Omnidirectional Autonomous Reentry Guidance Based on 3-D Analytical Glide Formulae Considering Influence of Earth’s Rotation

Abstract

Common Aero Vehicle (CAV) is a high-L/D hypersonic vehicle gliding in the region of the Earth’s atmosphere with altitude of 20–100 km. CAV is sent into a sub-orbital trajectory by a launch vehicle. After separating from the launch vehicle, CAV reenters the atmosphere with initial Mach number of about 20. As the maximum L/D (L/D_max) is up to 3, CAV can travel more than ten thousand kilometers, while its lateral maneuver range can also be up to thousands of kilometers. The flight of CAV can be roughly divided into entry and nosedive phases. In the entry phase, CAV manages the flight energy by performing proper lateral maneuvers, and eliminates the heading error by conducting several bank reversals.

Reprinted from ISA Transactions, Vol 65, Yu Wenbin, Chen Wanchun, Jiang Zhiguo, Liu Xiaoming, Zhou Hao, Omnidirectional autonomous entry guidance based on 3-D analytical glide formulas, Pages 487–503, Copyright (2016), with permission from Elsevier.

Appendices

Appendix 1: Generalized States of Motion

Before introducing the way of converting the conventional states of motion $\{ \lambda ,\phi ,H,V,\gamma ,\psi \}$ into the generalized states $\{ \tilde{\lambda },\tilde{\phi },\tilde{H},\tilde{V},\tilde{\gamma },\tilde{\psi }\}$, we need introduce two frames of reference: one is called the Geocentric Equatorial Rotating (GER) frame and the other is called the local North-East-Down (NED) frame [16].

The GER frame is a frame with origin at the Earth’s center E. The x_e and y_e axes are in the equatorial plane while the x_e axis intersects with the prime meridian. The z_e axis points towards the north polar. The GER frame rotates together with Earth.

The origin of the NED frame, denoted as o, is at the intersection of the Earth’s surface and the segment connecting the Earth’s center with the mass center of the vehicle, denoted as M. The x axis points to the local north, the y axis points to the local east, and the z axis points to the Earth’s center.

The coordinate transformation matrix from the GER frame to the NED frame can be calculated by

$$ {\mathbf{T}}_{{{\text{GER}}}}^{{\text{NEDP}}} = \left[ {\begin{array}{*{20}l} { - \cos (\lambda )\sin (\phi )} \hfill & { - \sin (\lambda )\sin (\phi )} \hfill & {\cos (\phi )} \hfill \\ { - \sin (\lambda )} \hfill & {\cos (\lambda )} \hfill & 0 \hfill \\ { - \cos (\lambda )\cos (\phi )} \hfill & { - \sin (\lambda )\cos (\phi )} \hfill & { - \sin (\phi )} \hfill \\ \end{array} } \right] $$

(13.87)

As shown in Sect. 13.3.3.2, the entry guidance updates the AGI frame once in each guidance cycle and uses the current conventional states to determine the initial generalized states appearing in the analytical glide formulas. Therefore, we have.

$$ \tilde{\lambda }_{0} = 0,\tilde{\phi }_{0} = 0,\tilde{H}_{0} = H $$

(13.88)

${\tilde{\mathbf{V}}}$ and V are the velocity vectors of the vehicle relative to the AGI frame and the rotating Earth, respectively. Since the AGI frame is an inertial frame, ${\tilde{\mathbf{V}}}$ is equal to the sum of V and the velocity vector due to the Earth’s rotation, as follows.

$$ {\tilde{\mathbf{V}}}_{0}^{{{\text{NED}}}} = \left[ {\begin{array}{*{20}c} {V\cos (\gamma )\cos (\psi )} & {V\cos (\gamma )\sin (\psi ) + \omega_{e} (R_{e} + H)\cos (\phi )} & { - V\sin (\gamma )} \\ \end{array} } \right]^{T} $$

(13.89)

where the superscript “NED” means the coordinates are with respect to the NED frame, and the superscript “T” represents the transform of vector or matrix. According to the definitions of the generalized states of motion shown in Sect. 13.3.3.2, from Eq. (13.86), we have

$$ \tilde{V}_{0} = \sqrt {V^{2} + 2V\omega_{e} (R_{e} + H)\cos (\phi )\cos (\gamma )\sin (\psi ) + \omega_{e}^{2} (R_{e} + H)^{2} \cos^{2} (\phi )} $$

