# Multidisciplinary Optimization of Aerocapture Maneuvers

Authors: Roberto Armellin; Michèle Lavagna
Journal: Journal of Artificial Evolution and Applications
 (2008)

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## Abstract

A multidisciplinary-multiobjective
optimization of aerocapture maneuvers is presented. The proposed approach
allows a detailed analysis of the coupling among vehicle's shape, trajectory
control, and thermal protection system design. A set of simplified models are
developed to address this analysis and a multiobjective particle swarm
optimizer is adopted to obtain the set of Pareto optimal solutions. In order to
deal with an unconstrained multiobjective optimization, a
two-point boundary value problem is formulated to
implicitly satisfy the constraints on the atmospheric exit conditions. The
trajectories of the most promising solutions are further optimized in a more
refined dynamical system by solving an optimal control problem using a direct
multiple shooting transcription method. Furthermore, a more complete vehicle
control is considered. All the simulations presented consider an aerocapture at
Mars with a polar orbit of 200 km of altitude as target orbit.


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## Body

## 1. Introduction

Aerocapture is a  state-of-the-art technology
considered to reduce the cost of planetary exploration. This technique, firstly
proposed by Cruz in 1979 [1], allows the reduction of fuel cost for planetary
insertion by using atmospheric drag to decrease the total orbital energy of the
vehicle. The aerocapture is designed to aerodynamically decelerate a spacecraft
from hyperbolic approach to a captured orbit within a single pass through the
atmosphere with no propulsion exploitation. Once the vehicle enters the
atmosphere, bank angle modulation is used to safely remain within the flight
corridor, preventing skip-out or planetary impact. Propulsion is used for
attitude control and periapsis raise only.Several missions—such as Magellan and Mars Global
Surveyor—have already employed aerobraking strategy: multiple atmospheric
passes over an extended period of time allow to gradually get the desired
orbit; but in these cases an impulsive maneuver is first required to make the
target planet capture the spacecraft. Future missions, like either robotic and
human missions to Mars or the Titan Explorer, are considering using a lifting
body to perform an aerocapture maneuver at the arrival planet: a lifting body
is less sensitive to variation in the entry angle and the drag is easily
modulated. As counterpart, sophisticated guidance algorithms are required to
successfully drive the vehicle in the atmospheric path.Many studies in the last thirty years focused on the
aerocapture maneuver optimization. This optimization was mainly conducted in
terms of trajectory: the shape and the aerodynamic characteristics were fixed.
Within this framework many techniques were developed to optimize an aerocapture
maneuver focusing on different control variables and on the minimization of the
path constraints [2–6].Bearing in mind space utilization and exploration, the
payload mass delivery's capability is an open issue that leads to a wider
analysis of aeroassisted maneuvers. The shape definition could be considered as
an additional degree of freedom to enhance the overall maneuver efficiency. In
this frame, parametric studies analyzing the influence of different shapes on
the considered maneuvers were accomplished [7, 8].The influence of variation of shape on this class of
problems shows the high sensitivity the aeroassisted maneuvers have, and
confirms the need of a multidisciplinary approach. In this context, the work of
Sudmeijer and Mooij underlined the relevance of the shape optimization process
to improve the performance of reentry probes [9].A multidisciplinary-multiobjective approach for the
coupled vehicle's shape and trajectory optimization of aerogravity assist
maneuvers has been recently proposed by the authors [10]. The coupling among
vehicle's shape, trajectory control, and heating rates has been exploited and a
tool useful for the preliminary design of aerogravity assist maneuvers has been
developed.A similar architecture is here proposed to address the   preliminary design of aerocapture maneuvers from a
multidisciplinary-multiobjective standpoint. It is important to note that the
aerocapture modeling and the optimization problem setup is completely different
form the aerogravity assist one. In fact, a capsule-like vehicle is considered,
a different control strategy is employed (exploiting both the angle of attach
and the bank angle), and a different set of ordinary differential equations is
carried out for the simplified dynamical model. Furthermore, the optimization
process is aimed in this case to minimize the thermal protection system mass
and to maximize the volumetric efficiency of the capsule, thus optimizing the
payload mass delivery capability. Moreover, the aerocapture optimization
problem is more challenging, as it involves the solution of a boundary value
problem within each objective function evaluation. This expedient is necessary
both to avoid equality constraints and to bound the propellant required for the
periapsis raise.A multiobjective particle swarm optimizer (MOPSO) is
applied to detect the set of Pareto optimal solutions of the analyzed problem.
The particle swarm optimization (PSO) method has been introduced for the first
time by Kennedy and Eberhart in 1995 [11, 12]. Since its introduction several works have been
carried out on PSO improving the original method and proving its efficiency; an
overview of the more important ones is given by Poli et al. in [13]. In recent years there have
been several proposals to extend particle swarm optimization to multiobjective
optimization problems [14–18]. The algorithm implemented in this work is based on
that proposed by Coello et al., the adoption of a variable inertia and the
application of the preservation of feasible solution method (FSM) being the two
main differences.The paper is organized as follows. The models
developed for the multidisciplinary treatment of the maneuver are presented
first. In particular the configuration, the aerodynamics, the thermal protection system (TPS), and the dynamics
models are described. Some considerations on the clashing requirements of
maximizing the vehicle volumetric efficiency and reducing the TPS mass follow.
Subsequently the guidelines of the MOPSO implementations are given. The coupled
vehicle's shape and trajectory optimization is then presented and results are
discussed. As a conclusion the trajectory refinement is analyzed.


