Automated Design of Multiphase Space MissionsUsing Hybrid Optimal Control

Christian M. Chilan∗ and Bruce A. Conway†

University of Illinois, Urbana, Illinois 61801

DOI: 10.2514/1.58766

A modern space mission is assembled from multiple phases or events such as impulsive maneuvers, coast arcs,

thrust arcs, and planetary flybys. Traditionally, amission plannerwould resort to intuition and experience to develop

a sequence of events for the multiphase mission and to find the space trajectory that minimizes propellant use by

solving the associated continuous optimal control problem.This strategy, however, will most likely yield a suboptimal

solution, as the problem is sophisticated for several reasons. For example, the number of events in the optimalmission

structure is not known apriori, and the system equations ofmotion change depending onwhat event is current. In this

work a framework for the automated design of multiphase space missions is presented using hybrid optimal control.

The method developed uses two nested loops: an outer-loop that handles the discrete dynamics and finds the optimal

mission structure in terms of the categorical variables, and an inner-loop that performs the optimization of the

corresponding continuous-time dynamical system and obtains the required control history. Genetic algorithms and

direct transcription with nonlinear programming are introduced as methods of solution for the outer-loop and

inner-loop problems, respectively.Automation of the inner-loop, continuous optimal control problem solver required

two new technologies. The first is a method for the automated construction of the nonlinear programming problems

resulting from the use of a transcriptionmethod for systems with different structures, including different numbers of

categorical events. The method assembles modules, consisting of parameters and constraints appropriate to each

event, sequentially according to the given mission structure. The other new technology is for a robust initial guess

generator required by the inner-loop nonlinear programming problem solver. The method, based on a real genetic

algorithm, approximates optimal control histories by incorporating boundary conditions explicitly using a

conditional penalty function. The solution of representative multiphase mission design problems shows the

effectiveness of the methods developed.

Nomenclature

A = adjacency matrix, or linear constraint matrixbL = vector of lower bounds for parameters and constraintsbU = vector of upper bounds for parameters and constraintsC = nonlinear constraint vectorc = propulsion engine exhaust velocity, or coast eventcc = exhaust velocity of the chemical rocket enginece = exhaust velocity of the low-thrust electric engineD = set of binary matrices with column-sum equal to oned = Euclidean norm of the constraint violationsF = discrete space of feasible event sequencesf = system equation vectorh = equality constrainti = impulsive eventJ = total cost of the missionK = infeasibility constantl = Lambert’s rendezvous eventm = spacecraft massm− = spacecraft mass immediately before a particular eventm� = spacecraft mass immediately after a particular eventNQ = cardinality of QNT = number of thrusting events in a particular qNT;max = maximum number of thrusting events for all qNs = number of switches in a particular q

Ns1 = Ns � 1, i.e., number of events in a particular qNs;max = maximum number of events for all qP = parameter vectorQ = categorical space of eventsq = event sequenceq = eventqj = event in the j� 1 place in the sequencer = spacecraft radial coordinateS = switching sets = boundary-free thrust arc eventt = boundary-specified thrust arc event, or timete = duration of a particular thrust arc eventtj = initial time for the event qj

ts = switching time between eventstol = feasibility tolerance for the conditional penalty

methodUD = set of discrete controllers that satisfy discrete con-

straintsu = control variable vectorvr = radial component of spacecraft velocityvθ = tangential component of spacecraft velocityx = state variable vectorx = state variableα = spacecraft thrust accelerationα0 = spacecraft thrust acceleration at the beginning of the

missionαf = spacecraft thrust acceleration at the end of the

missionα− = spacecraft thrust acceleration immediately before a

particular eventα� = spacecraft thrust acceleration immediately after a

particular eventβ = spacecraft thrust-pointing angleΔ = discrete controllerΔV = change in velocityϕ = cost functionθ = spacecraft angular coordinate

Received 25 April 2012; revision received 4 December 2012; accepted forpublication 12 December 2012; published online 20 June 2013. Copyright ©2012 by Christian M. Chilan and Bruce A. Conway. Published by theAmerican Institute of Aeronautics and Astronautics, Inc., with permission.Copies of this paper may be made for personal or internal use, on conditionthat the copier pay the $10.00 per-copy fee to the Copyright Clearance Center,Inc., 222 RosewoodDrive, Danvers,MA 01923; include the code 1533-3884/13 and $10.00 in correspondence with the CCC.

*Postdoctoral Research Associate, Department of Aerospace Engineering,104 South Wright St. MC-236. Member AIAA.

†Professor, Department of Aerospace Engineering, 104 South Wright St.MC-236. Associate Fellow AIAA.

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I. Introduction

M ANY interesting problems in numerical optimization arehybrid optimal control (HOC) problems. HOC problems

include both continuous-valued variables and categorical variables inthe problem formulation. For the types of problems envisioned here,the categorical variables will specify the structure or sequence ofevents that qualitatively describes a space trajectory or mission. Forexample, for an interplanetary spacecraft trajectory, a mission couldbe described by the sequence of categorical variables: Earthdeparture, low-thrust heliocentric arc, Mercury arrival. An equallyvalid and perhaps lower-cost sequence might be: Earth departure,low-thrust heliocentric arc, coast arc, low-thrust heliocentric arc,Mercury arrival. The identities of themission events and their order inthe sequence are the discrete or categorical variables of the HOCproblem. The time histories of the spacecraft position and velocityand of the spacecraft control (thrust-pointing angle) are thecontinuous-time variables of the HOC problem. The cost associatedwith a particular event sequence is found from the solution of thecorresponding continuous optimal control problem.Finding the sequence of events with minimum cost requires

searching a discrete space composed of the sequences resulting fromthe permutation of the mission events. Even by considering a catalogwith only a few events, there may be several thousand possiblemission designs. A simple approach to find the best mission structureis to perform a total enumeration of all the possible sequences.Although intuition on the part of the mission planner can reduce thesize of the search space, total enumeration would still consume asignificant amount of resources. A solver is needed for this type ofproblem, i.e., finding the best mission structure without performingtotal enumeration, andwith little or no intuition supplied (or required)on the part of the mission planner.Oneway to reduce the discrete search space is to apply pre-pruning

with respect to some criteria; in this way, the remaining missionsequences are the only ones considered of interest [1]. Vasile et al. [1]also present an approach to automatedmission planning, inwhich themission is composed of elementary blocks that represent events suchas Lambert maneuvers and flybys. Englander et al. [2] and Gad andAbdelkhalik [3] presented methods that find the flyby sequence andthe optimal trajectory using genetic algorithms (GA). In their work,the governing dynamics do not change during the trajectory legsbetween flybys, i.e., such legs can be modeled with a single phase.Ceriotti and Vasile [4] propose the use of a modified ant colonyoptimization (ACO) algorithm for the automated mission planningusing flybys and impulsive maneuvers.In this work, the problem of interest is the automated solution of

HOCproblems that consist of different events or phases. The problemconsists of finding the sequence composed of an unspecified numberof events (coast arcs, thrust arcs, and impulses) that minimizespropellant consumption for a space trajectory with given initial andboundary conditions, conducted in free or fixed final time.A recent approach to solving HOC problems is to use two nested

loops: an outer-loop that handles the discrete dynamics and findsa solution sequence in terms of the categorical variables, and aninner-loop that performs the optimization of the continuous-timedynamical system and obtains the required control history. Ross andD’Souza [5] present a general framework for the description of HOCproblems and the corresponding mathematical formalism.The nested-loop approach is qualitatively similar to the solution of

a discrete optimization problem using GA [6], where the objective isusually some known, analytic function of the discrete parameters ofthe GA that can be evaluated directly. In the HOC context, however,the cost cannot be found until an optimal control problem (withimportant event parameters supplied by the GA decision vector) issolved. This is much more difficult and time-consuming, e.g.,analytic functions always return a cost value, but the routine used tosolve the inner-loop problem inherits the difficulties associated withsolving optimal control problems.The inner-loop solver finds the optimal trajectories for the

continuous-time dynamical systems associated with the eventsequences generated by the outer-loop. Note that distinct event

sequences can constitute quite different optimal control problems.For instance, the equations of motion during a coast arc event are notthe same as the equations of motion during a continuous thrust arcevent. Similarly, the control parameters change in type and number;an impulse is defined bymagnitude and direction parameters, while athrust arc needs a flight time parameter and a continuous thrust-pointing angle (or angles). Because the outer-loop requests theevaluation of different sequences during the search, this workpresents a scheme that allows the inner-loop to solve trajectoryoptimization problems with variable structures without a prioriknowledge or experience. This is a challenging aspect of the problembecause the determination of just a single optimal space trajectorywith (possibly) multiple coast arcs and thrust arcs has not beenconsidered a simple or straightforward problem.The optimal trajectory for a particular mission sequence is found

using direct transcription with nonlinear programming (NLP) [7]. Asa gradient-based optimization method, NLP requires an initial guess,which after several numerical iterations by the NLP problem solvershould converge to a solution, i.e., satisfy specified tolerances onfeasibility and optimality. Traditionally, the mission planner has hadto resort to intuition and experience to generate such initial guesses.For example, solutions to Lambert’s problem [8] can provide goodinitial guesses for impulsive trajectories. If the solution process(outer-loop� inner-loop) is to be automated, methods that generatehigh-quality guesses are needed to yield a robust behavior of theNLPproblem solver. A new method was developed to generate theapproximate low-thrust trajectories to be used as the initial guessesfor the NLP solvers. The method, based on GA, directly approxi-mates optimal control histories by incorporating boundary conditionsexplicitly using a “conditional penalty” (CP) function [9].

