Sequential Gradient-restoration Algorithm Research Articles

Minimum fuel trajectories for round trip lunar missions

In this work, a study about minimum fuel trajectories in a round trip journey to the Moon is presented. It is assumed that the velocity changes are instantaneous, that is, the propulsion system is capable of delivering impulses such that the fuel consumption is represented by the total velocity increment applied to the space vehicle. It is also assumed that the velocity increments are applied tangentially to the terminal orbits, and, the outgoing trip and the return trip are analyzed separately such that the whole mission is performed with four impulses (two impulses in each trip). The mathematical models used to describe the motion of the space vehicle are three: the lunar patched-conic approximation; the classic planar circular restricted three-body problem, and, the planar bi-circular restricted four-body problem (PBR4BP). For computing the optimal trajectories, the Sequential Gradient-Restoration Algorithm with constraints is used. The influence of the Sun on round trip lunar missions is analyzed through the PBR4BP model. For all models, the trajectories studied are direct ascent maneuvers, and, both the outgoing and return trips are considered. The results obtained through the different models are compared with each other. The optimal results for the PBR4BP model show that a small reduction of the fuel consumption can be achieved if the initial phase angle of the Sun is chosen properly.

Application of the Sequential Gradient Restoration Algorithm to the Solution of the Low Thrust Transfer Problem

The Sequential Gradient Restoration Algorithm (SGRA) is an indirect optimisation method based on the iterative alternation of two phases: in the first one (gradient), the objective is the minimisation of the cost function; in the second phase (restoration) the objective is the fulfilment of the differential equations, equality and inequality path constraints and boundary conditions. This paper illustrates the application of the SGRA to the generation of very long transfer trajectories (up to some few thousand orbits during several months) in a typical GTO to GEO transfer scenario performed with electric propulsion. The reason for using electric propulsion in that case is the significant propellant mass savings if compared to the classical chemical propulsion strategies, at the expenses of an elongated transfer phase (from few days to some months). The electric propulsion system considered in this paper is characterised by thrust levels between 100 mN and 1 N (for some 3 to 5 tons launch mass) and high specific impulse, between 1000 s and 5000 s.

A study of the influence of the sun on optimal two-impulse Earth-to-Moon trajectories with moderate time of flight in the three-body and four-body models

In this work, a study about the influence of the Sun on optimal two-impulse Earth-to-Moon trajectories for interior transfers with moderate time of flight is presented considering the three-body and the four-body models. The optimization criterion is the total characteristic velocity which represents the fuel consumption of an infinite thrust propulsion system. The optimization problem has been formulated using the classic planar circular restricted three-body problem (PCR3BP) and the planar bi-circular restricted four-body problem (PBR4BP), and, it consists of transferring a spacecraft from a circular low Earth orbit (LEO) to a circular low Moon orbit (LMO) with minimum fuel consumption. The Sequential Gradient Restoration Algorithm (SGRA) is applied to determine the optimal solutions. Numerical results are presented for several final altitudes of a clockwise or a counterclockwise circular low Moon orbit considering a specified altitude of a counterclockwise circular low Earth orbit. Two types of analysis are performed: in the first one, the initial position of the Sun is taken as a parameter and the major parameters describing the optimal trajectories are obtained by solving an optimization problem of one degree of freedom. In the second analysis, an optimization problem with two degrees of freedom is considered and the initial position of the Sun is taken as an additional unknown.

Preliminary Analysis of Optimal Round Trip Lunar Missions

A study of optimal bi-impulsive trajectories of round trip lunar missions is presented in this paper. The optimization criterion is the total velocity increment. The dynamical model utilized to describe the motion of the space vehicle is a full lunar patched-conic approximation, which embraces the lunar patched-conic of the outgoing trip and the lunar patched-conic of the return mission. Each one of these parts is considered separately to solve an optimization problem of two degrees of freedom. The Sequential Gradient Restoration Algorithm (SGRA) is employed to achieve the optimal solutions, which show a good agreement with the ones provided by literature, and, proved to be consistent with the image trajectories theorem.