## 2. Models

## 2.1. Capsule

The capsules considered in this work are axisimmetric
vehicles whose geometry is described by means of five parameters(rn,rb,rr,θ,δ). These parameters are the vehicle nose radius, the
base radius, the rear-base radius, the front cone half-angle, and the rear-cone
half-angle, as shown in Figure 1.Figure 1
Shape parameters(rn,rb,rr,θ,δ) visualization.By allowing a quite wide search space for the
parameters, significantly different configurations can be obtained, from highly
blunted to slender vehicles, as reported in Figure2. The base area S=πrb2 is considered
as reference surface for aerodynamic coefficients. A constraint of 5 m3 on capsule volume is
considered. A density value ρv=230 kg/m3 is considered for a volume of
3.8 m3, equivalent to that of Mars
Express mission, whereas a reduced density value is considered for the
exceeding volume, as it serves only as structural mass.(a) Higly blunted vehicle and (b) slender
vehicle.
(a)(b)


## 2.2. Newtonian Flow

The aerodynamic properties for the capsule-like
vehicles described in the previous section are computed by summing up the
contributes given by each panel in which the geometry is discretized. The
Newtonian flow theory with Lee modification is applied for the estimate of the
pressure coefficient,Cp [19].The modified Newtonian flow is a local surface
inclination method in whichCp depends only on
the local surface deflection angle α; it does not depend on any surrounding flowfield.
Within this approximation, the classical expression for the pressure
coefficient can be derived as(1)Cp=Cpmax⁡sin⁡2α,where the maximum value of the
pressure coefficients, Cpmax, is evaluated at the stagnation point behind a normal
shock wave by(2)Cpmax⁡=p0−p(1/2)ρv2.In the former expression p, v, and ρ are the
asymptotic flow pressure, velocity, and density, respectively; p0 indicates the
stagnation point pressure. As the free stream collides only against the frontal
area of a body and it cannot curl around the body and collide against the back
surface, the shadow part of the body is characterized by free-stream pressure
and therefore Cp=0.In order to encompass high temperature effects, the
value ofp0 is computed
with the NASA Chemical Equilibrium with Application code [20]. As a result the computed
stagnation pressure takes into account of the high temperature effects that
characterize hypersonic flows.The aerodynamic code has been validated on the Viking
lander capsule hypersonic aerodynamics data from on-ground and on-board
measurements, which have been recently used by Edquist for comparison to the
LAURA Navier-Stokes code [21]. A high accuracy, average errors lower than 6%, is achieved for both drag and lift-to-drag
coefficients. The obtained accuracy is remarkably high and comparable to that
attained through the sophisticated CFD code LAURA.


## 2.3. Thermal Protection System

As aerodynamic efficiency is not a major issue for
aerocapture application, and the heating rate experienced are comparable to
previous Mars missions, ablative thermal protection systems are the
natural selection for this kind of maneuvers. In fact, ablative materials can
accommodate heating rates and heat loads through phase change and mass loss;
this represents the classical approach to TPS used for over 40 years in a broad
range of applications, and all NASA planetary entry probes (to date) have used
it [22].Therefore, the main issue for the aerocapture maneuver
stays in lowering the TPS mass fraction. In fact the mass saving gained by
reducing the propellant required for the achievement of the final orbit must
not be jeopardized by the need of a heavy heat shield. An analysis on past
NASA missions reveals a direct connection between the TPS mass fraction and the
total heatq0 experienced by
the vehicles at the stagnation point. Based on this consideration, the TPS mass
fraction is simply estimated through the power law fit curve:(3)TPS%=mTPSm=0.0091q00.51575.Neglecting the radiative
heating, the total heat is calculated integrating the well-known relation for
convective heating [23]:(4)q˙0=1.35(10−8)(ρrn)1/2v3.04(1−hwh0),in which hw is the wall
enthalpy, and h0 is the total
enthalpy.


