International Journal of Aerospace Engineering

Volume 2016 (2016), Article ID 3527460, 14 pages

http://dx.doi.org/10.1155/2016/3527460

## Steady Glide Dynamic Modeling and Trajectory Optimization for High Lift-to-Drag Ratio Reentry Vehicle

School of Astronautics, Beihang University, Beijing 100191, China

Received 13 March 2016; Accepted 30 June 2016

Academic Editor: Christian Circi

Copyright © 2016 Liang Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Steady glide trajectory optimization for high lift-to-drag ratio reentry vehicle is a challenge because of weakly damped trajectory oscillation. This paper aims at providing a steady glide trajectory using numerical optimal method. A new steady glide dynamic modeling is formulated via extending a trajectory-oscillation suppressing scheme into the three-dimensional reentry dynamics with a spherical and rotating Earth. This scheme comprehensively considers all factors acting on the flight path angle and suppresses the trajectory oscillation by regulating the vertical acceleration in negative feedback form and keeping the lateral acceleration invariant. Then, a study on steady glide trajectory optimization is carried out based on this modeling and pseudospectral method. Two examples with and without bank reversal are taken to evaluate the performance and applicability of the new method. A comparison with the traditional method is also provided to demonstrate its superior performance. Finally, the feasibility of the pseudospectral solution is verified by comparing the optimal trajectory with integral trajectory. The results show that this method not only is capable of addressing the case which the traditional method cannot solve but also significantly improves the computational efficiency. More importantly, it provides more stable and safe optimal steady glide trajectory with high precision.

#### 1. Introduction

Entry guidance plays an important role in generating the steering command to guide the vehicle from its initial condition to reach the destination safely and accurately. In general, traditional reentry guidance is divided into two parts. The first part is the generation of a feasible reference trajectory. The second part is the tracking of this reference trajectory [1]. This paper focuses on generating a feasible steady glide reference trajectory, especially for high lift-to-drag ratio reentry vehicle, using numerical optimal method. Previous researches in reentry trajectory optimization are summarized as follows. Scott applied the Legendre Pseudospectral method into the trajectory optimization of reentry vehicles. In Josselyn and Ross’s work [2], covector mapping theorem of Legendre Pseudospectral method was used to verify the first-order optimality condition arising in the path constraints trajectory optimization. Rao and Clarke [3] also studied the problem of reentry trajectory optimization using Legendre Pseudospectral method. The key features of the optimal trajectory and quality of trajectory obtained from the Legendre Pseudospectral method were discussed. Jorris and Cobb and Zhao and Zhou [4, 5] employed the Gauss Pseudospectral method to optimize the 2D and 3D reentry trajectory for Common Aero Vehicle (CAV), in which waypoint and no-fly zone constraints were considered as inner-point constraints in optimal process. Rahimi et al. [6] applied the particle swarm optimization into spacecraft reentry optimization. High-order polynomials were used to approximate the angle of attack and bank angle in problem formulation. The coefficients of both polynomials were considered as input variables in optimal process. It should be noted that, because of not considering the trajectory-oscillation suppressing scheme, the optimal trajectories generated from above methods are naturally oscillatory. In steady glide, the heating rate will not change sharply and the steady state will greatly release the burden of control system. Therefore, steady glide trajectory is the best reference trajectory for the reentry guidance. Actually, quasi-equilibrium-glide condition (QEGC) is a well-known “soft” path constraint that makes the trajectory change monotonously. However, complicated reentry dynamics, especially for high lift-to-drag ratio vehicle, are so sensitive to the “soft” path constraint that it is very difficult for numerical optimization method to converge when QEGC is considered in optimal process. Generally speaking, this constraint is suitable for the trajectory planning in which one or two parameters are searched by secant method so as to generate a feasible trajectory [7–9]. Therefore, steady glide trajectory optimization for the high lift-to-drag ratio vehicle is always a challenge for numerical optimization.