II. Hybrid Optimal Control Problem

The solution to the HOC problem consists of finding values for aset of discrete variables that minimize a cost function resulting fromthe trajectory optimization of a continuous-time dynamical system.One example of a HOC problem is the motorized traveling salesman[10]. The salesman drives a car in which the motion dynamics aredescribed by a system of differential equations. He must visit anumber of cities whose locations are specified in the problem.The objective is to find the ordered sequence of cities and thecorresponding optimal trajectory that minimizes the travel time ofthe salesman. The identities of the cities and their order of visitationare the discrete or categorical variables of theHOCproblem.The timehistories of the car position and velocity and of the car controls(acceleration and turn rate) are the continuous-time variables of theHOC problem.This section presents the general formulation of a HOC problem

using the formalism introduced by Ross and D’Souza [5].

A. Discrete Dynamics

The discrete events represent the qualitative states or phases of aHOC problem. They can be grouped together into a categorical spaceQ of finite cardinality NQ ∈ N. The task of the mission planner is toassemble a sequence q of events q ∈ Q that fulfills the missionobjectives and minimizes a cost function defined in the inner-loop.The construction of the sequenceq incorporates constraints that formthe finite dynamics of the problem. Such constraints can be modeledwith a finite state automaton in the formof a directed graph or digraphas shown in Fig. 1. The nodes and the directed edges constitute theevents and the allowed transitions between them, respectively.From the digraph, the categorical state space is

Q � fqa; qb; qcg

with cardinality NQ � 3. The subscript notation in qi identifies thestate in the categorical space. To encode the information contained inthe digraph, let a switching set be defined as the transition betweenstates at time ts ∈ R

S�q; q 0� � f�x;u;x 0;u 0; ts�g (1)

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where x ∈ RNx and u ∈ RNu are the continuous state and controlvectors, respectively. If a transition is allowed from the event q to theevent q 0, then the switching set S�q; q 0� ≠ ∅; otherwise, theswitching set is empty. All the possible switching sets can be encodedin the NQ × NQ adjacency matrix A, where

Aij ��1 if S�qi; qj� ≠ ∅0 if S�qi; qj� � ∅ (2)

Feasible event sequences can be obtained from the digraph. Oneexample of such a sequence is

q � �qa; qb; qa; qc; qa�

where the number of switches Ns � 4. The digraph also showstransitions that are not allowed. For example, the automaton cannotmove from the state qc to qb because there is not a directed edge inthat direction. Thus, the switching setS�qc; qb� is empty. The numberof events in a sequence, given by Ns1 � Ns � 1, is not fixed. Inpractice, however, an upper boundNs;max on the length of the feasiblesequences is given to constrain the problem to finite sequences and toreduce the computational complexity of the problem.Let �Q� be a 1 × NQ matrix whose components are the elements

qi ∈ Q. Let also the operation�be defined over theCartesian product�Q� × f0; 1g in the following way:

q�0 � ∅ and q�1 � q; ∀ q ∈ Q

Also, let D be defined by

Dn×m ≡nΔ ∈ f0; 1gn×m∶

Pni�1 Δij ∈ f0; 1g ∀ j � 1; : : : ; m

o(3)

where the column-sum property implies that the columns arecomposed mostly of zeros with only a single component equal to 1.The framework proposed originally does not consider the casewherean event must not appear more than once in the solution sequence [5].This constraint can be applied by requiring that D have fullcolumn rank.Finally, by including the following definitions:

q�∅ � q � ∅� q; ∅�∅ � ∅

the transition map �Q� → q can be characterized through the discretecontroller matrix Δ with Δ ∈ DNQ×Ns1 , as a matrix operation thatgenerates an event sequence of length Ns1

q � �Q� � Δ (4)

In constructing the discrete controller Δ, it is important to recall thatnot all transitions between the events of the categorical set areallowed. These constraints are encoded in the adjacencymatrixA as amodel for the finite state automaton. Therefore, the controllerΔmustsatisfy

Δi;j ∈ fAki; 0g for Δk;j−1 � 1; i � 1; : : : ; NQ; j � 1; : : : ; Ns

(5)

Let the superscript notation qj−1 identify the event in the j place in thesequence. Then, the condition in Eq. (5) indicates that if the currentevent qj−1 is qk, then the next event q

j can be qi if Aki is equal to 1. Ifthis were the case and the next event was chosen to be qi, then

Δk;j−1 � 1; and Δi;j � 1

where the rest of the elements in the columns j-1 and j of Δ are zerodue to the column-sum property in Eq. (3).The first and last columns of Δ are not directly constrained by the

adjacency matrix A, but they must satisfy discrete boundaryconditions. LetQ0 ⊆ Q be the set of all the allowed initial events, andQf ⊆ Q be the set of all permissible final events. Then, the initialevent q0 of all feasible sequences belongs toQ0. In the sameway, thelast event qNs of all feasible sequences belongs to Qf.Finally, letUD ⊆ D be the set of discrete controllersΔ that satisfy

adjacency and boundary constraints. Then, the problem to be solvedby the outer-loop is a feasible integer programming (FIP) problemthat can be stated as follows:

Find Δ; Ns1 subject to Δ ∈ UD ⊆ DNQ×Ns1 ;

Ns1 ≤ Ns;max ∈ N(6)

Assuming that the inner-loop handling the continuous-timedynamics can find the optimal trajectory for any feasible sequence,then each candidate q will have an associated cost. The objective ofthe outer-loop solver is finding the sequence of events that has theoptimal cost among all the feasible sequences withlength Ns1 ≤ Ns;max.The complexity of the search space for the FIP is �NQ�Ns1 because

Δ ∈ D. The fact that this type of problem is NP-complete [5,10]underscores the impracticality of total enumeration as ameans to findthe optimal feasible sequence. A more efficient approach known asbranch-and-bound optimization has been used in the outer-loop tosolve the FIP [10,11]. This work introduces use of the GA for thesame purpose [12].The GA [6] method is a relatively new technique (in comparison

with the calculus of variations or primer vector theory [13]) that hasbeen successfully applied to trajectory optimization problems[14,15]. TheGA requires a population of individuals; an individual isa set of values for optimization parameters that is encoded as a stringof binary digits (a chromosome). In a HOC problem, a typical stringmight consist of binary representations of states or events in asequence. GA methods have features that make them appealing foruse in an automated mission planner. For instance, planning intuitionin the form of an incumbent solution sequence is not required.Although GA does not ensure that the solution found represents aglobal optimum [16], the fact that the method is randomized allowsthe search to continue even when a local minimum is found, unlikegradient-based methods.

B. Continuous-Time Dynamics

A modern space mission is usually composed of several events.Some of them have finite duration, such as thrust arcs, while othersare instantaneous, such as impulsive maneuvers. The dynamicsgoverning the system during any of the events q ∈ Q is defined by aset of differential equations

_x � f�x;u; t; q� (7)

where x ∈ RNx and u ∈ RNu are the continuous state and controlvectors, respectively. The dependence on q means that the systemdynamics may change throughout the mission, e.g., a model couldrequire switching from a thrust arc to a coast arc.Let q � �q0; q1; : : : ; qNs� be a finite sequence of events

where qj ∈ Q for j � 0; 1; : : : ; Ns. Let t � �t0; t1; : : : ; tNs1 � be a

qa

qcqb

Fig. 1 Digraph for a finite state automaton.

