COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Topology Optimization of Spacecraft Structures Considering Attitude
Control Effort
Z. Kang*, C. Zhang
State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, 116024 China
Email: [email protected]
Abstract The topology optimization of spacecraft components for reducing the attitude control efforts is addressed.
Based on the derivation of the cold gas consumption of the three-axis stabilization actuators, it is pointed out that the
attitude control efforts associated with cold gas micro thrusters are closely related to the mass moment inertia of the
system. Therefore the need to restrict the mass moments of inertia of the structural components is highlighted in the
design of the load bearing structural components. The mathematical model of topology optimization for minimum
compliance under constraints of mass moment of inertia is presented and the numerical solution technique is discussed.
A Numerical example will be given for demonstration of the validity and applicability of the present problem
formulation.
Keywords: structural optimization, topology optimization, attitude control, moment of inertia
INTRODUCTION
A good performance of the attitude control is essential to the flight stability and maneuverability of outer spacecrafts.
For a certain type of three-axis stabilized miniature spacecrafts which are designed for outer-space missions and
required to possess both good qualities of attitude control and high maneuverability, this is implemented by active
control, typically, using cold gas thruster modules.
As the actuator of the active attitude control system, the cold gas thruster produces reaction forces by ejecting high
pressure compressed propellants such as nitrogen gas. These forces are needed by stabilization or attitude maneuver of
the spacecraft. Cold gas thrusters are suitable to generate pulse forces and are widely adopted in applications requiring
high precision attitude control. However, the spacecrafts using this type of actuators cannot function as expected any
longer once the propellant is exhausted. On the other hand, for a miniature spacecraft, the size and the weight of the
propellant storage subsystem and even of the whole system are closely related to the amount of the gas propellant it
carries in the mission. As will be exposed by the present study, the gas consumption for attitude control actuators using
compressed cool air is proportional to the mass moment of inertia of the whole system. This hence highlights the need
to account for the attitude control effort (the propellant consumption) in the design of the structure, particularly, in the
conceptual design stage. This paper intends to accomplish this task by extending the conventional topology
optimization problem into one with restrictions on mass moments of inertia.
Structural topology optimization has attracted intensive attentions both in theoretical research and practical
applications for the last two decades. It aims to seek the optimal material layouts of a load bearing structural
components. Since topology optimization is usually involved in the conceptual design stage, it becomes a rational
method for creative structural designs. Continuum topology optimization problems are conventionally formulated as to
minimize the structural weight (or material volume) or to optimize the structural performance (compliance, natural
frequencies, etc.) under material volume constraint. The numerical analysis-based topology optimization methods for
continuum can be broadly classified into two categories (Rozvany 2001, Eschenauer and Olhoff 2001) – macroscopic
approaches and microscopic approaches. The former include the Bubble Method and recently developed Level Set
method; examples of the latter are the Varied Thickness method, the Homogenization method (Bendsoe and Kikuchi
1988, Zhang and Sun 2005), the Evolutionary Structural Optimization method (ESO, Xie and Steven 1997) and the
SIMP method (Bendsoe 1997). In the SIMP method, an artificial material model is introduced. The relation of the
elastic stiffness of the material to the relative density is assumed to obey a power law, such that the intermediate density
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can be suppressed. Recent investigations reveal that this assumption may be physically meaningful, which further
strengthens the theoretical background of the SIMP method. Recently, some numerical techniques such as the filter
techniques, the Perimeter Control method (Fernandes et. al, 1999) have been developed to alleviate the mesh
dependency and checkerboard problems. For state-of-the-art in structural topology optimization, one is referred to
recent review papers by e.g. Rozvany (2001) and Eschenauer and Olhoff (2001).
In applications regarding structural optimization of spacecraft structures, topology optimization has been widely used
in design for minimum compliance or maximum fundamental frequency under material volume constraint. However,
to the authors’ knowledge, the structural topology optimization problem incorporating the attitude control effort of
spacecrafts using cold gas thrusters has not been addressed in literatures.
