To solve the multiobjective optimization problem on hypersonic glider vehicle trajectory design subjected to complex constraints, this paper proposes a multiobjective trajectory optimization method that combines the boundary intersection method and pseudospectral method. The multiobjective trajectory optimization problem (MTOP) is established based on the analysis of the feature of hypersonic glider vehicle trajectory. The MTOP is translated into a set of general optimization subproblems by using the boundary intersection method and pseudospectral method. The subproblems are solved by nonlinear programming algorithm. In this method, the solution that has been solved is employed as the initial guess for the next subproblem so that the time consumption of the entire multiobjective trajectory optimization problem shortens. The maximal range and minimal peak heat problem is solved by the proposed method. The numerical results demonstrate that the proposed method can obtain the Pareto front of the optimal trajectory, which can provide the reference for the trajectory design of hypersonic glider vehicle.

Boost-glide vehicle has become a research hotspot because of its unique advantages, such as increasing range and improving ability of penetration. Trajectory optimization technology is one of the key aircraft design technologies. Reentry trajectory optimization usually needs to consider multiple performance indicators. Some of these objectives are conflicting, such as velocity and heating value [

In recent years, there have been many studies on trajectory planning, such as pseudospectral method [

In this paper, the pseudospectral method is combined with boundary intersection method so that the multiobjective trajectory optimization problem can be converted to a set of single objective optimization subproblems, and then the subproblems are solved with pseudospectral method individually. In order to shorten the computing time, the solution of the subproblem is used as the initial value of the next subproblem. In this paper, the solving method of HGV multiobjective trajectory optimization problem has the following advantages:

Compared with the weight-added sum method, this method can avoid repeating design when engineering practice experience is lacking.

Compared with multiobjective trajectory optimization method based on evolutionary algorithm, this method can get sufficient accurate Pareto optimal solution with relatively less calculating quantity.

Content is arranged as follows: first, mathematical model of HGV trajectory optimization, typical objectives, and constraints are analyzed; then, multiobjective trajectory optimization algorithm based on pseudospectral method and the NBI method is proposed; finally, the multiobjective trajectory optimization problem on HGV maximal range-minimal peak heat flux is solved with the method in this paper and the Pareto front result is analyzed.

HGV is a lifting body with flat shape and it adopts bank-to-turn (BTT) swerve technique generally; therefore, flight sideslip angle can be set to zero. In this paper, the Earth rotation is ignored for simplification and three-degree-of-freedom motion model is established as (

Three-degree-of-freedom motion of HGV.

In the equations above,

From the perspective of the optimal control, the state variables of vehicle reentry dynamics system are location parameters and speed parameters; control variables are the angle of attack

Constraints of flight process, terminal parameters, and control variable must be considered in trajectory design. Flight process constraints are the constraints where trajectory parameters must be satisfied in the flight process. HGV reentry is a complex flight process, so heat flow, dynamic pressure, overload, and maneuvering capability of the aircraft must be considered. These elements should not exceed the affordability of aircraft. Terminal parameter constraints are the conditions that should be satisfied in the end of the aircraft trajectory. Control constraints refer to the limit of attack angle and bank angle.

Reentry trajectory design must consider the influence of aerodynamic thermal protection system (TPS). Material of TPS decides the limit of aircraft surface temperature and heat flux. To ensure the safety of aircraft to return, generally the stagnation heat flow restrictions along the trajectory of the flight are set as

The impact of aerodynamic load on the internal structure should be considered with the aircraft reentry. The horizontal aerodynamic load constraint of HGV reentry is not serious, so reentry trajectory design mainly considers vertical aerodynamic load

Considering dynamic pressure effect on the spacecraft attitude control system and stability, dynamic pressure is an important parameter. Aerodynamic torque and pneumatic torque are directly related to dynamic pressure. In addition, the dynamic pressure affects the stability and execution efficiency of the aircraft aerodynamic control surfaces. Postural stability (especially the lateral stability) requires that the dynamic pressure

