From 298eb38ff139c1d3cba3a26af5c9c7db323c7334 Mon Sep 17 00:00:00 2001 From: Connor Date: Tue, 15 Mar 2022 22:20:45 -0600 Subject: [PATCH] She did. Now I'm done! --- LaTeX/approach.tex | 48 +++--- LaTeX/conclusion.tex | 7 +- LaTeX/introduction.tex | 75 ++++----- LaTeX/results.tex | 42 +++-- LaTeX/thesis.tex | 17 +- LaTeX/trajectory_design.tex | 266 +++++++++++------------------- LaTeX/trajectory_optimization.tex | 42 ++--- 7 files changed, 198 insertions(+), 299 deletions(-) diff --git a/LaTeX/approach.tex b/LaTeX/approach.tex index f29b0b0..0cd7e93 100644 --- a/LaTeX/approach.tex +++ b/LaTeX/approach.tex @@ -1,11 +1,10 @@ \chapter{Algorithm Overview} \label{algorithm} - This thesis will attempt to develop an algorithm for the preliminary analysis of feasibility in - designing a low-thrust interplanetary mission to an outer planet by leveraging a monotonic basin - hopping algorithm. In this section, we will review the actual execution of the algorithm - developed. As an overview, the routine was designed to enable the determination of an optimized - spacecraft trajectory from the selection of some very basic mission parameters. Those parameters - include: + This thesis focuses on designing a low-thrust interplanetary mission to an outer planet by + leveraging a monotonic basin hopping algorithm. This section will review the actual execution of + the algorithm developed. As an overview, the routine is designed to enable the determination of + an optimal spacecraft trajectory that minimizes propellant usage and $C_3$ from the selection of + some very basic parameters. Those parameters include: \begin{itemize} \setlength\itemsep{-0.5em} @@ -25,7 +24,7 @@ \end{itemize} Which allows for an automated approach to optimization of the trajectory, while still providing - the mission designer with the flexibility to choose the particular flyby planets to investigate. + the designer with the flexibility to choose the particular flyby planets to investigate. This is achieved via an optimal control problem in which the ``inner loop'' involves solving a TPBVP to find the optimal solution given a suitable initial guess. Then an ``outer loop'' @@ -133,12 +132,11 @@ The following pseudo-code outlines the approach taken for the elliptical case. The approach is quite similar when $a<0$: - % TODO: Some symbols here aren't recognized by the font \begin{singlespacing} \begin{verbatim} i = 0 # First declare some useful variables from the state - sig0 = (position ⋅ velocity) / √(mu) + sig0 = dot(position, velocity) / √(mu) a = 1 / ( 2/norm(position) - norm(velocity)^2/mu ) coeff = 1 - norm(position)/a @@ -184,7 +182,7 @@ \label{laguerre_plot} \end{figure} - \subsection{Sims-Flanagan Propagator} + \subsection{Propagating with Sims-Flanagan Transcription} Until this point, we've not yet discussed how best to model the low-thrust trajectory arcs themselves. The Laguerre-Conway algorithm efficiently determines @@ -228,11 +226,9 @@ and the mass flow rate (a function of the duty cycle percentage ($d$), thrust ($f$), and the specific impulse of the thruster ($I_{sp}$), commonly used to measure efficiency)\cite{sutton2016rocket}: - \begin{equation} \Delta m = \Delta t \frac{f d}{I_{sp} g_0} \end{equation} - Where $\Delta m$ is the fuel used in the sub-trajectory, $\Delta t$ is the time of flight of the sub-trajectory, and $g_0$ is the standard gravity at the surface of Earth. From knowledge of the mass flow rate, we can then decrement the mass @@ -272,10 +268,9 @@ From this information, as can be seen in Figure~\ref{nlp}, we can formulate the mission in terms of a non-linear programming problem. Specifically, the variables describing the - trajectory contained within the Guess object can be represented as an input vector, - $\vec{x}$, the cost function produced by an entire trajectory propagation as $F$, and - the constraints that the trajectory must satisfy as another function $\vec{G}$ such that - $\vec{G}(\vec{x}) = \vec{0}$. + trajectory from the free variable, $\vec{x}$, the cost function produced by an entire + trajectory propagation, $F$, and the constraints that the trajectory must satisfy as + another function $\vec{G}$ such that $\vec{G}(\vec{x}) = \vec{0}$. This is a format that we can apply directly to the IPOPT solver, which Julia (the programming language used) can utilize via bindings supplied by the SNOW.jl @@ -333,12 +328,12 @@ \subsection{Random Trajectory Generation}\label{random_gen_section} - At a basic level, the algorithm needs to produce a guess (represented by all of the - values described in Section~\ref{inner_loop_section}) that contains random values within - reasonable bounds in the space. However, that still leaves the determination of which - distribution function to use for the random values over each of those variables, which - bounds to use, as well as the possibilities for any improvements to a purely random - search. + At a basic level, the algorithm needs to produce a guess for the free variable vector + (represented by all of the values described in Section~\ref{inner_loop_section}) that + contains random values within reasonable bounds in the space. However, that still leaves + the determination of which distribution function to use for the random values over each + of those variables, which bounds to use, as well as the possibilities for any + improvements to a purely random search. Currently, the first value set for the mission guess is that of $n$, which is the number of sub-trajectories that each arc will be broken into for the Sims-Flanagan @@ -372,18 +367,18 @@ missions with more flybys. Then, the internal components for each phase are generated. It is at this step, that - the mission guess generator splits the outputs into two separate outputs. The first + the trajectory guess generator splits the outputs into two separate outputs. The first is meant to be truly random, as is generally used as input for a monotonic basin hopping algorithm. The second utilizes a Lambert's solver to determine the appropriate hyperbolic velocities (both in and out) at each flyby to generate a - natural trajectory arc. For this Lambert's case, the mission guess is simply seeded + natural trajectory arc. For this Lambert's case, the trajectory guess is simply seeded with zero thrust controls and outputted to the monotonic basin hopper. The intention here is that if the time of flights are randomly chosen so as to produce a trajectory that is possible with a control in the vicinity of a natural trajectory, we want to be sure to find that trajectory. More detail on how this is handled is available in Section~\ref{mbh_subsection}. - However, for the truly random mission guess, there are still the $v_\infty$ values + However, for the truly random trajectory guess, there are still the $v_\infty$ values and the initial thrust guesses to generate. For each of the phases, the incoming excess hyperbolic velocity is calculated in much the same way that the launch velocity was calculated. However, instead of multiplying the randomly generate unit @@ -471,14 +466,11 @@ Because of this, the perturbation used in this implementation follows a bi-directional, long-tailed Pareto distribution generated by the following probability density function\cite{englander2014tuning}: - \begin{equation} 1 + \left[ \frac{s}{\epsilon} \right] \cdot \left[ \frac{\alpha - 1}{\frac{\epsilon}{\epsilon + r}^{-\alpha}} \right] \end{equation} - - \noindent Where $s$ is a random array of signs (either plus one or minus one) with dimension equal to the perturbed variable and bounds of -1 and 1, $r$ is a uniformly distributed random array with dimension equal