The rapid developments of artificial intelligence in the last decade are influencing aerospace engineering to a great extent and research in this context is proliferating. In this paper, the trajectory optimization of a three-stage launch vehicle in the powering phase subject to the sun-synchronous orbit is considered. To solve the optimal control problem, the Gauss pseudo-spectral method (GPM) is used to transform the optimization model to a nonlinear programming (NLP) problem and sequential quadratic programming is applied to find the optimal solution. However, the sensitivity of the initial guess may cost the solver significant time to do the Newton iteration or even lead to the local minimum. Aiming at this issue, a good initial guess generated by combination of harmony search and GPM is introduced to help the optimizer to faster and better solve the sun-synchronous orbit (SSO) trajectory optimization. Comparative simulation tests are conducted with the proposed algorithm and popular approaches, the results indicate that with the optimized initial guess, the proposed method could achieve better performance in terms of convergence ability and convergence rate.
The impact of artificial intelligence (AI) technology on our modern world is now more apparent than ever. The space sector is catching up to these developments, as more and more works are published that incorporate concepts related to AI. Applications of interest range from preliminary spacecraft design to mission operations, from guidance and control algorithms over navigation to the trajectory design and optimization of launch vehicles to only name a few. The SSO mission, as one of the most commonly used forms for space missions, is typically a near-circular low-Earth orbit with an altitude less than 1500 km. The list of past, present, and future Earth orbiting sun-synchronous missions is long and impressive. Classic examples of the SSO missions, such as the ESSA [
For SSO mission rockets, it is necessary to generate an optimal standard trajectory in advance to complete the binding of shooting elements. Since most satellites comanifest with a larger payload, leaving their launch schedule and trajectory at the mercy of the major payload, the optimal ascending trajectory leading to determination of the maximum payload mass in the desired orbit has been studied. Trajectory optimization problems are generally nonlinear, with optimal control problems with state and control constraints. Numerical methods for solving optimal control problems are generally classified into two major kinds, namely the indirect method and direct method [
The pseudo-spectral method uses global interpolation polynomials to approximate the state variables and control variables at a series of discrete points, and transforms the differential equation constraints into algebraic constraints by introducing a pseudo-spectral difference matrix similar to a finite difference matrix. Since the collocation point of the pseudo-spectral method is generally the root of an orthogonal polynomial, it is also called the orthogonal collocation method. Along with the technique improvement, Gauss-Legendre collocation [
Given this issue, this paper contributes to propose a fusion algorithm that employs the harmony search (HS) in Gauss pseudo-spectral framework to generate a good initial value for the subsequent gradient optimization solver. In this paper, the optimization model is firstly established with the maximum remaining mass at the shutdown point, taking into account of the constraints of the dynamic pressure, downrange, and angle of attack. And the sequences that best fit the mission requirements are resolved based on the proposed algorithm. And comparative simulation tests are carried out to demonstrate the effectiveness of our study.
The remainder of the paper is as follows: Section 2 presents the aerodynamic model of the launch vehicle and optimization model. Section 3 describes the method and process of solving the SSO trajectory problem. In Section 4, simulation tests are carried out to demonstrated the effectiveness of the proposed algorithm through a series of comparison. Section 5 concludes the paper.
A three-stage rocket that is being considered as the launcher, which is a rocket designed to place satellites in sun-synchronous orbit. The description of the structural, propulsive, aerodynamic characteristics and the trajectory model are given as follows. Mass distribution
To simplify the analysis at hand, the mass distribution of the rocket is depicted in the structural mass and propellant mass of each separate stage. Propulsive thrust
1 | 7000 | 74500 |
2 | 1950 | 18600 |
3 | 800 | 750 |
The propulsion system produces the required thrust to overcome the drag forces and gravity, to lift the launch vehicle from the ground and send it into the designed orbit. And the propulsive characteristic of the launch vehicle can be depicted in terms of thrust magnitude and specific impulse
1 | 7000 | 1188 |
2 | 3336 | 180 |
3 | 3010 | 6.5 |
Aerodynamic model
Aerodynamic forces arise as a result of the relative motion between the launch vehicle and the surrounding air. It can actually be broken down into three components, namely the lift, the drag and the side force. The lift force acts perpendicular to the direction of motion, the drag force is opposite of the motion direction, while the side force is mutually perpendicular to the plane where the lift and the drag lie in. As a valid assumption in the early design, only the drag force is considered in this paper. The expression for the drag force
Trajectory model
The complete description of a launch vehicle’s motion consists of three translational and three rotational degrees of freedom. Since the trajectory of the mass center of the vehicle is more concerned than its attitude motion in the early stage of design. Therefore, in this study, launch vehicle dynamics were modeled using three-degrees-of-freedom equations of motion and standard models of gravity and atmosphere were implemented in the trajectory optimization problem.