(13.90)

$$ \tilde{\gamma }_{0} = \arcsin \left( {\frac{V\sin (\gamma )}{{\tilde{V}_{0} }}} \right) $$

(13.91)

$$ \tilde{\psi }_{0} = \left\{ {\begin{array}{*{20}l} {\arccos \left( {\frac{{{\tilde{\mathbf{z}}}^{{{\text{GER}}}} \cdot {\tilde{\mathbf{V}}}_{H0}^{{{\text{GER}}}} }}{{||{\tilde{\mathbf{V}}}_{H0}^{{{\text{GER}}}} ||}}} \right);} \hfill & {({\tilde{\mathbf{V}}}_{H0}^{{{\text{GER}}}} \times {\tilde{\mathbf{z}}}^{{{\text{GER}}}} ) \cdot {\tilde{\mathbf{x}}}^{{{\text{GER}}}} \ge 0} \hfill \\ { - \arccos \left( {\frac{{{\tilde{\mathbf{z}}}^{{{\text{GER}}}} \cdot {\mathbf{V}}_{H}^{{{\text{GER}}}} }}{{||{\tilde{\mathbf{V}}}_{H0}^{{{\text{GER}}}} ||}}} \right);} \hfill & {({\tilde{\mathbf{V}}}_{H0}^{{{\text{GER}}}} \times {\tilde{\mathbf{z}}}^{{{\text{GER}}}} ) \cdot {\tilde{\mathbf{x}}}^{{{\text{GER}}}} < 0} \hfill \\ \end{array} } \right. $$

(13.92)

where ${\tilde{\mathbf{V}}}_{H0}^{{{\text{GER}}}}$ is the horizontal component of ${\tilde{\mathbf{V}}}_{0}^{{{\text{GER}}}}$ and calculated by the following equations, and ${\tilde{\mathbf{x}}}^{{{\text{GER}}}}$ and ${\tilde{\mathbf{z}}}^{{{\text{GER}}}}$ are the unit vectors along the positive directions of the $\tilde{x}$- and $\tilde{z}$-axes of the AGI frame, respectively. The superscript “GER” represents the coordinates are with respect to the GER frame.

$$ {\tilde{\mathbf{V}}}_{H0}^{{{\text{GER}}}} = \left( {{\mathbf{T}}_{{{\text{GER}}}}^{{{\text{NED}}}} } \right)^{T} {\tilde{\mathbf{V}}}_{H0}^{{{\text{NED}}}} $$

(13.93)

$$ {\tilde{\mathbf{V}}}_{H0}^{{{\text{NED}}}} = \left[ {\begin{array}{*{20}c} {V\cos (\gamma )\cos (\psi )} \\ {V\cos (\gamma )\sin (\psi ) + \omega_{e} (R_{e} + H)\cos (\phi )} \\ 0 \\ \end{array} } \right] $$

(13.94)

Now we determine the desired final generalized states of motion. As the dot product of two unit vectors is just equal to the cosine of the angle between the two vectors, the generalized longitude, latitude, and altitude of the predicted collision point P, as shown in Fig. 13.4, can be calculated by.

$$ \tilde{\lambda }_{P} = \arccos ({\hat{\mathbf{x}}}_{EP}^{{{\text{GER}}}} \cdot {\tilde{\mathbf{x}}}^{{{\text{GER}}}} ),\tilde{\phi }_{P} = 0,\tilde{H}_{P} = H_{T} $$

(13.95)

where ${\hat{\mathbf{x}}}_{EP}^{{{\text{GER}}}}$ is the unit vector pointing from E to P, and ${\tilde{\mathbf{x}}}^{{{\text{GER}}}}$ is the unit vector along the $\tilde{x}$-axis of the AGI frame.

The desired final generalized speed is denoted as $\tilde{V}_{{\text{TAEM}}}$. Apparently, due to the effect of the Earth’s rotation, there are $\tilde{V}_{{\text{TAEM}}} \ne V_{{\text{TAEM}}}$. As shown in Fig. 13.4, since the vehicle flies to the point P approximately along the generalized equator, we assume that the desired final generalized velocity vector ${\tilde{\mathbf{V}}}_{{\text{TAEM}}}$ is parallel to the generalized equator, where the unit vector along the generalized equator at the point P, denoted as ${\mathbf{x}}_{GEP}$, can be calculated using the following equations.

$$ {\mathbf{x}}_{1}^{{{\text{GER}}}} = {\hat{\mathbf{x}}}_{EP}^{{{\text{GER}}}} - {\tilde{\mathbf{x}}}^{{{\text{GER}}}} $$

