A method for identifying the accuracy of the reflector of a large circular antenna based on the sensed vertical rope tension

Abstract

A method for identifying the accuracy of the reflector of a large circular antenna based on sensing the vertical rope tension monitors the tension change on the vertical rope through a force sensor on the vertical rope of the large circular antenna; establishes a rope tension optimization model, combines the tension change on the vertical rope, and then solves the tension of the remaining ropes of the large circular antenna; then updates the rope length using the existing tension, further solves the force density, and solves the coordinates of each free node through the force density method; then judges the difference between the rope tension obtained by solving the optimization model and the initial rope tension; finally, solves the accuracy of the reflector of the large circular antenna through the free node coordinates; the present invention provides a data basis for the in-orbit reflector adjustment of the large circular antenna by establishing an optimization model for solving the tension of the remaining ropes in the circular antenna, obtaining the free node coordinates through the rope length equation and the force density method, and finally obtaining the accuracy of the reflector of the large circular antenna through continuous cyclic iteration, and has important engineering application value.

Description

Technical Field

[0001] The present invention relates to a method for identifying the accuracy of a reflecting surface of a large circular antenna, and particularly to a method for identifying the accuracy of a reflecting surface of a large circular antenna based on sensing the tension of vertical ropes. Technical Background

[0002] Space large circular mesh antennas are widely used in multi-field tasks such as deep space exploration because they can achieve a high folding ratio, a large aperture, and a light mass. With the increase in the antenna aperture, the requirement for the accuracy of its reflecting surface is getting higher and higher. When the antenna is under external loads during its service, the accuracy of the reflecting surface will be significantly affected. At this time, it is necessary to judge whether the antenna reflecting surface can still ensure normal electrical performance. Therefore, it is necessary to develop a method for identifying the reflecting surface of a space large circular mesh antenna.

[0003] Among the existing methods applicable to the identification of the accuracy of the reflecting surface of a space large circular mesh antenna, the relatively mature one is the digital photogrammetry method. However, this method is mostly used for the verification of the reflecting surface after the circular antenna is manufactured. By pasting indicating points on the antenna reflecting surface, and then taking pictures of the circular antenna reflecting surface with multiple external cameras, and combining the principle of stereo vision intersection to obtain the coordinates of the marked points, and then identifying the antenna reflecting surface through interpolation. Since it requires independent installation of multiple camera devices outside the circular antenna structure, it will damage the integrity of the circular antenna, and thus create difficulties for the deployment process of the circular antenna; on the other hand, the existence of independent devices will also increase the difficulty of solving the problem of anti-winding of the ropes in the cable net. In addition, operations such as pasting indicating points on the circular antenna in orbit also require other additional auxiliary devices such as robotic arms. Summary of the Invention

[0004] In order to overcome the above-mentioned deficiencies of the prior art, the purpose of the present invention is to provide a method for identifying the accuracy of a reflecting surface of a large circular antenna based on sensing the tension of vertical ropes. By embedding force sensors on the vertical ropes of the large circular antenna to sense the change in the tension of the vertical ropes, and then obtaining the tensions of all the ropes and the coordinates of each free node in the circular antenna through a reflecting surface identification algorithm, and finally obtaining the reflecting surface of the large circular antenna through an iterative method, so that the large circular antenna can identify the accuracy of the transmitting surface without independently installing external devices and without complex operations such as pasting indicating points.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] A method for identifying the accuracy of a reflecting surface of a large circular antenna based on sensing the tension of vertical ropes, comprising the following steps:

[0007] Step 1: Monitor the change in tension ΔF on the vertical ropes through the force sensors on the vertical ropes of the large circular antennav ;

[0008] Step 2: Establish a rope tension optimization model, and solve the tensions on the front cable net and the rear cable net of the large loop antenna through the tension change ΔF on the vertical ropes sensed in Step 1 v , and solve the tensions on the front cable net and the rear cable net of the large loop antenna;

[0009] Step 3: Update the rope lengths with the existing tensions obtained by solving in Step 2, and further solve the force density. On this basis, solve the coordinates of each free node by the force density method;

[0010] Step 4: Determine whether the difference between the rope tension obtained by solving the optimization model established in Step 2 and the initial rope tension meets the error requirement. If it meets, output the coordinates of the free nodes; if not, return to Step 2, update the optimization model with the coordinates of the free nodes obtained by solving in Step 3, and then loop through Steps 2 - 4 until the difference meets the error requirement;

[0011] Step 5: Solve the reflector accuracy of the large loop antenna through the coordinates of the free nodes obtained in Step 3 of the last loop.

