A novel linear uncertainty propagation method for nonlinear dynamics with interval process

Abstract

Interval process is a preferable model for time-varying uncertainty propagation of dynamic systems when only the range of uncertainties can be obtained. However, for nonlinear systems, except for Monte Carlo (MC) simulation, there are still few efficient uncertainty propagation methods under the interval process model. This paper develops a non-intrusive and semi-analytical uncertainty propagation method, named the “convex model linearization method (CMLM),” by constructing a linearization formulation of a nonlinear system in a non-probabilistic sense. First, the criterion to evaluate the difference between the original system and the linearization formulation is derived, represented by the discrepancy of middle point, radius and correlations of response. By minimizing these three parameters, the coefficients of linear equations will be optimized to obtain the linearization formulation of the original system. Then, analytical equations are built to calculate uncertainty response under the interval process, without time-consuming analysis of the original system. To further improve the efficiency of the linearization process, Chebyshev polynomial is introduced to approximate the nonlinear dynamic analysis. Two numerical examples of duffing oscillators and vehicle rides are set to test the proposed CMLM. Compared to the MC method, with comparable uncertainty response precision, the CMLM just needs 1–10% times of dynamic analyses of the nonlinear system. Furthermore, a practical launch vehicle ascent trajectory problem with black-box dynamics is solved by, respectively, the CMLM and MC method. The results verify the capacity of the CMLM to deal with black-box problems and show that the CMLM performs better in terms of accuracy, efficiency and robustness.

Appendices

Appendix 1

1.1 Derivation of CMLM criterion

By the weighted sum of e₁, e₂ and e₃, a comprehensive index e can be established. The weighting coefficients are first determined. The index e₁ is independent because it is only related to M and not to N. Thus, its weighting coefficient can be defined as 1, and the value does not affect the optimal value of M and N. Then, to achieve the weighted sum of e₂ and e₃ in an equivalent order of magnitude, they are compared in a similar form represented by covariances:

$$ \begin{gathered} e_{2} = \left( {f^{{\text{R}}} - \hat{f}^{{\text{R}}} } \right)^{2} = \left( {f^{{\text{R}}} } \right)^{2} + \left( {\hat{f}^{{\text{R}}} } \right)^{2} - 2f^{{\text{R}}} \hat{f}^{{\text{R}}} \\ = {\text{cov}} \left( {f^{{\text{I}}} ,f^{{\text{I}}} } \right) + {\text{cov}} \left( {\hat{f}^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right) - 2\sqrt {{\text{cov}} \left( {f^{{\text{I}}} ,f^{{\text{I}}} } \right){\text{cov}} \left( {\hat{f}^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right)} \\ e_{3} = \left( {1 - \rho \left( {f^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right)} \right) \\ = \frac{{\left( {\sqrt {{\text{cov}} \left( {f^{{\text{I}}} ,f^{{\text{I}}} } \right){\text{cov}} \left( {\hat{f}^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right)} - {\text{cov}} \left( {f^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right)} \right)}}{{\sqrt {{\text{cov}} \left( {f^{{\text{I}}} ,f^{{\text{I}}} } \right){\text{cov}} \left( {\hat{f}^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right)} }} \\ = \frac{{\left( {2\sqrt {{\text{cov}} \left( {f^{{\text{I}}} ,f^{{\text{I}}} } \right){\text{cov}} \left( {\hat{f}^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right)} - {\text{cov}} \left( {f^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right) - {\text{cov}} \left( {f^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right)} \right)}}{{2\sqrt {{\text{cov}} \left( {f^{{\text{I}}} ,f^{{\text{I}}} } \right){\text{cov}} \left( {\hat{f}^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right)} }} \\ \end{gathered} $$

(60)

From the comparison, it is obvious that e₃ should be weighted by

$$ 2\sqrt {{\text{cov}} \left( {f^{{\text{I}}} ,f^{{\text{I}}} } \right){\text{cov}} \left( {\hat{f}^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right)} = 2f^{{\text{R}}} \hat{f}^{{\text{R}}} $$

(61)