to the perturbed variable and bounds diff --git a/LaTeX/conclusion.tex b/LaTeX/conclusion.tex index 257f9b2..79c8594 100644 --- a/LaTeX/conclusion.tex +++ b/LaTeX/conclusion.tex @@ -10,11 +10,8 @@ In performing this examination, two results were selected for further analysis. These results are outlined in Table~\ref{results_table}. As can be seen in the table, both resulting trajectories have trade-offs in mission length, launch energy, fuel usage, and - more. However, both results show very interesting trajectories that could indicate some - favorable possibilities for such a mission profile. Each of these trajectories should be - within the capabilities of existing launch vehicles in terms of $C_3$. - - \section{Recommendations for Future Work}\label{improvement_section} + more. Each of these trajectories appear to be within the capabilities of existing launch + vehicles in terms of $C_3$. In the course of producing this algorithm, a large number of improvement possibilities were noted. This work was based, in large part, on the work of Jacob Englander in a number of diff --git a/LaTeX/introduction.tex b/LaTeX/introduction.tex index ca9831f..8c5b3c2 100644 --- a/LaTeX/introduction.tex +++ b/LaTeX/introduction.tex @@ -1,16 +1,16 @@ \chapter{Introduction} Continuous low-thrust engines utilizing technologies such as Ion propulsion, Hall thrusters, and - others can be a powerful system in the enabling of long-range interplanetary missions with fuel - efficiencies unrivaled by those that employ only impulsive thrust systems. The challenge in - utilizing these systems, then, is the design of trajectories that effectively utilize this - technology. Continuous thrust propulsive systems tend to be particularly suited to missions - which require very high total change in velocity ($\Delta V$) values and take place over a - particularly long duration. Traditional impulsive thrusting techniques can achieve these changes - in velocity, but typically have a far lower specific impulse and, as such, are much less fuel - efficient, costing the mission valuable financial resources that could instead be used for - science. Because of their inherently high specific impulse (and thus efficiency), low-thrust - propagation systems are well-suited to interplanetary missions. + others enable long-range interplanetary missions with fuel efficiencies unrivaled by those that + employ only impulsive thrust systems. The challenge in utilizing these systems, then, is the + design of trajectories that effectively utilize this technology. Continuous thrust propulsive + systems tend to be particularly suited to missions which require very high total change in + velocity ($\Delta V$) values and take place over a particularly long duration. Traditional + impulsive thrusting techniques can achieve these changes in velocity, but typically have a far + lower specific impulse and, as such, are much less fuel efficient, costing the mission valuable + financial resources that could instead be used for science. Because of their inherently high + specific impulse (and thus efficiency), low-thrust propulsion systems are well-suited to + interplanetary missions. The first attempt by NASA to use an electric ion-thruster for an interplanetary mission was the Deep Space 1 mission\cite{brophy2002}. This mission was designed to test the ``new'' technology, @@ -29,16 +29,15 @@ in October 2018 and is projected to perform a flyby of Earth, two of Venus, and six of Mercury before inserting into an orbit around that planet. - A common theme in mission design is that there always exists a trade-off between efficiency - (particularly in terms of fuel use) and the time required to achieve the mission objective. Low - thrust systems in particular tend to produce mission profiles that sacrifice the rate of - convergence on the target state in order to achieve large increases in fuel efficiency. Often a - low-thrust mission profile in Earth orbit will require multiple orbital periods to achieve the - desired change in spacecraft state. Interplanetary missions, though, provide a particularly - useful case for continuous thrust technology. The trajectory arcs in interplanetary space are - generally much, much longer than orbital missions around the Earth. Because of this increase, - even a small continuous thrust is capable of producing large $\Delta V$ values over the course - of a single trajectory arc. + A common theme in mission design is that there is a trade-off between efficiency (particularly + in terms of fuel use) and the time required to achieve the mission objective. Low thrust systems + in particular tend to produce mission profiles that sacrifice the rate of convergence on the + target state in order to achieve large increases in fuel efficiency. Often a low-thrust transfer + in Earth orbit will require multiple orbital periods to achieve the desired change in spacecraft + state. Interplanetary missions, though, provide a particularly useful case for continuous thrust + technology. The trajectory arcs in interplanetary space are generally much, much longer than + orbital missions around the Earth. Because of this increase, even a small continuous thrust is + capable of producing large $\Delta V$ values over the course of a single trajectory arc. Another technique often leveraged by interplanetary trajectory designers is the gravity assist. Gravity assists utilize the inertia of a large planetary body to ``slingshot'' a spacecraft, @@ -58,24 +57,22 @@ routine for producing unconstrained, globally optimal trajectories for realistic interplanetary mission development that utilizes both planetary flybys and efficient low-thrust electric propulsion techniques. Similar studies have also been performed by a number of researchers - including a team from JPL\cite{sims2006} as well as a Spanish team\cite{morante}, among several - others. + including a team from JPL\cite{sims2006}, among several others\cite{morante}. - This thesis will attempt to develop an algorithm for the optimization of low-thrust enabled - trajectories for initial feasibility analysis in mission design. The algorithm will utilize - a non-linear programming solver to directly optimize a set of control thrusts for the - user-provided flyby planets, for any provided cost function. A monotonic basin hopping algorithm - (MBH) will then be employed to traverse the search space in an effort to find additional local - optima. This approach differs from the work produced earlier by Englander and the other teams, - but is largely meant to explore the feasibility of such techniques and propose a few - enhancements. The approach defined in this thesis will then be used to investigate an example - mission to Saturn. + This thesis focuses on optimization of low-thrust enabled trajectories that use gravity assists. + The approach uses a non-linear programming solver to directly optimize a set of control thrusts + for the user-provided flyby planets, for any provided cost function. A monotonic basin hopping + algorithm (MBH) is then employed to traverse the search space in an effort to find additional + local optima. This approach differs from the work produced earlier by Englander and the other + teams, but is largely meant to explore the feasibility of such techniques and propose a few + enhancements. The approach defined in this thesis is then used to design low thrust trajectories + with gravity assits from the Earth to Saturn. - This thesis will explore these concepts in a number of different sections. Section - \ref{traj_dyn} will explore the basic dynamical principles of trajectory design, beginning the - with fundamental system dynamics, then exploring interplanetary system dynamics and gravity - flybys, and finally the dynamics that are specific to low-thrust enabled trajectories. Section - \ref{traj_optimization} will then discuss process of optimizing spacecraft trajectories in - general and the tool available for that. Section \ref{algorithm} will cover the implementation - details of the optimization algorithm developed for this paper. Finally, section \ref{results} - will explore the results of some hypothetical missions to Saturn. + This thesis is organized as follows: Section \ref{traj_dyn} will explore the basic dynamical + principles of trajectory design, beginning the with fundamental system dynamics, then exploring + interplanetary system dynamics and gravity flybys, and finally the dynamics that are specific to + low-thrust enabled trajectories. Section \ref{traj_optimization} will then discuss process of + optimizing spacecraft trajectories in general and the tool available for that. Section + \ref{algorithm} will cover the implementation details of the optimization algorithm developed + for this paper. Finally, section \ref{results} will explore the results of some hypothetical + missions to Saturn. diff --git a/LaTeX/results.tex b/LaTeX/results.tex index d2b8ec8..2c7f4fd 100644 --- a/LaTeX/results.tex +++ b/LaTeX/results.tex @@ -1,31 +1,29 @@ -\chapter{Sample Saturn Trajectory Analysis} \label{results} +\chapter{Application: Designing a Trajectory To Saturn} \label{results} - The algorithm described in this thesis is quite flexible in its design and could be used as - a tool for a mission designer on a variety of different mission types. However, to consider - a relatively simple but representative mission design objective, a sample mission to Saturn - was investigated. + To consider a relatively simple but representative mission design objective, a sample mission to + Saturn was investigated. - \section{Mission Constraints} + \section{Mission Scenario} - The sample mission was defined to represent a general case for a near-future low-thrust - trajectory to Saturn. No constraints were placed on the flyby planets, but a number of + The sample mission is defined to represent a general case for a near-future low-thrust + trajectory to Saturn. No constraints are placed on the flyby planets, but a number of constraints were placed on the algorithm to represent a realistic mission scenario. The first choice required by the application is one not necessarily designable to the - initial mission designer (though not necessarily fixed in the design either) and is that - of the spacecraft parameters. The application accepts as input a spacecraft object - containing: the dry mass of the craft, the fuel mass at launch, the number of onboard - thrusters, and the specific impulse, maximum thrust and duty cycle of each thruster. + initial mission designer (though not necessarily fixed in the design either) and is that of + the spacecraft parameters. The application accepts as input a spacecraft object containing: + the dry mass of the spacecraft, the fuel mass at launch, the number of onboard thrusters, + and the specific impulse, maximum thrust and duty cycle of each thruster. - For this mission, the spacecraft was chosen to have a dry mass of only 200 kilograms for - a fuel mass of 3300 kilograms. This was chosen in order to have an overall mass roughly - in the same zone as that of the Cassini spacecraft, which launched with 5712 kilograms - of total mass, with the fuel accounting for 2978 of those kilograms\cite{cassini}. The - dry mass of the craft was chosen to be extremely low in order to allow for a variety of - ''successful`` missions in which the craft didn't run out of fuel. That way, the - delivered dry mass to Saturn could be thought of as a metric of success, without - discounting mission that may have delivered just under whatever more realistic dry mass - one might set, in case those missions are in the vicinity of actually valid missions. + For this mission, the spacecraft was chosen to have a dry mass of only 200 kilograms for a + fuel mass of 3300 kilograms. This was chosen in order to have an overall mass roughly in the + same zone as that of the Cassini spacecraft, which launched with 5712 kilograms of total + mass, with the fuel accounting for 2978 of those kilograms\cite{cassini}. The dry mass of + the spacecraft was chosen to be extremely low in order to allow for a variety of + ''successful`` missions in which the spacecraft didn't run out of fuel. That way, the + delivered dry mass to Saturn could be thought of as a metric of success, without discounting + mission that may have delivered just under whatever more realistic dry mass one might set, + in case those missions are in the vicinity of actually valid missions. The thruster was chosen to have a specific impulse of 3200 seconds, a maximum thrust of 250 millinewtons, and a 100\% duty cycle. This puts the thruster roughly in line with @@ -308,6 +306,6 @@ \centering \includegraphics[width=\textwidth]{fig/c3} \caption{Plot of Delta IV and Atlas V launch vehicle capabilities as they relate to - payload mass \cite{c3capabilities} from a source from 2007} + payload mass \cite{c3capabilities} from Vardaxis, et al, 2007 } \label{c3} \end{figure} diff --git a/LaTeX/thesis.tex b/LaTeX/thesis.tex index ba01f91..df7f82f 100644 --- a/LaTeX/thesis.tex +++ b/LaTeX/thesis.tex @@ -22,14 +22,15 @@ Much work has been performed recently to utilize the increasingly viable technology of low-thrust electric propulsion systems on missions of interplanetary scope. This thesis analyzes - a technique for the initial analysis of feasibility of utilizing a combination of low-thrust - propulsion systems and natural gravity flybys for missions to the outer planets. First, a method - for finding local optima by utilizing an interior-point linesearch algorithm to directly - optimize the entire trajectory as a Non-Linear Programming problem is presented. Then, a - Monotonic Basin Hopping algorithm is utilized to traverse the search space, improve the local - optima determined by the internal optimizer, and determine the global optima. This allows for a - medium-fidelity, fully automated global optimization of the low thrust controls and flyby - parameters for a given mission objective. + a technique for designing trajectories for spacecraft with a low-thrust propulsion system that + also use natural gravity flybys for missions to the outer planets. Often, the goal is to find + feasible solutions that also minimize propellant mass requirements. First, locally optimal + solutions are constructed by using an interior-point linesearch algorithm, along with multiple + shooting techniques for optimization. Then, Monotonic Basin Hopping is utilized to traverse the + search space, improve the local optima determined by the internal optimizer, and determine the + global optima. This approach allows for a medium-fidelity, fully automated global optimization + of the low thrust controls and flyby parameters for a given target destination. As an + application of this method, two sample trajectories to Saturn are analyzed. } diff --git a/LaTeX/trajectory_design.tex b/LaTeX/trajectory_design.tex index 3c416bb..be54b72 100644 --- a/LaTeX/trajectory_design.tex +++ b/LaTeX/trajectory_design.tex @@ -16,24 +16,24 @@ very high-fidelity force models that account for aerodynamic pressure, solar radiation pressure, multi-body effects, and other forces may be too time intensive for a particular application. Initial surveys of the solution space often don't require such - complex