Computed elements of the CPT functions for G03, R01, E01, C06 satellites by * Iter* = 1000 and

_{MAX}

*= 5e-12 in loops {iteration numbers of 64Bit Python and 32Bit C++ software in windows are denoted as*thres

*and as*iter

*respectively}.*iter++

Open access peer-reviewed chapter

By Orhan Kurt

Submitted: October 11th 2017Reviewed: March 8th 2018Published: November 5th 2018

DOI: 10.5772/intechopen.76313

Downloaded: 723

To interpret and explain the mechanism of an engineering problem, the redundant observations are carried out by scientists and engineers. The functional relationships between the observations and parameters defining the model are generally nonlinear. Those relationships are constituted by a nonlinear equation system. The equations of the system are not solved without using linearization of them on the computer. If the linearized equations are consistent, the solution of the system is ensured for a probably global minimum quickly by any approximated values of the parameters in the least squares (LS). Otherwise, namely an inconsistent case, the convergence of the solution needs to be well-determined approximate values for the global minimum solution even if in LS. A numerical example for 3D space fixes coordinates of an artificial global navigation satellite system (GNSS) satellite modeled by a simple combination of first-degree polynomial and first-order trigonometric functions will be given. It will be shown by the real example that the convergence of the solution depends on the approximated values of the model parameters.

- nonlinear equation system
- objective function
- least squares
- convergence
- consistency

There are two main computing classes, these are hard and soft computing. Scientists and engineers generally prefer the first-class computing because they can easily establish an explicit mathematical relationship between the model parameters and their data (observations), not in the second class. The relationships between the parameters and data can be linear or nonlinear. If the relations are nonlinear, they should be linearized via Taylor expansion [1, 2, 3, 4, 5, 6, 7]. Therefore, the linear models can be solved by linear algebra [8, 9, 10, 11, 12, 13, 14, 15].

To overcome complicated real-life problems whose mathematical models are not known, the soft computing techniques have been developed in the last decades. We can count well-known techniques, some as artificial neural network (ANN), artificial intelligence (AI), machine learning (ML), deep learning (DP), fuzzy logic (FL) and genetic algorithms (GA) [16, 17, 18]. The techniques inspired by the human intelligence and learning processes can be very time-consuming according to the data given in run due to their processing based on the trial-and-error method. If these techniques are roughly defined, data (experimental outcomes and observations) are separated into two parts in them, learning (or training) data and test data. Mathematical (functional and/or stochastic) relations between data and model parameters are learned from the learning data. The handled model is tested by means of the test data. After that, the trained and developed model, if meets expectations, is used to estimate for producing unobserved data for the scientific (or engineering) problems [16, 17, 18].

In the soft computing techniques, the linear algebra is also a very effective tool to solve the problem as in the hard computing ones. For this reason, we should take a short overview on linear algebra used in science and engineering [16, 17, 18].

Advertisement## 2. Linear algebra and objective functions

ε n = A n , u δ u − l n , P n , n , E1A n , u = ∂ f y ̂ x ̂ ∂ x ̂ y ̂ , x ̂ = y , x and l n = f y x . B m , n ε n − A m , u δ u + l m = 0 P n , n E2B m , n = ∂ f y ̂ x ̂ ∂ y ̂ y ̂ , x ̂ = y , x 0 . ### 2.1. Objective functions

i ε T ↦ min L 1 − norm estimation Least absolute residuals , E3ε T ε ↦ min L 2 − norm estimation Least squares , E4ε max ↦ min L ∞ − norm estimation Minmax absolute residuals , E5i = 1 1 … 1 T . ### 2.2. Rank deficiencies in linear models

d s = n − r , Surjectivity “ onto ” mapping E6ad i = u − r . Injectivity “ one − to − one ” mapping E6bQ u , u = N + = S Λ + S T = V Σ + U T , Pseudoinverse of N E7aN u , u = S Λ S T = U Σ V T , For a positive definite symmetric matrix E7bΛ + = Σ + = Λ r − 1 0 r , d 0 r , d 0 d , d . ### 2.3. Hard computing

