design and attitude control of a satellite with variable ...a satellite with variable geometry would...
TRANSCRIPT
Design and Attitude Control of a Satellite with Variable
Geometry
Miguel Fragoso Garrido de Matos Lino
Thesis to obtain the Master of Science Degree in
Aerospace Engineering
Examination Committee
Chairperson: Professor Fernando Josรฉ Parracho Lau
Supervisor: Professor Paulo Jorge Soares Gil
Committee: Professor Joรฃo Manuel Gonรงalves de Sousa Oliveira
November 2013
i
Resumo
Exploramos um mรฉtodo de controlo de atitude de satรฉlites alternativo a mecanismos convencionais como
rodas de inรฉrcia ou thrusters โ controlo atravรฉs da mudanรงa da configuraรงรฃo do satรฉlite.
Um satรฉlite de geometria variรกvel รฉ concebido como sendo composto por partes mรณveis que podem ter
como รบnico objectivo controlar a orientaรงรฃo do satรฉlite ou podem ser componentes de outros sub-sistemas
que, alรฉm das suas respectivas funcionalidades, podem servir tambรฉm como ferramentas de controlo de
atitude.
Reescrevemos as equaรงรตes da dinรขmica de corpo rรญgido de modo a obter uma forma geral das equaรงรตes
de Euler nas quais a variaรงรฃo de geometria do corpo รฉ tida em conta. As influรชncias que a mudanรงa de
configuraรงรฃo tem na energia cinรฉtica e no momento angular do corpo sรฃo discutidas.
Apresentam-se vรกrios modelos simples de geometrias de satรฉlites que podem mudar a sua forma. Os
tensores de inรฉrcia de cada uma das geometrias รฉ obtido em funรงรฃo da configuraรงรฃo do satรฉlite em causa.
A capacidade de mudanรงa de configuraรงรฃo de um dos satรฉlites รฉ aplicada a um caso prรกtico. Considera-
se um satรฉlite que requer que a sua taxa de precessรฃo esteja sincronizada com a velocidade orbital, ao
longo de uma รณrbita elรญptica.
Tentamos explorar a capacidade de um dos satรฉlites concebidos variar os trรชs momentos principais de
inรฉrcia, bem como avaliar a estabilidade do movimento em torno dos eixos principais de inรฉrcia.
Palavras-chave:
Satรฉlite, Atitude, Controlo, Geometria-Variรกvel.
ii
Abstract
An alternative attitude control method to conventional orientation control mechanisms such as momentum
wheels or thrusters is explored - control by varying the configuration of the satellite.
A satellite with variable geometry would be composed by movable parts whose motion is controlled by the
action of actuators. The movable parts can have the single purpose of controlling the attitude of the satellite
or can be sub-systems, with certain primary purposes concerning the operation of the satellite, which could
also be used to control the attitude.
The rigid body dynamics equations are rewritten to obtain a general form of the Euler equations, where
shape changing capability of the body is taken into account. The influence of configuration changing on
kinetic energy and on angular momentum is also discussed.
Secondly, we envision several simple models for satellite geometries that are able to modify its shape. The
inertia tensors of the geometries are determined as function of the satellite's configuration.
The configuration changing capability of one of the satellite geometries is applied for a practical case. We
consider a satellite that requires its precession rate to be synchronized with the non-uniform orbital velocity
during a period of an elliptic orbit.
We explore the possibility of one of the envisioned satellites of changing the three moments of inertia and
assess the stability of rotation around the principal axes.
Keywords
Satellite, Attitude, Control, Variable-Geometry.
iii
TABLE OF CONTENTS
1 Introduction ...................................................................................................................................... 1
1.1 Objective .................................................................................................................................. 1
1.2 Attitude Control of Spacecraft ................................................................................................... 1
1.3 Satellites with Variable-geometry .............................................................................................. 2
1.4 Current Problems of Attitude Dynamics ..................................................................................... 2
1.4.1 Variable-Structure .............................................................................................................3
2 Attitude Dynamics of a Variable Geometry Body ............................................................................... 5
2.1 Angular Momentum .................................................................................................................. 5
2.2 Equations of Rotational Motion for a Variable-Geometry Body................................................... 6
2.3 General Form of Euler Equations .............................................................................................. 7
2.4 Alternative Derivation .............................................................................................................. 10
2.5 General Euler Equations in the Principal Frame ...................................................................... 11
2.6 Non-Rigid Terms .................................................................................................................... 14
2.6.1 Two-plane Symmetry ...................................................................................................... 14
2.7 Kinetic Energy of a Variable-Geometry Body ........................................................................... 16
2.7.1 Non-Rigid Terms ............................................................................................................. 17
2.7.2 Simplification of the Kinetic Energy Expression ............................................................... 19
2.7.3 Time-derivative of the Kinetic Energy .............................................................................. 20
2.8 Conservation of Angular Momentum ....................................................................................... 21
2.9 Geometrical Solutions for the Dynamics of a Variable Geometry Body .................................... 21
2.9.1 Ellipsoid of the Kinetic Energy ......................................................................................... 22
2.9.2 Ellipsoid of the Angular Momentum ................................................................................. 22
2.9.3 Geometrical Solution ....................................................................................................... 23
2.10 Modified Euler Equations in Terms of Euler Angles ................................................................. 26
2.10.1 Intermediate Frame for an Axisymmetric Body ................................................................ 27
2.11 Angular Momentum For a General Variable-Geometry Body ................................................... 29
2.11.1 Axisymmetric Variable-Geometry Body ........................................................................... 29
iv
3 Design of Variable Geometry Satellites ........................................................................................... 31
3.1 Connected Scissor Mechanism ............................................................................................... 31
3.1.1 Geometry ........................................................................................................................ 31
3.1.2 Body Frame .................................................................................................................... 31
3.1.3 Inertia Tensor .................................................................................................................. 32
3.1.4 Triangle Connected Scissor Mechanism .......................................................................... 36
3.1.5 Point Masses .................................................................................................................. 37
3.2 Octahedron............................................................................................................................. 40
3.2.1 Geometry ........................................................................................................................ 40
3.2.2 Body Frame .................................................................................................................... 41
3.2.3 Inertia Tensor .................................................................................................................. 41
3.2.4 Modified Octahedral ........................................................................................................ 42
3.3 Dandelion ............................................................................................................................... 43
3.3.1 Geometry ........................................................................................................................ 43
3.3.2 Body Frame .................................................................................................................... 44
3.3.3 Inertia Tensor .................................................................................................................. 44
3.4 Eight Point Masses ................................................................................................................. 45
3.4.1 Geometry ........................................................................................................................ 45
3.4.2 Body Frame .................................................................................................................... 46
3.4.3 Trajectory of the Masses ................................................................................................. 46
3.4.4 Geometric Relations with Axisymmetry ............................................................................ 48
3.5 Control Possibilities of axisymmetric satellites ......................................................................... 50
3.6 Compass ................................................................................................................................ 50
3.6.1 Geometry ........................................................................................................................ 50
3.6.2 Body Frame .................................................................................................................... 51
3.6.3 Inertia Tensor .................................................................................................................. 51
3.7 Discussion .............................................................................................................................. 52
4 Solutions of the Free Variable-Geometry Satellite Motion ............................................................... 53
4.1 Synchronous Rotation in an Elliptical Orbit with a Variable-Geometry Satellite ........................ 53
v
4.1.1 Analytical Approximation for Keplerโs Equation ................................................................ 53
4.1.2 Precession Rate as Function of Time .............................................................................. 55
4.1.3 Configuration Variation with Time .................................................................................... 55
4.1.4 Possible Solutions for Synchronous Rotation................................................................... 57
4.1.5 Example: Geosynchronous Orbit ..................................................................................... 58
4.2 Moments of Inertia Swap ........................................................................................................ 62
4.2.1 Relations between Moments of Inertia ............................................................................. 62
4.2.2 General Form of the Euler Equations for the Compass .................................................... 63
4.2.3 Approximated Solution .................................................................................................... 65
4.2.4 Stability Analysis ............................................................................................................. 69
5 Conclusions ................................................................................................................................... 73
5.1 Future Work ............................................................................................................................ 74
6 References..................................................................................................................................... 75
vi
List of Figures
Figure 1 โ Lateral perspective from the ๐ฅ axis of a Connected Scissor Mechanism of 6 bars of same
length, with a configuration ๐ผ1 on the left and a configuration ๐ผ2 at the right, being ๐ผ2 > ๐ผ1. .................. 32
Figure 2 โ Top view from the ๐ง axis of a Connected Scissor Mechanism with ๐ = 3 where the angle ๐ and
distances โ and ๐๐๐๐ ๐ผ are represented. ................................................................................................. 33
Figure 3 โ Connected Scissor Mechanism with 6 bars, 3 masses ๐1 at the center of the bars, displayed in
red, and 6 masses ๐2 at the ends of the bars, displayed in blue. ........................................................... 38
Figure 4 โ Octahedron at a configuration ๐ผ, where the body frame is displayed as well as the angle of 2๐ผ
between two opposite bars. .................................................................................................................... 41
Figure 5 โ Dandelion with ๐ = 4 bars at a configuration of ๐ผ and the body frame centered at the center of
mass for this particular configuration. ..................................................................................................... 43
Figure 6 โ Eight Point Masses in space. The trajectories of the masses so that ๐ด and ๐ต are kept constant
are visible in dashed curves. .................................................................................................................. 46
Figure 7 โ Compass for two different configurations, ๐ผ2 and ๐ผ1, with ๐ผ2 > ๐ผ1.The body frame is centered
at the center of mass.............................................................................................................................. 51
Figure 8 - Minimum ๐ผ0 as function of the eccentricity of the orbit for different number of bars. ................ 58
Figure 9 โ Configuration as function of time ๐ผ(๐ก) during two sidereal days in geosynchronous orbit with
๐ = 0.1 for different parameters. ............................................................................................................. 59
Figure 10 - Configuration as function of time ๐ผ(๐ก) during two sidereal days in geosynchronous orbit with
๐ = 0.01 for different parameters. ........................................................................................................... 59
Figure 11 - ๐ด/๐ถ and ๐ต/๐ถ ratios as function of the configuration ๐ผ for the Compass. ................................ 62
Figure 12 โ Perspectives from the ๐๐ฅ axis of the ellipsoids of the kinetic energy, at darker orange, and of
the angular momentum, at brighter orange, where the curve of the intersection of the ellipsoids is
displayed in blue. The configuration at the left is of ๐ผ0 and at the right is of ๐ผ. ........................................ 69
vii
List of Tables
Table 1- Sign of variables depending on the nutation angle and the bodyโs oblateness. .......................... 30
Table 2 โ First possible values for ๐. ...................................................................................................... 39
Table 3 - Configurations of stability for the Compass .............................................................................. 71
viii
1
1 INTRODUCTION
1.1 OBJECTIVE
Control of the rotational motion of a satellite is often necessary to fulfill the mission requirements. In this
work, we explore the possibility of a satellite with variable geometry and moments of inertia that can vary
according to a control law determined by the user. Some configurations are defined and some possibilities
are explored.
1.2 ATTITUDE CONTROL OF SPACECRAFT
Frequently, controlling the satelliteโs attitude is of critical importance. For example, it is fundamental for a
communication satellite to have its antennas correctly oriented. Other examples are spacecraft with solar
panels that are required to be facing the sun so that energy absorption can be optimized or a reconnaissance
satellite whose camera should be pointed to the target. Spacecraft attitude control is also necessary
because satellites are subject to space environmental torques. Examples of this are gravity gradient,
aerodynamic, magnetic or solar radiation pressure torques [1]. Even though the space industry did only
appear on the XX century, similar and simpler mechanical problems, such as the dynamics of tops, had
been solved before.
The first exact analytic solutions for rigid body dynamics problems were obtained around the XIX century.
Firstly, both the dynamics and kinematics of a torque-free rigid body, also called Euler-Poinsot problem,
were solved [2,3]. Classical problems concerning the dynamics and kinematics of tops with different
relations between their principal moments of inertia were solved taking into account the gravity torque [4,5].
Methods for obtaining the solution of the rotational motion of rigid bodies have been used in a broad range
of mechanical applications, including spacecraft control.
The control of the attitude of a spacecraft can be accomplished by several conventional ways, using
mechanisms such as momentum wheels or thrusters [6]. Momentum wheels have an electric motor attached
to a flywheel in such a way that the rotation speed of the wheel can be controlled by computer. This type of
control is precise and requires small amounts of energy, but it does not allow for large angle maneuvers or
changing the angular momentum. On the other hand, thrusters are able to vary the angular momentum and
can perform large changes on the satelliteโs orientation. Thrusters can perform small attitude changes as
well, being a more versatile control mechanism. However, the facts of being less precise in comparison with
other control methods and of requiring expenditure of fuel, which canโt be replenished during the mission,
are significant drawbacks [7].
Other alternative control methods include the use of solar sails or magnetic torques [7], as well as the
alternative explored in this work, which is the variable-geometry concept.
2
1.3 SATELLITES WITH VARIABLE-GEOMETRY
Configuration change as a technique to achieve physical equilibrium is well-known in sports. For example,
a ballet dancer or an ice skater that extends his arms towards or away from the body is using the non-rigid
properties of his body by changing its configuration. This technique might be executed in order to perform
a certain motion or to reach an equilibrium condition, for example. This concept can be transferred to the
problem of the attitude control of a satellite.
The change in configuration of a satellite could be performed using a controlled motion of its movable parts.
The movable parts, which could be solar panels, hardware storage containers or any other spacecraft
component that could be convenient to use for this purpose, would be articulated with other components of
the satellite that are fixed in the satelliteโs frame. The fixed parts only have rigid-body motion, unlike the
movable parts that also experience motion controlled by the actuators of the spacecraft.
1.4 CURRENT PROBLEMS OF ATTITUDE DYNAMICS
There are many interesting problems related to attitude dynamics. The dynamic problemsโ solutions are
more difficult to obtain when the body is not simple, when it has flexible appendages or moving fluids for
example, or when the external forces and torques are complicated functions.
Multi-body mechanics addresses the dynamical behavior of the degrees of freedom of a certain number of
interconnected parts that make up a whole system. Examples of multi-body structures are a tree
configuration [8] or a closed-loop structure [9].
The problem of the multi-bodyโs dynamics in space environment can be solved by the Lagrangian or by the
Eulerian approaches [10]. The Lagrangian approach is conceptually simple, since no compatibility equations
need to be satisfied, provided that the motion parameters selected are consistent with the compatibility
conditions. However, the computational implementation of the Lagrangian approach is prone to
programming errors because it tends to be computationally heavy [10]. Alternatively, in the Eulerian
approach, the dynamics of each individual part are considered and the compatibility conditions have to be
taken into account. An advantage of the Eulerian approach is that the equations have clearer physical
meaning than in the Lagrangian approach [10].
A particular case of connected rigid bodies is the dual-spin satellite. It is composed of a platform and a rotor
connected to each other, the rotor being able to rotate with respect to the platform. The platform and the
rotor spin around a common and fixed axis, in normal operating conditions [11]. The attitude stability of such
configuration has been analyzed [12], while an exact analytical solution is obtained when both the platform
and the rotor are asymmetric [13]. Recently, other similar problems have been addressed, such as the dual-
spin satellite with a spring-mass damper [14].
3
The problem of rigid-body attitude when time-varying torques are acting on the body has been considered
and approximate analytical solutions were obtained [15,16]. Many solutions for rigid-body dynamics
obtained on the literature rely on linearization techniques [17] and are only applicable for small orientation
changes. However, rigid-body dynamics were solved for large angle excursions from an initial orientation
using numerical simulations and the developed large-angle theory [17]. Other addressed problems were the
dynamics of a slightly asymmetric rigid body [18], and a fully asymmetric rigid body [19], when subjected to
constant torques in three axis. More recently, other exact analytical solutions have been found. For example,
the problem of a rigid body with a spherical ellipsoid of inertia subject to a constant torque was solved using
a parameterization of the rotation by complex numbers [20]. The solution for the dynamics of an
axisymmetric rigid body subject to torques in various initial conditions was also obtained [21]. Other
important issue is to find the time-optimal maneuver that takes the satellite from one orientation to another
[22,23].
The introduction of flexibility in the dynamics of the satellite increases the difficulty of solving the motion
equations. However, more accurate results could be expected from taking into account elastic behavior of
satellite parts such as antennas. Examples of such flexibility problems are: (i) the control of orbiting flexible
structure using a general Lagrangian formulation [24]; (ii) the dynamics of a dumbbell shaped spacecraft
that consists in two mass particles connected by an elastic link [25]; and (iii) the motion stability of a force-
free satellite with attached elastic appendages [26].
Flexibility and the variable-geometry explored in this work are different concepts. They differ in the fact that
the former is usually a consequence of the natural properties of the materials that one tries to cope with
while the latter is a technique to control the attitude of the spacecraft.
1.4.1 Variable-Structure
The consequences of the variation of inertia properties have been addressed before. A parametric study
has been performed to evaluate the influence of many properties, including the moments of inertia, and
parameters of the satellite on the angular momentum vector [27]. However, this study does not consider
any time-varying moments of inertia. Another problem with varying inertia properties is the dynamics of a
body that grows due to accretion of mass while maintaining rigid body motion [28]. The motion of a spinning
rocket that is ejecting mass through a cluster of nozzles was also addressed [29].
One of the methods of variable-structure control is sliding-mode control. This consists of multiple control
structures that can slide over the boundaries of other control structures. The control law of such motion is
not continuous in time since the input is modeled as a consequence of a discontinuous signal that takes the
system from a first continuous structure to a final continuous structure by a discontinuous transition [30].
Even though it is a variable-structure method used for attitude control, sliding-mode control is not a shape-
changing method of the spacecraft as studied at the present work. The sliding-mode technique has been
addressed in the context of large-angle maneuvers [31] and as a method to perform multi-axial maneuvers
[32]. Sliding-mode control can be also used to suppress vibrations on spacecraft [33].
4
The chaotic behavior of the motion of bodies with one time-varying principal moment of inertia was
addressed [34,35]. Unlike the present work, these works do not refer the spacecraft design that would allow
for such a variation of the moment of inertia or consider consequences of varying one moment of inertia in
other components of the inertia tensor. Other limitation is that the variation of one of the principal inertia
moments is considered as a mode of representing the satellite as a deformable rigid-body and not as a
method to control the satelliteโs attitude. In addition, the chaos phenomena in gyrostats [36] and the chaotic
modes of motion of an unbalanced gyrostat [37] have been addressed.
Other related topic concerns the dynamics of gyrostats with variable inertia properties, namely the
consequences in the phase space of such dynamic, without referring to the issue of chaotic behavior. The
changes in angular motion due to variable mass and variable inertia of gyrostat composed by multiple
coaxial bodies have already been investigated [38]. In addition, the free motion of a dual-spin satellite
composed of a platform and a rotor with variable moments of inertia has been addressed [39]. The present
work differs from these studies in the geometry studied, since they only consider the particular case of non-
variable geometry gyrostats.
The possibility of varying mass, via jets or thrusters, has been considered [29]. However, the interpretation
of additional terms that appear in the Euler equations due to the configuration variation is limited. The kinetic
energy of a variable-geometry body that is addressed in this work is not discussed in [29].
Other works consider two-tethered systems modelled as a variable length thin rod connecting two endpoint
masses. It has been proven that a length function for a tethered system in a Moon-Earth system exists so
that the rod has a constant angle with respect to the Moon-Earth direction [40]. A length function for an
uniform rotation for a two-tethered body in an elliptic orbit was obtained, as well as the motion stability
intervals for parameters such as eccentricity [41]. In contrast with the present work, where a configuration
function is obtained for a three-dimensional satellite made of an arbitrary number of bars to attain
synchronous rotation in an elliptic orbit, the geometry discussed in [41] is two-dimensional, belonging to the
orbital plane during the entire orbit and being composed of a single bar.
5
2 ATTITUDE DYNAMICS OF A VARIABLE GEOMETRY BODY
2.1 ANGULAR MOMENTUM
In this section we will follow Thomson [29] closely. We consider a generic variable-geometry body ฮฉ. The
system of mass ๐ consists of ๐ particles of mass ๐๐ with a certain motion within some specified
boundaries. The mass of the body ๐ is the sum of all the particleโs respective masses
๐ = โซ๐๐ฮฉ
. (2.1)
Two different frames are defined: ๐๐๐ is an inertial frame and the ๐ฅ๐ฆ๐ง is a body frame with an arbitrary
origin 0.
The time-derivatives depend whether the time-derivative is determined with respect to the inertial frame or
with respect to the body frame. The relationship between the two derivatives is given by Basic Kinematic
Equation or Transport Equation [11] in its general form,
๏ฟฝ๏ฟฝ โก๐
๐๐ก
๐ผ
(๐จ) =๐
๐๐ก
๐ต
(๐จ) +๐๐ต/๐ผ ร ๐จ, (2.2)
where ๐จ is an arbitrary vector and ๐๐ต/๐ผ , or ๐ as it will be named further on, is the angular velocity of the
body-frame with respect to the inertial frame. The notation of ๐
๐๐ก
๐ผ , which is equivalent to a dot, and
๐
๐๐ก
๐ต
refer to time-derivatives in the inertial and in the body frame, respectively. The angular velocity is arbitrary,
because the motion of the body frame with respect to the inertial frame is not specified at this point.
The position of a certain particle with respect to the inertial frame is given by
๐น = ๐น๐ + ๐ (2.3)
where ๐ is the position of that particle with respect to the body frame and ๐น๐ is the position of the origin of
the body frame.
