Samara State Aerospace University

Attitude dynamics and controlstabilization of spacecraft and gyrostat-satellites

Anton V Doroshindoraninboxru

Samara 2015

Outline

bull Part 1 - Introduction to attitude dynamics and control

bull Part 2 - Research results

- the magnetized dual-spin spacecraft attitude motion

- the regular and chaotic dynamics of dual-spin spacecraft

- the multi-rotor system and multi-spin spacecraft dynamics

bull Conclusion

1

Part 1 - Introduction to Attitude

Dynamics and Control

Most of the spacecraft have instruments or antennas that must be pointed

in specific directions

ndash The Hubble must point its main telescope

ndash Communications satellites must point their antennas

The orientation of the spacecraft in the space is called its attitude

To control the attitude the spacecraft operators must have the ability to

ndash Determine the current attitude (sensorshellip)

ndash Determine the error between the current and desired attitudes

ndash Apply torques to remove the error (with the help of actuatorshellip)

2

The Spacecraft Attitude Stabilizationbull mdash Spin Stabilization methodsbull mdash Gravity Gradient Stabilization methodsbull mdash Magnetic Stabilization methodsbull mdash Aerodynamic Stabilization methods

Actuatorsbull mdash Reaction Wheel Assemblies (RWAs)bull mdash Control Moment Gyros (CMGs)bull mdash Thrusters

3

Spin stabilization4

The TopSpin stabilized spacecraft

Spin-stabilization is a method of SC stabilizing in a fixed orientation using rotational motion around SC axis (usually symmetry axis ) ndash ldquothe gyroscopic effectrdquo

Spin StabilizationDual-Spin Spacecraft

4

TACSAT I was the largest and most powerful communications satellite at the time when it was launched into synchronous orbit by a Titan IIIC booster 9 February 1969 from Cape Canaveral Florida

TACSAT I The antenna is the platform and is intended to point continuously at the Earth spinning at one evolution per orbit The cylindrical body is the rotor providing gyroscopic stability through its 60 RPM spin

Spin StabilizationDual-Spin Spacecraft

4

The dual-spin construction-scheme is quite useful in the practice during all history of space flights realization and it is possible to present some examples of the DSSC which was used in real space-programs (most of them are communications satellites and observing geostationary satellites)

- This is long-continued and well successful project ldquoIntelsatrdquo (the Intelsat II series of satellites first launched in 1966) including 8th generation of geostationary communications satellites and Intelsat VI (1991) designed and built by Hughes Aircraft Company - The ldquoMeteosatrdquo-project by European Space Research Organization (initiated with Meteosat-1 in 1977 and operated until 2007 with

Meteosat-7) also used dual-spin configuration spacecraft - Spin-stabilized spacecraft with mechanically despun antennas was applied in the framework of GEOTAIL (a collaborative mission

of Japan JAXAISAS and NASA within the program ldquoInternational Solar-Terrestrial Physicsrdquo) launched in 1992 the GEOTAIL spacecraft and its payload continue to operate in 2013

- Analogously the construction scheme with despun antenna was selected for Chinese communications satellites DFH-2 (STW-3 1988 STW-4 1988 STW-55 1990)

- Well-known Galileo missionrsquos spacecraft (the fifth spacecraft to visit Jupiter launched on October 19 1989) was designed by dual-spin scheme

- Of course we need to indicate one of the worlds most-purchased commercial communications satellite models such as Hughes Boeing HS-376 BSS-376 (for example Satellite Business Systems with projects SBS 1 2 3 4 5 6 HGS 5 etc) they have spun section containing propulsion system solar drums and despun section containing the satellites communications payload and antennas

- Also very popular and versatile dual-spin models are Hughes HS-381 (Leasat project) HS-389 (Intelsat project) HS-393 (JCSat project)

The DSSC usually is used for the attitude stabilization by partial twist method only one of the DSSCs coaxial bodies (the laquorotorraquo-body) has rotation at the laquoquiescenceraquo of the second body (the laquoplatformraquo-body) ndash it allows to place into the laquoplatformraquo-body some exploratory equipment and to perform of space-mission tests without rotational disturbances

Part 1 - Introduction to Attitude

Dynamics and Control

Most of the spacecraft have instruments or antennas that must be pointed

in specific directions

ndash The Hubble must point its main telescope

ndash Communications satellites must point their antennas

The orientation of the spacecraft in the space is called its attitude

To control the attitude the spacecraft operators must have the ability to

ndash Determine the current attitude (sensorshellip)

ndash Determine the error between the current and desired attitudes

ndash Apply torques to remove the error (with the help of actuatorshellip)

2

The Spacecraft Attitude Stabilizationbull mdash Spin Stabilization methodsbull mdash Gravity Gradient Stabilization methodsbull mdash Magnetic Stabilization methodsbull mdash Aerodynamic Stabilization methods

Actuatorsbull mdash Reaction Wheel Assemblies (RWAs)bull mdash Control Moment Gyros (CMGs)bull mdash Thrusters

3

Spin stabilization4

The TopSpin stabilized spacecraft

Spin-stabilization is a method of SC stabilizing in a fixed orientation using rotational motion around SC axis (usually symmetry axis ) ndash ldquothe gyroscopic effectrdquo

Spin StabilizationDual-Spin Spacecraft

4

TACSAT I was the largest and most powerful communications satellite at the time when it was launched into synchronous orbit by a Titan IIIC booster 9 February 1969 from Cape Canaveral Florida

TACSAT I The antenna is the platform and is intended to point continuously at the Earth spinning at one evolution per orbit The cylindrical body is the rotor providing gyroscopic stability through its 60 RPM spin

Spin StabilizationDual-Spin Spacecraft

4

The dual-spin construction-scheme is quite useful in the practice during all history of space flights realization and it is possible to present some examples of the DSSC which was used in real space-programs (most of them are communications satellites and observing geostationary satellites)

- This is long-continued and well successful project ldquoIntelsatrdquo (the Intelsat II series of satellites first launched in 1966) including 8th generation of geostationary communications satellites and Intelsat VI (1991) designed and built by Hughes Aircraft Company - The ldquoMeteosatrdquo-project by European Space Research Organization (initiated with Meteosat-1 in 1977 and operated until 2007 with

Meteosat-7) also used dual-spin configuration spacecraft - Spin-stabilized spacecraft with mechanically despun antennas was applied in the framework of GEOTAIL (a collaborative mission

of Japan JAXAISAS and NASA within the program ldquoInternational Solar-Terrestrial Physicsrdquo) launched in 1992 the GEOTAIL spacecraft and its payload continue to operate in 2013

- Analogously the construction scheme with despun antenna was selected for Chinese communications satellites DFH-2 (STW-3 1988 STW-4 1988 STW-55 1990)

- Well-known Galileo missionrsquos spacecraft (the fifth spacecraft to visit Jupiter launched on October 19 1989) was designed by dual-spin scheme

- Of course we need to indicate one of the worlds most-purchased commercial communications satellite models such as Hughes Boeing HS-376 BSS-376 (for example Satellite Business Systems with projects SBS 1 2 3 4 5 6 HGS 5 etc) they have spun section containing propulsion system solar drums and despun section containing the satellites communications payload and antennas

- Also very popular and versatile dual-spin models are Hughes HS-381 (Leasat project) HS-389 (Intelsat project) HS-393 (JCSat project)

The DSSC usually is used for the attitude stabilization by partial twist method only one of the DSSCs coaxial bodies (the laquorotorraquo-body) has rotation at the laquoquiescenceraquo of the second body (the laquoplatformraquo-body) ndash it allows to place into the laquoplatformraquo-body some exploratory equipment and to perform of space-mission tests without rotational disturbances

The Spacecraft Attitude Stabilizationbull mdash Spin Stabilization methodsbull mdash Gravity Gradient Stabilization methodsbull mdash Magnetic Stabilization methodsbull mdash Aerodynamic Stabilization methods

Actuatorsbull mdash Reaction Wheel Assemblies (RWAs)bull mdash Control Moment Gyros (CMGs)bull mdash Thrusters

3

Spin stabilization4

The TopSpin stabilized spacecraft

Spin-stabilization is a method of SC stabilizing in a fixed orientation using rotational motion around SC axis (usually symmetry axis ) ndash ldquothe gyroscopic effectrdquo

Spin StabilizationDual-Spin Spacecraft

4

TACSAT I was the largest and most powerful communications satellite at the time when it was launched into synchronous orbit by a Titan IIIC booster 9 February 1969 from Cape Canaveral Florida

TACSAT I The antenna is the platform and is intended to point continuously at the Earth spinning at one evolution per orbit The cylindrical body is the rotor providing gyroscopic stability through its 60 RPM spin

Spin StabilizationDual-Spin Spacecraft

4

The dual-spin construction-scheme is quite useful in the practice during all history of space flights realization and it is possible to present some examples of the DSSC which was used in real space-programs (most of them are communications satellites and observing geostationary satellites)

- This is long-continued and well successful project ldquoIntelsatrdquo (the Intelsat II series of satellites first launched in 1966) including 8th generation of geostationary communications satellites and Intelsat VI (1991) designed and built by Hughes Aircraft Company - The ldquoMeteosatrdquo-project by European Space Research Organization (initiated with Meteosat-1 in 1977 and operated until 2007 with

Meteosat-7) also used dual-spin configuration spacecraft - Spin-stabilized spacecraft with mechanically despun antennas was applied in the framework of GEOTAIL (a collaborative mission

of Japan JAXAISAS and NASA within the program ldquoInternational Solar-Terrestrial Physicsrdquo) launched in 1992 the GEOTAIL spacecraft and its payload continue to operate in 2013

- Analogously the construction scheme with despun antenna was selected for Chinese communications satellites DFH-2 (STW-3 1988 STW-4 1988 STW-55 1990)

- Well-known Galileo missionrsquos spacecraft (the fifth spacecraft to visit Jupiter launched on October 19 1989) was designed by dual-spin scheme

- Of course we need to indicate one of the worlds most-purchased commercial communications satellite models such as Hughes Boeing HS-376 BSS-376 (for example Satellite Business Systems with projects SBS 1 2 3 4 5 6 HGS 5 etc) they have spun section containing propulsion system solar drums and despun section containing the satellites communications payload and antennas

- Also very popular and versatile dual-spin models are Hughes HS-381 (Leasat project) HS-389 (Intelsat project) HS-393 (JCSat project)

