Tie-System Calibration for the Experimental Setup of Large Deployable Reﬂectors

: The trade-off between the design phase and the experimental setup is crucial in satisfying the accuracy requirements of large deployable reﬂectors. Manufacturing errors and tolerances change the root mean square (RMS) of the reﬂecting surface and require careful calibration of the tie-rod system to be able to ﬁt into the initial design speciﬁcations. To give a possible solution to this problem, two calibration methods—for rigid and ﬂexible ring truss supports, respectively—are described in this study. Starting from the acquired experimental data on the net nodal co-ordinates, the initial problem of satisfying the static equilibrium with the measured conﬁguration is described. Then, two constrained optimization problems (for rigid or ﬂexible ring truss supports) are deﬁned to meet the desired RMS accuracy of the reﬂecting surface by modifying the tie lengths. Finally, a case study to demonstrate the validity of the proposed methods is presented.


Introduction
Satellite communications have become increasingly wide-spread, thanks to the possibility of having high transmitting capacities at a relatively low cost, compared to the establishment of a terrestrial broadcasting network. Deployable reflectors represent the most widely-used type of structure, due to their features, such as large scale, high packaging efficiency, high accuracy, and low weight. Typically, their architecture consists of a deployable ring truss support, two cable nets (facing each other and linked by a series of tension ties), and a RF mesh attached to the backside of the front net. The most representative kind of this type of reflector is the AstroMesh [1], and its components are shown in Figure 1.
The electromagnetic performance of these antennae is closely related to the shape of the reflector surface. In turn, this depends on the position of the nodes located on the front net. Therefore, it is clear that the measurement process of 3D node co-ordinates needs to be made with extreme accuracy, so as to avoid invalidating the real root mean square (RMS) value.
Several measurement systems have been developed, depending on the application area. Generally, they can be divided into three categories: • Photogrammetry, • Laser tracker, and • Laser radar.
Photogrammetry is a measurement technique that uses two-dimensional images of an object to obtain its dimensions.
As depicted in Figure 2, photogrammetry is based exclusively on angle measurements: Three-dimensional co-ordinates are calculated by an optical triangulation (or intersection) of two or more images taken from different positions. The object to be measured is identified by targets mounted on it, usually made of reflective material to produce a high contrast between the target and the background [2]. Typically, a calibrated scale bar is integrated into the object, in order to reproduce it in true scale. At the end of the measurements, a dedicated software calculates the 3D co-ordinates in the chosen Cartesian co-ordinate system (x, y, z). Laser trackers and laser radars, alternatively, are measurement systems based on the estimation of two angles and one length, as shown in Figure 3. Two high-resolution encoders measure the azimuth (θ) and elevation (ϕ) angles, whereas the radial co-ordinate (r) relative to the center of the target is measured by means of optical interference [3]. Several types of target can be mounted, but the most widely-used is the spherically mounted retro-reflector (SMR). Differing from the laser tracker, a laser radar does not require a retro-reflector. As a matter of fact, it is capable of measuring the surface of an object with just 1% of the reflected signal [4].
Each of these measurement systems, however, has advantages and disadvantages, which need to be assessed on the basis of various factors. Van Gestel et al. [4] identified influencing factors to be taken into account before making the measurement: Task requirements, part restrictions, and environmental restrictions. Regarding task requirements, the main element to be considered when measuring the position of the nodes is accuracy, since the permissible RMS error on the reflecting surface is less than 1 mm. Whilst the laser tracker has errors on the order of 1 µm, it is too costly due to the high price of each SMR and given the number of nodes (at least a hundred). The photogrammetry turned out to be the best choice [5,6], considering the affordable cost and accuracy of the measure. Furthermore, the possibility of obtaining multiple images from different angles makes it possible to overcome the lower accuracy of the angular encoders of laser trackers.
The acquisition of node co-ordinates is essential for the next calibration step, in order to meet the RMS design requirements of the reflecting surface. Generally, the main strategies followed by LDR developers and designers is to use the tie system, connecting front and rear nets, in order to locally move single nodes. This operation is long and delicate, but allows the adjustment of different error sources, such as manufacturing errors [7][8][9], material definition errors, clearance [10,11], friction [12,13], hysteresis [14], mechanical vibrations [15,16], and imperfect behaviour of the elastic properties of components. This paper describes a method for the tie-system calibration of LDRs with a rigid or flexible ring truss. To our knowledge, this topic has not been deeply investigated in the literature and the adjustment phase is entrusted with proprietary solutions of LDR companies. The outline of this paper is as follows. In Section 2, the problem of correcting the parameters to satisfy the static equilibrium in the deployed configuration is first addressed. Then, the method for finding the necessary corrections to the tie-system is discussed, for the two cases of LDR (i.e., with rigid or flexible ring truss). In Section 3, the method is applied to a LDR with an asymmetric ring truss, developed by Thales Alenia Space. A simulated error distribution is superimposed to the design configuration to represent a real experimental test. The tie-system corrections (expressed in terms of length elongation or shortening) necessary to meet RMS design requirement are obtained for both rigid and flexible ring truss cases. Finally, in Section 4, the conclusions of the paper are presented.