The position of center of mass with respect to the body frame is
๐ =
1
๐โซ๐๐๐ฮฉ
, (2.4)
while the velocity of the center of mass measured in the inertial frame is
๏ฟฝ๏ฟฝ =
1
๐โซ ๏ฟฝ๏ฟฝ๐๐ฮฉ
. (2.5)
6
The total angular momentum of the body about the arbitrary body frame origin 0 measured in the inertial
frame, since ๏ฟฝ๏ฟฝ is the velocity in the inertial frame of one particle, is
๐๐ = โซ๐ ร ๏ฟฝ๏ฟฝ๐๐
ฮฉ
. (2.6)
By differentiating (2.6) with respect to the inertial frame and using both the definition of total torque
๐ด๐ =
1
๐โซ๐ร ๐ญ๐๐ฮฉ
(2.7)
and Newtonโs Second Law it is possible to obtain [29]
๐ด๐ = ๏ฟฝ๏ฟฝ๐ + ๏ฟฝ๏ฟฝ๐ ร๐๏ฟฝ๏ฟฝ. (2.8)
2.2 EQUATIONS OF ROTATIONAL MOTION FOR A VARIABLE-GEOMETRY BODY
With the objective of obtaining angular momentum equations that include the inertia and mass distribution
properties, we first write the general expression for the acceleration in the inertial frame [29]
๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ0 + ๏ฟฝ๏ฟฝ ร ๐ +๐ร (๐ร ๐) + 2๐ร๐๐
๐๐ก
๐ต
+๐2๐
๐๐ก2
๐ต
. (2.9)
Equation (2.7) can be rearranged into
๐ด๐ = โซ๐ ร ๏ฟฝ๏ฟฝ๐๐
ฮฉ
, (2.10)
using Newtonโs Second Law. Then, (2.9) can be substituted into (2.10) yielding
๐ด๐ = โ๏ฟฝ๏ฟฝ๐ ร๐๏ฟฝ๏ฟฝ +โซ๐ร (๏ฟฝ๏ฟฝ ร ๐)๐๐ฮฉ
+โซ๐ร (๐ร (๐ ร ๐))๐๐ฮฉ
+2โซ ๐ร (๐ ร๐๐
๐๐ก
๐ต
)๐๐ฮฉ
+โซ ๐ร๐2๐
๐๐ก2
๐ต
๐๐ฮฉ
,
(2.11)
where the first term of the right-hand side, โซ ๐ ร ๏ฟฝ๏ฟฝ๐๐๐ฮฉ, was replaced by โ๏ฟฝ๏ฟฝ๐ ร๐๏ฟฝ๏ฟฝ [29].
The first term of (2.11) involves the acceleration of the origin of the body axes ๏ฟฝ๏ฟฝ๐ and the position of the
center of mass relative to the origin of the body frame ๏ฟฝ๏ฟฝ. It will be zero whenever one of three circumstances
occur: (i) the origin of the body axes is coincident with the origin of the inertial frame; (ii) ๏ฟฝ๏ฟฝ๐ and ๏ฟฝ๏ฟฝ are
parallel, implying ๏ฟฝ๏ฟฝ๐ ร ๏ฟฝ๏ฟฝ = 0; (iii) the origin of the body frame coincides with the center of mass of the
variable-geometry body.
7
The rotational inertia term ๐ฐ โ ๐ , written in vectorial notation [29], is given by
๐ฐ โ ๐ = โซ ๐ ร (๐ร ๐)๐๐๐
. (2.12)
The inertia tensor ๐ฐ, written in indicial notation, is
๐ผ๐๐ = โซ (๐ฟ๐๐๐2 โ ๐๐๐๐)๐๐
๐
. (2.13)
The second and third terms of (2.11) can be rearranged to incorporate this definition [29], yielding
๐ด๐ = ๐ฐ โ ๏ฟฝ๏ฟฝ +๐ ร (๐ฐ โ ๐) + 2โซ ๐ ร (๐ร
๐๐
๐๐ก
๐ต
)๐๐ฮฉ
+โซ ๐ร๐2๐
๐๐ก2
๐ต
๐๐ฮฉ
. (2.14)
Expression (2.14) is the general form of the Euler Equations, when written in vectorial notation. The first
two-terms on the right-hand side of (2.14) are present in the conventional Euler Equations because they are
related to the rigid-body motion.
2.3 GENERAL FORM OF EULER EQUATIONS
The third and fourth terms of the right-hand side of (2.14) account for non-rigid part of the body motion
because they contain ๐๐
๐๐ก
๐ต and
๐2๐
๐๐ก2
๐ต
in their expressions. These terms would be zero if the body was purely
rigid and if the selected body frame would have the same rotational motion of the body. We define
๐ฝ๐ฎ๐ป = 2โซ ๐ ร (๐ ร
๐๐
๐๐ก
๐ต
)๐๐ฮฉ
+โซ ๐ ร๐2๐
๐๐ก2
๐ต
๐๐ฮฉ
, (2.15)
where ๐ฝ๐ฎ๐ป stands for variable-geometry terms.
So that more physical interpretation can be obtained from ๐ฝ๐ฎ๐ป, we add and subtract โซ๐๐
๐๐ก
๐ตร (๐ ร ๐)๐๐
๐
to the right-hand side of (2.15):
๐ฝ๐ฎ๐ป = โซ
๐๐
๐๐ก
๐ต
ร (๐ ร ๐)๐๐๐
โโซ๐๐
๐๐ก
๐ต
ร (๐ ร ๐)๐๐๐
+โซ ๐ ร (๐ ร๐๐
๐๐ก
๐ต
)๐๐๐
+โซ ๐ร (๐ ร๐๐
๐๐ก
๐ต
)๐๐๐
+โซ ๐ร๐2๐
๐๐ก2
๐ต
๐๐๐
.
(2.16)
The first and third terms of the right-hand side of (2.16) will be rearranged separately. Using the vectorial
identity [42]
8
๐จ ร (๐ฉ ร ๐ช) = ๐ฉ(๐จ.๐ช) โ ๐ช(๐จ. ๐ฉ), (2.17)
on the sum of the integrand of those two terms, in indicial notation, and rearranging, yields
๐๐ (2๐๐
๐๐๐๐๐ก
๐ต
)โ ๐๐ (๐๐๐๐๐๐๐ก
๐ต
+ ๐๐๐๐๐๐๐ก
๐ต
). (2.18)
It is possible to simplify (2.18) to
๐๐
๐
๐๐ก
๐ต
(๐ฟ๐๐๐2 โ ๐๐๐๐), (2.19)
using
2๐๐
๐๐๐๐๐ก
๐ต
=๐(๐2)
๐๐ก
๐ต
, (2.20)
๐๐
๐๐๐๐๐ก
๐ต
+ ๐๐๐๐๐๐๐ก
๐ต
=๐(๐๐๐๐)
๐๐ก
๐ต
(2.21)
and
๐๐ = ๐ฟ๐๐๐๐ . (2.22)
After inserting the integrand (2.19) in the body integral and taking the derivative outside of the integral, the
derivative of the inertia tensor in indicial notation (2.13) is obtained.
๐๐
๐
๐๐ก
๐ต
(โซ (๐ฟ๐๐๐2โ ๐๐๐๐)๐๐
๐
) =๐๐ผ๐๐๐๐ก
๐ต
๐๐ . (2.23)
This term does not appear in the usual Euler Equations because, for a rigid body, the mass distribution does
not change with time with respect to the body frame. However, for a variable-geometry body, there is no
reference frame where the mass distribution is constant, due to the changes in configuration.
At this point, after including the result (2.23) in vectorial notation, ๐๐ฐ
๐๐ก
๐ตโ ๐, the variable geometry terms are
given by
9
๐ฝ๐ฎ๐ป =
๐๐ฐ
๐๐ก
๐ต
โ ๐+ โซ (๐ ร ๐) ร๐๐
๐๐ก
๐ต
๐๐๐
+โซ ๐ร (๐ร๐๐
๐๐ก
๐ต
)๐๐๐
+โซ ๐ ร๐2๐
๐๐ก2
๐ต
๐๐๐
,
(2.24)
where the sign of the second term was changed using a property of the external product [42]. The last three
terms of the right-hand side of (2.24) can still be rearranged, which is not performed in Thomson [29]. This
can be done by firstly performing the time-derivative in the inertial-frame of the expression
โซ ๐ ร
๐๐
๐๐ก
๐ต
๐๐๐
, (2.25)
which, using the Basic Kinematic Equation or Transport Equation (2.2), yields
โซ
๐๐
๐๐ก
๐ต
ร๐๐
๐๐ก
๐ต
๐๐๐
+โซ (๐ร ๐) ร๐๐
๐๐ก
๐ต
๐๐๐
+โซ ๐ ร๐2๐
๐๐ก2
๐ต
๐๐๐
+โซ ๐ร (๐ร๐๐
๐๐ก
๐ต
)๐๐๐
.
(2.26)
Since๐๐
๐๐ก
๐ตร
๐๐
๐๐ก
๐ต= 0 , expression (2.26) becomes the same sum that is present in (2.24). For that reason we
can now replace (2.26) into (2.24), obtaining
๐ฝ๐ฎ๐ป =
๐๐ฐ
๐๐ก
๐ต
โ ๐ + ๏ฟฝ๏ฟฝ, (2.27)
where we simplified the notation concerning the second non-rigid term by defining:
๐ธ = โซ ๐ ร
๐๐
๐๐ก
๐ต
๐๐๐
. (2.28)
With the variable-geometry terms ๐ฝ๐ฎ๐ป obtained, we can substitute (2.27) and (2.15) into (2.14) and write
the general form of the Euler equations, using a body frame centered at the center of mass, for the variable-
geometry body:
๐ด๐ = ๐ฐ โ ๏ฟฝ๏ฟฝ +๐ ร (๐ฐ โ ๐) +
๐
๐๐ก
๐ต
(๐ฐ) โ ๐ + ๏ฟฝ๏ฟฝ. (2.29)
In the case of a rigid body, (2.29) would simplify to the usual Euler Equations and the last two terms of the
right-hand side of (2.29) would vanish from the equation, because for a rigid body ๐
๐๐ก๐ฐ
๐ต= 0 and
๐๐
๐๐ก
๐ต= 0.
10
2.4 ALTERNATIVE DERIVATION
The general form of the Euler equations (2.29) can be derived through a different path. The velocity in the
inertial frame ๏ฟฝ๏ฟฝ is [29]
๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ๐ +๐ ร ๐+
๐๐
๐๐ก
๐ต
, (2.30)
which can be replaced in the expression of angular momentum (2.6), yielding
๐๐ = โซ๐ร ๏ฟฝ๏ฟฝ0๐๐
ฮฉ
+โซ๐ร (๐ ร ๐)๐๐ฮฉ
+โซ ๐ร๐๐
๐๐ก
๐ต
ฮฉ
๐๐. (2.31)
The first term of the right-hand side of (2.31) can be rearranged using the definition of center of mass (2.1),
while the second term is the rotational term (2.12). The general expression for the angular momentum of a
variable-geometry body is:
๐๐ = โ๏ฟฝ๏ฟฝ๐ ร๐๐ + ๐ฐ โ ๐+ ๐ธ (2.32)
After substituting (2.32) into (2.8), we obtain
๐ด๐ =
๐
๐๐ก
๐ผ
(โ๏ฟฝ๏ฟฝ๐ ร ๐๐ + ๐ฐ โ ๐ +โซ ๐ร๐๐
๐๐ก
๐ต
๐๐๐
) + ๏ฟฝ๏ฟฝ๐ ร๐๏ฟฝ๏ฟฝ
= โ๏ฟฝ๏ฟฝ๐ ร ๐๏ฟฝ๏ฟฝ โ ๏ฟฝ๏ฟฝ๐ ร๐๏ฟฝ๏ฟฝ +๐
๐๐ก
๐ผ
(๐ฐ โ ๐) +๐
๐๐ก
๐ผ
(โซ ๐ ร๐๐
๐๐ก
๐ต
๐๐๐
) + ๏ฟฝ๏ฟฝ๐ ร๐๏ฟฝ๏ฟฝ
= โ๏ฟฝ๏ฟฝ๐ ร ๐๏ฟฝ๏ฟฝ +๐
๐๐ก
๐ผ
(๐ฐ โ ๐) +๐
๐๐ก
๐ผ
(โซ ๐ ร๐๐
๐๐ก
๐ต
๐๐๐
).
(2.33)
The term โ๏ฟฝ๏ฟฝ๐ ร๐๏ฟฝ๏ฟฝ is present again, as it was present in (2.11), and vanishes when the body frame is
centered at the center of mass, which is the case we are considering. The term ๐
๐๐ก
๐ผ(๐ฐ โ ๐) can be developed
using (2.2), leading to
๐
๐๐ก
๐ต
(๐ฐ โ ๐) +๐ ร (๐ฐ โ ๐) =๐
๐๐ก
๐ต
(๐ฐ) โ ๐ + ๐ฐ โ๐
๐๐ก
๐ต
(๐) + ๐ร (๐ฐ โ ๐), (2.34)
which can be replaced in (2.33), having ๐
๐๐ก
๐ต(๐) = ๏ฟฝ๏ฟฝ, because the derivative of the angular velocity is the
same on both the frames. Using the definition (2.28) again on the last term of the right-hand side of (2.33),
the same general form of the Euler equations for a body frame centered at the center of mass is obtained:
11
๐ด๐ = ๐ฐ โ ๏ฟฝ๏ฟฝ +๐ ร (๐ฐ โ ๐) +
๐
๐๐ก
๐ต
(๐ฐ) โ ๐ + ๏ฟฝ๏ฟฝ. (2.35)
Result (2.35) confirms (2.29).
2.5 GENERAL EULER EQUATIONS IN THE PRINCIPAL FRAME
For a rigid body, the selected moving frame is usually one that has the same motion of the body, so that the
inertia tensor remains constant and the equations simplify. The orientation of the axes of this body frame
can be selected so that the inertia tensor is diagonal, which is called a principal frame. This is always
possible because the inertia tensor is positive-definite and symmetric.
In such conditions, when the body frame has the same motion of the body and this body frame is a principal
frame, the orientation of the principal axes with respect to the body remain constant because the bodyโs
mass distribution and hence its inertia tensor ๐ฐ remains unchanged.
However, for a general variable-geometry body, the behavior of the principal frame with respect to the body
is different: a variable-geometry body with a non-rigid motion, which principal frame has a certain orientation
at a certain point in time ๐ก, does not necessarily have the same orientation, relatively to the body, at time
๐ก + ฮ๐ก, with ฮ๐ก > 0. The reason for this is that the configuration change between ๐ก and ๐ก + ฮ๐ก in the body
frame generally changes the mass distribution of the body in such a way that the former principal frame, at
๐ก, is altered.
Since there is no reference frame with respect to which a variable-geometry body mass distribution is
constant during a configuration change, the body frame selected should be as convenient as possible. In
this work, we selected the principal frame to be the body frame, so the non-diagonal terms are zero.
The set of vector-base of the principal frame is defined as (๐๐ฅ , ๐๐ฆ , ๐๐ง), while the set of coordinates is (๐ฅ, ๐ฆ, ๐ง).
In a principal frame of a general variable-geometry, at a certain instant of time ๐ก, the inertia tensor in matrix
form is
๐ฐ โก [๐ด(๐ก) 0 00 ๐ต(๐ก) 00 0 ๐ถ(๐ก)
], (2.36)
where the principal moments of inertia are defined according to:
๐ด(๐ก) = โซ (๐ฆ2(๐ก) + ๐ง2(๐ก))๐๐
ฮฉ
, (2.37)
12
๐ต(๐ก) = โซ (๐ฅ2(๐ก) + ๐ง2(๐ก))๐๐
ฮฉ
, (2.38)
๐ถ(๐ก) = โซ (๐ฅ2(๐ก) + ๐ฆ2(๐ก))๐๐
ฮฉ
. (2.39)
Following the same notation, the time-derivative of the inertia tensor with respect to the body frame is
๐
๐๐ก
๐ต
(๐ฐ) โก [๏ฟฝ๏ฟฝ 0 00 ๏ฟฝ๏ฟฝ 00 0 ๏ฟฝ๏ฟฝ
], (2.40)
whereas its components are defined by:
๏ฟฝ๏ฟฝ(๐ก) =
๐
๐๐ก
๐ต
โซ (๐ฆ2(๐ก) + ๐ง2(๐ก))๐๐ฮฉ
, (2.41)
๏ฟฝ๏ฟฝ(๐ก) =
๐
๐๐ก
๐ต
โซ (๐ฅ2(๐ก) + ๐ง2(๐ก))๐๐ฮฉ
, (2.42)
๏ฟฝ๏ฟฝ(๐ก) =
๐
๐๐ก
๐ต
โซ (๐ฅ2(๐ก) + ๐ฆ2(๐ก))๐๐ฮฉ
. (2.43)
The total applied torque ๐ด๐ is written as
๐ด๐ = ๐๐ฅ๐๐ +๐๐ฆ๐๐ +๐๐ง๐๐, (2.44)
the angular velocity ๐ is
๐ = ๐๐ฅ๐๐ +๐๐ฆ๐๐ +๐๐ง๐๐ (2.45)
and the angular acceleration ๏ฟฝ๏ฟฝ is defined as
๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ๐ฅ๐๐ + ๏ฟฝ๏ฟฝ๐ฆ๐๐ + ๏ฟฝ๏ฟฝ๐ง๐๐. (2.46)
The derivative of ๐ธ with respect to the inertial frame is given by
๏ฟฝ๏ฟฝ =๐
๐๐ก
๐ผ
(โซ ๐ ร๐๐
๐๐ก
๐ต
๐๐๐
) = ๏ฟฝ๏ฟฝ๐ฅ๐๐ + ๏ฟฝ๏ฟฝ๐ฆ๐๐ + ๏ฟฝ๏ฟฝ๐ง๐๐ + ๐พ๐ฅ๏ฟฝ๏ฟฝ๐ + ๐พ๐ฆ๏ฟฝ๏ฟฝ๐ + ๐พ๐ง๏ฟฝ๏ฟฝ๐. (2.47)
13
The time-derivatives in the inertial frame of the base-vectors of the body frame are determined using (2.2),
considering that they are constant in the body-frame, which yields:
๏ฟฝ๏ฟฝ๐ฅ = ๐๐ง๐๐ โ๐๐ฆ๐๐, (2.48)
๏ฟฝ๏ฟฝ๐ฆ = ๐๐ฅ๐๐ โ๐๐ง๐๐, (2.49)
๏ฟฝ๏ฟฝ๐ง = ๐๐ฆ๐๐ โ๐๐ฅ๐๐. (2.50)
After replacing (2.48)-(2.50) in (2.47), we obtain
๏ฟฝ๏ฟฝ = (๏ฟฝ๏ฟฝ๐ฅ + ๐พ๐ง๐๐ฆ โ ๐พ๐ฆ๐๐ง)๐๐ + (๏ฟฝ๏ฟฝ๐ฆ + ๐พ๐ฅ๐๐ง โ ๐พ๐ง๐๐ฅ)๐๐ + (๏ฟฝ๏ฟฝ๐ง + ๐พ๐ฆ๐๐ฅ โ ๐พ๐ฅ๐๐ฆ)๐๐. (2.51)
Having every term of the general Euler equations (2.35) written in terms of three body frame coordinates,
we write it in vectorial form:
[
๐๐ฅ
๐๐ฆ
๐๐ง
] = [๐ด 0 00 ๐ต 00 0 ๐ถ
] . [
๏ฟฝ๏ฟฝ๐ฅ๏ฟฝ๏ฟฝ๐ฆ๏ฟฝ๏ฟฝ๐ง
] + [
๐๐ฅ๐๐ฆ๐๐ง
] ร ([๐ด 0 00 ๐ต 00 0 ๐ถ
] . [
๐๐ฅ๐๐ฆ๐๐ง
]) + [๏ฟฝ๏ฟฝ 0 00 ๏ฟฝ๏ฟฝ 00 0 ๏ฟฝ๏ฟฝ
] . [
๐๐ฅ๐๐ฆ๐๐ง
]
+ [
๏ฟฝ๏ฟฝ๐ฅ๏ฟฝ๏ฟฝ๐ฆ๏ฟฝ๏ฟฝ๐ง
] + [
๐พ๐ง๐๐ฆ โ ๐พ๐ฆ๐๐ง๐พ๐ฅ๐๐ง โ ๐พ๐ง๐๐ฅ๐พ๐ฆ๐๐ฅ โ ๐พ๐ฅ๐๐ฆ
].
(2.52)
Expression (2.52) is written in the three principal directions as:
๐๐ฅ = ๏ฟฝ๏ฟฝ๐๐ฅ +๐ด๏ฟฝ๏ฟฝ๐ฅ + ๐๐ฆ๐๐ง(๐ถ โ ๐ต) + ๏ฟฝ๏ฟฝ๐ฅ + ๐พ๐ง๐๐ฆ โ ๐พ๐ฆ๐๐ง , (2.53)
๐๐ฆ = ๏ฟฝ๏ฟฝ๐๐ฆ +๐ต๏ฟฝ๏ฟฝ๐ฆ +๐๐ฅ๐๐ง(๐ด โ ๐ถ) + ๏ฟฝ๏ฟฝ๐ฆ + ๐พ๐ฅ๐๐ง โ ๐พ๐ง๐๐ฅ , (2.54)
๐๐ง = ๏ฟฝ๏ฟฝ๐๐ง + ๐ถ๏ฟฝ๏ฟฝ๐ง + ๐๐ฅ๐๐ฆ(๐ต โ ๐ด) + ๏ฟฝ๏ฟฝ๐ง + ๐พ๐ฆ๐๐ฅ โ ๐พ๐ฅ๐๐ฆ . (2.55)
Equations (2.53)-(2.55) are valid for a general variable-geometry body when its dynamics are written in a
body frame that is a principal frame and whose origin is coincident with the position of the center of mass.