Then e₂ and e₃ can be weighted equivalently, and e is finally expressed as:

$$ \begin{gathered} e = e_{1} + e_{2} + e_{3} \cdot 2f^{{\text{R}}} \hat{f}^{{\text{R}}} \\ = \left( {f^{{\text{M}}} - \hat{f}^{{\text{M}}} } \right)^{2} + \left( {f^{{\text{R}}} - \hat{f}^{{\text{R}}} } \right)^{2} + \left( {1 - \rho_{{\text{s}}} \left( {f^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right)} \right) \cdot 2f^{{\text{R}}} \hat{f}^{{\text{R}}} \\ = \left( {f^{{\text{M}}} - \hat{f}^{{\text{M}}} } \right)^{2} + \left( {f^{{\text{R}}} } \right)^{2} + \left( {\hat{f}^{{\text{R}}} } \right)^{2} - 2\rho_{{\text{s}}} \left( {f^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right) \cdot f^{{\text{R}}} \hat{f}^{{\text{R}}} \\ \end{gathered} $$

(62)

where the SCC ρ_s is defined as:

$$\rho_{s} \left( {x_{i}^{I} ,x_{j}^{I} } \right) = \frac{{\mathop \sum \limits_{s = 1}^{{N_{s} }} \left( {x_{i}^{(s)} - x_{i}^{M} } \right)\left( {x_{j}^{(s)} - x_{j}^{M} } \right)}}{{\sqrt {\mathop \sum \limits_{s = 1}^{{N_{s} }} \left( {x_{i}^{(s)} - x_{i}^{M} } \right)^{2} \mathop \sum \limits_{s = 1}^{{N_{s} }} \left( {x_{j}^{(s)} - x_{j}^{M} } \right)^{2} } }} = \frac{{{\varvec{r}}_{i}^{\left( s \right)\;\text{T}} {\varvec{r}}_{j}^{\left( s \right)} }}{{\sqrt {{\varvec{r}}_{i}^{\left( s \right)\;\text{T}} {\varvec{r}}_{i}^{\left( s \right)} } \sqrt {{\varvec{r}}_{j}^{\left( s \right)\;\text{T}} {\varvec{r}}_{j}^{\left( s \right)} } }} $$

(63)

where x^I_i and x^I_j are two interval variables; and x _i^(s) and x _j^(s) are sampling points of x^I_i and x^I_j. Therefore, the SCC of the uncertain value of the nonlinear function and its approximate linear function can be calculated as:

$$ \rho_{{\text{s}}} \left( {f^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right) = \frac{{{\varvec{r}}_{f}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{{\hat{f}}}^{{\left( {\text{s}} \right)}} }}{{\sqrt {{\varvec{r}}_{f}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{f}^{{\left( {\text{s}} \right)}} } \sqrt {{\varvec{r}}_{{\hat{f}}}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{{\hat{f}}}^{{\left( {\text{s}} \right)}} } }} $$

(64)

where

$$ \begin{gathered} {\varvec{r}}_{f}^{{\left( {\text{s}} \right)}} = \left( {f\left( {{\varvec{x}}^{{\left( {\text{s}} \right)}} } \right) - f^{{\text{M}}} } \right) \hfill \\ {\varvec{r}}_{{\hat{f}}}^{{\left( {\text{s}} \right)}} = \left( {{\varvec{N}}^{{\text{T}}} {\varvec{r}}^{{\left( {\text{s}} \right)}} + {\varvec{M}}^{{\text{T}}} {\varvec{m}}^{{\left( {\text{s}} \right)}} - \hat{f}^{{\text{M}}} } \right)^{{\text{T}}} = {\varvec{r}}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{N}} \hfill \\ \end{gathered} $$

(65)

Thus,

$$ \rho_{{\text{s}}} \left( {f^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right) = \frac{{{\varvec{r}}_{f}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{N}}}}{{\sqrt {{\varvec{r}}_{f}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{f}^{{\left( {\text{s}} \right)}} } \sqrt {{\varvec{N}}^{{\text{T}}} {\varvec{r}}^{{\left( {\text{s}} \right)}} {\varvec{r}}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{N}}} }} = \frac{{{\varvec{r}}_{f}^{{\left( {\text{s}} \right)\;{\text{T}}}} \cdot \left[ {{\varvec{r}}_{1}^{{\left( {\text{s}} \right)}} , \cdots ,{\varvec{r}}_{n}^{{\left( {\text{s}} \right)}} } \right] \cdot {\varvec{N}}}}{{\sqrt {{\varvec{r}}_{f}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{f}^{{\left( {\text{s}} \right)}} } \sqrt {{\varvec{N}}^{{\text{T}}} {\varvec{r}}^{{\left( {\text{s}} \right)}} {\varvec{r}}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{N}}} }} $$