## 2.4. Dynamics

Two different sets of ordinary differential equations
are chosen to describe the vehicle dynamics: a simpler formulation to
facilitate the solution of the coupled trajectory and vehicle's shape
optimization problem, and a more complex set for trajectory further refinement.The complete dynamical model, written in a local
noninertial reference frame, reads(5)r˙=vsin⁡γ,ϑ˙=vcos⁡γcos⁡ψrcos⁡φ,φ˙=vcos⁡γsin⁡ψr,v˙=Dm−gsin⁡γ,vγ˙=Lcos⁡σm−gcos⁡γ+v2cos⁡γr,vψ˙=Lsin⁡σmcos⁡γ−v2tan⁡φcos⁡γcos⁡ψr,in which the state vector (r,ϑ,φ,v,γ,ψ) is made up by
the orbital radius, the longitude, the latitude, the velocity, the flight path
angle, and the heading angle of the spacecraft. Furthermore, L=(1/2)ρSCLv2 and D=(1/2)ρSCDv2 are the
classical expressions for the lift and the drag force, σ is the bank
angle and g the
gravitational acceleration. (Consult [10] for a complete description of the dynamics.)In the first phase of the optimization process, when
the coupled shape and trajectory's optimization are considered, we are only
concerned with parameters directly related to enter a closed orbit, no matter
whether a specific three dimensional orbit the spacecraft will be finally
placed on. For this reason, the variables of interest arer and v only. In order
to analyze the behavior of these variables, the equation of flight path angle
must also be taken into account. The simplified model for the aerocapture
maneuver is made up by a set of three differential equations. To facilitate the
analysis, the equations of motion are nondimensionalized [4]. The dimensionless arc
length s replaces the
time t:(6)ds=βrevdt.The altitude is
nondimensionalized with(7)y=ρρe=e−βh,in which the altitude h is relative to
the planet's atmosphere boundary, h<0 means the
vehicle is within the atmosphere. Note that a simple exponential model for the
planetary density ρ=e−βh is considered,
in which β is the inverse
scale height of the atmosphere, whose boundary is fixed for Mars at 100 km of
altitude. With the concerned speed, the following expression is
used:(8)x=ln⁡(vev)2.In the previous equations, and
throughout the paper, the e subscript
refers to properties at the atmosphere's boundary. The remaining required
nondimensional parameters are defined by(9)ε=βreρeSCDm,δ=μ/reve2,
in which μ is the
gravitational parameter of the the planet. The complete nondimensional set of
equations is then(10)y˙=−βreysin⁡γ,x˙=εy+2δexβresin⁡γ,γ˙=ε2(CLCD)cos⁡σ+cos⁡γβre(1−δex).The initial values of x and y are known from
the entry condition in the planetary sphere of influence, whereas the entire
final state is univocally defined if the apogee of the exit trajectory is
prescribed to minimize the required Δv for the
circularization. In order to have the same number of differential equations and
boundary conditions, two differential equations must be added to the system
(10). For this purpose, the arc length is scaled as s=Tτ where τ belongs to the
interval [0,1], and T is the unknown
final arc length, which is an unknown constant. One additional differential
equation is then T˙=0. Furthermore, if a constant bank angle control law is
considered, the differential equation σ˙=0 can be used to
match the number of equations and the number of boundary conditions. The
constant bank angle approximation represents the simplest control law that can
be adopted as far as the achievement of the tridimensional final orbit is not
of concern. In this work frame, the complete set of equations in the new
independent variable τ is(11)y˙=T(−βreysin⁡γ),x˙=T(εy+2δexβresin⁡γ),γ˙=T(ε2CLCDcos⁡σ+cos⁡γβre(1−δex)),T˙=0,σ˙=0.This set of equations is
suitable to define the two-point boundary value
problem (TPBVP) described in detail in the following section.