Euler equations written in a principal frame [29] are given by:
๐๐ฅ = ๐ด๏ฟฝ๏ฟฝ๐ฅ + ๐๐ฆ๐๐ง(๐ถ โ ๐ต), (2.56)
๐๐ฆ = ๐ต๏ฟฝ๏ฟฝ๐ฆ +๐๐ฅ๐๐ง(๐ด โ ๐ถ), (2.57)
๐๐ง = ๐ถ๏ฟฝ๏ฟฝ๐ง + ๐๐ฅ๐๐ฆ(๐ต โ ๐ด). (2.58)
14
We observe that the Euler equations are a particular case of the modified Euler Equations. This is so
because when the geometry is constant in the body frame that is centered at the center of mass, we have
that ๐๐
๐๐ก
๐ต= 0. This implies that not only does ๏ฟฝ๏ฟฝ vanish, but
๐
๐๐ก
๐ต(๐ฐ) โ ๐ vanishes as well.
In this work, we will consider the form of the general form of Euler equations (2.53)-(2.55) where the variable-
geometry body is not subject of any applied torques โ the satellite is considered a free body in space. In
addition, we will consider geometries where we have ๐ธ = 0 at all times, which significantly simplifies the
equations. The circumstances under which this assumption is valid will be discussed in section 2.6. Taking
this into account, the equations simplify to:
0 = ๏ฟฝ๏ฟฝ๐๐ฅ +๐ด๏ฟฝ๏ฟฝ๐ฅ +๐๐ฆ๐๐ง(๐ถ โ ๐ต), (2.59)
0 = ๏ฟฝ๏ฟฝ๐๐ฆ + ๐ต๏ฟฝ๏ฟฝ๐ฆ +๐๐ฅ๐๐ง(๐ด โ ๐ถ), (2.60)
0 = ๏ฟฝ๏ฟฝ๐๐ง + ๐ถ๏ฟฝ๏ฟฝ๐ง +๐๐ฅ๐๐ฆ(๐ต โ ๐ด). (2.61)
2.6 NON-RIGID TERMS
Of the two non-rigid terms of the general Euler equations (2.27), ๐
๐๐ก
๐ต(๐ฐ) โ ๐ expresses the influence of
varying the geometry of the body, and thus the mass distribution in space, in the body dynamics.
The other variable-geometry term that is present in the equations is ๏ฟฝ๏ฟฝ, has no direct relationship with the
inertia tensor ๐ฐ, unlike the term ๐
๐๐ก
๐ต(๐ฐ) โ ๐. The physical explanation for ๏ฟฝ๏ฟฝ is related with the orientation of
the principal frame. The geometry variation with time, by changing the mass distribution in space, generally
implies that the principal frame is no longer the same. For example, consider a body with a well-defined
principal frame, due to three perpendicular symmetry planes. If this body changes its geometry in a way that
all the symmetries are no longer present, the orientation of the principal frame with respect to the inertial
frame is no longer clear and the dynamics of it are affected by the non-zero term ๏ฟฝ๏ฟฝ. Thus, ๏ฟฝ๏ฟฝ accounts for
the consequence in the dynamics of the variable-geometry body having a non-rigid motion that is changing
the orientation of its principal axes.
2.6.1 Two-plane Symmetry
We study a particular variable-geometry body which geometry variation is such that ๏ฟฝ๏ฟฝ = 0 at all times โ
the variable geometry body with two perpendicular symmetry planes at all its configurations, whose
intersection contains the origin of the body frame.
15
Suppose the planes of symmetry are ๐ฅ0๐ง and ๐ฆ0๐ง, without loss of generality. We define that the position of
a general point of the body, of infinitesimal mass ๐๐, with respect to the body frame is
๐1 = ๐ฅ๐๐ + ๐ฆ๐๐ + ๐ง๐๐, (2.62)
having a non-rigid velocity, with respect to the body frame, of
๐๐1๐๐ก
๐ต
= ๐ฃ๐ฅ๐๐+ ๐ฃ๐ฆ๐๐ + ๐ฃ๐ง๐๐. (2.63)
Since the body has symmetry with respect to the ๐ฅ0๐ง and ๐ฆ0๐ง planes, the mirrored points of the first
infinitesimal mass with respect to them, with indexes 2 to 4 must be also infinitesimal masses with the same
mass ๐๐ that belong to the body. Their positions are:
๐2 = โ๐ฅ๐๐ + ๐ฆ๐๐ + ๐ง๐๐, (2.64)
๐3 = ๐ฅ๐๐ โ ๐ฆ๐๐ + ๐ง๐๐ (2.65)
๐4 = โ๐ฅ๐๐ โ ๐ฆ๐๐ + ๐ง๐๐. (2.66)
If the motion is symmetric with respect to these planes, then their velocities in the body frame are also
symmetric, so that the symmetry remains unchanged after the motion. For this reason, the velocities of
points 2, 3 and 4 are defined with respect to the velocity of point 1:
๐๐๐๐๐ก
๐ต
= โ๐ฃ๐ฅ๐๐ + ๐ฃ๐ฆ๐๐ + ๐ฃ๐ง๐๐, (2.67)
๐๐๐๐๐ก
๐ต
= ๐ฃ๐ฅ๐๐โ ๐ฃ๐ฆ๐๐ + ๐ฃ๐ง๐๐, (2.68)
๐๐๐๐๐ก
๐ต
= โ๐ฃ๐ฅ๐๐ โ ๐ฃ๐ฆ๐๐ + ๐ฃ๐ง๐๐. (2.69)
The total contribution of the 4 points to ๐ธ is
โ๐๐ ร
๐๐๐๐๐ก
๐ต4
๐=1
= ๐1 ร๐๐๐๐๐ก
๐ต
+ ๐2 ร๐๐๐๐๐ก
๐ต
+ ๐3 ร๐๐๐๐๐ก
๐ต
+ ๐4 ร๐๐๐๐๐ก
๐ต
= 0, (2.70)
using (2.62)-(2.69).
Since (2.70) is true for any set of 4 infinitesimal masses of the body,
16
๐ธ = โซ ๐ ร
๐๐
๐๐ก
๐ต
๐๐ฮฉ
= 0. (2.71)
This derivation proved that, for a body that permanently has at least two perpendicular symmetry planes
whose intersection contains the body frameโs origin, ๐ธ = 0 and hence ๏ฟฝ๏ฟฝ = 0 are valid for all configurations.
This result will be used throughout this work to justify ๐ธ = 0 for the proposed configurations.
It is important to note that if two planes of symmetry are constant with respect to the selected body frame,
then they are always perpendicular to their respective principal axes. Thus, the third principal axis is already
defined by being orthogonal to the plane where the other two principal axes lie on. Therefore, we see in this
particular case that ๏ฟฝ๏ฟฝ = 0 is true when the principal axes have a constant orientation with respect to the
selected body frame.
2.7 KINETIC ENERGY OF A VARIABLE-GEOMETRY BODY
The total kinetic energy of the variable-geometry body is obtained by integrating the square of the modulus
of the velocity in the inertial frame along the body [29]:
๐ = โซ
1
2|๏ฟฝ๏ฟฝ|
2
ฮฉ
๐๐ = โซ1
2๏ฟฝ๏ฟฝ. ๏ฟฝ๏ฟฝ
ฮฉ
๐๐ (2.72)
Inserting expression for ๏ฟฝ๏ฟฝ (2.30) in (2.72) yields
๐ =
1
2โซ ๏ฟฝ๏ฟฝ๐. ๏ฟฝ๏ฟฝ๐๐๐ฮฉ
+1
2โซ (๐ร ๐). (๐ ร ๐)๐๐ฮฉ
+โซ ๏ฟฝ๏ฟฝ๐. (๐ ร ๐)ฮฉ
๐๐
+1
2โซ
๐๐
๐๐ก
๐ต
.๐๐
๐๐ก
๐ต
ฮฉ
๐๐ +โซ ๏ฟฝ๏ฟฝ๐ .๐๐
๐๐ก
๐ต
ฮฉ
๐๐ +โซ (๐ ร ๐).๐๐
๐๐ก
๐ต
ฮฉ
๐๐.
(2.73)
We will interpret each one of the terms of (2.73) separately. The first term is the term of translational kinetic
energy, which can be rearranged considering the fact that ๏ฟฝ๏ฟฝ๐ is constant over the body resulting in
1
2โซ ๏ฟฝ๏ฟฝ๐. ๏ฟฝ๏ฟฝ๐๐๐ฮฉ
=1
2๐๏ฟฝ๏ฟฝ๐
๐. (2.74)
The second term of (2.73) can be rearranged using a vectorial property [42] to obtain the rotational kinetic
energy of the body,
1
2โซ (๐ ร ๐). (๐ ร ๐)๐๐ฮฉ
=1
2๐. (๐ฐ โ ๐), (2.75)
and, if the right-hand side of (2.75) is written in a body frame that is a principal frame, we have
17
1
2๐. (๐ฐ โ ๐) =
1
2(๐ด๐๐ฅ
2 +๐ต๐๐ฆ2 + ๐ถ๐๐ง
2). (2.76)
The third term of the right hand side of (2.73) can be rearranged taking into account that ๏ฟฝ๏ฟฝ๐ and ๐ are
constant over the body and inserting the definition of center of mass (2.4):
โซ ๏ฟฝ๏ฟฝ๐. (๐ ร ๐)ฮฉ
๐๐ = ๏ฟฝ๏ฟฝ๐. ๐ ร๐๏ฟฝ๏ฟฝ. (2.77)
The fact that the origin of the body frame is coincident with the center of mass implies that ๏ฟฝ๏ฟฝ = 0, making
(2.77) zero.
The other three terms of the right-hand side of (2.73) are terms that are related with non-rigid part of the
motion, since they would all be zero when ๐๐
๐๐ก
๐ต= 0. The first three terms would be already present for pure
rigid-body motion. We define the rigid body component of the kinetic energy
๐๐๐๐๐๐ =
1
2๐๏ฟฝ๏ฟฝ๐
๐ +1
2๐. (๐ฐ โ ๐), (2.78)
and the non-rigid component of the kinetic energy
๐๐๐๐โ๐๐๐๐๐ =
1
2โซ
๐๐
๐๐ก
๐ต
.๐๐
๐๐ก
๐ต
ฮฉ
๐๐ +โซ ๏ฟฝ๏ฟฝ๐ .๐๐
๐๐ก
๐ต
ฮฉ
๐๐ +โซ (๐ ร ๐) .๐๐
๐๐ก
๐ต
ฮฉ
๐๐. (2.79)
The third term in the right-hand side of (2.79) can be rearranged using the vectorial property [42]
(๐จ ร ๐ฉ). ๐ช = (๐ฉ ร ๐ช). ๐จ, (2.80)
taking into account that ๏ฟฝ๏ฟฝ๐ does not depend on the position and replacing the definition of ๐ธ (2.28), yielding
๐๐๐๐โ๐๐๐๐๐ =
1
2โซ
๐๐
๐๐ก
๐ต
.๐๐
๐๐ก
๐ต
ฮฉ
๐๐ + ๏ฟฝ๏ฟฝ๐ . โซ๐๐
๐๐ก
๐ต
ฮฉ
๐๐ +๐.๐ธ. (2.81)
2.7.1 Non-Rigid Terms
The first term of (2.81) is the pure non-rigid motion term. Its value is the same regardless of the rigid motion
of the body, both translation and rotation: it only depends on the configuration change and hence on the
distribution of the velocity with respect to the body frame ๐๐
๐๐ก
๐ต. This term is never negative because the
vectorial self-dot product is never negative.
18
Also, the rigid terms of the kinetic energy (2.78) and the first of the non-rigid terms on the right-hand side of
(2.81) are all non-negative:
1
2๐๏ฟฝ๏ฟฝ๐
๐ +1
2๐.(๐ฐ โ ๐) +
1
2โซ
๐๐
๐๐ก
๐ต
.๐๐
๐๐ก
๐ต
ฮฉ
๐๐ โฅ 0. (2.82)
However, the other two other terms, ๏ฟฝ๏ฟฝ๐. โซ๐๐
๐๐ก
๐ต
ฮฉ๐๐ and ๐.๐ธ, can be negative, because the sign of the result
of the dot products between these vectors is generally unknown. This contrasts with the terms already
referred in (2.82) that are non-negative. Moreover, the term ๏ฟฝ๏ฟฝ๐ . โซ๐๐
๐๐ก
๐ต
ฮฉ๐๐ displays a quadratic dependence
on velocity, as it is natural in translational kinetic energy.
To interpret ๏ฟฝ๏ฟฝ๐. โซ๐๐
๐๐ก
๐ต
ฮฉ๐๐ and ๐.๐ธ we firstly consider how the system and the problem have been defined.
The system of particles of the body has been looked upon as a mechanical system and its kinetic energy is
a component of the mechanical energy of the system. Nonetheless, the nature of the energy source of the
actuators of the system is not mechanical, it might be electrical or chemical for example, even though the
actuators belong to the system. This is the reason why the actuators can change the mechanical energy of
the whole system.
Varying the geometry of the body as a response to internal actuators of the body can either lead to an
increase or a decrease of the mechanical energy of the system, depending on whether the actuators are
performing positive or negative work. This depends on the direction of the forces generated by the actuator
with respect to the displacement of the particles they are acting on.
If the geometry of the body is not varying, then the kinetic energy will not vary either, because in that case
the body behaves like a rigid body. The kinetic energy will vary if external forces are applied do work on the
system. The actuators are no exception, because, from the point of view of the mechanical system, they
are external forces.
The term ๏ฟฝ๏ฟฝ๐. โซ๐๐
๐๐ก
๐ต
ฮฉ๐๐ in the kinetic non-rigid energy expression (2.81) depends on the translational state
of the body and the geometry variation. It will be positive if the projection of ๐๐
๐๐ก
๐ต onto ๏ฟฝ๏ฟฝ๐ has the direction of
๏ฟฝ๏ฟฝ๐ and negative if it has the opposite direction. This term is positive if the actuating internal forces are
increasing the translational energy and negative otherwise. It will be 0 if they are perpendicular and no
translational work is being performed to the body.
19
The term ๐.๐ธ of the kinetic non-rigid energy expression (2.81) has an equivalent interpretation. ๐ธ is related
with changes in symmetries and accounts for changes in the orientation of the principal axes. In addition, ๐ธ
can be understood as the angular momentum of the body measured in the body frame. When the body
geometry is varying, ๐ธ has the direction of the part of the total angular momentum that the body frame
rotation does not account for. Concerning the sign of this term, ๐.๐ธ is positive if the projection of ๐ธ onto ๐
is in the direction of ๐ and negative otherwise. The physical meaning of this is similar as for the first term:
๐.๐ธ is positive when the actuating forces are such that the positive rotational work is being done to the
system, speeding up the rotation and increasing its rotational kinetic energy. The negative and zero work
have a similar interpretation as for ๏ฟฝ๏ฟฝ๐. โซ๐๐
๐๐ก
๐ต
ฮฉ๐๐.
2.7.2 Simplification of the Kinetic Energy Expression
The total kinetic energy for a body frame centered in the bodyโs center of mass is
๐ =
1
2๐๏ฟฝ๏ฟฝ๐
๐ +1
2๐. (๐ฐ โ ๐) +
1
2โซ
๐๐
๐๐ก
๐ต
.๐๐
๐๐ก
๐ต
ฮฉ
๐๐ + ๏ฟฝ๏ฟฝ๐. โซ๐๐
๐๐ก
๐ต
ฮฉ
๐๐ +๐.๐ธ. (2.83)
The first term of the right-hand side of (2.83) will be not accounted for because the central point of this work
is attitude control regardless of the translation state of the body.
The fourth term of the right-hand side of (2.83) can be simplified using the definition of center of mass (2.4):
๏ฟฝ๏ฟฝ๐. โซ
๐๐
๐๐ก
๐ต
ฮฉ
๐๐ = ๏ฟฝ๏ฟฝ๐.๐
๐๐ก
๐ต
โซ ๐ฮฉ
๐๐ = ๏ฟฝ๏ฟฝ๐.๐(๐๏ฟฝ๏ฟฝ)
๐๐ก
๐ต
. (2.84)
The selected body frame is centered at the center of mass of the body to simplify the equations, implying
๏ฟฝ๏ฟฝ = 0 and making equation (2.84) zero.
Usually ๐.๐ธ is different from zero, but for the variable-geometry bodies studied in this work the symmetries
are such that ๐ธ = 0. Under these assumptions we have that the total kinetic energy (2.83) reduces to
๐ =
1
2๐. (๐ฐ โ ๐) +
1
2โซ
๐๐
๐๐ก
๐ต
.๐๐
๐๐ก
๐ต
ฮฉ
๐๐. (2.85)
Using the vectorial notation used so far, as well as (2.75), yields:
2๐ = ๐ด๐๐ฅ
2 + ๐ต๐๐ฆ2 + ๐ถ๐๐ง
2 +โซ ((๐๐
๐๐ก
๐ต
)๐ฅ
2
+ (๐๐
๐๐ก
๐ต
)๐ฆ
2
+ (๐๐
๐๐ก
๐ต
)๐ง
2
)ฮฉ
๐๐ (2.86)
20
In this work, the geometry changing mechanisms are intended to be simple. For that reason several
geometries have a single varying geometry degree of freedom. This is defined generally as ๐ผ, that is a
function of time ๐ก. In these conditions, the integrated terms in (2.86) are given by
(๐๐
๐๐ก
๐ต
)๐
2
= ๏ฟฝ๏ฟฝ2 (๐๐
๐๐ผ
๐ต
)๐
2
, (2.87)
with ๐ = ๐ฅ, ๐ฆ, ๐ง.
Additionally, we can assume that the geometry variation is slow. This is reasonable because, for attitude
maintenance maneuvers of spacecraft in orbits, the necessary attitude corrections are small. So, we have
that ๏ฟฝ๏ฟฝ2 โช 1 and that
๏ฟฝ๏ฟฝ2โซ ((
๐๐
๐๐ผ
๐ต
)๐ฅ
2
+ (๐๐
๐๐ผ
๐ต
)๐ฆ
2
+ (๐๐
๐๐ผ
๐ต
)๐ง
2
)ฮฉ
๐๐ (2.88)
is negligible when compared to the other terms of (2.86), which can be approximated by
2๐ โ ๐ด๐๐ฅ2 +๐ต๐๐ฆ
2 + ๐ถ๐๐ง2. (2.89)
The result (2.89) for the kinetic energy is the same as the expression of the kinetic energy for a rigid body
[29]. This proves that the kinetic energy of a variable-geometry body that is slowly changing its configuration
can be approximated by the kinetic energy of a rigid body. However, ๐ (2.89) is not a constant because the
moments of inertia depend on time.
2.7.3 Time-derivative of the Kinetic Energy
If we differentiate and rearrange (2.89), we obtain:
2๏ฟฝ๏ฟฝ = ๐๐ฅ(๏ฟฝ๏ฟฝ๐๐ฅ + 2๐ด๏ฟฝ๏ฟฝ๐ฅ) + ๐๐ฆ(๏ฟฝ๏ฟฝ๐๐ฆ + 2๐ต๏ฟฝ๏ฟฝ๐ฆ) + ๐๐ง(๏ฟฝ๏ฟฝ๐๐ง + 2๐ถ๏ฟฝ๏ฟฝ๐ง). (2.90)
If the terms inside the parenthesis in (2.90) are rearranged to have
2๏ฟฝ๏ฟฝ = ๐๐ฅ(๏ฟฝ๏ฟฝ๐๐ฅ +๐ด๏ฟฝ๏ฟฝ๐ฅ + ๐ด๏ฟฝ๏ฟฝ๐ฅ) + ๐๐ฆ(๏ฟฝ๏ฟฝ๐๐ฆ +๐ต๏ฟฝ๏ฟฝ๐ฆ + ๐ต๏ฟฝ๏ฟฝ๐ฆ) + ๐๐ง(๏ฟฝ๏ฟฝ๐๐ง + ๐ถ๏ฟฝ๏ฟฝ๐ง + ๐ถ๏ฟฝ๏ฟฝ๐ง), (2.91)
then the sums ๏ฟฝ๏ฟฝ๐๐ฅ + ๐ด๏ฟฝ๏ฟฝ๐ฅ, ๏ฟฝ๏ฟฝ๐๐ฆ +๐ต๏ฟฝ๏ฟฝ๐ฆ and ๏ฟฝ๏ฟฝ๐๐ง + ๐ถ๏ฟฝ๏ฟฝ๐ง can be replaced using equations (2.59)-(2.61), for a
free body whose non-rigid motion is such that ๐ธ = 0. Then, by using the time-derivative of the product on
the right-hand side of (2.91), we obtain
2๏ฟฝ๏ฟฝ =
1
2
๐
๐๐ก(๐ด๐๐ฅ
2 + ๐ต๐๐ฆ2 + ๐ถ๐๐ง
2) โ1
2(๏ฟฝ๏ฟฝ๐๐ฅ
2 + ๏ฟฝ๏ฟฝ๐๐ฆ2 + ๏ฟฝ๏ฟฝ๐๐ง
2). (2.92)
21
After identifying the kinetic energy as in (2.89) in the first term of the right-hand side of (2.92), we simplify
and obtain a simple expression for ๏ฟฝ๏ฟฝ that does not depend on the components of the angular acceleration:
๏ฟฝ๏ฟฝ = โ
1
2(๏ฟฝ๏ฟฝ๐๐ฅ
2 + ๏ฟฝ๏ฟฝ๐๐ฆ2 + ๏ฟฝ๏ฟฝ๐๐ง
2). (2.93)
Even though the square of the angular velocity components are always positive, the signal of ๏ฟฝ๏ฟฝ, ๏ฟฝ๏ฟฝ and ๏ฟฝ๏ฟฝ
are not known, so ๏ฟฝ๏ฟฝ could be positive, zero or negative.