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real-valued matrix associated with q. Then, the cost functional isgiven by

J�x�:�;u�:�; t;q;Δ; Ns�

�XNsj�0

�ϕ�x�tj�1�; tj�1; qj� �

Ztj�1

tj

L�x�t�;u�t�; qj� dt�

(8)

where ϕ and L correspond to the Mayer and Lagrange costs of eachphase. As with most optimal control problems of significantcomplexity, direct methods are preferred to avoid the difficultiesassociatedwith the derivation and solution of the two-point boundaryvalue problem (TPBVP) resulting from the Euler–Lagrangeconditions [17] of the calculus of variations (COV). There are twoprincipal schemes that directly transform continuous-time optimalcontrol problems into NLP problems: collocation and transcription.Both cause the equations ofmotion (EOM) to be satisfied by definingnonlinear constraint equations. Direct collocation describes suchconstraint equations with implicit rules, such as Hermite–Simpson[18] and Gauss–Lobatto [19], while direct transcription uses explicitrules, such as the fourth-degree Runge–Kutta (RK) method [7].Given thatNLP is a gradient-basedmethod, it requires an initial guessof the solution, i.e., the vector of solution parameters, which can beobtained by intuition or experience. The robustness and accuracy ofthemethod depends on the selected resolution level of the timemesh,the collocation or transcription scheme, and the quality of the initialguess, which means how closely the guess satisfies the constraintsand the optimality conditions. For the implementation of the inner-loop solver, direct transcription with RK integration rules and aparallel-shooting scheme [7,12] is selected.TheNLP parameters can then be arranged as a single vectorPT that

collects all the continuous variables. For example, the parametervector for a trajectory consisting of a single thrust arc becomes

PT � �ZT �

where ZT � �xT1 ;uT1 ;xT2 ;uT2 ; : : : ;xTN�1;uTN�1; tf�, xi ∈ RNx andui ∈ RNu for i � 1; 2; : : : ; N � 1 are parameter vectors thatrepresent the state and control variables at the nodes of the discretetime mesh, and N is the number of segments of the mesh [7].In the samemanner, the nonlinear constraints can be collected into

a vector CT. The optimal control problem can then be restated as aNLP problem of the form:

Minimize J�P�

subject to

bL ≤

8<:

PAPC�P�

9=; ≤ bU

whereAP is formed by all the linear constraints of the problem, andbL and bU are the lower and upper bounds of the parameters andconstraints. The upper and lower bounds for the great majority of thenonlinear constraints C�P� are usually set to zero, because thisforces the solver to choose values for the parameters that satisfy theEOMs when they are integrated forward using the RK rule withineach segment (there may be a small number of additional nonlinearconstraints, e.g., boundary conditions). Once the NLP problem isclearly defined, it can be solved using dense or sparse solverssuch as NPSOL and SNOPT [20]. SNOPT is selected becauseit can take advantage of the sparsity present in the constraintJacobian [18].

III. Multiphase Mission Design as a HybridOptimal Control Problem

The validity of the proposed GA� NLP approach for theimplementation of hybrid optimizers was shown by solving two

sample problems: the motorized traveling salesman [12] and theinterception of multiple asteroids [21]. However, these problemscontain several simplifications that make them qualitatively differentfrom multiphase missions. For instance, the categorical variablesrepresent targets such as cities and asteroids, and the length of thecategorical sequence is constant. This allowed a straightforwarddefinition of the GA chromosome and a static NLP structure, i.e., thestructure remained the same for any sequence, requiring onlychanging the value of the interception constraints in the inner-loop. Ina multiphase mission, the categorical variables represent events suchas coast arcs and thrust arcs, and the length of themission sequence isvariable. Therefore, approaches to accommodate variable-lengthsequences and to generate different NLP problem structures dynami-cally are needed.The problems of interest here are missions composed of an

unspecified number of events with given initial and boundaryconditions, and free or fixed final time. The discrete and continuous-time problems associated with the mission design will be describedalong with the proposed methods of solution.

A. Discrete Dynamics

The discrete events in a space mission plan can bemaneuvers suchas impulses, thrust arcs, and coast arcs, which can be groupedtogether into a categorical space Q of finite cardinality NQ ∈ N.Although it appears that only these three types of events need to bedefined, considerations regarding the robustness of the trajectoryoptimization in the inner-loop, to be described in the followingsections, warrant the definition of additional, more specificallydefined events as shown in Table 1.Therefore, the categorical space becomes

Q � fq0; q1; q2; q3; q4g � fc; i; s; l; tg (9)

with cardinalityNQ � 5. The subscripts identify the event within thecategorical space. The goal is to assemble a sequence q of eventsq ∈ Q that fulfills the mission objectives and minimizes a costfunction defined in the inner-loop. The construction of the sequenceq incorporates constraints that are consistent with the discretedynamics of the problem. For instance, to satisfy state boundaryconditions, only a Lambert’s rendezvous l or a boundary-specifiedthrust arc t can be the final event of the sequence [9]. Also, a coast isplaced at the beginning of the mission and between thrustingmaneuvers [9]; for example, sequences of the form �i; i� or �s; s� arenot allowed. These constraints help reduce the size of the discretesearch space.The discrete constraints can be modeled with a finite state

automaton in the form of a directed graph or digraph, as shown inFig. 2. The nodes and the directed edges constitute themission eventsand the allowed transitions between them, respectively.The directed edge that does not start from a node indicates that the

sequences start with event c; the double-circled nodes specify that thesequences end with events l or t. These constraints form the discreteboundary conditions

Q0 � fcg (10)

Table 1 Categorical events for mission design

Event (code) Description

Coast arc c A finite coasting arcImpulse i An instantaneous impulsive maneuverBoundary-free thrustarc s

A thrust arc without given boundary conditions,subject only to feasibility with respect to the EOMs

Lambert’srendezvous l

A rendezvous composed of an impulse, a coast arc,and another impulse that satisfies given boundaryconditions

Boundary-specifiedthrust arc t

A thrust arc with given boundary conditions

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Qf � fl; tg (11)

Feasible event sequences can be obtained from the digraph. Anexample of a feasible sequence is

q � �q0; q1; q2; q3; q4; q5; q6; q7� � �c; s; c; i; c; s; c; t�

where the number of switches Ns � 7. The superscripts specify theplace of the event in the sequence. The digraph also shows transitionsthat are not allowed. For example, the automaton cannot move fromthe event i to event s, because there is not a directed edge in thatdirection.The information contained in the digraph can be encoded

following the notation and analysis shown in Sec. II for the definitionof the switching sets and the adjacency matrix, which becomes, forthe digraph of Fig. 2

A �

266664

0 1 1 1 1

1 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 0 0 0 0

377775 (12)

The number of events in a sequence, given by Ns1 � Ns � 1, is notfixed. An upper bound Ns;max is given to constrain the problem tofinite sequences and to make the problem tractable.Let F be the set of all the feasible sequences, i.e., event sequences

that satisfy the adjacency constraints and the discrete initial andboundary conditions. The problem to be solved by the outer-loop canthen be stated as follows:

minimizeq

ϕ�q� subject to q ∈ F; Ns1 ≤ Ns;max ∈ N

The evaluation of the function ϕ is carried out by the inner-loopsolver, which will provide the outer-loop with the optimal cost of thetrajectory corresponding to the argument sequence.

B. Continuous-Time Problem

The continuous-time problem of multiphase mission design is tofind the optimal trajectory that minimizes the objective function,usually propellant consumption or time of flight, satisfies given initialand boundary conditions for the continuous-time state variables, andhas a given mission structure described by an event sequence. Thetotal mission time can be specified or free. Because a multiphasetrajectory can use an electric engine for providing a continuous thrustarc and a chemical rocket for impulses, parameters such as the initialthrust acceleration α0, and the exhaust velocities of the propulsionengines must be given.Using polar coordinates on a spacecraft-fixed basis, the equations

of motion become [9]

_r � vr_θ � vθ

r

_vr �v2θr−

1

r2� f�q�α sin�β�

_vθ � −vθvrr� f�q�α cos�β�

_α � f�q� α2

ce(13)

where

f�q� ��0 if q � c1 if q ∈ fs; tg

α is the thrust acceleration, ce is the exhaust velocity of the electricengine, and β is the control angle describing the direction of thethrust. Because the thrust magnitude provided by the electric engineis constant, and the rate of change of the thrust acceleration appears inthe EOMsgoverning the finite events, it is convenient to use the thrustacceleration as a measure of the spacecraft’s mass throughout thetrajectory.An impulsive maneuver is modeled as changing both mass and

velocity instantaneously.An equation relating such change is given in[22]

m− −m�

m− � 1 − e−ΔVccc (14)

where cc is the exhaust velocity of the chemical rocket, and thesuperscripts − and � refer to instants immediately before andimmediately following the event. Then,

m− −m�

m− � 1 −α−

α�(15)

using Newton’s second law. Substituting Eq. (15) into Eq. (14), thestates must satisfy the constraints [9]:

r� � r−

θ� � θ−

v�r � v−r � ΔVc sin�β�v�θ � v−θ � ΔVc cos�β�

α� � α−eΔVccc (16)

where ΔVc is the impulse magnitude, and β is the direction of theimpulse in a spacecraft fixed basis.An analytical expression for the thrust acceleration as a function of

the thrusting time te�� t� − t−� can be obtained from Eq. (13)

Zα�

α−

da

a2�Zte

0

dτ

ce

which yields

α� � 11α− −

tece

(17)

Minimizing the sum of the thrusting times can optimize a missionconsisting of multiple thrust arcs. However, this approach is notappropriate if electric engines with different exhaust velocities areused, because their difference in efficiency is not taken into account.For instance, consider a trajectory with n thrust arcs where ti and cirepresent the thrusting time and the exhaust velocity of the engineused during thrust arc i. Then, according to Eq. (17), the final thrustacceleration due to propellant consumption is given by [9]

i sl t

c

Fig. 2 Digraph of a finite state automaton for mission design.