Optimal round trip lunar missions based on the patched-conic approximation

A study of optimal bi-impulsive trajectories of round trip lunar missions is presented in this paper. The optimization criterion is the total velocity increment. The dynamical model utilized to describe the motion of the space vehicle is a full lunar patched-conic approximation, which embraces the lunar patched-conic of the outgoing trip and the lunar patched-conic of the return mission. Each one of these parts is considered separately to solve an optimization problem of two degrees of freedom. The parameters to be optimized are two: the phase angle of the point at which the space vehicle reaches the edge of the Moon’s sphere of influence and the initial velocity at departure. The Sequential Gradient Restoration Algorithm is employed to achieve the optimal solutions. Analytical and numerical derivatives of expressions describing the lunar patched-conic approximations are utilized to ensure the results. The results based on the patched-conic approximation show a good agreement with the ones provided by literature, and the solution trajectories proved to be consistent with the image trajectories theorem.

基于改进的序列梯度–修复算法的飞行器上升段轨迹优化

针对飞行器大气层内上升段轨迹优化问题，提出了一种改进的序列梯度–修复算法。根据序列梯度–修复算法仅可以处理等式约束的特点，给出了不等式过程约束的转化过程，同时为了解决飞行器大气层内上升段运动数学模型的强非线性带来的状态量求解更新困难的难题，引入了状态积分，改进了算法的更新方法，最后对改进的算法在飞行器大气层内上升段轨迹优化中的实用性进行了仿真分析。仿真结果表明改进的序列梯度–修复算法能够较好的获得满足末端约束和过程约束的大气层内上升段的最优轨迹，具有较强的适应性。 An improved sequential gradient-restoration algorithm is developed to solve endoatmospheric ascent trajectory optimization problem. First, the transformation process of trajectory constraints is introduced to accommodate the character that sequential gradient-restoration algorithm can deal with equation constraints only. Then the state integral method is introduced and the update method of the sequential gradient-restoration algorithm is improved to deal with the problem of state solve and update, due to the strong nonlinearity of endoatmospheric ascent dynamics. Finally, the simulation analysis of the improved algorithm ascent trajectory optimization in aircraft atmosphere of practicability is analyzed through simulation under five different scenarios. Simulation results demonstrate the validity and adaptability of the improved algorithm for trajectory optimization with terminal constraints and process constraints.

Optimization and Analysis on Trajectory with Multiple Constraints for Hypersonic Air-vehicle

The trajectory optimization technology is one of the key technologies for hypersonic air-vehicle. There are multiple constraints in the process of hypersonic flight, such as uncertainty of flight environment, thermal current, dynamic pressure and overload. The trajectory optimization of hypersonic air-vehicle is facing with a great challenge. This article studies the direct shooting method, the Gauss pseudo spectral method and sequential gradient-restoration algorithm, among which the direct shooting method simply makes the control variables discrete in the time domain, and obtains the status value by explicit numerical integration; Gauss pseu- do spectral method makes the status variable and control variable discrete in a series of Gauss points, and con- structs multinomial to approximate to the status and control variable by taking the discrete points as the nodes; sequential gradient-restoration algorithm uses iteration to meet the constraints and minimize the increment of initial value of control and status variable in order to constantly approximate to the optimal solution on condition that the constraints meet first order approximation. Finally this article conducts a numerical simulation by taking the diving segment of hypersonic air-vehicle as an example for comparative analysis on those three algorithms respectively from, such as, the initial value selection, constraint handling, convergence speed and calculation accuracy. The simulation result indicates Gauss pseudo spectral method is a method with fairly good comprehen- sive performance.

A near-optimal guidance for cooperative docking maneuvers

In this work we study the problem of minimum energy docking maneuvers between two Floating Spacecraft Simulators. The maneuvers are planar and conducted autonomously in a cooperative mode. The proposed guidance strategy is based on the direct method known as Inverse Dynamics in the Virtual Domain, and the nonlinear programming solver known as Sequential Gradient-Restoration Algorithm. The combination of these methods allows for the quick prototyping of near-optimal trajectories, and results in an implementable tool for real-time closed-loop maneuvering. The experimental results included in this paper were obtained by exploiting the recently upgraded Floating Spacecraft-Simulator Testbed of the Spacecraft Robotics Laboratory at the Naval Postgraduate School. A direct performances comparison, in terms of maneuver energy and propellant mass, between the proposed guidance strategy and a LQR controller, demonstrates the effectiveness of the method.

