ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS
DEVELOPMENT OF A RENDEZVOUS AND DOCKING SIMULATION
GRADUATION PROJECT
Utkan GÜDER
Department of Astronautical Engineering
Thesis Advisor: Dr. Öğr. Üyesi Cuma YARIM
MARCH, 2021
ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS
DEVELOPMENT OF A RENDEZVOUS AND DOCKING SIMULATION
GRADUATION PROJECT
Utkan GÜDER (110140129)
Department of Astronautical Engineering
Thesis Advisor: Dr. Öğr. Üyesi Cuma YARIM
MARCH, 2021
ii
Utkan Güder, student of ITU Faculty of Aeronautics and Astronautics student ID 110140129, successfully defended the graduation entitled “DEVELOPMENT OF A RENDEZVOUS AND DOCKING SIMULATION”, which he prepared after fulfilling the requirements specified in the associated legislations, before the jury whose signatures are below.
Thesis Advisor : Dr. Öğr. Üyesi Cuma YARIM ...... İstanbul Technical University
Jury Members :
Date of Submission : 10.06.2021 Date of Defense :
iii
FOREWORD
Even though my main interests have shifted towards software programming and computer graphics over the years of being an astronautical student, I have always been interested in orbital mechanics since I have began studying. Hence the topic of my thesis that I hope someone will find it useful or inspiring, although I have worked full-time while writing it.
March 2021 Utkan Güder
iv TABLE OF CONTENTS
Page
FOREWORD ...... iv TABLE OF CONTENTS ...... v LIST OF ABBREVIATIONS ...... vi LIST OF TABLES ...... vii LIST OF FIGURES ...... viii SUMMARY ...... ix 1. INTRODUCTION ...... 1 1.1 Purpose of Thesis ...... 1 1.2 Literature Review ...... 1 2. MANEUVERING IN SPACE ...... 5 2.1 Rendezvous Phase ...... 5 2.1.1 Orbital Insertions ...... 5 2.1.2 Rendezvous Calculations ...... 9 2.2 Docking Phase ...... 13 3. SIMULATION ...... 16 3.1 Development ...... 16 3.1.1 Tools ...... 16 3.1.2 Calculations ...... 17 3.2 Graphical Interface ...... 21 3.3 Example Scenario ...... 24 4. CONCLUSION ...... 26
v LIST OF ABBREVIATIONS
R&D : Rendezvous and Docking ISS : International Space Station EOR : Earth Orbit Rendezvous LOR : Lunar Orbit Rendezvous ECI : Earth-Centered Inertial
vi LIST OF TABLES
Page
Table 2.1 : ΔVs for the Hohmann transfer to ISS ...... 7 Table 2.2 : Final phase angles for different altititudes ...... 10
vii LIST OF FIGURES
Page
Figure 2.1 : Hohmann Transfer ...... 6 Figure 2.2 : Bi-elliptic Transfer ...... 7 Figure 2.3 : Bi-elliptic Transfer for an ISS Rendezvous Mission ...... 8 Figure 2.4 : Final Phase Angle and Lead Angle ...... 10 Figure 2.5 : Possible Phase Angles for a Bi-elliptic Transfer to ISS ...... 12 Figure 2.6 : Wait Times for Angle Differences to ISS ...... 13 Figure 2.7 : ISS Relocation Corridor for a Russian Vehicle ...... 14 Figure 3.1 : Matrix Multiplications for the Initial State Vectors ...... 18 Figure 3.2 : Obtaining Position or Velocity Vectors ...... 19 Figure 3.3 : Applying the Calculated Maneuver ...... 20 Figure 3.4 : Rendezvous Planning ...... 21 Figure 3.5 : Start and Custom Paramter Menu ...... 22 Figure 3.6 : Focus Panel ...... 22 Figure 3.7 : Time and Transfer Panels ...... 23 Figure 3.8 : Spacecraft Calculating for the Next Rendezvous ...... 25 Figure 3.8 : The Spacecraft in Docking Mode, Parked Near the ISS ...... 25
viii DEVELOPMENT OF A RENDEZVOUS AND DOCKING SIMULATION
SUMMARY
This thesis covers the development and usage of an orbital simulation in the Unity Engine where a rendezvous and docking (R&D) operation takes place to the International Space Station (ISS) from a circular orbit around the Earth. It discusses the usage and short history of R&D, its methods and how it is implemented to this specific software in which some constraints and assumptions on certain conditions are made. The main use of the program is to visualize a Hohmann transfer to the ISS’s orbit from a user defined circular orbit. It enables the user to plan an engine burn, with a start and end shown on the orbit, where the delta-V required for orbit matching and the time of the maneuver is calculated. This is done by finding the wait time between the two vehicles, for them to enter the correct phase, and is further optimized by taking the burn time into account and finding the closest approach via a simple iteration from its maneuver start position. However, the software does not find the most efficient maneuver, either in fuel consuption or duration, but rather shows how a transfer could be done from the given state of both the spacecraft and the ISS. The spacecraft in discussion is visualized as a Soyuz vehicle, which is one of the most frequent visitors to the ISS. This thesis could be divided into two parts where the first part touches on R&D and what aspects of it are applied to this project while the second part refers to the software, in which the development and use of the program is discussed. This includes information about the Unity Engine, a game engine that is used in a variety of fields and provides some useful tools to create the environment. The code is written in C#, an object oriented programming language, that Unity supports. This part also contains a guide on how to use the software and an example R&D mission.