Reentry vehicles should have sufficient capacity to meet the requirements of motorized guidance and control systems. When the trajectory height is too large, due to the thin air, the available lift is insufficient to balance gravity; once the aircraft is disturbed, vehicles cannot track the aircraft predetermined trajectory. Thus, it will affect the performance of guidance and even make the mission unable to be completed. To ensure controllable trajectory, the aircraft should keep the vertical upward force available to balance other forces:

During the flight, due to hardware limitations, the control variable such as attack angle and bank angle should not exceed the constraint value:

Terminal constraint is determined by the mission. For example, when the requirement is to hit the target, the trajectory terminal location parameters should be consistent with the target. And when the requirement is security reentry, the trajectory terminal velocity will be requested. According to practical requirements, the reentry vehicle usually requires landing speed and landing trajectory inclination angle:

According to the design requirements, trajectory optimization chooses different performance indicators. When analyzing the aircraft performance, the maximum range, maximum speed, and other parameters are usually taken as performance indicators. For example, the maximum terminal trajectory range as optimization indicator is taken as follows:

And when the end position of trajectory is determined, optimization indicators such as the shortest path, the shortest time cost, or the minimum total amount of heat may need to be considered. For example, the minimum total amount of heat as optimization indicator is chosen as follows:

The difference between multiobjective optimization problem and general trajectory optimization problem is the number of optimization objectives. These constraints are basically the same. With respect to single objective trajectory optimization, multiobjective trajectory optimization problem can be stated as follows [

The control function

For state equation constraints,

For boundary condition constraints,

For path constraints,

Equations (

To overcome the defect of weight-added sum method, Das and Dennis [

Calculation process of NBI method is as follows: first, find the single objective advantage; Then, create hyperplanes in the objective space through single objective optimal point (defined as Convex Hull of Individual Minima, CHIM) and build a group normal lines of this superplane. At last, solve lower left border intersection points of normal lines and reachable objectives set by single objective optimization methods to approximate Pareto front. Single objective optimization problem is converted by NBI method as

In (

Sometimes, for convenience, a set of lines from the ideal point can be used. In this case, the polymerization objectives are converted to find out the minimum distance spots from line. The spots are on a straight line to ideal spots in reachable space as follows:

The advantage of the NBI method is that it is not sensitive to the shape of Pareto front, and the resulting Pareto optimum is evenly distributed. But it needs to increase the number of equality constraints (consistent with the objective number), and more than one dominating solution may be found; then, the solution set should be filtered. Meanwhile, for some optimization problems with more than two goals, the solutions obtained by this method may not cover the entire Pareto frontier [

Schematic diagram of NBI method.

Translation

Rotation

In recent years, the pseudospectral method has become a common tool for solving complex optimal control problems because of its high accuracy solution. The basic idea is to use the roots of orthogonal polynomial as discrete points and then separate the state variables and control variables of continuous optimal control problem and then use Lagrange interpolation of full region to approximate the state and control variables. Thus, the optimal control problem is converted to the nonlinear programming problem to be solved [

Currently,

In this paper, hypersonic glide vehicle multiobjective trajectory optimization problem is converted to general multiobjective parameter optimization problem using

In the objective function, Mayer and Lagrange type indexes are outputted, respectively, corresponding to each objective.

The sparse matrix template is modified.

The normal boundary intersection method is added.

The subproblem solutions obtained are taken as the initial value of next subproblem solving process.

Normal boundary intersection method obtains different optimization subproblems by changing the direction of the vector. When these problems are solved, Pareto optimum solutions are obtained. In general, the subproblems of the adjacent vector direction are relatively close to the objective function, so it is considered that the optimal parameters are also close relatively. Therefore, optimal solution of adjacent subproblem is used as initial value of next subproblem in order to improve efficiency of solving the optimization problem.