models in order to gain valuable insight. + complex models in order to gain valuable preliminary insight. - Therefore, a common approach (and the one utilized in this implementation) is to first - use a lower-fidelity dynamical model that captures only the gravitational force due to - the primary body around which the spacecraft is orbiting. This approach can provide an + A common approach (and the one utilized in this implementation) is to first use a + lower-fidelity dynamical model that captures only the gravitational force due to the + primary body around which the spacecraft is orbiting. This approach can provide an excellent low-to-medium fidelity model that is useful as an underlying model in an algorithm for quickly categorizing a search space for initial mission feasibility explorations. In order to explore the Two Body Problem, we must first examine the full set of assumptions associated with the force model\cite{vallado2001fundamentals}. Firstly, we - are only concerned with the nominative two bodies: the spacecraft and the planetary body - around which it is orbiting. Secondly, both of these bodies are modeled as point masses - with constant mass. This removes the need to account for non-uniform densities and - asymmetry. Finally, for convenience in notation at the end, we'll also assume that the - mass of the spacecraft ($m_2$) is much much smaller than the mass of the planetary body - ($m_1$) and enough so as to be considered negligible. The only force acting on this - system is then the force of gravity that the primary body enacts upon the secondary. + are only concerned with the gravitational influence between the nominative two bodies: + the spacecraft and the planetary body around which it is orbiting. Secondly, both of + these bodies are modeled as point masses with constant mass. This removes the need to + account for non-uniform densities and asymmetry. Finally, for convenience in notation at + the end, we'll also assume that the mass of the spacecraft ($m_2$) is much much smaller + than the mass of the planetary body ($m_1$) and enough so as to be considered + negligible. \begin{figure}[H] \centering @@ -45,7 +45,6 @@ Under these assumptions, the force acting on the body due to the law of universal gravitation is: - \begin{align} F_2 &= - \frac{G m_1 m_2}{r^2} \frac{\vec{r}}{\left| r \right|} \\ F_1 &= \frac{G m_2 m_1}{r^2} \frac{\vec{r}}{\left| r \right|} @@ -53,7 +52,6 @@ And by Newton's second law (force is the product of mass and acceleration), we can derive the following differential equations for $r_1$ and $r_2$: - \begin{align} m_2 \ddot{\vec{r}}_2 &= - \frac{G m_1 m_2}{r^2} \frac{\vec{r}}{\left| r \right|} \\ m_1 \ddot{\vec{r}}_1 &= \frac{G m_2 m_1}{r^2} \frac{\vec{r}}{\left| r \right|} @@ -65,7 +63,6 @@ inertial frame. $G$ is the universal gravitational parameter, $m_1$ is the mass of the planetary body, and $m_2$ is the mass of the spacecraft. From these equations, we can then determine the acceleration of the spacecraft relative to the planet: - \begin{equation} \ddot{\vec{r}} = \ddot{\vec{r}}_2 - \ddot{\vec{r}}_1 = - \frac{G \left( m_1 + m_2 \right)}{r^2} \frac{\vec{r}}{\left| r \right|} @@ -76,27 +73,19 @@ negligible $m_2$ term. We can also introduce, for convenience, a gravitational parameter $\mu$ which represents the gravity constant for the system about the center of motion ($\mu = G (m_1 + m_2) \approx G m_1$). Doing so and simplifying produces: - \begin{equation} \ddot{\vec{r}} = - \frac{\mu}{r^2} \hat{r} \end{equation} - We may also wish to utilize the total orbital energy for a spacecraft within this model. Since the spacecraft is acting only under the gravitational influence of the planet and - no other forces, we can define the total specific mechanical energy as: - + no other forces, we can define the total specific mechanical energy as \cite{vallado2001fundamentals}: - \begin{equation} \label{energy} \xi = \frac{v^2}{2} - \frac{\mu}{r} \end{equation} - - \noindent Where the first term represents the kinetic energy of the spacecraft and the second term represents the gravitational potential energy. - \subsection{Kepler's Laws} - Now that we've fully qualified the forces acting within the Two Body Problem, we can concern ourselves with more practical applications of it as a force model. It should be noted, firstly, that the spacecraft's position and velocity (given an initial position and velocity @@ -105,6 +94,8 @@ one-dimensional equations (one for each component of the three-dimensional space) and three unknowns (the three components of the second derivative of the position). + \subsection{Kepler's Laws} + In the early 1600s, Johannes Kepler produced just such a solution, by taking advantages of what is also known as ``Kepler's Laws'' which are\cite{murray1999solar}: @@ -113,68 +104,61 @@ expanded to any orbit by re-wording as ``all orbital paths follow a conic section (circle, ellipse, parabola, or hyperbola) with a primary mass at one of the foci''. - Specifically the path of the orbit follows the trajectory equation: - + The conic trajectory equation explains this observation and offers a description + of the path as: \begin{equation} r = \frac{\sfrac{h^2}{\mu}}{1 + e \cos(\theta)} \end{equation} - - Where $h$ is the angular momentum of the satellite, $e$ is the + where $h$ is the angular momentum of the satellite, $e$ is the eccentricity of the orbit, and $\theta$ is the true anomaly, or simply - the angular distance the satellite has traversed along the orbit path. + the angular distance the satellite has traversed along the orbit path from + periapsis. \item The area swept out by the imaginary line connecting the primary and secondary bodies increases linearly with respect to time. This implies that the magnitude of the orbital speed is not constant. For the moment, we'll just take this value to be a constant: - \begin{equation}\label{swept} \frac{\Delta t}{T} = \frac{k}{\pi a b} \end{equation} - - Where $k$ is the constant value, $a$ and $b$ are the semi-major and + where $k$ is the constant value, $a$ and $b$ are the semi-major and semi-minor axis of the conic section, and $T$ is the period. In the following section, we'll derive the value for $k$. \item The square of the orbital period is proportional to the cube of the semi-major - axis of the orbit, regardless of eccentricity. Specifically, the relationship is: - + axis of the orbit, regardless of eccentricity. For an elliptical orbit this + observation connects to the following known expression for the orbit period: \begin{equation} T = 2 \pi \sqrt{\frac{a^3}{\mu}} \end{equation} - - Where $T$ is the period and $a$ is the semi-major axis. + where $T$ is the period and $a$ is the semi-major axis. \end{enumerate} \subsection{Kepler's Equation} Kepler was able to produce an equation to represent the angular displacement of an orbiting body around a primary body as a function of time, which we'll derive now for - the elliptical case\cite{vallado2001fundamentals}. Since the total area of an ellipse is - the product of $\pi$, the semi-major axis, and the semi-minor axis ($\pi a b$), we can - relate (by Kepler's second law) the area swept out by an orbit as a function of time, as - we did in Equation~\ref{swept}. This leaves just one unknown variable $k$, which we can - determine through use of the geometric auxiliary circle, which is a circle with radius - equal to the ellipse's semi-major axis and center directly between the two foci, as in - Figure~\ref{aux_circ}. + the elliptical case\cite{vallado2001fundamentals}. Because the total area of an ellipse + is the product of $\pi$, the semi-major axis, and the