An approximately analytical solution method for the cable-driven parallel robot in FAST
RResearch in Astronomy and Astrophysics manuscript no.(L A TEX: ms2020-0266.tex; printed on August 20, 2020; 1:01)
An approximately analytical solution method for the cable-drivenparallel robot in FAST
Jia-Ning Yin , , Peng Jiang and Rui Yao National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China; [email protected], [email protected] School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049,China; [email protected]
Received 20xx month day; accepted 20xx month day
Abstract
FAST is the largest single-dish aperture telescope with a cable-driven parallel robotintroduced to achieve the highest sensitivity in the world. However, to realize the high-precision, mechanical equations of such robot are always complicated, so that it is difficultto achieve real-time control by the traditional iterative method. In this regard, this paperproposes an approximately analytical solution method, which uses the approximately lin-ear relationship between the main parameters of FAST to bypass some iterations. With thecoefficients of the relationship extracted, static or quasi-static mechanical equations can beanalytically solved. In this paper’s example, this method saves at least 90% of the calculat-ing time and the calculated values are consistent with the experimental data. With such hugeefficiency improvements, real-time and high-precision control of FAST will no longer be adifficult work. Besides, all the work in this paper is expected to be used in the FAST.
Key words:
FAST — radio telescope — cable-driven parallel robot
FAST is the largest single-dish aperture telescope with the highest sensitivity in the world. To achieve it, thecable-driven parallel robot (CDPR) was introduced (Tang & Yao 2011), which mainly composed of somecables and the end effector connected to the cables. As shown in Figure 1, there are 6 cables connected tothe feed cabin in the center. The translation and rotation of the feed cabin can be controlled by adjustingthe length of the cables. This kind of robot not only has the advantages of high precision, high speed andhigh load, but also has a large working space. Therefore, CDRP is FAST’s perfect solution to solve the widerange movement of the feed cabin. Besides, CDPR is not only used in FAST, but also in many fields. Forexample, Kawamura et al. (1995) developed a robot for transport, called Falcon; Abbasnejad et al. (2016) a r X i v : . [ a s t r o - ph . I M ] A ug J.N. Yin, P. Jiang & R. Yao designed a robot for gait rehabilitation; Bruckmann et al. (2012) also invented a robot related to storagetechnology, and so on.According to the study by Ming & Higuchi (1994), CDPRs can be divided into three categories. Firstly,it is assumed that the number of cables is m and the degree of freedom of the end effector is n . Thus,if m = n + 1 , the system dynamics equation has a definite solution, so it is called completely restrainedpositioning mechanism (CRPM). if m > n +1 , the driving forces of the cables are redundant, and the systemdynamic equation has no definite solution, which is called redundantly restrained positioning mechanism(RRPM). If m < n + 1 , the system constraints are insufficient. It is called incompletely rested positioningmechanism (IRPM), which need to rely on external forces to maintain the stability of the mechanism. Asmentioned above, FASTs feed cabin is controlled by 6 cables. And the degree of freedom of the feed cabinis 6. Obviously, FAST is IRPM, so it needs to be stabilized by gravity.The research on the statics or dynamics of CDPRs must focus on the theoretical model of the cable,which determines the mechanical properties of the entire system. In this regard, many scholars use thestraight line as the cable model (Cui et al. 2019; Gonzalez-Rodriguez et al. 2017; ? ; Kawamura et al. 2000;Khosravi & Taghirad 2013), only considering the elastic deformation of the axial direction of the cables andignoring the weight influence of the cables. This model has analytical expressions and can be solved fast,so it is ideal for CDPRs with a small span. However, as for the case of FAST with a large span, the weightinfluence of the cables cannot be ignored, and the cable forces are extremely sensitive to the length of thecables. Obviously, the straight-line model is no longer applicable. In this regard, other scholars (Kozak etal. 2006; Merlet 2019; Yuan et al. 2015) introduced the catenary model derived by Irvine (1981). It has beenverified by Riehl et