Increasing Orbital Energy Via Tether Retrieval and Deployment

Home , Momentum exchange tether

INCREASING ORBITAL ENERGY VIA TETHER RETRIEVAL AND DEPLOYMENT

IN A SYNCHRONOUS CONFIGURATION

AIDA FERRO ARDANUY

B.S, Polytechnic University of Catalonia, Castelldefels, 2012

M.S, Polytechnic University of Catalonia, Terrassa, 2015

A thesis submitted to the Graduate Faculty of the

University of Colorado Colorado Springs

in partial fulfillment of the

requirements for the degree of

Master of Science

Department of Mechanical and Aerospace Engineering

2017

This thesis for the Master of Science degree by

Aida Ferro Ardanuy

has been approved for the

Department of Mechanical and Aerospace Engineering

Steven Tragesser, Chair

Peter Gorder

Radu Cascaval

Date: 8/1/2017

Ferro Ardanuy, Aida (M.S., Mechanical Engineering)

Increasing Orbital Energy via Tether Retrieval and Deployment in a Synchronous

Configuration

Thesis directed by Professor Steven Tragesser

ABSTRACT

The aim of this thesis is to propose a new control law for deployment and retrieval of a tethered satellite system in order to increase the orbital energy without using propellant.

The system is considered to be a non-rotating momentum exchange tether that does not release any of the end masses. Also, it is assumed that the system does not conserve total angular momentum and the cycle of retrieving and deploying is done under the equilibrium assumption maintaining a synchronous configuration around the Earth.

Therefore, the net tangent force due the different orbital altitude of the masses produces a net angular impulse used to increase the orbital energy.

iii

ACKNOWLEDGEMENTS

First of all, I want to say thank you to my family that from Barcelona had encourage and support me with anything I needed. Also, my friends, whom I could count on them at any time making this 5.240 mile to vanish. Thank you to the new friends done here at

Colorado Springs that made me feel like home, sharing adventures at the mountain, travelling and the countless hours at the library. Thank you to all the faculty and stuff of the department of Mechanical and Aerospace Engineering, especially the invaluable help of my advisor, Steven Tragesser, for the hours, dedication and orientation invested during these months. Finally, thank you a lot to the Balsells Fellowship that gave me the opportunity to learn and keep growing in a different world. Without you this thesis would not have been possible.

Per començar, vull donar les gràcies a la meva família, que desde Barcelona, m’han estat donant ànims i suport en tot el que he necessitat. També als meus amics de sempre, per fer que aquests 8.434 km no semblin tanta distància i poder comptar amb ells a cada moment. Gracies també, als nous amics d’aqui Colorado Springs que m’han fet sentir com a casa compartint aventures per la montanya, viatjant i les hores d’estudi a la biblioteca.

Agrair a tot el servei docent del department de “Mechanical and Aerospace Enginering”, sobretot la inestimable ajuda del tutor d’aquest projecte, Steven Tragesser, per les hores, dedicació i l’orientació rebuda al llarg d’aquets mesos. Per acabar, agrair a la beca

Balsells l’oportunitat que m’han donat per aprendre i continuar creixent en un món nou.

Sense vosaltres no hagués estat possible.