A n , u δ u = l n , P = I . E8λ ̂ n = Q n , n l n , Q n , n = A A T + . E9δ ̂ = A T λ ̂ = A T Q l , Σ δ ̂ = σ 0 2 A T Q Q A , E10ax ̂ = x + δ ̂ , Σ x ̂ = Σ δ ̂ , E10bx ̂ T x ̂ ↦ min . E10cδ ̂ u = Q A T l , Q u , u = A T A + , E11ax ̂ = x + δ ̂ , Σ x ̂ = σ ̂ 0 2 Q , E11bσ ̂ 0 2 = ε ̂ T ε ̂ n − r , A posteriori variance r = rank A E11cε ̂ = A δ ̂ − l , E11dε ̂ T ε ̂ ↦ min . L 2 − norm estimation Least Square E11e

Linear algebra has two basic problems. A solution of linear equations system is one of them; the other is the eigendecomposition. In this chapter, we will use both of them upon a linear equation system as a combined form (Eqs. (8)–(11)) in which we will solve the linear equations system by means of the singular value decomposition related with the eigendecomposition (or the matrix diagonalization) [8, 9, 13, 14].

Suppose an estimated unknown vector

Start with a linear or nonlinear functions vector

Mathematical model between data and unknowns can be established by Taylor expansion for any model. However, if

where

Most of science and engineering problems can be modeled as

A generalization for objective functions is * L*which is accepted as a robust estimation method in just linear models [9, 10, 11].

The second-degree objective function is * L*which is known as least squares (LS) method and widely used in hard and soft computations.

The last-degree objective function is _{∞}* -norm estimation*which is known as minmax method. In fact, the soft computing techniques use this objective while it applies the trial-and-error method in their learning stages. Eq. (1) under

While a rank is a number that indicates a linear independent column, the number of the coefficient matrix of unknowns in a linear equation system, a rank deficiency represents a linear dependent column number (if it is smaller than the row number) of the coefficient matrix. Inconsistency in the solution stage of a linear equation system results from the (rank) deficiencies. Defining the rank of

Denoting the rank defect

Objective functions are used to remove the surjectivity defect

* Datum defects (d-defects)*are closely related to the origin of the spatial system. The defect arises if the data do not carry any information to cover the absolute spatial position of the problem given.

* Configuration (Design) defects (c-defects)*occur from weak geometric relation among data and unknowns. To avoid the defect, we can be careful and planned when picking data (whose interval or/and place) and choosing the consistent mathematical model (can use auxiliary variables instead of original ones).

* Ill-conditioning defects (i-defects)*arise from the large intervals among the elements of the coefficient matrix of unknowns. Norming the matrix can reduce ill-conditioning defects but cannot remove it fully.

The defects lead to the failure of any given problem to be solved properly. Since the unknown coefficient matrix cannot be inverted by * regular (ordinary) inverse*methods, we should use

Since

Linearizing from nonlinear functions to their linear form by means of Taylor expansion, a linear equation system is to be handled as Eq. (1). To avoid complicated proofs in the solution of an equation system, the simplified mathematical model can be written in the following (statically rotation invariant [1]) numerical computation form.

Meet two states to solve Eq. (8),

And then

In the state (

Solution to the second state

End the solution if the condition ensured is

Relationships between nonlinearity and LS in a multidimensional surface have been shown by Teunissen et al. [1, 2]. The authors argued the relation on some simple examples and gave some analytical solutions for them. But, they highlighted that those types of analytical solutions have not been given for every problem and emphasized that suitable Taylor expansions have been useful to the solution not being transformed into the analytical ones.

These type functions can be used in defining the orbits of artificial satellites (and celestial bodies). Also, the numerical example part of this chapter, to estimate those type functions, will be inspected and applied on a real example. To foresee a model for any problem we should interpret the model parameter and comprehend the geometry of the model (Figure 1).

With respect to independent variable time * t*, a combination function of

where

In this chapter, the functions

For this propose, an easy transformation into any epoch (e.g., it can be taken as the first epoch

where

A solution of nonlinear Eq. (12) is realized in the following order. Linearizing Eq. (12) by Taylor expansion and omitting the terms greater than or equal to quadratic ones, the linear equation system as given by Eq. (8) is obtained. The explicit form of the Eq. (8) with respect to the approximate values of unknowns for a CPT is:

We can use a recursive solution for Eq. (14) instead of the batch solution as Eq. (11) because of its solution velocity.