ix ÖZET
Bu tez, Dünya etrafında dairesel bir yörüngeden, Uluslararası Uzay İstasyonuna bir randevu ve kenetlenme görevinin Unity motorunda geliştirilmesi ve kullanımı hakkındadır. Bu tip görevlerin tarihçesinden başlayarak, nasıl uygulandıklarını ve programa nasıl entegre edildiğine de değinir. Özetle program, kullanıcının tanımladığı dairesel bir yörüngeden uzay istasyonuna tek bir Hohmann transferi ile nasıl gidilebileceğini hesaplar ve gösterir. Bununla birlikte, daha yüksek bir yörüngeye bir manevra hesaplaması da mümkün. Görevin ilk aşaması gereken delta- v ve motorun çalıştırılması gereken anları gösterirken, ikinci aşamada kenetlenme için aracın istasyona doğru navigasyonunu kullanıcıya bırakır. Görseller için, Uluslararası Uzay İstasyonunu en sık ziyaret eden araçlardan biri olan Soyuz aracı tercih edildi. Tez iki genel olarak iki kısımdan oluşur. İlki yörünge hareketlerinden ve bir randevu görevi için yapılabilecek manevralardan bahseder. Bunlar tek ya da çift Hohmann transferlerinin farklı yörüngelerden uygulanmasını ve manevralar arası farkları ele alır. Ikinci bölüm ise programın geliştirilmesi ve yazılımın kullanılışının anlatıldığı kısımdır. Bu, çeşitli alanlarda kullanılan ve ortamı oluşturmak için bazı yararlı araçlar sağlayan bir oyun motoru olan Unity Engine hakkında bilgileri içerir. Kod, Unity'nin desteklediği nesne yönelimli bir programlama dili olan C# ile yazılmıştır. Ayrıca örnek bir görevin nasıl uygulanabileceği de mevcuttur.
x 1. INTRODUCTION
1.1 Purpose of Thesis
The main objective of this thesis is to create a readable visualization of a rendezvous and docking operation. This simulation does not cover rendezvous and docking in full detail but simply provides a way to see how a rendezvous maneuver could be done with user-defined parameters and roughly mimic the docking phase with a keyboard input. In addition, rather than being a real life simulation, the user can modify certain aspects like the altitude and angle of the orbit as well as the acceleration of the spacecraft, which provides experimentation of different circumstances. A more in-depth explanation about the capabilities of the software is discussed later.
1.2 Literature Review
Rendezvous and docking (R&D) defines the process of two spacecraft coming to a close approach, which is the rendezvous phase, and establishing a physical connection, which is the docking phase, in order to stay at the same orbit. The two spacecraft are referred to as the chaser and the target. Chaser vehicle tries to rendezvous with the target vehicle that is usually a station and is passive. Rendezvous phase consists of multiple orbital maneuvers that the chaser vehicle performs in order to reach the same orbit as the target vehicle and come to a close proximity for docking to begin. Docking phase is the two spacecraft joining where one of the spacecraft carefully approaches and triggers the mechanisms to achieve the structural link. [1] For the rendezvous, certain conditions have to be met in order to perform the set of orbital maneuvers required to reach the target. First, the two spacecraft must reside in the same orbital plane and secondly, the phases of the orbits must match. Docking also has its own constraints and complications like a requirement for slow approach, obstacle avoidance, approach cone and spacecraft’s engine system. This thesis has some critical assumptions about these topics, and their
1 integration to the program could be improved. The simulation assumes the chaser vehicle is already on the same orbital plane and in circular orbit. The target is fixed and is represented as the ISS, at an average altitude of 419 km. A Soyuz spacecraft represents the chaser vehicle with a user-defined altitude and true anomaly. The simulation is complete when the Soyuz spacecraft touches the port of the ISS with the correct orientation.
