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Low-Thrust Trajectory vs Impulsive Trajectory Sequence Design
As opposed to impulsive trajectories, continuous thrust trajectories have signi- cant burning time. It is not possible to parametrise the control exactly. We need to nd a control function u(t) in a Hilbert Space, that minimises a given objective function. For the sequence design, it is important to have a reliable approach.Indirect methods usually lack of robustness. The basin of attraction is small, and it is dicult to implement an automated program[Ber01]. Considering many possible sequences with an indirect approach can lead to wrong conclusions. If the algorithm fails to converge, we cannot conclude that there is no solution. Direct methods are fast and provide good solutions. Usually, direct methods should be favoured when considering many sequences.
Another approach, which can be even faster than direct methods, is the use of models. It is possible to build an approximation to the optimal control and continuous thrust dynamics with models. Modeling the control consists in choosing a family of functions that best ts the expected optimal control. This approach is useful to initiate exact algorithms. In addition, the algorithm presented in 2.4.2 can be used to eciently found solutions to transfer problems.
Current Models and Limitations
So far, exponential sinusoids [Pet02, PL01, Izz06] are used because they proved their eciency and their ease of use. However one can emphasise that the initial guess proposed by exponential sinusoid methods has sometimes a cost very far from that of the optimised trajectory. The dierences essentially come from a control that is continuously thrusting, with no bang-bang sequences. And themodel cannot correctly describe the rendezvous phase. The latter is of great importance and a major issue in most low thrust models. Only shaping methods through parametrised pseudo equinoctial elements [VD06] manage to solve the problem. But they present a violation of the dynamics and errors when propagated. Markopoulos [MC95] introduced a thrusting program, which includes a throttle parameter. In addition, his model allows multiple switching; however he didn’t treat the case with the zero throttling parameter which would be needed for coast segments. In [VSJ05], the authors survey the just mentioned low thrust models and propose a program for the global optimisation of multi-gravity assist low thrust trajectories. They also study the optimality of the exponential sinusoid. Their conclusion is that this model is far from satisfying the necessary condition of optimality, unlike the pseudo-equinoctial elements model. Curiously though, the exponential sinusoid model provides a cost closer to the optimum than the pseudo equinoctial model, whereas the latter provides more exibility. One of the most interesting models is the one of Pinkham, who models the thrust instead of the shape of the trajectory to get his low thrust model. Consequently, it is possible to construct a multi-level trajectory, however no such results have been found in the literature.
Earth – Venus – Mercury transfer
The following example is a low-thrust multi-gravity assist problem. This example has been selected for its diculty toward the rendezvous manoeuvre. Because of the high speed of planet Mercury, the nal rendezvous manoeuvre is usually important. A method to reduce it, is to allow multiple gravity-assists as in the MESSENGER mission, or BepiColombo with an EVVYY planet sequence (Earth – 2 Venus – 2 mercurY). However, such a multiple gravity assist scenario will not be considered here. Rather we seek the EVY trajectory (Earth Venus mercurY) allowing the lowest rendezvous manoeuvre. As a preliminary design tool we are mostly interested in a launch window, mission duration and overall consumption. The algorithm seeks the minimum overall V trajectory. The solution obtained can then be optimised with a tool using an indirect formulation.
Table of contents :
Contents
List of Figures
List of Tables
1 Introduction to Space Trajectory Problems
1.1 Space Propulsion Systems
1.1.1 General considerations
1.1.2 Chemical Propulsion
1.1.3 Electrical Propulsion Systems
1.1.4 Comparisons
1.2 Multi body dynamics
1.2.1 General Dynamical Equations
1.2.2 Sphere of Inuence
1.2.3 Patched Conic Approximation
1.3 Gravitational Assist (Gravity Assist, Swing-by)
1.3.1 Description
1.3.2 Simplied Model
1.3.3 Tisserand Criterion
1.4 Problem Statement
I Global Optimisation Method and Model
2 Automated Approach for Impulsive Interplanetary Trajectories
2.1 Energetics Approaches for promising Planet Sequences
2.1.1 GAP Plots
2.1.2 V1 plot
2.2 Chemical Trajectory Optimisation
2.2.1 Problem statement
2.2.2 Impulsive Trajectories
2.2.3 Impulsive Trajectories with Deep Space Manoeuvres
2.2.4 Multi-Gravity Assist Trajectory problems
2.2.5 Multi-Gravity Assist Trajectory problems with Deep Space Manoeuvres
2.3 Global Optimisation
2.3.1 Branch and Bound
2.3.2 Particle Swarm Optimisation
2.3.3 Brief synthesis
2.4 Simplifying the Search Space of Parameterised Trajectories for Global Optimisation
2.4.1 General problem and objectives
2.4.2 General Approach
2.4.3 Application: MGADSM Space trajectory
2.5 Conclusions
3 Automated Approach for Low-Thrust Interplanetary Trajecto- ries
3.1 Introduction
3.1.1 Low-Thrust Trajectory vs Impulsive Trajectory Sequence Design
3.1.2 Examples of Low-Thrust Global Optimisation Problems
3.2 Current Models and Limitations
3.3 A Continuous Thrust Model
3.3.1 Dynamics
3.3.2 Geometrical properties
3.3.3 Physical properties
3.3.4 Existence of solutions
3.4 Multi-thrust Segment Transfer
3.4.1 Construction of Coast – Thrust Control
3.4.2 Formulation of the multi-switch transfer
3.4.3 Number of switching points
3.5 Optimisation problem
3.5.1 Formulation for the parameterised trajectory problem
3.5.2 Algorithm
3.5.3 Local Solver for the Thrust Segments
3.5.4 Global Search for the Coast Segments
3.6 Conclusions
4 Applications
4.1 Earth – Mars rendezvous transfer
4.2 Earth – Venus – Mercury transfer
4.3 Earth – Mars – Vesta – Ceres
4.4 GTOC3 problem
II Optimal Control Methods
5 Review of Optimal Control Methods applied to Low-Thrust Interplanetary Trajectories
5.1 General Problem formulation
5.1.1 Problem Description
5.1.2 State of the art
5.2 Direct Problem Formulation
5.2.1 Formulation
5.2.2 Limitations
5.3 Indirect Problem Formulation
5.3.1 Examples of Low-Thrust Trajectory optimisation
5.3.2 The Maximum Principle
5.3.3 Numerical derivatives
5.3.4 Limitations
5.4 Summary, Conclusions
6 Algorithm for Optimising Low-Thrust Interplanetary Transfers in Multi-Body Dynamics
6.1 Introduction
6.1.1 Issues and Objectives
6.2 Solution Methods
6.2.1 General considerations
6.2.2 Modied Gradient Method
6.2.3 Optimal Control
6.2.4 Terminal State Constraints
6.2.5 Constraint on the Control
6.3 Convergence Issues
6.3.1 Minimisation
6.3.2 Constraints reduction, and Problem Feasibility
6.3.3 Improvement in Optimality
6.4 Numerical approach of the Continuous Problem Control
6.4.1 Continuous control issue
6.4.2 Optimal placement for Autonomous Systems
6.4.3 Mesh Placement Strategies
6.5 Algorithm and discussions
6.5.1 Presentation of the algorithm
6.6 Academic Examples
6.6.1 Goddard’s Problem
6.6.2 Orbital Transfer
6.7 Conclusion, discussions
7 Numerical Examples
7.1 Mars – Earth rendezvous transfer
7.2 Earth – Mars with capture and escape phases
7.3 GTOC3 Asteroid to Asteroid leg, with Automatic Swingby Design
Conclusions