[0012] The specific method of the above Step 1 is as follows:

[0013] Embed a force sensor group capable of sensing force changes on the vertical ropes of the large loop antenna. When the loop antenna is under the space load condition, the tension in the vertical ropes will change due to the influence of the load. At this time, the corresponding tension change ΔF in the vertical ropes is sensed by this force sensor group v .

[0014] The specific method of the above Step 2 is as follows:

[0015] According to the tension change ΔF of the vertical ropes obtained by solving in Step 1 v , the current tension value of the vertical ropes can be obtained through Equation (1):

[0016]

[0017] where F v is the tension vector of the vertical ropes at this time, is the initial tension vector of the vertical ropes, and ΔF v is the tension change vector of the vertical ropes;

[0018] In the cable net of the loop antenna, there are equilibrium equations for the tensions of each rope. For example, at node i, the tension balance in the x, y, and z directions is as shown in Equation (2),

[0019]

[0020] where node i and node j are adjacent nodes, and Fij is the tension on the rope connecting node i and node j, x i , y i , z i are the coordinates of node i in the x, y, and z directions, x j , y j , z j are the coordinates of node j in the x, y, and z directions, l ij is the length of the rope connecting node i and node j;

[0021] The rope length l ij can be expressed using the coordinates of adjacent nodes i and j, as shown in Equation (3),

[0022]

[0023] Introduce the topology matrix C. The topology matrix C can better describe the relationship between nodes and ropes, thus helping to simplify the equilibrium equations; the definition of the topology matrix C is shown in Equation (4),

[0024]

[0025] where the meaning of the topology matrix is that for the i-th rope, it is considered that the rope is connected to node a and node b in the cable net. Define the first node it connects as node a and the second node as node b. When j = a, the topology matrix takes the value +1; when j = b, the topology matrix takes the value -1; when j ≠ a and j ≠ b, the topology matrix takes the value 0;

[0026] The nodes in the loop antenna where the ropes are connected to the truss are called fixed nodes, and the rest are free nodes. The topology matrix C is partitioned according to fixed nodes and free nodes, as shown in Equation (5)

[0027] C = [C free | C fixed (5)

[0028] where C free is the topology matrix related to free nodes, and C fixed is the topology matrix related to fixed nodes;

[0029] Similarly, the coordinates of each node in the loop antenna are also separated according to fixed nodes and free nodes, as shown in Equation (6)

[0030]

[0031] where, x free , y free , z free are the coordinates of free nodes in the x, y, and z directions, xfixed , y fixed , z fixed are the coordinates of the fixed node in the x, y, and z directions;

[0032] In this way, the displacement difference part in the equilibrium equation can be expressed as shown in Equation (7)

[0033]

[0034] where u x = [u 1x , u 2x , …, u nx T , u y = [u 1y , u 2y , …, u ny T , u z = [u 1z , u 2z , …, u nz T are the displacement difference vectors in the x, y, and z directions, and n is the number of ropes in the loop antenna;

[0035] The tension equilibrium equation in the loop antenna cable net can be simplified to the form shown in Equation (8)

[0036]

[0037] where U x = diag(u x ), U y = diag(u y ), U z = diag(u z ), L = diag(l), l = [l1, …, l i , …, l n T , F = [F1, F2, …, F n T , l i is the length of the i-th rope;

[0038] The equilibrium equation at the free nodes in the loop antenna can be further simplified to the form shown in Equation (9)

[0039] A 3m×n F n×1 = 0 3m×1 (9)

[0040] where ​​​​​n is the number of ropes in the loop antenna, and m is the number of free nodes in the loop antenna;

[0041] Since the vertical rope tension can be sensed by the force sensor, the parts related to the vertical ropes and the parts related to the other ropes in the tension balance equation are separated. As shown in Equation (10):

[0042] A r F r = b v (10)

[0043] where b v = -A v F v A v and A r are the parts related to the vertical ropes and the parts related to the other ropes in matrix A, and F 3m×n and F v are the parts related to the vertical ropes and the parts related to the other ropes in vector F r ; n×1 After that, an optimization model for solving the tensions of the other ropes is established, as shown in Equation (11):