2.8 CONSERVATION OF ANGULAR MOMENTUM
We recall the expression of the applied torque (2.8)
๐ด๐ = ๏ฟฝ๏ฟฝ๐ +๐น๏ฟฝ๏ฟฝ ร๐๏ฟฝ๏ฟฝ. (2.94)
Having the body frameโs origin be the center of mass implies ๏ฟฝ๏ฟฝ = 0, simplifies (2.94) to
๐ด๐ = ๏ฟฝ๏ฟฝ๐ (2.95)
If no torques are acting upon the body, then ๐ด๐ = 0. The satellites will be also considered as a free-body in
this work. Therefore:
๏ฟฝ๏ฟฝ๐ = 0 (2.96)
We conclude from (2.96) that the angular momentum of a variable geometry free body with respect to his
center of mass is constant.
This result contrasts with the results regarding the kinetic energy. While varying the geometry of a certain
body generally changes its energy, it does not change the angular momentum. The reason for that
difference is that the forces responsible for changing the geometry of the body are internal to the mechanical
system considered but the energy sources responsible for that do not belong to the mechanical system โ
they are external. Consequently, the actuators are able to change the systemโs mechanical energy but not
its angular momentum.
2.9 GEOMETRICAL SOLUTIONS FOR THE DYNAMICS OF A VARIABLE GEOMETRY BODY
The geometrical solution consists in obtaining information about the rigid body dynamics using a geometrical
interpretation of two equations: kinetic energy equation and the angular momentum equation.
22
2.9.1 Ellipsoid of the Kinetic Energy
A general variable-geometry body that is free from torque and that has the body frame centered in the center
of mass is considered. Under the assumption that the bodyโs non-rigid motion is two-plane symmetric, ๐ธ =
0 is valid, and not considering the translation, the kinetic energy is given by
๐ =
1
2(๐ด๐๐ฅ
๐ + ๐ต๐๐ฆ๐ + ๐ถ๐๐ง
๐) +1
2โซ
๐๐
๐๐ก
๐ต
.๐๐
๐๐ก
๐ต
ฮฉ
๐๐. (2.97)
Expression (2.97) can be further rearranged in order to obtain an equation that allows for geometrical
interpretation:
1 =
๐๐ฅ2
2๐ โ โซ๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ด
+๐๐ฆ2
2๐ โ โซ๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ต
+๐๐ง2
2๐ โ โซ๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ถ
(2.98)
If we consider a geometrical space given by the coordinates (๐๐ฅ , ๐๐ฆ , ๐๐ง), (2.98) defines the surface of an
ellipsoid with semi-axes
โ2๐โโซ๐๐
๐๐ก
๐ต.๐๐
๐๐ก
๐ต๐๐
ฮฉ
๐ด, โ
2๐โโซ๐๐
๐๐ก
๐ต.๐๐
๐๐ก
๐ต๐๐
ฮฉ
๐ต , โ
2๐โโซ๐๐
๐๐ก
๐ต.๐๐
๐๐ก
๐ต๐๐
ฮฉ
๐ถ
(2.99)
along the directions of the ๐ฅ, ๐ฆ and ๐ง axes, respectively.
The existence of variable-geometry, represented in the term โซ๐๐
๐๐ก
๐ต .
๐๐
๐๐ก
๐ต๐๐
ฮฉ and ๐ด(๐ก), ๐ต(๐ก), ๐ถ(๐ก) ,
contributes to a decrease in the length of all three semi-axes of the ellipsoid, relatively to the ellipsoid for a
rigid-body. It also changes its shape because the ratios between the semi-axes are modified when the non-
rigid motion term is not zero.
The equation of the ellipsoid of a variable geometry-body reduces to the rigid-body equation of the ellipsoid,
or Poinsotโs ellipsoid [29], when the variable-geometry term โซ๐๐
๐๐ก
๐ต.
๐๐
๐๐ก
๐ต๐๐
ฮฉ is zero, since
๐๐
๐๐ก
๐ต= 0.
2.9.2 Ellipsoid of the Angular Momentum
The general expression for the angular momentum (2.31), written in the body frame with ๐ธ = 0 and ๏ฟฝ๏ฟฝ = 0,
is
๐๐ = ๐ด๐๐ฅ๐๐ + ๐ต๐๐ฆ๐๐ + ๐ถ๐๐ง๐๐. (2.100)
23
In order to obtain an expression with the angular moment that contains the squared components of the
angular velocity, we determine ๐๐. ๐๐:
๐๐. ๐๐ = โ2 = ๐ด2๐๐ฅ2 + ๐ต2๐๐ฆ
2 + ๐ถ2๐๐ง2 (2.101)
After rearranging, the expression for the angular momentum ellipsoid is obtained:
1 =
๐๐ฅ2
โ๐ด
+๐๐ฆ2
โ๐ต
+๐๐ง2
โ๐ถ
. (2.102)
The semi-axes are given by โ
๐ด, โ
๐ต and
โ
๐ถ, in the direction of the ๐ฅ, ๐ฆ and ๐ง axes respectively. It is important to
notice that, even though โ is constant, the shape of the ellipsoid of angular momentum varies, since ๐ด, ๐ต
and ๐ถ vary, changing the length of the semi-axes.
2.9.3 Geometrical Solution
Having both of the ellipsoids defined, the angular velocity of the variable-geometry body must belong to a
curve that is the intersection of both the ellipsoids at each instant of time. This curve is not constant because
๐ด, ๐ต and ๐ถ vary. Additionally, ๐ and โซ๐๐
๐๐ก
๐ต .
๐๐
๐๐ก
๐ต๐๐
ฮฉ depend on the geometry variation with time as well.
Only โ is constant.
This means that, unlike the geometric solution for a rigid-body, the ellipsoidโs shape depends on time
because the variable-geometry bodyโs inertia properties vary. Therefore, the angular velocity solution at a
particular instant, that lies on that instantโs ellipsoids intersection, has to lie on the next instantโs ellipsoids
intersection, which is different from the previous one.
We will verify that the intersection of the ellipsoids exists for a variable-geometry body. That intersection
would not exist, geometrically, if either one of two situations occur: (i) the angular momentum ellipsoid is
inside the kinetic energy ellipsoid, which implies that, for each of the directions, the kinetic energy ellipsoid
has larger semi-axes than the angular momentumโs ellipsoid:
โ2๐ โ โซ๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ด>โ
๐ด,โ2๐ โ โซ
๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ต>โ
๐ต,โ2๐ โ โซ
๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ถ>โ
๐ถ;
(2.103)
or (ii) the kinetic energy ellipsoid is inside the angular momentum ellipsoid, hence
24
โ2๐ โ โซ๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ด<โ
๐ด ,โ2๐ โ โซ
๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ต<โ
๐ต ,โ2๐ โ โซ
๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ถ<โ
๐ถ .
(2.104)
These geometric restrictions imply that a certain variable-geometry body canโt have arbitrary ๐ and โ.
We assume ๐ถ โฅ ๐ต โฅ ๐ด, which is an assumption that does not imply loss of generality because the derivation
would be equivalent if the relation between the different inertia moments would be different. Angular velocity
componentsโ ratios are defined:
๐๐ฅ = ๐1๐๐ง , (2.105)
๐๐ฆ = ๐2๐๐ง . (2.106)
Inserting the ratios (2.105)-(2.106) in (2.101) and dividing both sides of the equation by ๐ถ2, we obtain
โ2
๐ถ2= ๐๐ง
2 ((๐ด
๐ถ)2
๐12 + (
๐ต
๐ถ)2
๐22 + 1). (2.107)
If an equivalent derivation for the kinetic energy is done, we obtain
2๐ โ โซ
๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ถ= ๐๐ง
2 (๐ด
๐ถ๐12 +
๐ต
๐ถ๐22 + 1).
(2.108)
Solving (2.107) and (2.108) for ๐๐ง2 and taking the square root of both sides to obtain a semi-axes ratio on
the left-hand side yields:
โ๐ถ
โ2๐ โ โซ๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ถ
= โ(๐ด๐ถ)2
๐12 + (
๐ต๐ถ)2
๐22 + 1
๐ด๐ถ ๐1
2 +๐ต๐ถ ๐2
2 + 1
(2.109)
Since ๐ด
๐ถโค 1 and
๐ต
๐ถโค 1, the numerator of the fraction of the right-hand side of (2.109) is not larger than the
denominator. So:
โ
๐ถโคโ2๐ โ โซ
๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ถ
(2.110)
25
So that an inequality similar to (2.110) is obtained, but including ๐ด instead of ๐ถ, two other angular velocity
componentsโ ratios are defined:
๐๐ฆ = ๐2๐๐ฅ , (2.111)
๐๐ง = ๐3๐๐ฅ . (2.112)
Following a similar procedure used on the derivation of (2.110), we obtain
โ
๐ดโฅโ2๐ โ โซ
๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ด.
(2.113)
We can also write two inequalities concerning the semi-axes of the ellipsoids, using ๐ถ โฅ ๐ต โฅ ๐ด:
โ
๐ดโฅโ
๐ถ, (2.114)
โ2๐ โ โซ๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ดโฅโ2๐ โ โซ
๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ถ.
(2.115)
Finally, combining (2.110) and (2.113)-(2.115) results in
โ
๐ถโคโ2๐ โ โซ
๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ถโคโ2๐ โ โซ
๐๐๐๐ก
๐ต
.๐๐๐๐ก
๐ต
๐๐ฮฉ
๐ดโคโ
๐ด.
(2.116)
We conclude from (2.116) that the larger and smaller ellipsoid axes are axes of the angular momentum
ellipsoid. This means that the ellipsoid of angular momentum is always taper than the ellipsoid of kinetic
energy.
Rearranging (2.110) and (2.113) to obtain two inequalities for ๐ and then combining them allows to obtain
the upper and lower limits of ๐:
1
2(โ2
๐ถ+โซ
๐๐
๐๐ก
๐ต
.๐๐
๐๐ก
๐ต
๐๐ฮฉ
) โค ๐ โค1
2(โ2
๐ด+โซ
๐๐
๐๐ก
๐ต
.๐๐
๐๐ก
๐ต
๐๐ฮฉ
). (2.117)
When the geometry is varying with time, the kinetic energy is limited by two varying bounds. ๐ด and ๐ถ are
varying and the non-rigid term, which is generally dependent of time, varies as well. The both upper and
26
lower limits of ๐ are increased by 1
2โซ
๐๐
๐๐ก
๐ต.
๐๐
๐๐ก
๐ต๐๐
ฮฉ when compared to the rigid-body limits, that are
1
2
โ2
๐ถ and
1
2
โ2
๐ด, respectively.
As long as ๐ถ โฅ ๐ด, inequality (2.117) is always be true, because the upper limit is always at least as large as
the lower limit, since the term of configuration change is added on both the left and right-hand sides of the
inequality.
2.10 MODIFIED EULER EQUATIONS IN TERMS OF EULER ANGLES
The motion of the body frame could be defined according to many coordinate transformations with respect
to the inertial frame. In the particular case of satellites, using Euler angles is frequent because that allows
for easy interpretation of the orientation of the body. However, other representations could be used, such
as quaternions or Brian angles [7].
The considered Euler angles are: (i) the precession angle ๐; (ii) the nutation angle ๐; and (iii) the spin angle
๐. The orientation of the body frame is given by a first rotation of ๐ with respect to the ๐ axis, secondly by
a rotation of ๐ with respect to the ๐ฅโฒ axis of the second frame, and finally by a rotation of ๐ with respect to
the ๐งโฒโฒ axis of the third frame. The number of apostrophes on the axes is the number of rotations the frame
has experienced. The inertial frame has the orientation the body frame, after performing these three
rotations.
The time-derivatives of the Euler angles correspond to angular velocities: ๏ฟฝ๏ฟฝ is the precession rate; ๏ฟฝ๏ฟฝ is the
nutation rate; ๏ฟฝ๏ฟฝ is the spin rate. If these angular velocities, which are parallel to axes of their respective
frame, are projected into the body frame, the angular velocity ๐ components, ๐๐ฅ, ๐๐ฆ and ๐๐ง, are obtained
[29]:
๐๐ฅ = ๏ฟฝ๏ฟฝ sin๐ sin๐ + ๏ฟฝ๏ฟฝ cos๐ (2.118)
๐๐ฆ = ๏ฟฝ๏ฟฝ sin ๐ cos๐ โ ๏ฟฝ๏ฟฝ sin๐ (2.119)
๐๐ง = ๏ฟฝ๏ฟฝ cos๐ + ๏ฟฝ๏ฟฝ (2.120)
The general form of the Euler Equations can be written using Euler angles, in a body frame that is a principal
frame centered in the center of mass. This is done by substituting the components of the angular velocity
and of the angular acceleration into (2.53)-(2.55). The components of the angular acceleration are
determined by differentiating (2.118)-(2.120).
27
The equations obtained after doing this are a set 3 coupled non-linear second order equations. Some
simplifications occur if it is assumed: (i) the body is free from torques, ๐ด = 0; (ii) two plane-symmetry
simplification with ๐ธ = 0; and (iii) axisymmetry with ๐ด = ๐ต.
However, the equations still remain a set of three coupled non-linear second order equations.
2.10.1 Intermediate Frame for an Axisymmetric Body
The intermediate frame is obtained after the rotation of ๐ and then of ๐. Therefore, the rotation of the
intermediate frame with respect to the inertial frame is defined as
๐ = ๐โ ๏ฟฝ๏ฟฝ. (2.121)
In this frame, the equations are simpler because the expressions for the angular velocities in terms of the
Euler angles simplify. The system of axes of the intermediate frame is ๐ฅโฒโฒ๐ฆโฒโฒ๐งโฒโฒ. In addition, using the
intermediate frame is particularly useful when the body is axisymmetric. For a non-axisymmetric body, the
time-function of ๐ด and ๐ต would depend not only on the variation of the bodyโs geometry but on the
orientation of the body frame at each instant, being a major inconvenience. For an axisymmetric body, the
spin angle of the body frame does not change its inertia tensor.
The alternative derivation of the general form of the Euler equations allows choosing any auxiliary rotation
with the respect to the inertial frame when the time-derivation is performed. In (2.33), the auxiliary reference
frame used to calculate ๏ฟฝ๏ฟฝ๐ in the applied torque equation (2.8) was the body frame. In this case, the auxiliary
reference frame to be used will be the intermediate frame. Thus, the Basic Kinematic Equation becomes
๐
๐๐ก
๐ผ
๐จ =๐
๐๐ก
๐
๐จ +๐ ร ๐จ, (2.122)
where ๐
๐๐ก
๐ is the time-derivative measured in the intermediate frame. Inserting the expression of the angular
momentum (2.32) into the expression of the applied torque (2.8) and rearranging results in
๐ด๐ =
๐
๐๐ก
๐ผ
(๐ฐ๐) +๐
๐๐ก
๐ผ
(โซ ๐ ร๐๐
๐๐ก
๐ต
๐๐๐
)
=๐
๐๐ก
๐
(๐ฐ) โ ๐+ ๐ฐ โ๐
๐๐ก
๐
(๐) + ๐ร (๐ฐ โ ๐) + ๏ฟฝ๏ฟฝ,
(2.123)
(2.124)
where having the body frame centered in the center of mass eliminated the term โ๏ฟฝ๏ฟฝ๐ ร ๐๏ฟฝ๏ฟฝ.
28
We have that ๐
๐๐ก
๐(๐) = ๏ฟฝ๏ฟฝ because the derivative of the angular velocity is the same regardless of the
reference frame. ๐
๐๐ก
๐(๐ฐ) =
๐
๐๐ก
๐ต(๐ฐ), because the body is axisymmetric. In addition, we assume a simplifying
hypothesis: the body is two-plane symmetric and ๐ธ = 0. Taking this into account, (2.124) simplifies to
๐ด๐ = ๐ฐ โ ๏ฟฝ๏ฟฝ + ๐ ร (๐ฐ โ ๐) +
๐
๐๐ก
๐ต
(๐ฐ) โ ๐. (2.125)
The components of ๐ in the intermediate frame are
๐๐ฅโฒโฒ = ๏ฟฝ๏ฟฝ, (2.126)
๐๐ฆโฒโฒ = ๏ฟฝ๏ฟฝ sin๐, (2.127)
๐๐งโฒโฒ = ๏ฟฝ๏ฟฝ cos ๐ + ๏ฟฝ๏ฟฝ (2.128)
The components of ๐ in the intermediate frame components are given by:
ฮฉ๐ฅโฒโฒ = ๏ฟฝ๏ฟฝ, (2.129)
ฮฉ๐ฆโฒโฒ = ๏ฟฝ๏ฟฝ sin ๐, (2.130)
ฮฉ๐งโฒโฒ = ๏ฟฝ๏ฟฝ cos๐. (2.131)
The angular acceleration components written in the intermediate frame are obtained by differentiating
(2.126)-(2.128). The inertia tensor (2.36) and its time-derivative (2.40) are written having ๐ด = ๐ต and ๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ.
Having all the variables of (2.125) defined, we write (2.125) for a free-body in the three directions:
0 = ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ + ๐ด๏ฟฝ๏ฟฝ + (๐ถ โ ๐ด)๏ฟฝ๏ฟฝ2 sin๐ cos๐ + ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ sin ๐ (2.132)
0 = ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ sin ๐ + ๐ด(๏ฟฝ๏ฟฝ sin ๐ + 2๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ cos ๐) โ ๐ถ๏ฟฝ๏ฟฝ(๏ฟฝ๏ฟฝ + ๏ฟฝ๏ฟฝ cos ๐) (2.133)
0 = ๏ฟฝ๏ฟฝ(๏ฟฝ๏ฟฝ cos ๐ + ๏ฟฝ๏ฟฝ) + ๐ถ(๏ฟฝ๏ฟฝ cos๐ โ ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ sin ๐ + ๏ฟฝ๏ฟฝ) (2.134)
29
2.11 ANGULAR MOMENTUM FOR A GENERAL VARIABLE-GEOMETRY BODY
We have the inertial frameโs ๐ axis aligned with the angular momentum ๐. Then, ๐ can be projected onto
the body frame using Euler angles, yielding
๐ = โ sin ๐ sin๐ ๐๐ + โ sin ๐ cos๐ ๐๐ + โ cos ๐ ๐๐. (2.135)
The expression of the angular momentum, when the variable-geometry bodyโs symmetry is such that the
term ๐ธ is zero and ๏ฟฝ๏ฟฝ๐ = 0, is
๐ = ๐ด๐๐ฅ๐๐ +๐ต๐๐ฆ๐๐ + ๐ถ๐๐ง๐๐. (2.136)
Replacing the components of the angular velocity in terms of Euler angles (2.118)-(2.120) into (2.136) and
equaling (2.136) to (2.135) and separating by the body frame directions results in:
โ sin ๐ sin๐ = ๐ด(๏ฟฝ๏ฟฝ sin ๐ sin๐ + ๏ฟฝ๏ฟฝ cos๐), (2.137)
โ sin๐ cos๐ = ๐ต(๏ฟฝ๏ฟฝ sin๐ cos๐ โ ๏ฟฝ๏ฟฝ sin๐), (2.138)
โ cos๐ = ๐ถ(๏ฟฝ๏ฟฝ cos ๐ + ๏ฟฝ๏ฟฝ). (2.139)
If the system (2.137)-(2.139) is rearranged to separate the angular rates in different equations, we obtain:
๏ฟฝ๏ฟฝ = โ (
1
๐ด(๐ก)โ
1
๐ต(๐ก)) sin ๐ sin๐ cos๐, (2.140)
๏ฟฝ๏ฟฝ = โ(
sin2 ๐
๐ด(๐ก)+cos2 ๐
๐ต(๐ก)), (2.141)
๏ฟฝ๏ฟฝ = โ cos ๐ (
1
๐ถ(๐ก)โsin2๐
๐ด(๐ก)โcos2๐
๐ต(๐ก)). (2.142)
2.11.1 Axisymmetric Variable-Geometry Body
A variable geometry-body can still be axisymmetric even though its moments of inertia vary: as long as
๐ด(๐ก) = ๐ต(๐ก). Having ๐ด = ๐ต simplifies (2.140)-(2.142) to:
๏ฟฝ๏ฟฝ = 0, (2.143)
๏ฟฝ๏ฟฝ =
โ
๐ด(๐ก)=
โ
๐ต(๐ก), (2.144)
30
๏ฟฝ๏ฟฝ = โ cos๐ (
1
๐ถ(๐ก)โ
1
๐ด(๐ก)). (2.145)
Eq. (2.143) implies that the variable geometry body with ๐ธ = 0 has a constant nutation angle, regardless of
how the principal moments of inertia vary with time. A consequence of this is that an axisymmetric variable-
geometry bodyโs nutation angle cannot be controlled by configuration changing. The nutation angle is
unchangeable and it only depends on the initial conditions of the body dynamics.
We conclude from (2.144) that the precession rate ๏ฟฝ๏ฟฝ is inversely proportional to the moment of inertia ๐ด:
๐ด(๐ก)๏ฟฝ๏ฟฝ = โ = ๐๐๐๐ ๐ก๐๐๐ก (2.146)
As a consequence, the precession rate increases when the moment of inertia ๐ด decreases. So, eq. (2.146)
expresses conservation of angular momentum on the ๐ axis during the geometry variation of the body.
Since โ is the norm of the vector ๐, it is always positive, as well the moment of inertia ๐ด. Consequently, ๏ฟฝ๏ฟฝ
is never negative.