Continuation of the solution of Eq. (15) can be performed according to Eq. (11). The model given by Eq. (12) is a simple model to determine the satellite orbit motions. For more complicated models, the readers can utilize [19, 20, 21, 22, 23, 24, 25] resources.

For a nonlinear estimation of CPT functions, some numerical examples are chosen from GNSS {Global navigation satellite systems = GPS (USA) + GLONASS (RU), GALILEO (EU), COMPASS (CHN)} artificial satellite orbits whose coordinates are downloaded from the internet address ftp://ftp.glonass-iac.ru/MCC/PRODUCTS/17091/final/Sta19426.sp3 [26].

For this purpose, two estimation software have been developed in 64Bit Python (in accordance with the 2.7 and 3.6 version) and 32Bit C++ (Code::Blocks) environments to see the convergence rate of the mathematical model given in Eq. (12) [27, 28]. The computed elements of CPT functions for the selected four satellites R01 (GLONASS), G03 (GPS), E01 (GALILEO) and C06 (COMPASS) are summarized in Table 1 in which they are ordered from the nearest satellite to the farthest one.

The motions of the CPT functions estimated satellites (in Table 1) with respect to earth- (left column of Figure 3) and space-fixed (right column of Figure 3) coordinate systems are demonstrated in Figure 3. Moreover, coherence between the estimated CPT function (black solid line) of the C06 satellite and its data points given (colorful circles) is represented in detail in Figure 4.

We know that accuracy of precise SP3 file coordinates is about

Comparing the solution velocities (from the iteration numbers with respect to

If we chose the threshold as 51e-13, we can see the distinctions of solution convergences between 64Bit and 32Bit running on the estimations of C06 satellite from 1000 (X), 8 (Y), 7 (Z), 1000 (S) in 32Bit C++ and 10 (X), 8 (Y), 6 (Z), 28 (S) in 64Bit Python in Windows. In here, 1000 is the maximum iteration number. If the mathematical model would be more complicated and its data number would be bigger than the number used in the example part, we would see the state more prominently.

Approximated values and the loop element for the unknowns are computed following the order in all solutions of the satellites when running in 32Bit C++ and 64Bit Python platforms for the estimations in Table 1.

Maximum iteration number and threshold loop elements are

Taking the approximated values as

Since the estimated standard deviations

The added terms represent an amplitude

we can choose the approximate values of the new parameters as

from

The approximate values are substituted in the following linearized model as initial values for the loop in LS estimation.

After the evaluation, the improved solution for the C06 satellite is represented in Table 2. We can readily see the improvements upon the downs of the standard deviations from * =*{±56.552, ±38.018, ±51.901, ±0.491} (Table 1) into

Sat. | ||||||
---|---|---|---|---|---|---|

5 5 | 4 4 | 4 5 | — — | 10 9 | ||

R01 (RU) | _{φ}_{φ}_{φ}_{φ}_{φ}_{φ} | −11.860 0.028 25,474.503 −60°54′04.37″ 31°57′53.92″ 11 ^{h}05’44.37” | 2.420 0.003 11,178.453 −23°03′56.93″ −31°57′44.92″ 11 ^{h}15’47.54” | 2.771 0.052 22,965.361 −30°31′00.32″ −31°57′56.31″ 11 ^{h}15’43.53” | km km/h km deg deg/h h | 0.001 8.127 42°41′35.87″ −31°58′13.55″ 11 ^{h}15’37.46” |

2.902 | 1.266 | 2.653 | km | 0.211 | ||

6 6 | 4 4 | 4 4 | — — | 30 34 | ||

G03 (USA) | _{φ}_{φ}_{φ}_{φ}_{φ}_{φ} | −17.545 −0.132 24,800.602 66°34′39.73” 30°04′56.03” 11 ^{h}58’01.91” | 7.332 −0.045 17,959.166 −48°06′43.68” 30°04′52.50” 11 ^{h}58’3.31” | −1.824 −0.085 21,757.547 −7°51′47.16” 30°05′00.02” 11 ^{h}58’00.32” | km km/h km deg. deg./h h | −0.003 14.211 −72°24′19.52” −30°08′11.17” 11 ^{h}56’44.42” |