[0044] Find F

[0045] r

[0046]

[0047]

[0048] where is the tension vector of the other ropes except for the vertical rope tension of the loop antenna, F ri is the tension value of the i-th rope, n r is the number of ropes except for the vertical ropes of the loop antenna, f is the average value of the tension vectors of the ropes except for the vertical ropes of the loop antenna, b is the average value of the front cable net tension of the loop antenna, f is the average value of the rear cable net tension of the loop antenna, n b is the number of ropes in the front cable net of the loop antenna, n v is the number of ropes in the rear cable net of the loop antenna;

[0049] Finally, the optimization model is solved by the genetic algorithm, and the tension vector of all ropes F = [F T , F, F 0ij , F ij can be obtained. n .

[0050] The specific method of step 3 is as follows:

[0051] After all the cable tensions are sensed, the cable tensions are used to update the cables in the cable net of the loop antenna, as shown in Equation (12).

[0052]

[0053] where l 0ij is the initial length of the cable connecting adjacent nodes i and j, E is the Young's modulus of the cable, A is the cross-sectional area of the cable, α is the linear thermal expansion coefficient of the cable, and T and T0 are the current temperature and the initial temperature, respectively;

[0054] After that, the force density of each cable in the cable net of the loop antenna is calculated, as shown in Equation (13).

[0055]

[0056] where q ij is the force density of the cable connecting adjacent nodes i and j;

[0057] By the force density method, the coordinates of each free node in the cable net can be finally solved, including the coordinates corresponding to the nodes on the reflector surface, as shown in Equation (14).

[0058]

[0059] where Q = diag(q), q = [q1, q2,..., q n T is the force density vector formed by the force densities of each cable.

[0060] The specific method of step 4 is as follows:

[0061] In the k-th iteration, the cable tensions of the cables other than the vertical cables obtained by solving the optimization model are while the cable tensions of the cables other than the vertical cables obtained in the previous iteration are Calculate the difference in the cable tensions of the cables other than the vertical cables in the two iterations, as shown in Equation (15).

[0062]

[0063] If is greater than the set allowable error tol, then update matrix A with the latest free coordinates obtained in step 3 3m×n and repeat steps 2 - 4 until is less than the allowable error; if is less than the set allowable error tol, then proceed to the next step. ​

[0064] The specific method of step 5 is as follows:

[0065] For the coordinates x free , y free , z free of each free node of the rope obtained in step 3, only the coordinates of the free nodes in the front cable net are taken, and the accuracy calculation of the reflector surface of the large circular antenna is carried out through formula (16);

[0066]

[0067] where Δ rms is the accuracy of the reflector surface of the circular antenna, m f is the number of free nodes in the front cable net, and are the coordinates of the j-th free node in the identified reflector surface, and are the coordinates of the j-th free node in the ideal reflector surface.

[0068] Compared with the prior art, the present invention has the following beneficial effects:

[0069] The technical effect of the present invention is based on the vertical rope tension sensing. By establishing an optimization model for solving the tensions of the remaining ropes in the circular antenna and solving the optimization model by means of the genetic algorithm, the tensions of all the ropes of the circular antenna are obtained. Further, the coordinates of the free nodes are obtained through the rope length equation and the force density method. Finally, the accuracy of the reflector surface of the large circular antenna is obtained through continuous cyclic iteration. The method of the present invention replaces the complex operations such as independently installing other devices such as cameras in the circular antenna or pasting fiducial points in space by embedding force sensors in the vertical ropes, making it possible to identify the accuracy of the reflector surface of the large circular antenna in orbit. Thus, it provides a data basis for judging whether the large circular antenna can operate normally in orbit and for adjusting the reflector surface of the large circular antenna in orbit, and has important engineering application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 is a flowchart of the present invention.

[0071] Figure 2 is a program block diagram of the implementation of the present invention.

[0072] Figure 3 is a schematic diagram of each part of the large circular antenna, which is the research object in the embodiment of the present invention. Among them, Figure 3 (a) is a schematic diagram of the names of each part of the cable net in the embodiment of the large circular antenna; Figure 3 (b) is an exploded view of the large circular antenna, which can more clearly show the appearance of each part; Figure 3(c) Front view of an embodiment of a large circular antenna, including the aperture size of the embodiment.