The objective of this paper is to investigate the steady glide dynamic modeling and trajectory optimization for the high lift-to-drag ratio vehicle. A new steady glide dynamic modeling is formulated by extending a trajectory-oscillation suppressing scheme, which is presented by Yu and Chen in [10], into three-dimensional reentry dynamics. Firstly, a special fight path angle which is able to keep the vehicle flying in a steady glide is calculated from the command angle of attack, command bank angle, and current state. Then, the trajectory oscillation is suppressed via regulating the longitudinal acceleration in negative feedback form and keeping the lateral acceleration invariant. It should be noted that the negative feedback signal is the deviation between the special flight path angle and actual flight path angle. Simulation result shows that this scheme performs well in suppressing trajectory oscillation and guides the vehicle into steady glide as soon as possible. Additionally, a study on steady glide trajectory optimization is investigated based on this new modeling. The derivatives of command angle of attack and bank angle are chosen as the control variables. And the performance index is the weighted squares sum of those derivatives. The limits on actual angle of attack and bank angle are considered as the path constraints. In fact, steady glide trajectory optimization is a typical optimal control problem whose solutions change rapidly in certain regions. Therefore, Hp-adaptive Gaussian quadrature collocation method [11], which performs well in dealing with this kind of problem, is chosen to transfer the optimal control problem into a standard nonlinear programming problem and solve it. The notable difference from the Yu’s method is that the scheme is suitable for the three-dimensional reentry dynamics. Another notable difference is that the scheme comprehensively considers all factors (including the derivatives of reference angle of attack and reference bank angle) acting on the flight path angle. That makes it easy to integrate into the motion dynamics by the choice of those derivatives as the control variables. Two classical numeric optimal examples (with and without bank reversal) are taken to evaluate the performance of the steady glide trajectory optimization. In order to demonstrate the superior performance in applicability and computational efficiency, a comparison with the traditional method is also provided. Furthermore, a comparison between optimal trajectory and integral trajectory is carried out to verify the feasibility of the pseudospectral solution. The results show that the new method not only significantly improves the computational efficiency of trajectory optimization since using fewer nodes will achieve a higher accuracy for the steady glide reentry trajectory but also has an extensive applicability in considering more final constraints even with the bank reversal. Most importantly, it is capable of providing more stable and safe optimal steady glide trajectory with high precision, which would be a better choice for tracking guidance.

This paper is organized as follows: entry dynamics including entry trajectory constraints and vehicle model are described in Section 2; three-dimensional reentry trajectory-oscillation suppressing scheme is presented in Section 3; steady glide dynamic modeling and trajectory optimization are presented in detail in Section 4.

#### 2. Dynamics and Vehicle Description

##### 2.1. Entry Dynamics

The 3-DOF point mass dynamics of the reentry vehicle over a spherical, rotating Earth is described as follows [12]:where is the radial distance from the center of the Earth to the vehicle. In the latter, denotes the altitude. The radius of the Earth is 6378145 m. and are the longitude and latitude, respectively. is the Earth relative velocity. is the flight path angle of the Earth relative velocity. is the azimuth angle of the Earth relative velocity. is the mass of the vehicle. is the gravity acceleration, where is Earth’s gravitational constant. denotes the Earth self-rotation rate. The aerodynamic lift and drag are given as follows:where is the atmospheric density, where is the standard atmospheric pressure from the sea level. is the reference area of the vehicle. and are lift and drag coefficients which are dependent on the vehicle configuration.

##### 2.2. Entry Trajectory Constraints

Typical entry trajectory inequality path constraints for hypersonic vehicle includewhere (8) is the heating rate at a stagnation point on the surface of the vehicle. Equation (9) is the aerodynamic acceleration in the body-normal direction. Equation (10) is the dynamic pressure. The heating limit value is , the load factor limit value is , and the dynamic pressure limit value is . These are dependent on the vehicle configuration and mission. Those three constraints are considered to be “hard” constraints that should be enforced strictly.

##### 2.3. Vehicle Description and Model Assumption

Common Aero Vehicle (CAV) is one of the most representative hypersonic entry vehicles with high lift-to-drag ratio. Relying on aerodynamic control, this vehicle is able to glide without power through the atmosphere. There are two types of CAV in the report of Phillips [13], the low-lift CAV and the high-lift CAV. The high-lift CAV, namely, CAV-H, is modeled here to extend the trajectory optimization. The weight of CAV-H is 907 kg, the area reference is 0.4839 m^{2}, and the maximum lift-to-drag ratio is about 3.5. In order to make derivation analysis more intuitive and easier to follow, it is assumed that the lift and drag coefficients are only dependent on the angle of attack. They can be expressed in the form aswhere , , , , and . Moreover, because the angle of attack having the maximum lift-to-drag ratio is about 10 deg., the scope of angle of attack is extended to [5°, 20°]. The scope of bank angle is limited within [−60°, 60°]. The limiting values of heating rate, dynamic pressure, and normal load factor are 400 W/cm^{2}, 60 kpa, and 2 g, respectively.