Replacing โ (2.146) into (2.145) we obtain:
๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ=
๐ถ(๐ก)
๐ด(๐ก) โ ๐ถ(๐ก)sec ๐. (2.147)
Eq. (2.147) implies that the proportion between the precession rate and the spin depends not only on the
nutation angle but on the ratio of the inertia moments ๐ถ and ๐ด. Moreover, the ability of the body to change
the moments of inertia, while maintaining its axisymmetry, can change the ratio ๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ and even change the
signal of the ratio. For example, if the body changes from prolate, which implies ๐ด > ๐ถ, to oblate, which
means ๐ถ > ๐ด, while having a nutation angle between 0 and ๐
2, ๏ฟฝ๏ฟฝ that was positive when the body was
prolate, becomes negative when the body becomes oblate.
๐ ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ
๐ถ < ๐ด , 0 < ๐ <๐
2 + + โ
๐ถ < ๐ด , ๐
2< ๐ < ๐ + + +
๐ถ > ๐ด , 0 < ๐ <๐
2 + + โ
๐ถ > ๐ด , ๐
2< ๐ < ๐ + + +
Table 1- Sign of variables depending on the nutation angle and the bodyโs oblateness.
31
3 DESIGN OF VARIABLE GEOMETRY SATELLITES
After the derivation of the equations that express the relation between the geometry variation and the body
attitude dynamics, we envisioned satellites that could change its geometry in order to control its attitude.
The geometries studied are rough approximations of possible real variable-geometry satellites so that
insightful information about their dynamics may be obtained easily. For that reason, satellites are
approximated as being made of very simple components: uniform slender bars, springs or point masses. In
this first approach to the problem, we also avoided considering geometries with numerous degrees of
freedom of the shape-changing motion which would difficult the mathematical treatment โ geometries with
one degree of freedom were preferred. All the developed geometries have two-plane symmetry so that the
non-rigid term ๐ธ is zero, guaranteeing a major simplification of the dynamics.
3.1 CONNECTED SCISSOR MECHANISM
3.1.1 Geometry
The first developed concept for a variable-geometry satellite is one that is always axisymmetric from a
dynamical point of view and with constant transversal moments of inertia. The inertia tensor resulting from
this, with ๐ด = ๐ต = ๐๐๐๐ ๐ก๐๐๐ก, would be arguably the simplest, apart from the trivial spherical case. The control
would be achieved by a single degree of freedom that only changed the moment of inertia ๐ถ.
The Connected Scissor Mechanism was conceived as being composed by a certain even number of bars,
a total of 2๐. The bars are slender and are articulated to each other, held together on both ends by negligible
mass joints, connected in such a way that they form a closed loop of bars. The number of bars must be
such that ๐ โฅ 3, otherwise there would be no closed loop without superimposition of bars. Figure 1 shows
a Connected Scissor Mechanism with ๐ = 3, at two different configurations for clearer understanding.
The single degree of freedom of this geometry is the inclination of the bars: ๐ bars are inclined with a +๐ผ
angle and the other ๐ bars have an inclination of โ๐ผ with respect to the transversal symmetry plane of the
Connected Scissor Mechanism. In what concerns the mathematical description of the Connected Scissor
Mechanism, ๐ผ will be considered positive and will vary between the limit values ๐ผ = 0 and ๐ผ =๐
2.
3.1.2 Body Frame
The body frame used for the Connected Scissor Mechanism is oriented so that, from both the top
perspective and bottom perspective of the ๐ง axis, the bar structure is seen as a regular polygon of ๐ sides.
The ๐ง axis intersects the geometrical center of the top polygon formed by the slender bars upper ends and
intersects the geometrical center of the bottom polygon formed by the slender bars bottom ends.
32
Figure 1 โ Lateral perspective from the ๐ฅ axis of a Connected Scissor Mechanism of 6 bars of same length, with a
configuration ๐ผ1 on the left and a configuration ๐ผ2 at the right, being ๐ผ2 > ๐ผ1.
Both the ๐ฅ and ๐ฆ axes are freely oriented within the transversal symmetry plane of the geometry, as long
they are perpendicular to each other and make the body frame a right reference frame.
The selected body frame is centered at the center of mass to simplify terms in the equations, because ๏ฟฝ๏ฟฝ =
0. The geometric center and the center of mass coincide as a result of the bars have uniform density.
The limit value of ๐ผ = 0 corresponds to the situation when all the bars lie horizontally on the ๐ฅ0๐ฆ plane
while ๐ผ =๐
2 implies all the bars are vertically oriented within in the ๐ง axis. These extreme configurations for
the Connected Scissor Mechanism are not feasible in practice because even though the bars are modeled
as slender, having no thickness, the components of the satellite do have thickness and hence it is impossible
that they occupy the same points in space. Because of this, the configurations of the Connected Scissor
Mechanism should be distant of both the lower and upper limits of ๐ผ.
3.1.3 Inertia Tensor
We will determine the moments of inertia as a function of: (i) mass of each bar ๐; (ii) length of each bar ๐;
(iii) the number of bars 2๐; and (iv) the Connected Scissor Mechanismโs degree of freedom ๐ผ.
The top and the bottom views over the geometry are regular polygons. For that reason, ๐ symmetry planes
intersecting the sides of the polygon that are perpendicular to the ๐ฅ0๐ฆ plane exist. This adds up to a total of
๐ + 1 symmetry planes. Thus, at least two perpendicular symmetry planes exist permanently in the body
frame, containing the center of mass at their intersection. This implies ๐ธ = 0 for every configuration, as
intended.
33
The ๐ถ moment of inertia is the same regardless of the orientation of the ๐ฅ and ๐ฆ axes in the ๐ฅ0๐ฆ plane, as
a result of the distance of every point of the Connected Scissor Mechanism to the ๐ง axis being the same
independently of where the ๐ฅ and ๐ฆ planes are oriented to. Thus, the evaluation of how ๐ด and ๐ต depend on
the orientation of these axes can be regarded as a 2๐ท problem.
In the ๐ฅ0๐ฆ plane, a total of ๐ orientations of the ๐ฅ and ๐ฆ axes exist so that ๐ด and ๐ต donโt change between
the ๐ frames, because of the top/bottom polygonโs regularity. According to the Mohr Circle [43], for the two
dimensional inertia tensor, two orientations of the ๐ฅ and ๐ฆ axes allow for a maximum ๐ด and minimum ๐ต and
vice-versa, separated by an angle of amplitude of ๐
2. In this case, we have ๐ โฅ 3 equivalent inertia tensors
in the plane. Therefore, if ๐ orientations have the same inertia tensor then all possible frames in the ๐ฅ0๐ฆ
plane have the same inertia tensor. This implies the axisymmetry ๐ด = ๐ต , because the maximum and
minimum moments of inertia have to be equal for this to occur.
We will obtain the moments of inertia ๐ด = ๐ต and ๐ถ. If the top/bottom polygon is divided into ๐ geometrically
similar triangles, which are isosceles, we can obtain useful relations for angles and distances from them.
The angle ๐ is defined so that it is the double angle of the triangle. Using that the sum of the interior angles
of the triangle is ๐ and rearranging yields
๐ = ๐
๐ โ 2
2๐, (3.1)
being ๐ a function of ๐. A visualization of an example of ๐ for ๐ = 3 is available in Figure 2.
The distance between a midpoint of any of the sides of the 2D regular polygon and its geometrical center is
defined as
โ =
๐
2tan๐ cos๐ผ. (3.2)
Figure 2 โ Top view from the ๐ง axis of a Connected Scissor Mechanism with ๐ = 3 where the angle ๐ and distances โ
and ๐ ๐๐๐ ๐ผ are represented.
34
The Parallel axes theorem [43] is given by
๐ผ = ๐ผโฒ +๐๐2, (3.3)
where ๐ผโฒ is the moment of inertia with respect to an axis that intersects the center of mass of the body and
๐ผ is the moment of inertia with respect to any parallel axis. ๐ is the distance between the two axes.
At this point, we determine the expression for the moment of inertia of an uniform slender bar ฮฉ of mass ๐
and length ๐ around an arbitrary axis that belongs to a plane that contains both the axis and the bar. Let the
distance between a point of the bar and the axis be named ๐ and the infinitesimal bar length be ๐๐ . Thus,
the moment of inertia around the axis is
๐ผ = โซ๐2๐๐
ฮฉ
. (3.4)
The bar has an inclination of ๐ฝ with respect to the axis and so ๐ can be written in function of ๐, ๐ and ๐ฝ, being
the point ๐ = 0 corresponds to one of the two ends of the bar:
๐ = (๐ โ
๐
2) sin๐ฝ. (3.5)
Since the mass distribution along the length of the bar is uniform, we have ๐๐ = ๐๐๐
๐. Using this and ๐ from
(3.5) in (3.4) and integrating from 0 to ๐ along the bar gives the moment of inertia ๐ผ
๐ผ =
๐๐2
12sin2 ๐ฝ (3.6)
After the derivation of ๐ผ and the definitions concerning the polygon, (3.1) and (3.2), we are able to determine
the ๐ถ moment of inertia of the Connected Scissor Mechanism.
The symmetry of the geometry of the Connected Scissor Mechanism implies that the contribution of each
bar to ๐ถ is the same: they are all similar bars, equally inclined, with +๐ผ or โ๐ผ with respect to the ๐ฅ0๐ฆ plane,
having their midpoints all situated in the ๐ฅ0๐ฆ plane. Moreover, each of the bars and a virtual axis, being the
latter an axis that intersects the midpoint of the bars and is parallel to the ๐ง axis, belong to a plane containing
the bar and the axis.
We gather conditions to use expression (3.6) to obtain the individual contribution of each of the bars to the
total moment of inertia, ๐ถ๐โฒ, which is determined with respect to the respective virtual axis. Then, we use the
Parallel Axes Theorem (3.3), using โ (3.2) as ๐ in the expression, since it is the distance between the virtual
35
axis and the ๐ง axis. The inclination angle of the bars with respect to ๐ง is not ๐ผ but ๐
2โ ๐ผ , reason why
๐๐2
12sin2 ๐ผ becomes
๐๐2
12cos2 ๐ผ. Performing this yields:
๐ถ = 2๐(๐ถ๐โฒ + ๐โ
2) =๐๐๐2
6cos2 ๐ผ(1 + 3 tan2๐). (3.7)
To obtain ๐ด, we use a property of the polar moment of inertia [43]:
๐ด + ๐ต + ๐ถ = 2๐ผ0. (3.8)
Rearranging (3.8) using ๐ด = ๐ต yields
๐ด = ๐ผ0 โ
๐ถ
2, (3.9)
which allows to obtain ๐ด if ๐ผ0 is known.
The polar inertia moment of a single bar, defined as ๐ผ0๐, is firstly determined to then obtain the total polar
inertia moment. All the bars have the same ๐ผ0๐ because for every point in each bar we have a point on any
of the other bars that is equally distant to the origin, which happens for all the points of the bar. Thus:
๐ผ0 = ๐๐ผ0๐ . (3.10)
We consider a bar with an orientation that simplifies the calculation to obtain ๐ผ0: a bar that lies on a plane
that is perpendicular to the ๐ฅ axis. So we write
๐ผ0๐ = โซ๐ฅ2 + ๐ฆ2 + ๐ง2๐๐
ฮฉ
= โซ๐ฅ2๐๐ฮฉ
+โซ๐ฆ2 + ๐ง2๐๐ฮฉ
. (3.11)
๐ฅ is constant along this bar, being ๐ฅ = โ. Furthermore, โซ ๐ฆ2 + ๐ง2๐๐ฮฉ
is the moment of inertia of this bar, ๐ด๐,
which is ๐๐2
12 regardless of the inclination ๐ผ because, for all ๐ผ, the bar belongs to a perpendicular plane to
the ๐ฅ axis and is intersected by the ๐ฅ axis at its midpoint. Taking this into account simplifies (3.11) into
(๐ผ0)๐ = ๐โ2 +
๐๐2
12=๐๐2
12(3 tan2๐ cos2 ๐ผ + 1). (3.12)
Then, the total polar inertia moment is obtained by multiplying (3.12) by the total number of bars 2๐:
๐ผ0 =
๐๐๐2
6(3 tan2๐ cos2 ๐ผ + 1). (3.13)
After replacing ๐ผ0 (3.13) in (3.9) and simplifying the expression yields
36
๐ด =
๐๐๐2
12(cos2 ๐ผ (3 tan2๐ โ 1) + 2). (3.14)
3.1.4 Triangle Connected Scissor Mechanism
The dependence of ๐ผ in ๐ด vanishes when
tan2๐ =
1
3, (3.15)
obtaining a constant value for ๐ด, as intended:
๐ด =
๐๐๐2
6. (3.16)
Since ๐ is the double angle in the partial triangle of the regular top/bottom polygon, we have that ๐ is limited:
0 < ๐ <๐
2. ๐ is also limited: it is an integer such that ๐ โฅ 3. Taking these constraints into account and using
(3.1), (3.15) is rewritten:
tan(๐
๐ โ 2
2๐) = ยฑ
โ3
3. (3.17)
Firstly, the positive branch of (3.17) is obtained:
tan(๐
๐ โ 2
2๐) = tan(
๐
6)
โบ ๐๐โ 2
2๐=๐
6+ ๐๐, ๐ โ โค
โบ ๐ =2
23 + 2๐
, ๐ โ โค.
(3.18)
The only ๐ that satisfies ๐ being an integer number larger than 2 is ๐ = 0, making ๐ = 3 and ๐ = ๐/6.
Concerning the negative solution, we have:
tan(๐
๐ โ 2
2๐) = tan(โ
๐
6)
โบ ๐๐ โ 2
2๐= โ
๐
6+ ๐๐, ๐ โ โค
โบ ๐ =2
โ23 + 2๐
, ๐ โ โค
(3.19)
37
Eq. (3.19) has no solution, given the restrictions of ๐. So we conclude the only solution for having ๐ด = ๐ต =
๐๐๐๐ ๐ก๐๐๐ก, or equivalently ๐๐ด
๐๐ผ= 0, is ๐ = 3.
This implies that if the polygon is a regular triangle, then the Connected Scissor Mechanism has a constant
๐ด, depending only on ๐, ๐ and ๐, regardless of the degree of freedom of the Connected Scissor Mechanism
๐ผ. Thus, when ๐ = 3, the moments of inertia are
and
๐ถ = ๐๐2 cos2 ๐ผ (3.21)
The Triangle-Connected Scissor Mechanism is a simple geometry that allows, with a total of 6 bars, to have
an axisymmetric satellite that only varies its ๐ถ moment of inertia according to a very simple expression
(3.21). In addition to that, if a number of bars larger than 6 is selected, ๐ด also varies with the degree of
freedom ๐ผ, allowing for more options in what concerns attitude control.
3.1.5 Point Masses
It would be more unrestrictive if the Connected Scissor Mechanism geometry would allow to vary the ๐ถ
moment of inertia, leaving ๐ด = ๐ต = ๐๐๐๐ ๐ก๐๐๐ก, without the restraint of having 6 bars. In order to achieve this,
including masses on the joints of the bars was thought of. The masses are modeled as point, representing
structural reinforcements in the barsโ joints.
We define ๐1 as the mass of each of the ๐ point masses placed at the center of the bars, connecting every
pair of bars. ๐2 is the mass of the 2๐ point masses placed at the ends of the bars, connecting every pair of
bars.
๐ด =
๐๐2
2. (3.20)
38
Figure 3 โ Connected Scissor Mechanism with 6 bars, 3 masses ๐1 at the center of the bars, displayed in red, and 6
masses ๐2 at the ends of the bars, displayed in blue.
The ๐ง axis and the ๐ symmetry axes in the ๐ฅ0๐ฆ plane all remain symmetry axis with the addition of the point
masses. Thus, the body frame is still centered at the center of mass and is a principal frame, where ๐ด = ๐ต
remains valid, according to the explanation given in sub-section 3.1.3.
In order to simplify the calculations, two non-dimensional quantities with respect to the mass of the slender
bar are introduced:
๐1 =
๐1
๐; (3.22)
๐2 =๐2
๐. (3.23)
With the point masses, expression (3.14) is modified to:
๐ด =
๐๐๐2
12(cos2 ๐ผ ((3+ 3๐2 +
3
2๐1) tan
2๐ โ 1 โ 3๐2) + 2). (3.24)
The condition of having ๐ด constant regardless of the configuration of the Connected Scissor Mechanism is
having
tan2๐ =
1 + 3๐2
3 + 3๐2 +32๐1
(3.25)
If we consider extreme cases for the values of the masses ๐, ๐1 and ๐2 to satisfy (3.25), we have
39
0 โค ๐1 < โ, (3.26)
0 โค ๐2 < โ. (3.27)
Playing with all possible values of ๐1 and ๐2 in (3.25) yields
0 โค tan2๐ โค 1. (3.28)
However, ๐ cannot have continuous values, which results from (3.1), and ๐ is an integer larger than 2. The
first possibilities for ๐ are:
๐ ๐ ๐ญ๐๐ง๐
3 ๐
6
1
โ3
4 ๐
4 1
5 3๐
10 1.37638192047
6 ๐
3 โ3
โฆ โฆ โฆ
Table 2 โ First possible values for ๐.
The limit case of infinite bars,
lim๐โโ
(๐๐ โ 2
2๐) =
๐
2, (3.29)
corresponds to a Connected Scissor Mechanism which top/bottom polygon tends to a circle. We also have
lim๐โโ
(tan๐) = โ, (3.30)
being tan๐ a monotonously crescent function with ๐. Therefore, the only possible solutions are ๐ = 3 and
๐ = 4 because for ๐ > 4 the values of tan๐ fall out of the interval of (3.28). Only one additional alternative,
๐ = 4, becomes possible. Equation (3.25) can be solved to obtain the (๐1, ๐2) pairs to satisfy ๐ = 3 and ๐ =
4:
๐ = 3 โ tan2๐ =1
3โ {
๐1๐2= 4
or๐1 = ๐2 = 0
, (3.31)
40
๐ = 4 โ tan2๐ = 1 โ ๐1 = โ
4
3 (3.32)
Since the mass ratios cannot be negative, the alternative of ๐ = 4 is not possible.
We conclude that adding point masses does not allow for additional options of the number of bars. So, the
two possibilities of having ๐๐ด
๐๐ผ= 0 are: (i) 6 bars with no point masses; and (ii) 6 bars with masses such that
๐1
๐2= 4. The second alternative is not interesting because the first solution allows
๐๐ด
๐๐ผ= 0, without the point
masses.
3.2 OCTAHEDRON
3.2.1 Geometry
The objective of the concept behind this geometry is to have a variable-geometry axisymmetric body,
preferably with the possibility of selecting particular values for its properties so that ๐ด = ๐ต = ๐๐๐๐ ๐ก๐๐๐ก.
Consequently, the variability of the inertia tensor would be achieved by varying ๐ถ.
This configuration consists in eight slender rigid and uniform bars of mass ๐ and length ๐ that form an
octahedron with variable shape. The four top bars form the top quadrangular pyramid and the four bottom
bars form the bottom quadrangular pyramid, the pyramids being articulated at the four connections between
the four top bars with the four bottom bars.
The octahedron can change its shape by varying its single degree of freedom ๐ผ, which is half of the opening
2๐ผ between opposite bars connected at the top or bottom vertex. The degree of freedom ๐ผ is limited
between 0 and ๐/2.
41
Figure 4 โ Octahedron at a configuration ๐ผ, where the body frame is displayed as well as the angle of 2๐ผ between two
opposite bars.
3.2.2 Body Frame
The ๐ง axis is defined as the vertical axis that intersects the top vertex of the top pyramid and the bottom
vertex of the bottom pyramid. The ๐ฅ0๐ฆ plane is the transversal symmetry plane, which contains the base of
both pyramids, the orientation of the ๐ฅ and ๐ฆ axes being free as long as the body frame remains a right
reference frame and ๐ฅ and ๐ฆ are orthogonal axes. Since the symmetry of the body implies there are four
symmetry axes in the ๐ฅ0๐ฆ, the principal moments of inertia in the plane, ๐ด and ๐ต, are equal, as explained in
sub-section 3.1.3 for the Connected Scissor Mechanism.
The symmetry of the geometry also implies ๐ธ = 0, since the ๐ฅ0๐ฆ plane is perpendicular to each of the other
four symmetry planes of the octahedron. The body frame is centered at the center of mass of the origin, so
that ๏ฟฝ๏ฟฝ = 0.
3.2.3 Inertia Tensor
Performing a similar derivation to what was done in sub-section 3.1.3 for the Connected Scissor Mechanism
and in 3.6.3 for the Compass, we obtain the elements of the inertia tensor:
๐ด = ๐ต =
28
3๐๐2 cos2 ๐ผ, (3.33)
42
๐ถ =
8
3๐๐2 sin๐ผ2.
(3.34)
This geometry does not allow to select particular constants of the body such that ๐ด does not depend on ๐ผ.
3.2.4 Modified Octahedral
In order to modify the Octahedral so that ๐๐ด/๐๐ผ could be zero, we considered the possibility of adding an
uniform spring of mass ๐ and length ๐ that connected the top and bottom vertexes of the top and bottom.
The parameter ๐ is an additional degree of freedom to be controlled by actuators, totalizing two degrees of
freedom for this satellite. The objective of this is that ๐ could be controlled by a function of ๐ผ such that the
๐ด would no longer depend on ๐ผ.
With this modification, the four top bars are no longer connected to the four bottom bar, even though they
remain symmetric to each other with respect to the ๐ฅ0๐ฆ plane.
The addition of the central spring does not modify the symmetries of the geometry described on sub-section
3.2.2, because ๐ง axis is still a symmetry axis and the four symmetry axes in the ๐ฅ0๐ฆ plane remain symmetry
axes.