(66)

The SCC between the uncertain value of the nonlinear function and its uncertain inputs can be calculated as:

$$ \begin{gathered} {\varvec{\rho}}_{{\text{s}}} \left( {f^{{\text{I}}} ,{\varvec{x}}^{{\text{I}}} } \right) = \left[ {\rho_{{\text{s}}} \left( {f^{{\text{I}}} ,x_{1}^{{\text{I}}} } \right), \cdots ,\rho_{{\text{s}}} \left( {f^{{\text{I}}} ,x_{n}^{{\text{I}}} } \right)} \right]^{{\text{T}}} \hfill \\ \rho_{{\text{s}}} \left( {f^{{\text{I}}} ,x_{i} } \right) = \frac{{{\varvec{r}}_{f}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{i}^{{\left( {\text{s}} \right)}} }}{{\sqrt {{\varvec{r}}_{f}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{f}^{{\left( {\text{s}} \right)}} } \sqrt {{\varvec{r}}_{i}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{i}^{{\left( {\text{s}} \right)}} } }},i = 1,2, \ldots ,n \hfill \\ \end{gathered} $$

(67)

Then substitute (67) into (66) that

$$ \begin{gathered} \rho_{{\text{s}}} \left( {f^{{\text{I}}} ,\hat{f}^{{\text{I}}} } \right) = \frac{{\left[ {\rho_{{\text{s}}} \left( {f^{{\text{I}}} ,x_{1} } \right) \cdot \sqrt {{\varvec{r}}_{1}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{1}^{{\left( {\text{s}} \right)}} } , \cdots ,\rho_{{\text{s}}} \left( {f^{{\text{I}}} ,x_{n} } \right) \cdot \sqrt {{\varvec{r}}_{n}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{n}^{{\left( {\text{s}} \right)}} } } \right] \cdot {\varvec{N}}}}{{\sqrt {{\varvec{N}}^{{\text{T}}} {\varvec{r}}^{{\left( {\text{s}} \right)}} {\varvec{r}}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{N}}} }} \\ = \frac{{{\varvec{\rho}}_{{\text{s}}} \left( {f^{{\text{I}}} ,{\varvec{x}}} \right)^{{\text{T}}} }}{{\sqrt {{\varvec{N}}^{{\text{T}}} {\varvec{r}}^{{\left( {\text{s}} \right)}} {\varvec{r}}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{N}}} }} \cdot {\text{diag}}\left( {\sqrt {{\varvec{r}}_{i}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{i}^{{\left( {\text{s}} \right)}} } } \right) \cdot {\varvec{N}} \\ \end{gathered} $$

(68)

Through the substituting (69) into (62), e can be expressed as:

$$ e = \left( {f^{{\text{M}}} - \hat{f}^{{\text{M}}} } \right)^{2} + \left( {f^{{\text{R}}} } \right)^{2} + \left( {\hat{f}^{{\text{R}}} } \right)^{2} - 2\frac{{{\varvec{\rho}}_{{\text{s}}} \left( {f^{{\text{I}}} ,{\varvec{x}}^{{\text{I}}} } \right)^{{\text{T}}} }}{{\sqrt {{\varvec{N}}^{{\text{T}}} {\varvec{r}}^{{\left( {\text{s}} \right)}} {\varvec{r}}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{N}}} }}{\text{diag}}\left( {\sqrt {{\varvec{r}}_{i}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{r}}_{i}^{{\left( {\text{s}} \right)}} } } \right){\varvec{N}} \cdot f^{{\text{R}}} \hat{f}^{{\text{R}}} $$

(69)

Moreover,

$$ \frac{{\sqrt {{\varvec{r}}_{i}^{{\left( {\text{s}} \right)\,{\text{T}}}} {\varvec{r}}_{i}^{{\left( {\text{s}} \right)}} } }}{{\sqrt {{\varvec{N}}^{{\text{T}}} {\varvec{r}}^{{\left( {\text{s}} \right)}} {\varvec{r}}^{{\left( {\text{s}} \right)\;{\text{T}}}} {\varvec{N}}} }} = \frac{{x_{i}^{{\text{R}}} }}{{\hat{f}^{{\text{R}}} }} $$