The DSSC usually is used for the attitude stabilization by partial twist method only one of the DSSCs coaxial bodies (the laquorotorraquo-body) has rotation at the laquoquiescenceraquo of the second body (the laquoplatformraquo-body) ndash it allows to place into the laquoplatformraquo-body some exploratory equipment and to perform of space-mission tests without rotational disturbances

Spin stabilization4

The TopSpin stabilized spacecraft

Spin-stabilization is a method of SC stabilizing in a fixed orientation using rotational motion around SC axis (usually symmetry axis ) ndash ldquothe gyroscopic effectrdquo

Spin StabilizationDual-Spin Spacecraft

4

TACSAT I was the largest and most powerful communications satellite at the time when it was launched into synchronous orbit by a Titan IIIC booster 9 February 1969 from Cape Canaveral Florida

TACSAT I The antenna is the platform and is intended to point continuously at the Earth spinning at one evolution per orbit The cylindrical body is the rotor providing gyroscopic stability through its 60 RPM spin

Spin StabilizationDual-Spin Spacecraft

4

The dual-spin construction-scheme is quite useful in the practice during all history of space flights realization and it is possible to present some examples of the DSSC which was used in real space-programs (most of them are communications satellites and observing geostationary satellites)

- This is long-continued and well successful project ldquoIntelsatrdquo (the Intelsat II series of satellites first launched in 1966) including 8th generation of geostationary communications satellites and Intelsat VI (1991) designed and built by Hughes Aircraft Company - The ldquoMeteosatrdquo-project by European Space Research Organization (initiated with Meteosat-1 in 1977 and operated until 2007 with

Meteosat-7) also used dual-spin configuration spacecraft - Spin-stabilized spacecraft with mechanically despun antennas was applied in the framework of GEOTAIL (a collaborative mission

of Japan JAXAISAS and NASA within the program ldquoInternational Solar-Terrestrial Physicsrdquo) launched in 1992 the GEOTAIL spacecraft and its payload continue to operate in 2013

- Analogously the construction scheme with despun antenna was selected for Chinese communications satellites DFH-2 (STW-3 1988 STW-4 1988 STW-55 1990)

- Well-known Galileo missionrsquos spacecraft (the fifth spacecraft to visit Jupiter launched on October 19 1989) was designed by dual-spin scheme

- Of course we need to indicate one of the worlds most-purchased commercial communications satellite models such as Hughes Boeing HS-376 BSS-376 (for example Satellite Business Systems with projects SBS 1 2 3 4 5 6 HGS 5 etc) they have spun section containing propulsion system solar drums and despun section containing the satellites communications payload and antennas

- Also very popular and versatile dual-spin models are Hughes HS-381 (Leasat project) HS-389 (Intelsat project) HS-393 (JCSat project)

The DSSC usually is used for the attitude stabilization by partial twist method only one of the DSSCs coaxial bodies (the laquorotorraquo-body) has rotation at the laquoquiescenceraquo of the second body (the laquoplatformraquo-body) ndash it allows to place into the laquoplatformraquo-body some exploratory equipment and to perform of space-mission tests without rotational disturbances

Spin StabilizationDual-Spin Spacecraft

4

TACSAT I was the largest and most powerful communications satellite at the time when it was launched into synchronous orbit by a Titan IIIC booster 9 February 1969 from Cape Canaveral Florida

TACSAT I The antenna is the platform and is intended to point continuously at the Earth spinning at one evolution per orbit The cylindrical body is the rotor providing gyroscopic stability through its 60 RPM spin

Spin StabilizationDual-Spin Spacecraft

4

The dual-spin construction-scheme is quite useful in the practice during all history of space flights realization and it is possible to present some examples of the DSSC which was used in real space-programs (most of them are communications satellites and observing geostationary satellites)

- This is long-continued and well successful project ldquoIntelsatrdquo (the Intelsat II series of satellites first launched in 1966) including 8th generation of geostationary communications satellites and Intelsat VI (1991) designed and built by Hughes Aircraft Company - The ldquoMeteosatrdquo-project by European Space Research Organization (initiated with Meteosat-1 in 1977 and operated until 2007 with

Meteosat-7) also used dual-spin configuration spacecraft - Spin-stabilized spacecraft with mechanically despun antennas was applied in the framework of GEOTAIL (a collaborative mission

of Japan JAXAISAS and NASA within the program ldquoInternational Solar-Terrestrial Physicsrdquo) launched in 1992 the GEOTAIL spacecraft and its payload continue to operate in 2013

- Analogously the construction scheme with despun antenna was selected for Chinese communications satellites DFH-2 (STW-3 1988 STW-4 1988 STW-55 1990)

- Well-known Galileo missionrsquos spacecraft (the fifth spacecraft to visit Jupiter launched on October 19 1989) was designed by dual-spin scheme

- Of course we need to indicate one of the worlds most-purchased commercial communications satellite models such as Hughes Boeing HS-376 BSS-376 (for example Satellite Business Systems with projects SBS 1 2 3 4 5 6 HGS 5 etc) they have spun section containing propulsion system solar drums and despun section containing the satellites communications payload and antennas

- Also very popular and versatile dual-spin models are Hughes HS-381 (Leasat project) HS-389 (Intelsat project) HS-393 (JCSat project)

The DSSC usually is used for the attitude stabilization by partial twist method only one of the DSSCs coaxial bodies (the laquorotorraquo-body) has rotation at the laquoquiescenceraquo of the second body (the laquoplatformraquo-body) ndash it allows to place into the laquoplatformraquo-body some exploratory equipment and to perform of space-mission tests without rotational disturbances

Spin StabilizationDual-Spin Spacecraft

4

The dual-spin construction-scheme is quite useful in the practice during all history of space flights realization and it is possible to present some examples of the DSSC which was used in real space-programs (most of them are communications satellites and observing geostationary satellites)

- This is long-continued and well successful project ldquoIntelsatrdquo (the Intelsat II series of satellites first launched in 1966) including 8th generation of geostationary communications satellites and Intelsat VI (1991) designed and built by Hughes Aircraft Company - The ldquoMeteosatrdquo-project by European Space Research Organization (initiated with Meteosat-1 in 1977 and operated until 2007 with

Meteosat-7) also used dual-spin configuration spacecraft - Spin-stabilized spacecraft with mechanically despun antennas was applied in the framework of GEOTAIL (a collaborative mission

of Japan JAXAISAS and NASA within the program ldquoInternational Solar-Terrestrial Physicsrdquo) launched in 1992 the GEOTAIL spacecraft and its payload continue to operate in 2013

- Analogously the construction scheme with despun antenna was selected for Chinese communications satellites DFH-2 (STW-3 1988 STW-4 1988 STW-55 1990)

- Well-known Galileo missionrsquos spacecraft (the fifth spacecraft to visit Jupiter launched on October 19 1989) was designed by dual-spin scheme

- Of course we need to indicate one of the worlds most-purchased commercial communications satellite models such as Hughes Boeing HS-376 BSS-376 (for example Satellite Business Systems with projects SBS 1 2 3 4 5 6 HGS 5 etc) they have spun section containing propulsion system solar drums and despun section containing the satellites communications payload and antennas

- Also very popular and versatile dual-spin models are Hughes HS-381 (Leasat project) HS-389 (Intelsat project) HS-393 (JCSat project)

The DSSC usually is used for the attitude stabilization by partial twist method only one of the DSSCs coaxial bodies (the laquorotorraquo-body) has rotation at the laquoquiescenceraquo of the second body (the laquoplatformraquo-body) ndash it allows to place into the laquoplatformraquo-body some exploratory equipment and to perform of space-mission tests without rotational disturbances

Spin StabilizationDual-Spin Spacecraft

4

HS 376Class Communications Nation USA

Mass 654 kg at beginning-of-life in geosynchronous orbit

Spin stabilized at 50 rpm by 4 hydrazine thrusters with 136 kg propellant Star 30 apogee kick motor Solar cells mounted on outside of cylindrical satellite body provide 990 W of power and recharge two NiCd batteries 24 + 6 backup 9 W transmission beams

HS 376 Chronology- 15 November 1980 SBS 1 Program- hellip- 22 November 1998 BONUM-1 Program

Spin StabilizationDual-Spin Spacecraft

4

Hughes HS-389

Intelsat-6 [Boeing] SDS-2 [NRO]

Ordered Date

Intelsat 601 1982 29101991

Intelsat 602 1982 27101989

Intelsat 603 1982 14031990

Intelsat 604 1982 23061990

Intelsat 605 1982 14081991

SDS-2 1 08081989

SDS-2 2 15111990

SDS-2 3 02121992

SDS-2 4 03071996

Gravity Gradient Stabilization 5

UniCubeSat-GG is the first CubeSat mission of GAUSS (Gruppo di Astrodinamica dell Universita degli Studi ldquola Sapienzardquo) at the University of Rome (Universita di Roma ldquoLa Sapienzardquo Scuola di Ingegneria Aerospaziale) Italy

The Gravity-gradient stabilization is a method of SC stabilizing in a fixed orientation using only the orbited bodys mass distribution and the Earths gravitational field

SC is placed along the radius-vector of the Earth

Tether-satellitesPhysical pendulums

Magnetic Stabilization 6

Magnetic stabilization is a method of SC stabilizing using geomagnetic field of the Earth

SC is positioned along the magnetic lines of the geomagnetic field (ldquomagnetic compassrdquo)

One of Four Alinco-5 Permanent Magnet sets on board KySat-1

KySat-1 Passive Magnetic Stabilization System is used for antenna orientation and coarse camera pointing

KySat-1 the first satellite project by Kentucky Space is a 1-U CubeSat scheduled to launch in 2010 on a NASA mission

Aerodynamic Stabilization 7

The Aerodynamic stabilization is a method of SC stabilizing using aerodynamic force in rarefied atmosphere of low earth orbit

SC is positioned along the orbital velocity vector

Aerodynamically Stable CubeSat Design Concept

ldquogo like an arrow helliprdquo

SC with tail unit

Actuators 8Reaction Wheel Assemblies (RWAs)

RWAs are particularly useful when the spacecraft must be rotated by very small amounts such as keeping a telescope pointed at a star

This is accomplished by equipping the spacecraft with an electric motor attached to a flywheel which when rotated increasingly fast causes the spacecraft to spin the other way in a proportional amount by conservation of angular momentum

Actuators 9Control Moment Gyros (CMGs)