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αf �1

1α0−P

ni�1

tici

(18)

Minimizing the sum of the normalized thrusting times ti∕ci asdefined in Eq. (18) thus minimizes overall propellant consumptionfor the case when engines with different exhaust velocities areused.For impulsivemaneuvers, the change invelocityΔV has been used

as a measure of the propellant consumed. Therefore, the total amountof propellant used in a trajectory that uses multiple impulses isdescribed as the sum of the corresponding ΔVs. As with the low-thrust case, summingΔVs provided by rockets with different exhaustvelocities does not describe the total propellant used. Consider atrajectory with n impulses where ΔVi and ci represent the change invelocity of the spacecraft and the exhaust velocity of the rocket usedduring impulse i. According to Eq. (16), the total change in thrustacceleration due to propellant consumption is given by [9]

αf � α0e�P

n

i�1ΔVici�

(19)

Thus, the sum of the normalized change in velocityΔVi∕ci is a bettermeasure of propellant use, because values corresponding to differentrockets can be added to determine the total change in mass.Optimizing a system that uses both low-thrust and chemical

propulsion in the same trajectory requires a uniform metric tominimize. Equating the expressions for the thrust acceleration α� inEqs. (16) and (17) yields an expression that relates the normalizedthrusting time te∕ce of the electric engine to the normalized change invelocity ΔVc∕cc provided by the chemical rocket.

tece� 1

α−�1 − e−

ΔVccc � (20)

The mission planner can now choose which measure of propellant tominimize. In this work, the total normalized thrusting time is used asthe objective. Thus, if impulsive maneuvers are used, their ΔVs areconverted to normalized thrusting time using Eq. (20).Let q � �q0; q1; : : : ; qNs � be the finite sequence of events

provided by the outer-loop where qj ∈ Q for j � 0; 1; : : : ; Ns. Lett � �t0; t1; : : : ; tNs1 � be a real-valued time matrix associated with q.Then, the optimal control problem consists of minimizing the costfunctional given by

J�x�:�;u�:�; t;q; Ns� �XNsj�0

g�qj� (21)

where

g�qj� �

8>>>>><>>>>>:

0 if qj � c1α−j�1 − e−

ΔVc;jcc � if qj � i

tj�1−tjce

if qj ∈ fs; tgP2k�1

1α−j;k�1 − e−

ΔVc;j;kcc � if qj � l

subject to Eqs. (13) and (16) and

x�t0� � x0 (22a)

x�tNs1� � xf�tNs1� (22b)

tj−1 ≤ tj ≤ tj�1; for j � 1; : : : ; Ns (22c)

t0 � 0 (22d)

tNs1 free or fixed (22e)

where x ∈ R5 and u ∈ R are the continuous state and controlvectors, respectively.

In view of the fact that the inner-loop solver is expected to find theoptimal state trajectory for any given event sequence q, if it exists,robustness is more important than high accuracy. With this in mind,the proposed method of solution uses direct transcription with NLPusing RK integration rules and a parallel-shooting scheme [7]. Tomaximize robustness for a given number of segments, only one RKstep is used on each segment. The method used in the inner-loop togenerate a good initial guess and to solve the continuous optimalcontrol problem automatically with robustness and accuracy will bedescribed in Sec. V.

IV. Transcription of the Mission Event Sequence

The discrete component of the HOC problem consists of findingthe feasible categorical sequence that has minimum cost. In themission design context, this means finding the sequence of eventsthat achieves the mission objectives while minimizing propellantconsumption. The successful solutions of the sample problems in[12] suggest the use of GA for the implementation of the outer-loopsolver. In those problems, the discrete constraints were enforced byassigning a large constant cost to infeasible sequences. This approachis not practical for the multiphase problem because the discretedynamics are more complex. A new model that transforms theconstrained discrete optimization problem into an unconstrainedproblem [9] and searches for the solution only in the feasible discretespace is presented. The proposed approach also dealswith categoricalsequences of variable length.It is known that optimization algorithms improve their performance

if the search is carried out only in the feasible space defined by thegiven constraints [23]. In the mission design problem the potential forimprovement is significant, because the specified discrete constraintsdefine a feasible space that ismuch smaller than the total discrete space.Searching only in the feasible space requires the transformation of theconstrained optimization problem into an unconstrained problem. Theproposed GA model applies the specified constraints implicitly,allowing every individual to represent a feasible event sequence [9].According to Fig. 2 only the thrusting events need to be described in asequence because coast arcs are always to be placed between them andat the beginning of the sequence. Midcourse events can be modeledusing a binary digit, because there are only two allowed event types(0 for i, 1 for s). Similarly, a binarydigit can also represent the last eventin a sequence (0 for l, 1 for t). As a result, a feasible sequence can bemodeled using a binary string, inwhich every bit represents a thrustingevent. The remaining issue is to determine how to handle sequenceswith a variable number of events.A fixed-size binary string can represent sequences with variable

length by describing the number of thrusting eventsNT as the locationof the leading 1-bit in the chromosome [9]. The leading 1-bit does notrepresent an event itself; it is just a marker stating that only the bits thatfolloware tobe taken into account. In addition to the sequence events, acategorical variable specific to mission design, to be included in thechromosome, is the number of revolutions to be performed by thespacecraft. If themaximumnumber of revolutions allowed is three, it isnecessary to add two bits to the chromosome. Figure 3 shows a sample10-bit chromosome that can accommodate sequenceswith amaximumnumber of thrusting events NT;max equal to 7 (1 bit for the lengthmarker, 7 bits for thrusting events, and 2 bits for the number ofrevolutions). According to the convention of the previous paragraphs,the string in Fig. 3 is to be decoded as �i; s; t�. By adding the coast arcsthat are assumed to precede each thrusting event, the resulting eventsequence becomes �c; i; c; s; c; t� with two revolutions.Not all binary strings generated by GA are ready for decoding. For

instance, consider the following possible 8-bit strings: 00000000 and

0 0 0 0 1 0 1 1 1 0

Thrustingevents

Number ofrevolutions

Lengthmarker

Fig. 3 Sample chromosome for variable-length sequences.

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00000001. The former does not have the leading 1-bit, meaning thatits length is undefined, and the latter does have the leading 1-bit, butdoes not contain any thrusting event. This issue can be handled bypreprocessing every GA-generated binary string through a binaryaddition to 1000 before the decoding step, so that there is at least onethrusting event, and the number of revolutions is defined. The samplebinary strings would then become 00001000 and 00001001, whichrepresent the sequence �c; l� with 0 and 1 revolution, respectively.The outer-loop solver for the mission design problem will use this

model inwhich every string generated by theGA represents a feasiblesequence. The GA solver used is the MATLABGlobal OptimizationToolbox [24].

V. Method for the Solution of the Continuous-TimeOptimal Control Problem

The continuous-time component of the HOC problem consistsof finding the optimal space trajectory that satisfies initial andfinal boundary conditions and a given mission structure. In [12],two sample problems were solved using GA and NLP for theimplementation of the outer- and inner-loop solvers, respectively. Inthe motorized traveling salesman problem [12], the outer-loopsearched for the optimal visitation sequence of three cities. Duringthe search the outer-loop passed each candidate city sequence to theinner-loop, which solved the corresponding continuous-time optimalcontrol problem and returned the cost to the outer-loop. That is, theinner-loop must solve every optimal control problem required by theouter-loop, or the hybrid optimizer may not find the optimalsequence. Thismotorized traveling salesman problem is, however, anunusually straightforward HOC problem, because even though theinner-loop had to solve optimization problems with different visita-tion sequences, theNLPproblem structure of parameters, constraints,and systemEOMswas static, i.e., it did not change. The only requiredmodification was setting the interception constraints with thelocations of the cities corresponding to the given sequence.The mission design problem is qualitatively different from these

sample problems, as the discrete variables now represent events suchas impulses, coast arcs, and thrust arcs that, as will be shown, changethe structure of theNLP problem. In addition, the number of events inthe categorical sequence is not fixed; for example a mission mightconsist of a coast arc, an impulse, another coast arc, and a thrust arc.The same mission might be accomplished without the impulse, orwith an additional coast arc/thrust arc combination. Conventionalimplementations of NLP solvers expect a static problem structure.For the dynamical assembly of events required for the NLPdiscretization of the mission design problem, a scheme that definesevents as modules consisting of parameters and constraints ispresented. The method assembles the respective events sequentiallyin time according to the given mission structure [9].It was noted in the discussion of the solution process for the

example problems [12,21] that the continuous-time problem, aftertranscription, requires an initial guess, i.e., an approximate solution toinitialize theNLPproblem solver. If the guess is not sufficiently good,the NLP solver will not converge, and as mentioned previously, asuboptimal solution may be found or the HOC solution search couldstop. Because the mission design problems are sophisticated andchallenging, combining possibly lengthy sequences of events(coasts, impulses, thrust arcs), reliably finding an approximatesolution of good quality is challenging.Anewmethodwas developedthat approximates optimal low-thrust trajectories [9] and addressesthese issues. The method, based on GA, approximates optimalcontrol histories by incorporating boundary conditions explicitlyusing a CP function. The approximate solution from this method isgiven as an initial guess to an NLP problem solver to obtain anaccurate optimal trajectory.