In this paper, with the background of structural design problem for miniature spacecrafts utilizing three-axis attitude
control system, the consumption rate of cold gas versus the mass moments of inertia as well as the mass center position
is investigated. It is pointed out that, the control effort for the attitude stabilization of limit-circle-oscillating style is
promotional to the mass moment of inertia. Thus arises the structural topology optimization problem considering the
attitude control effort. The SIMP approach is used in the mathematical statement of the problem, where the material
density of the artificial material for each element is taken as the design variable. In the formulation of topology
optimization concerning moment of inertia, both the cases of the single-constraint problem and those of the
multiple-constraint one are addressed. In the former problem, a specified limit on the mass moment of inertia replaces
the material volume/weight restriction imposed on a conventional topology optimization problem, while in the latter
problem, constraints on other structural performance such as weight, position and mass center can be also present. The
numerical solution techniques are then discussed. For the solution of the problem with a single constraint on the
moment of inertia, the Optimality Criteria method is employed. To this end, we employ a design-variable updating
scheme derived based on the Kuhn-Tucker optimality condition. Therein a sensitivity filtering technique proposed by
Sigmund (1997) is used to alleviate numerical instabilities such as checkerboard patterns and mesh-dependency. For
problems with multiple constraints, in view of the fact that an optimality criterion is actually not available, the
gradient-based Mathematical Programming approach can be used. A numerical example regarding a typical load
bearing structure for a spacecraft is given and the applicability of the proposed model of topology optimization is thus
illustrated.
RELATION BETWEEN ATTITUDE CONTROL EFFORT AND STRUCTURAL PROPERTIES
We assume that the spacecraft is subject to no perturbation moments. In the principle axis system, the dynamic
equations governing the attitude control problem of a spacecraft is expressed as：
⎧ I xω& x + ( I z − I y )ω yω z = M cx
⎪
⎨ I yω& y + ( I x − I z )ω xω z = M cy
⎪ I ω& + ( I − I )ω ω = M
y
x
y x
cz
⎩ z z
(1)
where M cx , M cy and M cz are the control moments of the three-axis attitude control system in the direction of roll,
pitch and yaw, respectively; ω x , ω y and ωz are the velocity components; I x , I y and I z are the mass moments of
inertia of the vehicle with respect to the principle axis. If the attitude disturbance is small such that the coupling of the
attitude movements in different directions, the dynamic equations can be further simplified as
⎧ I xϕ&& = M cx
⎪ &&
⎨ I yθ = M cy
⎪ I ψ&& = M
cz
⎩ z
(2)
For three-axis stabilized spacecraft using cold gas thrusters as actuators, the attitude control algorithm is typically
nonlinear since the thrust cannot be continuously adjusted. In conjunction with the actual nonlinear control law
employed and the dynamic equation, the propellant flow rate can be determined. In what follows, as an example, the
gas consumption for the attitude control in pitch channel versus the mass moment of inertia and the position of the mass
center will be derived.
First of all, the attitude control actuators should be capable of producing enough thrust which satisfies the maneuverability
requirements of the spacecraft. To specify, the thrust of the actuators in the pitch channel is determined by
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F = I y aθ / l
(3)
where aθ is the required maximum angle acceleration in the pitch channel, l is the distance from the thruster nozzle to
the mass center of the spacecraft.