4.898 | 3.648 | 3.961 | km | 0.183 | ||

4 5 | 5 5 | 4 4 | — — | 32 93 | ||

E01 (EU) | _{φ}_{φ}_{φ}_{φ}_{φ}_{φ} | 1.866 0.052 21,349.011 −35°22′15.46″ −25°34′13.06” 14 ^{h}04’43.81” | 1.358 −0.048 26,031.918 85°57′22.67” −25°34′14.87” 14 ^{h}04’42.81” | −4.673 0.004 24,878.432 −16°23′29.47” 25°34′18.57” 14 ^{h}04’40.78” | km km/h km deg. deg./h h | −0.000 3.720 8°29′58.23” −25°42′10.68” 14 ^{h}00’22.19” |

1.796 | 1.428 | 1.058 | km | 0.236 | ||

7 10 | 6 7 | 6 7 | — — | 28 29 | ||

C06 (CHN) | _{φ}_{φ}_{φ}_{φ}_{φ}_{φ} | −407.495 61.945 41,716.806 −24°21′36.29” −15°07′02.75” 23 ^{h}48’48.86” | 228.037 −6.376 24,832.649 59°24′50.73″ −15°02′16.41” 23 ^{h}56’22.30” | 266.178 −7.892 34,235.695 67°36′9.42″ −15°02′36.74” 23 ^{h}55’49.94” | km km/h km deg. deg./h h | −0.358 227.599 18°32′27.85” −14°57′17.11” 24 ^{h}04’21.41” |

56.552 | 38.018 | 51.901 | km | 0.491 |

Sat. | ||||||
---|---|---|---|---|---|---|

27 | 158 | 53 | — | 19 | ||

C06 (CHN) | _{φ}_{φ}_{φ}_{φ}_{φ}_{φ}_{φ}_{φ} | 223.514 1.515 42,064.675 −25°07′54.78″ −15°02′24.75” 110.710 5°08′57.82” 30°10′58.43” | 149.296 0.224 66.183 11°15′24.23″ 30°04′30.44″ 24,808.034 30°28′55.61″ 15°02′17.24” | 182.012 0.060 91.527 3°15′58.76″ 30°04′02.09″ 34,217.994 −22°21′14.60″ −15°02′19.27” | km km/h km deg deg/h km deg. deg./h | 42,169.956 0.055 225.161 19°47′37.54″ −15°02′43.26″ 0.838 8°56′45.50″ −29°40′58.52” |

0.178 | 0.137 | 0.191 | km | 0.003 |

As another example, the expanded model has been trained on the coordinate components of R01 GLONASS satellite by means of Python 3.6 software on Windows. We can also see the improvements upon the downs of the standard deviations (* its iteration numbers*) from

Condition numbers computed from a rate of maximum and minimum eigenvalues or a rate of singular values under LS are very effective tools for determining the consistency as well. Therefore, a larger condition number can cause larger iteration number (related to convergence rate), We can see those states from the CPT estimations of the C06 satellite with the iteration (* iter*) and condition numbers (

In this chapter, the least squares (LS) estimations of the artificial satellite orbital movements by a combination of polynomial and trigonometric (CPT) functions have been given after a general overview has been made on the hard and soft computations. In practice, the orbital motions are modeled on Keplerian orbital elements. In contrary to this, the coordinate components have been selected for this chapter due to the nonlinear relations of the components and the unknowns which are the elements of CPT functions. The relations cause inconsistencies in the LS solutions. The inconsistencies result from the two injectivity defects, * c-defects*and

For the sake of simplicity for readers, a simple CPT function has been chosen at first. After the initial estimation of the function, the estimated errors vector has been found. We have seen that the errors have had a periodic characteristic in time. So, a new wave defining the error characteristic and been able to carry by the first wave has been planned for expanding the CPTs. It is shown that we can expand a CPT function until ensuring statically equivalency between a priory and a posteriori variances. For instance, one may secure the equivalency of the variances if one would expand more by a new wave in the last estimated model in the same manner.