So that there is no intersection between bars, for any ๐ผ, we have
๐
๐โฅ 2. (3.35)
Expression (3.36) is modified to
๐ด = ๐ต =
2๐๐2
3(2 cos2 ๐ผ +
๐
2๐(๐
๐)2
+ 12(๐
๐โcos๐ผ
2)2
), (3.36)
while the addition of the spring does not change (3.34) because the spring lies on the ๐ง axis:
๐ถ =
8
3๐๐2 sin๐ผ2. (3.37)
The dependence of ๐ผ in (3.36) cannot vanish by selecting of particular ๐/๐ ratios as a result of all the three
terms in the right-hand side being non-negative.
43
Other attempts to obtain ๐๐ด
๐๐ผ= 0 on octahedral configurations failed, because the addition of other elements
in the geometry such as springs and movable masses added a positive term to the right-hand side of (3.36)
that does not depend on ๐ผ.
3.3 DANDELION
3.3.1 Geometry
The concept of the Dandelion was to envision a satellite that, being axisymmetric, would allow a broader
range of geometry possibilities, similarly as the concept of the Connected Scissor Mechanism, being made
of an arbitrary number of bars. An additional objective was to explore the possibility of selecting particular
values for its properties so that ๐ด = ๐ต = ๐๐๐๐ ๐ก๐๐๐ก. Consequently, the attitude control would be achieved by
varying ๐ถ.
This geometry consists in a central bar of mass ๐ and length ๐ฟ and by ๐ other bars of mass ๐ and length
๐ that have their top end articulated to the top end of the central bar. The ๐ bars are equivalently spaced
and form the same angle ๐ผ with the central bar. The attitude control is made by changing ๐ผ, the single
degree of freedom of this geometry. ๐ผ is limited between 0 and ๐.
Figure 5 โ Dandelion with ๐ = 4 bars at a configuration of ๐ผ and the body frame centered at the center of mass for this
particular configuration.
44
3.3.2 Body Frame
We select a body frame which ๐ง axis is aligned with the central bar and is centered at the center of mass of
the geometry. Since there are ๐ symmetry axes on the ๐ฅ0๐ฆ plane, because the bars are equally spaced,
the orientation of the ๐ฅ and ๐ฆ axes can be any and ๐ด = ๐ต is valid for all the orientations, taking into account
the discussion in sub-section 3.1.3. In addition, the symmetry of the geometry implies ๐ธ = 0.
For this geometry, the position of the center of mass, with respect to the articulated point at the top end of
the central bar, depends on ๐ผ. The distance between the center of mass and the top point of the central bar
is
๐ง๐ถ๐(๐ผ) =
๐๐ฟ + ๐๐๐ cos ๐ผ
2(๐ + ๐๐). (3.38)
The origin of the body frame has the motion along the z axis prescribed by (3.38).
3.3.3 Inertia Tensor
The moment of inertia ๐ถ can be obtained without taking into account the motion of the frame depending on
๐ผ, since the distance of the ๐ bars to the ๐ง axis is the same regardless of where the ๐ฅ0๐ฆ plane is with respect
to the body. Consequently:
๐ถ =
๐๐๐2
3sin2 ๐ผ. (3.39)
The motion of the origin of the body frame has to be accounted for in order to determine ๐ด. Firstly, we
determine ๐ด for the central bar with respect to an axis that intersects its center of mass, yielding
๐ด๐โฒ =
๐๐ฟ2
12 (3.40)
and the contribution of the ๐ bars with respect to an axis that belongs to the plane that intersects their
centers of mass and is parallel to ๐ฅ0๐ฆ
๐ด๐โฒ =
๐๐๐2
12cos2 ๐ผ. (3.41)
The parallel axis theorem can be used on (3.40) and on (3.41) separately, because the distance between
the ๐ฅ0๐ฆ plane and the axis with respect to where (3.40) and on (3.41) were calculated are different and both
depend on ๐ง๐ถ๐. The former distance is given by
๐0 = ๐ง๐ถ๐ โ
๐ฟ
2, (3.42)
45
while the latter is
๐๐ = ๐ง๐ถ๐ โ
๐
2cos ๐ผ. (3.43)
Using the parallel axis theorem (3.3) and then adding the contributions of the central bar and the central
and the set of ๐ bars to the total ๐ด yields:
๐ด =
๐๐ฟ2
12+ ๐(๐ง๐ถ๐ โ
๐ฟ
2)2
+๐๐๐2
12cos2 ๐ผ + ๐๐ (๐ง๐ถ๐ โ
๐
2cos๐ผ)
2
. (3.44)
We replace (3.38) in (3.44) and then it is possible to rearrange the expression to obtain
๐ด =
๐๐ฟ2(๐ + 4๐๐) + ๐๐๐ cos๐ผ (๐ cos๐ผ (4๐ + ๐๐) โ 6๐๐ฟ)
12(๐ +๐๐). (3.45)
We conclude from (3.45) that it is not possible select constants such that ๐๐ด
๐๐ผ= 0 for all configurations.
3.4 EIGHT POINT MASSES
3.4.1 Geometry
This concept consists in having the attitude control of the satellite be performed by prescribing the variation
of one of the principal moments of inertia with time while keeping the other two constant. We selected ๐ถ(๐ก)
as the variable moment of inertia, without loss of generality. Firstly, we consider the general case when ๐ด
and ๐ต are allowed to be different and then the axisymmetric case.
The variation of ๐ถ is achieved by the controlled motion of eight masses ๐ along a specified trajectory within
their respective tubes. The tubes are regarded as having negligible mass comparing with the masses. The
eight masses and the structure required for it are an installation in the satellite to control its attitude rather
than a model for a satellite itself.
The eight-mass system as a whole is conceived as having a single degree of freedom because the position
of the masses are given by the mirrored positions of the other masses, for all configurations.
46
Figure 6 โ Eight Point Masses in space. The trajectories of the masses so that ๐ด and ๐ต are kept constant are visible in
dashed curves.
3.4.2 Body Frame
The selected body frame for this geometry has the coordinate planes ๐ฅ0๐ฆ, ๐ฆ0๐ง and ๐ฅ0๐ง be the symmetry
planes of the masses motion, which implies the body frame is principal and that the reference frame is
centered at the center of mass, having ๏ฟฝ๏ฟฝ = 0. The symmetries implied by this body frame also ensure ๐ธ =
0 for all configurations.
The eight masses can be regarded as vertexes of a parallelepiped that can change its shape. Each of the
faces of the parallelepiped is intersected at the respective geometric center by a coordinate axis, as can be
observed in Figure 6.
3.4.3 Trajectory of the Masses
The shape of the tubes or, equivalently, the trajectory of the masses, is obtained by imposing that ๐ด and ๐ต
are constant regardless of their position. At a first instant ๐ก1, the eight mirrored positions of the eight point
masses are given by
๐๐(๐ก1) = ยฑ๐ฅ1๐๐ ยฑ ๐ฆ1๐๐ ยฑ ๐ง1๐๐ with ๐ = 1,2, . . . ,8 , (3.46)
where ๐ฅ1, ๐ฆ1 and ๐ง1 are the coordinates of the mass whose motion is confined to the first octant, being
therefore positive values for all configurations. The positions of the eight masses at a different instant ๐ก2 are
also defined with respect to the positive coordinates of the mass of the first octant:
๐๐(๐ก2) = ยฑ๐ฅ2๐๐ ยฑ ๐ฆ2๐๐ ยฑ ๐ง2๐๐ with = 1,2, . . . ,8 . (3.47)
47
The total moments of inertia at ๐ก1 are given by
๐ด(๐ก1) = 8๐(๐ฆ12 + ๐ง1
2), (3.48)
๐ต(๐ก1) = 8๐(๐ฅ12 + ๐ง1
2), (3.49)
๐ถ(๐ก1) = 8๐(๐ฅ12 + ๐ฆ1
2), (3.50)
while the moments of inertia at ๐ก2 are
๐ด(๐ก2) = 8๐(๐ฆ22 + ๐ง2
2), (3.51)
๐ต(๐ก2) = 8๐(๐ฅ22 + ๐ง2
2), (3.52)
๐ถ(๐ก2) = 8๐(๐ฅ22 + ๐ฆ2
2). (3.53)
The prescribed change in the moment of inertia ๐ถ is defined:
ฮ๐ถ = ๐ถ(๐ก2) โ ๐ถ(๐ก1) (3.54)
After equaling (3.48) to (3.51) and (3.49) to (3.52), so that ๐ด(๐ก1) = ๐ด(๐ก2) and ๐ต(๐ก1) = ๐ต(๐ก2), we can solve a
system of 3 equations to obtain the coordinates at ๐ก2 as function of the coordinates at ๐ก1and ฮ๐ถ, which is
the known intended change in the variable moment of inertia. Performing this yields:
๐ฅ2 = ยฑโ๐ฅ12 +
ฮ๐ถ
16๐, (3.55)
๐ฆ2 = ยฑโ๐ฆ12 +
ฮ๐ถ
16๐, (3.56)
๐ง2 = ยฑโ๐ง12 โ
ฮ๐ถ
16๐. (3.57)
It can be concluded from (3.57) that the ฮ๐ถ is limited by ๐ and ๐ง1. Rearranging the expression inside the
square in (3.57) and having, ๐ง1 โฅ 0, as a result of the negative solutions of the coordinates being already
present in the double sign, yields the lower and upper limits of ฮ๐ถ
โ8๐(๐ฅ12 + ๐ฆ1
2) โค ฮ๐ถ โค 16๐๐ง12, (3.58)
48
where the lower limit corresponds to the limit situation ฮ๐ถ = โ๐ถ(๐ก1) when the eight masses are moved to
the ๐ง axis to have ๐ถ = 0. The upper limit is the situation where the eight point masses are located in the ๐ฅ0๐ฆ
plane.
Inequalities (3.58) can be rearranged to obtain the range of possible values of ๐ถ, giving
0 โค ๐ถ โค 8๐(๐ฅ12 + ๐ฆ1
2 + 2๐ง12). (3.59)
Thus, ๐ถ is limited by the initial coordinates of the point masses and the mass ๐.
3.4.4 Geometric Relations with Axisymmetry
Conditions ๐ฅ1 = ๐ฆ1 and ๐ฅ2 = ๐ฆ2 must be verified in order to have the set of eight masses be axisymmetric,
with ๐ด = ๐ต.
We define the degree of freedom
๐ผ(๐ก) = ๐ฅ(๐ก) = ๐ฆ(๐ก), (3.60)
for any instant ๐ก, while the third coordinate is ๐ง(๐ก).
The moments of inertia as function of ๐ง and ๐ผ are obtained by inserting (3.60) in (3.48)-(3.53), replacing
the instants ๐ก1 and ๐ก2 by a general instant of time ๐ก:
๐ด = ๐ต = 8๐(๐ผ(๐ก)2 + ๐ง(๐ก)2), (3.61)
๐ถ = 16๐๐ผ(๐ก)2. (3.62)
We now perform an adimensionalisation of the variables, defining
๐ =
1
8 (3.63)
and
๐ด = ๐ต = 1. (3.64)
Inserting (3.63) and (3.64), for a certain instant ๐ก, into the general expressions (3.61) and (3.62) and solving
them for ๐ผ and ๐ง as function of ๐ถ yields
๐ผ = โ๐ถ
2 (3.65)
49
and
๐ง = โ1 โ๐ถ
2= โ1 โ ๐ผ2. (3.66)
The range of possible values for ๐ผ, ๐ง and ๐ถ is obtained from (3.65) and (3.66):
0 โค ๐ผ โค 1, (3.67)
0 โค ๐ง โค 1, (3.68)
0 โค ๐ถ โค 2. (3.69)
From (3.69) we conclude that the maximum of ๐ถ is
๐ถ๐๐๐ฅ = 2๐ด = 2๐ต. (3.70)
Expression (3.66) is arranged to obtain a geometrical relation between ๐ผ and ๐ง:
๐ผ2 + ๐ง2 = 1, (3.71)
defining a circumference of radius 1 in a plane with ๐ผ and ๐ง as axis.
It can be concluded from (3.71) that the allowed trajectories for the eight masses belong to the planes ๐ฅ = ๐ฆ
and ๐ฅ = โ๐ฆ form an ellipse in three-dimensional space, one for each of plane. In these planes, the distance
that is perpendicular to the ๐ง axis is such that coordinate ๐ผ, equal in the ๐ฅ and ๐ฆ axes, projected into the
mentioned planes, becomes โ2๐ผ. Therefore the geometrical ellipse on the planes ๐ฅ = ๐ฆ and ๐ฅ = โ๐ฆ is
defined by
(๐ผ
โ2)2
+ ๐ง2 = 1. (3.72)
Thus, the vertical semi-axis is 1 while the horizontal semi-axis is โ2, making the ellipse tapered in the
horizontal direction along ๐ฅ = ๐ฆ or ๐ฅ = โ๐ฆ. This can be observed in Figure 6.
50
3.5 CONTROL POSSIBILITIES OF AXISYMMETRIC SATELLITES
A downside of having variable-geometry axisymmetric satellites as the Connected Scissor Mechanism, the
Octahedron, the Dandelion and the Eight Point Masses with ๐ด = ๐ต , is that the nutation angle is not
controllable. In addition, for the Connected Scissor Mechanism geometries, which was the single geometry
where it was possible to obtain ๐๐ด
๐๐ผ= 0 for all ๐ผ, the precession rate ๏ฟฝ๏ฟฝ is not controllable as well, which was
concluded from (2.146). Only the spin rate is controllable for this geometry, given by
๏ฟฝ๏ฟฝ = โ cos๐
๐ด โ ๐ถ(๐ผ)
๐ด๐ถ(๐ผ), (3.73)
which resulted from combining (2.146) and (2.147).
The attitude control possibilities offered by the Connected Scissor Mechanism with constant ๐ด are limited.
However, for the Connected Scissor Mechanism with variable ๐ด, the Octahedron and the Dandelion, both
the precession rate and the spin rate are modified by a change in ๐ผ, since ๐๐ด
๐๐ผ and
๐๐ถ
๐๐ผ are different from zero.
They are:
๏ฟฝ๏ฟฝ =
โ
๐ด(๐ผ), (3.74)
๏ฟฝ๏ฟฝ = โ cos๐
๐ด(๐ผ) โ ๐ถ(๐ผ)
๐ด(๐ผ)๐ถ(๐ผ).
(3.75)
Consequently, these geometries offer a wider range of attitude control possibilities. Nevertheless, it is
possible to control also the nutation angle of satellite if the satellite is not axisymmetric.
The next envisioned geometry, described in section 3.6, has variable moments of inertia such that ๐ด โ ๐ต โ
๐ถ, in order to allow for a less limited attitude control than the already discussed geometries.
3.6 COMPASS
3.6.1 Geometry
The Compass consists of two bars articulated at one joint, where one end of each bar coincides. Its degree
of freedom is the opening angle around the joint, 2๐ผ. In this work, we limited the range of ๐ผ to 0 โค ๐ผ โค ๐,
but it would be interesting to explore the control possibilities of having an unlimited range of ๐ผ. The bars are
slender and have a mass ๐ and a length ๐, having an uniform mass distribution.
51
3.6.2 Body Frame
The frame selected for the Compass is such that the ๐ฆ0๐ง plane contains both the bars. The ๐ง axis is a
symmetry axis for the compass, having its positive direction pointing towards the joint.
So that ๏ฟฝ๏ฟฝ = 0, the body frame is centered at the center of mass. The position of the center of mass with
respect to the joint depends on the aperture ๐ผ and is fixed with respect to the inertial frame when the
compass has no translational movement.
Figure 7 โ Compass for two different configurations, ๐ผ2 and ๐ผ1, with ๐ผ2 > ๐ผ1.The body frame is centered at the center
of mass.
The position of the center of mass with respect to the joint is
๐ง๐ถ๐ = โ
๐ cos๐ผ
2 . (3.76)
The motion of the origin of the body frame with respect to the joint as function of the configuration is defined
by (3.76). This definition of the body frame implies that the ๐ฆ axis is intersecting the midpoint of both bars
for every allowed configuration of the Compass.
3.6.3 Inertia Tensor
The selected body frame is a principal frame because the ๐ง axis is a symmetry axis and the body lies on
the ๐ฆ0๐ง plane for all configurations. Moreover, the non-rigid motion term ๐ธ is zero for every ๐ผ, because ๐ฅ0๐ง
and ๐ฆ0๐ง are perpendicular symmetry planes for every possible configuration of the Compass.
Expression (3.6) can be used to calculate ๐ต of the two bars, having the inclination ๐ฝ equal to ๐
2โ ๐ผ, yielding
52
๐ต =
๐๐2
6cos2 ๐ผ (3.77)
We use the parallel axes theorem (3.3), having ๐ =๐
2sin๐ผ, and combine with the expression for the moment
of inertia of the inclined slender bar (3.6) to obtain
๐ถ = 4
๐๐2
6sin2 ๐ผ (3.78)
for the Compass.
Since the compass is contained in the ๐ฆ0๐ง plane we use the two-dimensional expression for the polar
moment of inertia ๐ผ0, which is in this case, the same as ๐ด, resulting in
๐ด =
๐๐2
6(3 sin2 ๐ผ + 1). (3.79)
3.7 DISCUSSION
The suggested variable-geometries form a wide range of attitude control possibilities for spacecraft.
If the only objective is to control the spin rate of the satellite, then the axisymmetric solutions that only allow
for ๐ถ to vary can be used. These are the triangular Connected Scissor Mechanism, which could be regarded
as simplified model of a satellite composed of movable parts, and the Eight Point Masses, that could be an
installation to add to the satellite itself.
The other axisymmetric satellites envisioned in the present work are such that varying ๐ถ implies varying the
transverse moments of inertia as well. For these geometries, we verified if it would be possible to select
particular values for the constants, such as length, mass or number of bars, that would allow for the satellite
to vary ๐ถ while keeping ๐ด constant. This was only achieved on the Connected Scissor Mechanism.
Nevertheless, the satellites where this was not possible, such as the Octahedron and the Dandelion, allow
for a wider range of possibilities of attitude control since they can also change the precession rate, being
interesting possibilities.
The compass is the geometry that allows for more control possibilities because the three Euler angular
velocities can vary.
In what concerns the applicability of these models, both the Connected Scissor Mechanism and the
Dandelion are the most versatile solutions since they can be made of an arbitrary number of bars without
losing dynamic dynamical axisymmetry.
53
4 SOLUTIONS OF THE FREE VARIABLE-GEOMETRY SATELLITE
MOTION
4.1 SYNCHRONOUS ROTATION IN AN ELLIPTICAL ORBIT WITH A VARIABLE-GEOMETRY
SATELLITE
An interesting application of the variable-geometry satellites is the use of the configuration change so that
the satellite could maintain the same orientation with respect to the central body, throughout an elliptic orbit.
Since the velocity of a satellite is not constant during the orbital period in an elliptic orbit, the configuration
of the geometry as function of time would be such that the precession rate is synchronized with the motion
of the satellite in the orbital plane.
This attitude control could be useful to spacecraft that requires its instrumentation to be pointing to the
central body.
We will consider small eccentricity orbits, with ๐ โค 0.1, which require small geometry variations from the
satellite, because conventional attitude control, by momentum wheels or thrusters for example, would be
more suitable for large-angle maneuver for the synchronous rotation in very eccentric orbits.
We selected the Connected Scissor Mechanism of the set of envisioned satellites to explore this application
of variable-geometry spacecraft. This geometry, being axisymmetric, can change its moment of inertia ๐ด
and therefore is able to control the precession rate as intended. In addition, it is versatile and has simpler
expressions for the moment of inertia ๐ด (3.14) when compared to the equivalent expression for the
Dandelion (3.45).
The condition for having the rotation of the satellite be synchronous with the movement in the orbital plane
is satisfied by having
๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ (4.1)
during all instants of an orbital period, where ๐ is the true anomaly. Finding the configuration of the satellite
as function of time ๐ผ(๐ก) that satisfies (4.1) is the objective of section 4.1.
4.1.1 Analytical Approximation for Keplerโs Equation
We will consider the solution of the two-body problem in the present section.
The time-derivative of the true anomaly [29] is
54
๏ฟฝ๏ฟฝ = (1 + ๐ cos๐)2โ๐
๐3(1 โ ๐2) , (4.2)
where ๐ is the eccentricity of the orbit, ๐ is the gravitational constant and ๐ is the semi-major axis.
Inserting the definition of mean angular velocity [29]
๐ = โ
๐
๐3 (4.3)
into (4.2) so that expression (4.2) becomes function of only two orbital parameters, ๐ and ๐, and then
substituting cos ๐ by the expression that relates the mean anomaly with the eccentric anomaly ๐ธ,
cos๐ =
cos๐ธ โ ๐
1 โ ๐ cos๐ธ, (4.4)
yields:
๏ฟฝ๏ฟฝ = (1 + ๐
cos๐ธ โ ๐
1 โ ๐ cos ๐ธ)2 ๐
โ1 โ ๐2. (4.5)
In order to determine ๏ฟฝ๏ฟฝ as function of time, ๐ธ(๐ก) also needs to be obtained. The relationship between the
mean anomaly ๐ and the eccentric anomaly is
๐ = ๐ธ โ ๐ sin๐ธ, (4.6)
where the mean anomaly, unlike the eccentric anomaly, can be obtained directly from time:
๐ = ๐(๐ก โ ๐ก0), (4.7)
being ๐ก0 the instant of time of passing the periapsis point of the orbit.
Equation (4.6) is a transcendental equation whose known analytical solutions are series, which can be
obtained by Lagrange Inversion Theorem or with Bessel functions [44].
Numerical approaches, such as the Newtonโs Method, can also be used to obtain ๐ธ from ๐ in (4.6).
However, in this work, we use the analytical approximation of the inversion of (4.6).
