Hybrid Method for Constrained and Unconstrained Trajectory Optimization of Space Transportation

ABSTRACT In this research, a new method named δ to solve non-linear constrained and un constrained optimal control problems for trajectory optimization was proposed. The main objective of this method was defined as solving optimal control problems by the combination of the orthogonal functions, the heuristic optimization techniques, and the principles of optimal control theory. Three orthogonal functions Fourier, Chebyshev, and Legendre were considered to approximate the control variables. Also, GA-PSO and imperialist competition algorithms were considered as heuristic optimization techniques. Moreover, the motivation of the mentioned method belonged to a novel combination of zero Hamiltonian in the optimal control theory, optimality conditions, and newly proposed criteria. Furthermore, lunar landing, asteroid rendezvous, and low-thrust orbital transfer with respect to minimum-time and minimum-fuel criteria were investigated to show the ability of the proposed method in regard to constrained and un constrained optimal control problems. Results demonstrated that the δ method has high accuracy in the optimal control theory for non-linear problems. Hence, the δ method allows space trajectory and mission designers to solve optimal control problems with a simple and precise method for future works and studies.


INTRODUCTION
Solving optimal control problems for achieving optimal trajectories was the main focus of many studies up to now. Three steps of solving trajectory optimization problems were classified as mathematical modeling, defining the main criterion, and proposing the method of solving. In this way, one of the most important parts of the mentioned steps was the approach of achieving the optimal solutions regarding the introduction of a precise method. Hence, various methods to solve the optimal control problems were presented with respect to direct, indirect, and the combination of direct and indirect methods in the optimal control theory (Ben-Asher 2010; Chen and Tang 2018;Han et al. 2019;Naidu 2003;Shirazi et al. 2018). In this study, a combination of direct and indirect optimal control methods was demonstrated. The main contribution of this work belonged to introducing the precise approximation of optimal control by heuristic optimization techniques and orthogonal functions. Regarding the approximation of the optimal control, new augmented criteria based on the optimal control theory were introduced to minimize by optimization techniques. The mentioned augmented criteria was named by the different indices of δ that belonged to the optimal control was proposed based on the magnitude of the velocity without any changes in directions. Further , a GA was applied to achieve the minimum changes in the magnitude of velocity. Furthermore, a new method by global optimization technique as GA was proposed to avoid an initial population with several initial guesses (Mohammadi and Naghash 2019). It should be noted, different space missions such as constraint and non-constrained Lunar landing, rendezvous, and low-thrust orbital transfer for different criteria were studied by the proposed method without any initial guesses and optimal control and state equations discretization.
Open-loop and closed-loop methods were two different branches of the optimal control theory. Closed-loop optimal controls were robust against disturbances, however, the open-loop optimal controls were sensitive to the disturbances. Also, the model predictive control could enhance the ability of the system against unknown disturbances. The mentioned method in this research belonged to the open-loop branch of optimal control theory. It should be noted that, usually, to introduce the closed-loop optimal control, first, the open-loop optimal control theory was studied (Naidu 2003). In this way, the application of artificial intelligence and robust methods were studied in the references. In Chai et al. (2021) the problem of trajectory optimization for autonomous ground vehicles with the consideration of irregularly placed on-road obstacles and multiple maneuver phases were studied by introducing a series of event sequences. Also, fast trajectory planning by reinforcement learning in an unknown environment was studied in Chai et al. (2022a). The application of neural networks to predict the optimal control for the autonomous motion of ground vehicles regarding parking maneuvers was studied by Chai et al. (2022b) in. In this way, Chai et al. (2022c) see the other reference for deep learning trajectory planning and control for autonomous ground vehicles . In Chai et al. (2022d), optimal time-varying with system constraints and considering disturbances for attitude tracking of a spacecraft was demonstrated. The optimal control in Chai et al. (2022d) was established by introducing a non-linear robust model predictive control and a dual-loop cascade tracking control framework. Also, the other reference was suggested for model predictive control for a reentry vehicle in Chai et al. (2022e).
In this work, a new method will be introduced as a δ method with combination characters of direct and indirect methods.
First, this article that follows introduces δ method for optimal control problems regarding optimal control theory, orthogonal functions, and GA-PSO, and Imperialist Competition Algorithm (ICA) optimization techniques. Next, two non-linear problems in space trajectory optimization, the Lunar landing and the asteroid rendezvous are studied. Finally, a low-thrust orbital transfer is investigated regarding minimum-time and minimum-fuel criteria. Minimum-fuel criterion is considered a constrained problem with constant thrust magnitude and multiple on-off optimal controls. Emphasizing on the minimum-time and minimum-fuel criteria and different space missions show the ability of this method for different types of optimal control problems.