A CMG consists of a spinning rotor and one or more motorized gimbals that tilt the rotorrsquos angular momentum As the rotor tilts the changing angular momentum causes a gyroscopic torque that rotates the spacecraft

Microsatellite with four control moment gyroscopes

Actuators 10Thrusters

A thruster is a small propulsive device used by spacecraft for attitude control in the reaction control system or long-duration low-thrust acceleration

Liquefied gases in high-pressure balloons

Main Equations of Angular Motion of Rigid Body(Attitude Motion of SC)

11

Euler dynamical equations

Euler kinematical equations

The study of the angular motion of the SC attitude dynamics is one of the main problems of rigid body systems dynamics in the classical mechanics

And vice versa

Tasks of analysis and synthesis of the rigid bodiesrsquo motion have important applications in the space-flight dynamics

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

Louis Poinsot interpreacutetation

Inertia tensor with polhodes

--- notation by CDHall

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

If we consider rigid body angular motion on the base of Hamilton dynamics than for rigid body motion we take pendulum phase structure

x=tetta (nutation) and

y = impulse (tetta) ~ d(tetta)dt

Pendulum phase space

Part 2 ndash Research Results

1 The magnetized dual-spin spacecraft attitude motion

2 The regular and chaotic dynamics of dual-spin spacecraft

3 The multi-rotor system and multi-spin spacecraft dynamics

13

14

Letrsquos consider an angular motion (about point O) of spider-type multiple rotor rigid bodies systems and

coaxial bodies systems

Mechanical models of multiple-rotor systems

Coaxial bodies system(dual-spin spacecraft)

Multirotor spiders(gyrostat-spacecrafts)

Number of rotors = N

Number of rotors = 6 Number of rotors = 1

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

Magnetic Stabilization 6

Magnetic stabilization is a method of SC stabilizing using geomagnetic field of the Earth

SC is positioned along the magnetic lines of the geomagnetic field (ldquomagnetic compassrdquo)

One of Four Alinco-5 Permanent Magnet sets on board KySat-1

KySat-1 Passive Magnetic Stabilization System is used for antenna orientation and coarse camera pointing

KySat-1 the first satellite project by Kentucky Space is a 1-U CubeSat scheduled to launch in 2010 on a NASA mission

Aerodynamic Stabilization 7

The Aerodynamic stabilization is a method of SC stabilizing using aerodynamic force in rarefied atmosphere of low earth orbit

SC is positioned along the orbital velocity vector

Aerodynamically Stable CubeSat Design Concept

ldquogo like an arrow helliprdquo

SC with tail unit

Actuators 8Reaction Wheel Assemblies (RWAs)

RWAs are particularly useful when the spacecraft must be rotated by very small amounts such as keeping a telescope pointed at a star

This is accomplished by equipping the spacecraft with an electric motor attached to a flywheel which when rotated increasingly fast causes the spacecraft to spin the other way in a proportional amount by conservation of angular momentum

Actuators 9Control Moment Gyros (CMGs)

A CMG consists of a spinning rotor and one or more motorized gimbals that tilt the rotorrsquos angular momentum As the rotor tilts the changing angular momentum causes a gyroscopic torque that rotates the spacecraft

Microsatellite with four control moment gyroscopes

Actuators 10Thrusters

A thruster is a small propulsive device used by spacecraft for attitude control in the reaction control system or long-duration low-thrust acceleration

Liquefied gases in high-pressure balloons

Main Equations of Angular Motion of Rigid Body(Attitude Motion of SC)

11

Euler dynamical equations

Euler kinematical equations

The study of the angular motion of the SC attitude dynamics is one of the main problems of rigid body systems dynamics in the classical mechanics

And vice versa

Tasks of analysis and synthesis of the rigid bodiesrsquo motion have important applications in the space-flight dynamics

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

Louis Poinsot interpreacutetation

Inertia tensor with polhodes

--- notation by CDHall

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

If we consider rigid body angular motion on the base of Hamilton dynamics than for rigid body motion we take pendulum phase structure

x=tetta (nutation) and

y = impulse (tetta) ~ d(tetta)dt

Pendulum phase space

Part 2 ndash Research Results

1 The magnetized dual-spin spacecraft attitude motion

2 The regular and chaotic dynamics of dual-spin spacecraft

3 The multi-rotor system and multi-spin spacecraft dynamics

13

14

Letrsquos consider an angular motion (about point O) of spider-type multiple rotor rigid bodies systems and

coaxial bodies systems

Mechanical models of multiple-rotor systems

Coaxial bodies system(dual-spin spacecraft)

Multirotor spiders(gyrostat-spacecrafts)

Number of rotors = N

Number of rotors = 6 Number of rotors = 1

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

Aerodynamic Stabilization 7

The Aerodynamic stabilization is a method of SC stabilizing using aerodynamic force in rarefied atmosphere of low earth orbit

SC is positioned along the orbital velocity vector

Aerodynamically Stable CubeSat Design Concept

ldquogo like an arrow helliprdquo

SC with tail unit

Actuators 8Reaction Wheel Assemblies (RWAs)

RWAs are particularly useful when the spacecraft must be rotated by very small amounts such as keeping a telescope pointed at a star

This is accomplished by equipping the spacecraft with an electric motor attached to a flywheel which when rotated increasingly fast causes the spacecraft to spin the other way in a proportional amount by conservation of angular momentum

Actuators 9Control Moment Gyros (CMGs)

A CMG consists of a spinning rotor and one or more motorized gimbals that tilt the rotorrsquos angular momentum As the rotor tilts the changing angular momentum causes a gyroscopic torque that rotates the spacecraft

Microsatellite with four control moment gyroscopes

Actuators 10Thrusters

A thruster is a small propulsive device used by spacecraft for attitude control in the reaction control system or long-duration low-thrust acceleration

Liquefied gases in high-pressure balloons

Main Equations of Angular Motion of Rigid Body(Attitude Motion of SC)

11

Euler dynamical equations

Euler kinematical equations

The study of the angular motion of the SC attitude dynamics is one of the main problems of rigid body systems dynamics in the classical mechanics

And vice versa

Tasks of analysis and synthesis of the rigid bodiesrsquo motion have important applications in the space-flight dynamics

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

Louis Poinsot interpreacutetation

Inertia tensor with polhodes

--- notation by CDHall

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

If we consider rigid body angular motion on the base of Hamilton dynamics than for rigid body motion we take pendulum phase structure

x=tetta (nutation) and

y = impulse (tetta) ~ d(tetta)dt

Pendulum phase space

Part 2 ndash Research Results

1 The magnetized dual-spin spacecraft attitude motion

2 The regular and chaotic dynamics of dual-spin spacecraft

3 The multi-rotor system and multi-spin spacecraft dynamics

13

14

Letrsquos consider an angular motion (about point O) of spider-type multiple rotor rigid bodies systems and

coaxial bodies systems

Mechanical models of multiple-rotor systems

Coaxial bodies system(dual-spin spacecraft)

Multirotor spiders(gyrostat-spacecrafts)

Number of rotors = N

Number of rotors = 6 Number of rotors = 1

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

Actuators 8Reaction Wheel Assemblies (RWAs)

RWAs are particularly useful when the spacecraft must be rotated by very small amounts such as keeping a telescope pointed at a star

This is accomplished by equipping the spacecraft with an electric motor attached to a flywheel which when rotated increasingly fast causes the spacecraft to spin the other way in a proportional amount by conservation of angular momentum

Actuators 9Control Moment Gyros (CMGs)

A CMG consists of a spinning rotor and one or more motorized gimbals that tilt the rotorrsquos angular momentum As the rotor tilts the changing angular momentum causes a gyroscopic torque that rotates the spacecraft

Microsatellite with four control moment gyroscopes

Actuators 10Thrusters

A thruster is a small propulsive device used by spacecraft for attitude control in the reaction control system or long-duration low-thrust acceleration

Liquefied gases in high-pressure balloons

Main Equations of Angular Motion of Rigid Body(Attitude Motion of SC)

11

Euler dynamical equations

Euler kinematical equations

The study of the angular motion of the SC attitude dynamics is one of the main problems of rigid body systems dynamics in the classical mechanics

And vice versa

Tasks of analysis and synthesis of the rigid bodiesrsquo motion have important applications in the space-flight dynamics

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

Louis Poinsot interpreacutetation

Inertia tensor with polhodes

--- notation by CDHall

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

If we consider rigid body angular motion on the base of Hamilton dynamics than for rigid body motion we take pendulum phase structure

x=tetta (nutation) and

y = impulse (tetta) ~ d(tetta)dt

Pendulum phase space

Part 2 ndash Research Results

1 The magnetized dual-spin spacecraft attitude motion

2 The regular and chaotic dynamics of dual-spin spacecraft

3 The multi-rotor system and multi-spin spacecraft dynamics

13

14

Letrsquos consider an angular motion (about point O) of spider-type multiple rotor rigid bodies systems and

coaxial bodies systems

Mechanical models of multiple-rotor systems

Coaxial bodies system(dual-spin spacecraft)

Multirotor spiders(gyrostat-spacecrafts)

Number of rotors = N

Number of rotors = 6 Number of rotors = 1

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

Actuators 9Control Moment Gyros (CMGs)

A CMG consists of a spinning rotor and one or more motorized gimbals that tilt the rotorrsquos angular momentum As the rotor tilts the changing angular momentum causes a gyroscopic torque that rotates the spacecraft

Microsatellite with four control moment gyroscopes

Actuators 10Thrusters

A thruster is a small propulsive device used by spacecraft for attitude control in the reaction control system or long-duration low-thrust acceleration

Liquefied gases in high-pressure balloons

Main Equations of Angular Motion of Rigid Body(Attitude Motion of SC)

11

Euler dynamical equations

Euler kinematical equations

The study of the angular motion of the SC attitude dynamics is one of the main problems of rigid body systems dynamics in the classical mechanics

And vice versa

Tasks of analysis and synthesis of the rigid bodiesrsquo motion have important applications in the space-flight dynamics

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

Louis Poinsot interpreacutetation

Inertia tensor with polhodes

--- notation by CDHall

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

If we consider rigid body angular motion on the base of Hamilton dynamics than for rigid body motion we take pendulum phase structure

x=tetta (nutation) and

y = impulse (tetta) ~ d(tetta)dt

Pendulum phase space

Part 2 ndash Research Results

1 The magnetized dual-spin spacecraft attitude motion

2 The regular and chaotic dynamics of dual-spin spacecraft

3 The multi-rotor system and multi-spin spacecraft dynamics

13

14

Letrsquos consider an angular motion (about point O) of spider-type multiple rotor rigid bodies systems and

coaxial bodies systems

Mechanical models of multiple-rotor systems

Coaxial bodies system(dual-spin spacecraft)