## 3. Volumetric Efficiency and TPS Mass Percentage Analysis

To better
understand the results of the following section, the effects of the vehicle's
shape on the trajectory minimum altitude, on the heat shield percentage, and on
the volumetric efficiency are analyzed firstly. The simplified model of the
dynamic (11) is considered.The selected control law imposes both bank angleσ and the angle
of attack α to be constant;
the CL/CD is, therefore,
determined by the angle of attack α. The goal is to use the atmospheric path to reduce
the Δv required to
achieve a circular target orbit of 200 km of altitude. Within this frame, the
atmospheric path is used to model the incoming hyperbola into an elliptical
trajectory having the apogee at an altitude of 200 km, without considering the
achievement of the 3D target orbit. The achieved orbital plane as the vehicle
leaves the atmosphere, is assumed to be the target orbit plane: that matching
condition is assured by a proper choice of the entry plane in the planet's
atmosphere (i.e., the proper selection of the pericenter radius of the incoming
hyperbola). The satisfaction of this constraint is addressed within the
solution refinement process. The atmospheric path is computed by solving the
TPBVP after the vehicle aerodynamic coefficients, the volume, and the mass have
been computed. Some details of the TPBVP formulation are now illustrated.The conditions at the edge of the atmosphere are
labeled with the subscripte, those at Mars sphere of influence by ∞, and the entry and exit conditions with the
superscripts − and +, respectively.
The atmospheric boundary is settled at 100 km over Mars mean radius. The
2-body problem and (11) completely describes the system dynamics
outside and inside the atmosphere respectively [24]. The velocity magnitude at
the boundary of the sphere of influence is assigned to v∞−=5 km/s. At the
atmospheric entry, the initial radius re− is re and the entry
velocity ve− is computed by
the solution of the Vis-Viva equation:(12)12ve−2−μre−=12v∞−2.In order to apply a tangential Δv correction,
which represents the minimum propellant strategy, the apogee ra of the
trajectory obtained after the atmospheric phase is constrained to lie on the
target orbit. Thus, the values of γe+ and Δv required for
the circularization can be computed as a function of the final velocity ve+ through the
following procedure.(1)
Evaluate the
energy of the exit trajectory:(13)E=12ve+2−μre+.(2)
Compute the
semimajor axis and the eccentricity:(14)a=−μ2E,e=raa−1.(3)
Obtain the
angular momentum:(15)Γ=a(1−e2)μ.(4)
Finally,
compute the exit flight path angle and theΔv burn:(16)γe+=a⁢cos⁡(Γve+re+),Δv=μra−−μa+2μra. The procedure
admits solutions of Figure 3 for values of ve+ in the range [825.9,3527.4] m/s.Figure 3
Theve+-γe+ constraint and
the associated Δv.A subset of exit trajectories with the apogee on the
target orbit are sketched in Figure4. The Δvmax corresponds to
the unrealistic vertical exit from the atmosphere. In that case the spacecraft
would arrive at the apogee at zero velocity, resulting in a Δv=3453.7 m/s: the
circular target orbit velocity. The minimum Δv correction is
24.4 m/s and it is associated to the maximum exit velocity ve+=3527.4 m/s and to the
minimum exit flight path angle γe+=0 deg. The exit
trajectory can be identified with an Hohmann transfer connecting the circular
orbit at the atmosphere boundary to the target orbit, with the main difference
being only the apogee impulse to be required.Figure 4
Sketch of the periapsis raiseΔv.To facilitate the TPBVP numerical solution, a 3500 m/s
constraint onve+ is posed; the
resultant γe+ and Δv are 1.73 deg
and 52.6 m/s, respectively. It's worth noting that the computed trajectories
are not optimal from the Δv standpoint,
although the selected constraints assure them to be close to the theoretical
minimum (considering the required additional propellant). The proposed
procedure assures a complete knowledge of the final state x, y, and γ; the remaining unknowns, that is, the total arc
length T, the entry flight path angle γe− and the bank
angle σ, are computed by the TPBVP solver. A linear
multipoint method [25–27] is applied to solve the problem and the second order
solution derived by Vihn et al. [4] is used as first guess solution. The algorithm shows
quadratic convergence typical of Newton's method and converges on average
within 10 iterations. The integration interval is transcribed using 100 nodes
to assure absolute and relative accuracy of 10−8.TheΔv correction is
univocally determined once the constraints on the final conditions are
satisfied, therefore no Δv optimization is
required. As the Δv minimization is
substituted by a constraint satisfaction problem, the attention can be focused
on two other important aspects of the aerocapture design: the vehicle
volumetric efficiency maximization and the TPS mass fraction minimization. A
high volumetric efficiency has two main benefits: a smaller launcher fairing is
required as the vehicle is more bulky, and the vehicle's volume can be better
exploited for the spacecraft accommodation. The minimization of the TPS mass
ratio leads to a maximization of the mission outcome, as the percentage of
payload mass increases. It needs to be remarked that the TPS mass must be lower
than the propellant required to achieve the target trajectory without using the
aerocapture in order to consider the aerocapture beneficial. These goals are
monitored by the TPS mass fraction of (3) and(17)η=lmax⁡V,where lmax⁡ is the maximum
linear dimension of the capsule. The main difference between these two indexes
is that η is simply a
geometric value, whereas TPS% depends on the vehicle's geometry and on the
atmospheric trajectory.It is now shown that these indexes have clashing
behaviors, thus justifying the multiobjective optimization approach described
in the following section. The vehicles of Figures5 and 6 are considered to
clarify this concept: the first one is more compact and it is characterized by η=0.59m−2, the second has a bigger base
area and being shorter, resulting in η=0.71m−2.  Both the shapes have a volume of 5m3, and the same trim angle of
attack α=−20 deg. The bank
angle that guarantees the satisfaction of the constraint on the exit conditions
is σ=163.05 deg for the
first shape and σ=115.7 deg for the
second. Figures 7 and 8 describe the atmospheric path and the velocity history.
Shape 1 requires a steeper entry, a longer maneuver, and a lower altitude in
order to achieve the required kinetic energy loss. As a result, the vehicle
experiences a greater value of total heat load, therefore a greater TPS mass is
needed. More specifically, the TPS mass percentage are 9.48 and 7.44 for Shape
1 and Shape 2, respectively. In general, for
a given mass, a more compact vehicle (i.e., high volumetric efficiency) must
fly deeper into the atmosphere to lose its kinetic energy, thus experiencing
high thermal loads (i.e., high TPS mass). It should be underlined that within
the framework of constant vehicle volume and density, the performance index η plays a role
similar to that of the ballistic coefficient, defined as m/(SCD), and frequently used in literature. A high volumetric
efficiency translates into a high ballistic coefficient (see Table 1), thus a
great value of atmosphere density is required to produce a significant amount
of drag. Furthermore, for a fixed shape, the ballistic coefficient almost
linearly increases with the vehicle's dimension. That is the reason why special
devices like parachutes or ballutes are considered when the aerocapture
maneuver for human missions to Mars is studied.Table 1
Numerical results for the two analyzed
shapes.
ParameterShape 1Shape 2TPS%—9.487.44ηm−20.560.71αdeg−20−20σdeg163.18115.7CL/CD—0.150.31m0/(SCD)kg/m2135.9969.62hminkm43.0148.6γe−deg−8.57−8.11Figure 5
Shape 1, volumetric efficiency oriented
vehicle.Figure 6
Shape 2, TPS% oriented vehicle.Figure 7
Shape 1 and Shape 2 altitude
profile comparison.Figure 8
Shape 1 and Shape 2 velocity comparison.