As a rule, a miniature spacecraft which utilizes cold gas thrusters as the attitude control actuators employs a control law
based on feedback of both its position (angles) and angular velocities. Noting the effect of the thruster dead zone, the
area in which the applied power is insufficient to create a desirable response of the thruster), we consider a relay
control law in this study. Therein the output force moment of the thruster is expressed as
⎧− Fl if θ >θ 0 , θ& >-θ&0
⎪⎪
M cy (θ , θ&) = ⎨0
if -θ 0 ≤ θ ≤ θ 0 , θ& ≥ θ&0
⎪
if θ < −θ 0 , θ& < θ&0
⎪⎩ Fl
(4)
where, θ 0 and θ&0 are the angle and angular velocities at which the valve of the thruster is opened or closed,
respectively; F is the control force provided by the thruster and is assumed to be constant when the thruster is
working, l is the distance from the thruster to the mass center of the system. It can be easily proved that the
stabilization of the spacecraft attitude under any initial attitude perturbation corresponds to a limit cycle oscillation in
the phase space and this limit cycle passes through the point ( θ 0 , θ&0 ).
When the attitude is disturbed, the spacecraft will quickly reach the state of stable limit cycle oscillation after a
relatively short transient process. Thus it is reasonable to assume that the high pressure cold gas is mostly consumed in
the limit cycle oscillation process. In what follows, the cold gas flow rate is to be determined with reference to the limit
cycle oscillation mode. To this end, we consider the solution of the following equation under initial
conditions θ = θ 0 , θ& = θ&0 :
⎧− Fl if θ >θ 0 , θ& >-θ&0
⎪⎪
if -θ 0 ≤ θ ≤ θ 0 , θ& ≥ θ&0
I yθ&& = ⎨0
⎪
if θ < −θ 0 , θ& < θ&0
⎪⎩ Fl
(5)
The integral of Eq. (5) over the first cycle yields
⎧
⎪θ
⎪
⎪⎪θ
⎨
⎪θ
⎪
⎪
⎪⎩θ
Fl 2
t ,
= θ 0 + θ&0t −
2I y
= θ − θ& (t − t ),
0
θ& = θ&0 −
θ& = −θ&0
0 −1
0
Fl
t
Iy
if 0 ≤ t ≤ t0 −1
if t0 −1 < t < t1− 2
Fl
Fl
= −θ 0 − θ&0 (t − t1− 2 ) +
(t − t1− 2 ) 2 , θ& = −θ&0 + (t − t1− 2 ) if t1− 2 ≤ t ≤ t2 − 3
Iy
2I y
= −θ 0 + θ&0 (t − t2 − 3 ),
if t2 − 3 < t < t3− 0
θ& = θ&0
(6)
where,
t0 −1 =
2θ&0 I y
Fl
, t1− 2 =
2θ&0 I y
Fl
+
4θ& I
4θ& I
2θ 0
2θ
4θ
, t 2 − 3 = 0 y + 0 , t3 − 0 = 0 y + 0
&
&
θ0
θ0
θ&0
Fl
Fl
(7)
The period of the limit circle oscillation is
T=
4θ&0 I y
Fl
+
4θ 0
θ&
(8)
0
The limit circle is schematically depicted as in Fig. 1, where 0,1,2,3 denote the points in the phase plane where the
valves of the thrusters are opened or shutted. From Eq. (7), the impulse width of the thrusters Tp can be expressed as
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Tp = t0 −1 =
2θ&0 I y
(9)
Fl
It can be deduced from Eqs. (8) and (9) that the ideal average gas consuming rate over a limit cycle period is
2Tp F /( I s g )
F 2Tp2l
&
=
m=
T
I s g (Tp2 Fl + 2θ 0 I y )
(10)
Substituting Eq. (3) into Eq. (10), one obtains
m& =
Tp2 aθ2 I y
(11)
I s gl (Tp2 aθ + 2θ 0 )
θ&
3
θ&0
−θ 0
2
0
0
−θ&0
θ0
θ
1
Figure 1: Phase diagram of limit cycle oscillation for the attitude stabilization process using cold gas thrusters
where I s is the specific thrust of the attitude control actuators, g is the gravity constant, l is the length of the force
arm and is associated with the positions of the thruster and the mass center. Obviously, for specified requirements on
the maneuverability, given thruster performance and force arms, the gas consuming rate is proportional to the mass
moments of inertia of the whole spacecraft. Since the inertia of the load bearing structure contributes most to the inertia
of the whole system of the miniature spacecraft particularly studied in this paper, its gas consuming rate increases with
the mass moment of inertia of the structure.