#### 3. Trajectory-Oscillation Suppressing Scheme

The objective of trajectory-oscillation suppressing scheme is to make the vehicle flying in a steady glide so that the heating rate will not change sharply, which also will significantly release the burden of control system. In this section, a trajectory-oscillation suppressing scheme using the flight path angle feedback is extended. Firstly, the command angle of attack and bank angle are used to calculate the special flight path angle that keeps the two-order derivative of flight path angle zero. Then, the trajectory oscillation is suppressed via regulating the longitudinal acceleration in negative feedback form and keeping the lateral acceleration invariant. The negative feedback signal is the deviation between the special flight path angle and actual flight path angle. Simulation results show that the scheme is capable of guiding the vehicle into a steady condition in which the command angle of attack and bank angle can generate enough vertical lift to sustain the vehicle glide.

##### 3.1. Generic Theory for the Oscillation Suppressing Scheme

Let us pay attention to (5). While the flight path angle is small and varies relatively slow, it is assumed that and . Therefore, (5) without considering the earth rotation is formulated aswhere and . Substitute them into (12), then, (13) is the derivative of (12) with respect to time. Considerwhere

Substitute (1), (4), and the derivative of the atmospheric density into (14). Then, substitute (14) into (13). The two-order derivative of flight path angle can be rewritten aswhere the derivative of lift coefficient is formulated as follows:

For reentry vehicle, the Mach number is much larger than five. Therefore, the aerodynamic coefficients are often assumed to be not dependent on the Mach number. Then, the last term in (16) can be neglected.

Now, consider that the vehicle is in a certain condition of reentry process; if the command angle of attack and bank angle are given, the special flight path angle that keeps the two-order derivative of flight path angle zero can be formulated as follows:where superscript denotes the lift and drag coefficients dominated by the command angle of attack and bank angle. The notable difference from Yu’s work is that the special fight path angle is derived from the three-dimensional reentry dynamics and comprehensively considers all factors including the derivatives of command angle of attack and bank angle. In the later section, the special flight path angle is easily calculated in the trajectory optimization when considering the derivatives of command angle of attack and bank angle as control variables. Then, trajectory oscillation is suppressed via regulating the longitudinal acceleration in negative feedback form and keeping the lateral acceleration invariant. The flight path angle also tends to the special flight path angle as soon as possible. Consider where is the actual lift coefficient and is the command lift coefficient dominated by the command angle of attack. is the actual bank angle. is the command bank angle. is a negative feedback gain; it should be noted that a better will perform well in suppressing the trajectory oscillation. Then, a numerical simulation of entry process is carried out to evaluate the performance of the proposed scheme.

##### 3.2. Performance of the Trajectory-Oscillation Suppressing Scheme

In this section, the proposed scheme is applied into reentry simulation with constant command angle of attack and bank angle. The simulation model, aerodynamic data, and some key parameters are stated in Section 2.3. The command angle of attack is 10 deg. The command bank angle is also 10 deg. If the assumptions mentioned in Section 2.3 are held, the actual angle of attack and bank angle suppressing the trajectory oscillation can be calculated as follows:

The initial states are km, deg., deg., m/s, deg., and deg. All programs run on a personal computer with a 3.3 Ghz processor and MATLAB 2008b. The solver of integral is ODE-45. The simulation stops when the altitude reduces to 30 km. Another worthy note is the negative feedback gain. After some attempts, it is easy to find that the trajectory oscillation will be suppressed perfectly when is −16. Figure 1 shows the three-dimensional view of reentry trajectories. Note that the reference trajectory is the integral trajectory using the constant angle of attack and bank angle. It is obvious that the vehicle glides through the atmosphere in a great performance. Moreover, the glide trajectory is similar to the reference trajectory except that the trajectory oscillation is suppressed.