[0073] Figure 4 Comparison of the identified tension values and the true tension values of all ropes in the large circular antenna obtained by the optimization algorithm in the embodiment of the present invention, where Figure 4 (a) Comparison of the identified tension value of the large circular antenna and the true value of the rope tension; Figure 4 (b) Solution deviation of the identified tension value based on the true value of the rope tension in the large circular antenna.

[0074] Figure 5 Results of the displacement deviation between each point on the identified reflector net and each point on the ideal reflector net under different loads obtained by the identification method in the embodiment of the present invention, where Figure 5 (a) Displacement deviation between each point on the identified reflector net and each point on the ideal reflector net at -100 °C; Figure 5 (b) Displacement deviation between each point on the identified reflector net and each point on the ideal reflector net at 100 °C; Figure 5 (c) Displacement deviation between each point on the identified reflector net and each point on the ideal reflector net after the stiffness degradation of the rope. Detailed implementation manners

[0075] The technical solutions of the present invention will be described in detail below with reference to the accompanying drawings, but the protection scope of the present invention is not limited to the described embodiments.

[0076] A method for identifying the accuracy of the reflector surface of a large circular antenna based on sensing the vertical rope tension, and the specific operations are as follows:

[0077] Step 1: Set k = 0, which is the 0th iteration process, and obtain the initial vertical rope tension vector of the large circular antenna The initial non-pre-tension length l of each rope 0 , the Young's modulus E of each rope, the cross-sectional area A of each rope, the initial coordinates of each free node in the cable net and the coordinates of each fixed node Number all the ropes and all the nodes in the cable net of the large circular antenna in the research embodiment; collect the tension change data of all the force sensors in the vertical cable net of the large circular antenna at the same moment, and represent it in vector form ΔF in the order of the numbers v ;

[0078] Step 2: According to the initial vertical rope tension vector of the large circular antenna obtained in Step 1 and the vertical rope tension change vector ΔF sensed in the force sensor v , calculate through Equation (1) to obtain the vertical rope tension vector F at this moment v,

[0079]

[0080] There are equilibrium equations in the loop antenna cable net, as shown in Equation (2).

[0081]

[0082] Among them, the length part of the ropes in the cable net can be expressed by the coordinates of both ends of the ropes, as shown in Equation (3).

[0083]

[0084] According to Equation (4),

[0085]

[0086] Write the topological matrix C of the large loop antenna, and divide the topological matrix C into blocks according to the free node and fixed node numbers according to Equation (5)

[0087] C = [C free | C fixed (5)

[0088] Similarly, according to Equation (6),

[0089]

[0090] The cable net node coordinates are divided into free node coordinates and fixed node coordinates; by substituting the topological matrix C free , C fixed and the free node coordinates and the fixed node coordinates into Equation (7),

[0091]

[0092] the displacement difference vector of the ropes in the loop antenna cable net can be obtained This displacement difference describes the span of the corresponding ropes in the loop antenna in the x, y, and z directions. Substitute the above results into Equation (9),

[0093] A 3m×n F n×1 = 0 3m×1 (9)

[0094] A v and A r can be solved, and thus the constraint conditions in the optimization model as shown in Equation (10) can be obtained:

[0095] A rF r = b v (10)

[0096] Establish an optimization model as shown in Equation (11)

[0097] Find F r

[0098]

[0099]

[0100] And solve it through the genetic algorithm. The general settings of the genetic algorithm used in this optimization model are as follows: the population size is set to 100, the crossover rate is 0.7, the mutation rate is 0.0015, and the number of iterations is 300 times. In different calculation models, the basic parameters may be changed accordingly. Finally, the tensions of all ropes can be obtained

[0101] Step 3: Use the tensions F of all ropes obtained by solving in Step 2 k , through Equation (12)

[0102]

[0103] Update the rope length l k+1 , substitute the calculated l k+1 into Equation (13),

[0104]

[0105] Calculate the force density values of each rope, and substitute the calculated force density vector into the equation

[0106]

[0107] The free node coordinates can be solved

[0108] Step 4: In the k-th loop, the tensions of the remaining ropes except the vertical ropes obtained by solving using the optimization model are while the tensions of the remaining ropes except the vertical ropes obtained by solving in the previous time are Adopt Equation (15)