al. (2010) that the catenary model has high accuracy. However, at the same time, thecatenary model needs to be solved by iteration due to its complex nonlinear nature. In order to optimize theiteration time, Merlet (2019) proposed to simplify the iteration by changing variables based on the catenarymodel. On the contrary, Ferravante et al. (2019) abandoned the catenary model and calculated it throughfinite element method.However, by now the modeling and solving efficiency of CDPRs has been low, because it is inefficientto use the catenary model in real-time control. For example, the CDPR control of FAST has to adopt theclosed-loop method to save time, which is not conducive to increasing its control precision. This means thatthe previous methods cannot achieve real-time and high-precision control at the same time. And this paperprecisely has a breakthrough at this point.Based on the static catenary model, with FAST as the research object, this paper proposes an approxi-mately analytical solution method. This method uses an approximately linear relationship between the mainparameters for solution, which is found by numerical analysis. After the coefficients of the relationship isextracted, static or quasi-static equations of the CDPR can be simplified and solved analytically, whichgreatly improves the calculation efficiency of FAST.The approximately analytical solution method for static or quasi-static equations of CDPRs will beintroduced in detail. For the convenience of description, the following approximately analytical solutionmethod is abbreviated as the AAS method. Firstly, the static equations of the FAST cable model will beestablished, which is the catenary model, and then the relationship between the mechanism parameters and AS method for the CDPR in FAST 3
Fig. 1: Schematic diagram of the FAST coordinate system.the solution parameters will be analyzed. With this relationship, the CDPR’s equations can be simplified andsolved analytically. Finally, there are some example comparisons between the AAS method and the iterationmethod. The solution accuracy and time of the AAS method are obtained. Also, there is an experiment,which compares the calculated values of the AAS method with the measured values during the actualoperation of FAST to test the rationality of the AAS method.
This paper takes FAST as the research object, which controls the movement and attitude of the central feedcabin by pulling 6 cables through 6 towers. It is a typical cable-driven parallel robot (CDPR), as shown inFigure 1.Firstly, the global Cartesian coordinate system O − xyz is established. The lower vertex of the sphericalreflection surface is the origin O . The direction from the origin O towards the tower B is the x -axis andthe upward direction perpendicular to the ground is z -axis, as shown in Figure 1.Simultaneously, the local Cartesian coordinate system O (cid:48) − x (cid:48) y (cid:48) z (cid:48) of the feed cabin is also established.The center of the anchor points plane of the feed cabin is the origin O (cid:48) . The local Cartesian coordinatesystem is bound to the feed cabin and rotates with the attitude of the feed cabin. When the feed cabin is inthe center, the local Cartesian coordinate system is totally parallel to the global Cartesian coordinate system.Wherein, the anchor points A [ i ] of the feed cabin is evenly distributed on the circle with the radius r a .The center of the circle just is the origin O (cid:48) of the local Cartesian coordinate system. The 6 towers B [ i ] areevenly distributed on the circle with the radius r b and each tower height is H . Every 2 towers are connectedto an anchor point by 2 cables, as shown in Figure 2.Then, the local Cartesian coordinate system O (cid:48)(cid:48) − x (cid:48)(cid:48) y (cid:48)(cid:48) z (cid:48)(cid:48) of every cable is established with the corre-sponding anchor point A [ i ] as the origin O (cid:48)(cid:48) . For the convenience of calculation, these coordinate systemsare always required to be parallel with the global Cartesian coordinate system. In these coordinate systems,the coordinates of the cable lower and upper end are set to the origin O (cid:48)(cid:48) and ( X [ i ] , Y [ i ] , Z [ i ]) , respectively,as shown in Figure 3. So, the following geometric relationship can be derived: X [ i ] Y [ i ] Z [ i ] = r b cos( π ( i − / r b sin( π ( i − / H − ( R · r A [ i ] + r p ) (1)where r A [ i ] is given by J.N. Yin, P. Jiang & R. Yao
Fig. 2: Schematic diagram of the FAST structure.Table 1: Specific values of parameters.