semi-minor axis ($\pi a b$), we + can relate (by Kepler's second law) the area swept out by an orbit as a function of + time, as we did in Equation~\ref{swept}. This leaves just one unknown variable $k$, + which we can determine through use of the geometric auxiliary circle, which is a circle + with radius equal to the ellipse's semi-major axis and center directly between the two + foci, as in Figure~\ref{aux_circ}. \begin{figure}[H] \centering \includegraphics[width=0.8\textwidth]{fig/kepler} - \caption{Geometric Representation of Auxiliary Circle}\label{aux_circ} + \caption{Geometric representation of auxiliary circle}\label{aux_circ} \end{figure} In order to find the area swept by the spacecraft\cite{vallado2001fundamentals}, $k$, we can take advantage of the fact that that area is the triangle $k_1$ subtracted from the elliptical segment $PCB$: - \begin{equation}\label{areas_eq} k = area(seg_{PCB}) - area(k_1) \end{equation} - - \noindent Where the area of the triangle $k_1$ can be found easily using geometric formulae: - \begin{align} area(k_1) &= \frac{1}{2} \left( ae - a \cos E \right) \left( \frac{b}{a} a \sin E \right) \\ &= \frac{ab}{2} \left(e \sin E - \cos E \sin E \right) @@ -186,7 +170,6 @@ can find the area for the elliptical segment $PCB$ by first finding the circular segment $POB'$, subtracting the triangle $COB'$, then applying the fact that an ellipse is merely a vertical scaling of a circle by the amount $\frac{b}{a}$. - \begin{align} area(PCB) &= \frac{b}{a} \left( area(POB') - area(COB') \right) \\ &= \frac{b}{a} \left( \frac{a^2 E}{2} - \frac{1}{2} \left( a \cos E \right) @@ -197,26 +180,20 @@ By substituting the two areas back into Equation~\ref{areas_eq} we can get the $k$ area swept out by the spacecraft: - \begin{equation} k = \frac{ab}{2} \left( E - e \sin E \right) \end{equation} - Which we can then substitute back into the equation for the swept area as a function of time (Equation~\ref{swept}) for period of time since the spacecraft left periapsis: - \begin{equation} \frac{\Delta t}{T} = \frac{t_2 - t_{peri}}{T} = \frac{E - e \sin E}{2 \pi} \end{equation} - Which is, effectively, Kepler's equation. It is commonly known by a different form: - \begin{equation} M = \sqrt{\frac{\mu}{a^3}} \Delta t = E - e \sin E \end{equation} - - Where we've defined the mean anomaly as $M$ and used the fact that $T = - \sqrt{\frac{a^3}{\mu}}$. This provides us a useful relationship between Eccentric Anomaly + where we've defined the mean anomaly as $M$ and used the fact that $T = + \sqrt{\frac{a^3}{\mu}}$. This provides us a useful relationship between eccentric anomaly ($E$) which can be related to spacecraft position, and time, but we still need a useful algorithm for solving this equation in order to use this equation to propagate a spacecraft. @@ -224,34 +201,25 @@ \subsection{LaGuerre-Conway Algorithm}\label{laguerre} For this thesis, the algorithm used to solve Kepler's equation was the general numeric - root-finding scheme first developed by LaGuerre in the 1800s and first applied to - Kepler's equation by Bruce Conway in 1985\cite{laguerre_conway}. In his paper, Conway - makes a compelling argument for utilizing the less common LaGuerre method over higher - order Newton or Newton-Raphson methods. - - The Newton-Raphson methods, while found to generally have quite impressive convergence - rates (generally successfully solving Kepler's equation correctly within 5 iterations), - were prone to failures in convergence given certain specific initial conditions. - Therefore LaGuerre's algorithm is proposed as an alternative. - - The algorithm can be relatively easily derived by examining the polynomial equation with - $m$ roots: + root-finding scheme first developed by LaGuerre in the 1800s and first applied to Kepler's + equation by Bruce Conway in 1985\cite{laguerre_conway}. In his paper, Conway makes a + compelling argument for utilizing the less common LaGuerre method over higher order Newton + or Newton-Raphson methods. The Newton-Raphson methods, while found to generally have quite + impressive convergence rates (generally successfully solving Kepler's equation correctly + within 5 iterations), were prone to failures in convergence given certain specific initial + conditions. Therefore LaGuerre's algorithm is proposed as an alternative. + The algorithm can be derived by examining the polynomial equation with $m$ roots: \begin{equation} g(x) = (x - x_1) (x - x_2) ... ( x - x_m) \end{equation} - - \noindent We can then generate some useful convenience functions as: - \begin{align} \ln|g(x)| &= \ln|(x - x_1)| + \ln|(x - x_2)| + ... + \ln|( x - x_m)| \\ \frac{d\ln|g(x)|}{dx} &= \frac{1}{x - x_1} + \frac{1}{x - x_2} + ... + \frac{1}{x - x_m} = G_1(x) \end{align} - and - \begin{align} \frac{-d^2\ln|g(x)|}{dx^2} &= \frac{1}{(x - x_1)^2} + \frac{1}{(x - x_2)^2} + ... + \frac{1}{(x - x_m)^2} = G_2(x) @@ -259,42 +227,32 @@ Now we define the targeted root as $x_1$ and make the approximation that all of the other roots are equidistant from the targeted root, which means: - \begin{equation} x - x_i = b, i=2,3,...,m \end{equation} - - \noindent We can then rewrite $G_1$ and $G_2$ as: - \begin{align} G_1 &= \frac{1}{a} + \frac{n-1}{b} \\ G_2 &= \frac{1}{a^2} + \frac{n-1}{b^2} \end{align} - - \noindent Which may be solved for $a$ in terms of $G_1$, $G_2$: - \begin{equation} a = \frac{n}{G_1 \pm \sqrt{(n-1)(nG_2 - G_1^2)}} \end{equation} - - \noindent With corresponding iteration function: - \begin{equation} x_{i+1} = x_i - \frac{n g(x_i)}{g'(x_i) \pm \sqrt{(n-1)^2 f'(x_i)^2 - n (n-1) f(x_i) f''(x_i)}} \end{equation} This iteration scheme can be shown to be globally convergent, regardless of the initial - guess. More relevantly, Conway also showed that the application of this method to - Kepler's equation was shown to converge with similar speed to many of the best common - higher order Newton-Raphson solvers. However, LaGuerre's method was also found to be - incredibly robust, converging to the correct value for every one of Conway's 500,000 - tests. Because of this robustness, it is very useful for propagating spacecraft states. + guess. Conway also showed that the application of this method to Kepler's equation was shown + to converge with similar speed to many of the best common higher order Newton-Raphson + solvers. However, LaGuerre's method was also found to be incredibly robust, converging to + the correct value for every one of Conway's 500,000 tests. Because of this robustness, it is + useful for solving Kepler's equation. - \section{Interplanetary Considerations}\label{interplanetary} + \section{Interplanetary Trajectories}\label{interplanetary} In interplanetary travel, the primary body most responsible for gravitational forces might be a number of different bodies, dependent on the phase of the mission. In fact, at some @@ -346,14 +304,15 @@ This effectively breaks the trajectory into a series of arcs each governed by a distinct Two-Body problem patched together by distinct transition points. These transition points - occur along the spheres of influence of the planets nearest to the spacecraft. + occur along the spheres of influence of the planets nearest to the spacecraft. A + conceptual example of this process, labeled the method of patched conics, appears in + Figure~\ref{patched_conics_fig}. Therefore, we must understand how to convert our spacecraft's state from the Sun frame to the planetary frame as it crosses this boundary. An elliptical orbit about the sun will have enough orbital energy to represent a hyperbolic orbit around the planet. So we first need to determine the velocity of