A. Methods for the Approximate Optimization of Multiphase

Impulsive Trajectories

The proposed method to approximate an optimal solution is basedon real GA. Real GA still uses binary representations and operationsfor the evolutionary processes but at a level that is transparent to the

user. The approximate optimal trajectory is to be used as the initialguess for a more accurate optimization method, such as directtranscription with NLP.Because experience shows that penalty methods do not handle

explicit constraints well in general, such constraints should behandled implicitly whenever possible, i.e., posing the optimizationproblem in such a way that it appears to be unconstrained from thestandpoint of the GA. For instance, a compound event named“Lambert’s rendezvous” has been defined for trajectories that mustsatisfy rendezvous conditions using impulsive maneuvers. Itsimplementation is based on algorithms for Lambert’s problem,whichconsist of the determination of an orbit that connects two positionvectors and has a specified transfer time. Battin [8] and Prussing [25]present algorithms for Lambert’s problem using single andmultirevolution trajectories, respectively. Such algorithms yield thesemimajor axis and eccentricity of the transfer orbit, which issufficient to allow the determination of the velocity vectors at thebeginning and the end of the transfer. Although the original definitionof Lambert’s problem does not consider rendezvous maneuvers,it is possible to match required velocity vectors by computing therespectiveΔVs using the resulting terminal velocity vectors. Becausethe interception or rendezvous constraints are handled by theLambert’s problem algorithm, the GA can find optimal values for therespective parameters for the states and transfer time withoutexplicitly dealing with any constraint.For an entirely impulsive trajectory, the following events constitute

the categorical space Q:1) A coast arc c is the most basic event to represent. It is

characterized by only one GA parameter, the flight time. Itsevaluation consists of integrating the EOMs in Eq. (13) from theinitial state of the event for the duration of the flight time. The state atthe end of the integration becomes the boundary state of the event.2) An impulse i is also straightforward; it consists of two GA

parameters, the direction, and magnitude of the impulse. Its evalua-tion applies the vector operation in Eq. (16) to the initial state of theevent. The resulting vector is the boundary state of the event.3) The Lambert’s rendezvous l is a compound event consisting of

an impulse, a coast arc, and another impulse that is placed at the endof the mission to satisfy given boundary conditions. Although thiscompound event is equivalent to the sequence �i; c; i�, the Lambert’srendezvous event was introduced because the coast arc c and theimpulse i events would need explicit constraints to satisfy the givenboundary conditions. The only GA parameter required is the transfertime, because the initial and target state vectors for the event arespecified. The evaluation of the event yields the terminal velocitiesand hence the impulses required to perform the maneuver.Assembling the respective events, in this case, i, c, and l,

sequentially in time constitutes a multiphase mission. Given that afew parameters represent each event, a GA individual cancharacterize an entire mission by collecting the parameters of all theconstitutive events. For example, the sequence q � �q0; q1; q2; q3�� �c; i; c; l� can be represented by the chromosome

�Coast Time0; Direction1;Magnitude1;Coast Time2; TransferTime3�

where the superscripts identify which event in the sequence therespective parameter belongs to. The cost determination of thisindividual starts by evaluating each of the component events from leftto right, successively. Every event has a defined initial state at themoment of evaluation that corresponds to the boundary state ofthe previous event. The assessment of the last event concludes theevaluation of the individual and precedes the computation of thesequence cost. For the sample chromosome, it would be the sumof allthe explicit impulse magnitudes, in this case just magnitude1, andthe two impulse magnitudes resulting from solving Lambert’sproblem. The addition of events that use low-thrust propulsion ismore complex because they require continuous control histories. Thenext section presents a method to approximate optimal trajectoriesincluding these types of events.

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B. Conditional Penalty Method for the Approximate Optimization

of Multiphase Low-Thrust Trajectories

AGA-based approach, the CP method [9], has been developed forthe approximate optimization of multiphase low-thrust trajectories.In principle, any evolutionary algorithm such as particle swarmalgorithms or genetic algorithms can be used with the CPformulation. Instead of using traditional penalty methods to satisfyexplicit boundary conditions, a conditionally valued fitness functionis introduced to first find a feasible trajectory, and then refine it into atrajectory that is also optimal. The approximate solution can then beused as an initial guess for a direct solver that converts the continuousoptimal control problem into aNLPproblem. Themethod inherits theparallel scalability of GA for use in parallel computing systems [26].An important feature of this method is that it is not constrained to aparticular choice of coordinate system. Also, the method allowsconsidering actualmission features, such as the use of constant thrust,and the thrust acceleration increase as propellant is consumed.A spacemission can be accomplished successfully by thrusting for

the entire duration of the flight. However, primer vector theory showsthat such a strategy is likely not optimal. Trajectories consisting ofmultiple low-thrust and coasting arcs, i.e., multiphase missions, canoften satisfy similar boundary conditions using less propellant. Theconditional penalty method described here optimizes this multiphasetype of trajectory without a priori knowledge. It is illustrated bymeans of a sampleminimum-propellant rendezvous using low thrust,whose state trajectory is shown in cartoon form in Fig. 4.Assuming that initial conditions are given, the state point A is

specified. For a rendezvous, the state point D is also given or isdeterminable as a function of time. Intermediate points B and C arenot specified directly but can be obtained by integrating the systemwith a control history during the first thrust arc and by coasting,respectively. Assuming that the mission events can be evaluated inchronological order, some general observations can be made atthis point:1) Every arc has initial conditions before its evaluation.2) The thrust arcAB does not have terminal boundary conditions.3) The thrust arc CD has given boundary conditions.4) The flight times of each arc, including the coast, are free.

A GA method can then optimize this mission by finding values for1) control parameter vector uAB, 2) control parameter vector uCD,3) flight time of thrust arcAB, 4) flight time of coast arcBC, and 5)flight time of thrust arc CD, that satisfy the rendezvous conditionswhile minimizing the total thrusting time. The control vectors consistof parameters that determine the thrust-pointing angle history duringeach thrust arc.The integration of the EOMs in a thrust arc requires many

integration steps, which in turn calls for a rather large number ofparameters to represent the respective control history. Such a largenumber of parameters in theGAmodel of an individualwould requirethe use of very large populations, reducing the effectiveness of themethod. A reduction in the number of parameters can beaccomplished by representing the control history at only a few pointsin time. Interpolating the few control parameters using polynomialssuch as the Hermite cubic or a Fourier transform [9,27,28] yields the

higher time resolution of the control history required for an accurateintegration.A design of a chromosome for GA individuals, for the example

shown in Fig. 4, would require the following vector P of parameters:

P � �tAB;uAB; tBC; tCD;uCD�

The determination of the cost of each individual is accomplished byevaluating the mission phases successively using the parametervalues. For instance, the first thrust arc has a defined initial point Athat, with flight time tAB and control vector uAB, are used to obtainpointB. Then, pointB is usedwith tBC to find pointC at the end of thecoast. Finally, the evaluation of the second thrust arc requires theintegration of the system starting from C, using flight time tCD andcontrol vector uCD, to obtain point x�tD�. Although this approachensures every individual satisfies the EOMs throughout thetrajectory, the specified terminal conditions may not be achieved. Anapproach to address this issue is to use penalty methods in the fitnessfunction. For example, the constrained optimization problem

Minimize J�P� subject to hi�P� � 0; i � 1; 2; : : : ; n

would become

minimize J�P� � kXni�1

�hi�P��

where Φ is the penalty function, and k is the penalty coefficient.Experience shows that using a linear combination to minimize thecost and constraint violations simultaneously is not an effectivetechnique [29]. Because the inner-loop in the HOC problem solvershould be able to optimize a trajectory with any structure, analternate, more robust method is needed. Coello Coello [29] presentsa method that handles the constraints based on evolutionarymultiobjective optimization (MOO). Themethod explicitly ranks theGA individuals in the population according to prescribed rules ontheir feasibility and optimality. The CP method [9] used in this workhas a simpler implementation, but it is effective in finding solutions toconstrained optimization problems.In theCPmethod, the constraint violations aremapped into a scalar

distance d, i.e., the Euclidean distance between the boundary statex�tD� and the boundary conditionsD. Figure 5 shows two boundary-specified thrust arcs identified by subscripts (1, 2) corresponding tothe last thrust arcs of multiphase missions represented by differentGA individuals. The last thrust arcs for individuals 1 and 2 terminateat x1�tD� and x2�tD�, respectively. If the distance d is greater than thespecified tolerance, as is the case for individual 1, the trajectory isconsidered to be infeasible; the cost assigned to the individual is thenthe addition of a large infeasibility constant K and the distance d,which provides the search with directionality information even frominfeasible individuals. If the trajectory is feasible, as results from thelast thrust arc of individual 2, the cost assigned to the individual is nolonger related to the distance d, but instead it is based on the original

A B

C

D

X1

X2

thrustcoast

thrust

Fig. 4 State trajectory of multiphase low-thrust mission.