## 4. Multidisciplinary Optimization

## 4.1. Multiobjective Particle Swarm Optimizer

As shown in
the previous section, it is difficult to identify a single objective function
when designing an aerocapture maneuver. In order to use classic optimization
codes (i.e., gradient based methods) to solve a multiobjective optimization
problem a common practise is to merge the different objective functions into a
single scalar objective function by means of weighting factors. This technique
requires an accurate selection of the weights, and it has, as major drawback,
the identification of a single optimal solution per run. On the contrary,
population-based optimizers can be more easily modified to deal with a vector
of objective functions delivering the entire set of Pareto optimal solutions.
Furthermore, particle swarm optimization seems particularly suitable for
multiobjective optimization mainly because of the high speed of convergence
that the algorithm presents for single-objective optimization [28]. In a multiobjective
optimization problem, the objective function is an M-dimensional
vector:(18)f(x)=(f1(x),f2(x),…,fM(x)).In this frame, a criterion to
compare vectors is necessary to identify the optimal solution set. The Pareto
dominance is the appropriate criterion to serve this aim, enabling the
solutions ranking [29].The MOPSO implemented for the solution of the problem
at hand is based on the following algorithmic flow.(1)
Randomly initialize a number of individuals or
particlesN within the
design space.(2)
Evaluate the objective function:(19)yi=f(xi)fori=1,…,N.(3)
Update the personal best
solutionpbest. The solutions are compared using the Pareto
dominance criterion. Thus, for each particle we have(20)pbest={xi,ifxidominatespbest,pbest,ifpbestdominatesxi,xiorpbest,randomly in the other cases.fori=1,…,N,(4)
Update global best listGbest. In the multiobjective problem, Gbest is the
analogous of the scalar global best gbest and it
represents the entire set of nondominated solutions. This list is updated by
processing the subset of nondominated solutions xj with j=1,…,N*≤N.(i)
Ifxj is dominated by
one of the solution belonging to the list, do not updated the list.(ii)
Ifxj dominates one
or more solutions belonging to the list, then add xj to the Gbest list and delete
the dominated solutions.(iii)
Ifxj neither
dominates nor is dominated by any solution belonging to the Gbest list, then
simply add xj to the
list.(5)
Update the global best solutiongbest. Note that the gbest is univocally
defined for a scalar objective function, whereas it must be opportunely chosen
within the Gbest list in the
multiobjective case. The selection of the gbest plays a key
role in obtaining a uniform set of Pareto optimal solutions. For this purpose,
a uniform 30-cell grid in the objective space is
defined at each iteration and the number of solutions belonging to each grid
cell is calculated. Based on this number, a roulette-wheel method is then
applied to promote the selection of gbest in a
low-populated grid cell.(6)
Compute the new particles position
by(21)xik+1=xik+vik+1Δtfori=1,…,N,in which vik+1 is the velocity
of the ith particle at
the (k+1) iteration,
given by(22)vik+1=wvik+c1r1xik−pbestΔt+c2r2xik−gbestΔt.(7)
Repeat (2)–(6) until the
convergence criterion is satisfied or the maximum number of iterations is
reached.The parametersc1 and c2 of (22) are
considered constant and equal to 2 during the optimization, assuring a balance
between local and global terms. A linear decrease of w with the
iteration number in the interval [0.4,1.4] is adopted. In
particular a greater value of the inertia enables a better exploration of the
search domain in the first phase of the optimization, whereas a lower value
allows a better analysis of the most promising areas of research space in the
subsequent phases. Note that if the position of a particle goes outside the
search space, the violated component of the decision vector takes the value of
the corresponding boundary and its velocity component is multiplied by a random
number between [−1,0].The maximum numbers of particle belonging to theGbest is fixed to 100
units. The same procedure adopted for selecting the gbest is used to
delete those solutions belonging to a highly populated grid-cell, if the
maximum list size is exceeded.The problem addressed with implemented MOPSO is
characterized by the presence of inequality constraints necessary to guarantee
a minimum aerodynamic performance and vehicle's length. As the feasible domain
inside the search space is sufficiently large the FSM is adopted for the
constraints handling [30].
More specifically the swarm initialization is performed randomly, but only
feasible solutions are retained. This implies that the first step of the
algorithm generally requires the evaluation of a number of solutions greater
than the population size. Furthermore, only feasible solutions are counted for
the gbest and pbest values during
the optimization. The initial velocity of the particle is set to be 0.The convergence criterion adopted is based on the
comparison of the average position of the non dominated solutions in the
objective space with the same average position of the previous 20 iterations.
If the componentwise difference of this two vectors is lower than1% the Pareto
set of optimal solutions is assumed to have been found. Furthermore, a maximum
number of iterations of 100 and a 20-particle swarm are considered. These
values are chosen, on the basis of several experiments, to assure an acceptable
repeatability of the Pareto optimal solution set with a limited computational
time.


