Introduction to Attitude
Dynamics and Control
Chris Hall
Chris Hall
Aerospace and Ocean Engineering
Aerospace and Ocean Engineering
cdhall@vt.edu
cdhall@vt.edu
What is spacecraft attitude?
And why should we care about it?
• Most spacecraft have instruments or antennas that
• Most spacecraft have instruments or antennas that
must be pointed in specific directions
must be pointed in specific directions
–
– Hubble must point its main telescope
Hubble must point its main telescope
–
– Communications satellites must point their antennas
Communications satellites must point their antennas
• The orientation of the spacecraft in space is called its
• The orientation of the spacecraft in space is called its
attitude
attitude
• To control the attitude, the spacecraft operators
• To control the attitude, the spacecraft operators
(which could be the spacecraft’s computer in the case
(which could be the spacecraft’s computer in the case
of an autonomous “ADCS”) must have the ability to
of an autonomous “ADCS”) must have the ability to
–
– Determine the current attitude
Determine the current attitude
–
– Determine the error between the current and desired
Determine the error between the current and desired
attitudes
attitudes
– Apply torques to remove the error
– Apply torques to remove the error
Spacecraft Attitude Determination
and Control
• So, the spacecraft needs an Attitude
• So, the spacecraft needs an Attitude
Determination and Control System (ADCS)
Determination and Control System (ADCS)
• To do the determination function requires
• To do the determination function requires
knowledge of kinematics
knowledge of kinematics
• Attitude is determined using sensors
• Attitude is determined using sensors
• To do the control function requires
• To do the control function requires
knowledge of kinetics and kinematics
knowledge of kinetics and kinematics
(dynamics)
(dynamics)
• Attitude is controlled using actuators
• Attitude is controlled using actuators
Attitude Determination
Determine the attitude, or orientation, or
Determine the attitude, or orientation, or
pointing direction of a reference frame fixed in
pointing direction of a reference frame fixed in
the body, with respect to a known reference
the body, with respect to a known reference
frame, usually an inertial frame. That is, where
frame, usually an inertial frame. That is, where
is the spacecraft pointing?
is the spacecraft pointing?
• Generally involves finding a rotation matrix,
• Generally involves finding a rotation matrix,
or its equivalent
or its equivalent
• Requires two or more attitude sensors
• Requires two or more attitude sensors
– Sun sensor, Earth horizon sensor, Moon sensor, star
– Sun sensor, Earth horizon sensor, Moon sensor, star
tracker, magnetometer
tracker, magnetometer
• Requires an algorithm
• Requires an algorithm
The Differential Equation
• Every good dynamics course must begin with a differential
• Every good dynamics course must begin with a differential
equation
equation
• For attitude dynamics and control, the equation of choice is
• For attitude dynamics and control, the equation of choice is
r r
&
h=g Euler (1707-1783)
• This is the rotational equivalent of
• This is the rotational equivalent of
r r r r
ma = f or m&& = f
r Newton (1643-1727)
• Other notation used in other books and papers:
• Other notation used in other books and papers:
r r
& r r
&
L=N H=M
• Why doesn’t everybody get together and agree on a specific
• Why doesn’t everybody get together and agree on a specific
notation?
notation?
Euler’s Equations
• Euler’s vector differential equation
• Euler’s vector differential equation
r r
& h is angular momentum
h=g g is torque
• Becomes a matrix differential equation when
• Becomes a matrix differential equation when
expressed in a body-fixed reference frame
expressed in a body-fixed reference frame
I is inertia matrix
Iω = − ω×Iω + g
& ω is angular velocity
• And when expressed in a principal reference frame, it
• And when expressed in a principal reference frame, it
becomes
becomes
ω1 = I 2 I−1I 3 ω 2ω 3 + g11
& I
I 3 − I1
ω2 =
& I2 ω1ω 3 + g
I
2
2
I1 − I 2
ω3 =
& I3 ω1ω 2 + g
I
3
3
Rigid Body Spin Stability
Z • Ixx > Iyy > Izz
• Ixx > Iyy > Izz
• Major axis spin is stable
• Major axis spin is stable
MINOR •
• Minor axis spin is stable
Minor axis spin is stable
•
• Intermediate axis spin is
Intermediate axis spin is
INTERMEDIATE
unstable
unstable
Y
• Energy dissipation changes
• Energy dissipation changes
these results
these results
→ Minor axis spin becomes
→ Minor axis spin becomes
unstable
unstable
X MAJOR
• This is called the Major- Axis
• This is called the Major--Axis
Major-
Major
Rule
Rule
Sputnik & Explorer I
•
• Sputnik was launched in 1957
Sputnik was launched in 1957
•
• Professor Ronald Bracewell, a radio
Professor Ronald Bracewell, a radio
astronomer at Stanford, deduced that Sputnik
astronomer at Stanford, deduced that Sputnik
was spinning about a symmetry axis, and that
was spinning about a symmetry axis, and that
it must be the major axis
it must be the major axis
•
• He called JPL to make sure that the Explorer II
He called JPL to make sure that the Explorer
design was taking this into account, but security
design was taking this into account, but security
prevented him from getting through
prevented him from getting through
•
• Explorer II was designed as a minor axis
Explorer was designed as a minor axis
spinner, launched in 1958
spinner, launched in 1958
Spin-Stabilized Satellites
Explorer I (1958) was supposed to be
spin-stabilized about its minor axis.
It went into a flat spin due to Telstar I (1962) was spin-stabilized
energy dissipation. about its major axis, spinning at
about 200 RPM.