the spacecraft relative to the planet as it crosses the SOI, which we can determine by subtraction \cite{vallado2001fundamentals}: - \begin{equation} \vec{v}_{sc/p} = \vec{v}_{sc/sun} - \vec{v}_{planet/sun} \end{equation} @@ -361,8 +320,8 @@ Since the orbit around the planet is hyperbolic, in order to characterize the hyperbola we must determine the velocity of the spacecraft when it has infinite distance relative to the planet. Since this never occurs, a further approximation is made that the - velocity that the spacecraft has (relative to the planet) as it crosses the SOI can be - modeled as the $\vec{v}_\infty$ of that hyperbolic arc. + velocity of the spacecraft (relative to the planet) as it crosses the SOI can be modeled + as the $\vec{v}_\infty$ of that hyperbolic arc. As an example, we may wish to determine the velocity relative to the planet that the spacecraft has at the periapsis of its hyperbolic trajectory during the flyby. This @@ -371,14 +330,12 @@ around its target planet. For a given incoming hyperbolic $\vec{v}_\infty$, we can first determine the specific mechanical energy of the hyperbola at infinite distance by using Equation~\ref{energy}: - \begin{equation} \xi = \frac{v^2}{2} - \frac{\mu}{r} = \frac{v_\infty^2}{2} \end{equation} We can then leverage the conservation of energy to determine the velocity at a particular point, $r_{ins}$: - \begin{align} \xi_{ins} &= \frac{v_{ins}^2}{2} - \frac{\mu}{r_{ins}} \\ \xi_{ins} &= \xi_\infty = \frac{v_\infty^2}{2} \\ @@ -387,14 +344,13 @@ \subsection{Launch Considerations} - Generally speaking, an interplanetary mission begins with launch. For a satellite of - given size, a certain amount of orbital energy can be imparted to the satellite by the - launch vehicle. In practice, this value, for a particular mission, is actually - determined as a parameter of the mission trajectory to be optimized. The excess velocity - at infinity of the hyperbolic orbit of the spacecraft that leaves the Earth can be used - to derive the launch energy. This is usually qualified as the quantity $C_3$, which is - actually double the kinetic orbital energy with respect to the Sun, or simply the square - of the excess hyperbolic velocity at infinity\cite{wie1998space}. + For a satellite of given size, a certain amount of orbital energy can be imparted to the + satellite by the launch vehicle. In practice, this value, for a particular mission, is + actually determined as a parameter of the mission trajectory to be optimized. The excess + velocity at infinity of the hyperbolic orbit of the spacecraft that leaves the Earth can + be used to derive the launch energy. This is usually qualified as the quantity $C_3$, + which is actually double the kinetic orbital energy with respect to the Sun, or simply + the square of the excess hyperbolic velocity at infinity\cite{wie1998space}. This algorithm will assume that the initial trajectory at the beginning of the mission will be some hyperbolic orbit with velocity enough to leave the Earth. That initial @@ -405,12 +361,12 @@ what the maximum mass any launch provider is capable of imparting that specific $C_3$ to. - A similar approach is taken at the end of the mission. This algorithm doesn't attempt to - exactly match the velocity of the planet. Instead, the excess hyperbolic velocity is - also treated as a parameter that can be minimized by the cost function. If a mission is - to then end in insertion, a portion of the mass budget can then be used for an impulsive - thrust engine, which can provide a final insertion burn. This approach also allows - flexibility for missions that might end in a flyby rather than insertion. + A similar approach is taken at the end of the trajectory. This algorithm doesn't attempt + to exactly match the velocity of the planet. Instead, the excess hyperbolic velocity is + also treated as a parameter that can be minimized by the cost function. If a trajectory + is to then end in insertion, a portion of the mass budget can then be used for an + impulsive thrust engine, which can provide a final insertion burn. This approach also + allows flexibility for missions that might end in a flyby rather than insertion. \subsection{Gravity Assist Maneuvers} @@ -441,7 +397,7 @@ \begin{figure}[H] \centering \includegraphics[width=0.8\textwidth]{fig/flyby} - \caption{Visualization of velocity changes during a gravity assist} + \caption{Velocity changes during a gravity assist} \label{grav_assist_fig} \end{figure} @@ -451,7 +407,7 @@ turning angle of this bend. In doing so, one can effectively achieve a (restricted) free impulsive thrust event. - \subsection{Flyby Periapsis} + \subsection{Flyby Periapsis Altitude} Now that we understand gravity assists, the natural question is then how to leverage them for achieving certain velocity changes\cite{cho2017b}. But first, we must consider @@ -460,7 +416,6 @@ mentioned in the previous section, given an excess hyperbolic velocity entering the planet's sphere of influence ($\vec{v}_{\infty, in}$) and a target excess hyperbolic velocity as the spacecraft leaves the sphere of influence ($\vec{v}_{\infty, out}$): - \begin{equation}\label{turning_angle_eq} \delta = \arccos \left( \frac{\vec{v}_{\infty,in} \cdot \vec{v}_{\infty,out}}{|\vec{v}_{\infty,in}| |\vec{v}_{\infty,out}|} \right) @@ -470,12 +425,10 @@ that we must target in order to achieve the required turning angle. The periapsis of the flyby, however, can provide a useful check on what turning angles are possible for a given flyby, since the periapsis: - \begin{equation}\label{periapsis_eq} r_p = \frac{\mu}{v_\infty^2} \left[ \frac{1}{\sin\left(\frac{\delta}{2}\right)} - 1 \right] \end{equation} - - Cannot be lower than some safe value that accounts for the radius of the planet and + cannot be lower than some safe value that accounts for the radius of the planet and perhaps its atmosphere if applicable. \subsection{Multiple Gravity Assist Techniques} @@ -511,7 +464,9 @@ less than 180 degrees, which we classify as a Type I trajectory, and the second will have a $\Delta \theta$ of greater than 180 degrees, which we call a Type II trajectory. They will also differ in their direction of motion (clockwise or - counter-clockwise about the focus). This can be seen in Figure~\ref{type1type2}. + counter-clockwise about the focus). This can be seen in Figure~\ref{type1type2}, + where both of the Lambert's solutions are presented for sample points in an orbit + around the Sun. \begin{figure}[H] \centering @@ -523,7 +478,6 @@ The iteration used in this thesis will start by first calculating the change in true anomaly, $\Delta \theta$, as well as the cosine of this value, which can be found by: - \begin{align} \cos (\Delta \theta) &= \frac{\vec{r}_1 \cdot \vec{r}_2}{|\vec{r}_1| |\vec{r}_2|} \\ \Delta \theta &= \arctan(y_2/x_2) - \arctan(y_1/x_1) @@ -532,7 +486,6 @@ The direction of motion is then chosen such that counter-clockwise orbits are considered, as travelling in the same direction as the planets is generally more efficient. Next, the variable $A$ is defined: - \begin{equation} A = DM \sqrt{|r_1| |r_2| (1 - \cos(\Delta \theta))} \end{equation} @@ -547,7 +500,6 @@ time of flight matches the expected value to within a provided tolerance. In order to calculate the time of flight at each step, we must first calculate some useful coefficients: - \begin{equation}\label{loop_start} c_2 = \begin{cases} \frac{1-\cos(\sqrt{\psi})}{\psi} \quad &\text{if} \, \psi > 10^{-6} \\ @@ -555,7 +507,6 @@ 1/2 \quad &\text{if} \, 10^{-6} > \psi > -10^{-6} \end{cases} \end{equation} - \begin{equation} c_3 = \begin{cases} \frac{\sqrt{\psi} - \sin \sqrt{\psi}}{\psi^{3/2}} \quad &\text{if} \, \psi > 10^{-6} \\ @@ -563,23 +514,18 @@ 1/6 \quad &\text{if} \, 10^{-6} > \psi > -10^{-6} \end{cases} \end{equation} - - \noindent Where the conditions of this piecewise function represent the elliptical, hyperbolic, and parabolic cases, respectively. Once we have these, we can calculate another variable, $y$: - \begin{equation} y = |r_1| + |r_2| + \frac{A (c_3 \psi - 1)}{\sqrt{c_2}} \end{equation} We can then finally calculate the variable $\chi$, and from that, the time of flight: - \begin{equation} \chi = \sqrt{\frac{y}{c_2}} \end{equation} - \begin{equation} \Delta t = \frac{c_3 \chi^3 + A \sqrt{y}}{\sqrt{c_2}} \end{equation} @@ -592,22 +538,17 @@ The resulting $f$ and $g$ functions (and the derivative of $g$, $\dot{g}$) can then be calculated: - \begin{align} f &= 1 - \frac{y}{|r_1|} \\ g &= A \sqrt{\frac{y}{\mu}} \\ \dot{g} &= 1 - \frac{y}{|r_2|} \end{align} - And from these, we can calculate the velocities of the transfer points as: - \begin{align} \vec{v}_1 &= \frac{\vec{r}_1 - f \vec{r}_2}{g} \\ \vec{v}_2 &= \frac{\dot{g} \vec{r}_2 - \vec{r}_1}{g} \end{align} - - \noindent - Fully constraining the connecting orbit. + Fully describing the connecting path with the specified flight time. \subsubsection{Planetary Ephemeris} @@ -620,8 +561,8 @@ The primary use of SPICE in this thesis, however, was to determine the planetary ephemeris at a known epoch. Using the NAIF0012 and DE430 kernels, ephemeris in the - ecliptic plane J2000 frame (ICRF) could be easily determined for a given epoch, provided as - a decimal Julian Day since the J2000 epoch. + International Celestial Reference Frame could be easily determined for a given + epoch, provided as a decimal Julian Day since the J2000 epoch. \subsubsection{Porkchop Plots} @@ -641,10 +582,9 @@ Using porkchop plots such as the one in Figure~\ref{porkchop}, mission designers can quickly visualize which natural trajectories are possible between planets. Using the fact that incoming and outgoing $v_\infty$ magnitudes must be the same for a flyby, - a savvy mission designer can even begin to work out what combinations of flybys - might be possible for a given timeline, spacecraft state, and planet selection. + a mission designer can even begin to work out what combinations of flybys might be + possible for a given timeline, spacecraft state, and planet selection. - %TODO: Create my own porkchop plot \begin{figure}[H] \centering \includegraphics[width=\textwidth]{fig/porkchop} @@ -652,13 +592,7 @@ \label{porkchop} \end{figure} - However, this is an impulsive thrust-centered approach. The solution to Lambert's - problem assumes a natural trajectory. A natural trajectory is unnecessary when the - trajectory can be modified by a continuous thrust profile along the arc. Therefore, - for the hybrid problem of optimizing both flyby selection and thrust profiles, - porkchop plots are less helpful, and an algorithmic approach is preferred. - - \section{Low Thrust Considerations} \label{low_thrust} + \section{Modeling Low Thrust Control} \label{low_thrust} In this section, we'll discuss the intricacies of continuous low-thrust trajectories in particular. There are many methods for optimizing such profiles and we'll briefly discuss @@ -666,7 +600,7 @@ as introduce the concept of a control law and the notation used in this thesis for modelling low-thrust trajectories more simply. - \subsection{Specific Impulse} + \subsection{Engine Model} The primary advantage of continuous thrust methods over their impulsive counterparts is in their fuel-efficiency in generating changes in velocity. Put specifically, all @@ -678,45 +612,34 @@ This efficiency is often captured in a single variable called specific impulse, often denoted as $I_{sp}$. We can derive the specific impulse by starting with the rocket thrust equation\cite{sutton2016rocket}: - \begin{equation} F = \dot{m} v_e + \Delta p A_e \end{equation} - - \noindent Where $F$ is the thrust imparted, $\dot{m}$ is the fuel mass rate, $v_e$ is the exhaust velocity of the fuel, $\Delta p$ is the change in pressure across the exhaust opening, and $A_e$ is the area of the exhaust opening. We can then define a new variable $v_{eq}$, such that the thrust equation becomes: - \begin{align} v_{eq} &= v_e + \frac{\Delta p A_e}{\dot{m}} \\ F &= \dot{m} v_{eq} \label{isp_1} \end{align} - - \noindent And we can then take the integral of this value with respect to time to find the total impulse, dividing by the weight of the fuel to derive the specific impulse: - \begin{align} I &= \int F dt = \int \dot{m} v_{eq} dt = m_e v_{eq} \\ I_{sp} &= \frac{I}{m_e g_0} = \frac{m_e v_{eq}}{m_e g_0} = \frac{v_{eq}}{g_0} \end{align} - Plugging Equation~\ref{isp_1} into the previous equation we can derive the following formula for $I_{sp}$: - \begin{equation} \label{isp_real} I_{sp} = \frac{F}{\dot{m} g_0} \end{equation} - - \noindent Which is generally taken to be a value with units of seconds and effectively represents the efficiency with which a thruster converts mass to thrust. \subsection{Sims-Flanagan Transcription} - This thesis chose to use a model well suited for modeling low-thrust paths: the + In this thesis the following approach is used for modeling low-thrust paths: the Sims-Flanagan transcription (SFT)\cite{sims1999preliminary}. The SFT allows for flexibility in the trade-off between fidelity and performance, which makes it very useful for this sort of preliminary analysis. @@ -752,7 +675,7 @@ continuous low-thrust trajectory within the Two-Body Problem, with only linearly-increasing computation time\cite{sims1999preliminary}. - \subsection{Low-Thrust Control Laws} + \subsection{Low-Thrust Control Vector Description} In determining a low-thrust arc, a number of variables must be accounted for and, ideally, optimized. Generally speaking, this means that a control law must be determined @@ -765,24 +688,23 @@ The methods for determining this direction varies greatly depending on the particular control law chosen for that mission. Often, this process involves first determining a useful frame to think about the kinematics of the spacecraft. In this case, we'll use a - frame often used in these low-thrust control laws: the spacecraft $\hat{R} \hat{\theta} - \hat{H}$ frame. In this frame, the $\hat{R}$ direction is the radial direction from the - center of the primary to the center of the spacecraft. The $\hat{H}$ hat is - perpendicular to this, in the direction of orbital momentum (out-of-plane) and the - $\hat{\theta}$ direction completes the right-handed orthonormal frame. + frame often used in these low-thrust control laws: the spacecraft-centered $\hat{R} + \hat{\theta} \hat{H}$ frame. In this frame, the $\hat{R}$ direction is the radial + direction from the center of the primary to the center of the spacecraft. The $\hat{H}$ + hat is perpendicular to this, in the direction of orbital momentum (out-of-plane) and + the $\hat{\theta}$ direction completes the right-handed orthonormal triad. This frame is useful because, for a given orbit, especially a nearly circular one, the $\hat{\theta}$ direction is nearly aligned with the velocity direction for that orbit at that moment. This allows us to define a set of two angles, which we'll call $\alpha$ and $\beta$, to represent the in and out of plane pointing direction of the thrusters. This - convention is useful because a $(0,0)$ set represents a thrust force more or less - directly in line with the direction of the velocity, a commonly useful thrusting - direction for most effectively increasing (or decreasing if negative) the angular - momentum and orbital energy of