R&D has been mostly used for carrying supplies and crew to space stations as well as repair and maintenance missions. The first successful docking operation was executed in 1966 by the Gemini 8 by NASA. It is a predeccosor project that allowed the development and testing of many methods required to get to the Moon via the Apollo Program. [2] Main goal of Apollo missions differ from most of the rendezvous operations done around Earth, refered to as Earth-orbit rendezvous (EOR), in which a Lunar-orrbit rendezvous (LOR) was executed. Three approaches were thought of when the Moon was the next chosen target for humanity. A direct ascent, EOR or LOR. The direct ascent method was not a practical approach since it needed an enormous rocket for both leaving the Earth, landing on the Moon and coming back, though proposals like the huge booster named Nova was made within NASA. The thought behind the EOR approach was to have assemble a direct ascent vehicle in the orbit of Earth. However, LOR was decided upon. It featured a three staged vehicle launching from Earth on top of a Saturn V booster to reach the Moon and utilize its three stages as a command module, descent module and an ascent module around the Moon. Despite the complexity of LOR, its advantages were hard to ignore. It needed less fuel, had half the payload and could be done with more familiar technology which allowed a sooner Moon landing as planned. [3] Russian Soyuz missions were also taking place when the Apollo 9 tested its docking capabilities in low Earth orbit in 1969. Two Soyuz spacecraft, Soyuz 4 and Soyuz 5, docked in Earth orbit and successfully transferred crew members. [4] An Apollo- Soyuz international mission was done as well in 1975 that relaxed the tension between the United States and the Soviet Uniton but it was also a breakthrough in which two very different vehicles docked in low Earth orbit without them being built specifically for the operation. [5] Russian Salyut missions, the first space stations, were crewed and supplied by Soyuz spacecrafts from 1971 to 1986, which were later followed by Mir, the first modular station, from 1986 to 2001. [6] Skylab, which
2 operated between 1973 and 1974, had three crewed missions to it with Apollo spacecraft for maintenance. [7] Hubble Space Telescope received servicing missions featuring repairs, maintenance and upgrades via the space shuttles between 1993 and 2009, playing a critical role in its success. [8] Chinese prototype station, Tiangong-1 that served from 2011 to 2018, was visited by Shenzhou spacecrafts multiple times which is a preperation for a future modular station. [9] Finally, since 1998, the ISS is being serviced by a multitude of vehicles for crew transfer and supplies and expeditions are still being held or planned. [10]
By checking the methods used in above missions and the information in Howard D. Curtis’s book, constraints of a rendezvous mission can be decided. Starting with the launch window, most missions optimize the rendezvous time and resources by launching at the correct time window which helps lower the time of wait in between maneuvers. In order for spacecraft to match the orbit, both spacecrafts have to be on the same plane. In the case of reaching ISS, the rocket launch is usually scheduled accordingly to minimize this maneuver so that the insertion orbit’s inclination is near that of the ISS’s orbit. Small corrections are still required because of the slight unpredictable nature of a rocket launch. Just before the rendezvous burn, a manuever is executed to change the inclination ever so slightly, so their inclinations are near identical but not exactly equal so the risk of a collision is eliminated. Inclination correction is one of the missing features in this simulation since both vehicles start on the same plane however this does not mean the spacecraft is locked on to the plane. It is just the automated calculation that cannot use the inclination.
All the rendezvous missions paved the way to more efficient maneuvering operations, seen in the Soyuz to ISS missions. Instead of a simple Hohmann transfer, a bi-elliptic transfer method is used in some circumstances. If the wait time is too high for the correct phase angle between the orbits, then using the bi-elliptic transfer can reduce this travel time substantially. While a Hohmann transfer has two engine burns, the bi-elliptic transfer uses three where an intermediate orbit is targeted and used as a step so the desired phase angle is achieved faster. Currently, the software mentioned in the thesis can only calculate the delta-V requirements and wait time for a basic Hohmann transfer but the user can actually execute a bi-elliptic transfer manually if the calculations are made before hand and the engine burns are performed correctly.
3 The docking phase is much more simplified in the program, creating a sandbox environment that is open for testing. In real life, the chaser vehicle’s approach is carefully planned and calculated for obstacle avoidance. The ISS has different layers of regions with velocity limits depending on the distance of the spacecraft. The actions of the chaser depends on what the target vehicle is and manipulates its attitude accordingly. For example, when a Soyuz spacecraft is docking the ISS, the spacecraft’s tip is pointed at the corner of a solar array when approaching, acting as a reference point. Only later in proximity, the available docking port is prioritized. The methods for rendezvous and docking maneuvers can change for different circumstances, like in the Apollo missions, but in principal all calculations are derived from already known methods. The next chapter will discuss how these calculations are obtained from Keplerian orbit calculations.