TABLE OF CONTENTS

CHAPTER

I. INTRODUCTION ...... 1

1.1 State of art ...... 1

1.2 Outline of the thesis ...... 5

II. THEORETICAL DEVELOPMENT ...... 7

2.1 Model ...... 7

2.2 Equations of motion ...... 8

2.2.1 Translational movement ...... 9

2.2.2 Rotational movement ...... 10

2.2.3 General equations ...... 11

2.3 Control law for equilibrium orientation ...... 12

2.4 Mission concept ...... 16

2.4.1 Cycle for orbit pumping ...... 16

2.4.2 Analytic development of change in orbit energy ...... 18

III. Results ...... 20

3.1 Validation ...... 20

3.1.1 Linearly increasing case ...... 20

3.1.2 Quasi-equilibrium case ...... 23

3.2 Optimal retrieval and deployment angles ...... 24

3.3 Optimal mass and length factors ...... 28

3.4 Mission results ...... 30

IV. CONCLUSIONS ...... 33

REFERENCES ...... 34

APPENDICES ...... 36

A. Tether length and rate of change development ...... 36

LIST OF FIGURES

FIGURE

1. Gravity force depending on tether orientation ...... 5

2. Model representation ...... 8

3. Net tangent force ...... 17

4. Libration angle vs Time. Rate of change km/s. Mantri ...... 21

5. Libration angle vs Time. Rate of change l = − km/s ...... 21

6. Libration angle vs Time. Rate of change l = − km/s. Mantri ...... 22

7. Libration angle vs Time. Rate of change l = − km/s ...... 22

8. Comparison between new control law andl =classic − control law ...... 24

9. Libration angle vs. Time, retrieval ...... 25

10. Tether length vs. Time, retrieval ...... 26

11. Semi-major axis vs. Time, retrieval ...... 26

12. Tangent force vs. Tether length ...... 27

13. Effect of the mass and length factor on the orbit pumping ...... 29

14. Tether length variation vs. Time ...... 31

15. Libration angle variation vs. Time ...... 31

16. Semi-major axis variation vs. Time ...... 32

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CHAPTER I

INTRODUCTION

1.1 State of art

The tether satellite system (TSS) consists of a long thin cable that couples two masses such as satellites, spacecraft, space stations, astronauts or even asteroids. Its main purpose is to provide space transportation without using propellant. Nevertheless, it is also used in other types of missions such as experiments of the upper atmosphere, cargo transfer or as a physical connection between the astronaut and the spacecraft.

The inception of the tethered satellite system derived from the space tether, introduced by the scientist Tsiolkovsky back in 1895. He visualized a long tether anchored to the Earth’s surface going up to space for traveling purposes. Then, it seemed like an idea worthy of a Jules Verne novel, so it was not until 65 years later, when the scientist

Artsutanov developed the concept on a Sunday Pravda supplement [1]. He defined it as a synchronous tether with a geostationary mother ship and two cables deploying towards and away from Earth respectively, thus the centrifugal force acting on the upper part of the structure will compensate for the gravitational force acting on the lower part.

During the following years the theory of the space elevator was never given much credence, but in the 1960s, using the idea of an anchored satellite, the scientist Colombo proposed a system of two satellites connected by a tether to be used for low-orbital-altitude research [2]. Also, on a report published by NASA, Rupp contributed to the study of the dynamics of a system being deployed along its local vertical assuming the tether maintains

1 equilibrium [3]. From then on, the TSS got the attention of the space researchers and publications about this topic increased drastically. New lines of investigation appeared, mostly focused on how to take advantage of different types of tethers and studies about the dynamics and control of each one of those systems for a better understanding of their behavior. By that time, some experiments had been performed and a collection of the literature was published by Cosmo and Lorenzini [4].

The two big areas of study are the electrodynamic tethers and the momentum exchange tethers. Estes, Lorenzini and Santangelo [5] did a thorough overview on the first ones explaining their applications and experimental research up to that time. Later on,

Sanmartin, Lorenzini and Martinez [6] updated the information explaining the missions that had taken place until then while also giving an overview of the deployment dynamics.

The literature on electrodynamic tether (EDT) is broad because just by interacting with the magnetic field of the Earth, or any other planet, it is possible to create energy without using propellant. The tether must be made of a conductive metal and when it interacts with the magnetic field it creates an electromotive force (EMF) used by the tether. The current traveling through the tether can either be used to supply electricity to the system (decay) or to create a force to boost it.

The other main group of tether is the momentum exchange tether which is inert and does not require current. Kumar [7] did an extended overview of this type of tether which uses a gain of momentum to do an orbit transfer. He divided them in two big groups depending on their way to gain momentum. One category gains momentum by releasing a subsatellite and the other gains momentum by retrieving and deploying the tether, the second category being the subject of this thesis.

As for the first type, a smaller payload can be deployed from the mothership in low

Earth orbit (LEO) and be launched into deep space or a geosynchronous (GEO) orbit.

When the masses have a huge difference in magnitude, the bigger mass barely notices

2 the change of orbit either because it is decaying or has been boosted. Since the objective of the mission is to increase the altitude, the subsatellite is deployed from the mothership away from the Earth. There are two types of deployment; controlled along the vertical or a time-varying orientation. At the end of the deployment aligned with the local vertical, both masses have the same angular velocity. In this case, when the subsatellite is released it is thrown to a new higher energy elliptic while the mothership ends up in a lower elliptical orbit. This happens because the upper mass has more linear velocity than the lower one [8]. When the deployment is done with a time-varying orientation and the tether swings around the local vertical, it is possible to boost the subsatellite even more.

When it crosses the local vertical, it has an extra velocity that allows it to increase the orbit even if it is using the same tether length as before [4], [9].