Optimal guidance of spacecraft rendezvous via free initial velocity vector

In this research, we focus on an optimal trajectory of a spacecraft rendezvous operation. This optimal trajectory, which is studied in rendezvous problem consists of a controlling distribution, parametric and permitted forms of performance index of minimum fuel-time free; this method will guide the chaser spacecraft toward the target spacecraft in optimal trajectory by using multiple-subarc sequential gradient–restoration algorithm. During trajectory analysis, we will define two problems, P1 and P2. In P1, initial position vector and initial velocity vector are given and fixed and in P2, initial position vector is given and fixed, while initial velocity vector varies and will be free. Previous studies show that the condition of optimal fuel in a solution will be obtained with four subarcs and it shows that for distribution of optimal thrust, the amount of maximum thrust or zero during each subarc will be needed. The main result is that, the new method proposed by this study leads to reduction in performance index and also fuel consumption will be in an appropriate amount. This event takes place in proportion of initial free velocity (problem P2) to initial fixed velocity (problem P1). Another result is that the reduction in fuel consumption and performance index is associated with a remarkable reduction in central processing unit time up to 1650 order. The result of this investigation for rendezvous mission between target spacecraft and chaser spacecraft named OCAT1.1 was extracted and examined on SuperCluster computer.

Collision Avoidance for an Aircraft in Abort Landing: Trajectory Optimization and Guidance

For a host aircraft in the abort landing mode under emergency conditions, the best strategy for collision avoidance is to maximize wrt to the controls the timewise minimum distance between the host aircraft and an intruder aircraft. This leads to a maximin problem or Chebyshev problem of optimal control. At the maximin point of the encounter, the distance between the two aircraft has a minimum wrt the time; its time derivative vanishes and this occurs when the relative position vector is orthogonal to the relative velocity vector. By using the zero derivative condition as an inner boundary condition, the one-subarc Chebyshev problem can be converted into a two-subarc Bolza-Pontryagin problem, which in turn can be solved via the multiple-subarc sequential gradient-restoration algorithm. Optimal Trajectory. In the avoidance phase, maximum angle of attack is used by the host aircraft until the minimum distance point is reached. In the recovery phase, the host aircraft completes the transition of the angle of attack from the maximum value to that required for quasisteady ascending flight. Guidance Trajectory. Because the optimal trajectory is not suitable for real-time implementation, a guidance scheme is developed such that it approximates the optimal trajectory results in real time. In the avoidance phase, the guidance scheme employs the same control history (maximum angle of attack) as that of the optimal trajectory so as to achieve the goal of maximizing wrt the control the timewise minimum distance. In the recovery phase, the guidance scheme employs a time-explicit cubic control law so as to achieve the goal of recovering the quasisteady ascending flight state at the final time. Numerical results for both the optimal trajectory and the guidance trajectory complete the paper.

Rendezvous Guidance Trajectories via Multiple-Subarc Sequential Gradient-Restoration Algorithm

We consider the three-dimensional rendezvous between a target spacecraft in a circular orbit and a chaser spacecraft with an initial separation distance and an initial separation velocity. We assume that the chaser spacecraft has variable mass and that its trajectory is governed by three controls, one determining the thrust magnitude and two determining the thrust direction. We employ the Clohessy–Wiltshire equations, describing the relative motion of the chaser vis-à-vis the target, and the multiple-subarc sequential gradient-restoration algorithm to produce first optimal trajectories and then guidance trajectories for the following problems: P1—minimum time, fuel free; P2—minimum fuel, time free; P3—minimum time, fuel given; P4—minimum fuel, time given; and P5—minimum time×fuel, time and fuel free. Clearly, P1 and P2 are basic problems, while P3, P4, and P5 are compromise problems. Problem P1 leads to a two-subarc solution including a max-thrust subarc followed by another max-thrust subarc. Problem P2 leads to a four-subarc solution including two coasting subarcs alternating with two max-thrust subarcs. Problems P5 leads to a three-subarc solution including two max-thrust subarcs alternating with one coasting subarc. Problems P3 and P4 include P1, P2, and P5 as particular cases and lead to two-, three-, or four-subarcs solutions depending on the prescribed value of fuel or time. For all problems, the thrust magnitude control is saturated at one of its extreme values: in optimization studies, we determine the best thrust direction controls; in guidance studies, we force the thrust direction controls to be constant in each subarc and determine the best thrust direction parameters. Of course, the time lengths of all the subarcs must also be determined. The computational results show that, for Problems P1–P5, the performance index of the multiple-subarc guidance trajectory approximates well the performance index of the multiple-subarc optimal trajectory: the pairwise relative differences in performance index are less than 1∕100 in all cases. To sum up, the produced guidance trajectories are highly efficient and yet quite simple in implementation.