The process of hypersonic glide vehicle multiobjective trajectory optimization based on normal boundary intersection method and pseudospectral method is shown as flow chart in Figure

Multiobjective optimization problem is constructed. Optimization objectives are chosen according to the task. Differential equations of reentry dynamics are established. The constraints of state variables and control variables needed to be satisfied are considered. Then, the mathematical model of multiobjective optimization problem is constructed. It should be noted that suboptimization objectives selected should be conflicting. For example, in trajectory optimization problem, minimize heat flux and maximize range or the minimum total absorption heat and maximum range, and so forth.

Multiobjective optimization problem is dispersed. Multiobjective mathematical model is dispersed by normal boundary intersection method. Weight vector

Single objective optimization subproblem is solved with pseudospectral method. The single objective optimization subproblems are solved with pseudospectral method and sequential quadratic programming method. Then, the subproblem solutions obtained are taken as the initial value of next subproblem solving process so that the computing time is shortened. Multiobjective Pareto front and corresponding solutions of optimal controls and states are obtained.

Process of multiobjective trajectory optimization.

No rotating spherical Earth model or exponent atmospheric model is adopted. Specific parameters are as follows.

Lift coefficient ^{3}, reference height ^{2}, aircraft quality

The initial states (reentry points parameters) are taken as follows: geocentric distance

Terminal constraint does not limit flight time but considers the height, speed, and trajectory angle requirements. Set parameters

Process constraints such as heat constraint, overload constraint, and dynamic pressure constraint are taken into account. Attack angle constraint and bank angle constraint are set as

Maximum terminal range and minimum peak heat flux are taken as optimization objectives.

The first cost function is terminal range:

In the equation above,

Because

The second cost function is aerodynamic heating on the vehicle wing leading edge [

In the equations above,

For optimization objectives,

Pareto front of multiobjective trajectory problem is solved as in Figure ^{−3}. Total computing time with iteration method in this paper is 391.67 minutes. Total computing time of all the subproblems solved severally by

Important node in Pareto front of multiobjective optimization solution.

Pareto front | Minimum peak heat solution | Maximum terminal range solution |
---|---|---|

Range (km) | 5597 | 6102 |

^{2}) |
67.92 | 128.3 |

Pareto front of multiobjective trajectory optimization solution.

From the results, Pareto front points are evenly distributed. In maximum terminal range solution, the value of terminal range is 6102 km, and its peak heat flux is 128.3 W/cm^{2}. In minimum peak heat solution, the value of terminal range is 5597 km, and its peak heat flux is 67.92 W/cm^{2}. The remaining objective points are between these two limits; this feature means that these two objectives appear to be in strong conflict. In addition, as can be seen from Pareto front in Figure

Solution curves of minimum peak heat flow and maximum terminal range are shown in Figure

Trajectory parameters of the minimum peak heat trajectory and maximum terminal range trajectory in Pareto optimal solution.

Altitude-time curve

Speed-time curve

Range-time curve

Peak heat-time curve

Angle of attack-time curve

Angle of bank-time curve

The simulation results show that the method based on boundary intersection method and pseudospectral method in this paper can effectively solve the multiobjective glide vehicle trajectory optimization problem, and this method can provide a useful reference to trajectory design.

In hypersonic glide vehicle trajectory design, multiple conflicting optimization objectives usually need to be considered using multiobjective optimization method. To solve this problem, this paper presents multiobjective hypersonic glide vehicle trajectory optimization method based on normal boundary intersection method and pseudospectral method. Multiobjective method optimization problem is converted to multiple single objective optimization problems with normal boundary intersection method, and then the optimal control problems are converted to parameter optimization problems with pseudospectral method, so that it can be solved with nonlinear programming algorithm. Hypersonic glide vehicle trajectory multiobjective optimization problem about maximum range and minimum peak heat is numerically simulated, and Pareto front solution is evenly distributed relatively. The varying range of trajectory conflicting objectives can be obtained from Pareto front, and it provides series of candidate solutions. This paper has reference value for hypersonic glide vehicle trajectory design.

The authors declare that they have no competing interests.