MAE 525 Spacecraft Dynamics & Control Guidelines. Engineering Essay

Satellite Formation Flying Control via Countercurrent Exchange
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Satellite Formation Flying Control via Countercurrent Exchange
Satellite formation flying refers to the notion of having various satellites working together in a group to attain the goal of one, larger, normally more expensive satellite. Satellites are objects placed in space to gather information and facilitate communication between two or more objects through space (Damjanov, 2017). In this space system, multiple elements fly in short relative distances in an orbit where they communicate, and interact to achieve a common goal. Coordination of these smaller satellites has several benefits such as simpler designs that make it easy to repair, faster build times, cheap to replace, creating a higher redundancy, results into an unexpected high resolution for better image quality, and the ability to observe research targets from different angles and at various times. Besides, the system also allows for concurrency between satellites, and the scalability and flexibility in design and deployment of a system. These features have made satellites suitable tools for astronomy, communication and meteorological purposes. The formations have found extremely useful applications in aerial vehicles, with their applications being associated with improve fuel economy, enhanced efficiency in air traffic control, and facilitation of corporative task allocation. The precise control of formation flight in the case of space vehicles will also enable future large-aperture space telescopes, and the robotic assembly of space structures.
The main characteristic of a satellite formation flying is that it is composed of a small number of satellites that communicate between each other. The system operates under the concept of formation flying, which entails the distribution of the functionality of bigger spacecraft into smaller, less expensive ones to reduce the complications that may face the larger systems. The smaller satellites are organized around a centralized control and are governed by a mother-daughter relationship. Just like in a mother-daughter relationship, where a mother is expected to know what is best for her children and has command over them, the centralized control understands its satellites and controls their motion. The cluster also operates under the leader-follower relationship, in which the leader moves anywhere he or she wants and the followers pursue him. Similarly, the central control system moves anywhere and the satellites have to follow. The motion occurs along predefined trajectories and there is a constant communication between the satellites. There are various forms of trajectories which include the A-train formation, the relative circular orbit, the projected circular orbit, and the docking trajectories, among others. The A-train formation comprises of five satellites that flying in close proximity. Satellites are put in this form to allow scientists obtain a better understanding of the key parameters relevant to hurricanes as well as understand climate change information. This formation also enables simultaneous coordinated measurements where data from different satellites can be used to obtain more comprehensive information regarding the atmospheric components. Its appearance is as shown in the figure below:
A-train formation flying
A relative circular orbit, on its part, is characterized by satellites who path traces a circle. Under such a system, the gravity of a central body dominates the orbits around it, making them elliptical in nature. The ellipse of zero eccentricity is a special case of this circular orbit. The velocity of satellites under this formation are calculated using the formula for the velocity of a body located at a distance r from the center of the mass and is given by the formula below.
v2r = GMr2
Hence,
v=GMr
Where G is the gravitational force, which is equal to 6.67384 x 10-11 m3/(kg.s2)
Mass Exchange Dynamics
The concept of mass dynamics was born out of the fact that it is possible to control satellite formation flying without spending on fuel because of the close relative motion. Under the concept of momentum exchange, one satellite ejects a separable mass and creates momentum. The adjacent satellite captures the mass and redirects it back. The mass transfer will be initiated by a laser propulsion mechanism where a laser system, which will be ground-based, will emit a beam that will excite the mass of photons in space to form a propulsion system. This photon-propelled system was invented by Eugene Sanger in collaboration with Gyorgy Marx, a Hungarian physicist. There are two main ways that the system can transfer momentum to a spacecraft. One of the ways is through photon radiation pressure, which drives the transfer of the momentum. This approach has been used in solar sails and laser sails. The second approach involves the use of laser to assist in ejecting mass from the spacecraft in a manner that resembles a conventional rocket. In this process, the laser beans are ejected from the source at a very high velocity towards the target object. The beams then strike the surface at an angle, and based on the principle of energy transfer, the energy contained in the beam is transferred to the mass, ejecting it from the surface of the object. While this approach is the most popular, its main drawback has been its limitations in final spacecraft velocities by the rocket equation.
