本論文研究了飛彈導引中的純比例導引理論對於飛彈攔截移動及靜止目標的撞擊時間估算。首先定義了LOS(Line-of-sight) 座標系統，之後推導了動力方程式及帶入純比例導引理論，最後以來獲得撞擊時間
估算的方程式。使用此座標系統能夠得到較簡單的撞擊時間預測方程式。此方程式為一積分式，但此方程式之解析解收斂速度太慢。因此本研究會將撞擊時間預測方程式之被積分式使用Chebyshev polynomials
數值方法來近似，使其容易積分藉此得到撞擊時間估算方程式之解析解。
  為了能夠縮短計算時間及簡化方程式，使用曲線擬合計算Chebyshev polynomials 之係數。最後使用迭代方式來計算出自適應導引增益來實現可控之撞擊時間。自適應導引增益可以使用在多枚飛彈同時攔截同一目標。

  This paper analyzes the impact time of PPN guidance law for nonmaneuvering target and stationary target. First, define the LOS (Line-of-Sight) coordinate system and derives the dynamic equations to implement the PPN guidance law. Then, through some definitions and by using the rotation angle of relative velocity, the function of time-to-go is obtained. Although using the LOS coordinate system can yield a simple time-to-go function, the analytical solution of integration is in hypergeometric form, which the convergence rate to achieve our desired accuracy is too slow. To address this issue, this research employs Chebyshev polynomials to estimate the integrand of the integration, transforming the integrand into a polynomial form for
easier integration and obtaining an analytical solution.

  To increase computational speed and simplify the function, curve fitting is utilized to compute the coefficients of Chebyshev polynomials. Finally, through iterative method, the research computes adaptive navigation gain to control the impact time. Adaptive navigation gain can be applied on launching multiple missiles to intercept a single target simultaneously.

Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Guidance Law and Time-to-go 4
2.1 LOS(Line-of-sight) coordinate system [1] . . . . . . . . . . . . . 4
2.2 PPN guidance law [1] . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Equations of motion [1] . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Time-to-go analysis [1] . . . . . . . . . . . . . . . . . . . . . . . 6
2.4.1 Nonmaneuvering target . . . . . . . . . . . . . . . . . . . 6
2.4.2 Stationary target . . . . . . . . . . . . . . . . . . . . . . 10
3 Time-to-go Approximation 11
3.1 Time-to-go approximation for nonmaneuvering target . . . . . . 11
3.1.1 ¯ρ approximation . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.2 τf approximation . . . . . . . . . . . . . . . . . . . . . . 12
3.1.3 Modified shifted second kind Chebyshev polynomials approximation
. . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.4 Approximation results . . . . . . . . . . . . . . . . . . . 16
3.2 Time-to-go approximation for stationary target . . . . . . . . . . 17
3.2.1 Modified shifted second kind Chebyshev polynomials approximation
. . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Taylor series approximation . . . . . . . . . . . . . . . . 21
3.2.3 Approximation result . . . . . . . . . . . . . . . . . . . . 26
3.2.4 Analytical solution for Taylor series and modified shifted
second kind Chebyshev polynomials . . . . . . . . . . . . 31
3.3 Second kind Gauss-Chebyshev quadrature . . . . . . . . . . . . . 31
4 Curve fitting 35
4.1 Curve fitting for nonmaneuvering target . . . . . . . . . . . . . . 35
4.1.1 Polynomial regression . . . . . . . . . . . . . . . . . . . . 35
4.1.2 Curve fitting result . . . . . . . . . . . . . . . . . . . . . 36
4.1.3 τf approximation using curve fitting . . . . . . . . . . . . 39
4.2 Curve fitting for stationary target . . . . . . . . . . . . . . . . . 41
4.2.1 Curve fitting result . . . . . . . . . . . . . . . . . . . . . 41
4.2.2 τf approximate result using curve fitting . . . . . . . . . 42
5 Adaptive β 46
5.1 Iterative β formula . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Example 2: multiple missiles cooperative attack . . . . . . . . . . 50
6 Conclusion 52
Appendix 54
Appendix A: Relative dynamics . . . . . . . . . . . . . . . . . . . . . 54
Appendix B: Angular velocity of LOS coordinate system . . . . . . . . 55
Appendix C: Details of (6.1) . . . . . . . . . . . . . . . . . . . . . . . 56
Appendix D: Details of (3.6) . . . . . . . . . . . . . . . . . . . . . . . 57
Appendix E: Modified shifted second kind chebyshev polynomials . . . 58
Appendix F: Coefficients of curve fitting . . . . . . . . . . . . . . . . . 60
List of Figures
2.1 XYZ: Inertial fixed coordinate, (er, et, eΩ): LOS fixed coordinate 4
2.2 Figure of α in the (er, et)-plane . . . . . . . . . . . . . . . . . . . 8
3.1 Modified shifted second kind Chebyshev polynomials approximation
result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 || sin(α)M−1 − (b0U0(α) + b2U2(α) + b4U4(α))|| . . . . . . . . . . 15
3.3 1
(sin α)γ approximation using 5 terms modified shifted second kind
Chebyshev polynomials (β = 1.7, 2, 3, 4) . . . . . . . . . . . . . . 19
3.4 1
(sin α)γ approximation using 5 terms modified shifted second kind
Chebyshev polynomials (β = 5, 6, 7, 8) . . . . . . . . . . . . . . . 19
3.5 ∥ 1
sin(α)γ
− [b0U0(α) + b2U2(α) + b4U4(α)]∥(β = 1.7, 2, 3, 4) . . . . 20
3.6 ∥ 1
sin(α)γ
− [b0U0(α) + b2U2(α) + b4U4(α)]∥(β = 5, 6, 7, 8) . . . . . 20
3.7 1
(sin α)γ approximation using Taylor series in [0, 0.1π] (β = 1.7) . 21
3.8 1
(sin α)γ approximation using Taylor series in [0.9π, π] (β = 1.7) . 22
3.9 1
(sin α)γ approximation using Taylor series in [0, 0.1π] (β = 3) . . 22
3.10 1
(sin α)γ approximation using Taylor series in [0.9π, π] (β = 3) . . 23
3.11 1
(sin α)γ approximation using Taylor series at [0, 0.1π] (β = 5) . . 23
3.12 1
(sin α)γ approximation using Taylor series in [0.9π, π] (β = 5) . . 24
3.13 1
(sin α)γ approximation using Taylor series at [0, 0.1π] (β = 8) . . 24
3.14 1
(sin α)γ approximation using Taylor series in [0.9π, π] (β = 8) . . 25
3.15 Error of 1 term Taylor series . . . . . . . . . . . . . . . . . . . . 26
3.16 τf approximation result using (3.15) (β = 1.7, 2, 3, 4) . . . . . . . 28
3.17 τf approximation result using (3.15) (β = 5, 6, 7, 8) . . . . . . . . 28
3.18 The difference between the τf obtained by (3.15) & (2.22) (β =
1.7, 2, 3, 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.19 The difference between the τf obtained by (3.15) & (2.22) (β =
5, 6, 7, 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.20 The difference between the τf obtained by (3.15) & (2.21) for β = 3 30
3.21 Error of using interpolation at navigation gain = 3 [2] . . . . . . 31
4.1 Curve fitting for b0 . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Curve fitting for b2 . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Curve fitting for b4 . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.4 Curve fitting result for b0, b2 and b4 . . . . . . . . . . . . . . . . 41
4.5 τf approximation result by using curve fitting in (3.15) (β =
1.7, 2, 3, 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.6 τf approximation result by using curve fitting in (3.15) (β = 5, 6, 7, 8) 43
4.7 The difference between the τf obtained by using curve fitting in
(3.15) & (2.22) (β = 1.7, 2, 3, 4) . . . . . . . . . . . . . . . . . . 44
4.8 The difference between the τf obtained by using curve fitting in
(3.15) & (2.22) (β = 5, 6, 7, 8) . . . . . . . . . . . . . . . . . . . 44
5.1 ¯ρ v.s. τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Corresponding β at every τ of Figure 5.1 . . . . . . . . . . . . . 48
5.3 Iteration steps at every τ of Figure 5.1 . . . . . . . . . . . . . . . 48
5.4 ¯ρ v.s. τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.5 Corresponding β at every τ of Figure 5.4 . . . . . . . . . . . . . 49
5.6 Iteration steps at every τ of Figure 5.4 . . . . . . . . . . . . . . . 50
5.7 Multiple missiles intercept one target simulation . . . . . . . . . 51
6.1 First five terms of modified shifted second kind Chebyshev polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
List of Tables
3.1 Coefficients of modified shifted second kind Chebyshev polynomials
approximation for Figure 3.1 . . . . . . . . . . . . . . . . . . . . 16
3.2 Approximate result (β = 3, ν = 0.3, α0 = 0.3π,CM = 0.97678,CT =
0.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Approximate result (β = 4, ν = 0.3, α0 = 0.3π,CM = 0.97678,CT =
0.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Approximate result (β = 3, ν = 0.3, α0 = 0.7632π,CM = 0.97943,CT =
0.74) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Approximate result (β = 4, ν = 0.3, α0 = 0.7632π,CM = 0.97943,CT =
0.74) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.6 Coefficients of modified shifted second kind Chebyshev polynomials
for stationary target at β = 1.7, 3, 5, 8 . . . . . . . . . . . . . . . 18
3.7 Coefficients of modified shifted second kind Chebyshev polynomials
for nonmaneuvering targets . . . . . . . . . . . . . . . . . . . . . 33
3.8 Coefficients of modified shifted second kind Chebyshev polynomials
for stationary targets . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1 RMSE for given range . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Range for given RMSE . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Approximate result (β = 3, ν = 0.3, α0 = 0.3π,CM = 0.97678,CT =
0.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Approximate result (β = 4, ν = 0.3, α0 = 0.3π,CM = 0.97678,CT =
0.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 Approximate result (β = 3, ν = 0.3, α0 = 0.7632π,CM = 0.97943,CT =
0.74) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.6 Approximate result (β = 4, ν = 0.3, α0 = 0.7632π,CM = 0.97943,CT =
0.74) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.7 CPU time (CPU: i5-10400F, memory: 32GB) . . . . . . . . . . . 41
4.8 RMSE for given range . . . . . . . . . . . . . . . . . . . . . . . . 42
4.9 CPU time (CPU: i5-10400F, memory: 32GB) . . . . . . . . . . . 45

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