Multirotor spiders(gyrostat-spacecrafts)

Number of rotors = N

Number of rotors = 6 Number of rotors = 1

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

Actuators 10Thrusters

A thruster is a small propulsive device used by spacecraft for attitude control in the reaction control system or long-duration low-thrust acceleration

Liquefied gases in high-pressure balloons

Main Equations of Angular Motion of Rigid Body(Attitude Motion of SC)

11

Euler dynamical equations

Euler kinematical equations

The study of the angular motion of the SC attitude dynamics is one of the main problems of rigid body systems dynamics in the classical mechanics

And vice versa

Tasks of analysis and synthesis of the rigid bodiesrsquo motion have important applications in the space-flight dynamics

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

Louis Poinsot interpreacutetation

Inertia tensor with polhodes

--- notation by CDHall

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

If we consider rigid body angular motion on the base of Hamilton dynamics than for rigid body motion we take pendulum phase structure

x=tetta (nutation) and

y = impulse (tetta) ~ d(tetta)dt

Pendulum phase space

Part 2 ndash Research Results

1 The magnetized dual-spin spacecraft attitude motion

2 The regular and chaotic dynamics of dual-spin spacecraft

3 The multi-rotor system and multi-spin spacecraft dynamics

13

14

Letrsquos consider an angular motion (about point O) of spider-type multiple rotor rigid bodies systems and

coaxial bodies systems

Mechanical models of multiple-rotor systems

Coaxial bodies system(dual-spin spacecraft)

Multirotor spiders(gyrostat-spacecrafts)

Number of rotors = N

Number of rotors = 6 Number of rotors = 1

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

Main Equations of Angular Motion of Rigid Body(Attitude Motion of SC)

11

Euler dynamical equations

Euler kinematical equations

The study of the angular motion of the SC attitude dynamics is one of the main problems of rigid body systems dynamics in the classical mechanics

And vice versa

Tasks of analysis and synthesis of the rigid bodiesrsquo motion have important applications in the space-flight dynamics

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

Louis Poinsot interpreacutetation

Inertia tensor with polhodes

--- notation by CDHall

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

If we consider rigid body angular motion on the base of Hamilton dynamics than for rigid body motion we take pendulum phase structure

x=tetta (nutation) and

y = impulse (tetta) ~ d(tetta)dt

Pendulum phase space

Part 2 ndash Research Results

1 The magnetized dual-spin spacecraft attitude motion

2 The regular and chaotic dynamics of dual-spin spacecraft

3 The multi-rotor system and multi-spin spacecraft dynamics

13

14

Letrsquos consider an angular motion (about point O) of spider-type multiple rotor rigid bodies systems and

coaxial bodies systems

Mechanical models of multiple-rotor systems

Coaxial bodies system(dual-spin spacecraft)

Multirotor spiders(gyrostat-spacecrafts)

Number of rotors = N

Number of rotors = 6 Number of rotors = 1

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

Louis Poinsot interpreacutetation

Inertia tensor with polhodes

--- notation by CDHall

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

If we consider rigid body angular motion on the base of Hamilton dynamics than for rigid body motion we take pendulum phase structure

x=tetta (nutation) and

y = impulse (tetta) ~ d(tetta)dt

Pendulum phase space

Part 2 ndash Research Results

1 The magnetized dual-spin spacecraft attitude motion

2 The regular and chaotic dynamics of dual-spin spacecraft

3 The multi-rotor system and multi-spin spacecraft dynamics

13

14

Letrsquos consider an angular motion (about point O) of spider-type multiple rotor rigid bodies systems and

coaxial bodies systems

Mechanical models of multiple-rotor systems

Coaxial bodies system(dual-spin spacecraft)

Multirotor spiders(gyrostat-spacecrafts)

Number of rotors = N

Number of rotors = 6 Number of rotors = 1

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

Main Properties of the Free Angular Motion of Rigid Body(Attitude Motion of SC)

12

If we consider rigid body angular motion on the base of Hamilton dynamics than for rigid body motion we take pendulum phase structure

x=tetta (nutation) and

y = impulse (tetta) ~ d(tetta)dt

Pendulum phase space

Part 2 ndash Research Results

1 The magnetized dual-spin spacecraft attitude motion

2 The regular and chaotic dynamics of dual-spin spacecraft

3 The multi-rotor system and multi-spin spacecraft dynamics

13

14

Letrsquos consider an angular motion (about point O) of spider-type multiple rotor rigid bodies systems and

coaxial bodies systems

Mechanical models of multiple-rotor systems

Coaxial bodies system(dual-spin spacecraft)

Multirotor spiders(gyrostat-spacecrafts)

Number of rotors = N

Number of rotors = 6 Number of rotors = 1

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

Part 2 ndash Research Results

1 The magnetized dual-spin spacecraft attitude motion

2 The regular and chaotic dynamics of dual-spin spacecraft

3 The multi-rotor system and multi-spin spacecraft dynamics

13

14

Letrsquos consider an angular motion (about point O) of spider-type multiple rotor rigid bodies systems and

coaxial bodies systems

Mechanical models of multiple-rotor systems

Coaxial bodies system(dual-spin spacecraft)

Multirotor spiders(gyrostat-spacecrafts)

Number of rotors = N

Number of rotors = 6 Number of rotors = 1

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

14

Letrsquos consider an angular motion (about point O) of spider-type multiple rotor rigid bodies systems and

coaxial bodies systems

Mechanical models of multiple-rotor systems

Coaxial bodies system(dual-spin spacecraft)

Multirotor spiders(gyrostat-spacecrafts)

Number of rotors = N

Number of rotors = 6 Number of rotors = 1

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

1cos sin

cos

sin cos

sincos sin

cos

p q

p q

r p q

(1)

(2)

15

I1 The angular motion general equations

I The coaxial bodies system motion

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

cos sinP

P Q M Q

The coaxial top

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

I2 The angular motion equations of the magnetic DSSC and the coaxial Lagrange top

16

The single dipole model of the Earthrsquos magnetic field and the constant magnetic field vector corresponding to the circle equatorial orbit

The magnetic DSSC coordinate frames

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

17I3 The equations

1

1

1

1

x

y

z

Ap C B qr qC M

Bq A C pr pC M

Cr C B A pq M

C r M

2 2 2

2 1

0

T

x y zOx y z

T

orb orb

M M M

B m B m

oθ

θ

rbM m B

M

1 2

2 2

3 2

cos

cos

cos

OZ Ox

OZ Oy

OZ Oz

i k

j k

k k

2 2

2 1

2

0

Ap C B qr q Q

Bq A C pr p Q

C r B A pq M

orbQ B m

1

22 2 2 2 22

1 2 1 2 1 2

sin cos 1

2 2

T P

LI L l lT

A A A B C C

H H

cos sinP

P Q M Q

(6)

(5)

(4)

(3)

(2)

(1)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

18I4 The integrals and solutions

2

2

2

1

2

23

x

y

z

K Ap

K KK Bq

K KK C r

K K

2

2

2

2 3

2

const

cos

z

z

L K C r

I I K

K C rL

K K K

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

2

2 2 22 2

1

2 2Ap Bq C r E C r TC

22 2 2 2 22 2A p B q C r K DT

2 1 2

22 21

2

22 2

Bq A C f q f q

C B C Ff q V E

A A B A A B

H B A B qf q V V

C A C

- AV Doroshin Exact Solutions in Attitude Dynamics of a Magnetic Dual-Spin Spacecraft and a Generalization of the Lagrange Top WSEAS Transactions on Systems Issue 10 Volume 12 (2013)

471-482

(11) (10)

(9)

(8) (7)

19I5 The analytical solution

The polhodes ellipsoid

2

2 2

22 2

2 1

1

1

q t H C A C x t SB A B

p t C B C x t FA A B

EAr t x t S t r t

A C C

2 20 0

2 20 0

sn

sn

i

i

R P c N t t I kx t e

R P c N t t I k

S E

The numerical integration (lines) and analytical (points) results

A2=15 B2=8 C2=6 A1=5 C1=4 [kgm2] p0=075 q0=2 r0=583 [1s] K=50

[kgm2s] Q=100 [kgm2s2]

5

(13)

(12)

20I6 The analytical solution for Euler angles

orbB

K2K + ω timesK = mtimesK

K + ΩtimesK = 0

orbB

K 2Ω ω m

γ + Ωtimes γ = 0

1

2

3 2

t Ap t K

t Bq t K

C r t K

2cos

tg

t C r t K

Ap tt

Bq t

0

0

2 2

0 2 2 2 2

01

( ) ( )

( ) ( )

( )

t

t

t

t

Ap t Bq tt K dt

A p t B q t

t r t dtC

5

10 20 3003 052 08

The numerical integration (lines) and analytical (points) results for the directional cosines

A2=15 B2=8 C2=6 A1=5 C1=4 [kgm2] p0=075

q0=2 r0=583 [1s] K=50 [km2s]

Q=100 [kgm2s2]

(19)

(20)

(18) (17)

(16)

(15)

(14)

21I7 Hamilton form of equations in the Andoyer-Deprit variables

(21)

(25)

(22)

(23)

(24)

1 C r

22

3 23

T T

T

L I Kl

I L I

K k K s K

K k

2

2

2

2 22

2 22

2

sin

cos

x

y

z

K Ap I L l

K Bq I L l

K C r L

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

22 2 2 2 22

1 2 1 2 1 2 2

sin cos 1

2 2

LI L l l LQ

A A A B C C I

H

22

(26)

Hamilton form of equations 2 2 2 1 1A B C A C

Equations (21) have heteroclinic solutions

Perturbations

0 0 L lM t f l t L t g t t dt

Melnikov function

(28)

The Melnikov function has the infinite number of simple roots This proves the fact of the motion chaotization at the presence of small harmonic perturbations

22 2 2 2

2 1

C B C EB

p t y t q t s k y t E r t y t t r tA A B B C C

0

sin cosN

n p n pn

Q t a n t b n t

22 2 2 2 2

20 0

1 2 1 2 1 2 2 20 1 0 1

sin cos 1

2 2

LI L l l L LQ Q Q t

A A A B C C I I

H H H H H

0 0 1 1

L L

L l L l

l l

L f l L gf f g g

l L l Ll f l L g

H H H H

0 0 00

cos sinN

ns n p n p

n

M t J a n t b n t

(27)