The inversion of (4.6) is performed by using both sides of the equation as arguments of sine and using the
properties of the sine function on the right-hand side:
sin(๐) = sin(๐ธ) cos(๐ sin๐ธ) โ sin(๐ sin๐ธ) cos(๐ธ). (4.8)
55
Writing the functions cos(๐ sin๐ธ) and sin(๐ sin๐ธ) as Taylor series and neglecting equal or larger than
second order on the eccentricity, since the eccentricities considered in this work are small, we obtain
sin(๐) โ sin(๐ธ) โ ๐ sin(๐ธ) cos(๐ธ) (4.9)
We multiply equation (4.9) by ๐ and substitute ๐ sin(๐ธ) by ๐ธ โ๐, as given by (4.6). Neglecting the second
order term ๐2 sin(๐ธ) cos(๐ธ) yields the approximated expression for ๐ธ(๐):
๐ธ โ ๐ + ๐ sin๐ (4.10)
For ๐ < 0.1, the maximum difference between the solutions for ๐ธ(๐ก) throughout an orbit obtained with
Newtonโs Method and by (4.10) is smaller than 3.7ยฐ.
We replace (4.10) into (4.5), using (4.7) to have time explicitly in the equation, obtaining
๏ฟฝ๏ฟฝ = (1 + ๐
cos(๐(๐ก โ ๐ก0) + ๐ sin ๐(๐ก โ ๐ก0)) โ ๐
1 โ ๐ cos(๐(๐ก โ ๐ก0) + ๐ sin ๐(๐ก โ ๐ก0)))
2๐
โ1 โ ๐2. (4.11)
4.1.2 Precession Rate as Function of Time
The precession rate for an axisymmetric body is given by (2.146), which becomes
๏ฟฝ๏ฟฝ =
โ
๐๐๐2
12[cos2 ๐ผ (3 tan2 (๐
๐ โ 22๐
) โ 1) + 2] ., (4.12)
when ๐ด(๐ผ) of the Connected Scissor Mechanism (3.14) and ๐ as function of ๐ (3.1) are substituted.
4.1.3 Configuration Variation with Time
In order to obtain ๐ผ(๐ก), we equal (4.12) to (4.11), which yields
โ๐๐๐2
12
cos2 ๐ผ (3 tan2 (๐๐ โ 22๐
) โ 1) + 2.= (1 + ๐
cos(๐(๐ก โ ๐ก0) + ๐ sin๐(๐ก โ ๐ก0)) โ ๐
1 โ ๐ cos(๐(๐ก โ ๐ก0) + ๐ sin ๐(๐ก โ ๐ก0)))
2๐
โ1 โ ๐2.
(4.13)
By rearranging (4.13), we obtain the analytical expression of ๐ผ:
๐ผ = arccos
โ โ
๐๐๐2
12(1 + ๐
cos(๐๐ก + ๐ sin๐๐ก) โ ๐1 โ ๐ cos(๐๐ก + ๐ sin ๐๐ก)
)2 ๐
โ1โ ๐2.
โ 2
3 tan2 (๐๐ โ 22๐
) โ 1.
(4.14)
56
We observe that (4.14) is impossible for ๐ = 3, which corresponds to the Triangular Connected Scissor
Mechanism. The reason for this is that the ๐๐ด
๐๐ผ= 0 for ๐ = 3, being ๐ด constant and ๏ฟฝ๏ฟฝ constant as well,
making impossible to have ๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ for an elliptic orbit.
From (4.14), we conclude also that the existence of the square root on the right-hand side may imply that
combinations of parameters exist such as they will not allow to have ๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ. For an increasing ratio ๐๐๐2
12/โ,
the positive term on the numerator on the left-hand side of (4.14) becomes smaller and, consequently, in
the limit situation when this ratio tends to zero, (4.14) becomes impossible, implying that ๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ is not
possible.
Having that ๐๐๐2
12~๐ผ and โ~๐ผ๐, we have that the ratio
๐๐๐2
12/โ is of order ๐. Thus, the smaller the angular
velocity, the closest is the satellite to be near condition where the precession rate is not able to follow orbit
motion. This is reasonable because in a limit situation where ๐ = 0, no configuration change could make
the satellite have angular momentum or have a prescribed precession rate.
Equation (4.14) can be rewritten taking into account that โ can be expressed in terms of a reference
precession rate ๏ฟฝ๏ฟฝ0 and reference configuration ๐ผ0, using (2.144)
โ = ๐ด(๐ผ0)๏ฟฝ๏ฟฝ0, (4.15)
being โ constant.
We arbitrate that the reference conditions in (3.62) are the conditions of the satellite at the periapsis when
๐ก = ๐ก0, this being the first instant when the ๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ begins to be satisfied. Nevertheless, we could select any
other point in the orbit to use its conditions as reference in (3.62).
Having the reference conditions defined in the periapsis implies
๏ฟฝ๏ฟฝ0 = ๏ฟฝ๏ฟฝ(๐ก0) = ๏ฟฝ๏ฟฝ(๐ก0), (4.16)
which, by replacing (4.11), becomes
๏ฟฝ๏ฟฝ0 = ๐
(1 + ๐)2
โ1 โ ๐2. (4.17)
We insert (3.62) into (4.15) and use ๐ด(๐ผ) of the Connected Scissor Mechanism as given in (3.14):
โ =
๐๐๐2
12[cos2 ๐ผ0 (3 tan
2 (๐๐ โ 2
2๐) โ 1) + 2] ๐
(1 + ๐)2
โ1 โ ๐2. (4.18)
57
With the angular momentum given, we replace (3.62) into (4.14) and rearrange to obtain
๐ผ = arccos
โ [cos2 ๐ผ0 (3 tan
2 (๐๐ โ 22๐
) โ 1) + 2] (1 + ๐)2
(1 + ๐cos(๐๐ก + ๐ sin ๐๐ก) โ ๐1 โ ๐ cos(๐๐ก + ๐ sin ๐๐ก)
)2 โ 2
3 tan2 (๐๐ โ 22๐
) โ 1,
(4.19)
where ๐ก0 was omitted, because the instant at the periapsis was considered as ๐ก0 = 0.
4.1.4 Possible Solutions for Synchronous Rotation
We will verify under which conditions (4.19) can be satisfied.
Defining
๐ = 3 tan2 (๐
๐ โ 2
2๐) โ 1, (4.20)
which is a constant value that depends only on the number of bars selected for the Connected Scissor
Mechanism, we simplify (4.19) to
๐ผ = arccosโโ2
๐+๐ cos2 ๐ผ0 + 2
๐๐(๐ก), (4.21)
where the time-dependent function is
๐(๐ก) = (
๐ cos(๐๐ก + ๐ sin(๐๐ก)) โ 1
๐ โ 1)
2
. (4.22)
Taking into account that the square root must have a non-negative argument and that the arccosine must
have an argument smaller or equal to |1| so that (4.19) is possible, the condition for possible solution is
0 โค
2
๐(๐(๐ก) โ 1) + cos2 ๐ผ0 ๐(๐ก) โค 1. (4.23)
We conclude from (4.22) that ๐(๐ก) is bounded by
1 โค ๐(๐ก) โค (
1 + ๐
1 โ ๐)2
, (4.24)
58
which implies that neither of the terms of (4.23) are negative. Thus, the left inequality in (4.23) is always
valid.
In what concerns the right inequality in (4.23), we rearrange (4.23) to obtain ๐ผ0 limited by ๐ and ๐, taking
the maximum value of ๐(๐ก) so that (4.23) is satisfied during the orbital period:
cos2 ๐ผ0 โค(1 โ ๐)2 โ
8๐๐
(1 + ๐)2. (4.25)
We conclude from (3.62) that the angle ๐ผ0 can be allowed to have any value larger than a certain minimum
๐ผ0 that depends on ๐ and ๐.
It is possible to assess how the minimum ๐ผ0 varies with ๐ and ๐ by plotting different curves of ๐ผ0, as function
of ๐ for different number of bars.
Figure 8 - Minimum ๐ผ0 as function of the eccentricity of the orbit for different number of bars.
We observe from Figure 8 that the range of possible ๐ผ0 becomes more limited with the increase in
eccentricity. On the other hand, increasing the number of bars decreases the minimum ๐ผ0 allowed, since
the curves in Figure 8 get lower with larger ๐, until the theoretical limit of ๐ = โ. Consequently, the most
limiting situation is having the minimum number of bars, ๐ = 4 and having the highest eccentricity ๐ = 0.1,
which corresponds to a minimum ๐ผ0 of 54.4ยฐ, which still leaves almost half of the total range of ๐ผ available.
4.1.5 Example: Geosynchronous Orbit
We will obtain ๐ผ(๐ก) graphically for a geosynchronous orbit, with ๐ = 7.292 ร 10โ5rad/s [11].
59
In order to observe the influence of the parameters ๐ผ0, ๐ and ๐ in the configuration as function of time ๐ผ(๐ก),
we selected two different values for each of the parameters: (i) ๐ผ0 = 45ยบ and ๐ผ = 60ยบ; (ii) ๐ = 6 and ๐ = 10;
and (iii) ๐ = 0.1 and ๐ = 0.01. This selection adds up to a total of eight different ๐ผ(๐ก), which will be plotted
during two sidereal days using the selected values according to (4.21). The curves are be displayed
separately for ๐ = 0.1 in Figure 9 and for ๐ = 0.01 in Figure 10.
Figure 9 โ Configuration as function of time ๐ผ(๐ก) during two sidereal days in geosynchronous orbit with ๐ = 0.1 for
different parameters.
Figure 10 - Configuration as function of time ๐ผ(๐ก) during two sidereal days in geosynchronous orbit with ๐ = 0.01 for
different parameters.
60
We observe the solution exists for every point in the orbit for the arbitrated values. This was expected
because the most limitative set of parameters, which corresponds to having maximum eccentricity and
minimum number of bars, according to section 4.1.4, requires having ๐ผ0 larger than a certain minimum,
which was respected in the selection of ๐ผ0 for the curves.
The most limitative set of parameters is ๐ = 0.1 and ๐ = 6, which imply a minimum ๐ผ0 of 37.5ยฐ, according to
(3.62). Since 37.5ยฐ is below the arbitrated values of ๐ผ0 = 45ยฐ and ๐ผ0 = 60ยฐ, there must exist solution for all
points in the orbit.
We observe in Figure 9 and in Figure 10 that ๐ผ(๐ก) is a periodic function with a period equal to the orbital
period, which is in this case a sidereal day.
Additionally, it is noticeable that ๐ผ(๐ก) has the maximum value at the periapsis and the minimum value at the
apoapsis. This is explained by the fact that at the periapsis the satellite experiences maximum orbital
velocity while at the apoapsis the satellite has the minimum orbital speed. For the synchronous rotation to
occur, the maximum and minimum of the precession rates must coincide with the extreme points of the
orbital speed. Consequently, the precession rate is maximum at the periapsis and minimum at the apoapsis,
which implies minimum ๐ด at the periapsis and maximum ๐ด at the apoapsis, according to (2.144). Since ๐ด of
the Closed Scissor Mechanism increases when ๐ผ decreases (3.14), the minimum ๐ผ has the maximum at
the periapsis and the maximum at the apoapsis, which is confirmed by Figure 9 and Figure 10.
๐ผ(๐ก) has the maximum value at the periapsis, which is ๐ผ0 at the initial instant because we arbitrated that the
synchronous rotation would start at the periapsis. If any other point in the orbit had been selected, the shape
of function ๐ผ(๐ก) would be the same, but would be shifted horizontally.
The influence of the eccentricity is clear: higher eccentricities imply larger configuration changes during an
orbital period, as can be observed by comparing curves of Figure 9 to Figure 10 that are displayed in the
same color. This is explained by the fact that a more eccentric orbit has larger orbital velocity variations
during a period, implying therefore greater changes in the inertia tensor so that the precession rate is
synchronized with the orbital motion.
Increasing the number of bars implies that the required amplitude of the configuration change becomes
smaller, which is visible if curves with the same eccentricity and the same ๐ผ0 are compared in both in Figure
9 and Figure 10.
Moreover, changing the number of bars of the Connected Scissor Mechanism has more influence in the
configuration function over time ๐ผ(๐ก) for more eccentric orbits. For example, the two top curves of ๐ผ0 = 60ยฐ
61
have a much smaller difference in Figure 10 where the eccentricity is lower than the difference the analogue
curves in Figure 10 have, where the eccentricity is higher.
The solution does not depend linearly on the initial configuration ๐ผ0, which can be realized graphically. For
example, if we compare the two curves in Figure 9 of ๐ = 10 and ๐ = 0.1, which differ only in the parameter
๐ผ0 by 15ยฐ, we obtain that the absolute difference between the two curves is a periodic function with minimum
15ยฐ at the periapsis and maximum 22.2ยฐ at the apoapsis. Consequently, a solution with a different ๐ผ0, having
all the other parameters equal to a known solution ๐ผ(๐ก), cannot be obtained by simply adding a constant to
๐ผ(๐ก).
62
4.2 MOMENTS OF INERTIA SWAP
As referred in section 3.7, the Compass is, of the envisioned models for variable-geometry satellites, the
geometry that offers the widest range of control possibilities by configuration changing because the three
moments of inertia vary with the configuration.
4.2.1 Relations between Moments of Inertia
An interesting feature of this geometry concerns the ratios between the moments of inertia. Using the
expressions (3.77)-(3.79) obtained in sub-section 3.6.3 for the moments of inertia, we obtain
๐ด
๐ถ=
1
4 sin2 ๐ผ+3
4 (4.26)
and
๐ต
๐ถ=
1
4 sin2 ๐ผโ1
4. (4.27)
Using (4.26) and (3.62), we plot ๐ด/๐ถ and ๐ต/๐ถ.
Figure 11 - ๐ด/๐ถ and ๐ต/๐ถ ratios as function of the configuration ๐ผ for the Compass.
The obtained plots are symmetric with respect to the vertical line ๐ผ = ๐/2 due to symmetry of the geometry
with respect to the selected body frame. We observe in Figure 11 that ๐ด/๐ถ โฅ 1 for any ๐ผ. In addition, the
curve of the ๐ด/๐ถ ratio is always on top of ๐ต/๐ถ, implying that ๐ด is the largest of the three moments of inertia
regardless of the configuration, which is reasonable since ๐ด = ๐ต + ๐ถ as explained in sub-section 3.6.3.
Since ๐ถ tends to zero for ๐ผ = 0 or ๐ผ = ๐, the curves of Figure 11 tend to infinity at the bounds of the plot.
If we subtract (4.27) from (4.26), we obtain a constant value:
63
๐ด โ ๐ต
๐ถ= 1 (4.28)
However, the ratio ๐ต/๐ถ can be larger or smaller than 1 depending on ๐ผ. We define the critical configuration,
๐ผ๐ถ, such that
๐ต(๐ผ๐ถ) = ๐ถ(๐ผ๐ถ). (4.29)
If we solve (4.27) for ๐ต/๐ถ = 1, we obtain
๐ผ๐ถ = arcsin(
1
โ5), (4.30)
which has the numerical solutions of 26.6ยฐ and 153.4ยฐ. Consequently:
๐ต(๐ผ)
๐ถ(๐ผ)> 1, for 0 < ๐ผ < ๐ผ๐ถ โจ ๐ โ ๐ผ๐ < ๐ผ < ๐ (4.31)
and
๐ต(๐ผ)
๐ถ(๐ผ)< 1, for ๐ผ๐ถ < ๐ผ < ๐ โ ๐ผ๐ถ . (4.32)
4.2.2 General Form of the Euler Equations for the Compass
We use
๐๐ผ
๐๐ก=๐๐ผ
๐๐ผ
๐๐ผ
๐๐ก= ๏ฟฝ๏ฟฝ
๐๐ผ
๐๐ผ (4.33)
in the general form of the Euler Equations (2.59)-(2.61) for a free variable-geometry body with ๐ธ = 0, which
is the case for the selected body frame for the Compass, as explained in sub-section 3.6.3, which yields:
๐๐ด
๐๐ผ(๐ผ(๐ก))๏ฟฝ๏ฟฝ(๐ก)๐๐ฅ +๐ด(๐ผ(๐ก))๏ฟฝ๏ฟฝ๐ฅ = (๐ต(๐ผ(๐ก)) โ ๐ถ(๐ผ(๐ก)))๐๐ฆ๐๐ง , (4.34)
๐๐ต
๐๐ผ(๐ผ(๐ก))๏ฟฝ๏ฟฝ(๐ก)๐๐ฆ +๐ต(๐ผ(๐ก))๏ฟฝ๏ฟฝ๐ฆ = (๐ถ(๐ผ(๐ก)) โ ๐ด(๐ผ(๐ก)))๐๐ฆ๐๐ง , (4.35)
๐๐ถ
๐๐ผ(๐ผ(๐ก))๏ฟฝ๏ฟฝ(๐ก)๐๐ง + ๐ถ(๐ผ(๐ก))๏ฟฝ๏ฟฝ๐ง = (๐ด(๐ผ(๐ก)) โ ๐ต(๐ผ(๐ก)))๐๐ฅ๐๐ฆ , (4.36)
64
where the dependence of ๐ผ by the moments of inertia and the respective derivatives is displayed explicitly,
as well as the dependence of ๐ผ and ๏ฟฝ๏ฟฝ on time.
We perform the differentiation of (3.77)-(3.79) with respect to ๐ผ, obtaining:
๐๐ด
๐๐ผ= ๐๐2 sin๐ผ cos๐ผ, (4.37)
๐๐ต
๐๐ผ= โ
1
3๐๐2 sin๐ผ cos๐ผ, (4.38)
๐๐ถ
๐๐ผ=4
3๐๐2 sin๐ผ cos๐ผ, (4.39)
and we replace (3.77)-(3.79) and (4.37)-(3.62) into (4.34)-(4.36), which yields
6๏ฟฝ๏ฟฝ๐๐ฅ sin๐ผ cos๐ผ + ๏ฟฝ๏ฟฝ๐ฅ(3 sin2 ๐ผ + 1) = ๐๐ฆ๐๐ง(1 โ 5 sin2 ๐ผ) (4.40)
โ2๏ฟฝ๏ฟฝ๐๐ฆ sin ๐ผ cos ๐ผ + ๏ฟฝ๏ฟฝ๐ฆ(1 โ sin2 ๐ผ) = ๐๐ฅ๐๐ง(sin
2 ๐ผ โ 1) (4.41)
2๏ฟฝ๏ฟฝ๐๐ง sin๐ผ cos๐ผ + ๏ฟฝ๏ฟฝ๐ง sin2 ๐ผ = ๐๐ฅ๐๐ฆ sin
2 ๐ผ (4.42)
where ๐๐2 was divided in all equations. It can be observed that for ๐ผ = ๐/2 equation (4.41) becomes 0 = 0,
which is justified by the fact that, for that configuration, both the bars of the Compass lie on the ๐ฆ axis,
having therefore zero moment of inertia around that axis. Consequently, the Compass is allowed to have
any ๐๐ฆ , making indeterminate the system of differential equations.
A similar situation happens for (4.42), which becomes 0 = 0 when ๐ผ = 0 or ๐ผ = ๐.
Taking this into account, the range of operation for attitude control of the Compass, according to this model,
should be selected as either 0 < ๐ผ < ๐/2 or ๐/2 < ๐ผ < ๐, preferably with configurations significantly distant
from the limit configurations of those intervals.
It is important to notice that the mentioned singularities come from the assumption of the bars being slender,
which are modelled as infinitely thin by definition. With a more sophisticated model for the bars, where the
thickness would be accounted for, no singularities would exist because there would be no configuration for
the Compass such that a moment of inertia is zero.
65
Equations (4.40)-(4.42) can be further simplified, dividing (4.40) and (4.42) by sin2 ๐ผ and (4.41) by cos2 ๐ผ:
6๏ฟฝ๏ฟฝ๐๐ฅ cotan๐ผ + ๏ฟฝ๏ฟฝ๐ฅ(4 + cotan2 ๐ผ) = ๐๐ฆ๐๐ง(cotan
2 ๐ผ โ 4), (4.43)
โ2๏ฟฝ๏ฟฝ๐๐ฆ tan๐ผ + ๏ฟฝ๏ฟฝ๐ฆ = โ๐๐ฅ๐๐ง , (4.44)
2๏ฟฝ๏ฟฝ๐๐ง cotan๐ผ + ๏ฟฝ๏ฟฝ๐ง = ๐๐ฅ๐๐ฆ . (4.45)
4.2.3 Approximated Solution
The difficulty of finding a solution for the system of differential equations given by (4.43)-(4.45) depends on
the nature of the function ๐ผ(๐ก), which simplest form, apart from ๐ผ(๐ก) = ๐๐๐๐ ๐ก๐๐๐ก, is a linear function,
๐ผ = ๐ + ๐๐ก, (4.46)
where ๐ and ๐ are constants. Even with the configuration as function of time given by (3.62), the system
(4.43)-(4.45) has no obvious analytic solution.
However, we will obtain an approximate solution for the dynamics of the Compass under two assumptions:
(i) ๐ผ(๐ก) is close to the critical configuration, which is having ๐ผ โ ๐ผ๐ถ; (ii) ๐๐ง~1 rad/s is larger than the other
two components of the angular velocity, so that
๐๐ง2 โซ ๐๐ฅ๐๐ฆ , (4.47)
being ๐๐ฅ , ๐๐ฆ~0.1 rad/s.