X1

X2

C1

D

x1(tD)

d1 > tol

d2 toltolerance region

C2

x2(tD)

Fig. 5 Feasibility determination of boundary-specified thrust arcs.

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cost metric, i.e., the amount of propellant used in the entire mission.This method can be implemented using the following conditionalfitness function:

J ��

K� d if d > tolPNsj�0 g�qj� if d ≤ tol

(23)

The CP method requires the tuning of only two parameters. Theinfeasibility constant can be trivially set to a value that is higher thanthe expected cost of any feasible solution. In general, the toleranceshould be set to low-accuracy values in the order of 10−1 to 10−2 fornormalized problems, because evolutionary metaheuristics are lessaccurate than gradient-based deterministic methods such as NLP.High-accuracy tolerances are likely to cause overpenalization,i.e., convergence to a feasible solution with a cost that is significantlyhigher than the optimal cost.

C. Methods of Solution for the Optimization of a Multiphase

Space Trajectory

Direct methods have become a popular choice for trajectoryoptimization problems because they avoid the difficulties related tothe derivation and solution of the TPBVP resulting from theEuler–Lagrange equations of the COV. Unlike GA solutions,trajectories found using NLP satisfy optimality conditions to aspecified tolerance, which yields accurate solutions. Schemes for thetransformation include Hermite–Simpson collocation [18], Runge–Kutta transcription with parallel shooting [7], and Gauss–Lobattocollocation [19], which vary in their degree of accuracy androbustness. Regardless of the scheme chosen, the NLP trans-formation consists of discretizing time andmodeling the continuous-time state and control variables at several points in time. The systemdynamics and boundary conditions are satisfied by definingnonlinear constraints that relate nodalNLPparameters. Creating sucha NLP structure of parameters and constraints is an involved processthat the mission planner has to perform each time the optimizationproblem is formulated with a different structure. It is clear that thisapproach is not practical for a mission “automaton,” because theproblem statement for the inner-loop calls for the optimization oftrajectories with a variety of structures. Therefore, a modular schemeis needed for the automatic construction of NLP problems duringruntime.The basic module in this work is the categorical event, which

consists of a set of NLP parameters and constraints [9]. A property ofevery event is the presence of terminal nodes that allows representinginitial and boundary states. This property, alongwith the definition ofcontinuity constraints on the states during the event transitions,allows assembling the events successively in time regardless of theirinternal dynamics in a fashion similar to that described by von StrykandGlocker [10] as shown in Fig. 6. The internal constitution of eachevent module depends on the type of event it represents [9].The NLP representation of impulses is implemented as follows:1) An impulse is represented by a knot, i.e., two nodes

corresponding to the states immediately before and following theimpulsive maneuver.2) Two additional parameters are used for the direction and

magnitude of the impulse.3) Five nonlinear constraints ensure that the parameters satisfy the

change in velocity and mass described in Eq. (16).In a similar way, the NLP representation for coast and thrust arcs is

the following:1) A coast/thrust arc is represented by a standardmesh for parallel-

shooting using RK integration rule. One RK step per segment hasbeen selected for maximum robustness.2) An additional parameter is added for the flight time.3) Five nonlinear constraints on eachmesh segment ensure that the

parameters satisfy the EOMs in Eq. (13).4) Additional constraints may be imposed depending on the

desired maneuver, e.g., interception or rendezvous.Note that there is no particular event definition for a Lambert’s

rendezvous, because the NLP problem does not employ any of thealgorithms to solve Lambert’s problem. When the inner-loop is

assigned to optimize a trajectory containing this type of event, itassembles an impulse, a coast arc, and another impulse, and definesthe nonlinear constraints required to satisfy the boundary conditions.By parsing the events provided by the outer-loop, this modular

approach allows the inner-loop to set the vector of NLP parameterswith suitable bounds and to compute the appropriate nonlinearconstraints. The modeler has the discretion to define the upper andlower bounds for the parameters in a way that is appropriate for theproblem in consideration; the bounds for theGAandNLPeventsmayor may not be similar as long as the GA search space is a subset of theNLP search space. This is a necessary condition for the GA solutionto be a feasible initial guess for the NLP problem. Regarding theterminal constraints, numerical experimentation shows that the NLPsolver finds the optimal solution more robustly if the interceptionconstraints, i.e., position matching, are transformed to Cartesiancoordinates instead of directly using polar coordinates.The cost function is a measure of the propellant used throughout

the trajectory. In this work, it has been constructed as the sum of thenormalized thrusting times corresponding to the thrust arcs and thoseresulting from the transformation of the impulse magnitudes asdefined in Eq. (20). It is important to note that a feasible trajectorymay not exist for some values of the categorical variables, e.g.,number of revolutions, and the constraints for the continuousvariables, e.g., boundary conditions and flight time. For these cases,the inner-loop assigns a large constant cost to the event sequenceprovided by the outer-loop.

D. Transformation of the Approximate GA Solution

into the NLP Initial Guess

The approximate solution for the trajectory optimization problemfound using the conditional penalty method [9] is a good initial guessfor the NLP problem solver, because the EOMs are satisfied and thepropellant consumption has been minimized heuristically. However,such a solution cannot be used directly by theNLP solver, because theGA and NLP parameter representations are different. For example,the GA model does not use parameters to represent state variables;they are computed during the evaluation of each event taking as initialstate the boundary state of the previous event. Therefore, a procedureis required to generate guess values for the respective NLPparameters from the approximate GA solution.For an impulse, the GAvalues for the initial state and the impulse

direction and magnitude can be used directly as guess values for theNLP parameters corresponding to the state at initial node and thedirection and magnitude of the impulse. The vector operations inEq. (16) describe the impulse dynamics and generate guess values forthe state at the boundary node of the event.For a coast arc, the GA values for the initial state and flight time

can be used directly as guess values for the NLP parameterscorresponding to the state at the initial node and the flight time.Performing an integration of the system in Eq. (13) yields guessvalues for the state parameters at the inner and boundarymesh nodes.For a thrust arc, theGAvalues for the initial state and the flight time

can again be used as guess values for the NLP parameters for the stateat the initial node and the flight time. To obtain guess values for thestates at the inner nodes, the few GA control parameters are

t0= 0 tNs1= tf

event q0

t1 t2 titi+1

event q1 event qi

x = f(x, u, q0 ) x = f(x, u, q1) x = f(x, u, qi )

x(0)

x(tf)

Fig. 6 Assembly of events in a multiphase trajectory.

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interpolated to achieve a resolution consistent with the transcriptionmesh; this provides the guess values for all the control parameters forthe thrust arc. Then, the system in Eq. (13) can be integrated to obtainthe state values for the inner and boundary nodes. This is acomputationally practical approach, because the interpolation of thecontrol history involves only a few GA parameters, and the integ-ration of the EOMs is an initial value problem (IVP) with a number ofsteps given by the resolution of the discrete time mesh.Finally, for a Lambert’s rendezvous, the GA model provides an

initial state and a flight time; a target state is given in the problemdefinition. A Lambert’s problem algorithm can find the terminalvelocities needed to perform the rendezvous, which in turn yield themagnitude and direction of the first and second impulsivemaneuvers.With this information, guess values can be generated for theparameters of the events that compose the Lambert’s rendezvousin the NLP model, i.e., a first impulse, a coast arc, and a secondimpulse.The next paragraph shows an outline of the algorithm for the

optimization of trajectories with different structures using NLP. Aswith the CP approximation method, the events are evaluated one at atime, and the locations of the respective event parameters andconstraints in the GA vector, NLP parameter vector, and NLPconstraint vector are determined through the use of an offset variableon each vector. Each offset variable indicates the location of the firstelement, e.g., parameter or constraint, for the event that is currentlyevaluated; the rest of the elements for the current event are recognizedby their relative position with respect to the first element. After theevaluation of the event is complete, the number of elements(parameters or constraints) of the current event in the respectivevector is added to the corresponding offset variable so that it indicatesthe location of the first element of the next event. This processcontinues successively until all events in the given sequence havebeen evaluated.