## 4.2. Optimization Architecture

Since the minimization ofη and TPS% are
two clashing requirements, a multiobjective optimization architecture is
adopted. The overall optimization architecture is then given, the scheme of
Figure 9 can be used as a visual aid.
The optimization
variables are the four geometric parameters rn∈[0.1,0.9],rr∈[0.2,0.9],θ∈[20,70]deg,δ∈[5,60]deg, and the angle of attack α∈[−30,30]deg. Note that the vehicle nose radius and the rear
radius are expressed as fraction of the base radius, whose value is computed
to satisfy the constraint of 5m3 for the capsule's volume. The
sphere of influence entry velocity v∞−=5 km/s and the
vehicle's density ρv=230kg/m3 are user provided constants
necessary to define the optimization problem.Figure 9
Aerocapture
optimization scheme.For each set of optimization variables the vehicle's
aerodynamic performance at the trim angle of attack are computed. At this point
the TPBVP can be formulated and solved, delivering the total heat load
experienced by the vehicle. Theγe+ is constrained
to 1.73 deg as in Section 3, resulting in the same Δv of 52.6 m/s for
all the solutions. The two performance indexes are then simply evaluated and
the optimizer iterates until the convergence criterion is satisfied or the
maximum iteration number reached. Two simple inequality constraints are also
considered on the lift-to-drag ratio, |L/D|≥0.3, and vehicle length, l≥0.8 m. The fist one
is necessary to provide the vehicle with a means to accurately control the
atmospheric path, the second removes solutions with extremely short length from
the search space. A function evaluation takes on average 1.26 s on a Intel
Pentium 4, 2.53 GHz Desktop.