(13.96)

$$ {\mathbf{x}}_{2}^{{{\text{GER}}}} = {\mathbf{x}}_{1}^{{{\text{GER}}}} - ({\mathbf{x}}_{1}^{{{\text{GER}}}} \cdot {\hat{\mathbf{x}}}_{EP}^{{{\text{GER}}}} ){\hat{\mathbf{x}}}_{EP}^{{{\text{GER}}}} $$

(13.97)

$$ {\mathbf{x}}_{GEP}^{{{\text{GER}}}} = \frac{{{\mathbf{x}}_{2}^{{{\text{GER}}}} }}{{||{\mathbf{x}}_{2}^{{{\text{GER}}}} ||}} $$

(13.98)

$$ {\mathbf{x}}_{GEP}^{{{\text{NEDP}}}} = {\mathbf{T}}_{{{\text{GER}}}}^{{{\text{NEDP}}}} {\mathbf{x}}_{GEP}^{{{\text{GER}}}} $$

(13.99)

where the superscript “NEDP” represents the NED frame at the point P, and ${\mathbf{T}}_{{{\text{GER}}}}^{{{\text{NEDP}}}}$ is the transform matrix from the GER frame to the NEDP frame (Fig. 13.29).

Fig. 13.29

${\tilde{\mathbf{V}}}_{{\text{TAEM}}}$ is assumed to be tangent to the generalized equator

${\mathbf{V}}_{e}^{P}$ is the velocity vector due to the Earth’s rotation at the point P. ${\mathbf{V}}_{e}^{P}$ points towards the local East and has a magnitude of

$$ V_{e}^{P} = \omega_{e} (R_{e} + H_{{\text{TAEM}}} )\cos (\phi_{P} ) $$

(13.100)

Using the assumption that ${\tilde{\mathbf{V}}}_{{\text{TAEM}}}$ is parallel to ${\mathbf{x}}_{GEP}$, we have

$$ {\tilde{\mathbf{V}}}_{{\text{TAEM}}} = \tilde{V}_{{\text{TAEM}}} {\mathbf{x}}_{GEP} $$

(13.101)

$$ \cos (\theta_{P} ) = {\mathbf{x}}_{GEP}^{{{\text{NEDP}}}} |_{y} $$

(13.102)

Then we have

$$ ({\tilde{\mathbf{V}}}_{{\text{TAEM}}} - {\mathbf{V}}_{e}^{p} )^{2} = ({\mathbf{V}}_{{\text{TAEM}}} )^{2} $$

(13.103)

Expanding the above equation yields

$$ \tilde{V}_{{\text{TAEM}}}^{2} - 2\tilde{V}_{{\text{TAEM}}} V_{e}^{P} \cos (\theta_{P} ) + (V_{e}^{P} )^{2} = V_{{\text{TAEM}}}^{2} $$

(13.104)

Solving the above equation for $\tilde{V}_{{\text{TAEM}}}$ yields

$$ \tilde{V}_{{\text{TAEM}}} = V_{e}^{P} \cos (\theta_{P} ) + \sqrt {(V_{e}^{P} )^{2} (\cos^{2} (\theta_{P} ) - 1) + V_{{\text{TAEM}}}^{2} } $$

(13.105)

Further, the desired final absolute specific energy can be calculated by

$$ \tilde{E}_{{\text{TAEM}}} = \frac{{\tilde{V}_{{\text{TAEM}}}^{2} }}{2} - \frac{\mu }{{R_{e} + H_{{\text{TAEM}}} }} $$

(13.106)

Appendix 2: Generalized Aerodynamic Forces

Fig. 13.30

Relationship between the conventional and generalized aerodynamic forces

In this appendix, we show the relationship between the conventional aerodynamic forces $\{ L_{1} ,\;L_{2} ,\;D\}$ and the generalized aerodynamic forces $\{ \tilde{L}_{1} ,\;\tilde{L}_{2} ,\;\tilde{D}\}$. To reduce the complexity of derivation, we assume that $\gamma = 0$. In Fig. 13.30, ${\mathbf{V}}_{e}$ is the velocity vector due to the Earth’s rotation and $\theta_{V}$ is the angle between ${\tilde{\mathbf{V}}}$ and V, which can be calculated by

$$ \sin (\theta_{V} ) = \frac{{({\mathbf{V}} \times {\tilde{\mathbf{V}}}) \cdot {\tilde{\mathbf{z}}}}}{{||{\mathbf{V}}|| \cdot ||{\tilde{\mathbf{V}}}||}} = \frac{{\omega_{e} (R_{e} + H)\cos (\psi )\cos (\phi )}}{{\tilde{V}_{0} }} $$