The second method of momentum exchange consists on deploying and retrieving the tether without releasing any of the end masses. Using an electrical power source that gets its energy from the sun is possible to retrieve and deploy the system, converting this electrical energy into mechanical energy. Therefore, the system is self-sufficient, propellant-less and it opens the door to a new type of mission. So far, it does not exist automatic refueling of satellites and the mission lifetime is limited by the propellant they have on-board in order to re-boost themselves into the original orbit. The momentum exchange tether can be used as an alternative re-boosting tool and extend the operational lifetime of the missions. There are two different groups under this type of tether. The first group assumes that the variation of the total momentum is equal to zero and the orbit pumping is due to the exchange between the rotational angular momentum (of the tether and end masses about the system’s center of mass) and the orbital angular momentum

(of the system’s translation about the center of the Earth). Landis [10] studied this case and found out that the system was limited by the material constraints as there is a maximum tip velocity for which the tether breaks. Furthermore, Gratus and Tucker [11],

3 using the same control cycle as Landis, gave an analytical formula that predicts a variation rate of 300 m/h by varying a tether length of 50 km in a low Earth orbit. Baoyin, Yu and Li

[12] simulated a sinusoidal control law varying the tether length between 4 km to 10 km and increases the orbit’s altitude 8 km over 14 periods which is around 50 m/h of rate of change. The major drawback of the method above is that the initial rotational angular momentum must be generated, which would require additional cost, mass, time and possibly even fuel. Another way to approach the orbit transfer without the requirement of a large initial rotational momentum is to generate a transverse gravitational force (i.e perpendicular to the radial direction) in order to increase the total system angular momentum [13]. Breakwell and Gearhart [14] designed a rotating tether that uses the oblateness of the Earth to generate this transverse force. The proposed problem assumes that the tether is in a circular high inclined orbit, near polar, and the cycle deploys the tether when is at a local horizontal position and retrieves it when is at a local vertical position. The rotation of the tether is the same as the mean motion, so for one orbit the tether completes one cycle that consists on two deployments and two retrievals. Breakwell and Gearhart demonstrated an increase in the semi-major axis of around 100 m/year, significantly less than the rotating systems [7]. This thesis develops an alternative way to generate a transverse gravity force – by maintaining a non-vertical equilibrium position of the tether. Since the end masses are at different orbit altitudes, the gravity force acting on them is also different and therefore there is a net force acting on the center of masses.

The direction of the net force varies depending on the position of the masses with respect to the local vertical, so when they are not aligned the net force has a component perpendicular to the radius vector [10], [15].

tangent gravity tangent gravity force force

orbital motion orbital motion

Figure 1. Gravity force depending on tether orientation

The angle between the local vertical and the masses is known as a libration angle.

When this angle is positive, i.e in the same sense as the orbital angular velocity, the net force follows the direction of orbital motion contributing on the gain of angular momentum, as shown in Figure 1. Otherwise, when the libration angle is negative it decreases the angular momentum of the system. These non-zero tether libration angles are maintained by deployment and retrieval of the tether. The concept investigated herein is to see if the tether can be retrieved at one positive libration angle to maximize energy decrease and deployed at a different negative libration angle to minimize energy loss, so that there is a net increase of orbital energy for every retrieval/deployment cycle.

1.2 Outline of the thesis

The aim of this thesis is to achieve a new control law for deployment and retrieval that optimizes the increase of orbital energy over time. This allows for an initial assessment of whether momentum exchange with a synchronous system is higher performing than existing concepts.

To do so Chapter II: Theoretical development first is based on finding the equations of motion for a general model of a TSS. The model of the problem is presented and the equations of motion are studied separately breaking them between the translational movement and the rotational movement of the system. After that, the general set of

5 equations is put together. Also, this chapter finds a new control law for deployment and retrieval under the statement of quasi-equilibrium as proposed by Landis [10]. Finally, the mission concept is developed along with an attempt to represent the orbit with an analytical expression.

In Chapter III: Results, there is the application of the equations found in the previous chapter. First, a validation of the equations is done by comparing it with existing literature.

After the optimal angles of retrieval and deployment are found, plus the optimal factors of mass and length. At last, a full study of the entire retrieval/deployment cycle is carried out.

Finally, in Chapter IV: Conclusions, the important facts and results found in this thesis are discussed and an outline for future work is given.

CHAPTER II

THEORETICAL DEVELOPMENT

2.1 Model

The scenario under study pretends to be an accurate simplified representation of a tethered satellite system. It follows a similar model discussed by Mandri on his doctoral dissertation and on other researches like Landis and Gratus and Trucker.

