Spacecraft trajectory optimization problems belong to a general class of problems called hybrid optimal control problems [114–116]. The defining characteristic of an HOCP is the inclusion of both discrete and continuous variables in the parameterization of the problem. Specifically, the values of the discrete variables define the configuration of the sub-problem described by the continuous variables. When applied to an MGA problem, this means that an HOCP seeks to simultaneously solve both the pathfinding as well as the pathsolving problems that were discussed in section 1.2.1. Consider the problem of transferring a spacecraft from the Earth to another body in the solar system. A typical discrete variable set that characterizes this problem is the flyby sequence (including the option to not use gravity assists) that the spacecraft will use to complete the transfer to the target body. These discrete variables describe a particular transfer structure that is then locally optimized by varying continuous variables such as the launch epoch, the engine throttle time history and the TOF of each flyby leg. The first description of an HOC solution framework for aerospace mission/trajectory design problems can be found in the pioneering by Ross and D’Souza [117]. The first application of the HOC method to the interplanetary MGA problem was by Conway et al. [93], followed shortly by an extension to multi-objective HOC by Vavrina and Howell [51]. Recently, the incorporation of additional variables such as the spacecraft’s solar array size, thruster type, launch vehicle employed and other systems-level discrete optimization variables have also been successfully incorporated into the HOCP structure [100, 101].
1.3.1 Hybrid Optimal Control Solver Structure
One method for solving HOCPs, such as the notional mission described above, is to use a nested optimization loop structure. This method utilizes an outer-loop to select values for the discrete variables. These in turn define a particular problem space in which the real-valued variables may be varied by the inner-loop. The purpose of the inner-loop solver is to evaluate a particular choice of outer-loop variable values based on a cost metric (e.g. the amount of spacecraft mass that was delivered to the target) in order to compare it against other discrete variable configurations. Examples of typical outer-loop solvers include genetic algorithms GAs, BB or even total enumeration (grid searches) for small problems. The inner-loop problem has been solved using a variety of methods, several of which are discussed later in sections 3.1, 3.2, 4.2 and 5.1. The most popular solution method is to discretize the problem and convert it into a nonlinear programming problem [118]. The specific nested loop solver implementation referenced in this work is the EMTG [46, 48],
however, this nested loop solver structure has enjoyed widespread success, and several research efforts have exploited it for spacecraft trajectory design [110, 111, 119].
1.3.2 Outer-Loop Solver
In the context of the MGA problem, one of the primary roles of the outer-loop solver is to perform flyby sequence pathfinding. For example, if the specified mission is to transfer a spacecraft from Earth to Jupiter, on a given iteration of the outer-loop one or more sequences will be constructed from a set of possible gravity-assist target bodies (e.g. {Venus-Earth-Mars, Earth-Venus-Venus-Earth}) for the inner-loop to evaluate.
The inner-loop will then solve the continuous optimization problem, and compute the resulting cost function value. This cost function information is, in turn, used by the outer-loop to assess the efficacy of any one flyby route.
Several algorithms have been shown to be effective at solving the HOC outer-loop problem. Branch and bound is perhaps the most straight-forward to both conceptualize and implement and forms the basis for most exhaustive MGA solution techniques. The BB method casts the set of all possible event sequences as a tree structure. For a multi-stage optimization problem, the total enumeration of the states that the problem variables may occupy are represented as nodes in the tree. Possible subsequent decisions are represented as child nodes that branch from the current parent node. Since traversing the tree of a problem with high combinatorial complexity, such as the MGA problem, rapidly becomes computationally intractable, partial solutions are pruned away at each level of the tree according to some bounding law. For example, the beam search tree traversal algorithm only branches a specified number or percentage of child nodes at each level of the tree according to the cost function value of the partial solutions.
Since typical cost functions for the MGA problem do not obey the law of superposition, that is each flyby-to-flyby phase of the problem is not independent of previous and subsequent phases, BB will not guarantee the identification of the globally optimal solution, and also means that the cost function value of partial solutions cannot be meaningfully compared to known full-length solutions (solutions that have traversed the entire depth of the tree) or even other partial-length solutions. Nevertheless, BB has been employed to explore the solutions spaces of challenging mission design problems such as the GTOC 6 problem, and is often the only feasible choice when the complexity of the problem becomes too great to effectively employ other outer-loop algorithms such as GAs. This technique for solving the MGA sequencing problem used in chapter 6 where it is applied to the planetary satellite tour problem.
Genetic algorithms have become a popular choice for solving the interplanetary MGA problem as first
demonstrated by Gage et al. [83]. A GA was first incorporated into an MGA HOC solution framework by Conway et al. [93]. Genetic algorithms operate by first generating a population of random decision vectors, which are referred to as chromosomes. At each generation, the fitness of each of these chromosomes is determined by evaluating the problem’s cost function. After evaluation, a selection operator is applied that selects the members of the population that will become the parents for the next generation. A crossover operator is then applied that creates the next population from the selected parents, replacing a certain portion of the overall population. Finally, with some small propability, random mutation occurs amongst the population. In order to guard the highest-valued members of the population from being discarded due to random mutation, a certain number of elite individuals are guaranteed to be preserved and carried over to the next generation. The genetic algorithm will proceed for a certain number of pre-specified generations, or until some user-specifed convergence criterion is satisfied.
The most useful class of genetic algorithm, as far as preliminary mission design is concerned, are the so-called multi-objective genetic algorithms. These routines are capable of mapping Pareto surfaces in a problem’s design space as opposed to optimal point-solutions, and are of more use to mission design engineers who must react to constantly evolving spacecraft hardware designs and requirements. Recent advances by Vavrina et al. [100] and Englander et al. [101] employ the NSGA-II [102] and represent the current state-of-the-art in system-level spacecraft trajectory optimization research.