4 2. MANEUVERING IN SPACE
2.1 Rendezvous Phase
In order to perform a rendezvous operation, the active chase vehicle has to match the orbit of the passive target vehicle at the correct time. The first orbital element to match is the inclination. This ensures the spacecraft are on same plane and simplifies the rendezvous process [11]. It is achieved either by launching the spacecraft at an angle or by performing a burn at one of the two intersections of the two planes. Since it requires extra delta-V on top of the transfer maneuvers, former method is used for spacecraft that will rendezvous with the ISS.
The next phase mostly requires a series of burns that the spacecraft performs to change its orbit while meeting up with the spacecraft. This maneuver could be a Hohmann transfer, a bi-elliptic transfer or a low energy transfer, which is not covered in this thesis since they are usually used in planetary transfers. Keep in mind that this paper assumes that the rendezvous maneuver takes place for two circular orbits, more specifically a spacecraft that is launched from Earth to the ISS.
2.1.1 Orbital Insertions
Any maneuver that affects the orbit more than a simple correction could be called an orbital insertion. In this rendezvous case, a Hohmann transfer orbit is used. This orbit simply connects the two orbits via an intermediate trajectory. Hohmann transfer maneuver is best for energy efficiency for a spacecraft to use when going from one circular orbit to another on the same plane (Curtis, 2005). In our case, the spacecraft is required to execute two burns when performing the maneuver. The first burn is for putting the spacecraft in the elliptical transfer orbit, shown as the Hohmann transfer ellipse in Figure 1. Later the spacecraft performs the second burn to increase the periapsis of its orbit.
5
Figure 2.1 : Hohmann Transfer, Curtis, 2005
The delta-V required for these burns can be calculated from applying the vis-viva equation (2.1) for each orbit. These are the initial orbit, the elliptical orbit and the target orbit.
( )
(2.1) - is the speed of the spacecraft, - which is the standard gravitational parameter (for Earth, equals to ), - is the distance of the spacecraft from focus, - is the semi-major axis of the orbit.
By inserting the parameters for each orbit in the equation, and assuming the velocities are the delta-V for each burn can be calculated as (2.2),
(2.2)
6 Table 2.1 : ΔVs for the Hohmann transfer to ISS.
Altitude (km) ΔV1 (m/s) ΔV2 (m/s) 100 88.40 87.41 150 73.19 72.49 200 58.16 57.72 250 43.33 43.09 300 28.70 28.59 350 14.25 14.23
The bi-elliptic transfer is actually very similar in the vein that both are actually Hohmann transfers, however bi-elliptic transfer uses two intermediate transfer orbits instead of one. This method is known for some cases where the delta-V required is less than a single Hohmann maneuver (Gobetz, 1969). It is mostly used by rising to a higher transfer orbit in the first burn from point A in Figure 2.2. Then a second burn to match the periapsis of the current orbit to target orbit’s apsis. The third burn is to slow down at C, since the first Hohmann ellipse’s apoapsis is higher than both orbits.
Figure 2.2 : Bi-elliptic Transfer, Curtis, 2005
However, a bi-elliptic transfer in a rendezvous operation, especially for a spacecraft targeting the ISS for rendezvous and at a lower orbit than its target, uses this method to reduce the wait time instead of delta-V efficiency. This wait time is the duration where the spacecraft waits in its phasing orbit to catch up to its target before
7 maneuvering and may be considerably long for a single Hohmann transfer. This wait time is touched upon in more detail later in the next topic.
This method’s utilisation for a rendezvous mission to ISS can be seen in Figure 2.3 where both transfer ellipses, from A to B and B to C, have lower semi-major axis than the target orbit. Vis-viva equation is used to find the required delta-V at each point, but additional calculations are done for the three instances of delta-V. This also negates the duration of bi-elliptic transfer’s main drawback, the time of flight, since point B resides at a lower altitude which takes up most of the time in the former figure.
Figure 2.3 : Bi-elliptic Transfer for an ISS Rendezvous Mission
Since we are performing Hohmann transfers, we can also easily find the duration of these transfers, the travel time from the a burn to another. This simply equates to the half a period of the transfer ellipse for each maneuver (2.3). This transfer time is going to be essential when calculating the wait time for the rendezvous.
√
(2.3)
- is the transfer time, - is the period of the orbit - which is the standard gravitational parameter, - is the semi-major axis of the orbit.
8 For a bi-elliptic transfer however, the transfer time is equal to sum of two half Hohmann transfer durations. These transfers are related in a way that the orbit in between can have a varying altitude between the chaser and target vehicles’ altitudes. This provides a flexible time window to some extent, though the transfer time is slightly longer.
√ √ (√ √ ) ( ( ) ( ) ) √
(2.4)
Where,
- and are the distances of spacecraft to the main focus,
- is the transfer time, - which is the standard gravitational parameter, - is the distance from main focus to the intermediate orbit
at (2.4) has to be between and . This does increase the time of flight but compared to the wait time to the next rendezvous, it can be ignored. This difference comes from where the interdiction, or rendezvous, happens. A traditional Hohmann carries the chase vehicle for a total of 180° around the main body while the latter completes a 360°.