Experimental Settings for Calibration
Once the antenna has been manufactured, it is necessary to carry out some experimental tests in order to check the RMS of the reflector in the deployed configuration. To do this, the first operation consists of measuring the position of all nodes of the nets with respect to a reference system through one of the methods described above. Due to different sources of errors, such as manufacturing errors or assembly errors, the deployed configuration will be different from the design configuration and the RMS will usually be greater than the design requirement. Even the measurement operation of the node co-ordinates will be affected by errors. Laser trackers and photogrammetry have errors on the order of 1 µm; while laser radar reaches 0.1 µm. Errors related to the measurement systems are, regardless, significantly below the preceeding mechanical errors. In the following, two methods for the experimental setting of the rigid and flexible ring truss supports are described.

Rigid Ring Truss Support: Construction Length Determination
Tie-cable regulation is usually independent of the actuators used for the LDA deployment [38][39][40] and from the control system [41][42][43][44]. Here, the method to regulate tie cable tension in an antenna with rigid ring truss support is first described. Once the deployment has been carried out and all node co-ordinates of the net have been measured, the reflective surface will typically deviate from the design configuration due to mechanical errors. As a consequence of this deviation, the system of equilibrium equation is not satisfied for the current configuration. Then, denoting with E ij , A ij , Lc ij , and Lc 0 ij the Young modulus, the cross-section area, the measured length, and the construction length of cable ij, respectively, and with k ij , Lt ij , and Lt 0 ij the spring constant, the measured length, and the construction length of tie ij, respectively, the system of nonlinear equations for each free node i, with j adjacent nodes, is not satisfied: where x l , y l , and z l are the measured Cartesian co-ordinates of the l-th node. It is noteworthy to remark that the only measured parameters are the node co-ordinates; while the measured length is derived using the Euclidian norm. The remaining parameters of the previous system are design parameters, instead; each affected by different types of error. Here, denoting by L ij and L 0 ij the measured and construction length for both cable and tie, respectively, we choose to gather all these errors inside the construction length L 0 ij , defined as the distance between the centers of the eyelets i and j belonging to the same cable, as shown in Figure 4. This choice is motivated by the fact that the construction length of the cables is affected by two main sources of errors, such as the manufacturing errors generated during the cutting operation of (Computer Numerical Control) CNC machines [45], and the assembly errors coming from the bad placement of the eyelets (which are necessary to connect two or more cables). In order to restore the equilibrium condition in System (1), only the design parameters can be adjusted, while the measured parameters describe the real configuration of equilibrium: The measured configuration is already in equilibrium and, thus, System (1) is to be satisfied in this configuration without changing the node co-ordinates.
Then, as each cable works in traction force only, the constraint L ij ≥ L 0 ij must be imposed. To solve the system of nonlinear equations, Matlab c provides the command fsolve, but it does not allow the inclusion of any constraints. To overcome this problem, the nonlinear programming solver fmincon can be used by giving a constant objective function and setting (1) as the nonlinear equality constraint, in addition to the linear inequality constraint L ij ≥ L 0 ij . The resulting constrained optimization problem is described below: Hence, a first check is required. As the springs used in tension ties require a pre-stress value, here denoted by F 0 , representing the minimum value necessary for their activation, the condition F ij ≥ F 0 ij for each tension tie cable must be verified, where is the spring force of tie ij. If the condition is satisfied, the next step can be conducted; otherwise, the spring belonging to the tie which has failed the test must be replaced and the algorithm starts again by calculating the array L 0 of all construction lengths.
L Figure 4. Layout of a cable: The construction length L 0 is affected by manufacturing errors generated during the cutting operation and assembly errors coming from a bad placement of the eyelets necessary to connect two or more cables.