1.3.3 Inner-Loop Solver
The objective of the HOC inner-loop solver is to compute the locally optimal trajectory solution for a given configuration of the outer-loop optimization problem, i.e. it performs trajectory pathsolving. The specific problem formulation varies depending on the level of fidelity required as well as whether a chemical or electric propulsion system is being employed. Several trajectory models are considered in this work.
As with the outer-loop, there are also many methods available for solving the inner-loop problem. If the trajectory problem is formulated as a Hamiltonian boundary value problem, then primer vector techniques can be applied to obtain solutions that satisfy the necessary conditions of optimality. Metaheuristic tech-niques such as PSO and DE are attractive in that they are relatively straightforward to implement and do not require gradient information to perform their local search. The inner-loop solution technique that is the topic of discussion for much of this work is nonlinear programming, a gradient-based technique that will be introduced in section 2.1. Regardless of the strategy that is employed to solve the trajectory optimization problem, it is essential to the successful operation of the overall HOC solver that the inner-loop operate in
a robust fashion.
1.3.4 Accurate and Efficient Execution of the Inner-Loop
The goal of the inner-loop solver in an HOC framework is to return the cost function value associated with a particular selection of outer-loop decision variables. If the inner-loop fails to return as solution (assuming one exists), or even if it returns a solution of poor quality (i.e. not locally optimal), then the effectiveness of the outer-loop will be compromised. For example, if a genetic algorithm is used as the outer-loop solver, and the inner-loop returns an inaccurate cost function value for a particular flyby sequence, then that sequence will be scored lower than it otherwise would have been. As a result of this, during the selection phase, that member of the population could be removed from the population pool prematurely, thus spoiling the overall integrity of the genetic pool.
In addition to robustness concerns, the second most important characteristic of a high-quality inner-loop solver is fast execution. The vast majority of the execution time associated with an HOC solver is attributed to the inner-loop. Whereas the outer-loop typically only needs to perform a few basic initialization operations during each of its iterations, the costliest typically being sorting routines, the inner-loop must evaluate the trajectory cost function. This routine typically includes the evaluation of the equations of motion, calls to hardware modeling routines, ephemerides and even costly numerical integration for higher fidelity models.
In order to ensure that the local decision space is thoroughly explored, and the local minimum identified, many solution paradigms evaluate the inner-loop multiple times for each iteration of the outer-loop. For all of these reasons, inner-loop execution usually dominates the runtime of the overall optimization framework, and any improvements to the efficiency of its operation are important for reducing the time-to-solution.
One of the most significant applications of the contributions described in this work to a practical prob-lem was the optimization of the Phase-A trajectory for the Lucy mission. Flight dynamics analysis tasks supporting this mission heavily rely on the inner loop of the optimization framework that is a focal point of this work. Lucy is a NASA Discovery program project that will perform flybys of one main belt asteroid and six Jupiter Trojan small bodies. It is scheduled to launch in October of 2021. It was selected in January 2017 to proceed to Phase-B study. The original design was performed by Brian Sutter and Chelsea Welch from Lockheed Martin, however, the final optimization was performed by Jacob Englander using EMTG.
The optimization of the trajectory decreased the propellant cost function, reduced the small body encounter relative velocities by a significant amount and made the Polymele and Leucus encounters possible. The Lucy mission itinerary is summarized in Table 1.1 and the trajectory is shown in Fig. 1.4 and 1.5 [120].
Table 1.1: Lucy small body targets and nominal encounter dates.
Event/Encounter Encounter Date
Launch October 16 2021
DSM 1 April 19 2022
EGA 1 October 16 2022
DSM 2 February 2 2024
EGA 2 December 13 2024
52246 Donaldjohanson April 20 2025
DSM 3 April 3 2027
3548 Eurybates August 12 2027 15094 Polymele September 15 2027
DSM 4 September 29 2027
11351 Leucus April 18 2028
DSM 5 July 23 2028
21900 Orus November 11 2028
EGA 3 December 26 2030
617 Patroclus + Menoetius March 2 2033
Figure 1.4: Lucy optimized trajectory showing the inclined orbits of the small body targets.
Figure 1.5: Lucy optimized trajectory. Red dots are bipropellant maneuvers, purple diamonds are small body intercepts and blue dots are Earth flybys.
The analytic derivative methods described in chapters 2 and 3, which significantly increase the robustness of the local optimizer, directly contributed to the improved Lucy solution. The development of the high-thrust optimization transcription described in chapter 3, as well as its analyic gradients, have also been directly applied to Lucy missed-maneuver analysis activities by the author and other members of the Lucy flight dynamics team.
1.3.5 Trajectory Design Heritage of Hybrid Optimal Control
Trajectory design and optimization using HOC is a relatively new practice. Despite this, HOC has al-ready accrued design heritage on several mission proposals as well as a phase-B flight project. This was accomplished with the EMTG mission design software, which represents the cumulative efforts of several researchers including Dr. Jacob Englander, Mr. Matthew Vavrina, Dr. Alexander Ghosh, Mr. Ryne Beeson as well as the author.
To the author’s knowledge, the first example of a HOC automaton being successfully applied to a mission
proposal effort is Asteroid Robotic Redirect Mission (ARRM) [100]. This effort was particularly noteworthy as the use of the HOC method was instrumental in ARRM Option B being selected in favor of Option A before the entire project was ultimately canceled by the United States congress. Since the work on ARRM, EMTG has been used on several other proposals, including several in the New Frontiers 4 round of selection.
The most successful of these was the design of the trajectory for the CAESAR mission, which was selected to advance to Phase A in December 2017.