2.1.2 Rendezvous Calculations
In a rendezvous, the Hohmann transfer has to be performed at a suitable angle by the chaser to successfully achieve close proximity to the target. This angle can be calculated by knowing the time of flight of transfer and the time it takes to reach the desired angle, which is the phase angle, φfinal, between the two spacecraft. In order to find this angle we must find how much the target moves in the amount of time it takes for the chaser to complete its transfer. [12]
9
Figure 2.4 : Final Phase Angle and Lead Angle [12]
This angle is the α lead in the Figure 2.4. It can be found from the angular velocity of target and the transfer time.
(2.5)
- is the leading angle for the chase vehicle,
- is the transfer time,
- is the angular velocity of the target.
By knowing this angle, we can also determine the phase angle since the transfer ellipse sweeps and angle of 180° or π.
(2.6)
- is the leading angle for the chase vehicle,
- is the phase angle right before transfer burn.
And the angular velocity of a spacecraft in circular orbit is,
√
(2.7) - is the radius of the orbit, - which is the standard gravitational parameter
10 Table 2.2 : Final phase angles for different altitudes.
Altitude (km) Angle (degrees) 100 5.94 150 4.96 200 3.97 250 2.98 300 1.99 350 0.99
Table 2.2 shows the final phase angles required with a single Hohmann transfer, for a target at 400 km, like the ISS, calculated from (2.6). If for example a 5.94 degrees is missed for a spacecraft at 100 km, the next rendezvous opportunity can only happen after another synodic period is complete, which is,
(2.8)
- and are the periods of both orbits.
- is the period at which the orbits return to the same configuration.
This issue can be softened in a very useful way with bi-elliptic transfer where the final phase angle can be chosen from an available window depending on all available orbits between the two spacecraft. With two Hohmann transfers the angle can be modified together with the height of the intermediate orbit to while keeping the rendezvous position same. These phase angles can differ because of the flexible intermediate orbit height, in equation (2.4). The available angle scales with the distance to the target orbit from the initial orbit. These window of opportunities can be seen in Figure 2.5, for a target at 400 km. Each line represents a different initial orbit with varying altitudes. This figure is obtained by calculating the angular velocities for each altitude using (2.7). Then by calculating the time of flight from
(2.4) for , where , we can obtain the phase angle for the corresponding . The y-axis, Hohmann altitude, in the figure corresponds to the altitude of the intermediate orbit from the Earth’s surface.
11
Figure 2.5 : Possible phase angles for a bi-elliptic transfer to ISS
Notice that an orbit of 100 km can have a varying phase angle between 6 and 18. The low-bound phase angle is actually where , which is a single Hohmann transfer.
Most of the time the interceptor will have to wait to get into position. The wait time,
Twait, can be found by how long it takes for both spacecraft to cover the remaining angle to reach the phase angle. The relation between the initial and final phase angle comes from the angular velocity difference of the spacecraft, determining the speed at which the angle changes. Since both spacecraft are in circular orbits, meaning the mean motion is near constant, the wait time for the interceptor is,
( ) (2.9)
Solving for ,
(2.10)
- is the wait time,
- is the final phase angle,
12 - is the initial phase angle,
- is the angular velocity of the target vehicle,
- is the angular velocity of the chase vehicle.
However, this result can be negative as we can see from the equation. In order to prevent this outcome, we should iterate the equation once more by adding 2π to the initial phase angle for the next available opportunity. This can be iterated to find further available transfers as well.
Figure 2.6 shows the how long it takes for the corresponding angle to be covered, obtained from the equation (2.10).
Figure 2.6 : Wait Times for Angle Differences to ISS
Notice how the wait time increases considerably for higher angles, which forces planned launched windows for a shorter mission.