MAIN PROGRAMinitialize offsets on GA solution vector and NLP initial guess and bounds

vectorsFOR each event

transform solution values of the GA event parameters into initial guessvalues for the NLP event parameters, andset them in the NLP initial guess vector

set bounds for the respective NLP event parameters and constraints in theNLP bounds vectors

advance offsets on GA parameters vector andNLP initial guess and boundsvectors to the next event

ENDcall NLP solver with initial guess vector, bounds vectors, and names for the

OBJECTIVE FUNCTION and the NONLINEAR CONSTRAINTSFUNCTION

OBJECTIVE FUNCTIONinitialize offset on NLP parameter vectorFOR each eventaccumulate the cost contribution of the eventadvance offset on NLP parameter vector to the next event

ENDreturn accumulated cost

NONLINEAR CONSTRAINTS FUNCTIONinitialize offsets on NLP parameters and constraints vectorsFOR each eventcompute values for the nonlinear constraints for the event and set them in

the NLP constraints vectorIF event is NOT first

compute values for state continuity constraints and set them in theNLP constraints vector

ENDIF event is terminal

compute values for terminal constraints and set them in the NLPconstraints vector

ENDadvance offsets onNLPparameters and constraints vectors to the next event

ENDreturn NLP constraints vector

VI. Example Problem with Specified Final Time

A minimum-propellant time-fixed rendezvous between circularorbits was selected as a testbed for the GA� NLP hybrid optimizer[9]. The GA and NLP solvers used are the MATLAB GlobalOptimization Toolbox [24] and TOMLAB/SNOPT [30], respec-tively. The computing system used to solve this problem is a 1.66-GHz Intel Core Duo. This problem was developed and solved byPrussing and Chiu [31] using primer vector theory [13]. In theproblem, two spacecraft orbit a planet in coplanar circular orbits ofunit radius. The target spacecraft has an initial lead angle of π radwith respect to the controlled spacecraft as shown in Fig. 7, and thetransfer time is 2.3 orbit periods. Prussing and Chiu [31] found alocally optimal trajectory with four impulses and a cost of ΔV �0.1891 DU∕TU, where a DU is equal to the radius of the circle orbitand a TU is the time to advance one radian in the unit circle orbit.Colasurdo and Pastrone [32] found another solutionwith a lower costof ΔV � 0.1638 DU∕TU also using four impulses, which is shownin Fig. 8; the triangles show the locations where the impulses areapplied. The optimality of this mission structure can be verified usingthe Pontryagin minimum principle through a switching function [33]shown in Fig. 9. Primer vector theory requires the application of athrusting maneuver each time the switching function is greaterthan or equal to zero, and a coast otherwise. The existence ofknown solutions for this nontrivial problem makes it a good choicefor evaluating the effectiveness of the methods presented in thiswork.Although the problem as constructed by Prussing and Chiu [31]

allows only impulsive maneuvers, in this work it is modified to allowlow-thrust arcs [34]. In addition, the problem is solved accounting forthe difference in efficiency of the propulsion systems, i.e., that theelectric low-thrust engine is many times more efficient than achemical rocket in its use of propellant mass. The controlledspacecraft position and velocity are defined in a polar coordinatesystem, and canonical units are used, i.e., a DU is the radius of thecircular orbit and one TU is the time required to advance one radian inthe unit circle. At the initial time, the polar coordinates of thecontrolled spacecraft are r � 1 DU, and θ � 0 rad. Assuming thatthe circular orbit is geosynchronous and that the propulsion systemsused are the advanced space engine rocket (Isp � 476 s) and astandard ion electric engine (Isp � 3000 s), the exhaust velocitiesbecome 1.52 DU∕TU and 9.57 DU∕TU, respectively. The initialthrust acceleration is assumed to be 0.03 DU∕TU2 and an upperbound on the final thrust acceleration is set to 0.30 DU∕TU2. Thecontinuous optimal control problem is described in Eqs. (21) and (22)with the following initial and boundary conditions and timeconstraints:

x0 � � 1 0 0 1 0.03 �T

xf � � 1 �0.8� n�2π 0 1 free �T

n ∈ f 0; 1; 2; 3 gt0 � 0; tNs1 � 4.6π

The goal of the hybrid optimizer is to find the mission structure thatminimizes the amount of propellant required to perform therendezvous. The maximum length of the event sequences Ns;max ischosen to be 12. Because the cardinality of the categorical space is 5,the total size of the discrete search space then becomes4P

12i�1 5

i � 1; 220; 703; 120. The transcription approach for theouter-loop problem described in Sec. IVallows reducing the discretesearch space by considering only the event sequences that satisfy thediscrete constraints. In this case, the mission sequences can use up tosix thrusting events, so the chromosome for the outer-loop problem isa string of six binary digits plus 1 bit for the specification of thesequence length and 2 bits for the representation of the number ofrevolutions. The size of the discrete search space has been reduced to512. The GA population size is chosen to be 25, and the evolutionaryprocess is set to run for 20 generations.For the inner-loop problem, the timemesh for each coast/thrust arc

consists of 30 segments to represent the time history of 5 continuous-

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time states and 1 control variable (an additional parameter representsthe flight time). Consequently, each thrust arc uses 217 NLPparameters and each coast arc uses 156 NLP parameters; there are150 nonlinear constraints on each arc. For impulses, each knot uses12 NLP parameters and has 5 constraints. There are 5 state continuityconstraints for each event transition, and 5 final constraints for theboundary conditions and time constraint. The NLP feasibility andoptimality tolerances were set to 10−5. The CPmethod is used for thegeneration of the initial guesses for the NLP solver. The CP tolerancewas set to 10−1 and the control history during a thrust arc isrepresented using 3 parameters and Hermite cubic interpolation. Thereal GA used in the CPmethod is configuredwith a population size of500 individuals. The evolutionary process is set to stop after reaching500 generations or when 50 generations have had the same bestindividual. The real GA search space for the magnitude of theimpulsive maneuvers is � 0.002; 1.8 � and for the thrust pointingangle parameters is �−π; π�; the search space for the duration of thecoast/thrust arcs is � 0.5; 12 � with the exception of the first arc,which can collapse to 10−3. Because the geometrical configuration ofthe system is not affected by an initial coast, the guess for the durationof the initial coast is set to 10−3.The mission structure that minimizes the propellant consumed is�c; s; c; s; c; s; c; t� with two revolutions, which has a normalizedthrusting time of 0.6541 TU2∕DU; the actual thrusting time(6.2594 TU) can be found by multiplying the normalized metric bythe respective exhaust velocity. Finding the best event sequencerequired the cumulative evaluation of 65 sequences in the outer-loopas shown in Table 2. The optimal solution is found in the sixthgeneration; 14 more generations are run (only 4 of these are shown)but no improvement is seen. Note that for improved efficiency of theouter-loop GA method, the evaluated sequences and their corres-ponding costs are cached in memory to avoid solving the sameproblems repeatedly if the sequences appear again in subsequentgenerations. This explains why the number of “evaluated sequences”in the table rapidly diminishes as the search converges to the solution.The total computational time for the hybrid optimizer was approxi-mately 16 days. As an example, finding the solution trajectory for thebest sequence in the inner-loop required 262min for the CP to find aninitial guess and then 108 min for the NLP solver to converge fromthis guess to the solution. The optimal trajectory and control historyare shown in Figs. 10 and 11, respectively. The optimality of themission structure can be verified through a switching function [7,33],which can be computed using the NLP parameters corresponding tothe states and Lagrange multipliers from the NLP solution [7].Figure 12 shows that the mission structure found is optimal; primervector theory requires the application of a thrust arc when the

Fig. 8 Optimal trajectory using four impulses and two revolutions for

the circle-to-circle rendezvous, Colasurdo and Pastrone solution [32].

Fig. 9 Switching function history for optimal trajectory using four

impulses and two revolutions for the circle-to-circle rendezvous.

Fig. 7 Initial spacecraft configuration for the circle-to-circle rendezvous.

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switching function is greater than zero and a coast otherwise (thus theduration of the first arc of the sequence is collapsed to the lowerbound, 10−3). The fact that the hybrid optimizer found that theoptimalmission sequence uses only low-thrust propulsionmeans thatthe solver effectively took into account themuch greater efficiency ofthat engine vs the conventional rocket motor. As a comparison, notethat total cost for the four-impulse strategy found by Colasurdo and

Pastrone [32] in normalized thrusting time is 3.6065 TU2∕DU(normalized ΔV of 0.1078) for the given propulsion parameters.However, the low-thrust strategy used, placing the spacecraft into aninner orbit to reduce the lead angle of the target, is similar to that of theimpulsive case shown in Fig. 8. It can be said that the trajectory shownin Fig. 10 is the low-thrust analogue of the impulsive trajectory. Thesimilarity of the trajectories is also shown by their switchingfunctions in Figs. 9 and 12.