The convergence rates (upon the iteration numbers) of the LS estimation have been inspected according to the threshold (* thres = 5e-12*) which is a good value for the estimation of the nonlinear CPT function. An algorithm compiled by different compilers and run in different architectures (with 32 Bit or 64 Bit) changes the convergence rate of the estimations in such as the inconsistent scientific problems. It is also observed that the iteration numbers change when the 64-bit Python software is run on Linux platform which has a different framework than Windows. But, the numbers have not been given in the example part of the chapter. Contrary to inconsistency model, namely in a consistent one, the iteration numbers can take equivalence values in all circumstances. Another way to determine the inconsistency of a model is to obtain its condition number which is computed from a rate of maximum and minimum eigenvalues or of singular values under LS. If the condition number is close to one, the projected model is accepted as a consistent model.

We can use the Soft Computing Methods (SCM) if not an exact mathematical relationship between the data and unknowns. The mathematical model is established by the trial-and-error method in training part of SCM by means of arbitrary weights and activation functions depending on SCM expert forecasts. For the solution of the SCM model during the training, we can use least absolute residuals (LAR) and minmax absolute residuals (MAR) objective functions by the linear programming or the LS estimation as in hard computing method (HCM). In the state, the inconsistency problem can erase whatever the solution method (LAR, MAR or LS) is. The inconstancy can be removed by means of experiences gained from HCMs.

Prior information is very important to select a suitable mathematical model for a scientific problem. For example, comparing a priori variance with a posteriori variance at the end of the estimation is a useful warning to the user to determine the correct mathematical model as seen from the expanded model in the example section of the chapter.

In numerical computation, there are two main phenomena which are the mathematical model (as a combination of functional and stochastic models) and objective function. The solution strategy is of no importance if the same mathematical model and objective function are preferred in the same problem of hard computing. All solution strategies always give same results, only their solution time spans can be distinct from each other (Table 3).

Advertisement## Acknowledgments

I wish to thank * TÜRKSAT A.S*(https://www.turksat.com.tr/en) supporting this study.

h_{j}−t_{0} [ | km_{j} [ | km_{j} [ | km_{j} [ | μsec_{j} [ | |
---|---|---|---|---|---|