Symbol Significance Specific value(unit) r a Feed cabin anchor point distribution radius . r b Tower distribution radius H Tower height E Cable elastic modulus . × (Pa) A Cable cross-sectional area with the cable not stressed . × − (m ) ρ Cable linear density with the cable not stressed . / m) g the acceleration of gravity − . / s ) r e Position vector from the origin O (cid:48) of the feed cabin [0 , , . T (m) local coordinate system to the feed cabin mass center m Feed cabin mass R Reflecting surface radius D Reflecting surface projection diameter r A [ i ] = r a cos( π/ π (cid:98) ( i − / (cid:99) / r a sin( π/ π (cid:98) ( i − / (cid:99) / (2)Among the formulas above, [ i ] represents the i -th cable corresponding to the i -th tower, R is the rotationmatrix of the local Cartesian coordinate system O (cid:48) − x (cid:48) y (cid:48) z (cid:48) of the feed cabin relative to the global Cartesiancoordinate system O − xyz , r A [ i ] is the position vector of the anchor point connected to the i -th cable in thelocal Cartesian coordinate system O (cid:48) − x (cid:48) y (cid:48) z (cid:48) of the feed cabin, (cid:98) (cid:99) is a mathematical symbol, which meansrounding down, and r p is the position vector of the origin O (cid:48) of the local Cartesian coordinate system ofthe feed cabin in the global Cartesian coordinate system.For the specific values of the above and other necessary parameters, please refer to Table 1. The cable model of this paper is the static catenary model, and the coordinates are shown in Figure 3.Because the equations of the 6 cables’ model are the same, for the convenience, the cable number i is AS method for the CDPR in FAST 5 generally not specified in this section unless it is necessary. Let the forces in the three directions of thecable lower end be F x , F y and F z , respectively, as shown in Figure 4.Fig. 3: Local coordinate system of the cable.
Fig. 4:
Schematic diagram of the cable force.
Where the length of the cable is p , T is the cable force at that point and s is the length of this cablewithout tension, called original length of the cable. The length of the cable p and the cable force T areboth the functions of the original length s . In addition, let ρ be the linear density when the cable is nottensioned, and g be the acceleration of gravity. Then, according to the equilibrium equation, the followingcan be obtained: T d x d p + F x = 0 (3) T d y d p + F y = 0 (4) T d z d p + F z + ρgs = 0 (5)where T is given by T ( s ) = (cid:113) F x + F y + ( F z + ρgs ) (6)Then, according to the elastic equation, there is T ( s ) = EA (cid:18) d p d s − (cid:19) (7)where EA is the cable elastic modulus multiplied by the cross-sectional area when the cable is nottensioned.Combined with equation (3-7), the following can be obtained: d x d s = − F x EA EA (cid:113) F x + F y + ( F z + ρgs ) (8) d y d s = − F y EA EA (cid:113) F x + F y + ( F z + ρgs ) (9) d z d s = − F z + ρgsEA EA (cid:113) F x + F y + ( F z + ρgs ) (10) J.N. Yin, P. Jiang & R. Yao
According to the boundary conditions x (0) = 0 , y (0) = 0 and z (0) = 0 , shown in Figure 4, thesolutions are equation (11-13). x ( s ) = − F x EA s − F x ρg sinh − F z + ρgs (cid:113) F x + F y − sinh − F z (cid:113) F x + F y (11) y ( s ) = − F y EA s − F y ρg sinh − F z + ρgs (cid:113) F x + F y − sinh − F z (cid:113) F x + F y (12) z ( s ) = − F z EA s − ρg EA s − ρg (cid:20)(cid:113) F x + F y + ( F z + ρgs ) − (cid:113) F x + F y + F z (cid:21) (13)Let the original length of the whole cable be s , and know that the coordinates of the cable upper endare ( X, Y, Z ) , then equation (14-16) can be obtained, where the unknown variables are F x , F y , F z and s . X = − F x EA s − F x ρg sinh − F z + ρgs (cid:113) F x + F y − sinh − F z (cid:113) F x + F y (14) Y = − F y EA s − F y ρg