The system is composed of two punctual masses connected by a momentum exchange tether, which is a non-electromagnetic cable and just subject to the gravity gradient. All the other external torques such as drag, solar wind, etc. are neglected. The center of masses and the center of gravity are supposed to be close enough that both points are considered to be the same. Furthermore, the center of masses is initially on a near circular orbit, therefore it is at distance from the center of the Earth and it is

traveling with the mean velocity . Due to the equilibrium hypothesis, all of the system is moving in a synchronous configuration� with respect to the Earth.

The tether is assumed to be massless since Mandri proved that the dynamics of the system are barely affected by the mass [16]. Also, it is considered to be a rigid thin rod, therefore the effect of elasticity, slack or vibrations are not studied. Its function is to create a physical connection between the two masses, and , and to change its length increasing or decreasing the distance between the masses. Consequently, the position of the masses with respect to the center of Earth, and , and the center of masses and

, are variable.

To simplify the model, and because the literature has proof that the out-of-plane disturbances are small, the system rotates in the plane. Hence, there is a libration in-plane angle between the local vertical and the tether position, .

�

Figure 2. Model representation

2.2 Equations of motion

The system has three degrees of freedom , , ; two of them determine the position of the center of masses (CM) and the last one� determines� the libration angle.

Then, the EOMs consist on a set of three non-linear ordinary-differential equations (ODE) that represents the translational movement ( , ) and the rotational movement of the system ( ), respectively. The equations are developed � using polar coordinates and follow the model� shown in Figure 2.

2.2.1 Translational movement

The translational movement is obtained by Newton’s second law. The only external force acting on the system is gravity, so the total force (i.e the summation of the forces applied on each mass) is equal to the product of the total mass of the system by the acceleration of the CM.

(1)

The forces follow Newton’s law�̅ of+ �̅ gravitation = ̅ so they depend on the standard gravitational parameter, the mass and the position vector between the point mass and the center of Earth.

(2) � � ̅ |�̅ | ̅ |�̅ | The end mass position vectors� = − are ̅the result� =of −adding̅ the position vector of the CM

( ) and the vector from the CM to each mass ( , ). Their vector expressions with respect̅ to the rotating frame are: ̅ ̅

cos � (3) ̅ = ̅ + ̅ = [ ] + [ ] sin � − cos � ̅ = ̅ + ̅ = [ ] + [ ] Since the tether is considered a rigid body of total− length sin � , it is possible to define the length of each part of the tether as a function of the masses and the total length = and . The left side of Eq. (1) is defined when substituting the equations above: =

� � � � (4) cos � ( − ) − ( + ) ̅ ̅ |̅ | |̅ | |̅ | |̅ | � + � = [ � � ] sin � ( − ) |̅| |̅|

To find the right part of Eq. (1), the transport theorem is applied to the position vector of the CM, which allows the calculation of the time derivatives of the rotating reference frame with respect to the inertial reference frame. The rotating frame has an angular velocity equal to the orbital angular velocity , therefore applying the transport theorem once, the velocity of CM is found, and applying� it twice the acceleration of CM is found.

(5)

̅ = [ ] (6) � ̅ = ̅ + �̅ × ̅ = [�] (7) − � � ̅ = ̅ + �̅ × ̅ = [� + �] When solving Eq. (1) by substituting (4) and (6) at each side of the equation, the first two EOMs ( , ) are found.

� 2.2.2 Rotational movement

The rotational movement is obtained by applying Euler’s second law of motion for a rigid body, which states that the rate of change of angular momentum is equal to the sum of the external moments about the system center of mass.

(8)

The left handed term is the total external�̅ = ℎ̅ torque defined as the summation of the cross product between the position, Eq. (3), and the force, Eq. (2), of each point mass. As the force and position vectors are on the same plane, the torque is perpendicular to the orbit and has the following expression:

(9)

when substituting, �̅ = �̅ + �̅ = ̅ × �̅ + ̅ × �̅

(10) � � �̅ = sin � ( − ) ̂ The right handed term of Eq. (8) is the time|̅| derivative|̅| of the angular momentum which is calculated by the cross product of the tether length with respect to the center of masses within the inertial reference frame and its linear momentum of each mass.

(11)

ℎ = ∑ ̅ × ̅ = (12) ̂ ̂ Since the angular velocityℎ = of the (� system+ �) +and the ( �angular+ �) momentum are in the same plane, the cross product is equal to zero. Therefore, when applying the transport theorem, the time derivative of the angular momentum with respect to the inertial frame is equal to the time derivative of itself with respect to the local reference frame.

