geodesics between arbitrary two points on the paraboloidal surface. This method is very efficient and convenient, because of no needs to solve the second order ordinary differential equation. (2) The dynamic boundary adjustment method is presented to make the three-band geodesics intersect in the topological nodes, which can form the geodesic cable net with the prescribed topology, shortest total length and higher surface accuracy. (3) Three types of geodesic cable net have different performances in the aspects of the geometric design and form-finding analysis because of their different boundaries. The 1st geodesic cable net has the larger efficient area. The 2nd geodesic cable net has shorter total length and the larger efficient area. And the 3rd geodesic cable net has the shortest total length.
GetAreaðP1; P2; P3Þ ¼
ðA:1Þ
ðA:2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρðρ aÞðρ bÞðρ cÞ
ðA:3Þ
Appendix B: Intersecting point of two discrete curves For two discrete curves on the paraboloidal surface, we propose a numerical method for calculating the intersecting point. The method is denoted by GetPoint ðC1 ; C2 Þ and the detailed steps are as follows. 1) Projecting the discrete curves onto the x–y plane in the coordinate system, we obtain two planar curves: C1 ¼ fX1 ; Y 1 g and C2 ¼ fX2 ; Y 2 g. 2) As shown in Fig. B.1(a), we can obtain the intersecting region R as follows.
R ¼ fX; Y g ¼ x A X; yA Y xm1 r x rxm2 ; ym1 ry r ym2 ðB:1Þ where
xm1 ¼ max minfX1g; minfX2g
xm2 ¼ minfmaxfX1g; maxfX2gg
ym1 ¼ max minfY1g; minfY2g ym2 ¼ minfmaxfY1g; maxfY2gg And then we get the x-coordinates in the region R
Xp ¼ ðfXg \ fX1 gÞ [ ðfXg \ fX2 gÞ
ðB:2Þ
Acknowledgment This project is supported by National Natural Science Foundation of China (Grant no. 51375360).
Appendix A: Calculation of the triangle area For three spatial points Piðx; y; zÞ; i ¼ 1; 2; 3, the corresponding triangle area can be calculated by Heron's
3) As shown in Fig. B.1(b), by the linear interpolation, we
redraw the two curves according to the above Xp as:
C1p ¼ Xp ; Y 1p and C2p ¼ Xp ; Y 2p . 4) If there is a coincident point between two curves, it is the intersecting point, and then goes to the step 7). If not, go to the
next step.
5) fΔY g ¼ Y 1p Y 2p . If Δyk Δyk þ 1 o 0, we choose four points: p1 xk ; y1k , p2 xk þ 1 ; y1k þ 1 , p3 xk ; y2k and p4 xk þ 1 ; y2k þ 1 , as shown in Fig. B.1(c), where k is the point number.
Fig. B.1. Intersection of two discrete curves. (a) intersecting region R; (b) new curves; (c) intersecting point.
H. Deng et al. / Acta Astronautica 119 (2016) 13–21
6) According to the four points, we obtain the intersecting point by the diagonal equations. x y
! ¼
y1k þ 1 y1k
xk xk þ 1
y2k þ 1 y2k
xk xk þ 1
!1
! y1k þ 1 y1k xk þ xk xk þ 1 y1k y2k þ 1 y2k xk þ xk xk þ 1 y2k
ðB:3Þ 7) Projecting the planar intersecting point back to the paraboloidal surface with Eq. (6), we get the finial intersecting point as follows. GetPoint ðC1 ; C2 Þ ¼ x; y; x2 þ y2 =4f ðB:4Þ
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