(13.107)

From Fig. 13.30, we have

$$ L_{1} = \tilde{L}_{1} $$

(13.108)

$$ L_{2} = \tilde{L}_{2} \cos (\theta_{V} ) - \tilde{D}\sin (\theta_{V} ) $$

(13.109)

$$ D = \tilde{D}\cos (\theta_{V} ) + \tilde{L}_{2} \sin (\theta_{V} ) $$

(13.110)

Then we can convert $\widetilde{{{\text{L}}_{{1}} {\text{/D}}}}$ and $\widetilde{{{\text{L}}_{2} {\text{/D}}}}$ into ${\text{L}}_{{1}} {\text{/D}}$ and ${\text{L}}_{2} {\text{/D}}$, as follows

$$ {\text{L}}_{{1}} {\text{/D}} = \frac{{\widetilde{{{\text{L}}_{{1}} {\text{/D}}}}}}{{\cos (\theta_{V} ) + \widetilde{{{\text{L}}_{2} {\text{/D}}}}\sin (\theta_{V} )}} $$

(13.111)

$$ {\text{L}}_{2} {\text{/D = }}\frac{{\widetilde{{{\text{L}}_{2} {\text{/D}}}}\cos (\theta_{V} ) - \sin (\theta_{V} )}}{{\cos (\theta_{V} ) + \widetilde{{{\text{L}}_{2} {\text{/D}}}}\sin (\theta_{V} )}} $$

(13.112)

Conversely, we can also convert ${\text{L}}_{{1}} {\text{/D}}$ and ${\text{L}}_{2} {\text{/D}}$ into $\widetilde{{{\text{L}}_{{1}} {\text{/D}}}}$ and $\widetilde{{{\text{L}}_{2} {\text{/D}}}}$, as follows

$$ \widetilde{{{\text{L}}_{{1}} {\text{/D}}}} = \frac{{{\text{L}}_{{1}} {\text{/D}}}}{{\cos (\theta_{V} ) - \sin (\theta_{V} ){\text{L}}_{2} {\text{/D}}}} $$

(13.113)

$$ \widetilde{{{\text{L}}_{2} {\text{/D}}}} = \frac{{\cos (\theta_{V} ){\text{L}}_{2} {\text{/D + }}\sin (\theta_{V} )}}{{\cos (\theta_{V} ) - \sin (\theta_{V} ){\text{L}}_{2} {\text{/D}}}} $$

(13.114)

Now we investigate the relationship between ${\text{L/D}}$ and $\widetilde{{\text{L/D}}}$. From Eqs. (13.107–13.108), we have

$$ {\text{L/D}} = \frac{{\sqrt {\left( {\widetilde{{{\text{L}}_{{1}} {\text{/D}}}}} \right)^{2} + \left( {\widetilde{{{\text{L}}_{2} {\text{/D}}}}\cos (\theta_{V} ) - \sin (\theta_{V} )} \right)^{2} } }}{{\cos (\theta_{V} ) + \widetilde{{{\text{L}}_{2} {\text{/D}}}}\sin (\theta_{V} )}} $$

(13.115)

From Eqs. (13.109–13.110), we have

$$ \widetilde{{\text{L/D}}} = \frac{{\sqrt {\left( {{\text{L}}_{{1}} {\text{/D}}} \right)^{2} + \left( {{\text{L}}_{2} {\text{/D}}\cos (\theta_{V} ) + \sin (\theta_{V} )} \right)^{2} } }}{{\cos (\theta_{V} ) - {\text{L}}_{2} {\text{/D}}\sin (\theta_{V} )}} $$

(13.116)

Because $\theta_{V}$ is near zero, we approximate the numerator of Eq. (13.112) by the first-order Taylor series at $\theta_{V} = 0$, as follows

$$ \widetilde{{\text{L/D}}} \approx \frac{{{\text{L/D}} + \sin (\sigma )\theta_{V} }}{{\cos (\theta_{V} ) - {\text{L/D}}\sin (\sigma )\sin (\theta_{V} )}} $$

(13.117)

Thus, we get the approximation relation between ${\text{L/D}}$ and $\widetilde{{\text{L/D}}}$.