(26)

00 0 2

2

00 1 2 02

4 exp

exp 4

M aa E y t

ky t

Mt aE y a a a

k

23

2 l L I

I8 Numerical modeling Poincareacute sections mod 2 0pt

2 22 2 2 1 1 2 1 3 5

0 0 0 0

15 10 6 5 4[ ] 20 5 [ ] 075 [1 ] 1 5 20 [1]

) 5 004 ) 5 01 ) 10 01 ) 15 02

pA B C A C kg m I kg m s s a a a

a Q b Q c Q d Q

(a) (b)

(c) (d)

II The spider-type systems motion 24

m r K K K

1 2

3 4

5 6

4 2 m r

Ap p

Bq J I q I

Cr r

K K

ed

dt

Kω K M

12 56 34

34 12 56

56 34 12

ex

ey

ez

Ap I C B qr I q r M

Bq I A C pr I r p M

Cr I B A pq I p q M

4 2

4 2 4 2

iji j A A J I

B B J I C C J I

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i ex x

i e i ey y

i e i ez z

I p M M I p M M

I q M M I q M M

I r M M I r M M

(1)

(2)

(3)

(4)

II1 Angular momentum of the system in projections onto axes of the frame Oxyz

II2 The motion equations of the multirotor system

25

1 1

1 1

4 2

4 2 4 2

N Nij

il jl ll l

N N

l ll l

A A J NI

B B J NI C C J NI

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i el l lx l l lx

i e i el l ly l l ly

i e i el l lz l l lz

I p M M I p M M

I q M M I q M M

I r M M I r M M

0

1

2

3

cos 2

cos sin2

cos sin2

cos sin2

2 λ Θ λ

0

1

2

3

0

0

0

0

p q r

p r q

q r p

r q p

λ Θ

(4rsquo)

II3 The Euler parameters

(5)

26II4 The conjugate spinups and captures

Def1 Conjugate rotors are paired rotors located in the same layer on the opposite rays For example rotor 3N and rotor 4N are conjugate rotors (also rotor 12 and rotor 22 etc)

Def2 Conjugate spinup mean a process of spinning up conjugate rotors in opposite directions up to a desired value of relative angular velocity with the help of internal moments from main body Velocities of conjugate rotors will be equal in absolute value and opposite in sign

Def3 Rotor capture is an immediate deceleration of rotor relative angular velocity with the help of internal moment from the main body So rotor capture means an ldquoinstantaneous freezingrdquo of rotor with respect to the main body The capture can be performed with the help of gear meshing friction clutch or other methods

27

(6)

II6 Analytical solutions of motion equations

0e e ex y zM M M

A B C D

12 121

12 122

0

0

0

0

si

si

M if t tM

otherwise

M if t tM

otherwise

12 12 12sS M t I

1 12 2 12S S

1 1 12c c st t t

1 0

12

122

ISp

A IAS

A I

1 0 Conjugate spinup of rotors 1 2

Rotor 1 capture

Rotor 2 capture

2 2 1c c ct t t

1 2

121 2

0 0

c c

c c

t t tp IS

P t t tA I

Two serial captures of conjugate rotors bring to piecewise constant angular velocity of the main body

The main body performed the rotation about Ox axis by finite angle 12 2 1

c c

x

IS t t

A I

28

1 3 5 2 4 6

0 1 2 3

1 0

c c c c c c c cstart finish

c c c cstart start start start

t t t t t t t t

t t t t

0 1

2 3

2 2 2

cos sin2 2

sin sin2 2

c cstart start

c cstart start

c c c cstart finish finish start

t t t tPt t

t t t tQ Rt t

P Q R t t t T t t

1 2 t t t p P q Q r R

Letrsquos assume synchronically captures of coresponded conjugate rotors 1 2 3 4 5 6 and coincidence of the frame Oxyz initial position and the fixed frame OXYZ

Angular velocities between conjugate captures of rotors

Solutions of kinematic equations (5)

cos cos cosT

P Q R

T

e

The main body performed finite rotation about vector e by angle χ

(18)

(20)

(19)

(21)

29II7 Numerical simulation of reorientation process

const 0 1 135 246

i s cj jj j j

i s cj jj j j

jj

M M H t H t t H t t

M M H t H t t H t t

M j j

Two sets of results of numerical simulation

1 4 5(0) (0) (0) 0 (0) 100 1 s ω

ν = 300 Nms

H(t) is Heaviside function

FigA - zero value of angular momentum and different inertia moments of main body

Fig A Fig B

Fig4 - reorientation process at nonzero system angular momentum

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36

20I6 The analytical solution for Euler angles

orbB

K2K + ω timesK = mtimesK

K + ΩtimesK = 0

orbB

K 2Ω ω m

γ + Ωtimes γ = 0

1

2

3 2

t Ap t K

t Bq t K

C r t K

2cos

tg

t C r t K

Ap tt

Bq t

0

0

2 2

0 2 2 2 2

01

( ) ( )

( ) ( )

( )

t

t

t

t

Ap t Bq tt K dt

A p t B q t

t r t dtC

5

10 20 3003 052 08

The numerical integration (lines) and analytical (points) results for the directional cosines

A2=15 B2=8 C2=6 A1=5 C1=4 [kgm2] p0=075

q0=2 r0=583 [1s] K=50 [km2s]

Q=100 [kgm2s2]

(19)

(20)

(18) (17)

(16)

(15)

(14)

21I7 Hamilton form of equations in the Andoyer-Deprit variables

(21)

(25)

(22)

(23)

(24)

1 C r

22

3 23

T T

T

L I Kl

I L I

K k K s K

K k

2

2

2

2 22

2 22

2

sin

cos

x

y

z

K Ap I L l

K Bq I L l

K C r L

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

22 2 2 2 22

1 2 1 2 1 2 2

sin cos 1

2 2

LI L l l LQ

A A A B C C I

H

22

(26)

Hamilton form of equations 2 2 2 1 1A B C A C

Equations (21) have heteroclinic solutions

Perturbations

0 0 L lM t f l t L t g t t dt

Melnikov function

(28)

The Melnikov function has the infinite number of simple roots This proves the fact of the motion chaotization at the presence of small harmonic perturbations

22 2 2 2

2 1

C B C EB

p t y t q t s k y t E r t y t t r tA A B B C C

0

sin cosN

n p n pn

Q t a n t b n t

22 2 2 2 2

20 0

1 2 1 2 1 2 2 20 1 0 1

sin cos 1

2 2

LI L l l L LQ Q Q t

A A A B C C I I

H H H H H

0 0 1 1

L L

L l L l

l l

L f l L gf f g g

l L l Ll f l L g

H H H H

0 0 00

cos sinN

ns n p n p

n

M t J a n t b n t

(27)

(26)

00 0 2

2

00 1 2 02

4 exp

exp 4

M aa E y t

ky t

Mt aE y a a a

k

23

2 l L I

I8 Numerical modeling Poincareacute sections mod 2 0pt

2 22 2 2 1 1 2 1 3 5

0 0 0 0

15 10 6 5 4[ ] 20 5 [ ] 075 [1 ] 1 5 20 [1]

) 5 004 ) 5 01 ) 10 01 ) 15 02

pA B C A C kg m I kg m s s a a a

a Q b Q c Q d Q

(a) (b)

(c) (d)

II The spider-type systems motion 24

m r K K K

1 2

3 4

5 6

4 2 m r

Ap p

Bq J I q I

Cr r

K K

ed

dt

Kω K M

12 56 34

34 12 56

56 34 12

ex

ey

ez

Ap I C B qr I q r M

Bq I A C pr I r p M

Cr I B A pq I p q M

4 2

4 2 4 2

iji j A A J I

B B J I C C J I

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i ex x

i e i ey y

i e i ez z

I p M M I p M M

I q M M I q M M

I r M M I r M M

(1)

(2)

(3)

(4)

II1 Angular momentum of the system in projections onto axes of the frame Oxyz

II2 The motion equations of the multirotor system

25

1 1

1 1

4 2

4 2 4 2

N Nij

il jl ll l

N N

l ll l

A A J NI

B B J NI C C J NI

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i el l lx l l lx

i e i el l ly l l ly

i e i el l lz l l lz

I p M M I p M M

I q M M I q M M

I r M M I r M M

0

1

2

3

cos 2

cos sin2

cos sin2

cos sin2

2 λ Θ λ

0

1

2

3

0

0

0

0

p q r

p r q

q r p

r q p

λ Θ

(4rsquo)

II3 The Euler parameters

(5)

26II4 The conjugate spinups and captures

Def1 Conjugate rotors are paired rotors located in the same layer on the opposite rays For example rotor 3N and rotor 4N are conjugate rotors (also rotor 12 and rotor 22 etc)

Def2 Conjugate spinup mean a process of spinning up conjugate rotors in opposite directions up to a desired value of relative angular velocity with the help of internal moments from main body Velocities of conjugate rotors will be equal in absolute value and opposite in sign

Def3 Rotor capture is an immediate deceleration of rotor relative angular velocity with the help of internal moment from the main body So rotor capture means an ldquoinstantaneous freezingrdquo of rotor with respect to the main body The capture can be performed with the help of gear meshing friction clutch or other methods

27

(6)

II6 Analytical solutions of motion equations

0e e ex y zM M M

A B C D

12 121

12 122

0

0

0

0

si

si

M if t tM

otherwise

M if t tM

otherwise

12 12 12sS M t I

1 12 2 12S S

1 1 12c c st t t

1 0

12

122

ISp

A IAS

A I

1 0 Conjugate spinup of rotors 1 2

Rotor 1 capture

Rotor 2 capture

2 2 1c c ct t t

1 2

121 2

0 0

c c

c c

t t tp IS

P t t tA I

Two serial captures of conjugate rotors bring to piecewise constant angular velocity of the main body

The main body performed the rotation about Ox axis by finite angle 12 2 1

c c

x

IS t t

A I

28

1 3 5 2 4 6

0 1 2 3

1 0

c c c c c c c cstart finish

c c c cstart start start start

t t t t t t t t

t t t t

0 1

2 3

2 2 2

cos sin2 2

sin sin2 2

c cstart start

c cstart start

c c c cstart finish finish start

t t t tPt t

t t t tQ Rt t

P Q R t t t T t t

1 2 t t t p P q Q r R

Letrsquos assume synchronically captures of coresponded conjugate rotors 1 2 3 4 5 6 and coincidence of the frame Oxyz initial position and the fixed frame OXYZ