MATHEMATICAL MODEL OF STRUCTURAL TOPOLOGY OPTIMIZATION CONSIDERING
ATTITUDE CONTROL EFFORT
As aforementioned, a spacecraft with a smaller mass moment of inertia demands less gas supply for the attitude control
thrusters. This will prolong the service life accordingly, or further decrease the weight as well as the size of the
spacecraft itself by allowing for smaller compressed gas storage devices. In this context, we point out here that an
effective way of reducing the attitude control efforts is to implement the optimal design of the load bearing structures
under constraints on mass moments of inertia, particularly, by employing the topology optimization techniques in the
conceptual design stage.
Conventional topology optimization of continuum structures is formulated as a minimum compliance problem under a
material volume constraint, as expressed by
N
min
c(x) = UT KU = ∑ ( xe ) p uTe k 0u e
subject to
V (x)
= fV
V0
x
e =1
(12)
0 < xmin ≤ xe ≤ xmax = 1, e = 1, 2, ..., N
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Where c represents the strain energy, x is the vector of design variables, V is the material volume, V0 is the design
domain volume, f v is the prescribed volume fraction, K is the global stiffness matrix, U and F are the vectors of
nodal displacement and applied force, respectively, u e denotes the elemental displacement vector, k 0 is the stiffness
matrix of an element with unit density, N is the total number of the elements, p is the penalty factor, which is usually
set to p=3. In order to avoid numerical difficulties caused by non-positive definite stiffness matrix, a lower limit
x min >0 is imposed to the material density.
The problem of structural topology optimization under a constraint on the mass moment of inertia can be
mathematically stated as
N
min
x
subject to
c(x) = UT KU = ∑ ( xe ) p uTe k 0u e
e =1
I (x) = I max
(13)
0 < xmin ≤ xe ≤ xmax = 1, e = 1, 2, ..., N
where I and I max are the actual and upper limit of the mass moments of inertia with respect to a specified axis,
respectively. This problem differs from the conventional one in that it replaces the constraint on the material volume
with the constraint on the mass moment of inertia.
In applications regarding design of spacecrafts, other structural performances such as the structural weight, the
position of the mass center and the eigenfrequencies are usually also subject to specified restrictions. A topology
optimization problem under constraints both on mass moment of inertia and other performance functions is thus
proposed as
N
min
x
subject to
c(x) = UT KU = ∑ ( xe ) p uTe k 0u e
e =1
I (x) ≤ I max
g k (x) ≤ 0 k = 1, 2, ..., m
(14)
0 < xmin ≤ xe ≤ xmax = 1, e = 1, 2, ..., N
For the multiple constraint problem expressed by Eq. (14), a heuristic resizing scheme based on the optimality
condition is hardly available. Consequently, the optimization model is suitably solved by mathematical programming
methods (Kang et. al 2005).
The MMA algorithm (Svanberg 1987) has been reported to behave well in the solution of continuum topology
optimization problems. Therefore, we employ this method for solving the problem with constraints on both moment of
inertia and material volume.
NUMERICAL EXAMPLE
An example problem with regard to design of the load bearing structure of a spacecraft module is solved to test the
proposed topology optimization method. The structure is a cylinder shell with weight-saving holes, the number and
layouts of which are to be determined using topology optimization method so that the mass moment of inertia of the
structure can be controlled.
Fig. 2 shows the geometrical dimensions of the design domain. The structure is subject to an inertia force induced by an
acceleration of 98N/s 2 and external forces distributed along four line segments, which are actually caused by inertia
forces of the lumped mass attached to the structure. The mass density, the Young’s modulus and the Poisson’s ratio of
the material are 2.8 × 103 Kg/m3 , E = 7.9 × 1010 N/m 2 and ν = 0.3 , respectively. The thickness of the shell is
t = 4mm . A constraint on the mass moment of inertia about the pitch axis I ≤ 4.0 ×10−2 Kg ⋅ m 2 , and another on the
material volume fraction V (x)/V0 ≤ 0.5 , are to be observed.