0 | 0.00 | −17531.307506 | 21541.792054 | 31834.209680 | 691.175390 |

1 | 0.25 | −19992.147900 | 20678.913485 | 30924.464057 | 691.487182 |

2 | 0.50 | −22367.333395 | 19727.437365 | 29882.214843 | 691.798804 |

3 | 0.75 | −24646.589754 | 18691.353300 | 28711.797065 | 692.110527 |

4 | 1.00 | −26820.032098 | 17575.021170 | 27418.102912 | 692.422088 |

5 | 1.25 | −28878.208475 | 16383.154007 | 26006.563346 | 692.733923 |

6 | 1.50 | −30812.142166 | 15120.799311 | 24483.127129 | 693.045689 |

7 | 1.75 | −32613.372313 | 13793.318783 | 22854.237453 | 693.357416 |

8 | 2.00 | −34273.992670 | 12406.366506 | 21126.806228 | 693.668853 |

9 | 2.25 | −35786.688260 | 10965.865676 | 19308.186097 | 693.980453 |

10 | 2.50 | −37144.769753 | 9477.983947 | 17406.140275 | 694.291853 |

11 | 2.75 | −38342.205344 | 7949.107471 | 15428.810302 | 694.603337 |

12 | 3.00 | −39373.649964 | 6385.813755 | 13384.681845 | 694.914789 |

13 | 3.25 | −40234.471624 | 4794.843425 | 11282.548651 | 695.226157 |

14 | 3.50 | −40920.774721 | 3183.071022 | 9131.474825 | 695.537828 |

15 | 3.75 | −41429.420160 | 1557.474971 | 6940.755568 | 695.849062 |

16 | 4.00 | −41758.042128 | −74.893170 | 4719.876568 | 696.160359 |

17 | 4.25 | −41905.061383 | −1706.940006 | 2478.472204 | 696.471627 |

18 | 4.50 | −41869.694990 | −3331.562017 | 226.282782 | 696.782836 |

19 | 4.75 | −41651.962330 | −4941.677516 | −2026.889008 | 697.094239 |

20 | 5.00 | −41252.687387 | −6530.258698 | −4271.222183 | 697.405373 |

21 | 5.25 | −40673.497209 | −8090.363611 | −6496.921900 | 697.716659 |

22 | 5.50 | −39916.816531 | −9615.167884 | −8694.263981 | 698.028206 |

23 | 5.75 | −38985.858544 | −11097.996030 | −10853.639319 | 698.339619 |

24 | 6.00 | −37884.611845 | −12532.352178 | −12965.597898 | 698.651055 |

25 | 6.25 | −36617.823589 | −13911.950035 | −15020.892223 | 698.962566 |

26 | 6.50 | −35190.978917 | −15230.741942 | −17010.519896 | 699.273893 |

27 | 6.75 | −33610.276761 | −16482.946839 | −18925.765115 | 699.585363 |

28 | 7.00 | −31882.602129 | −17663.077000 | −20758.238873 | 699.896674 |

29 | 7.25 | −30015.495009 | −18765.963379 | −22499.917620 | 700.208082 |

30 | 7.50 | −28017.116067 | −19786.779434 | −24143.180195 | 700.519378 |

31 | 7.75 | −25896.209299 | −20721.063292 | −25680.842806 | 700.830889 |

32 | 8.00 | −23662.061860 | −21564.738140 | −27106.191889 | 701.142117 |

33 | 8.25 | −21324.461276 | −22314.130731 | −28413.014646 | 701.453707 |

34 | 8.50 | −18893.650283 | −22965.987908 | −29595.627135 | 701.764972 |

35 | 8.75 | −16380.279535 | −23517.491071 | −30648.899728 | 702.076478 |

36 | 9.00 | −13795.358449 | −23966.268499 | −31568.279851 | 702.387737 |

37 | 9.25 | −11150.204459 | −24310.405495 | −32349.811865 | 702.699185 |

38 | 9.50 | −8456.390949 | −24548.452300 | −32990.154024 | 703.010540 |

39 | 9.75 | −5725.694161 | −24679.429736 | −33486.592421 | 703.321863 |

40 | 10.00 | −2970.039352 | −24702.832610 | −33837.051887 | 703.633117 |

41 | 10.25 | −201.446497 | −24618.630817 | −34040.103800 | 703.944417 |

42 | 10.50 | 2568.024186 | −24427.268222 | −34094.970801 | 704.255873 |

43 | 10.75 | 5326.326597 | −24129.659289 | −34001.528424 | 704.567076 |

44 | 11.00 | 8061.482740 | −23727.183555 | −33760.303648 | 704.878433 |

45 | 11.25 | 10761.636054 | −23221.677952 | −33372.470443 | 705.189684 |

46 | 11.50 | 13415.103835 | −22615.427078 | −32839.842340 | 705.501070 |

47 | 11.75 | 16010.428511 | −21911.151470 | −32164.862120 | 705.812547 |

48 | 12.00 | 18536.427516 | −21111.993965 | −31350.588697 | 706.124190 |