Angular velocities between conjugate captures of rotors

Solutions of kinematic equations (5)

cos cos cosT

P Q R

T

e

The main body performed finite rotation about vector e by angle χ

(18)

(20)

(19)

(21)

29II7 Numerical simulation of reorientation process

const 0 1 135 246

i s cj jj j j

i s cj jj j j

jj

M M H t H t t H t t

M M H t H t t H t t

M j j

Two sets of results of numerical simulation

1 4 5(0) (0) (0) 0 (0) 100 1 s ω

ν = 300 Nms

H(t) is Heaviside function

FigA - zero value of angular momentum and different inertia moments of main body

Fig A Fig B

Fig4 - reorientation process at nonzero system angular momentum

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36

21I7 Hamilton form of equations in the Andoyer-Deprit variables

(21)

(25)

(22)

(23)

(24)

1 C r

22

3 23

T T

T

L I Kl

I L I

K k K s K

K k

2

2

2

2 22

2 22

2

sin

cos

x

y

z

K Ap I L l

K Bq I L l

K C r L

2

2

2

0

Ap C B qr q QBq K

Bq A C pr p QAp K

C r B A pq M

22 2 2 2 22

1 2 1 2 1 2 2

sin cos 1

2 2

LI L l l LQ

A A A B C C I

H

22

(26)

Hamilton form of equations 2 2 2 1 1A B C A C

Equations (21) have heteroclinic solutions

Perturbations

0 0 L lM t f l t L t g t t dt

Melnikov function

(28)

The Melnikov function has the infinite number of simple roots This proves the fact of the motion chaotization at the presence of small harmonic perturbations

22 2 2 2

2 1

C B C EB

p t y t q t s k y t E r t y t t r tA A B B C C

0

sin cosN

n p n pn

Q t a n t b n t

22 2 2 2 2

20 0

1 2 1 2 1 2 2 20 1 0 1

sin cos 1

2 2

LI L l l L LQ Q Q t

A A A B C C I I

H H H H H

0 0 1 1

L L

L l L l

l l

L f l L gf f g g

l L l Ll f l L g

H H H H

0 0 00

cos sinN

ns n p n p

n

M t J a n t b n t

(27)

(26)

00 0 2

2

00 1 2 02

4 exp

exp 4

M aa E y t

ky t

Mt aE y a a a

k

23

2 l L I

I8 Numerical modeling Poincareacute sections mod 2 0pt

2 22 2 2 1 1 2 1 3 5

0 0 0 0

15 10 6 5 4[ ] 20 5 [ ] 075 [1 ] 1 5 20 [1]

) 5 004 ) 5 01 ) 10 01 ) 15 02

pA B C A C kg m I kg m s s a a a

a Q b Q c Q d Q

(a) (b)

(c) (d)

II The spider-type systems motion 24

m r K K K

1 2

3 4

5 6

4 2 m r

Ap p

Bq J I q I

Cr r

K K

ed

dt

Kω K M

12 56 34

34 12 56

56 34 12

ex

ey

ez

Ap I C B qr I q r M

Bq I A C pr I r p M

Cr I B A pq I p q M

4 2

4 2 4 2

iji j A A J I

B B J I C C J I

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i ex x

i e i ey y

i e i ez z

I p M M I p M M

I q M M I q M M

I r M M I r M M

(1)

(2)

(3)

(4)

II1 Angular momentum of the system in projections onto axes of the frame Oxyz

II2 The motion equations of the multirotor system

25

1 1

1 1

4 2

4 2 4 2

N Nij

il jl ll l

N N

l ll l

A A J NI

B B J NI C C J NI

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i el l lx l l lx

i e i el l ly l l ly

i e i el l lz l l lz

I p M M I p M M

I q M M I q M M

I r M M I r M M

0

1

2

3

cos 2

cos sin2

cos sin2

cos sin2

2 λ Θ λ

0

1

2

3

0

0

0

0

p q r

p r q

q r p

r q p

λ Θ

(4rsquo)

II3 The Euler parameters

(5)

26II4 The conjugate spinups and captures

Def1 Conjugate rotors are paired rotors located in the same layer on the opposite rays For example rotor 3N and rotor 4N are conjugate rotors (also rotor 12 and rotor 22 etc)

Def2 Conjugate spinup mean a process of spinning up conjugate rotors in opposite directions up to a desired value of relative angular velocity with the help of internal moments from main body Velocities of conjugate rotors will be equal in absolute value and opposite in sign

Def3 Rotor capture is an immediate deceleration of rotor relative angular velocity with the help of internal moment from the main body So rotor capture means an ldquoinstantaneous freezingrdquo of rotor with respect to the main body The capture can be performed with the help of gear meshing friction clutch or other methods

27

(6)

II6 Analytical solutions of motion equations

0e e ex y zM M M

A B C D

12 121

12 122

0

0

0

0

si

si

M if t tM

otherwise

M if t tM

otherwise

12 12 12sS M t I

1 12 2 12S S

1 1 12c c st t t

1 0

12

122

ISp

A IAS

A I

1 0 Conjugate spinup of rotors 1 2

Rotor 1 capture

Rotor 2 capture

2 2 1c c ct t t

1 2

121 2

0 0

c c

c c

t t tp IS

P t t tA I

Two serial captures of conjugate rotors bring to piecewise constant angular velocity of the main body

The main body performed the rotation about Ox axis by finite angle 12 2 1

c c

x

IS t t

A I

28

1 3 5 2 4 6

0 1 2 3

1 0

c c c c c c c cstart finish

c c c cstart start start start

t t t t t t t t

t t t t

0 1

2 3

2 2 2

cos sin2 2

sin sin2 2

c cstart start

c cstart start

c c c cstart finish finish start

t t t tPt t

t t t tQ Rt t

P Q R t t t T t t

1 2 t t t p P q Q r R

Letrsquos assume synchronically captures of coresponded conjugate rotors 1 2 3 4 5 6 and coincidence of the frame Oxyz initial position and the fixed frame OXYZ

Angular velocities between conjugate captures of rotors

Solutions of kinematic equations (5)

cos cos cosT

P Q R

T

e

The main body performed finite rotation about vector e by angle χ

(18)

(20)

(19)

(21)

29II7 Numerical simulation of reorientation process

const 0 1 135 246

i s cj jj j j

i s cj jj j j

jj

M M H t H t t H t t

M M H t H t t H t t

M j j

Two sets of results of numerical simulation

1 4 5(0) (0) (0) 0 (0) 100 1 s ω

ν = 300 Nms

H(t) is Heaviside function

FigA - zero value of angular momentum and different inertia moments of main body

Fig A Fig B

Fig4 - reorientation process at nonzero system angular momentum

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36

22

(26)

Hamilton form of equations 2 2 2 1 1A B C A C

Equations (21) have heteroclinic solutions

Perturbations

0 0 L lM t f l t L t g t t dt

Melnikov function

(28)

The Melnikov function has the infinite number of simple roots This proves the fact of the motion chaotization at the presence of small harmonic perturbations

22 2 2 2

2 1

C B C EB

p t y t q t s k y t E r t y t t r tA A B B C C

0

sin cosN

n p n pn

Q t a n t b n t

22 2 2 2 2

20 0

1 2 1 2 1 2 2 20 1 0 1

sin cos 1

2 2

LI L l l L LQ Q Q t

A A A B C C I I

H H H H H

0 0 1 1

L L

L l L l

l l

L f l L gf f g g

l L l Ll f l L g

H H H H

0 0 00

cos sinN

ns n p n p

n

M t J a n t b n t

(27)

(26)

00 0 2

2

00 1 2 02

4 exp

exp 4

M aa E y t

ky t

Mt aE y a a a

k

23

2 l L I

I8 Numerical modeling Poincareacute sections mod 2 0pt

2 22 2 2 1 1 2 1 3 5

0 0 0 0

15 10 6 5 4[ ] 20 5 [ ] 075 [1 ] 1 5 20 [1]

) 5 004 ) 5 01 ) 10 01 ) 15 02

pA B C A C kg m I kg m s s a a a

a Q b Q c Q d Q

(a) (b)

(c) (d)

II The spider-type systems motion 24

m r K K K

1 2

3 4

5 6

4 2 m r

Ap p

Bq J I q I

Cr r

K K

ed

dt

Kω K M

12 56 34

34 12 56

56 34 12

ex

ey

ez

Ap I C B qr I q r M

Bq I A C pr I r p M

Cr I B A pq I p q M

4 2

4 2 4 2

iji j A A J I

B B J I C C J I

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i ex x

i e i ey y

i e i ez z

I p M M I p M M

I q M M I q M M

I r M M I r M M

(1)

(2)

(3)

(4)

II1 Angular momentum of the system in projections onto axes of the frame Oxyz

II2 The motion equations of the multirotor system

25

1 1

1 1

4 2

4 2 4 2

N Nij

il jl ll l

N N

l ll l

A A J NI

B B J NI C C J NI

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i el l lx l l lx

i e i el l ly l l ly

i e i el l lz l l lz

I p M M I p M M

I q M M I q M M

I r M M I r M M

0

1

2

3

cos 2

cos sin2

cos sin2

cos sin2

2 λ Θ λ

0

1

2

3

0

0

0

0

p q r

p r q

q r p

r q p

λ Θ

(4rsquo)

II3 The Euler parameters

(5)

26II4 The conjugate spinups and captures

Def1 Conjugate rotors are paired rotors located in the same layer on the opposite rays For example rotor 3N and rotor 4N are conjugate rotors (also rotor 12 and rotor 22 etc)

Def2 Conjugate spinup mean a process of spinning up conjugate rotors in opposite directions up to a desired value of relative angular velocity with the help of internal moments from main body Velocities of conjugate rotors will be equal in absolute value and opposite in sign

Def3 Rotor capture is an immediate deceleration of rotor relative angular velocity with the help of internal moment from the main body So rotor capture means an ldquoinstantaneous freezingrdquo of rotor with respect to the main body The capture can be performed with the help of gear meshing friction clutch or other methods

27

(6)