sinh − F z + ρgs (cid:113) F x + F y − sinh − F z (cid:113) F x + F y (15) Z = − F z EA s − ρg EA s − ρg (cid:20)(cid:113) F x + F y + ( F z + ρgs ) − (cid:113) F x + F y + F z (cid:21) (16)In the local coordinate system of the cable, let the resultant force on the O (cid:48)(cid:48) − x (cid:48)(cid:48) y (cid:48)(cid:48) plane be F l = (cid:113) F x + F y . Refer to equation (14) and (15) and the following can be obtained: F x = − X √ X + Y F l (17) F y = − Y √ X + Y F l (18)If the cable is straight, the cable length must be √ X + Y + Z , and let k = √ X + Y + Z .Because EA is of a large magnitude, the actual original cable length s generally does not exceed theinterval [0 . k, . k ] . Now with FAST as the object, when the feed cabin is at a random position, bysolving the numerical value of equation (14-16), the change trends of F l and F z can be obtained with thecable length s in the interval above, as shown in Figure 5.There is an obvious feature in Figure 5. When the cable length shrinks to a certain value, the sensitivityof the cable force to the original length s of the whole cable rises rapidly, but later it quickly remains stable.This is a complex form of function, which leads to difficulties in iteration. However, it is observed that thetrends F l and F z are highly consistent, so another figure of F z on F l is considered, as shown in Figure 6.Obviously, F z has a strong linear relationship with F l , which is much simpler than F z ’s case on s . Aftermassive calculation with taking all the position of the feed cabins working space in Figure 2, it is foundthat minimum value of the determination coefficient of this linear relationship is 0.999999999750919, sothe linear relationship can be considered always to exist and be independent of the original length s of thewhole cable. AS method for the CDPR in FAST 7
Fig. 5: F l and F z change with s . Fig. 6: F z changes with F l . However, it should be noted that the linear relationship is related to the spatial structure and physicalproperties of the research object. For each research object, the relationship needs to be verified by numericalcalculation in the CDPR’s workspace. In this paper, FAST has such a good linear relationship.Therefore, in the actual calculation, it is only necessary to take two kinds of s in the equation (14-16).For example, s = 0 . k and s = 1 . k . Then, the linear expression of F z about F l can be determined: F z = aF l + b (19)However, FAST has 6 cables, so there are 6 groups of equation (14-16), which means there are a totalof 24 unknown variables with only 18 equations. So, another 6 equations are needed to solve the equation.Fortunately, the feed cabin balance equations just meet this: − (cid:88) i =1 F x [ i ] = 0 (20) − (cid:88) i =1 F y [ i ] = 0 (21) mg − (cid:88) i =1 F z [ i ] = 0 (22) (cid:88) i =1 ( − r y [ i ] F z [ i ] + r z [ i ] F y [ i ]) + mge y = 0 (23) (cid:88) i =1 ( r x [ i ] F z [ i ] − r z [ i ] F x [ i ]) + mge x = 0 (24) (cid:88) i =1 ( − r x [ i ] F y [ i ] + r y [ i ] F x [ i ]) = 0 (25)where [ i ] represents the i -th cable and m is the feed cabin mass, and [ r x [ i ] , r y [ i ] , r z [ i ]] T = R · r A [ i ] . e x and e y are the projection distances of the position r e shown in Table 1, respectively in the x -axis directionand the y -axis direction.Substitute equation (17-19) into equation (20-25), which can be reduced to the following matrix form: A · F l = B (26) J.N. Yin, P. Jiang & R. Yao where A , F l and B are given by equation (27-29). A = X [1] √ X [1] + Y [1] · · · X [6] √ X [6] + Y [6] Y [1] √ X [1] + Y [1] · · · Y [6] √ X [6] + Y [6] a [1] · · · a [6] r z [1] Y [1]+ r y [1] a [1] √ X [1] + Y [1] √ X [1] + Y [1] · · · r z [6] Y [6]+ r y [6] a [6] √ X [6] + Y [6] √ X [6] + Y [6] r z [1] X [1]+ r x [1] a [1] √ X [1] + Y [1] √ X [1] + Y [1] · · · r z [6] X [6]+ r x [6] a [6] √ X [6] + Y [6] √ X [6] + Y [6] r y [1] X [1] − r x [1] Y [1] √ X [1] + Y [1] · · · r y [6] X [6] − r x [6] Y [6] √ X [6] + Y [6] (27) F l = (cid:104) F l [1] F l [2] F l [3] F l [4] F l [5] F l [6] (cid:105) T (28) B = (cid:104) mg − (cid:80) i =1 b [ i ]) ( mge y − (cid:80) i =1 r y [ i ] b [ i ]) ( mge x − (cid:80) i =1 r x [ i ] b [ i ]) 0 (cid:105) T (29)Therefore, it is easy to get the resultant force F l of each cable on the respective O (cid:48)(cid:48) − x (cid:48)(cid:48) y (cid:48)(cid:48) plane, whichis also on the global plane O − xy , because the local Cartesian coordinate system O (cid:48)(cid:48) − x (cid:48)(cid:48) y (cid:48)(cid:48) z (cid:48)(cid:48) of eachcable is parallel with the global Cartesian coordinate system O − xyz . F l = A − · B (30)Then according to the equation (17-19), the forces F x , F y and F z of the lower end of each cable can beobtained.It can be seen that the form of equation (26) is very similar to the straight-line models. The differenceis in the matrix A and the array B . New parameters a [ i ] and b [ i ] are introduced, so that the expressionnot only corresponds to the geometric relationship, but also the mechanical parameters of the cable and theattitude of the feed cabin. In a sense, a [ i ] and b [ i ] are equivalent to the correction parameters used to correctthe error between the linear model and the catenary model, which depend on the mechanical and geometricproperties of the entire system.So far, the process of solving the static or quasi-static equations of CDPRs by the approximately ana-lytical solution method (AAS) has been very clear, see Figure 7 for details.Obviously, the process can solve all the required parameters just in one loop. Compared to the traditionaliterative operation, there is no step of loop calculation and selecting step size. For this reason, AAS methodcan greatly improve the static solution speed of the CDPR in FAST. In this section, a comparison between the approximately analytical solution method and the traditionaliterative method will be shown, based on MAPLE programming. Under the condition of the same feedcabin trajectory, the same static or quasi-static equations of FAST’s CDPR are solved by the two methodsrespectively. Finally, the cable force values of the lower ends of the six cables and the time required for thesolution will be compared.
AS method for the CDPR in FAST 9
Fig. 7: Process of approximately analytical solution method.Fig. 8: Schematic diagram of the feed cabin trajectory.Table 2: Comparison with the feed cabin hovering at the lower vertex of the workspace.
Method Cable force at the lower end/kN Time cost/s1 2 3 4 5 6Iteration 159.1762 159.1670 159.1587 159.1587 159.1670 159.1762 4.984AAS 158.9899 158.9808 158.9725 158.9725 158.9808 158.9899 0.938Relative error 0.1170% 0.1170% 0.1170% 0.1170% 0.1170% 0.1170% (1) The feed cabin is hovering at the lower vertex (0 , , of the working area, which means the feedcabin remains stationary at point K in Figure 8. Because the feed cabin is in the center, according to theprinciple of symmetry, the six cables should be subjected to the same force. Table 2 shows the calculationresults.In the case of high symmetry, the relative error between AAS and iteration method is very small. It canbe considered that the two methods have similar accuracy, but the time cost of AAS is obviously much lessthan the iteration method.(2) The feed cabin is hovering at a point that is not specific in the working area, such as the point K . , . , . , shown in Figure 8. This point is closer to the B , B , B and B towers, sothe cable tension of the four towers should be larger. Table 3 shows the calculation results. Table 3: Comparison with the feed cabin hovering at a point that is not specific.