Neglecting the term of the right-hand side of (4.45), according to (3.62), and rearranging yields:
๐๏ฟฝ๏ฟฝ๐๐ง
= โ2๏ฟฝ๏ฟฝ cot ๐ผ. (4.48)
We integrate (4.65) from an instant ๐ก0 to a time ๐ก , which correspond to a configuration ๐ผ0 and to a
configuration ๐ผ, respectively:
โซ
๏ฟฝ๏ฟฝ๐ง๐๐ง๐๐ก
๐ก
๐ก0
= โซ โ2๏ฟฝ๏ฟฝ cot๐ผ ๐๐ก๐ก
๐ก0
= โซ โ2cot ๐ผ ๐๐ผ๐ผ
๐ผ0
. (4.49)
The integrations are solved to obtain
๐๐ง(๐ผ) = ๐๐ง(๐ผ0)
sin2 ๐ผ0sin2 ๐ผ
. (4.50)
66
Taking into account that ๐ผ โ ๐ผ๐ถ and that, according to (4.30), cot๐ผ๐ถ = 4, we can neglect the term of the
right-hand-side of (4.43) because it has a product of two small quantities, ๐๐ฆ and cotan2 ๐ผ โ 4, with respect
to the terms on the left-hand side. Rearranging the approximated (4.43) and integrating yields
โซ๏ฟฝ๏ฟฝ๐ฅ๐๐ฅ
๐๐ก๐ก
๐ก0
= โซ โ6cot ๐ผ
4 + cot2๐ผ๐๐ผ
๐ผ
๐ผ0
. (4.51)
We perform the integrations to obtain
๐๐ฅ(๐ผ) = ๐๐ฅ(๐ผ0)
1 + 3 sin2 ๐ผ01 + 3 sin2 ๐ผ
. (4.52)
Even though we obtained approximated expressions for ๐๐ฅ(๐ผ) and ๐๐ง(๐ผ) from (4.43) and from (4.45)
respectively, ๐๐ฆ cannot be obtained easily from (4.44) because the term of the right-hand side of (4.44) is
not negligible.
However, we can use the conservation of angular momentum (2.101) to obtain ๐๐ฆ(๐ผ) as function of the
initial conditions of the angular velocity and of the configuration. Having
โ2
๐๐2/6= (1 + 3 sin2(๐ผ))2๐๐ฅ
2(๐ผ) + cos4(๐ผ) ๐๐ฆ2(๐ผ) + 16 sin4(๐ผ) ๐๐ง
2(๐ผ) (4.53)
at any configuration ๐ผ and
โ2
๐๐2/6= (1 + 3 sin2(๐ผ0))
2๐๐ฅ2(๐ผ0) + cos4(๐ผ0)๐๐ฆ
2(๐ผ0) + 16 sin4(๐ผ0)๐๐ง2(๐ผ0) (4.54)
for the initial configuration, we equal (4.53) to (4.54) and replace the obtained components of the angular
velocity (4.50) and (4.52), which eliminates the contributions of the other angular velocities and can be
simplified to:
๐๐ฆ(๐ผ) = ๐๐ฆ(๐ผ0)
cos2(๐ผ0)
cos2(๐ผ). (4.55)
The expressions obtained for the components of angular momentum, (4.50), (4.52) and (4.55), show that,
under the mentioned conditions for the approximation, there is no exchange between the three components
of the angular momentum, since each of the components is conserved separately.
67
4.2.3.1 Kinetic Energy
Once the angular components are determined, we can obtain the kinetic energy as function of the
configuration according to (2.86) after determining
1
2โซ
๐๐
๐๐ก
๐ต .
๐๐
๐๐ก
๐ต
ฮฉ๐๐ as function of ๐ผ as well.
We define ๐ข as the variable that runs along the length of both bars, starting at 0 at the joint up to the end
point ๐ข = ๐. The position of each point of the compass with respect to the joint ๐๐ is given by
๐๐(๐ข, ๐ผ) = ยฑ๐ข sin๐ผ ๐๐ โ ๐ข cos ๐ผ ๐๐, (4.56)
where the positive branch of the first term of right-hand side concerns the right bar and the negative
concerns the left bar of the compass. In order to obtain ๐๐
๐๐ก
๐ต, we add the motion of the center of mass ๐ง๐ถ๐
as given in (3.76) to the ๐๐ component of (4.56), which yields
๐(๐ข, ๐ผ) = ยฑ๐ข sin๐ผ ๐๐ + (๐ข โ
๐
2) cos๐ผ ๐๐. (4.57)
Differentiating (3.62) results in
๐๐
๐๐ก
๐ต
(๐ข, ๐ผ) = ๏ฟฝ๏ฟฝ(ยฑ๐ข cos ๐ผ ๐๐ + (๐
2โ ๐ข) sin๐ผ ๐๐. (4.58)
With the velocity of each point of the compass obtained, we perform the integration using the symmetry of
the bars and determine the non-rigid term of the kinetic energy:
1
2โซ
๐๐
๐๐ก
๐ต
.๐๐
๐๐ก
๐ต
ฮฉ
๐๐ = โซ ๏ฟฝ๏ฟฝ2 (๐ข2 + sin2 ๐ผ (๐2
4โ ๐ข๐))
๐
๐
๐
0
๐๐ข =๐๐2
12๏ฟฝ๏ฟฝ2(4 โ 3 sin2 ๐ผ ). (4.59)
We can write the kinetic energy as function of the configuration using the reference configuration ๐ผ0 by
replacing (4.59) and the moments of inertia (3.77)-(3.79) in (2.86):
2๐(๐ผ)
๐๐2
6
=(1 + 3 sin2 ๐ผ0)
2
1 + 3 sin2 ๐ผ๐๐ฅ2(๐ผ0) +
(1 โ sin2 ๐ผ0)2
1 โ sin2 ๐ผ๐๐ฆ2(๐ผ0) +
sin4 ๐ผ0
sin2 ๐ผ๐๐ง2(๐ผ0) +
๏ฟฝ๏ฟฝ2(4 โ 3 sin2 ๐ผ )
2. (4.60)
68
4.2.3.2 Ellipsoids of the Kinetic Energy and of the Angular Momentum
Having the kinetic energy determined, it is possible to obtain the semi-axes of the ellipsoid of the kinetic
energy as function of the configuration and initial conditions. We insert the moments of inertia of the
Compass (3.77)-(3.79) in (2.99):
โ
2๐(๐ผ)๐๐2
6
โ๏ฟฝ๏ฟฝ2(4 โ 3 sin2 ๐ผ )
2
1 + 3 sin2 ๐ผ,
โ
2๐(๐ผ)๐๐2
6
โ๏ฟฝ๏ฟฝ2(4 โ 3 sin2 ๐ผ )
2
1 โ sin2 ๐ผ,
โ
2๐(๐ผ)๐๐2
6
โ๏ฟฝ๏ฟฝ2(4 โ 3 sin2 ๐ผ )
2
4 sin2 ๐ผ
(4.61)
The initial angular velocity is used on (4.60) to determine ๐(๐ผ), which eliminates the contribution of ๏ฟฝ๏ฟฝ to the
length of the semi-axes of the ellipsoid of the kinetic energy in (4.61).
The semi-axes of the ellipsoid of the angular momentum are
โ
๐๐2
6(1 + 3 sin2 ๐ผ)
,โ
๐๐2
6(1 โ sin2 ๐ผ)
,โ
๐๐2
6 4 sin2 ๐ผ, (4.62)
as obtained by replacing (3.77)-(3.79) in (2.102). โ is obtained according to (4.54) as function of certain
initial components of the angular velocity, ๐๐ฅ(๐ผ0), ๐๐ฆ(๐ผ0) and ๐๐ง(๐ผ0).
We obtained the ellipsoids for an example with an initial configuration of ๐ผ0 = 0.45, which is near the critical
angle ๐ผ๐ถ = 0.46, that changes to a configuration of ๐ผ = 0.47.
The initial components of the angular velocity also respect the assumptions made on this section: ๐๐ฅ(๐ผ0) =
๐๐ฆ(๐ผ0) = 0.1 rad/s and ๐๐ง(๐ผ0) = 1 rad/s.
69
Figure 12 โ Perspectives from the ๐๐ฅ axis of the ellipsoids of the kinetic energy, at darker orange, and of the angular
momentum, at brighter orange, where the curve of the intersection of the ellipsoids is displayed in blue. The
configuration at the left is of ๐ผ0 and at the right is of ๐ผ.
For ๐ผ and for ๐ผ0, Figure 12 shows nearly circular shapes, because the semi-axes of the ellipsoids on the
๐๐ฆ and on the ๐๐ง directions are very close, since ๐ผ and ๐ผ0 are near the critical configuration ๐ผ๐ถ, for which
the Compass has ๐ต = ๐ถ.
4.2.4 Stability Analysis
Since the relation between the moments of inertia depends on the configuration, as discussed in section
4.2.1, it is interesting to explore how this influences the stability of the motion of the Compass in each of the
three axes.
4.2.4.1 Constant Configuration
When the Compass is not changing its configuration, the terms on ๏ฟฝ๏ฟฝ vanish in (4.43)-(4.45). We will check
if the motion of the Compass around the ๐ฅ axis is stable and in for which configurations it is stable.
The angular velocities are:
๐๐ฅ = ๐0๐ฅ + ๐ฟ๐๐ฅ, (4.63)
๐๐ฆ = ๐ฟ๐๐ฆ, (4.64)
๐๐ง = ๐ฟ๐๐ง, (4.65)
being ๐0๐ฅ โซ ๐ฟ๐๐ฅ , ๐ฟ๐๐ฆ , ๐ฟ๐๐ง and ๐ฟ๐๐ฅ2, ๐ฟ๐๐ฆ
2 and ๐ฟ๐๐ง2 negligible. Replacing (4.63)-(3.62) in (4.43)-(4.45) with
๏ฟฝ๏ฟฝ = 0 yields:
70
๐ฟ๐๐ฅ (4 + cotan2 ๐ผ(๐ก)) = 0 , (4.66)
๐ฟ๐๐ฆ = โ๐0๐ฅ๐ฟ๐๐ง, (4.67)
๐ฟ๐๐ง = ๐0๐ฅ๐ฟ๐๐ฆ, (4.68)
We differentiate eqs. (4.67) and (4.68) and separate the infinitesimal angular velocities into two equations:
๐ฟ๐๐ฆ = โ๐0๐ฅ2 ๐ฟ๐๐ฆ. (4.69)
๐ฟ๐๐ง = โ๐0๐ฅ2 ๐ฟ๐๐ง, (4.70)
The solutions of (4.69) for ๐ฟ๐๐ฆ and of (4.70) for ๐ฟ๐๐ง are stable regardless of the configuration because
โ๐0๐ฅ2 is negative and, therefore, induced perturbations on the angular velocity are never amplified. Thus, if
the Compass is spinning around the ๐ฅ axis, the motion is stable.
If we perform a similar procedure concerning a situation where the Compass is spinning around the ๐ฆ axis,
defining
๐๐ฅ = ๐ฟ๐๐ฅ, (4.71)
๐๐ฆ = ๐0๐ฆ + ๐ฟ๐๐ฆ, (4.72)
๐๐ง = ๐ฟ๐๐ง, (4.73)
being ๐0๐ฆ โซ ๐ฟ๐๐ฅ , ๐ฟ๐๐ฆ , ๐ฟ๐๐ง and neglecting second order infinitesimal angular speeds, we obtain:
๐ฟ๐๐ฅ = ๐0๐ฆ
2cotan2 ๐ผ โ 4
cotan2 ๐ผ + 4๐ฟ๐๐ฅ, (4.74)
๐ฟ๐๐ง = ๐0๐ฆ
2cotan2 ๐ผ โ 4
cotan2 ๐ผ + 4๐ฟ๐๐ง. (4.75)
In contrast with the ๐ฅ axis, the stability for the ๐ง axis depends on the configuration. The condition for stability
is having
cotan2 ๐ผ โ 4
cotan2 ๐ผ + 4< 0, (4.76)
which is satisfied by having
๐ผ๐ถ < ๐ผ < ๐ โ ๐ผ๐ถ, (4.77)
according to the definition of (4.30), while the instability occurs for 0 < ๐ผ โค ๐ผ๐ถ and ๐ โ ๐ผ๐ถ โค ๐ผ < ๐.
71
The derivation to verify the condition stability around the ๐ง axis leads to condition (4.76) with an opposite
sign on the left-hand side. Consequently, the configurations that allow for a stable spin around ๐ง are
0 < ๐ผ < ๐ผ๐ถ and ๐ โ ๐ผ๐ถ < ๐ผ < ๐. (4.78)
The stability for the Compass without configuration change is resumed on Table 3. We verified that the axis
around which the Compass can rotate stably are the axes that correspond to either the maximum moment
of inertia or to the minimum moment of inertia for the current configuration, as it happens for a rigid body
[29].
Spin around ๐ axis Spin around ๐ axis Spin around ๐ axis
0 < ๐ผ < ๐ผ๐ถ ๐ด > ๐ต > ๐ถ stable unstable stable
๐ผ๐ถ < ๐ผ < ๐ โ ๐ผ๐ถ ๐ด > ๐ถ > ๐ต stable stable unstable
๐ โ ๐ผ๐ถ < ๐ผ < ๐ ๐ด > ๐ต > ๐ถ stable unstable stable
Table 3 - Configurations of stability for the Compass
4.2.4.2 Varying Configuration
If the configuration is changing then all terms of (4.43)-(4.45) must be considered, because ๏ฟฝ๏ฟฝ is no longer
zero. The stability of rotation around each of the three principal axes depends on the function ๐ผ(๐ก), which
particular case of ๐ผ(๐ก) = 0 was verified in sub-sub-section 4.2.4.1. Consequently, we will be able to obtain
three systems of differential equations for a general prescribed function ๐ผ(๐ก), one for the rotation around
each of the principal axes.
We will firstly assess the stability around the ๐ฅ axis. Using the definitions of (4.73)-(4.75) on (4.43)-(4.45),
and differentiating the equations on the ๐ฆ and ๐ง directions yields:
๐ฟ๐๐ฆ โ 2๐ฟ๐๐ฆ
๏ฟฝ๏ฟฝ tan๐ผ โ 2๐ฟ๐๐ฆ (๏ฟฝ๏ฟฝ tan ๐ผ +๏ฟฝ๏ฟฝ2
cos2 ๐ผ) = โ๐0๐ฅ๐ฟ๐๐ง , (4.79)
๐ฟ๐๐ง + 2๐ฟ๐๐ง ๏ฟฝ๏ฟฝ cotan๐ผ + 2๐ฟ๐๐ง (๏ฟฝ๏ฟฝ cotan๐ผ โ
๏ฟฝ๏ฟฝ2
sin2 ๐ผ) = ๐0๐ฅ๐ฟ๐๐ฆ . (4.80)
In contrast with what occurs for the case of constant configuration, the equations on the two directions are
coupled and the system of equations has non-linear factors.
The system can be written in State-System Representation. We define
72
{๐} = [๐ฟ๐๐ง ๐ฟ๐๐ฆ ๐ฟ๐๐ง ๐ฟ๐๐ฆ ]๐, (4.81)
add the equations ๐ฟ๐๐ฆ = ๐ฟ๐๐ฆ and ๐ฟ๐๐ง = ๐ฟ๐๐ง to the set of equations (4.79)-(4.80) and rearrange to have
the system written in the form
{๐} = โ[๐]{๐}, (4.82)
which does not allow for trivial solution of exponential of [๐] for {๐}(๐ก) because [๐] is not constant.
For the particular case of the rotation around the ๐ฅ axis we obtain
[๐]๐ฅ =
[
0 0 1 00 0 0 1
2(๏ฟฝ๏ฟฝ2
sin2 ๐ผโ ๏ฟฝ๏ฟฝ cotan๐ผ) 0 โ2๏ฟฝ๏ฟฝ cotan๐ผ ๐0๐ฅ
0 2(๏ฟฝ๏ฟฝ tan๐ผ +๏ฟฝ๏ฟฝ2
cos2 ๐ผ) โ๐0๐ฅ 2๏ฟฝ๏ฟฝ tan๐ผ
]
. (4.83)
We determine the dynamics matrix [๐] for the other two directions, defining the state-vector for rotation
around ๐ฆ , [๐ฟ๐๐ฅ ๐ฟ๐๐ง ๐ฟ๐๐ฅ ๐ฟ๐๐ง ]๐
and the state-vector for the rotation around ๐ง , [๐ฟ๐๐ฅ ๐ฟ๐๐ฆ ๐ฟ๐๐ฅ ๐ฟ๐๐ฆ ]๐
,
obtaining:
[๐]๐ฆ =
[
0 0 1 00 0 0 1
6๏ฟฝ๏ฟฝ2
sin2 ๐ผโ 6๏ฟฝ๏ฟฝ cotan๐ผ
4 + cotan2๐ผโ
2๐0๐ฆ๏ฟฝ๏ฟฝ cotan๐ผ
sin2 ๐ผ4 + cotan2๐ผ
2๏ฟฝ๏ฟฝ cotan๐ผsin2 ๐ผ
โ 6๏ฟฝ๏ฟฝ cotan๐ผ
4 + cotan2 ๐ผ
๐0๐ฆ(4 โ cotan2 ๐ผ)
4 + cotan2 ๐ผ
0 2(๏ฟฝ๏ฟฝ2
sin2๐ผโ ๏ฟฝ๏ฟฝ cotan๐ผ) ๐0๐ฆ โ2๏ฟฝ๏ฟฝ cotan๐ผ
]
, (4.84)
[๐]๐ง =
[
0 0 1 00 0 0 1
6๏ฟฝ๏ฟฝ2
sin2 ๐ผโ 6๏ฟฝ๏ฟฝ cotan๐ผ
4 + cotan2๐ผโ
2๐0๐ง๏ฟฝ๏ฟฝ cotan๐ผsin2 ๐ผ
4 + cotan2๐ผ
2๏ฟฝ๏ฟฝ cotan๐ผsin2 ๐ผ
โ 6๏ฟฝ๏ฟฝ cotan๐ผ
4 + cotan2 ๐ผ
๐0๐ง(4 โ cotan2 ๐ผ)
4 + cotan2๐ผ
0 2(๏ฟฝ๏ฟฝ tan๐ผ +๏ฟฝ๏ฟฝ2
cos2 ๐ผ) โ๐0๐ง 2๏ฟฝ๏ฟฝ tan๐ผ
]
.
(4.85)
We conclude from (4.83)-(4.85) that assessing the stability of the rotation of the Compass around each of
the principal axes is difficult, given that the terms in the dynamics matrixes are not only variable but contain
non-linear functions of the configuration as well.
73
5 CONCLUSIONS
The attitude dynamics of a variable-geometry body has been described. We concluded that the added
complexity of the configuration changing is in general large, even though we could obtain some simplifying
assumptions to obtain either approximations or to make the terms related with configuration changing
vanish.
The physical meaning of the terms that are not present in rigid-body motion is also discussed in this work.
We addressed the determination of the kinetic energy of a variable geometry body and concluded that the
kinetic energy varies depending on the relationship between the actuating forces and both the translation
and rotational state of the body.
We obtained a variety of simple geometries that eliminate the extra-terms related to variable geometry to
simplify the problem, but still change their inertia tensors in different ways.
It was concluded that it is possible to obtain an axisymmetric geometry made of connected slender bars
where only one moment of inertia varies, even though we realized that such an inertia tensor does not allow
for either precession rate or nutation rate controls. Other conceived geometries are axisymmetric satellites
with variable transversal moments of inertia, which allow for precession rate control. The most unrestrictive
geometry from the attitude control point of view is a geometry that has three different and variable principal
moments of inertia, in a general configuration.
We developed an application for attitude control using the geometry variation method suggested in this work
โ a satellite in an elliptic orbit that requires synchronous rotation. This situation requires attitude control so
that the satelliteโs precession rate varies accordingly with the non-uniform orbital velocity, which is more
critical in orbits of higher eccentricities where the velocity variations are larger.
Having selected one of the envisioned geometries to exemplify the variable-geometry control method, we
determined the configuration function as function of time required for the synchronous rotation for small
eccentricities, depending on several parameters: the mean angular velocity and the eccentricity of the orbit,
the number of bars of the satellite, and the initial configuration.
We concluded that the solution for the synchronous rotation is possible in most situations, being not
achievable only in unrestrictive conditions. Interesting physical insight on how the configuration function
depends with various parameters has been discussed by comparing graphics in different situations.
74
In the case of the satellite with three variable principal moments of inertia, it was observed that an interesting
feature of the satellites with such a capability is to be able to swap the order of the values of the moments
of inertia, being therefore possible to alter the rotation around principal axes from stable to unstable or
unstable to stable. Having this feature in mind, we obtained the equations of the dynamics of this satellite,
for which we could find an approximate solution around the configuration where the smaller and intermediate
moments of inertia of the satellite swap.
We determined the stable configurations for the rotation around each of the principal axes, when the
geometry is not varying. For a geometry variation situation, a state-space formulation was obtained for the
rotation around the principal axes, even though we concluded the solution is very difficult to determine due
to the non-linearities of the system.
5.1 FUTURE WORK
We used simple models for the envisioned satellites as a first approach. However, for a precise description
of the attitude dynamics, more detailed spacecraft models would be required.
It would be interesting to assess which components of the satellite sub-systems could be used more
efficiently to perform the secondary objective of controlling the attitude. Also, a feasibility study could be
conducted in order to determine how the variable geometry attitude control method compares with
conventional control methods, in terms of precision and energy expense, depending on how large the
required attitude maneuvers are.
In this work, the suggested geometries had symmetries so that a variable-geometry term vanishes, in order
to simplify the problem. A different approach to this would be to study the dynamics of bodies that wouldnโt
have such symmetries and evaluate the consequences on the attitude solutions.
We have faced in the present work the difficulties related to obtaining analytical solutions for variable
geometry satellites, even for simple configuration functions and simple satellite models. Consequently, it
would be interesting to assess the applicability of numerical methods to obtain approximate accurate
solutions for the motion of the satellite.
Swapping the order of the values of the principal moments of inertia, as a consequence of configuration
changing, can be studied in the future as a method to induce controlled instability in the rotation of an axis,
which could result in an intended attitude maneuver to achieve large angle changes on the orientation of
spacecraft.
75
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