2.2 Docking Phase
The phase where the spacecraft is now 1-2 kilometers away from the target could be referred to as the docking phase. It requires small and precise adjustments to the
13 rotation and velocity of the vehicle. This phase is usually performed without using the main engines and relying on the attitude control systems and small thrusters of the spacecraft. After the spacecraft is in close proximity that is 100-200 meters away, the vehicle relocates itself toward the port that it will connect to. There are two safety concerns to take into account. The spacecraft colliding with a debris or the spacecraft colliding with each other. To prevent these risks, but mostly the latter, the target spacecrafts usually have control zones asigned. [13]
For the ISS, there are already certain protocols and guidelines when approaching the station. These divide the docking phase into smaller regions ,or control zones, such as the approach sphere, keep-out sphere and approach/departure corridors. There are certain requirements and specifications for each region that ensure the docking is successful. For the approach sphere, the trajectory of the vehicle must not intercept this region for 24 hours before the docking phase. This is to prevent any possible collisions with other incoming or outgoing spacecraft. The keep-out sphere is actually a no-fly zone for the spacecraft, meaning the movement inside this region is analyzed and regulated by the authorities and can only be entered via the defined approach corridor. This approch corridor is designed depending on the capabilities of the spacecraft. Each type of vehicle may have a unique approach corridor. The spacecraft has to follow this approach corridor at slow speeds. [14]
Figure 2.7 : ISS Relocation Corridor for a Russian Vehicle [14]
14 Figure 2.7 shows the relocation corridor which is used to switching up already docked spacecraft or departing. It is a combination of approach/departure corridor and the fly-around corridor.
When the Soyuz craft is approaching the ISS, before entering the keep-out sphere, the spacecraft points at one of the solar arrays to keeps its orientation in control. When closer, Soyuz will follow the pre-defined path or corridor to an available docking port, switching its focus on the docking port itself. The spacecraft has to be at the right orientation and velocity while the final phase of the docking is performed.
15
3. SIMULATION
This part of the thesis touches the tools used in development and how the program works. It also contains a tutorial and an example scenario along with samples of codes from certain parts that contains the frequently used calcluations.
3.1 Development
3.1.1 Tools
The program is developed with the Unity Engine and all code is written in C# programming language, object oriented programming language that is integrated with the engine. Unity Engine is a game engine that provides ready to use tools for 3D applications for games, architecture and engineering. It was a favourable choice because of its ease of use and the considerable amount of personal experience. The engine works as a component based object system meaning that the behaviours are contained in a component in the object that exists in the scene. There are useful graphical components like mesh and line renderers or particle systems since it is a game engine after all but if a desired component is not found, it can be created in C# as a class that inherits from a class that Unity provides, “Monobehaviour”. Even though some components are provided, there were still some obstacles to overcome, like the engine itself storing positional information in floats, storing 7 significant digits inside its “Transform” component, which is not ideal for a simulation that covers lengths of thousands of kilometres with a resolution of centimetres. So an additional component is required that stores this information in doubles. This extra class is called the Advanced Transform in the project and stores the basic Cartesian coordinates with high-precision along with velocity and rotation. The data inside this component is used for all components. However, this is not enough on its own to visualize in the scene, so a manager object resizes and moves these objects according to the static camera and user input, enabling free movement with high-precision.
16 For a realistic simulation, the built-in “Rigidbody” component is not adequate for the same reasons as the position, so an extra class is created called “Advanced Mass” which stores the mass of the object and handles the interaction between large masses such as the Earth. This is achieved by simply applying Newton’s law of universal gravitation. It also contains behaviors that enable the user to simulate the object forward in time for estimating its position or velocity. Another essential component is the “Orbit” component that transforms Earth-centered inertial (ECI) coordinates into Keplerian elements and vice versa. In addition to providing the visualization needed with line renderers, this component contains all calculations related to Keplerian elements including the conversion of anomalies.
These custom components are used throughout the actual calculations for rendezvous planning. The target and the chaser, both have all Advanced Transform, Advanced Mass and Orbit components with the initial values that match the Soyuz and the ISS.
3.1.2 Calculations
The initial states of the spacecraft in ECI are calculated from the Keplerian elements,
- : Semi-major axis
- : Eccentricity
- : True anomaly
- : Inclination
- Ω : Longitude of the ascending node
In order to find the position and velocity vectors, we first obtain the parameter p and the distance from the focus, r.
( )
(3.1)
- : Specific angular momentum
- : Standard gravitational parameter
17
(3.2)
Figure 3.1 : Matrix Multiplications for the Initial State Vectors [15]
The above connections are used to initialize the spacecraft in their current orbit. The modification done by the user in the beginning, which is the angle and the altitude of the chase vehicle, modifies the semi-major axis and the true anomaly in these equations.
The position and velocity vectors are updated with every time step for each object in the scene. The time step is constant and 0.02 seconds but can be modified inside the project. The velocity is manipulated by the Newton’s law of universal gravitation by applying the current gravitation force from a dominant body or bodies. This force is equal to,
(3.3)
- is the gravitational constant (6.67430E-11 m3kg-1s-2),
- and are the masses of the bodies in context,
- is the distance between the masses
Before mentioning the trajectory-planning phase, we also need to be able to find the position and velocity of an object after a certain time. This is provided by methods inside the custom components that return new position and velocity vectors.
18
Figure 3.2 : Obtaining Position or Velocity Vectors
Figure 3.2 shows the simple iteration required for getting the state vectors at any given time for any given time.