(70)

Finally, e can be calculated as:

$$ e = \left( {f^{{\text{M}}} - {\varvec{Mm}}} \right)^{2} + \left( {f^{{\text{R}}} } \right)^{2} + {\varvec{N}}^{{\text{T}}} {\varvec{C}}_{{{\varvec{x}}^{{\text{I}}} {\varvec{x}}^{{\text{I}}} }} {\varvec{N}} - 2{\varvec{\rho}}_{{\text{s}}} \left( {f^{{\text{I}}} ,{\varvec{x}}^{{\text{I}}} } \right)^{{\text{T}}} {\text{diag}}\left( {f^{{\text{R}}} x_{i}^{{\text{R}}} } \right){\varvec{N}} $$

(71)

Appendix 2

2.1 Approach to taking samples in a MEM domain

The characteristic matrix G to describe a MEM can be decomposed by eigenvalue decomposition as:

$$ \varvec{G = C}^{{ - 1}} \varvec{ = Q}^{{\text{T}}} {\varvec{DQ}} $$

(72)

where D is eigenvalue matrix; Q is eigenvector matrix; and Q^TQ = I. The samples, x^(s), uniformly distributed in a “multidimensional ellipsoid” can be obtained through a transformation (73) from the sampling points, U.^(s), uniformly distributed in a unit hyper-sphere[28]

$$ {\varvec{x}}^{{\left( {\text{s}} \right)}} = {\varvec{x}}^{{\text{M}}} + {\varvec{Q}}^{{\text{T}}} {\varvec{D}}^{{ - \frac{1}{2}}} {\varvec{U}}^{{\left( {\text{s}} \right)}} $$

(73)

where x^M is a vector of the middle points of the interval variables. For a k-dimensional problem, every single set of sampling points U^(s) can be calculated in the spherical coordinates (r, θ₁, θ₂, …, θ_k-1) as:

$$ {\varvec{U}}^{{\left( {\text{s}} \right)}} = \left[ {\begin{array}{*{20}c} {r^{{\left( {\text{s}} \right)}} \cos \theta_{1}^{{\left( {\text{s}} \right)}} } \\ {r^{{\left( {\text{s}} \right)}} \sin \theta_{1}^{{\left( {\text{s}} \right)}} \cos \theta_{2}^{{\left( {\text{s}} \right)}} } \\ {r^{{\left( {\text{s}} \right)}} \sin \theta_{1}^{{\left( {\text{s}} \right)}} \sin \theta_{2}^{{\left( {\text{s}} \right)}} \cos \theta_{3}^{{\left( {\text{s}} \right)}} } \\ \vdots \\ {r^{{\left( {\text{s}} \right)}} \sin \theta_{1}^{{\left( {\text{s}} \right)}} \sin \theta_{2}^{{\left( {\text{s}} \right)}} \cdots \sin \theta_{k - 2}^{{\left( {\text{s}} \right)}} \cos \theta_{k - 1}^{{\left( {\text{s}} \right)}} } \\ {r^{{\left( {\text{s}} \right)}} \sin \theta_{1}^{{\left( {\text{s}} \right)}} \sin \theta_{2}^{{\left( {\text{s}} \right)}} \cdots \sin \theta_{k - 2}^{{\left( {\text{s}} \right)}} \sin \theta_{k - 1}^{{\left( {\text{s}} \right)}} } \\ \end{array} } \right] $$

(74)

where r^(s) are the sampling points lie in a one-dimensional IM with range [0, 1] and [θ₁^(s), θ₂^(s), …, θ_k-1^(s)] are the sampling points lying in a (k-1)-dimensional IM with range [0, 2π] of every dimension.

Finally, the sampling points in the MEM domain can be obtained by taking samples in the above IMs conventionally. Incidentally, through the approach, the scanning method and MC simulation can also be achieved by taking samples orthogonally and randomly, respectively.