Rear Node Determination
In the event that only the co-ordinates of the nodes of the front net are known by measurement, before implementing the algorithm described above, the rear node co-ordinates have to be estimated. Their co-ordinates are initialized with the design data and System (1) is implemented for each free node of the front and rear net. Note the dualism between the two methods: In the former, the unknowns are the construction lengths; in the latter, they are the rear node co-ordinates. Next, the construction lengths are obtained as a consequence of using the Euclidian norm. We checked that both methods lead to the same results (except for negligible errors) if the experimental configuration is not too distant from the design one.

Rigid Ring Truss Support: Tie Calibration
Once the configuration satisfying the static equilibrium has been found, the algorithm continues with the estimation of the values of stretching or shortening for each tension tie-cable, which ensures that the surface accuracy of the reflector can be met. Figure 5 shows the screw adjustment system for a tie. One fixed part is connected to a node of the front net, while one mobile part-adjustable with a screw-is connected to a node of the rear net. Now, the system of nonlinear equations of each free node i can be written as follows: Figure 5. Screw adjustment in the tie system: (left) Tie system before adjustment; and (right) tie system after the screw regulation.
This system, similar to (1), is used to determine the vector L 0 , with the difference that the variables to be found are the stretching/shortening values δLt ij and the co-ordinates of the free nodes of the front and rear net. The values δLt ij can be positive or negative: Here, we assume positive values for tie-shortening and negative for tie-stretching. Additionally, this system is subject to some constraints: One condition for all net cables is Lc ≥ Lc 0 . For the tension tie-cables, for which the values δLt ij are to be considered, the constraint condition is that the final force F ij must be one of traction: Finally, the third constraint is related to the surface accuracy, since the RMS error must be lower than the desired design value RMS target . This is a typical constrained optimization problem, largely used in the design optimization of complex systems [46][47][48][49] and mechanisms [50][51][52]. The optimization problem can be summarised as follows: where x 1 , y 1 , and z 1 are the free node co-ordinates and δLt is the array containing all corrections δLt ij . The RMS is calculated by measuring the minimum distance of the free nodes of the front net, compared to the ideal surface of the paraboloid [53]. The initial condition for the free nodes is represented by their experimental measurement, while the guess value for δLt is set equal to zero.

Flexible Ring Truss Support
The truss support is generally manufactured out of carbon fiber and it is, therefore, reasonable to consider truss deformation under the effect of the tension of the cable net.
This implies the displacement of the nodes connected to the truss support, also called vertices, when the net system is tensioned. By considering the stiffness of the structure, the nonlinear system (4), described in the previous section, becomes as follows: where c is the total number of cables connected to the i-th node; r is the number of rods; x v , y v , and z v are the unknown co-ordinates of the j-th vertex; and x 0 v , y 0 v , and z 0 v are the initial co-ordinates of the vertex itself. The stiffness of the truss support can be represented as a three-dimensional bushing, with stiffnesses k xx ij , k yy ij , and k zz ij , connecting the rods to the vertex, as shown in Figure 6.
The constrained optimization (6) still applies to this flexible ring truss case and the overall algorithm is summarised in the flowchart in Figure 7. Start -Measure all node co-ordinates; -Measure construction lengths of all cables Compute construction lengths L 0 ij to satisfy the equilibrium of the net by Equation (1) Compute F 0 ij for all ties by Equation ij for all ties?
Compute the tie correction values δLt ij , the position of verices x v , y v and z v and the RMS by using Equation (7) Is the desired RMS reached?

