1 Samara State Aerospace University (SSAU) Modern methods of analysis of the dynamics and motion control of space tether systems Practical lessons Yuryi Zabolotnov Mikhailovich, Oleg Naumov Nikolaevich, Samara 2015

2 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 system Lesson 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

3 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 angles from the vertical, - the center of mass of the STS 4

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

6 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

8 Simulation of the motion of the STS It solve the initial problem for a system of ordinary differential 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 :

9 3. It is calculation maneuver descent payload to orbit using space tether systems Fig. 2 Deploying STS payload 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) 10

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 byare described Equation (16) has an integral energy : (17) If the tether is bent at an angleat the end of deployment STS, then (18) Then, at the time of the passage of tether, an additional vertical deorbit burn (19) 11

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 speeds when 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) 13

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 14

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

16 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 position in the endpoint deployment STS. The final boundary conditions : (29) 17

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. Whenthe position of equilibrium (29) is asymptotically stable. Trajectories for development of the STS Fig. 8 WhenFig. 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 20 Fig.10 The program deployment STS consisting 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,