The two equations in
In this paper, the maximum payload is set as the optimization purpose to determine the thrust direction subject to the dynamic constraints, path constraints, event constraints and boundaries. The objective function is to regulate the control vector to maximize the mass at the end of the last phase. The initial settlings are the position and velocity coordinates in the beginning of the mission, which are defined as: Event constraints. The terminal constraints are applied to the endpoint of the last phase, which is corresponding to the target sun-synchronous orbit are The dynamic constraints are a set of ordinary differential equations in state-space form, which utilizes the three-degree-of-freedom motion equations modeled in earth centered inertial frame. The path constraints are generally equality or inequality, which are defined for restricting the values’ range in the form of separate or combined functions of the state and/or control variables. In this paper, the thrust direction
As one of the critical load conditions for the launch vehicle, the dynamic pressure
In order to avoid unrealistic trajectories regarding the structural loads, the maximum angle of attack
In this paper, we also take consideration of the scatter degree of the structural jettisons, so that the downrange of the first stage is constrained as:
To solve the SSO trajectory optimization problem, the Gauss pseudo-spectral method is used to transform the optimal control problem to a NLP problem. Although the number of variables and constraints for the original optimal control problem in this paper is not very large, the obtained NLP problem after discretization has large scale and high complexity. In order to quickly find the resolution that satisfies the dynamic constraints, path constraints and terminal constraints, an efficient and accurate optimization algorithm is needed. The commonly used optimization algorithms can be divided into gradient optimization algorithm and intelligent optimization algorithm. Both kinds of the algorithms have their strengths. In this paper, a serial optimization strategy combining harmony search (HS) algorithm with sequential quadratic programming (SQP) algorithm is adopted to do the optimization. The flowchart of the employed optimization resolve is show in
To solve this discrete Bolza problem requires the use of numerical solution methods. In this paper, a direct spectral collocation method--GPM, is employed to perform the trajectory optimization. In which, the states and the control of an optimal control problem are discretized with some optimal variables at a series of Legendre-Gauss points (LG points) in the time span. And the Lagrange interpolation polynomial is constructed to approximate the state variables and the control variables by using the discrete points as nodes. The terminal state is obtained by the integral of the initial state and the right function in the whole process. By the above transformation, the optimal control problem can be transformed into NLP problem with a series of algebraic constraints.
First of all, the time interval
Then, the continuous Bolza problem through the time transformation is expressed as follows:
In which, the state and control variables are approximated using a basis of
Based on the characteristics of Lagrangian interpolation in
To obtain the derivative
Based on the above-mentioned state differential approximation, the dynamic differential equation is configured on the LG point (collocation point):
Similarly, the configuration of the kinetic equation on the collocation point, instead of the discrete point, can obtain a higher precision costate variable. And to ensure the consistency, the control variables are also allocated at the LG point. After all, the cost function of the Gauss pseudo-spectral method can be obtained.
Harmony search is a music-based metaheuristic optimization algorithm. It was inspired by the observation that the aim of music is to search for a perfect state of harmony. It was first proposed by Geem et al. [
The harmony memory is important, as it is to choose the best-fit harmony, it can be initialized through a uniform distribution. And a randomly chosen harmony from the HS is initial solution. Besides, relative parameters, namely the harmony memory size, the number of improvisations
Generating a new harmony is called improvisation. The new harmony vector is generated based on the following rules: memory consideration, pitch adjustment and randomization.
In the memory consideration, it is important to assign a HMCR parameter
The pitch adjustment in music means to change the frequencies; in harmony search, the pitch adjustment is implemented by:
Randomization is to increase the diversity of the solutions. Different from the pitch adjustment, randomization is able to drive the system further to explore various diverse solutions so as to find the global optimality, while the pitch adjustment generally limits to certain local mutation and thus corresponds to a local search.
In terms of the obtain value of the fitness function, if the new harmony is better than minimum harmony in harmony memory, include the new harmony in HM, and exclude the minimum harmony from HM.
If stopping criteria (maximum number of improvisations) are not satisfied, go to Step 2; otherwise, terminate the iteration.
To illustrate the feasibility of presented method in the preceding sections, a number of numerical simulations were carried out. To test the initial guess generator, comparative studies of the harmony search with other intelligent algorithms such as the genetic algorithm (GA) and particle swarm optimization (PSO) are presented. And the three-stage rocket is employed to perform the test, whose vehicle model, optimization model, and constraints establishment are given in Section 2. And the convergence analysis of the proposed method and other algorithms are compared through simulation, and the iteration time is set as 50 for a comprehensive consideration.
As it is seen from
For this study, finding good initial value is to do a good trajectory optimization for the SSO target launch vehicle. After Gauss pseudo-spectral method and the intelligent algorithms, the initial guesses are provided to SQP to solve the optimization problem, the optimized trajectories with different initial generation algorithms are plotted in
Time histories of altitude and speed are given in
Based on the intelligent harmony search initial generator, latitude
In this paper, we solve the trajectory optimization problem for a multistage launch vehicle in consideration of various combinations of dynamic constraints in powering phase. The application of Gauss pseudo-spectral discrete method to three dimensional SSO trajectory optimization problem has been introduced. The optimization problem is modeled and the launch vehicle parameters are present. In order to tackle with the sensitivity of the initial guess for current gradient-based NLP solvers, the intelligent harmony search algorithm is employed. Experimental results show that by applying the initial guess generated using the harmony search, the convergence accuracy for the NLP solver can be improved significantly. Besides, this work could also provide insight into the preferred constraints settings in trajectory optimization for multistage system.