BRIEF INTRODUCTION TO OPTIMAL CONTROL THEORY
Optimal Control Problems lead to solving a system of first-order differential equations (Eq. 1). The main goal of the mentioned equations is deriving an optimal control u(t) to reduce a cost function.
(1) where x(t) is the state vector. Therefore, the main cost function J m is demonstrated by Eq. 2: (2) In the equation above , ϕ(x(t f )) is the final condition of the main cost function. With respect to the optimal control theory, Hamiltonian of the system in Eq. 1 is introduced by Eq. 3.
(3) λ(t) is considered as the co-state vector and achieved as (Eq. 4): (4) Also, the optimality condition regarding the Hamiltonian of the system is introduced in Eq. 5 (Naidu 2003).
Optimal control problems with non-differentiating control functions (constrained control) cover a wide range of optimal trajectory designing. Also, some of the constrained optimal control problems are called "bang-bang" because of the on-off control functions. Hence, optimal control u(t) is obtained with respect to satisfying the following inequality (Eq. 6). H(t) is the Hamiltonian function and x(t), λ(t), u(t) are states, co-states, and optimal controls respectively. Also, '*' is referred to as the optimal parameter. Since u(t) is discrete in the bang-bang problems, the optimality condition ∂H/(∂u(t))=0 cannot be used. So, u(t) is considered by Eq. 7.
The sign of switch function ζ s (t) has an important role in the bang-bang problems.

METHOD
In the δ method proposed in this study, a combination of direct and indirect optimal control methods was considered. In the mentioned method, the precise approximation of optimal control by heuristic optimization techniques and orthogonal functions is introduced. Regarding the approximation of the optimal control, new augmented criteria based on the optimal control theory were introduced to minimize by the optimization techniques. The mentioned augmented criteria were named by the different indices of δ that belonged to the optimal control theory. The main criterion (such as minimum-time or minimum-fuel) was added by new criteria introduced by δ based on the necessary and sufficient conditions in the optimal control theory to improve the precision and simple achievement of the results.
In the proposed δ method, switch function is considered through orthogonal functions by Eq. 8: where ζ i (t) and π i are the orthogonal functions and multipliers of the estimated switch function respectively. Switch function f s can be achieved regarding optimal determining ζ i (t) and v, where i is the index of the components. Heuristic optimizer determines the optimal multipliers π i for the best f s (t). Also, based on the positive or negative value of f s (t), control variable u*(t) is achieved. When f s (t) (switch function) and u*(t) (optimal control) are approximated by the orthogonal functions, states and co-states dx/dt(t)=(∂H(t))/(∂λ(t)),(λ)/dt(t)=-(∂H(t))/(∂x(t)) could be integrated with respect to initial conditions. An augmented cost function (see Eq. 9) is defined based on the terminal conditions and states at the end of the trajectory.
where j is the index of terminal conditions and k is the weight coefficients. Hence, the optimizer minimize the J a and the finalconditions are satisfied.

5
The main cost function J m (such as minimum-time or minimum-fuel), augmented cost function J a (Eq. 9) that belongs to the satisfaction of state equations at the end) will be obtained. Next, criteria δ from δ method, will be explained to construct the total cost function J total .
Main process of δ method to solve optimal control problems is considered via powerful heuristic optimizers. Regarding Fig. 1, optimizer chooses a suitable switch function through the best π i which has reduced J total . This process is continued through iterative loop of optimizer to reduceJ total .