The use of momentum from layers was initially investigated by Bae for its relative control of satellites. He fronted a proposal to send a laser beam to and fro between two space systems. To achieve the back and forth movement, he proposed the use of mirrors. The transfer of momentum generates a repulsive force. Tragesser further investigated this concept, where he proposed the creation of a repulsive force using a constant stream of mass moving between the satellites. His study addresses the aspect of equilibrium positions that exist between the rotating orbital frames during the momentum exchange. Implementation of the momentum interchange was first proposed by Joslyn and Ketsdever, based on the liquid droplet streams to generate the force. According to this approach, every satellite has a liquid film surrounding it and as it rotates, it generates a droplet stream, which is then projected through space. The satellite in the recipient end captures the droplet stream and returns it to the one that produced it at first. However, the main challenge about this perspective is the presence of the perturbations in the orbit that are bound to deflect the droplets from their normal path.
The most essential part of this concept involves the mechanism of ejection or mass collection. In this regard, the mechanism may be a robotic arm that is installed on the satellites. Examples of these include the satellite Cather used to clean up the space debris. The systems are connected to the satellites at various lengths and serve as collectors. Their movement is facilitated by electrical drives or magnetic forces that do not require any fuel. Their rate of rotation can be adjusted by adjusting the lengths of these a, which intern affects the speed of ejection of the mass. The scheme below demonstrates the mass exchange control concept.
* (II)
(III) (IV)
Image (I) shows the mass prior to the mass exchange. Image (II) shows how the first satellite ejects the mass. The third image (III) shows the mass hitting the second satellite and the final diagram (IV) shows the motion following the mass transfer. It is imperative to explain the tolerable errors in position and velocity at the moment of ejection to enable this exchange of mass to occur in such a manner that the mass transferred is not lost. The errors that may occur courtesy of control inaccuracies are further investigated and eliminated.
Statement of the Problem
The topic of flying control has been extensively discussed in recent times, given their numerous applications both in the transport sector and in the telecommunication sector. In the transport sector, the formation of flying control has been praised for several advantages that have included but not limited to saving fuel, improved efficiency in controlling traffic, and the allocation of cooperative tasks. In the communication sector, the formation flying control has facilitated the operation of large space telescopes, space interferometers, and the robotic assembly of space structures. These formation flying missions require a relative motion control. Recent scientific research has focused on control approaches that do not rely on fuel (Hartjes, Visser & van Hellenberg Hubar, 2019). The approaches that have been assessed have been based on the drag effect of the atmosphere, the magnetic effect, the electrostatic effect and the Lorentz forces. This paper seeks to explore a formation flying control technique that depends on the exchange of mass that may occur between any two satellites systems. This paper studies the feasibility of this approach from a mathematical perspective. For clarity purposes in using this perspective, we shall consider the linearized Hill-Clohessy- Wiltshire computations that explain the comparative movement between two space systems, in which both satellites move in an orbit (Djojodihardjo, 2014). The equations will take the following form:
ẍ = - 2nẑ
ӱ = -n2y
ż = 2nẋ + 3n2z (1)
The alphabetical letters x, y, and z represent the coordinates of the secondary space system relative to the orbital orientation Oxyz. The center of mass of the main satellite is represented by letter O and its radius as r0. The angular velocity for the main satellite will be obtained using the formula
The center of the satellite is connected to the center of the earth via axis Oz, with a normal vector that joins the orbit to the plane marked as Oy. The product of vectors Oy and Oz give Ox. The layout of the structure is as shown in the figure below.
The above equation will be solved using the following expression.
X = C4 – 3C1nt + 2C2cosnt – 2C3sinnt
Y = C5sinnt + C6cosnt,
Z = 2C1 + C2sinnt + C3cosnt (2)
In the above expressions, C1, C2, C3, C4, C5, C6, are all constants given at the time t0 = 0. Their respective values will be given by the following equations.
C1=2z0+ ẋ(0)n , if C1 = 0 (relative trajectory closed)
C2= ż(0)n
C3= -3z0-2ẋ(0)n
C4=x0- 2ż(0)n
C5= ẏ(0)n
C6=y(0) (3)
Drawing from equation (2) above, the value of C1 accounts for the comparative drift of the space systems. In this case, the magnitude of the swinging of the trajectories along the x axis and the z axis is defined byC22+C32, while the magnitude of the movement along axis y will be given byC52+C62. The shift along the track direction will be represented by C4. All these coefficients represent the magnitude of the relative course of motion. Therefore, it is appropriate to apply these figures in the study of relative motion.