(13) ℎ = ℎ� + �̅× ℎ̅ (14) ℎ̅ = [(� + �)( + ) + (� + �) + ]̂

Substituting (10) and (14) to the Eq. (8) and rearranging it to find the last EOM that completes this problem ( ).

� 2.2.3 General equations

The equations of motion that are governing the model are a set of three non-linear ordinary-differential equations. They are the same as the ones developed by Mantri [16] under the long massless tether analysis.

(15) = cos � − − + + �

(16) � = sin � − − �

(17) − sin � � = − (� + �) − � Where

(18) � = |̅| (19) � = |̅| 2.3 Control law for equilibrium orientation

The state variables of this system are radii of the orbit, the true anomaly, the libration angle and their respective derivatives. Therefore, once these variables are determined the only remaining parameter that can be controlled is the tether length and its derivative.

Following the concept of orbit pumping, there is the need to find a control law that maintains the tether out of the local vertical in order to provide a continuous tangent force for a finite amount of time. So, if the system is in equilibrium, without rotating about the

CM, there is going to be a net tangent force. Setting EOM (17) and the rate of change of the libration angle equal to zero, , it should be possible to find a tether length deployment law that satisfies this �condition. = � = Unfortunately, the equation still depends on the second time-derivative of the true anomaly, which is a time-varying term. Therefore, an exact equilibrium solution for a nonzero orientation of tether does not exist. Instead an

12 under quasi-equilibrium solution is sought. Since the true anomaly changes really slowly it is fair to assume its double derivative is almost zero which allows to find the control law for a quasi-equilibrium system. Developing it from� = the third equation of the set Eq.

(17).

(20)

The terms of c’s are given = by expressions− sin � (18) − and� (19) so they depend on the distance from the masses to the center of Earth, which means that by trigonometry, applying the law of cosines, the position vectors depend on the tether length and orientation.

(21) ⁄ ⁄ |̅| = ( + + cos �) = ( + + cos �) (22) ⁄ ⁄ |̅| = ( + − cos �) = ( + − cos �) Substituting it to the definition and dividing it by the term yields an expression for the c´s terms that can be expand using a binomial series.

� � = = ⁄ |̅ | (23) ( + + cos �) − ⁄ � = + ( ) + cos �

� � = = ⁄ |̅ | (24) ( + − cos �) − ⁄ � = + ( ) − cos �

Applying the binomial expansion since the term makes all the other terms small: �� 13

(25) − + � ≈ + � + � + ⋯ ! (26) � = ( ) + cos � (27) � = ( ) − cos � (28) − = ⁄

� = − cos � − ( ) + ( ) cos � + ( ) cos � (29) + ( )

� = + cos � − ( ) + ( ) cos � − ( ) cos � (30) + ( )

Since the small order terms of Eq. (26) and Eq. (27) have a quadratic term of , �� the approximation has to have up to the second order term, that is why the truncation is done for the third order numbers. The resulting parameters have the following expressions.

(31) � = − cos � − ( ) + ( ) cos � (32) � = + cos � − ( ) + ( ) cos � To substitute them to the general equations, is easier to calculate the differences between them.

(33) � − = ( − cos � − − cos �) (34) � − = (cos � − − cos � − )

Rearranging Eq. (20) using the new term Eq. (33):

(35) − = � sin � ( − cos � − cos �) The control law comes from the integrating Eq. (35), solved by following standard mathematical tables [17] (See Appendix A). The analytical time-dependent tether length equation for quasi-equilibrium is:

(36)

�� cos � = �� where cos � − � −

(37)

� = − � � � cos � (38)

− � = − cos � It is also needed to find the analytical time-dependent expression for the rate of change of the tether, because when working with on-board systems the calculation is faster and ensures a solution. Therefore, the derivative of Eq. (36) is:

(39) �� cos � − �� cos � = �� If the masses are equal then( costhe �term − � − and the) expression (36) becomes the same as the one developed by Arnold [15].� A= comparison between this new result and the existing quasi-equilibrium solution is presented in the Results chapter.

2.4 Mission concept

2.4.1 Cycle for orbit pumping

The aim of this thesis is to study a new cycle of retrieval/deployment for a TSS and determine if it is beneficial to implement this for orbit maneuvering. The cycle proposed begins with a deployed tether with an initial length that is retrieved until a minimum length and then deployed again back to its initial length. This thesis does not cover the first �deployment, when the system is launched to space and has to deploy to start the cycle, because the study is focused on the long term effect of executing the cycle several times.