By using the equations mentioned in Chapter 2 and the methods above, the rendezvous planning for a single Hohmann transfer can be performed. The first values to find are the current and final phase angles for the current wait time. The initial phase angle is the difference between their current mean anomalies. Even though the orbits are near circular, meaning the mean anomaly is very close to true anomaly, the simulation does all calculations depending on the mean motion of the spacecraft for a more precise phase calculation. This conversion requires the calculation of eccentric anomaly as well, which can be calculated from the following. ( ) ( ) ( )
(3.4) Where is the true anomaly. The mean anomaly from this eccentric anomaly can be obtained by,
( )
(3.5) Where is the eccentricity.
19 After finding the required phase angle, the wait time will be calculated from (2.10). However since this can return a value smaller than zero, we will run an iteration where 360 degrees is added to the initial phase angle until the wait time is non- negative. This could be modified into finding the wait time for the next rendezvous opportunities as well, hence the reason for the iteration. The burn time is also subtracted from the calculated time to be taken into account. By requesting the new positions and velocities of both spacecraft after the wait time, the maneuver nodes are placed on the visualized orbit. Now the maneuver can be applied.
Figure 3.3 : Applying the Calculated Maneuver
Keep in mind that the maneuver is applied to another placeholder object in the simulation so the original state of the spacecraft can is not lost.
The next step is a slight fine-tuning for the closest approach where the bodies will be simulated as if the engine applied its thrust over time. This fine-tuning repeatedly simulates as if the spacecraft started the maneuver earlier or later to check if a closer approach can be found. The main reason behind this optimization was to make up for the non-perfect circular orbits, which is not a huge problem but it definitely provides a closer approach to the target. Without this optimization, in most cases, the closest approach tends to be larger than 1 kilometer and varies greatly, yielding an inconsistent result.
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Figure 3.4 : Rendezvous Planning
The docking part utilizes the transform and mass components where the user input is fed to transformational and rotational functions. These add velocities and angular velocities in all three axes to the object, which is applied and updated in each time step. They are used in combination with the Unity’s transform component since they provide functions that handle the rotation of the object in a Unity scene. For the sake of simplicity, a button for zeroing the relative velocity and rotation to the target is implemented as well since the relative motion to the ISS is always known.
3.2 Graphical Interface
When the application is first opened, the user will be greeted with a panel on the left with everything else covered with a darker panel. This panel contains the custom parameters, which are the altitude and relative angle to the target of the starting orbit for the chaser and the engine thrust in delta-V per second. Hitting “Start” will initialize the spacecraft with the entered parameters. Information about the current session can also be found in this panel along with the controls.
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Figure 3.5 : Start and Custom Parameter Menu
After starting, the spacecraft will be positioned in their relative orbits. A blue placeholder for the Earth will be visible in the middle. The Earth can be shown with its real radius from the top right, focus panel by setting the toggle next to its name. Clicking on the buttons in this panel will lock the camera on the corresponding object.
Figure 3.6 : Focus Panel
The camera can be rotated by clicking with the mouse and dragging along the screen. The focused object can be zoomed in and out with the mouse wheel. The current scale in the camera frame can be seen at the bottom left as well.
Below that is the time panel, which has varying ways to modify the time. The time can be sped up by the multiplier buttons and the active time scale will be highlighted
22 green. It can be paused by either the button or pressing “Space”. The bottom row enables skipping time by different amounts. It is usually not necessary to use this panel except for skipping forward for waiting the time of flight, or when in approaching distance to fast forward in time.
Figure 3.7 : Time and Transfer Panels
The transfer right below the time panel is used for two operations. First, a time in seconds can be entered along with the desired altitude and the program will calculate the required delta-V for the Hohmann transfer required. The second operation is the rendezvous operation that can target the ISS, which is enabled by checking the box. Hitting “Calculate” will begin the process, where the optimal angle and delta-V is found for a close approach. This maneuver can later be applied automatically or the user can jump forward manually and try to initiate the burn manually. This will put the spacecraft in an intersecting orbit with the ISS and the user will have to apply the second delta-V to match the orbit of the target when in close approach.
When approaching the ISS, the distance and the relative speed to the target will be displayed on the ISS as shown. After the orbits are matched, the docking sequence is initiated by pressing “R”. This enables translational and rotational input with the keyboard. Using the keys “WASDQE” for translation and “IJKLUO” for rotation the user has to guide the spacecraft to the docking port of the ISS marked with a colored cross. Note that both the spacecraft and docking port indicators has to match rotations as well.
23 All delta-V uses are tracked and can be displayed from the main panel by pressing “Esc”. Information about the controls and the certain statistics of the current operation can also be found. If the positioning of the spacecraft is successful, the simulation will end the user can see how long the operation took and how much delta-V is applied. These two numbers not only depend on the initial parameters but also the user’s efficieny at docking and matching the speed of the target. As a reference the Soyuz spacecraft has a total delta-V capability of 390 m/s.