II6 Analytical solutions of motion equations

0e e ex y zM M M

A B C D

12 121

12 122

0

0

0

0

si

si

M if t tM

otherwise

M if t tM

otherwise

12 12 12sS M t I

1 12 2 12S S

1 1 12c c st t t

1 0

12

122

ISp

A IAS

A I

1 0 Conjugate spinup of rotors 1 2

Rotor 1 capture

Rotor 2 capture

2 2 1c c ct t t

1 2

121 2

0 0

c c

c c

t t tp IS

P t t tA I

Two serial captures of conjugate rotors bring to piecewise constant angular velocity of the main body

The main body performed the rotation about Ox axis by finite angle 12 2 1

c c

x

IS t t

A I

28

1 3 5 2 4 6

0 1 2 3

1 0

c c c c c c c cstart finish

c c c cstart start start start

t t t t t t t t

t t t t

0 1

2 3

2 2 2

cos sin2 2

sin sin2 2

c cstart start

c cstart start

c c c cstart finish finish start

t t t tPt t

t t t tQ Rt t

P Q R t t t T t t

1 2 t t t p P q Q r R

Letrsquos assume synchronically captures of coresponded conjugate rotors 1 2 3 4 5 6 and coincidence of the frame Oxyz initial position and the fixed frame OXYZ

Angular velocities between conjugate captures of rotors

Solutions of kinematic equations (5)

cos cos cosT

P Q R

T

e

The main body performed finite rotation about vector e by angle χ

(18)

(20)

(19)

(21)

29II7 Numerical simulation of reorientation process

const 0 1 135 246

i s cj jj j j

i s cj jj j j

jj

M M H t H t t H t t

M M H t H t t H t t

M j j

Two sets of results of numerical simulation

1 4 5(0) (0) (0) 0 (0) 100 1 s ω

ν = 300 Nms

H(t) is Heaviside function

FigA - zero value of angular momentum and different inertia moments of main body

Fig A Fig B

Fig4 - reorientation process at nonzero system angular momentum

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36

23

2 l L I

I8 Numerical modeling Poincareacute sections mod 2 0pt

2 22 2 2 1 1 2 1 3 5

0 0 0 0

15 10 6 5 4[ ] 20 5 [ ] 075 [1 ] 1 5 20 [1]

) 5 004 ) 5 01 ) 10 01 ) 15 02

pA B C A C kg m I kg m s s a a a

a Q b Q c Q d Q

(a) (b)

(c) (d)

II The spider-type systems motion 24

m r K K K

1 2

3 4

5 6

4 2 m r

Ap p

Bq J I q I

Cr r

K K

ed

dt

Kω K M

12 56 34

34 12 56

56 34 12

ex

ey

ez

Ap I C B qr I q r M

Bq I A C pr I r p M

Cr I B A pq I p q M

4 2

4 2 4 2

iji j A A J I

B B J I C C J I

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i ex x

i e i ey y

i e i ez z

I p M M I p M M

I q M M I q M M

I r M M I r M M

(1)

(2)

(3)

(4)

II1 Angular momentum of the system in projections onto axes of the frame Oxyz

II2 The motion equations of the multirotor system

25

1 1

1 1

4 2

4 2 4 2

N Nij

il jl ll l

N N

l ll l

A A J NI

B B J NI C C J NI

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i el l lx l l lx

i e i el l ly l l ly

i e i el l lz l l lz

I p M M I p M M

I q M M I q M M

I r M M I r M M

0

1

2

3

cos 2

cos sin2

cos sin2

cos sin2

2 λ Θ λ

0

1

2

3

0

0

0

0

p q r

p r q

q r p

r q p

λ Θ

(4rsquo)

II3 The Euler parameters

(5)

26II4 The conjugate spinups and captures

Def1 Conjugate rotors are paired rotors located in the same layer on the opposite rays For example rotor 3N and rotor 4N are conjugate rotors (also rotor 12 and rotor 22 etc)

Def2 Conjugate spinup mean a process of spinning up conjugate rotors in opposite directions up to a desired value of relative angular velocity with the help of internal moments from main body Velocities of conjugate rotors will be equal in absolute value and opposite in sign

Def3 Rotor capture is an immediate deceleration of rotor relative angular velocity with the help of internal moment from the main body So rotor capture means an ldquoinstantaneous freezingrdquo of rotor with respect to the main body The capture can be performed with the help of gear meshing friction clutch or other methods

27

(6)

II6 Analytical solutions of motion equations

0e e ex y zM M M

A B C D

12 121

12 122

0

0

0

0

si

si

M if t tM

otherwise

M if t tM

otherwise

12 12 12sS M t I

1 12 2 12S S

1 1 12c c st t t

1 0

12

122

ISp

A IAS

A I

1 0 Conjugate spinup of rotors 1 2

Rotor 1 capture

Rotor 2 capture

2 2 1c c ct t t

1 2

121 2

0 0

c c

c c

t t tp IS

P t t tA I

Two serial captures of conjugate rotors bring to piecewise constant angular velocity of the main body

The main body performed the rotation about Ox axis by finite angle 12 2 1

c c

x

IS t t

A I

28

1 3 5 2 4 6

0 1 2 3

1 0

c c c c c c c cstart finish

c c c cstart start start start

t t t t t t t t

t t t t

0 1

2 3

2 2 2

cos sin2 2

sin sin2 2

c cstart start

c cstart start

c c c cstart finish finish start

t t t tPt t

t t t tQ Rt t

P Q R t t t T t t

1 2 t t t p P q Q r R

Letrsquos assume synchronically captures of coresponded conjugate rotors 1 2 3 4 5 6 and coincidence of the frame Oxyz initial position and the fixed frame OXYZ

Angular velocities between conjugate captures of rotors

Solutions of kinematic equations (5)

cos cos cosT

P Q R

T

e

The main body performed finite rotation about vector e by angle χ

(18)

(20)

(19)

(21)

29II7 Numerical simulation of reorientation process

const 0 1 135 246

i s cj jj j j

i s cj jj j j

jj

M M H t H t t H t t

M M H t H t t H t t

M j j

Two sets of results of numerical simulation

1 4 5(0) (0) (0) 0 (0) 100 1 s ω

ν = 300 Nms

H(t) is Heaviside function

FigA - zero value of angular momentum and different inertia moments of main body

Fig A Fig B

Fig4 - reorientation process at nonzero system angular momentum

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36

II The spider-type systems motion 24

m r K K K

1 2

3 4

5 6

4 2 m r

Ap p

Bq J I q I

Cr r

K K

ed

dt

Kω K M

12 56 34

34 12 56

56 34 12

ex

ey

ez

Ap I C B qr I q r M

Bq I A C pr I r p M

Cr I B A pq I p q M

4 2

4 2 4 2

iji j A A J I

B B J I C C J I

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i ex x

i e i ey y

i e i ez z

I p M M I p M M

I q M M I q M M

I r M M I r M M

(1)

(2)

(3)

(4)

II1 Angular momentum of the system in projections onto axes of the frame Oxyz

II2 The motion equations of the multirotor system

25

1 1

1 1

4 2

4 2 4 2

N Nij

il jl ll l

N N

l ll l

A A J NI

B B J NI C C J NI

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i el l lx l l lx

i e i el l ly l l ly

i e i el l lz l l lz

I p M M I p M M

I q M M I q M M

I r M M I r M M

0

1

2

3

cos 2

cos sin2

cos sin2

cos sin2

2 λ Θ λ

0

1

2

3

0

0

0

0

p q r

p r q

q r p

r q p

λ Θ

(4rsquo)

II3 The Euler parameters

(5)

26II4 The conjugate spinups and captures

Def1 Conjugate rotors are paired rotors located in the same layer on the opposite rays For example rotor 3N and rotor 4N are conjugate rotors (also rotor 12 and rotor 22 etc)

Def2 Conjugate spinup mean a process of spinning up conjugate rotors in opposite directions up to a desired value of relative angular velocity with the help of internal moments from main body Velocities of conjugate rotors will be equal in absolute value and opposite in sign

Def3 Rotor capture is an immediate deceleration of rotor relative angular velocity with the help of internal moment from the main body So rotor capture means an ldquoinstantaneous freezingrdquo of rotor with respect to the main body The capture can be performed with the help of gear meshing friction clutch or other methods

27

(6)

II6 Analytical solutions of motion equations

0e e ex y zM M M

A B C D

12 121

12 122

0

0

0

0

si

si

M if t tM

otherwise

M if t tM

otherwise

12 12 12sS M t I

1 12 2 12S S

1 1 12c c st t t

1 0

12

122

ISp

A IAS

A I

1 0 Conjugate spinup of rotors 1 2

Rotor 1 capture

Rotor 2 capture

2 2 1c c ct t t

1 2

121 2

0 0

c c

c c

t t tp IS

P t t tA I

Two serial captures of conjugate rotors bring to piecewise constant angular velocity of the main body

The main body performed the rotation about Ox axis by finite angle 12 2 1

c c

x

IS t t

A I

28

1 3 5 2 4 6

0 1 2 3

1 0

c c c c c c c cstart finish

c c c cstart start start start

t t t t t t t t

t t t t

0 1

2 3

2 2 2

cos sin2 2

sin sin2 2

c cstart start

c cstart start

c c c cstart finish finish start

t t t tPt t

t t t tQ Rt t

P Q R t t t T t t

1 2 t t t p P q Q r R

Letrsquos assume synchronically captures of coresponded conjugate rotors 1 2 3 4 5 6 and coincidence of the frame Oxyz initial position and the fixed frame OXYZ

Angular velocities between conjugate captures of rotors

Solutions of kinematic equations (5)

cos cos cosT

P Q R

T

e

The main body performed finite rotation about vector e by angle χ

(18)

(20)

(19)

(21)

29II7 Numerical simulation of reorientation process

const 0 1 135 246

i s cj jj j j

i s cj jj j j

jj

M M H t H t t H t t

M M H t H t t H t t

M j j

Two sets of results of numerical simulation

1 4 5(0) (0) (0) 0 (0) 100 1 s ω

ν = 300 Nms

H(t) is Heaviside function

FigA - zero value of angular momentum and different inertia moments of main body

Fig A Fig B

Fig4 - reorientation process at nonzero system angular momentum

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36

25

1 1

1 1

4 2

4 2 4 2

N Nij

il jl ll l

N N

l ll l

A A J NI

B B J NI C C J NI

1 1 1 2 2 2

3 3 3 4 4 4

5 5 5 6 6 6

i e i el l lx l l lx

i e i el l ly l l ly

i e i el l lz l l lz

I p M M I p M M

I q M M I q M M

I r M M I r M M

0

1

2

3

cos 2

cos sin2

cos sin2

cos sin2

2 λ Θ λ

0

1

2

3

0

0

0

0

p q r

p r q

q r p

r q p

λ Θ

(4rsquo)