As outlined before, the retrieval and deployment are done under the quasi-equilibrium hypothesis, therefore the tether maintains an off-set libration angle with respect to the local vertical during both stages. Focusing on Eq. (39), the expression of the rate of change of the tether and Eq. (37) which is the expression of , it can be seen that an angle between

0 and 90 degrees makes the term negative, leading� to a negative rate of change (i.e tether retrieval). Similarly, when the �libration angle is between 0 and -90 degrees, the term

become positive and the rate of change is also positive (i.e tether deployment).

�

Figure 3. Net tangent force

Analyzing the geometry of the configuration, Figure 3, when the libration angle is positive the net force of the system has a horizontal component following the direction of movement. This force increases the angular momentum which increases the orbital energy, which can also be seen as an increase of the semi-major axis of the orbit for the following relationship;

(40) � � = − So, the system retrieves when the equilibrium angle is between 0 and 90 degrees and this stage gives an increase of the semi-major axis. On the contrary, the system deploys when the equilibrium angle is between 0 and -90 degrees which leads to a loss of orbital energy.

The objective of the cycle is to maximize the gain of the semi-major axis when retrieving the system and then minimize the loss when deploying it, in order to obtain a net gain over a cycle of retrieving and deployment. Nevertheless, it is also important to minimize the time of each of the stages in order to have a higher gain in less time,

17 otherwise to have a decent gain of the semi-major axis over a long period of time is not effective. The next chapter presents an attempt to find the optimal deployment and retrieval angles to maximize the next increase in orbital energy per unit time.

2.4.2 Analytic development of change in orbit energy

The increase of the semi-major axis is proportional to the gain of angular momentum of the system which is due to the net external horizontal force from gravity. Taking the force acting along the axis of Eq. (4) and substituting the terms and based on the binomial series expansion,̂� the first order expression for the force is:

(41) This force produces a total�� = external torque� sinthat� cosleads � to a variation of the angular momentum of the system. Euler’s second law of motion states that the rate of change of angular momentum of a body with respect to a fixed point of the inertial reference frame is equal to the sum of all the external torques acting on that body. Therefore integrating the total torque over time the variation of the angular momentum is found. To keep it simple the expression used for the tether length is the equation developed by Arnold.

(42) �∆� = (43) �� ∆ℎ = ∫� � = � sin � cos � ∫� = �( − ) The variation of the semi-major axis is:

(44) ℎ √ = → = ℎ Considering a very small� eccentricity − , :√ � − ≪ (45) ∆ = √ ∆ℎ �

Substituting Eq. (43) into Eq. (45):

(46) ∆ = √ ( − ) As can be seen in the Eq. (46) it turns out that the analytical expression to calculate the semi-major axis change is independent of the libration angle. Since the tether is designed to retrieve and then deploy to the same initial length, then, when Eq. (46) is applied the total semi-major axis change after a cycle is zero.

Consequently, the analytical calculation cannot be accurately done with the simple equilibrium expression for the tether length. Rather than attempt the very long and cumbersome solution for the more accurate equilibrium expressions developed in this thesis, the computations of the semi-major axis variation are determined by numerical simulation.

CHAPTER III

Results

3.1 Validation

The equations developed in Chapter II can be validated by comparing with the existing literature. The simulations are done by numerical integration using a Runge-Kutta of 4th order method, from to , where is the period of the orbit and is a real positive number. All the cases = consider = � an initial� circular orbit, initial true anomaly of 0 degrees and . The initial libration angle, the control law and the mass factor depend on each = one of � the problems. The mass factor is defined as the relationship between and , being 0.5 when the lower mass is half of the upper mass.

3.1.1 Linearly increasing case

To ensure that the obtained EOMs are correct, this section compares them with the existing literature done by Mantri [16]. The case under study is the deployment of the tether using a linearly increasing control law:

(47)

The radii of the initial orbit is set at GEO = + (42,250 km), the libration angle is zero and the system has equal masses. Two cases are studied depending on the deployment rate

and . On the following plots it can be seen how the libration − − angle = varies / during the = deployment / .

Figure 4. Libration angle vs Time. Rate of change km/s. Mantri − l =

Figure 5. Libration angle vs Time. Rate of change km/s − l =

Figure 6. Libration angle vs Time. Rate of change km/s. Mantri − l =

Figure 7. Libration angle vs Time. Rate of change km/s − l =

The tether starts aligned with the local vertical and swings out following a decreasing sinusoidal-exponential performance. It gets almost stabilized around a small negative angle and the fastest rate of change gets stabilized first. The fact that the system gets a quasi-stable position around a negative value of the libration angle is due the orbit difference of the masses. The fact that the masses are connected by the tether forces them to move at the same angular velocity of the CM. During the deployment, the lower mass gains more angular velocity so the CM is retaining it. Instead, the upper mass has less angular velocity than the CM so this is pulling it. Therefore, the system stabilizes with a negative off-set of the libration angle.