the trajectory. + convention is useful because, in a near-circular path, a $(0,0)$ set represents a thrust + force more or less directly in line with the direction of the velocity, a commonly + useful thrusting direction for most effectively increasing (or decreasing if negative) + the angular momentum and orbital energy of the trajectory. Using these conventions, we can then redefine our thrust vector in terms of the $\alpha$ and $\beta$ angles in the chosen frame: - \begin{align} F_r &= F \cos(\beta) \sin (\alpha) \\ F_\theta &= F \cos(\beta) \cos (\alpha) \\ @@ -791,12 +713,12 @@ \subsubsection{Thrust Magnitude} - However, there is actually another variable that can be varied by the majority of - electric thrusters. Either by controlling the input power of the thruster or the duty - cycle, the thrust magnitude can also be varied, limited by the maximum thrust available - to the thruster. Not all control laws allow for this fine-tuned control of the thruster. + There is another variable that can be varied by the majority of electric thrusters. + Either by controlling the input power of the thruster or the duty cycle, the thrust + magnitude can also be varied, limited by the maximum thrust available to the thruster. + Not all control laws allow for this fine-tuned control of the thruster. - The algorithm used in this thesis does vary the magnitude of the thrust control. In + The approach used in this thesis does vary the magnitude of the thrust control. In certain cases it actually can be useful to have some fine-tuned control over the magnitude of the thrust. Since the optimization in this algorithm is automatic, it is relatively straightforward to consider the control thrust as a 3-dimensional vector in diff --git a/LaTeX/trajectory_optimization.tex b/LaTeX/trajectory_optimization.tex index a3e403f..120266a 100644 --- a/LaTeX/trajectory_optimization.tex +++ b/LaTeX/trajectory_optimization.tex @@ -6,10 +6,11 @@ highly non-linear, unpredictable systems such as this. The field that developed to approach this problem is known as Non-Linear Programming (NLP) Optimization. - A Non-Linear Programming Problem is defined by an attempt to optimize a function + A Non-Linear Programming Problem involves finding a solution that optimizes a function $f(\vec{x})$, subject to constraints $\vec{g}(\vec{x}) \le 0$ and $\vec{h}(\vec{x}) = 0$ where $n$ is a positive integer, $x$ is any subset of $R^n$, $g$ and $h$ can be vector - valued functions of any size, and at least one of $f$, $g$, and $h$ must be non-linear. + valued functions of any size, and at least one of $f$, $\vec{g}$, and $\vec{h}$ must be + non-linear. There are, however, two categories of approaches to solving an NLP. The first category, indirect methods, involve declaring a set of necessary and/or sufficient conditions for @@ -20,10 +21,10 @@ The other category is the direct methods. In a direct optimization problem, the cost function itself provides a value that an iterative numerical optimizer can measure - itself against. The optimal solution is then found by varying the inputs $\vec{x}$ until the - cost function is reduced to a minimum value, often determined by its derivative - jacobian. A number of tools have been developed to optimize NLPs via this direct method - in the general case. + itself against. The optimal solution is then found by varying the inputs $\vec{x}$ until + the cost function is reduced to a minimum value, often determined by its derivative + jacobian. A number of tools have been developed to formulate NLPs for optimization via + this direct method in the general case. Both of these methods have been applied to the problem of low-thrust interplanetary trajectory optimization \cite{Casalino2007IndirectOM} to find local optima over @@ -40,7 +41,7 @@ Therefore, a direct optimization method was leveraged by transcribing the problem into an NLP and using IPOPT to find the local minima. - \subsubsection{Non-Linear Solvers} + \subsection{Non-Linear Solvers} One of the most common packages for the optimization of NLP problems is SNOPT\cite{gill2005snopt}, which is a proprietary package written primarily using a @@ -63,7 +64,7 @@ libraries that port these are quite modular in the sense that multiple algorithms can be tested without changing much source code. - \subsubsection{Interior Point Linesearch Method} + \subsection{Interior Point Linesearch Method} As mentioned above, this project utilized IPOPT which leveraged an Interior Point Linesearch method. A linesearch algorithm is one which attempts to find the optimum @@ -74,7 +75,7 @@ step the initial guess, now labeled $x_{k+1}$ after the addition of the ``step'' vector and iterates this process until predefined termination conditions are met. - \subsubsection{Shooting Schemes for Solving a Two-Point Boundary Value Problem} + \subsection{Shooting Schemes for Solving a Two-Point Boundary Value Problem} One straightforward approach to trajectory corrections is a single shooting algorithm, which propagates a state, given some control variables forward in time to @@ -82,31 +83,22 @@ iterative process, using the correction scheme, until the target state and the propagated state matches. - As an example, we can consider the Two-Point Boundary Value Problem (TPBVP) defined - by: - + As an example, we can consider the one-dimensional Two-Point Boundary Value Problem + (TPBVP) defined by: \begin{equation} y''(t) = f(t, y(t), y'(t)), y(t_0) = y_0, y(t_f) = y_f \end{equation} - - \noindent We can then redefine the problem as an initial-value problem: - \begin{equation} - y''(t) = f(t, y(t), y'(t)), y(t_0) = y_0, y'(t_0) = x + y''(t) = f(t, y(t), y'(t)), y(t_0) = y_0, y'(t_0) = \dot{y}_0 \end{equation} - - \noindent With $y(t,x)$ as a solution to that problem. Furthermore, if $y(t_f, x) = y_f$, then the solution to the initial-value problem is also the solution to the TPBVP as well. Therefore, we can use a root-finding algorithm, such as the bisection method, Newton's Method, or even Laguerre's method, to find the roots of: - \begin{equation} F(x) = y(t_f, x) - y_f \end{equation} - - \noindent To find the solution to the IVP at $x_0$, $y(t_f, x_0)$ which also provides a solution to the TPBVP. This technique for solving a Two-Point Boundary Value Problem can be visualized in Figure~\ref{single_shoot_fig}. @@ -114,7 +106,7 @@ \begin{figure}[H] \centering \includegraphics[width=\textwidth]{fig/single_shoot} - \caption{Visualization of a single shooting technique over a trajectory arc} + \caption{Single shooting over a trajectory arc} \label{single_shoot_fig} \end{figure} @@ -133,8 +125,8 @@ each of these points we can then define a separate control, which may include the states themselves. The end state of each arc and the beginning state of the next must then be equal for a valid arc (with the exception of velocity discontinuities - if allowed for maneuvers at that point), as well as the final state matching the - target final state. + if allowed for maneuvers or gravity assists at that point), as well as the final + state matching the target final state. \begin{figure}[H] \centering @@ -144,7 +136,7 @@ \end{figure} In this example, it can be seen that there are now more constraints (places where - the states need to match up, creating an $x_{error}$ term) as well as control + the states need to match up, creating an $\vec{x}_{error}$ term) as well as control variables (the $\Delta V$ terms in the figure). This technique actually lends itself very well to low-thrust arcs and, in fact, Sims-Flanagan Transcribed low-thrust arcs in particular, because there actually are control thrusts to be optimized at a