Generating Switch Function
Investigating Total Cost Function

Optimizer
Calculating States and Co-states  Components of criteria δ are explained step by step in the following (Eq. 10).

First, δ S
H is defined in be low equations. From Eq. 6: So, δ S H is defined as (Eq. 11): In the above formula, (i -1) indicates the previous iteration of the optimization process. Optimizer selects the better multipliers of orthogonal functions in order to achieve the best control regarding satisfying inequality (Eq. 10). So, the Hamiltonian is reduced and will be smaller in every iteration when δ S H → 0 (see Eqs. 10 and 11).
Second, δ S H regarding the switch function is defined. In this way, Hamiltonian can be stated as follow (Eq. 12): where η(t) is the other part of Hamiltonian that has u(t) (Eq. 13).
So, f s l (t) is considered as (Eq 14): The best switch function is achieved regarding minimizing the difference between f s (t) and f s l (t). Hence, δ S l is defined in Eq.
15 to be minimized to zero δ S l → 0.
So, δ S l and δ S H are taken into account as sub-criteria in the process of optimization. Therefore, optimizer not only minimize J total as well asJ a but also, it minimize δ S H and δ S l to zero.
Third, δ u regarding changing in the Hamilton ian with respect to un constrained control variable u(t) is defined. Hamiltonian is differentiable when a control variable acts continuously as u(t). Therefore, (∂H(t))/(∂u(t)) should converge to zero. So, δ u is defined by Eq. 16: As δ u is minimized to zero, the new criterion is also taken into account. Finally, the total criterion (or cost function) is expressed by Eq. 17 or 18: Fourth, Free-Final-Time problems: Solving the optimal control problem is more difficult when the final-time is free and the control variable is constrained. To make this problem easier, final-time is supposed t d . In this way, t d is considered as the approximation of final time t f by the path planning designer. So t d is determined t d = Ct f and the optimizer will find the best coefficient of c. In a case of freefinal-time of optimal control problems, the Eq. 19 can be used for the problems from transversality in the optimal control theory. Φ(t) indicates terminal section of the main criterion. If (∂Φ(t))/∂t = 0, the Hamiltonian will be equal to zero in the final-time H(t) = 0. The Hamiltonian variations with respect to the time will equal to zero in the optimal control problems, when the state equations are not explicitly functions of time (see Naidu 2003). Therefore, Eqs. 20 and 21 are resulted. (20) Since H(t) = cte and H(t) = 0, then it can be concluded that the Hamiltonian equals to zero over the time of optimal control problem so, H(t) = 0. In the problems of free-final-time, the Eq. 22 criterion is added to the total cost function.
As δ H is minimized to zero, some other principles of optimal control theory are considered in solving free-final-time problems regarding the δ method.
So, the total cost function will be considered by Eq. 25: The mentioned issues represent a novel method for optimal control problems with un constrained and constrained control variables.