Method Cable force at the lower end/kN Time cost/s1 2 3 4 5 6Iteration 181.5009 190.7462 200.7024 119.5531 119.8599 191.0149 4.594AAS 181.3146 190.5478 200.6720 119.3861 119.6912 190.9943 0.969Relative error 0.1026% 0.1040% 0.0151% 0.1397% 0.1407% 0.0108%
Table 4: Comparison with the feed cabin moving slowly in a straight line.
Maximum relative error of cable force at the lower end Time cost/s1 2 3 4 5 6 AAS Iteration0.1140% 0.1020% 0.1020% 0.1140% 0.1197% 0.1197% 20.844 343.203
In the case of no special position, the relative error of AAS with the iteration is still very small. It canbe considered that the two methods have similar accuracy. And AAS is obviously much faster than theiteration.(3) The feed cabin slowly moves in a straight path from G , , to G − , , ,as shown in the blue line in Figure 8. Because the feed cabins movement is very slow, it can be consideredthat the system is quasi-static during the whole process. In the solution, the trajectory is evenly divided into101 nodes. The static equations of the CDPR of each node are solved by the two methods. Figure 9 showsthe change of the forces of the six cables in the whole process, and the abscissa is the distance traveled bythe feed cabin.The cable forces solved by the two methods are almost identical. Table 4 lists the maximum relative errorof each cable force during the calculation process and the solution time. With the same solution accuracy,AAS takes much less time than the iteration, which is important for FAST to achieve real-time control andimprove accuracy. (a) Cable forces solved by AAS (b) Cable forces solved by the iterative Fig. 9:
Comparison with the feed cabin moving slowly in a straight line. (4) The feed cabin slowly moves in a circular path with the point (0 , , as the center and as the radius, keeping the height unchanged from G , , to G − , , , see the redline in Figure 8. As in the previous case, it can be considered that the entire system is quasi-static, and the AS method for the CDPR in FAST 11
Table 5: Comparison with the feed cabin moving slowly in a circular path.
Maximum relative error of cable force at the lower end Time cost/s1 2 3 4 5 6 AAS Iteration0.1308% 0.1016% 0.1017% 0.1308% 0.1308% 0.1308% 22.750 372.000 trajectory is divided into 101 nodes to solve one by one. Figure 10 shows the change of the forces of the sixcables in the whole process, and the abscissa is the distance traveled by the feed cabin.Like the case of the straight line, the cable forces solved by the two methods are almost identical. Table 5lists the maximum relative error of each cable force during the calculation process and the solution time.With the same solution accuracy, AAS is still much faster than the iteration. (a) Cable forces solved by AAS (b) Cable forces solved by the iterative
Fig. 10:
Comparison with the feed cabin moving slowly in a circular path.
It can be seen from the comparison above that the calculation accuracy of the AAS method for solvingFAST’s cable forces is comparable to the iterative method, and the calculation time is greatly reduced.However, because the applied catenary model is a static model, this method is best applied to static orquasi-static situations. Whether this theory can be applied to dynamic calculations requires in-depth analysiscombined with the actual model and further research.The following is a comparison between the AAS method and the method currently used in FAST. Byletting the feed cabin run the same trajectory, the cable forces calculated by the AAS method are comparedwith the cable forces measured by the sensors when FAST is actually controlled. These sensors are respec-tively installed on 6 cables as close as possible to the anchor points A [ i ] shown in Figure 2. The trajectory isshown in Figure 11, with (0 , , . as the center, . as the radius, and making a full circle from G , , , . while maintaining the same height. It should be noted that the running process isslow and the system can be considered as quasi-static.Figure 12(a) shows the theoretically calculated cable forces as the feed cabin moves under this trajectory,while Figure 12(b) shows the actual cable forces measured during real-time control. And the root meansquare errors between them are shown in Table 6. It can be seen that the theoretical and experimentalnumerical trends are consistent, but there are still considerable discrepancies. Considering that the attitudechange of the feed cabin has a huge influence on the cable force, it is necessary to use the feed cabin attitudemeasured in real-time control when using the AAS method for calculation. Fig. 11: Schematic diagram of the feed cabin trajectory.Table 6: Root mean square errors between AAS method and the real-time control.
