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Samara State Aerospace University (SSAU)

Modern methods of analysis of the dynamics and motion control of space tether systems

Practical lessons

Yuryi Zabolotnov Mikhailovich, yumz@yandex.ru

Oleg Naumov Nikolaevich, oleg_naumov63@mail.ru

Samara 2015

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Topics of practical lessons

Lesson 1. It is construction of mathematical models of the motion of space tether systems (STS) in the mobile orbital coordinate systemLesson 2. It is construction of mathematical models of controlled motion STS geocentric fixed coordinate system

Lesson 3. It is calculation maneuver descent payload to orbit using space tether systems

Lesson 4. It is calculation maneuver launch small satellites into a higher orbit by a space tether systems

Lesson 5. It is construction nominal program deployment STS final vertical position

Lesson 6. It is construction nominal program deployment STS with a deviation from the vertical in the end position

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1. It is construction of mathematical models of the motion of the STS in the mobile orbital coordinate system

Method of construction - Lagrange formalism

(1)

where - kinetic energy, - potential energy,

and - generalized coordinates and velocities,

- degree of freedom, - nonpotential generalized forces.

- time,

Lagrange equations

Kinetic and potential energy must be expressed in terms of generalized coordinates and velocity to derive the equations of motion from (1).

(2)

Example choice of generalized coordinates

Fig.1 Coordinate systems (CS)

Generalized coordinates

- orbital movable CS,

- tether CS

- tether length,

here

- tether  deflection anglesfrom the vertical,

- the center of mass of the STS

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Example of calculating the kinetic energy potential

Assumption: SC mass is much larger than the mass of the cargo

Kinetic energy :(3)

where

- the angular velocity of the spacecraft in a circular orbit.

The potential energy of the gravitational field in the center of Newton :

(4)

where

Differentiation :

- Earth's gravitational parameter.5

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The equations of motion of the STS

where - the force of tether tension.

The force of the tether tension is determined based on the selected law deploying STS.

(5)

2. It is construction of mathematical models of controlled

motion STS geocentric fixed coordinate system

Model: two material points connected by an elastic-sided communication

The equations of motion : (6)

Gravitational force : where - weight endpoints.

The tension tether : (7)

where

- modulus of elasticity, - cross-sectional area tether.

Aerodynamic force :

where

(8)

- coefficient of resistance, - density,

- the characteristic area, - velocity relative to the atmosphere.7

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Simulation of the motion of the STS

It solve the initial problem for a system of ordinarydifferential Equations (DE)

(9)

where - the state vector of the STS,

- vector function of the right sides DE.

An example of a method of numerical integration :

(10)

where

- integration step.

Local integration error :

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3. It is calculation maneuver descent payload to orbit using space tether systems

Fig. 2 Deploying STSpayload during the descent from orbit

Fig. 3 Addition of velocities in the the cargo compartment of the spacecraft

The algorithm for calculating the descent maneuver payloads to orbit with the help of the STS

- the final tether length.

The initial speed of the separation of the cargo : (11)

Portable velocity : (12)

where

relative velocity : (13)

where

Module initial velocity : (14)

where

Effective deorbit burn

(15)

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The algorithm for calculating the descent maneuver payloads to orbit with the help of the STS

Increased braking burn due to the deviation from the vertical tether  

(16)

Fluctuations in orbit at STS by are described

Equation (16) has an integral energy :

(17)

If the tether is bent at an angle at the end of deployment STS, then

(18)

Then, at the time of the passage of tether , an additional verticaldeorbit burn

(19)

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.

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4. It is calculation maneuver launch small satellites into a higher orbit with the help of the STS

schemes launch

Fig. 4 Starting a small spacecraft into an elliptical orbit

Fig. 5 Starting a small spacecraft into a circular orbit

The algorithm for calculating launch small satellites into a higher orbit

Fig. 6 Addition of speedswhen you start a small spacecraft to a

higher orbit

Calculation of the parameters when you start a small spacecraft into an elliptical

orbit

The initial velocity the separation of the STS

(20)

where

The initial angle of the trajectory of the separation of the STS

(21)

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The algorithm for calculating launch small satellites into a higher orbit

Calculation of parameters of an elliptical orbit:

(22)

where

(23)

(24)

Effective deorbit burn :

The equation of the orbit :

where - orbital parameter, - eccentricity,

- true anomaly.

Calculation formulas :

where

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The algorithm for calculating launch small satellites into a higher orbit

Calculation of parameters circular orbit

(25)

where

(26)

(27)

Effective deorbit burn :

The initial velocity :

The initial flight path angle :

The other of the formula coincides with the elliptic orbit.15

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The trajectory of the spacecraft relative to the base of small satellites

Fig. 7 The trajectory of a small spacecraft with the launch of an elliptical orbit

5. It is construction nominal program deployment STS to final vertical position

Equation (5) stored in the mobile orbital coordinate system and shown on the slide 6 are used to build the program deployment.It is to build the program uses the equation of motion of the STS in the orbital plane, then the equation (5) takes the form

(28)

Formulation of the problem: it is necessary to find the law tension control tether the condition of the existence of asymptotically stable equilibrium positionin the endpoint deployment STS.

The final boundary conditions :

(29)

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It is solution of the problem of constructing a nominal STS program deployment to the final vertical position

Deployment program STS obtained from (29) in the form

(30)

where - options program.

When the position of equilibrium (29) is asymptotically stable.

Trajectories for development of the STS

Fig. 8 When Fig. 9 When 18

6. It is construction nominal program deployment STS with a deviation from the vertical in the end position

The equation of plane motion (28) used to build the program.

Deploying STS is split into two phases : 1) Deployment of STS final vertical position - law (30);2) Deployment of the STS with a deviation from the vertical in the final position.

In the second phase deployment program is used close to the relay

(31)

where - switching time s onto

- smoothing parameter relay law.

Program parameters are chosen from the condition of the implementation of the boundary conditions :

(32)19

It is example of constructing software deployment STS with the deviation from the vertical in the end position

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Fig.10 The program deployment STSconsisting of two stages

Fig.11 The trajectory of the deployment of the STS

The first phase of 3 km, while 6,000 s

The second stage of 27 kilometers, while in 2155 s,