Optimal Trajectories for Spacecraft Rendezvous

The efficient execution of a rendezvous maneuver is an essential component of various types of space missions. This work describes the formulation and numerical investigation of the thrust function required to minimize the time or fuel required for the terminal phase of the rendezvous of two spacecraft. The particular rendezvous studied concerns a target spacecraft in a circular orbit and a chaser spacecraft with an initial separation distance and separation velocity in all three dimensions. First, the time-optimal rendezvous is investigated followed by the fuel-optimal rendezvous for three values of the max-thrust acceleration via the sequential gradient-restoration algorithm. Then, the time-optimal rendezvous for given fuel and the fuel-optimal rendezvous for given time are investigated. There are three controls, one determining the thrust magnitude and two determining the thrust direction in space.

Guidance Trajectories for Spacecraft Rendezvous

In a previous paper of Miele et al. (J. Optim. Theory Appl. 132(1), 2007), we employed the single-subarc sequential gradient-restoration algorithm to optimize the three-dimensional rendezvous between a target spacecraft in a planar circular orbit and a chaser spacecraft with an initial separation distance and separation velocity. The achieved continuous solutions are characterized by two, three, or four subarcs depending on the performance index (time, fuel) and the constraints. In this paper, based on the solutions in Miele et al. (J. Optim. Theory Appl. 132(1), 2007), we employ the multiple-subarc sequential gradient-restoration algorithm to produce pieced guidance trajectories implementable in real time via constant control components. In other words, in this investigation, we force the controls to behave as parameters in each subarc. With the above understanding, we investigate four problems: (P1) minimum time, fuel free; (P2) minimum fuel, time free; (P3) minimum time, fuel given; (P4) minimum fuel, time given.

Optimal Trajectories and Guidance Schemes for Ship Collision Avoidance

The best strategy for collision avoidance under emergency conditions is to maximize wrt the controls the timewise minimum distance between the host ship and the intruder ship. In a restricted waterway area, two main constraints must be satisfied: the lateral deviation of the host ship from the original course is to be contained within certain limits; the longitudinal distance covered by the host ship is to be subject to a prescribed bound. At the maximin point of the encounter, the time derivative of the relative distance vanishes; this yields an inner boundary condition (orthogonality between the relative position vector and the relative velocity vector) separating the main phases of the maneuver: the avoidance and recovery phases. In this way, the optimal trajectory problem (a Chebyshev problem) can be converted into a Bolza problem with an inner boundary condition. Numerical solutions are obtained via the multiple-subarc sequential gradient-restoration algorithm (SGRA). Because the optimal trajectory is not suitable for real-time implementation, a guidance scheme approximating the optimal trajectory in real time is to be developed. For ship collision avoidance, the optimal trajectory results show that the rudder angle time history has a bang-bang form characterized by the alternation of saturated control subarcs of opposite signs joined by rapid transitions. Just as the optimal trajectory can be partitioned into three phases (avoidance phase, recovery phase, steady phase), a guidance trajectory can be constructed in the same way. For the avoidance and recovery phases, use of decomposition techniques leads to an algorithm computing the time lengths of these phases in real time. For the steady phase, a feedback control scheme is used to maneuver the ship steadily. Numerical results are presented.