Appendix 3

3.1 Model of LV flight dynamics

The detailed expansion of the LV flight dynamics model is expressed as

$$ \begin{gathered} \left[ \begin{gathered} \dot{v}_{x} \hfill \\ \dot{v}_{y} \hfill \\ \dot{v}_{z} \hfill \\ \end{gathered} \right] = \frac{1}{m}{\varvec{G}}_{{\text{B}}} \left[ {\begin{array}{*{20}c} {P - X_{{\text{c}}} } \\ {Y_{{\text{c}}} } \\ {Z_{{\text{c}}} } \\ \end{array} } \right] + \frac{1}{m}{\varvec{G}}_{{\text{V}}} \left[ {\begin{array}{*{20}c} { - X} \\ Y \\ Z \\ \end{array} } \right] \\ + \frac{{g_{r} }}{r}\left[ \begin{gathered} x + R_{ox} \hfill \\ y + R_{oy} \hfill \\ z + R_{oz} \hfill \\ \end{gathered} \right] + \frac{{g_{\omega e} }}{{\omega_{e} }}\left[ \begin{gathered} \omega_{ex} \hfill \\ \omega_{ey} \hfill \\ \omega_{ez} \hfill \\ \end{gathered} \right] - {\varvec{A}}\left[ {\begin{array}{*{20}c} {x + R_{ox} } \\ {y + R_{oy} } \\ {z + R_{oz} } \\ \end{array} } \right] - {\varvec{B}}\left[ {\begin{array}{*{20}c} {v_{x} } \\ {v_{y} } \\ {v_{z} } \\ \end{array} } \right] \\ \left[ \begin{gathered} {\dot{x}} \hfill \\ {\dot{y}} \hfill \\ {\dot{z}} \hfill \\ \end{gathered} \right] = \left[ \begin{gathered} v_{x} \hfill \\ v_{y} \hfill \\ v_{z} \hfill \\ \end{gathered} \right] \\ \end{gathered} $$

(75)

where m is the mass of LV; [P, 0, 0]^T is the component of propulsion P; [X_c, Y_c, Z_c]^T are the components of control force F_c; and [X, Y, Z]^T are the components of aerodynamic force R, which can be calculated as:

$$ \left\{ \begin{gathered} X = C_{x} qS_{{\text{R}}} \hfill \\ Y = C_{y}^{\alpha } qS_{{\text{R}}} \alpha \hfill \\ Z = - C_{y}^{\alpha } qS_{{\text{R}}} \beta \hfill \\ \end{gathered} \right. $$

(76)

where α and β are the angle of attack and sideslip angle, respectively. C_x is the drag coefficient, and C_y^α is the derivative of the lift coefficient with respect to α; S_R is the reference surface area; and q is the dynamic pressure, which is calculated as:

$$ q = \frac{1}{2}\rho v^{2} $$

(77)

where ρ is the atmospheric density and v is the resultant velocity of the LV flight as

$$ v = \sqrt {v_{x}^{2} + v_{y}^{2} + v_{z}^{2} } $$

(78)

Then, G_B and G_V are coordinate transform matrixes as:

$$ \begin{gathered} {\varvec{G}}_{B} = \left[ {\begin{array}{*{20}c} {\cos \varphi cos\psi } & { - \sin \varphi } & {\cos \varphi \sin \psi } \\ {\sin \varphi \cos \psi } & {\cos \varphi } & {\sin \varphi \sin \psi } \\ { - \sin \psi } & 0 & {\cos \psi } \\ \end{array} } \right] \hfill \\ {\varvec{G}}_{{\text{V}}} = \left[ {\begin{array}{*{20}c} {\cos \theta cos\sigma } & { - \sin \theta } & {\cos \theta \sin \sigma } \\ {\sin \theta \cos \sigma } & {\cos \theta } & {\sin \theta \sin \sigma } \\ { - \sin \sigma } & 0 & {\cos \sigma } \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(79)

where φ and ψ are, respectively, the pitch angle and yaw angle, which describe the flight attitude of the LV. Meanwhile, θ and σ are, respectively, flight path angle and flight path azimuth angle, which describe the flight direction of the LV. These angles can be derived as:

$$ \left\{ \begin{gathered} \theta = \arctan \frac{{v_{y} }}{{v_{x} }} \hfill \\ \sigma = - \arcsin \frac{{v_{z} }}{v} \hfill \\ \varphi = \theta + \alpha \hfill \\ \psi = \sigma + \beta \hfill \\ \end{gathered} \right. $$