3.1.2 Quasi-equilibrium case

The control law for a quasi-equilibrium orientation can also be validated by comparing it with the research done by Arnold. As it is said in that section, the new control law has an extra term, , that depends on the relationship of the masses. Consequently, when the masses are� equal, , the new control law is exactly the same as the one developed by Arnold. But� when = the masses are not equal the new control law shows an improvement on the stability of the orientation. To compare both control laws the radius is set to 8000 km with an initial libration angle of 45 degrees and a factor mass of 0.25. The initial tether length is of 100 km and the simulation runs from to . This interval of time had been chosen in order to see how the classic =control law = . loses � the equilibrium position and how it affects the tether length, while the new control law maintains its quasi-equilibrium. As it can be seen in Figure 8. Comparison between new control law and classic control law Figure 8, the divergence of the libration angle starts around 0.8 periods, but the tether length stays 1 km of difference until 1.1 periods. At this

23 point, the classic control law is not valid anymore because the tether starts deploying due to the loss of a stable position of the libration angle.

Figure 8. Comparison between new control law and classic control law

3.2 Optimal retrieval and deployment angles

As discussed in section 2.4 when the libration angle has a positive value between 0 and 90 degrees the tether is retrieved, consequently a retrieval libration angle that yields the maximum semi-major axis increase per unit time must be in that interval. Similarly, the optimal deployment angle that provides the minimum semi-major axis decay must be between the interval of 0 and -90 degrees.

As it has been seen the semi-major axis change is proportional to the force, the optimal libration angles must be at a stationary point of the force function Eq. (41). Making the first derivative of the equation equal to zero:

(48)

�� = � cos � − sin � = → � = ±º Both angles �gives a maximum value of the force, the positive one for retrieval and the negative for deployment. Since the retrieval stage is the only one that needs to be maximized, the optimal retrieval angle is set at 45 degrees. To corroborate this, the next figures show the tether performance during the retrieval stage for different retrieval angles.

The simulation assumes an initial tether length of 100 km and a mass factor of 1.

Figure 9. Libration angle vs. Time, retrieval

Figure 10. Tether length vs. Time, retrieval

Figure 11. Semi-major axis vs. Time, retrieval

As it can be seen in Figure 9 a libration angle of 45 degrees gives a bigger change of the semi-major axis. Even if this configuration stays stable for less time, it can be seen that during this period the tether length almost decreases linearly and presents a higher rate of change. After a certain period of time, when the tether becomes smaller, the control law is not satisfied anymore, Figure 10, because the tether starts deploying even if the libration angles are negative. Therefore, there is a finite period of time for which the control law is valid, or another way of looking at it is that there is a minimum tether length for which the control law is valid. The proportion between the minimum tether length and the initial tether length is called length factor.

To check that 45 degrees is the libration angle that gives the largest variation of the tangent force, it has been simulated Eq. (41) for a set of tether lengths and different libration angles. Figure 12, shows that the angle 45 gives at any tether length a bigger tangent force.

Figure 12. Tangent force vs. Tether length

Going back to find the optimal deployment angle, it is known that the minimum horizontal force leads to the minimum semi-major axis change. Looking at Eq. (41), when the libration angle is either 0 “vertical” or -90 “horizontal”, the force is equal to zero.

Therefore, deployment orientations near zero are chosen for the mission design.

3.3 Optimal mass and length factors

Once the libration angles are set, the remaining variables of the model are the mass factor, the length factor and the initial tether length. To study the effect of the factors an initial tether length of 100 km is chosen. The rest of the parameters are the same and the retrieval angle and the deployment angle are 45 degrees and -1 degrees respectively.

The semi-major axis variation is found by subtracting the initial value from the final one obtained from the simulation, . To calculate these semi-major axis

values, the definitions of orbital elements∆ = are− applied using the values of the state variables that have been propagated.

(49) |ℎ̅| = � − |̅|

Where

(50)

ℎ̅ = ̅ × ̅

(51)

� ̅ = (̅ × ℎ̅ − ̅) �

By dividing the variation of the semi-major axis by the time of the propagation,

, the rate of change of the variation of the semi-major axis, is found. During∆ the =

− ∆ 28 simulation it is possible to collect the times and semi-major axis variation for both stages, retrieval and deployment.