SOFT LANDING
This simple benchmark investigates the soft landing of a Moon lander. Soft landing of a vehicle with discontinuous optimal control is investigated with the δ method and the mentioned problem is formulated as a constrained optimal control problem.
At the end of the mission to the Moon, the lander has a soft landing through the reverse thrust. In Naidu (2003) and Udupa et al. (2018) this problem is formulated with respect to minimum-time criterion with on-off optimal control. With reference to the Cartesian coordinate, dynamic equations of the Moon lander are given by Eqs. 26-28: with initial conditions (Eq. 29): where v(t) is the vertical velocity and g Moon is the gravitational acceleration of the moon. κ is a constant that refers to the exhaust coefficient of the thruster and m is the mass of the lander. Thrust, vector T is upward to the Moon surface and its magnitude is |T → |=κ(dm/dt) and dm/dt is the mass reduction rate. Also, the on-off control is represented by u. For a soft landing, the Eq. 30 terminal conditions are satisfied.
The main part of solving this optimal control problem is determining discontinuous optimal control regarding three orthogonal functions: Fourier, Chebyshev, and Legendre. Figures 2-5 represent the Fourier, Chebyshev and Legendre solutions for the optimal control, velocity, height, and mass. In this way, the optimizer chooses the best coefficients for the mentioned three orthogonal functions to construct control switch functions.       Naidu (2003). Moreover, Fig. 5 represents the mass reduction and it shows the reduction in mass when the thruster is operating based on the switching time. Figure 6 demonstrates on-off optimal control regarding the switch function achieved from δ method.   The novel parts w ere introduced in this method as criteria a re δ H , δ s H and δ s I . Based on the δ method, these parameters are minimized to zero to satisfy the principles of δ method and optimal control theory. Hence, converging to zero rapidly. Finally, in Fig. 10 the reduction of the total cost function is illustrated by three orthogonal functions. It is concluded that the Fourier series has rapid convergence and is considered the best orthogonal function for the δ method. Moreover, to investigate the robustness of the algorithm, ICA is considered for the next optimization technique. So, the next figures are investigating two optimizers GA-PSO and ICA to show the accuracy of the δ method and converging speed. Also, the parameters of the optimizers are illustrated in Table 1. For more references in ICA optimization technique see Abdollahi et al. (2013), Ardalan et al. (2015), Maheri and Talezadeh (2018) and Rabiee et al. (2018). It should be noted, parameters of optimizers in Table 1 are considered by try and errors regarding converging the total cost (see Fig. 10).         Table 2

Fixed Final-Time Problem: Asteroid Rendezvous
The A steroid Rendezvous problem is about finding the best solution to an interplanetary trajectory problem. The mentioned problem is considered as designing a planar optimal trajectory that starts from an asteroid and intercepts with another asteroid. The mathematical formulation is demonstrated in the s olar-centric polar coordinate. The control variable is represented by β(t). The control variable is the angle between the bang-bang thruster and the local horizon. The system of equations for this orbital transfer is represented by five state equations: r, θ, u, v and m. States r and θ are belonged to the position in polar coordinate and u, v are radial and tangential velocities (see Fig. 19). Also, the mass for this transfer is m and it is reduced by m˙ = -T/(I sp g 0 ).
The magnitude of the thruster is in the form of bang-bang and considered T min and T max . The specific impulse is I sp = 3000(s) and the initial mass of the spacecraft is m 0 = 1500(kg)m 0 . The main cost function of the problem is minimum-fuel with fixed-               All δ criteria act dynamically to enhance the robustness of the δ method and they are converging rapidly and precisely. In this problem, there are two optimal controls, β(t) is considered as continuous control and the other as the bang-bang thruster (bangbang optimal controller for fuel consumption) namely discontinuous optimal control. Based on the δ method, δ u for continuous optimal controls (regarding satisfying necessary condition of optimality ∂H(t)/(∂u=0)) are considered and should be minimized to zero. Figures 28-30 show the components of the criteria vector δ' = [δ S I δ u δ S H ] that are converging to zero rapidly. Therefore, the novel-introduced principles in this method are satisfied precisely. Also, Fig. 31 shows the rapid converging of the total cost function in the 6 th iteration and it is one of the advantages of this method. Results as the switch function (for constrained control) and thrust direction (unconstrained control) are achieved in Eqs. 40 and 41. It should be noted, the mentioned asteroid rendezvous has non-linear equations with multi-input controls. (40)