[0109]

[0110] Calculate the difference between the two And further calculate If is greater than the set allowable error tol, then use the latest free coordinates obtained by solving in Step 3 for matrix A 3m×nUpdate and repeat steps 2 - 4 until is less than the allowable error tol; if is less than the set allowable error tol, then proceed to the next step;

[0111] Step 5: Output the free node coordinates x free , y free , z free calculated in step 3 of the last loop, and extract the coordinate values corresponding to the free nodes of the front cable net in the large loop antenna, and substitute them into Equation (16),

[0112]

[0113] Finally, the reflector surface accuracy Δ of the large loop antenna can be calculated rms .

Claims

1. A method for identifying the accuracy of a large circular antenna reflector based on sensing the vertical rope tension, characterized in that: Specifically, it includes the following steps: Step 1. Monitor the change in tension ΔF on the vertical rope through the force sensor on the large circular antenna vertical rope v ; Step 2: Establish a cable tension optimization model, and obtain the tension change ΔF of the vertical cables sensed in Step 1 v , and solve the tensions of the front cable net and the rear cable net of the large loop antenna; specifically: according to the tension change ΔF of the vertical cables obtained by solving in Step 1 v , the current tension value of the vertical cables can be obtained through Equation (1): Among which F v is the vertical rope tension vector at this time, is the initial tension vector of the vertical rope, and ΔF v is the vertical rope tension change vector; There are balance equations for the tensions of each rope in the circular antenna cable net. For example, at node i, the tension balance in the x, y, and z directions is as shown in Equation (2). where node i and node j are adjacent nodes, F ij is the tension on the rope connecting node i and node j, x i , y i , z i are the coordinates of node i in the x, y, and z directions, x j , y j , z j are the coordinates of node j in the x, y, and z directions, and l ij is the length of the rope connecting node i and node j; The length l of the rope ij It is represented by the coordinates of adjacent nodes i and j, as shown in Equation (3). Introduce the topology matrix C. The topology matrix C better describes the relationship between nodes and ropes, thus helping to simplify the balance equations. The definition of the topology matrix C is as shown in Equation (4). The meaning of the topology matrix is that for the i-th rope, it is considered that the rope is connected to node a and node b in the cable net. Define the first node it is connected to as node a and the second node as node b. When j = a, the topology matrix takes the value +1; when j = b, the topology matrix takes the value -1; when j ≠ a and j ≠ b, the topology matrix takes the value 0. The nodes where the ropes in the circular antenna are connected to the truss are called fixed nodes, and the rest are free nodes. The topology matrix C is partitioned according to fixed nodes and free nodes, as shown in Equation (5). C = [C free |C fixed (5) Among them, C free is the topological matrix related to free nodes, and C fixed is the topological matrix related to fixed nodes; Similarly, the coordinates of each node in the circular antenna are also separated according to fixed nodes and free nodes, as shown in Equation (6). Among them, x free , y free , z free are the coordinates of the free node in the three directions of x, y, and z, and x fixed , y fixed , z fixed are the coordinates of the fixed node in the three directions of x, y, and z; In this way, the displacement difference part in the balance equation is expressed as shown in Equation (7). where u x = [u 1x , u 2x , …, u nx T , u y = [u 1y , u 2y , …, u ny T , u z = [u 1z , u 2z , …, u nz T is the displacement difference vector in the x, y, and z directions, and n is the number of ropes in the loop antenna;​​​ The tension balance equation in the circular antenna cable net is simplified into the form as shown in Equation (8). where U x = diag(u x ), U y = diag(u y ), U z = diag(u z ), L = diag(l), l = [l1, ···, l i , …, l n T , F = [F1, F2, ···, F n T , l i is the length of the i-th rope;​​ The balance equation at the free nodes in the circular antenna is further simplified into the form as shown in Equation (9). A 3m×n F n×1 = 0 3m×1 (9) wherein n is the number of ropes in the loop antenna, and m is the number of free nodes in the loop antenna; Since the tension of the vertical rope is sensed by the force sensor, the parts related to the vertical rope and the parts related to the remaining ropes in the tension balance equation are separated, A 3m×n = [A v |A r , As shown in Equation (10): A r F r = b v (10) where b v = -A v F v , A v and A r is the part of the matrix A 3m×n related to the vertical ropes and the part related to the remaining ropes, F v and F r is the part of the vector F n×1 related to the vertical ropes and the part related to the remaining ropes; After that, an optimization model for solving the tensions of the remaining ropes is established, as shown in Equation (11): Among them is the vector of the tension of the ropes other than the vertical rope tension of the loop antenna, F ri is the tension value of the i-th rope, n r is the number of ropes other than the vertical rope of the loop antenna, is the average value of the tension vectors of the ropes other than the vertical rope of the loop antenna, is the average value of the tension of the front cable net of the loop antenna, is the average value of the tension of the rear cable net of the loop antenna, n f is the number of ropes of the front cable net of the loop antenna, n b is the number of ropes of the rear cable net of the loop antenna; Finally, the genetic algorithm is used to solve the optimization model, and the complete rope tension vector F = [F f , F b , F v T ;​ Step 3: Update the rope lengths using the existing tensions obtained by solving in Step 2, and further solve the force density. On this basis, solve the coordinates of each free node by the force density method. Specifically: after sensing the tensions of all ropes, update the cables in the circular antenna cable net using the rope tensions, as shown in Equation (12). where l 0ij is the initial length of the rope connecting adjacent nodes i and j, E is the Young's modulus of the rope, A is the cross-sectional area of the rope, α is the linear thermal expansion coefficient of the rope, and T and T0 are the current temperature and the initial temperature, respectively; After that, calculate the force density of each rope in the circular antenna cable net, as shown in Equation (13). where q ij is the force density of the rope connecting adjacent nodes i and j; Finally, solve the coordinates of each free node in the rope net by the force density method, including the coordinates corresponding to the nodes on the reflector surface, as shown in Equation (14). where \(Q = \text{diag}(q)\), \(q = [q_1, q_2, \cdots, q\) n T is the force density vector formed by the force densities of each cable;​ Step 4: Judge whether the difference between the tensions of the remaining ropes except the vertical ropes obtained by solving the optimization model in the k-th cycle of Step 2 and the tensions of the remaining ropes except the vertical ropes obtained in the previous cycle meets the error requirement. If it meets, output the coordinates of the free nodes; if it does not meet, return to Step 2, update the optimization model with the coordinates of the free nodes obtained in Step 3, and then cycle through Steps 2 - 4 until the difference meets the error requirement. Step 5: Solve the accuracy of the reflector surface of the large circular antenna using the coordinates of the free nodes obtained in Step 3 of the last cycle.