The optimal structural layout obtained by the topology optimization is shown in Fig. 3. In the figure, the material with a desity
lower than a prescribed threshold has been removed from the structure. For this design, the mass moment of inertia and the
material volume fraction are 4.0 × 10−2 Kg ⋅ m 2 and 0.5 , respectively. The corresponding strain energy is 0.354N ⋅ m .
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Figure 2: Dimension and loading condition of the load bearing structure
(a) top view
(b) side view
(c) from arbitrary angle
Figure 3: Optimal layout under constraints on mass moment of inertia and volume fraction
When compared with the optimal solution obtained under a single constraint on material volume, the structural layout
as in Fig. 3 shows a significant difference.
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CONCLUDING REMARKS
This paper addresses the topology optimization of spacecraft components for reducing the attitude control efforts. The
driving torque requirements for attitude control of a spacecraft and the gas consumption rate are shown to be closely
related to the mass moment inertia of the system. In such a circumstance, it becomes meaningful to restrict the mass
moment of inertia of the structural components in order to reduce the attitude control efforts. The present paper
proposes a formulation for structural topology optimization of this type of spacecrafts, in which the mass moments of
inertia are involved as design constraints. A numerical example shows the feasibility of the present problem statement.
It is worth remarking here that the topology optimization problem exposed in this study may also present in other
engineering applications. For example, it is often desirable to reduce the moments of inertia of a robot arm with respect
to the pivot axes, so that the arm can move rapidly back and forth under a given driving torque of the electrical motor.
Similar requirements are often encountered in the design of rotating parts of some other mechanisms, such as the head
arms in a hard disk drive. These designs raise the optimal material lay-out problems in which the moment of inertia
should be accounted for in the constraint conditions or in the objective function. The present study can be easily
extended to these applications.
Acknowledgements
The supports of Natural Science Foundation of China under Grant 90305019, 10421202 are gratefully acknowledged
by the authors.
REFERENCES
1.
Bendsoe MP. Optimization of Structural Topology, Shape and Material. Springer, Berlin, Germany, 1997.
2.
Bensdoe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenization method.
Computer Methods in Applied Mechanics and Engineering, 1988; 71: 197-224.
3.
Eschenauer HA, Olhoff N. Topology optimization of continuum structures: a review. Applied Mechanics
Reviews, 2001; 54(4): 331-389.
4.
Fernandes P, Guedes JM, Rodrigues H. Topology optimization of three-dimensional linear elastic structures
with a constraint on “perimeter”. Computers and Structures, 1999; 73: 583-594.
5.
Kang Zhan, Zhang Chi, Cheng Gengdong. Structural topology optimization considering mass moment of inertia.
Proc. 6th World Congress on Structural and Multidisciplinary Optimization (WCSMO6), Rio de Janeiro, Brazil,
30 May - 3 June, 2005.
6.
Rozvany G. Aims, scope, method, history and unified terminology of computer-aided topology optimization in
structural mechanics. Structural Optimization, 2001; 21(2): 90-108.
7.
Sigmund O. On the design of compliant mechanisms using topology optimization. Mech. Struct. Mach., 1997;
25: 495-526.
8.
Svanberg K. The method moving asymptotes ⎯ a new method for structural optimization. International Journal
for numerical Method in Engineering, 1987; 24: 359-373.
9.
Xie YM, Steven GP. Evolutionary Structural Optimization. Springer, Berlin, Germany, 1997.
10. Zhang WH, Sun SP. Scale-related topology optimization from unit cell homogenization to superelement model.
Proc. 6th World Congress on Structural and Multidisciplinary Optimization (WCSMO6), Rio de Janeiro, Brazil,
30 May - 3 June, 2005.
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