Optimal Planetary Orbital Transfers via Chemical Engines and Electrical Engines

Because the orbital periods for planetary orbital transfers are of order hour, the primary objective of an optimal trajectory is to minimize the propellant consumption. In this paper, we present a systematic investigation of optimal trajectories for planetary orbital transfer. Major results on thrust control, propellant consumption, and flight time are presented with particular reference to LEO-to-GEO transfer. The following results were obtained with the sequential gradient-restoration algorithm in either single-subarc form or multiple-subarc form: (i) For minimum propellant consumption, the thrust direction is tangent to the flight path. The thrust magnitude has a three-subarc form: powered flight with maximum thrust is followed by coasting flight, which is followed by powered flight with maximum thrust. (ii) The flight time is determined mainly by the thrust-to-weight ratio. A transfer via chemical engines is relatively short: usually, it requires less than one cycle to achieve the mission, which involves a large portion of coasting flight. A transfer via electrical engines is relatively long: usually, it requires a multicycle spiral trajectory to achieve the mission, which involves a large portion of powered flight, mostly in the first subarc. (iii) The propellant consumption is determined mainly by the specific impulse: the electrical engine is more efficient than the chemical engine, resulting in lower propellant consumption and higher payload.

Computation of optimal Mars trajectories via combined chemical/electrical propulsion, Part 3: Compromise solutions

The success of the solar-electric ion engine powering the DS1 spacecraft has paved the way toward the use of low-thrust electrical engines in future planetary/interplanetary missions. Vis-à-vis a chemical engine, an electrical engine has a higher specific impulse, implying a possible decrease in propellant mass; however, the low-thrust aspect discourages the use of an electrical engine in the near-planet phases of a trip, since this might result in an increase in flight time. Therefore, a fundamental design problem is to find the best combination of chemical propulsion and electrical propulsion for a given mission, for example, a mission from Earth to Mars. With this in mind, this paper is the third of a series dealing with the optimization of Earth–Mars missions via the use of hybrid engines, namely the combination of high-thrust chemical engines for planetary flight and low-thrust electrical engines for interplanetary flight. We look at the deep-space interplanetary portion of the trajectory under rather idealized conditions. The two major performance indexes, the propellant mass and the flight time, are in conflict with one another for the following reason: any attempt at reducing the former causes an increase in the latter and vice versa. Therefore, it is natural to consider a compromise performance index involving the scaled values of the propellant mass and flight time weighted respectively by the compromise factor C and its complement 1 - C . We use the compromise factor as the parameter of the one-parameter family of compromise trajectories. Analyses carried out with the sequential gradient-restoration algorithm for optimal control problems lead to results which can be highlighted as follows. Thrust profile. Generally speaking, the thrust profile of the compromise trajectory includes three subarcs: the first subarc is characterized by maximum thrust in conjunction with positive (upward) thrust direction; the second subarc is characterized by zero thrust (coasting flight); the third subarc is characterized by maximum thrust in conjunction with negative (downward) thrust direction. Effect of the compromise factor. As the compromise factor increases, the propellant mass decreases and the flight time increases; correspondingly, the following changes in the thrust profile take place: (a) the time lengths of the first and third subarcs (powered phases) decrease slightly, meaning that thrust application occurs for shorter duration; also, the average value of the thrust direction in the first and third subarcs decreases, implying higher efficiency of thrust application wrt the spacecraft energy level; as a result, the total propellant mass decreases; (b) the time length of the second subarc (coasting) increases considerably, resulting in total time increase. Minimum time trajectory. If C = 0 , the resulting minimum time trajectory has the following characteristics: (a) the time length of the coasting subarc reduces to zero and the three-subarc trajectory degenerates into a two-subarc trajectory; (b) maximum thrust is applied at all times and the thrust direction switches from upward to downward at midcourse. Minimum propellant mass trajectory. If C = 1 , the resulting minimum propellant mass trajectory has the following characteristics: (a) the thrust magnitude has a bang-zero-bang profile; (b) for the powered subarcs, the thrust direction is tangent to the flight path at all times.