(80)

Moreover, A and B, in(75), are the matrixes to describe inertial force caused by the rotation of the earth as:

$$ \begin{gathered} {\varvec{A}} = \left[ {\begin{array}{*{20}c} {\omega_{{{\text{e}}x}}^{2} - \omega_{{\text{e}}}^{2} } & {\omega_{{{\text{e}}x}} \omega_{{{\text{e}}y}} } & {\omega_{{{\text{e}}x}} \omega_{{{\text{e}}z}} } \\ {\omega_{{{\text{e}}x}} \omega_{{{\text{e}}y}} } & {\omega_{{{\text{e}}y}}^{2} - \omega_{{\text{e}}}^{2} } & {\omega_{{{\text{e}}y}} \omega_{{{\text{e}}z}} } \\ {\omega_{{{\text{e}}x}} \omega_{{{\text{e}}z}} } & {\omega_{{{\text{e}}y}} \omega_{{{\text{e}}z}} } & {\omega_{{{\text{e}}z}}^{2} - \omega_{{\text{e}}}^{2} } \\ \end{array} } \right] \hfill \\ {\varvec{B}} = \left[ {\begin{array}{*{20}c} 0 & { - 2\omega_{{{\text{e}}z}} } & {2\omega_{{{\text{e}}y}} } \\ {2\omega_{{{\text{e}}z}} } & 0 & { - 2\omega_{{{\text{e}}x}} } \\ { - 2\omega_{{{\text{e}}y}} } & {2\omega_{{{\text{e}}x}} } & 0 \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(81)

where ω_e is the earth rotation rate and [ω_ex, ω_ey, ω_ez]^T are the components of the vector ω_e. Afterward, [R_0x, R_0y, R_0z]^T, in (75), are the components of the vector R₀ to describe the position of the launch point. Next, g_r and g_ωe are the components of gravitational acceleration and can be calculated as:

$$ \left\{ \begin{gathered} g_{r} = - \frac{\mu }{{r^{2} }}\left[ {1 + J\left( {\frac{{a_{{\text{e}}} }}{r}} \right)^{2} \left( {1 - 5\sin^{2} \phi } \right)} \right] \hfill \\ g_{\omega e} = - 2\frac{\mu }{{r^{2} }}J\left( {\frac{{a_{{\text{e}}} }}{r}} \right)^{2} \sin \phi \hfill \\ \end{gathered} \right. $$

(82)

where μ and J are the constant characteristics of gravity. a_e is the length of the semimajor axis of the earth under an ellipsoid model; besides, the semiminor axis is symbolled by b_e. And, r, the geocentric distance of the LV, is calculated as:

$$ r = \sqrt {(x + R_{ox} )^{2} + (y + R_{oy} )^{2} + (z + R_{oz} )^{2} } $$

(83)

Meanwhile, ϕ, the geocentric latitudinal, can be derived from:

$$ \sin \phi = \frac{{(x + R_{ox} )\omega_{ex} + (y + R_{oy} )\omega_{ey} + (z + R_{oz} )\omega_{ez} }}{{r\omega_{e} }} $$

(84)

In addition, the flight height of the LV can also be obtained by r and ϕ as:

$$ h = r - \frac{{a_{e} b_{e} }}{{\sqrt {a_{e}^{2} \sin^{2} \phi + b_{e}^{2} \cos^{2} \phi } }} $$

(85)

Finally, these equations can be solved according to a given flight program angle, and generally, they are provided in the form:

$$ \left\{ \begin{gathered} \varphi^{*} = \varphi_{{{\text{PR}}}} \left( t \right) \hfill \\ \psi^{*} = 0 \hfill \\ \end{gathered} \right. $$

(86)

Accordingly, to achieve the flight program, the corresponding α and β are expressed as

$$ \left\{ \begin{gathered} \alpha = A_{\varphi } [(\varphi_{{{\text{PR}}}} - \omega_{ez} t - \theta )] \hfill \\ \beta = A_{\psi } [(\varphi_{ex} \sin \varphi - \omega_{ey} \cos \varphi )t - \sigma ] \hfill \\ \end{gathered} \right. $$

(87)

where A_φ and A_ψ are both constant coefficients.