II3 The Euler parameters

(5)

26II4 The conjugate spinups and captures

Def1 Conjugate rotors are paired rotors located in the same layer on the opposite rays For example rotor 3N and rotor 4N are conjugate rotors (also rotor 12 and rotor 22 etc)

Def2 Conjugate spinup mean a process of spinning up conjugate rotors in opposite directions up to a desired value of relative angular velocity with the help of internal moments from main body Velocities of conjugate rotors will be equal in absolute value and opposite in sign

Def3 Rotor capture is an immediate deceleration of rotor relative angular velocity with the help of internal moment from the main body So rotor capture means an ldquoinstantaneous freezingrdquo of rotor with respect to the main body The capture can be performed with the help of gear meshing friction clutch or other methods

27

(6)

II6 Analytical solutions of motion equations

0e e ex y zM M M

A B C D

12 121

12 122

0

0

0

0

si

si

M if t tM

otherwise

M if t tM

otherwise

12 12 12sS M t I

1 12 2 12S S

1 1 12c c st t t

1 0

12

122

ISp

A IAS

A I

1 0 Conjugate spinup of rotors 1 2

Rotor 1 capture

Rotor 2 capture

2 2 1c c ct t t

1 2

121 2

0 0

c c

c c

t t tp IS

P t t tA I

Two serial captures of conjugate rotors bring to piecewise constant angular velocity of the main body

The main body performed the rotation about Ox axis by finite angle 12 2 1

c c

x

IS t t

A I

28

1 3 5 2 4 6

0 1 2 3

1 0

c c c c c c c cstart finish

c c c cstart start start start

t t t t t t t t

t t t t

0 1

2 3

2 2 2

cos sin2 2

sin sin2 2

c cstart start

c cstart start

c c c cstart finish finish start

t t t tPt t

t t t tQ Rt t

P Q R t t t T t t

1 2 t t t p P q Q r R

Letrsquos assume synchronically captures of coresponded conjugate rotors 1 2 3 4 5 6 and coincidence of the frame Oxyz initial position and the fixed frame OXYZ

Angular velocities between conjugate captures of rotors

Solutions of kinematic equations (5)

cos cos cosT

P Q R

T

e

The main body performed finite rotation about vector e by angle χ

(18)

(20)

(19)

(21)

29II7 Numerical simulation of reorientation process

const 0 1 135 246

i s cj jj j j

i s cj jj j j

jj

M M H t H t t H t t

M M H t H t t H t t

M j j

Two sets of results of numerical simulation

1 4 5(0) (0) (0) 0 (0) 100 1 s ω

ν = 300 Nms

H(t) is Heaviside function

FigA - zero value of angular momentum and different inertia moments of main body

Fig A Fig B

Fig4 - reorientation process at nonzero system angular momentum

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36

26II4 The conjugate spinups and captures

Def1 Conjugate rotors are paired rotors located in the same layer on the opposite rays For example rotor 3N and rotor 4N are conjugate rotors (also rotor 12 and rotor 22 etc)

Def2 Conjugate spinup mean a process of spinning up conjugate rotors in opposite directions up to a desired value of relative angular velocity with the help of internal moments from main body Velocities of conjugate rotors will be equal in absolute value and opposite in sign

Def3 Rotor capture is an immediate deceleration of rotor relative angular velocity with the help of internal moment from the main body So rotor capture means an ldquoinstantaneous freezingrdquo of rotor with respect to the main body The capture can be performed with the help of gear meshing friction clutch or other methods

27

(6)

II6 Analytical solutions of motion equations

0e e ex y zM M M

A B C D

12 121

12 122

0

0

0

0

si

si

M if t tM

otherwise

M if t tM

otherwise

12 12 12sS M t I

1 12 2 12S S

1 1 12c c st t t

1 0

12

122

ISp

A IAS

A I

1 0 Conjugate spinup of rotors 1 2

Rotor 1 capture

Rotor 2 capture

2 2 1c c ct t t

1 2

121 2

0 0

c c

c c

t t tp IS

P t t tA I

Two serial captures of conjugate rotors bring to piecewise constant angular velocity of the main body

The main body performed the rotation about Ox axis by finite angle 12 2 1

c c

x

IS t t

A I

28

1 3 5 2 4 6

0 1 2 3

1 0

c c c c c c c cstart finish

c c c cstart start start start

t t t t t t t t

t t t t

0 1

2 3

2 2 2

cos sin2 2

sin sin2 2

c cstart start

c cstart start

c c c cstart finish finish start

t t t tPt t

t t t tQ Rt t

P Q R t t t T t t

1 2 t t t p P q Q r R

Letrsquos assume synchronically captures of coresponded conjugate rotors 1 2 3 4 5 6 and coincidence of the frame Oxyz initial position and the fixed frame OXYZ

Angular velocities between conjugate captures of rotors

Solutions of kinematic equations (5)

cos cos cosT

P Q R

T

e

The main body performed finite rotation about vector e by angle χ

(18)

(20)

(19)

(21)

29II7 Numerical simulation of reorientation process

const 0 1 135 246

i s cj jj j j

i s cj jj j j

jj

M M H t H t t H t t

M M H t H t t H t t

M j j

Two sets of results of numerical simulation

1 4 5(0) (0) (0) 0 (0) 100 1 s ω

ν = 300 Nms

H(t) is Heaviside function

FigA - zero value of angular momentum and different inertia moments of main body

Fig A Fig B

Fig4 - reorientation process at nonzero system angular momentum

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36

27

(6)

II6 Analytical solutions of motion equations

0e e ex y zM M M

A B C D

12 121

12 122

0

0

0

0

si

si

M if t tM

otherwise

M if t tM

otherwise

12 12 12sS M t I

1 12 2 12S S

1 1 12c c st t t

1 0

12

122

ISp

A IAS

A I

1 0 Conjugate spinup of rotors 1 2

Rotor 1 capture

Rotor 2 capture

2 2 1c c ct t t

1 2

121 2

0 0

c c

c c

t t tp IS

P t t tA I

Two serial captures of conjugate rotors bring to piecewise constant angular velocity of the main body

The main body performed the rotation about Ox axis by finite angle 12 2 1

c c

x

IS t t

A I

28

1 3 5 2 4 6

0 1 2 3

1 0

c c c c c c c cstart finish

c c c cstart start start start

t t t t t t t t

t t t t

0 1

2 3

2 2 2

cos sin2 2

sin sin2 2

c cstart start

c cstart start

c c c cstart finish finish start

t t t tPt t

t t t tQ Rt t

P Q R t t t T t t

1 2 t t t p P q Q r R

Letrsquos assume synchronically captures of coresponded conjugate rotors 1 2 3 4 5 6 and coincidence of the frame Oxyz initial position and the fixed frame OXYZ

Angular velocities between conjugate captures of rotors

Solutions of kinematic equations (5)

cos cos cosT

P Q R

T

e

The main body performed finite rotation about vector e by angle χ

(18)

(20)

(19)

(21)

29II7 Numerical simulation of reorientation process

const 0 1 135 246

i s cj jj j j

i s cj jj j j

jj

M M H t H t t H t t

M M H t H t t H t t

M j j

Two sets of results of numerical simulation

1 4 5(0) (0) (0) 0 (0) 100 1 s ω

ν = 300 Nms

H(t) is Heaviside function

FigA - zero value of angular momentum and different inertia moments of main body

Fig A Fig B

Fig4 - reorientation process at nonzero system angular momentum

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36

28

1 3 5 2 4 6

0 1 2 3

1 0

c c c c c c c cstart finish

c c c cstart start start start

t t t t t t t t

t t t t

0 1

2 3

2 2 2

cos sin2 2

sin sin2 2

c cstart start

c cstart start

c c c cstart finish finish start

t t t tPt t

t t t tQ Rt t

P Q R t t t T t t

1 2 t t t p P q Q r R

Letrsquos assume synchronically captures of coresponded conjugate rotors 1 2 3 4 5 6 and coincidence of the frame Oxyz initial position and the fixed frame OXYZ

Angular velocities between conjugate captures of rotors

Solutions of kinematic equations (5)

cos cos cosT

P Q R

T

e

The main body performed finite rotation about vector e by angle χ

(18)

(20)

(19)

(21)

29II7 Numerical simulation of reorientation process

const 0 1 135 246

i s cj jj j j

i s cj jj j j

jj

M M H t H t t H t t

M M H t H t t H t t

M j j

Two sets of results of numerical simulation

1 4 5(0) (0) (0) 0 (0) 100 1 s ω

ν = 300 Nms

H(t) is Heaviside function

FigA - zero value of angular momentum and different inertia moments of main body

Fig A Fig B

Fig4 - reorientation process at nonzero system angular momentum

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36

29II7 Numerical simulation of reorientation process

const 0 1 135 246

i s cj jj j j

i s cj jj j j

jj

M M H t H t t H t t

M M H t H t t H t t

M j j

Two sets of results of numerical simulation

1 4 5(0) (0) (0) 0 (0) 100 1 s ω

ν = 300 Nms

H(t) is Heaviside function

FigA - zero value of angular momentum and different inertia moments of main body

Fig A Fig B

Fig4 - reorientation process at nonzero system angular momentum

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36

30

__________________________________________________________

The main properties of the attitude stabilization and control of SC (and multirotor systems) have been examined

Research into attitude motion of the one-body-SC dual-spin SC and spider-type-SC is very complicated

Nontrivial and chaotic modes are possible in the SC attitude motion

Conclusion

• Slide 1
• Outline
• Part 1 - Introduction to Attitude Dynamics and Control
• Slide 4
• Spin stabilization
• Spin Stabilization Dual-Spin Spacecraft
• Spin Stabilization Dual-Spin Spacecraft (2)
• Spin Stabilization Dual-Spin Spacecraft (3)
• Spin Stabilization Dual-Spin Spacecraft (4)
• Gravity Gradient Stabilization
• Magnetic Stabilization
• Aerodynamic Stabilization
• Actuators
• Actuators (2)
• Actuators (3)
• Main Equations of Angular Motion of Rigid Body (Attitude Motion
• Main Properties of the Free Angular Motion of Rigid Body (Attit
• Main Properties of the Free Angular Motion of Rigid Body (Attit (2)
• Part 2 ndash Research Results
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36