Maximin Approach to the Ship Collision Avoidance Problem via Multiple-Subarc Sequential Gradient-Restoration Algorithm

The ideal strategy for ship collision avoidance under emergency conditions is to maximize wrt the controls the timewise minimum distance between a host ship and an intruder ship. This is a maximin problem or Chebyshev problem of optimal control in which the performance index being maximinimized is the distance between the two ships. Based on the multiple-subarc sequential gradient-restoration algorithm, a new method for solving the maximin problem is developed.

Reflections on the Hohmann Transfer

Walter Hohmann was a civil engineer who studied orbital maneuvers in his spare time. In 1925, he published an important book (Ref. 1) containing his main result, namely, that the most economical transfer from a circular orbit to another circular orbit is achieved via an elliptical trajectory bitangent to the terminal orbits. With the advent of the space program some three decades later, the Hohmann transfer maneuver became the most fundamental maneuver in space. In this work, we present a complete study of the Hohmann transfer maneuver. After revisiting its known properties, we present a number of supplementary properties which are essential to the qualitative understanding of the maneuver. Also, we present a simple analytical proof of the optimality of the Hohmann transfer and complement it with a numerical study via the sequential gradient-restoration algorithm. Finally, as an application, we present a numerical study of the transfer of a spacecraft from the Earth orbit around the Sun to another planetary orbit around the Sun for both the case of an ascending transfer (orbits of Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto) and the case of a descending transfer (orbits of Mercury and Venus).

On feasibility of launch vehicle designs

We study various launch vehicle designs for a spacecraft to be launched vertically and ascending to an Earth orbit located at Space Station altitude ( h = 463 km). We optimize the trajectory from the maximum payload viewpoint with respect to the angle of attack time history and thrust setting time history. We consider single-stage, double-stage, and triple-stage configurations. To perform the study, we employ a newly developed member of the family of sequential gradient-restoration algorithms for optimal control problems, the multiple-subarc sequential gradient-restoration algorithm. Assuming that the engine specific impulse has the best value available today ( I sp = 450 s), we study the effect of the structural factor ε (ratio of the structural mass to sum of structural mass and propellant mass) on the payload and then develop simple relations connecting the structural factor to the payload. With these relations, we are able to predict the limiting value of the structural factor for which the payload vanishes. To two significant digits, the limiting value is approximately ε ≅ 0.11 for a single-stage configuration, ε ≅ 0.27 for a double-stage configuration, and ε ≅ 0.33 for a triple-stage configuration. Based on these results, we conclude that, given the current values of the engine specific impulse and spacecraft structural factor, a single-stage configuration is not feasible at this particular time, while double-stage and triple-stage configurations are feasible. The above limiting values refer to multistage configurations designed with uniform structural factor. Higher payloads and higher limiting values for the last stage structural factor are possible for configurations designed with nonuniform structural factor, that is, with a last-stage structural factor much higher than the structural factors of the previous stages. For example, a double-stage configuration with ε 1 = 0.10, ε 2 = 0.25 can deliver to orbit a payload mass equal to 4.7% at the takeoff mass; a triple-stage configuration with ε 1 = ε 2 = 0.10, ε 3 = 0.25 can deliver to orbit a payload mass equal to 6.8% of the takeoff mass.

Low thrust transfer optimisation of satellites formations to heliocentric Earth trailing orbits through a gradient restoration algorithm

When dealing with mission requirements calling for highly stable gravitational and thermal environments, one of the common options always considered is that of the heliocentric Earth trailing orbits (HETO). This is the case, for instance, of the Laser Interferometer Space Antenna (LISA), a joint ESA-NASA effort aimed at detecting gravitational radiation from deep-space sources, hence testing the fundamental gravitational theories. This article is about the low-thrust transfer trajectory optimisation for a three-satellites flotilla to a HETO. To this end, and after a brief presentation of the selected reference mission (LISA), a five impulses transfer strategy is conceived to build a solution guess for the ignition of the Sequential Gradient Restoration Algorithm (SGRA), which solves the full optimal control problem (FOCP) of finding the optimal low thrust profiles injecting each SC to its respective target orbit while minimising a given functional. An inequality path constraint on the Solar Aspect Angle (SAA) is imposed on the low-thrust vector of each SC. Finally several relevant conclusions are derived from the presented results, among them the direct relationship between the departure velocity from Earth, the final SC masses and the transfer duration.