12. 3-D Reentry Guidance with Real-Time Planning of Reference using New Analytical Solutions Based on Spectral Decomposition Method

In the entry guidance, when the analytical solutions are used to plan the L ₁/ D profile and bank reversals, the states of motion relative to the AGI frame are required. Therefore, in this appendix, we show how to get the states of motion relative to the AGI frame from that relative to the GER frame. As stated in Sect.  12.4.1, because the AGI frame is fixed with respect to the inertial space, the velocity vector relative to the AGI frame ( \( {\tilde{\mathbf{V}}} \)) is the vector sum of the velocity relative to the Earth and the velocity due to the Earth’s rotation. Because the entry guidance uses the current states as the initial states in the analytical solutions, the initial value of \( {\tilde{\mathbf{V}}} \) used in the current guidance cycle is where the superscript “NED” represents that the coordinates of a vector are in the NED frame. From Eq. ( 12.114), the initial values of \( \tilde{V} \) and \( \tilde{\gamma } \) in the current guidance cycle are As shown in Fig.  12.2, because the tangent line to the generalized prime meridian at o is parallel to \( {\tilde{\mathbf{z}}}^{\text{GER}} \), which is the unit vector along the \( \tilde{z} \)-axis of the AGI frame, the initial value of \( \tilde{\psi } \) is just equal to the angle between the local horizontal component of \( {\tilde{\mathbf{V}}}_{0} \) (denoted as \( {\tilde{\mathbf{V}}}_{H0}^{\text{GER}} \)) and \( {\tilde{\mathbf{z}}}^{\text{GER}} \), as shown in Eq. ( 12.117). Here, the superscript “GER” represents that the coordinates of a vector are in the GER frame. where \( {\tilde{\mathbf{y}}}^{\text{GER}} \) is the unit vector along the \( \tilde{y} \)-axis of the AGI frame, and \( {\tilde{\mathbf{V}}}_{H0}^{\text{GER}} \) is calculated by From Fig.  12.2, we can find that the generalized longitude, latitude and altitude of P are where \( {\hat{\mathbf{x}}}_{EP}^{\text{GER}} \) is the unit vector pointing from E to P, and \( {\hat{\mathbf{x}}}_{EP}^{\text{GER}} \) is the unit vector along the \( \tilde{x} \)-axis of the AGI frame. Section  12.3 shows the terminal conditions relative to the rotating Earth. Now we need to convert these conditions into that relative to the AGI frame, as the analytical solutions are implemented in this frame. As observed from the AGI frame, the horizontal distance between the end point of the glide trajectory and P is approximately \( S_{\text{TAEM}} \). To facilitate the derivation, we assume that the end point of the glide trajectory is over P because \( S_{\text{TAEM}} \) is much smaller than the total glide range and Earth’s radius. As shown in Fig.  12.41, \( {\mathbf{x}}_{1} \) is the unit vector parallel to the tangent line to the generalized equator at P, \( {\mathbf{V}}_{\text{TAEM}} \) is the desired final velocity vector relative to the rotating Earth, \( {\tilde{\mathbf{V}}}_{f} \) is the desired final velocity vector relative to the AGI frame, and \( {\mathbf{V}}_{\text{e}}^{P} \) is the velocity due to the Earth rotation and points to the local east. We assume that \( {\tilde{\mathbf{V}}}_{f} \) is parallel to \( {\mathbf{x}}_{1} \) as the entry guidance approximately steers the vehicle to P along the generalized equator, and the generalized flight path angle is small. Note that, although this assumption may be inaccurate, the entry guidance will eliminate any resulting error using the trajectory simulations to correct the last bank reversal. The magnitude of \( {\mathbf{V}}_{\text{e}}^{P} \) is

The magnitude of \( {\tilde{\mathbf{V}}}_{f} \) is denoted as \( \tilde{V}_{f} \). Using geometry, we can obtain where \( \theta_{P} \) is the angle of \( {\mathbf{x}}_{1} \) measured counterclockwise from the local east. Then, the desired final specific absolute energy, required by the analytical solutions in the entry guidance, can be estimated by