3.3 Example Scenario
For the example scenario a basic resupply crew transfer mission is chosen that features the Soyuz spacecraft. This is a mission with a duration between 6 hours and 2 days, depending on the launch window and the method of transfer. Because of the capabilities of the simulation it is assumed that the Soyuz is already in phasing orbit where the crew is waiting for the right angle to begin the Hohmann transfer. The initial parameters are set as followed.
Current Altitude: 300 km
Current Angle: -15 relative to target
Delta V per second: 5 m/s
Pressing the start button will place the Soyuz spacecraft behind the ISS by 15 degrees. Time flow will be paused by default and can be toggled on or off or sped up via the menu on the right. Since our objective is to meet with the ISS, we mark the ISS from the right and start the calculation. The program will find the required phase angle depending on the current altitude of the orbit and try to optimize the approach distance by acutally simulating both bodies forward in time and taking burn time into account.
After the iteration, the program will place certain nodes on both the chaser and target orbits. These indicate the start and end of the manuever as well as the position of both spacecraft after the maneuver and transfer time. From this point, the user can either fast-forward to the maneuver and execute the burn manually or let the program handle it. Manually performing the maneuver may put the spacecraft in a different orbit than the calculated one depending on how well the timing is done.
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Figure 3.8 : Spacecraft Calculating for the Next Rendezvous
Assuming the maneuver was done successfully, the user is now at the transfer stage, where the spacecraft will follow the Hohmann transfer orbit. After a while, an approach indicator will appear on the ISS, showing the distance and relative speed to the station’s docking port. When the distance is suitable, the user can perform the second burn, which will match the orbits of the spacecraft, until the relative speed is low enough to transition into docking mode, disabling the main engine.
Figure 3.9 : The Spacecraft in Docking Mode, Parked Near the ISS
In order to complete the simulation, the spacecraft has to be guided towards the ISS carefully. When closer than 200 meters, the docking port should now be visible as the next target. Matching the orientation of the docking mechanism and slowly touching the docking port of the station will complete the simulation. The resulting statistics contains the performed delta-V in total, time and the relative speed at the collision.
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4. CONCLUSION
The approach to a development of a rendezvous and docking simulation was covered in this thesis. It touched the transfer methods considered for the implementation and the docking protocols of the ISS while letting the user have some control over the starting variables.
I would, once more, like to point out that this simulation considers near-perfect circular orbits for its calculations and does not find the most efficient ways of transfer to the ISS but merely visualizes it. It is still open to upgrades and further development. There were features which I wanted to implement but couldn’t because of time constraints. For example the bi-elliptic transfer could be implented as either an option or another window opportunity. The targets or main bodies could also be open to customization, simulating rendezvous around different celestial bodies, with different spacecraft. A more detailed docking sequence can also be considered since the physics of the spacecraft in close proximity to the station are handled just adequately. The program should also have a friendlier user interface which is a more trivial task but is still important for the ease of use.
A much more advanced version could account for debris avoidance and a docking navigation helper, presenting the discussed guidelines to the user and ranking their performance depending on how they handle the mission manually.
The source codes for both Matlab scripts and the Unity project is available online along with the latest version.
26 REFERENCES
Curtis H. D. (2005). Orbital Mechanics for Engineering Students, Embry-Riddle Aeronautical University, Daytona Beach, Florida
Belbruno, E. (2004). Capture Dynamics and Chaotic Motions in Celestial Mechanics: With Applications to the Construction of Low Energy Transfers, Princeton University Press
Gobetz, F.W., Doll, J.R. (1969). A Surver of Impulsive Trajectories, AIAA Journal, American Institue of Aeronautics and Astronautics
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[2] A Journey to the Moon
[3] Hansen, J. R. (1992). The Rendezvous That Was Almost Missed, retrieved from https://www.nasa.gov
[4] Soyuz 4
[5] Apollo-Soyuz
[6] Mir FAQs - Facts and History
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[8] Treat, J., Scalamogna A., Conant E. (n.d.) Hubble Timeline, date retrieved 04.04.2021, from https://www.nationalgeographic.com
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[10] Current Happenings in Orbit
[11] Soyuz Rendezvous and Docking Explained
[12] Maneuvering In Space
27 [13] Yazhong, L. Jin, Z. Guojin T. (2014) Survey of Orbital Dynamics and Control of Space Rendezvous, Chinese Journal of Aeronautics
[14] International Rendezvous System Interoperability Standards (2019)
[15] Peet, M, M. (n,d) Spacecraft Dynamics and Control
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