## 4.3. Experiments and Results

The values of the MOPSO parameters given in Section4.1 are the result of a tuning process based on several experiments in which
the swarm size, the maximum number of iterations, and the value of the inertia
have been changed. The average behavior of the algorithm is shown in Figure 10
with a plot of 30 simulations in which the solutions belonging to the Gbest list are
interpolated by means of cubic splines to avoid cluttering the plot. The result is satisfactory from the
preliminary design point of view. The algorithm is very effective in computing
the Pareto optimal solutions for low TPS mass vehicles; on the other hand
suboptimal solutions appear in the high volumetric efficiency region. This
behavior is mainly due to the difficulty in finding feasible solutions of the TPBVP
when compact capsules are considered. In these cases, is not always possible to
decelerate the vehicle's as required to satisfy the constraints imposed in the
TPBVP formulation. Furthermore, note that the poor behavior in the flat part of
the Pareto front is due more to the interpolation process than to the actual
values of the solutions. Only 5 simulations out of 30 stop for the satisfaction
of the convergence criteria, whereas the reamaining ones reach the maximum
number of iterations. Nevertheless, it has been noticed that increasing the
iterations does not significantly improve the quality of the solution, whereas
augmenting the computational time. These considerations underline the
difficulty of defining an appropiate convergence criteria in multiobjective
optimization.Figure 10
Cubic spline interpolated Pareto optimal sets.Thex in
Figure 11 shows the mean values of the two objective functions for each of the
Pareto set analyzed before. It is worth noting that the mean values are all
within a range of 5%. Their values also show that, as already pointed out,
the left branch of the Pareto front is more densely sampled than the right one.
The □ and • describe a
typical behavior of the algortithm when the constant value w=0.4 is employed for
the iniertia. In these cases the algorithm tend to locally converge to one of
the two branches of the Pareto set, as clearly enlighten in Figure 12.Figure 11
Mean
values of the Pareto optimal solutions.Figure 12
Two examples of local convergences (w=0.4).In Figure13 the Gbest list after 100
iterations of one of the 30 simulations performed is plotted.   The Pareto front confirms the considerations reported in Section 3: the
volumetric efficiency oriented solutions (bottom right) are compact capsules
with higher value of TPS mass; as opposite the TPS oriented solutions tend to
have high base radius and to be very short.Figure 13
Aerocapture Pareto optimal solutions.Table2 summarizes the numerical data of the five
solutions highlighted in the previous figure. All the solutions tend to the
minimum value allowed for the lift-to-drag ratio in order to decelerate the
capsule at higher altitudes, thus reducing the TPS mass ratio. Soln 2 to Soln 5
are characterized by almost the same value of the front cone half-angle which
allows to consider large nose radii. Soln 1 belongs to a different class of
vehicles: the front cone half-angle is the lowest allowed, and the forebody is
longer than the aftbody. The difference between these two classes of vehicles
is highlighted also by the different sign of the lift-to-drag ratio: Soln 1 is
the only shape that produces negative CL for negative
angles of attack. Eventually, note that TPS oriented solutions have higher
values of minimum altitude, shallower entry angles, and lower ballistic
coefficient.Table 2
Numerical
results for five solutions belonging to the Pareto optimal set.
ParameterSoln 1Soln 2Soln 3Soln 4Soln 5TPS%—13.2897.97.56.1ηm−20.460.550.630.680.89αdeg−29.45−21.13−19.3−20.3−19.27σdeg57.4116.9117.2115.8115CL/CD—−0.310.320.300.310.30m/(SCD)kg/m2299.72120.2686.1377.1145.02hminkm36.543.946.747.752.3γe−deg−9.08−8.50−8.25−8.19−7.73rbm1.161.371.591.692.19rnm0.731.221.421.521.94rrm0.580.470.450.480.82θdeg2168.517069.6470δdeg4156.594042.075.2The maximum of TPS% is 13.30, which results in
126 kg of TPS mass in the worse case solution. In order to establish the
aerocapture effectiveness, the TPS mass must be compared with the propellant
required for the circularization without employing aerocapture. This value is
computed in a 2-body approximation and the spacecraft is assumed to switch on
the thrusters at the hyperbola pericenter, located at 200 km of altitude. TheΔv required to
circularize the orbit is given by(23)Δv=μra+v∞−2−μraand the propellant fraction by rocket equation(24)mpm=1−e−(Δv/Ispg0),in which Isp is the thruster
specific impulse, and g0=9.81m/s2 represents the Earth
gravitational acceleration at sea level. For a value of Isp=400 s it results mp/m=0.594, which means that more than the 59% of the spacecraft
should be propellant, largely greater than the 13.30% of TPS. Therefore the
aerocapture represents an effective means to reduce the propellant compared to
a classical orbital circularization, thus significantly increasing the mass
specifically devoted to the payload. In the following section, the refinement
of the Soln 2 and Soln 3 is addressed, as they show a good compromise between
volumetric efficiency and TPS mass ratio.