The Boundary Value Challenge
Given a boundary value predicament, where one is expected to calculate the expected velocity of the mass, the first step would be to determine the certain relative velocity at a static point of the trajectory where the mass splits from the secondary satellite. Eventually, the mass will affect the primary satellite, which implies that the trajectory of the mass has to cross the primary satellite location. Solving the problem will require derivation of the mass separation velocity, which is determined as follows.
Let time t0 = 0, then based on the section with coordinates
x0 = x(0); y0 = y(0); z0 = z(0) (4)
The velocity of the mass mb located at the deputy satellite is separated with velocity
ẋb = ẋ0 + ∆ẋ; ẏ = ẏ0 + ∆ẏ; zb = ż0 + ∆ż (5)
The relative velocity components ∆ẋ, ∆ẏ, ∆ż, of the mass that leaves the secondary satellite located at ẋ0, ẏ0, ż0 represent the components of the speed of the secondary satellite, whereas the speed of the mass in the orbital orientation are represented by components ẋb, ẏb, żb. Equations (4) and (5) highlighting the initial conditions explain the relative course of the mass in totality and it is possible to compute constants (3).
At a given time t1, the mass should cross the location of the primary satellite. Given that the orbital orientation structure is linked to the primary satellite, it is located in the starting point of the frame, which means that;
x1=xt1=0, y1=yt1=0, z1=zt1=0
Therefore, it is possible to derive an expression that establishes the relative speed by which the mass leaves the secondary satellite, resulting to the mass impacting the chief satellite. Substitution of the original conditions captured in equations (4) and (5) to (3), and applying the second equation (2), incorporating the boundary conditions specified in equation (6) results into three equations for the three undetermined variables. Solving the equations logically produces the following values for the relative speed of departure ∆ẋ, ∆ẏ, ∆ż;
∆ẋ= - x0A-nx0sinnt1-z0(14n-14ncosnt1-6n2t1sin⁡(nt1)A,
∆ẏ=-ẏ0-y0ncot(nt1),
∆ż=-ż0A+x0(2n-2ncosnt1+z0(3n2t1cosnt1-4nsinnt1)A (7)
Where A=8cosnt1+3nt1sinnt1-8
In this case, A should not be equal to zero and in cases where it is equal to zero, then the boundary value question cannot be deciphered, and hence impossible to influence the primary satellite by mass. Besides, if nt1=πk, k=0,1,2… and y0 is not equal to zero simultaneously, then ∆ẏ is nonexistent and it is impossible to effect the mass exchange. Therefore, it is possible to use equation (7) to derive the required relative mass velocity by explaining the relative trajectory of the formation flying, the initial time t0 when the mass separated, and the time of impact t1.
Variation of the comparative course that the secondary satellite assumes following the mass exchange
Following a single mass exchange, it is possible to determine the change in trajectory of the deputy satellite. Drawing from motion equations (2), the changes in trajectory affect values C 1-C6, which makes it rational to establish their variations following the mass change.
According to the law of momentum conservation, a mass separation from the secondary space system with relative speed defined by equation (7) is accompanied by a variation in the speed of the secondary satellite. The initiating force of thrust is directed at the center of mass of the satellite and does not have any impact on the angular motion. Therefore, we take md to represent the mass of the secondary satellite minus the dissociated mass. We then let the separated mass be represented by mb. The speed of the secondary satellite before the impact is v0, whereas the velocity of the mass separation is ∆v, the equation below can then be used to calculate the final velocity v1 of the deputy satellite.
v1=v0-∆vmbmd (8)
Equations (4) and (8) highlight the initial conditions that specify the course of the secondry satellite following separation of the mass. When the impact on the chief satellite is inelastic, its velocity varies in accordance with the following equation
vc=mbmc+mbvb(t1) (9)
Where mc represents the mass of the chief satellite, whereas vb(t1) represents the mas speed at time t1 just before the effect. The equations operate on the assumption that the primary satellite attains the force employed to its center of mass. The trajectory of the chief satellite is then defined by the initial conditions highlighted in equations (6) and (9). Ultimately, the mass exchange between the secondary and the primary satellites makes the two satellites to travel along the altered courses within the frame of reference. The courses followed by the satellites can...