Steady Glide Dynamics and Guidance of Hypersonic Vehicle

Hypersonic Glide Vehicle (HGV) is generally a near-space vehicle that can fly at a velocity of more than Mach number 5. It is characterized by high velocity, long flight distance and high heating rate. Therefore, through traditional methods, it is difficult to obtain the optimal trajectory satisfying various constraints for HGV.

Regular perturbation method (i.e. direct expansion method) is a technique used for solving the asymptotic power series of solutions to regular perturbation problems. By assuming that the solutions are in the form of power series with respect to the basis function and substituting the power series into the regular perturbation equation, we can obtain the regular equation of the problem. Then collect coefficients of like powers of \( \varepsilon \). Since the system is an identity for \( \varepsilon \), each coefficient of \( \varepsilon \) vanishes independently. Thus by setting these coefficients to zero, we can get a set of equations about the coefficients. After solving the coefficients, we can obtain the asymptotic power series of the solutions [1, 2].

In order to solve the optimization problem of multi-phase trajectory and obtain the optimal trajectory, dynamic equations for hypersonic vehicle are established first. On this basis, further analysis and solution are carried out later.

The entry process of hypersonic gliding vehicle is constrained by path constraints such as heating rate, load factor, dynamic pressure, control variables and terminal constraints such as position, velocity, flight path angle and heading angle.

Gliding problem of hypersonic aircrafts is a high-sensitivity two-point boundary value problem, it is difficult to solve with a single method. The initial value of the common state variable has no actual meaning, which is difficult to predict. Therefore, an optimization algorithm is proposed to solve two-point boundary value problem in this chapter, combining genetic algorithm, local optimization algorithm and neighboring extremum method. The key to solve optimal ballistic problem is to solve the two-point boundary value problem, that is, to find the initial value of a suitable set of common state variables to satisfy the terminal constraints. The initial value of the common state variable is very sensitive to results, and it is not necessarily continuous. Therefore, traditional search algorithms can only find local optimal solution, and are related to the selection of initial value. Solving the problem directly with a gradient algorithm sometimes gives no solution. Therefore, the selection of initial value of general optimization algorithm is a difficult problem. Genetic algorithm is an adaptive global optimization probability search algorithm, which simulates the genetic and evolutionary processes of living things in natural environment. It only needs to estimate the range of optimization variables, and it seeks global approximate optimal solution. However, genetic algorithm is computationally intensive, and often simply converge to a rough global optimal solution. Therefore, based on genetic algorithm, the global optimal solution is further obtained by algorithms with strong local search ability. That is to say, the global approximate optimal solution is obtained by genetic algorithm, and then the optimal solution is further obtained by other algorithms with strong local search ability. Then based on strict constraints, the neighboring extremum method is used to solve two-point boundary value problem, and the optimal solution satisfying all constraints is obtained. Feasibility of this method has been proved in practice.

Direct method has been widely applied in trajectory optimization. Direct trajectory optimization software—SOCS, CAMTOS and GPOPS—adopts direct collocation method, direct multiple shooting algorithm and Gauss Pseudo-spectral Method in solving process respectively. Direct method is less complex than indirect method. It is unnecessary to derive adjoint equations, and the number of differential equations is less than half of that in indirect method. However, the scale of discrete optimization variables is very large. As a result, higher requirements of the calculation efficiency for these optimization algorithms are imposed.

The equilibrium glide [1] proposed by Sänger, is an important reentry flight model. As its vertical forces, including the gravity, centrifugal force and vertical component of the lift, are approximately balanced, the corresponding trajectory has small altitude variation, low peaks of heating rate and dynamic pressure. The equilibrium glide condition is widely used in reentry guidance laws.

With the development of the computer technology, the optimal control methodology usually is the first choice in designing a glide trajectory of a hypersonic vehicle.

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 divides into two parts. The first part is the generation of a feasible reference trajectory. The second part is the tracking of this reference trajectory. This chapter 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.

In the previous chapter, the trajectory optimation for maximum range of hypersonic glider with no constraints has been finished by collocation method and sequential quadratic programming method. Then, a guidance law can be designed for tracking the optimal trajectory. However, in practice, the off-line calculations are very time consuming, and the original optimal trajectory may lose its optimality due to the changes in flight conditions caused by various disturbances or unexpected external forces.

A Common Aero Vehicle (CAV) [1] is a hypersonic gliding vehicle that is boosted to the speed of about Mach 20 by a launch vehicle and reenters the atmosphere without power. The flight of CAV can be roughly divided into two phases: the entry and terminal guidance phases. The entry phase starts shortly after the CAV is separated from the launch vehicle and ends at a specified distance from the target. In the entry phase, the CAV performs lateral maneuvers under the heating rate, dynamic pressure and load factor constraints to manage its energy. In the terminal guidance phase, the CAV attacks the ground target from a near-vertical orientation. In this paper, we study the guidance problem in the entry phase.

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.

Common Aero Vehicle (CAV) [1] is a Hypersonic Glide Vehicle (HGV) featuring a hypersonic-L/D of up to 3, whereas the in-flight L/Ds of the space shuttle and X-33 Reusable Launch Vehicle (RLV) are only about 1.

Model predictive control (MPC) is a form of control in which the current control is obtained by on-line solving a finite horizon open-loop optimal control problem, using the current state of the plant as the initial state in the calculation [1]. It has had a tremendous impact on industrial application development in the last decades. Now, an increasing number of researchers focus their attention on the development of “fast MPC”. Different from the off-line control policy, which is devoted to obtain the control by solving a full nonlinear optimal control problem. MPC usually involves employing the neighboring optimal control problem based on linearized dynamics. In general, a two-point boundary value problem (TPBVP) is formulated to calculate the current control.

Since the successful flights of Apollo capsule and Space Shuttle, researches on entry guidance algorithm have made impressive and remarkable progress.

The terminal guidance problem of a hypersonic gliding vehicle (Phillips in A Common Aero Vehicle (CAV) Model, Description, and Employment Guide, Schafer Corporation for AFRL and AFSPC, Arlington, 2003 [1]) is studied in this paper. In this problem, in order to let the seeker have a good field of view, the guidance law needs to steer the vehicle to destination from a near-vertical orientation.