VII. Example Problem with Free Final Time

A minimum-propellant Earth–Mars transfer was chosen as anexample of a problem with an unspecified (free) final time [34].Typical missions to Mars have durations of less than a year, e.g., theHohmann transfer from Earth requires 9 months approximately;therefore, an upper bound of 5 years was set on the total mission time.The problem uses the J2000 heliocentric ephemeris [35] for thecomputation of the position of the planets, and a coplanar rendezvousis studied for the sake of simplicity. The spacecraft position andvelocity are defined in a heliocentric polar coordinate system, andcanonical units are used, i.e., 1 DU is equal to 1 astronomical unit(au), and a TU is the corresponding time to advance one radian in a 1-au circular orbit. The initial time is the J2000 epoch, and the initialthrust acceleration is assumed to be 0.025 DU∕TU2; an upper boundon the final thrust acceleration is set to 0.300 DU∕TU2. The initialand boundary states are obtained from the orbital elements resultingfrom the ephemeris computation. The propulsion systems usedare a nuclear thermal rocket (Isp � 850 s) and a standard ionelectric engine (Isp � 3000 s), so the exhaust velocities become0.280 DU∕TU and 0.988 DU∕TU, respectively. As before, the goalof the hybrid optimizer is to find the mission structure that minimizesthe amount of propellant required to complete the mission. Thespacecraft can perform 0 through 3 revolutions. The configuration ofthe outer- and inner-loop solvers is the same as that for the previous

Fig. 10 Optimal trajectory for the sequence �c;s;c;s;c;s;c;t� with two

revolutions for the circle-to-circle rendezvous. The bold segments

represent thrust arcs.

Fig. 11 Optimal control history for the sequence �c;s;c;s;c;s;c;t� withtwo revolutions for the circle-to-circle rendezvous.

Fig. 12 Switching function history for the sequence �c;s;c;s;c;s;c;t�with two revolutions for the circle-to-circle rendezvous.

Table 2 Progress of the outer-loop GA solver for the circle-to-circle rendezvous

Generation Evaluatedsequences

Cumulativeevaluations

Best normalized thrustingtime (TU2∕DU)

Best normalizedΔV(dimensionless)

Best sequence

1 25 25 0.6977 0.0212 �c; s; c; i; c; s; c; s; c; t� 2 rev2 14 39 0.6977 0.0212 �c; s; c; i; c; s; c; s; c; t� 2 rev3 10 49 0.6688 0.0203 �c; s; c; s; c; s; c; s; c; s; c; t� 2 rev4 9 58 0.6688 0.0203 �c; s; c; s; c; s; c; s; c; s; c; t� 2 rev5 4 62 0.6688 0.0203 �c; s; c; s; c; s; c; s; c; s; c; t� 2 rev6 3 65 0.6541 0.0198 �c; s; c; s; c; s; c; t� 2 rev7 1 66 0.6541 0.0198 �c; s; c; s; c; s; c; t� 2 rev8 0 66 0.6541 0.0198 �c; s; c; s; c; s; c; t� 2 rev9 0 66 0.6541 0.0198 �c; s; c; s; c; s; c; t� 2 rev10 0 66 0.6541 0.0198 �c; s; c; s; c; s; c; t� 2 rev

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example, and the conditional penalty method is used to generate theinitial guesses for the NLP solver. The real GA search space for themagnitude of the impulsive maneuvers is � 0.003; 0.4 � and forthe thrust-pointing angle parameters is �−π; π�; the search space forthe duration of the coast/thrust arcs is � 0.5; 25 � with the exceptionof the first arc, which can collapse to 10−3.

The best mission sequence found is �c; s; c; s; c; t� with tworevolutions and a normalized thrusting time of 6.8625 TU2∕DU(actual thrusting time is 6.7802 TU); the duration of the mission is2.83 years. Finding this solution required the cumulative evaluationof 40 sequences as shown in Table 3; the optimal solution is found inthe second generation. The total computational time for the hybridoptimizer was approximately 15 days. As an example, finding thesolution trajectory for the best sequence in the inner-loop required235min for theCP to find an initial guess and then 73min for theNLPsolver to converge from this guess to the solution. The correspondingoptimal trajectory and control history are shown in Figs. 13 and 14.The switching function [7,33] in Fig. 15 shows the optimality of themission structure. The first and the second thrust arcs occur nearperiapse, primarily in the direction of motion, which raises theapoapse. In thisway, the apoapse of the last transfer trajectory reachesthe orbit of Mars. Shortly before entering Mars orbit, the third andfinal thrust arc begins to raise the periapse and complete therendezvous.

VIII. Conclusions

In this work, the problem of interest is the automated solution ofhybrid optimal control (HOC) problems. A HOC problem is anoptimization problem defined in terms of discrete and continuousvariables. Although in this work the nested-loop approach has beenfollowed, the solution of complex problems, such as the automateddesign of multiphase space missions, requires new sophisticatedmethods, because the problem structure is no longer static as in otherHOC problems. The discrete variables now represent events such asimpulses, coast, and thrust arcs, which means that the spacecraftdynamics are described by different equations of motion during thedifferent phases of the trajectory.Analysis was performed to reduce the discrete search space in the

outer-loop to only sequences that satisfy given discrete constraints.The strategy of using a leading bit as a lengthmarkerwas successfullyused in the outer-loop genetic algorithm (GA)when searching among

Table 3 Progress of the outer-loop GA solver for the Earth–Mars mission

Generation Evaluatedsequences

Cumulativeevaluations

Best normalized thrustingtime (TU2∕DU)

Best normalizedΔV(dimensionless)

Best sequence

1 25 25 6.8705 0.1885 �c; s; c; s; c; s; c; t� 2 rev2 15 40 6.8625 0.1882 �c; s; c; s; c; t� 2 rev3 12 52 6.8625 0.1882 �c; s; c; s; c; t� 2 rev4 5 57 6.8625 0.1882 �c; s; c; s; c; t� 2 rev5 5 62 6.8625 0.1882 �c; s; c; s; c; t� 2 rev6 2 64 6.8625 0.1882 �c; s; c; s; c; t� 2 rev7 1 65 6.8625 0.1882 �c; s; c; s; c; t� 2 rev8 0 65 6.8625 0.1882 �c; s; c; s; c; t� 2 rev9 1 66 6.8625 0.1882 �c; s; c; s; c; t� 2 rev10 0 66 6.8625 0.1882 �c; s; c; s; c; t� 2 rev

Fig. 13 Optimal trajectory for the sequence �c;s;c;s;c;t� with two

revolutions for the Earth–Mars mission. The bold segments represent

thrust arcs.

Fig. 14 Optimal control history for the sequence �c;s;c;s;c;t� with two

revolutions for the Earth–Mars mission.

Fig. 15 Switching function history for the sequence �c;s;c;s;c;t� withtwo revolutions for the Earth–Mars mission.

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variable-length sequences using a fixed-length chromosome. Inaddition, the strategy of caching the cost of evaluated event sequencesin memory provided significant savings in computation; theevaluation of a single event sequence is computationally intensive,because it requires solving an optimal control problem.The automated construction of nonlinear programming (NLP)

problems of arbitrary structure in the inner-loop was made possiblebecause of the introduction of a new modular scheme. The basicmodule consists of a mission event, which is represented by a set ofNLP parameters and constraints. For the generation of initial guessesfor the NLP problem, a new method based on real GA approximatesoptimal control histories by incorporating boundary conditionsexplicitly using a conditional penalty (CP) function. The quality ofthe solutions found by the CP method allows the NLP solver torobustly converge to a solution. However, multiphase trajectoryoptimization problems have difficulties not found in single-phaseoptimization problems. For instance, if every event is allowed tocollapse, some of themmay collapse early during the search, yieldinga convergent solution that is feasible but suboptimal. To prevent thisproblem, a small lower boundwas set on themagnitude of the events;the definition of what is “small” will depend on the nature and thescaling of the problem. This strategy does not pose a major loss ofgenerality, because if an event should collapse to yield the optimalsolution, the outer-loop GA should call for the evaluation of a similarsequence without the offending event. A second strategy used intandem is the definition of a low-accuracy tolerance in the CPmethodto prevent overpenalization. Statistics show that the computationaltimes for the CP and the NLP methods are of the same order. Theimplementation of these methods in future studies to find three-dimensional trajectories is straightforward but should have longercomputational times due to the presence of additional searchparameters.The solved problems included the minimum-propellant, time-

fixed circle-to-circle orbit rendezvous. The second solved problemwas a minimum-propellant time-free, coplanar Earth–Mars mission.The application of normalized cost metrics and transformationsallowed the optimization of trajectories that employ differentpropulsion systems. The optimality of the multiphase structuresfound was verified using the Pontryagin minimum principle througha switching function.In this work, methods were developed that can be used for the

automated design of multiphase space missions but can also apply tothe solution of any dynamical HOC problem. Similarly, the CPmethod, developed here for robustly finding approximate solutions tobe used as initial guesses for the NLP solver in the inner-loop of theautomaton, could also be used independently, e.g., when initialguesses are needed for solutions of general optimal control problems.

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