49 | 12.25 | 20982.241560 | −20221.504257 | −30400.681303 | 706.435990 |

50 | 12.50 | 23337.381077 | −19243.621724 | −29319.381091 | 706.747258 |

51 | 12.75 | 25591.770667 | −18182.656653 | −28111.490271 | 707.058650 |

52 | 13.00 | 27735.791345 | −17043.269964 | −26782.348927 | 707.370038 |

53 | 13.25 | 29760.320445 | −15830.451549 | −25337.809640 | 707.681516 |

54 | 13.50 | 31656.769034 | −14549.497344 | −23784.210083 | 707.992915 |

55 | 13.75 | 33417.116707 | −13205.985271 | −22128.343727 | 708.304435 |

56 | 14.00 | 35033.943650 | −11805.750146 | −20377.428827 | 708.615955 |

57 | 14.25 | 36500.459878 | −10354.857705 | −18539.075847 | 708.927258 |

58 | 14.50 | 37810.531573 | −8859.577863 | −16621.253481 | 709.238773 |

59 | 14.75 | 38958.704447 | −7326.357314 | −14632.253449 | 709.549980 |

60 | 15.00 | 39940.224086 | −5761.791632 | −12580.654219 | 709.861492 |

61 | 15.25 | 40751.053248 | −4172.596961 | −10475.283826 | 710.172943 |

62 | 15.50 | 41387.886081 | −2565.581420 | −8325.181952 | 710.484439 |

63 | 15.75 | 41848.159196 | −947.616367 | −6139.561422 | 710.796071 |

64 | 16.00 | 42130.059795 | 674.392412 | −3927.769279 | 711.107495 |

65 | 16.25 | 42232.530676 | 2293.533436 | −1699.247596 | 711.419275 |

66 | 16.50 | 42155.272132 | 3902.918094 | 536.505839 | 711.730566 |

67 | 16.75 | 41898.740894 | 5495.708947 | 2769.976807 | 712.042086 |

68 | 17.00 | 41464.146141 | 7065.147654 | 4991.673762 | 712.353663 |

69 | 17.25 | 40853.442432 | 8604.582424 | 7192.166646 | 712.665021 |

70 | 17.50 | 40069.319938 | 10107.494927 | 9362.125250 | 712.976412 |

71 | 17.75 | 39115.191840 | 11567.526548 | 11492.356997 | 713.287987 |

72 | 18.00 | 37995.179006 | 12978.503915 | 13573.844046 | 713.599417 |

73 | 18.25 | 36714.092016 | 14334.463621 | 15597.779572 | 713.910828 |

74 | 18.50 | 35277.410608 | 15629.676058 | 17555.603122 | 714.221952 |

75 | 18.75 | 33691.260665 | 16858.668299 | 19439.034930 | 714.533584 |

76 | 19.00 | 31962.388796 | 18016.245942 | 21240.109087 | 714.845281 |

77 | 19.25 | 30098.134631 | 19097.513875 | 22951.205467 | 715.156764 |

78 | 19.50 | 28106.400907 | 20097.895874 | 24565.080300 | 715.468390 |

79 | 19.75 | 25995.621474 | 21013.152993 | 26074.895308 | 715.779824 |

80 | 20.00 | 23774.727299 | 21839.400671 | 27474.245306 | 716.091334 |

81 | 20.25 | 21453.110591 | 22573.124521 | 28757.184187 | 716.402951 |

82 | 20.50 | 19040.587166 | 23211.194732 | 29918.249189 | 716.714457 |

83 | 20.75 | 16547.357150 | 23750.879046 | 30952.483389 | 717.026029 |

84 | 21.00 | 13983.964160 | 24189.854253 | 31855.456321 | 717.337474 |

85 | 21.25 | 11361.253075 | 24526.216175 | 32623.282652 | 717.649172 |

86 | 21.50 | 8690.326544 | 24758.488074 | 33252.638847 | 717.960633 |

87 | 21.75 | 5982.500334 | 24885.627475 | 33740.777748 | 718.272113 |

88 | 22.00 | 3249.257698 | 24907.031342 | 34085.541002 | 718.583548 |

89 | 22.25 | 502.202871 | 24822.539587 | 34285.369282 | 718.895203 |

90 | 22.50 | −2246.986126 | 24632.436884 | 34339.310236 | 719.206528 |

91 | 22.75 | −4986.605239 | 24337.452758 | 34247.024111 | 719.518326 |

92 | 23.00 | −7704.972522 | 23938.759921 | 34008.787012 | 719.830016 |

93 | 23.25 | −10390.476491 | 23437.970866 | 33625.491749 | 720.141620 |

94 | 23.50 | −13031.624539 | 22837.132665 | 33098.646226 | 720.452998 |

95 | 23.75 | −15617.091194 | 22138.719998 | 32430.369356 | 720.764247 |

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© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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