Method Root mean square error of cable force/kN1 2 3 4 5 6AAS 22.554 17.517 20.078 18.687 17.167 18.190AAS with actual attitudes 23.942 14.226 19.531 14.210 17.534 16.058AAS with actual attitudes and corrected coordinates 20.060 12.683 16.422 10.018 16.457 12.053 (a) Cable forces solved by AAS (b) Cable forces measured during real-time control
Fig. 12:
Comparison between the AAS method and the real-time control.
Figure 13(a) shows the cable forces calculated by the AAS method after considering the measuredattitude of the feed cabin, while Figure 13(b) shows the relative errors between these calculated cable forcesand the actual measured cable forces in Figure 12(b). And the root mean square errors between them areshown in Table 6. It can be seen that the theoretical and experimental numerical trends are more consistent,but there are still some deviations. After careful inspection, it was found that the mass center coordinates ofthe feed cabin had a large deviation. After iterative calculation, it is finally determined that the mass centeris near (0 . , . , . , which is far away from the theoretical coordinates (0 , , . . This alsoleads to larger deviations of cable forces. Therefore, it is necessary to correct the mass center coordinatesto recalculate the cable forces by the AAS method.Figur 14(a) shows the cable forces calculated by the AAS method after considering the measured at-titude of the feed cabin and correcting the mass center coordinates, while Figure 14(b) shows the relativeerrors between these calculated cable forces and the actual measured cable forces in Figure 12(b). And theroot mean square errors between them are shown in Table 6. It can be found that the theoretical and experi- AS method for the CDPR in FAST 13 (a) Cable forces solved by AAS with actual attitudes (b) Relative errors between AAS and real-time control
Fig. 13:
Comparison between the AAS method with actual attitudes and the real-time control. mental numerical trends are very close, and the relative errors are already acceptable. There are still manyreasons for these errors. (a) Cable forces solved by AAS with actual attitudes and corrected coordinates (b) Relative errors between AAS and real-time control
Fig. 14:
Comparison between the AAS method with actual attitudes and corrected coordinates and the real-time control.
The first is the coordinate deviation of the mass center. Even a slight error after correction can have ahuge impact on the cable forces. Moreover, the structure of the feed cabin actually changes during operation,which also causes the change in the mass center coordinates more or less.The second is the fact that many wires and sensors are added to the cables, which results that the cablesare not of uniform quality assumed by theory. This causes the deviations in the cable forces.Third, the object of comparison is the result of the existing model combined with PID control. Althoughit has been verified and can be used, it still has errors compared with true values.In addition, the sensors on the cables also have a measurement error of about 3%, which causes devia-tions in the cable forces as well.In summary, the theoretical values calculated by AAS after considering the measured attitudes and thecorrected mass center are consistent with the trend of the experimental data. It is expected to replace theexisting model of FAST in the future and combine with PID control or even machine learning to achievemore efficient and accurate control.
In this paper, a fast method, called AAS, for solving the static or quasi-static equations of the CDPR ofFAST is proposed to achieve FASTs real-time control. By extracting the necessary geometric and physicalcoefficients, the static or quasi-static equations can be solved analytically. In the comparison example with the traditional iterative method, AAS can save at least 75% of the time in the calculation of single cablesforce at a certain moment and even can save 90% of the time in the calculation of single cables force duringthe CDPR slowly moves. Also, it is verified through the experiment that the values calculated by AAS areconsistent with the measured data. Obviously, the difficulty of using the catenary model to control FASTin real-time is solved. Presumably in the future, FAST can be controlled with higher precision and moreefficient to complete more and more difficult observation tasks, and this method may be extended to otherCDPRs.
Acknowledgements
This work was financially supported by the National Natural Science Foundation ofChina (Nos. 11673039,11973062); the Youth Innova-tion Promotion Association CAS; the Open ProjectProgram of the Key Laboratory of FAST, NAOC, Chinese Academy of Sciences.