Results
In order to verify the validity of the proposed method, a case study of an asymmetric large deployable reflector, designed by Thales Alenia Space [53], is described. The relevant parameters and geometric data are listed as follows: • The value of the spring constant for tie cables ranges from 2 × 10 3 N/m to 68 × 10 3 N/m, with a radial step of 11 × 10 3 N/m starting from the centre (central node) to the outer ring cables. The initial RMS error on the front net was simulated by introducing an additional value to each node proportionally to the length of the tie connected to it. By imposing the equilibrium in System (1), we can determine the construction lengths described in Section 2. Nevertheless, from Figure 8 it can be noted that the maximum error e L 0 obtained was about 1 mm, representing only 1% of the total cables; the largest percentage (86%) showed an error between 0.6 mm and 1 mm,.
In Figure 9 the local faceting error for the chosen initial configuration is shown. The local faceting error is calculated by considering the centroid of the triangular facets into which the surface can be decomposed [64][65][66][67]. As can be observed, the faceting error follows the shape of the asymmetric reflector since the chosen error is proportional to the tie lengths. As a matter of fact, the central zone is the one with the shortest cables and, therefore, with the lowest faceting error. On the contrary, the tie lengths, and consequently the errors, grow; moving from the centre to the outer perimeter of the net. The optimization method described in the previous sections is first applied to the rigid ring truss support. Figure 10 shows the error of each free node of the front net with respect to the ideal surface, coupled with the stretching/shortening value necessary to reach the desired surface accuracy. The reason why the correction values are all positive is that the initial error is simulated by positioning all nodes of the front net above the ideal surface, so there is a need to shorten the tie lengths to satisfy the RMS design value. The bars are grouped by spring constant value.
The corresponding faceting error is shown in Figure 11. As can be observed, the RMS of the faceting error decreases until the required value of 0.2036 mm is furnished, as per specification.  Then, the same analysis has been performed for the flexible ring truss support. It can be noted that, in Figure 12, the correction values are lower than the corresponding values for the rigid truss support: This is because the additional tensioning of ties further deforms the shape of the truss support, resulting in a closure of the support itself [68]. As a result, this causes a slight lowering of the front net, thus reducing the shortening action of the tie lengths. Finally, Figure 13 shows the faceting error distribution on the front net. Even in this case, the final RMS faceting error reached 0.2045 mm, representing the design value of the RMS. Comparing Figures 11 and 13, it can be observed that the faceting error distribution is more uniform for the flexible ring truss support. This result can be explained by considering that the deformation of the truss support relaxes the front and rear tension truss systems, making the tension distribution more uniform.
Moreover, other simulations with different initial RMS errors revealed that, with the given design data, the maximum initial RMS error which can be fixed is about 1 mm. Beyond this limiting value, a revision of the design data is needed. This demonstrates the validity of the method only for small RMS errors.

Conclusions
In this paper, a method for the tie-system calibration of large deployable reflectors (LDRs) is provided. The LDRs are very sensitive to errors and usually require a careful experimental setup to meet the design requirements for surface accuracy. Due to manufacturing errors, clearance, friction, and the imperfect behaviour of materials, the real configuration moves away from the design configuration and a fine calibration is needed to improve the quality of the reflecting surface, expressed in terms of closeness to the ideal paraboloidal geometry. The proposed method follows two steps: The determination of the parameters satisfying the static equilibrium in the real deployed configuration; and a fine calibration to meet the RMS design requirements. For the first step, a constrained optimization problem was proposed, in order to find the cable construction lengths once all node co-ordinates had been measured. The second step was developed by acting on the system of screw-adjustable ties. A further constrained optimization problem was formulated to find the length corrections of each tie. Using the same approach, the cases of LDR with a rigid or flexible tension truss were studied. Finally, the method was applied to a LDR with an asymmetrical ring truss designed by Thales Alenia Space. Considering an initial RMS of 0.58 mm, the results (not yet validated by experimental test) seemed comforting in reaching the design RMS. The convergence of the method depends on the starting and desired RMS. Here, the convergence was insured up to a reasonably high initial RMS value of 1 mm. Beyond this value, the tie system was not able to reach the equilibrium and satisfy the constraints, and a different solution should be adopted.