## 5. Trajectory Refinement

The
dynamical model applied for the solution of the TPBVP considers the evolution
of the velocity, the altitude, and the flight path angle only, and terms of
orderΔr/re are neglected.In order
to address the problem of achieving the 3D target orbit the complete dynamical
model (3) is considered. The constant bank angle approximation is lifted and an
optimal control problem is formulated to optimize the bank profile. In the
trajectory refinement the vehicle's shape is fixed and the constant angle of
attack strategy is retained, taking advantage of the values computed in the
coupled trajectory-shape optimization. Thus, all the variables considered in
the former optimization are fixed parameters. The state and control vectors are x=(r,θ,φ,v,γ,ψ)T and u=σ, respectively. The initial conditions are expressed
by(25)re−=re,v∞−(xe−)=v∞−.The scalar constraint assures that the maneuver begins at the planet atmosphere
interface. The vectorial constraint states that the incoming velocity at the
planet sphere of influence corresponding to the initial state xe−, that is, v∞−(xe−), must be equal to the spacecraft incoming velocity v∞−, imposed by the heliocentric trajectory analysis. The
optimizers exploits the two degrees of freedom on the initial state to choose
the proper entry plane in the Martian atmosphere. The following three scalar
constraints are enforced on the final position:(26)re+=re,a⁢sin(Γe,z+Γe+)=π2,−μve+2−(2μ/re+)(1+‖ve+×Γe+μ−re+re+‖)=raThe first constraint assures that the trajectory ends at the atmospheric
boundary, whereas the remaining two guarantee the exit trajectory to be polar
and with an apogee lying on the target orbit. If the additional constraint γe+=0 is considered,
thus achieving the theoretical minimum Δv=24.4 m/s as
explained in Section 3, the local optimizer fails to converge. More
specifically, the tight relative tolerance of 10−8 on the
constraints satisfaction is violated. This problem is avoided if the constraint
on the final flight path angle is dropped and the Δv minimization is
addressed. Thus the objective function is(27)J=Δv=μra−ve+2−2μ(1re+−1ra),in which the first term represents the target orbit velocity and the second one
the vehicle's velocity at the apogee.For each
shape the TPS% is bounded by the value of the coupled shape-trajectory
optimization. In order to evaluate the constraint, the stagnation heat load is
added to the state vector and it is integrated along with the dynamics. The
trajectory is split into four multiple shooting intervals, and the integration
is performed adopting a8th order
fixed-step Runge-Kutta scheme with absolute and relative tolerances of 10−8. The optimal solution is found using a sequential
quadratic programming (SQP) optimizer.The
results obtained for the Soln 2 and Soln 3 are shown in Figures14–17. For
both the solutions the value of the minimum altitude is slightly lower than the
one found in the previous section due to a steeper entry flight path
angle. The optimizer
completely changes the bank control law; the bank modulation is exploited to
minimize the Δv and mainly to
match the target orbital plane. Note that the maximum value of the aerodynamic
forces on the trajectory is reached during the deepest phase in the atmosphere,
when the bank angle is not far from the value found in the coupled
shape-trajectory optimization. Furthermore, if most of the points describing
the optimal bank angle spline are substituted by the constant bank angle value
computed in the shape-trajectory optimization, only a slight change in the
trajectory is obtained. Moreover the trajectory refinement has a high
convergence rate when the constant bank angle law is used as first guess
solution. These considerations prove that the assumptions adopted in Section 4
deliver sufficiently accurate results.Figure 14
Soln 2 and Soln 3 refined solution: height profile.Figure 15
Soln 2 and Soln 3 refined solution:
velocity magnitude.Figure 16
Soln 2 and Soln 3
refined solution: flight path angle.Figure 17
Soln 2 and Soln 3 refined solution:
bank profile.TheΔv is 39.22 m/s
for Soln 2 and 38.25 m/s for Soln 3, close to the theoretical minimum of 24.4 m/s, and slightly lower than the 52.6 m/s constraint of the shape-trajectory
optimization. Furthermore, the TPS mass ratio is lower compared to the one
found in the shape-trajectory optimization for both of the solutions. This
result is a consequence of the shorter permanence of the vehicles in the lower
layers of Mars atmosphere. Note that, although the trajectory refinement
changes the value of the TPS mass, the trend highlighted in the previous
section still holds, that is, a higher ballistic vehicle requires a higher TPS
mass. The main results of the trajectory refinement are summarized in Table 3.Table 3
Aerocapture
trajectory refinement: numerical results.
hmin [km]γe− [deg]TPS%Δv [m/s]Soln 240.85−9.167.9839.22Soln 344.8−9.387.0538.25


## 6. Conclusions

The use
of an MOPSO for the optimization of an aerocapture maneuver at Mars from a
multidisciplinary-multiobjective standpoint is presented. More specifically,
the interaction among vehicle's shape, trajectory control, and TPS design is
taken into account in the preliminary design of the maneuver. The aerocapture
multiobjective optimization emphasizes the conflict between volumetric
efficiency maximization and thermal protection system mass ratio minimization.
Solutions that show a compromise between the two objective goals have shapes
similar to those adopted in previous landing missions to Mars. Aerocapture
maneuver is demonstrated to be a valid means for lowering the propellant
required to accomplish interplanetary missions if compared to classical
circularization maneuvers. The models adopted are suitable for Phase-A studies
of future aerocapture missions, and the proposed method properly matches the
requirements of concurrent engineering, which is nowadays the leading approach
in aerospace field. The MOPSO, although of simple implementation, effectively
compute the Pareto optimal solution set of a complex engineering problem using
low number of function evaluations. The advantage of having a set of Pareto
optimal solutions is of vital importance in aerospace field, where most of the
design processes are characterized by the interaction of several subsystems and
disciplines, and it is often impossible to identify a single performance index.
Based on these considerations, in the authors belief, MOPSO could be
successfully applied to a broad set of aerospace engineering problems,
especially in system design and trajectory optimization fields.



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*Source: 248798-2008-04-07.xml*