Low thrust Orbital Transfer
The proposed method as δ has been able to overcome the complex non-linear free and fix-final-time problems. In the following, the problem of the optimal orbital transfer, which is a completely complex problem with nonlinear equations, is considered regarding orbital parameters. The mentioned orbit transfer is considered a continuous orbital transfer from LEO to GEO. The thrust force is expressed in terms of the Eqs. 42-44 three-dimensionally based on two control angles ψ(t), φ(t) and an on-off thruster. (42) In the Eqs. 42-44, three components of the thrust vector are demonstrated as q, s and w. Also, Th is the magnitude of the thruster. The Eqs. 45-52 modified equatorial are used to solve the mentioned problem where which P(t), e x (t), e y (t), h x (t), h y (t), L(t), m(t) are state variables in the modified equatorial coordinate. The boundary conditions of the problem are also considered as follows (Eqs. 53 and 54) regarding orbital elements as semi-major axes, eccentricity, inclination, the argument of periapsis, the longitude of the ascending node, true anomaly, and mass. It should be noted that two boundary conditions, true anomaly and mass, are considered free at the final. (53) Two main cost functions minimum-time and minimum-fuel are considered. Results are obtained by the optimal on-off switching thruster. This on-off thruster can operate in space by two un constrained control angles ψ(t), φ(t) and bring the space vehicle to its final destination. Therefore, the mentioned optimal control problem is considered with three controls, with respect to one on-off thruster and two un constrained controls. Figures 32-36 describe the state variables in the modified equatorial coordinate regarding the two mentioned main cost functions.           Since two control angles ψ(t), φ(t) are un constrained, the following optimality conditions (∂H(t))/(∂ψ(t))=0, (∂H(t))/(∂φ(t))=0 can be considered, and the δ parameters as δ 1 , δ 1 ' are minimized to zero (see Figs. 42 and 43).           Results of the switch function for constrained control and two un constrained control angles ψ(t), φ(t) for minimum-fuel criterion are illustrated in Eqs. 55-57 regarding the Fourier series and GA-PSO optimizer.
(55) (56) (57) Also, the results of two unconstrained control angles ψ(t), φ(t) for the minimum-time criterion are illustrated in Eqs. 58 and 59. (58) In Table 3, the GA-PSO optimizer parameters are presented for the two mentioned problems.  Table 4 summarizes the final-time of minimum-time and minimum-fuel criteria. From the results, the reduction of final-time in the problem of the min-time is clear, but the fuel consumption is higher in front of the minimum-fuel criterion. The above problem is a complex non-linear problem in space missions and trajectory design, which provides precise results based on the δ method. Regarding the solved problems, the δ method can be considered a novel and innovative solution in the field of constrained and un constrained non-linear problems. Also, optimal controls can be achieved simply in the form of time series. Therefore, the mentioned method can be presented to space mission designers as an efficient method with desirable accuracy.

CONCLUSION
In this work, a novel method named as δ method for constrained and un constrained optimal control problems is introduced.
The mentioned method investigated space trajectory problems. This novel method does not need any initial guess and has highspeed convergence with the aid of heuristic optimization techniques such as GA-PSO and imperialist competition algorithm, orthogonal functions (Fourier, Chebyshev, and Legendre), and the principles of the optimal control theory. Regarding the introduced method, three case studies of soft Lunar landing, asteroid rendezvous, and low-thrust orbit transfer are considered to solve this method. Two powerful optimization techniques (GA-PSO and imperialist competition algorithm) and the three orthogonal functions (Fourier, Chebyshev, and Legendre) have precise results. However, from processing time and converging speed, the combination of the Fourier series and GA-PSO optimizer is the candidate. Regarding the approximation of the optimal control, new augmented criteria based on the optimal control theory were introduced to minimize the optimization techniques.
The main criterion (such as minimum-time or minimum-fuel) was added by new criteria introduced by δ based on the necessary and sufficient conditions in the optimal control theory to improve the precision and simple achievement of the results. One of the advantages of this study belonged to the independence of the proposed δ method to the initial guesses. Emphasizing the minimumtime and minimum-fuel criteria and different space missions show the ability of this method for different types of optimal control problems. From processing time and rapidly converging, the combination of the Fourier series and the GA-PSO optimizer is a candidate. Also, the vectors of optimization variables are achieved in the defined ranges. Furthermore, the results show that the boundary conditions are satisfied exactly. All δ criteria act dynamically to enhance the robustness of the δ method and they are converging rapidly and precisely. It should be noted, the mentioned method overcomes the non-linear equations with multiinput controls. Moreover, the achieved results show the accuracy and simplicity of this new method versus common complicated methods in optimal control theory. So, it will be a novel method for space mission analyzers and designers for future studies.

CONFLICT OF INTEREST
The authors declare no conflict of interest.

DATA AVAILABILITY STATEMENT
All data sets were generated or analyzed in the current study.

FUNDING
Not applicable.