(52)

∆�� + ∆ ∆ = �� Figure 13 consists of a contour that∆ represents+ ∆ the rate of change of the semi-major axis variation for a full cycle combining several mass and length factors. The interval of the mass factor goes from 0.2 to 1. It has skipped the factors from 0 to 0.2 for faster computational times. The interval of the length factor goes from 0.1 to 0.8. Again, the numbers out of this interval had been neglected in order to gain time on the computation.

Figure 13. Effect of the mass and length factor on the orbit pumping

It can be seen that the optimal configuration is using a mass factor of 1 and a length factor of 0.15, which leads to a maximum average of orbit pumping of 6.14 m/h. That the mass factor is 1 can be seen from the force Eq. (41) because it depends on the

29 multiplication of each mass divided by their summation. This factor is the biggest when both masses are the same. Instead the length factor is constrained by the time. As it will be seen in the next section 3.4 Mission results the retrieval stage is fast paced, but the deployment stage takes longer times. This means that if the minimum tether length is small the deployment stage will take longer penalizing the rate of change of the semi-major axis

Eq. (52). At the same time, if the minimum tether length is big, the change in the semi- major axis will be small. Consequently, there is the need to find a compromised minimum tether length that will maximize the change of the semi-major axis during retrieval and minimize the time of deployment, and it has been found that the best solution for this case is 15 km.

3.4 Mission results

The initial conditions of the problem are set as an initial orbit of 8000 km, an initial tether length of 100 km with a tether length factor of 0.15 and with a mass factor of 1. The simulation starts the retrieval stage using the optim =al retrieval � angle of 45 degrees. When the tether length gets to the minimum tether length, the variables are updated to start the deployment stage. These updates consist of setting the initial tether length equal to the minimum length of the tether, setting the initial time equal to the final time of retrieval and determining the new libration angle as the optimal deployment one,

-1 degrees.

Figure 14. Tether length variation vs. Time

Figure 15. Libration angle variation vs. Time

Figure 16. Semi-major axis variation vs. Time

It is clear from the three figures, that the retrieval is much shorter than the deployment. In fact, the retrieval stage is complete after gaining of the semi-major axis. This means that the average rate. of � change during retrieval is

. The deployment takes to complete while decreasing

∆�� = .. Then, /ℎ the rate of change of deployment � is . Consequently,

when adding the two stages the total semi-major axis∆ variation= −. sets /ℎ an increase of

after , which leads to the rate of change of the

∆variation�� = of the semi-major ∆axis�� that= has. been� seen before of .

∆�� = . /ℎ

CHAPTER IV

CONCLUSIONS

This thesis has found a new control law for the tether deployment and retrieval under the equilibrium hypothesis. The retrieval stage works well since it increases the semi- major axis in a short time, but the deployment stage takes too long in order to make all of the system effective. Overall, the average gain of orbital increase is better than in other studies that apply the same type of tether. The cycle presented here increases the semi- major axis about 6 m/h in comparison with Breakwell which increases 100 m/yr. Instead, the new control law works worse when compared with the rotational tethers, which has been found to increase 300 m/h. Although the performance on gaining orbital energy with a non-rotating tether is worse than the rotational ones, they do not have to deal with the need to induce an initial rotating impulse or the limitations of material. These limitations come from the fact that the system is rotating around the CM and produces high tension at the tip of the tether, therefore the material of the tether has to be strong enough to support these forces without breaking.

A future study furthering this method would be to find a non-equilibrium deployment law that involves less time. Therefore, by combining it with the retrieval proposed in this thesis, it could lead to improved results that may be competitive with the rotating tether.

Also, it would be interesting to include the study of the transition stages when the tether passes from one equilibrium configuration to the next one. The analysis of these new stages will add time to the process and probably a decay of the semi-major axis, making the system worse but at the same time more realistic.

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APPENDICES

A. Tether length and rate of change development

The equation (19) stand for the rate of change of the tether when solving the third EOM for quasi-equilibrium case:

− = � sin � − cos � + − cos � Rearranging it with factors:

�� where, = � − cos �

� = − � sin � cos � − � = − cos � From Standard Mathematical Tables and Formulae, pag 427, form 108 [17]:

� = + � + � = −

� � + − √− ∫ = ln � √− � + + √−

This case:

�� = � − cos � = = � �� = − cos � = −�

∫ = ∫ �� − cos � −� � ln ( )| = |� � −� + cos �

−� + cos � ln = �∆ −� + cos � �∆� −� + cos � = −� + cos � �∆� �∆� (−� + cos � + � ) = cos �

�∆� cos � = �∆� cos � − � −