2. The method for identifying the accuracy of the reflector of a large circular antenna based on sensing the vertical rope tension according to claim 1, characterized in that: The specific method of the said Step 1 is as follows: Force sensor groups capable of sensing force changes are embedded in the vertical ropes of a large loop antenna. When the loop antenna is under space payload conditions, the tension in the vertical ropes will change due to the influence of the payload. At this time, the corresponding tension change ΔF occurring in the vertical ropes is sensed by the force sensor groups v .

3. The method for identifying the accuracy of the reflector of a large circular antenna based on sensing the vertical rope tension according to claim 1, characterized in that: The specific method of the said Step 4 is as follows: The tension of the ropes other than the vertical ropes obtained by solving using the optimization model in the k-th cycle is The tension of the ropes other than the vertical ropes obtained by solving in the previous cycle is Calculate the difference in the tension of the ropes other than the vertical ropes between the two times as shown in Equation (15) If is greater than the set allowable error tol, then update the matrix A with the latest free coordinates obtained by solving in step 3 3m×n and repeat steps 2 - 4 until is less than the allowable error; if is less than the set allowable error tol, then proceed to the next step.

4. The method for identifying the accuracy of the reflector of a large circular antenna based on sensing the vertical rope tension according to claim 1, characterized in that: The specific method of the said Step 5 is as follows: For the free node coordinates x of the ropes obtained in step 3 free , y free , z free , only take the free node coordinates in the front cable net, and calculate the accuracy of the reflector surface of the large loop antenna through Equation (16): where Δ rms is the accuracy of the reflector surface of the loop antenna, m f is the number of free nodes in the front cable net, and are the coordinates of the j-th free node in the identified reflector surface, and are the coordinates of the j-th free node in the ideal reflector surface.

Images