Gravity-Gradient Stabilization
• Gravitational attraction:
• Gravitational attraction:
ff = µm/r22
= µm/r f2
• Top: ff11 > ff22 ⇒ torque is out
• Top: > ⇒ torque is out f1
of the page
of the page
• Bottom: ff11 > ff22 ⇒ torque is
• Bottom: > ⇒ torque is
into the page
into the page
• In both cases, the torque is a
• In both cases, the torque is a f1
restoring torque, tending to
restoring torque, tending to f2
make the satellite swing like
make the satellite swing like
a pendulum
a pendulum
Gravity-Gradient Stabilization
• In the 60s was viewed as “free”
• In the 60s was viewed as “free”
attitude control
attitude control
• In general, “G22” is not accurate
• In general, “G ” is not accurate
enough, spacecraft can even flip over
enough, spacecraft can even flip over
• Not really free, because of boom mass
• Not really free, because of boom mass
• However, OrbComm and TechSat 21
• However, OrbComm and TechSat 21
use gravity gradient with flexible solar
use gravity gradient with flexible solar
panels on an extensible wrapper
panels on an extensible wrapper
around the boom
around the boom
• The Moon is gravity-gradient
• The Moon is gravity-gradient
stabilized; Lagrange (1736-1813)
stabilized; Lagrange (1736-1813)
showed this
showed this
TechSat 21
TechSat 21
Augmented G2 Stabilization
• Problem: with G2 there is practically no yaw
• Problem: with G2 there is practically no yaw
stability
stability
• Solution: Add a small momentum wheel
• Solution: Add a small momentum wheel
spinning about the pitch axis
spinning about the pitch axis
• In effect, the wheel is a spin-stabilized s/c,
• In effect, the wheel is a spin-stabilized s/c,
with its angular momentum vector aligned
with its angular momentum vector aligned
with the orbital angular momentum vector
with the orbital angular momentum vector
• Called pitch wheel or yaw wheel
• Called pitch wheel or yaw wheel
• Can still flip over! (Polar Bear)
• Can still flip over! (Polar Bear)
Roll, Pitch & Yaw
r
v
• Same as for aircraft
• Same as for aircraft
(usually)
(usually)
• Roll is rotation about
ˆ
o1
• Roll is rotation about
the velocity vector
the velocity vector
• Pitch is rotation about
• Pitch is rotation about r ˆ
the orbit normal vector
the orbit normal vector
− r o3
• Yaw is rotation about
• Yaw is rotation about ˆ
o2
the nadir vector
the nadir vector
• Keep these color codes
• Keep these color codes r
in mind
−w
in mind
Effect of Rotor on Spin Stability
• A spinning rotor can
Z stabilize the intermediate
axis, destabilize others
• Stability condition
ωR
IR ωR > (Ixx-Iyy)ωy
R R xx yy y
R
Y • As with rigid body, energy
dissipation changes
stability results
→ some stable spins
X Platform become unstable
Two Spacecraft With Rotors
Defense Support Program Global Positioning System
One large rotor Four momentum wheels
(120 RPM) (several thousand RPM)
Dual-Spin Stabilization
• Spin-stabilized satellites must be major axis spinners:
• Spin-stabilized satellites must be major axis spinners:
“short and fat”
“short and fat”
• Spin axis must in orbit normal direction (well, usually)
• Spin axis must in orbit normal direction (well, usually)
• Two problems:
• Two problems:
–
– launch vehicles are “tall and skinny”
launch vehicles are “tall and skinny”
–
– antennas need to point at earth
antennas need to point at earth
• In mid-60s, two engineers invented a solution
• In mid-60s, two engineers invented a solution
–
– Vernon Landon at RCA
Vernon Landon at RCA
–
– Tony Iorillo at Hughes
Tony Iorillo at Hughes
• Make the spacecraft with two parts: one spins
• Make the spacecraft with two parts: one spins
relatively fast, the other spins slowly or not at all
relatively fast, the other spins slowly or not at all
• The major axis rule generalizes to make it possible to
• The major axis rule generalizes to make it possible to
spin stably about the minor axis
spin stably about the minor axis
• Solves both problems: fits in launch vehicle, points
• Solves both problems: fits in launch vehicle, points
the despun platform at the Earth
the despun platform at the Earth
Dual-Spin-Stabilized
Satellites
TACSAT I (1969) was the first
satellite to successfully spin
about its minor axis.
The antenna is the platform, and
is intended to point
continuously at the Earth,
spinning at one revolution per
orbit.
The cylindrical body is the rotor,
providing gyric stability through
its 60 RPM spin.
Gimbaled Momentum Wheels
• Gimbal axis is fixed in
• Gimbal axis is fixed in
the body frame
the body frame Gimbal
• Spin axis is controlled by
• Spin axis is controlled by motor
gimbal motor
gimbal motor Wheel
motor
• Spin rate is controlled
• Spin rate is controlled
by wheel motor
by wheel motor
• Fixed gimbal angle gives
• Fixed gimbal angle gives a
t
a
momentum wheel
momentum wheel s Transverse
Spin axis
(MW) or reaction wheel
(MW) or reaction wheel axis
(RW)
(RW)
• Fixed wheel speed gives
• Fixed wheel speed gives
control moment gyro
control moment gyro a Gimbal
g
axis
(CMG)
(CMG)
Three-Axis Stabilization
• Instead of keeping the spin axis pointing in a specific direction,
• Instead of keeping the spin axis pointing in a specific direction,
keep all 3 axes pointed in specified directions
keep all 3 axes pointed in specified directions
• Can be done with thrusters, reaction wheels, momentum wheels,
• Can be done with thrusters, reaction wheels, momentum wheels,
control moment gyros, or combination
control moment gyros, or combination
Magnetic Stabilization
•
• Spacecraft is moving through Earth’s magnetic field B
Spacecraft is moving through Earth’s magnetic field B
•
• Passing a current through a conductor creates a magnetic
Passing a current through a conductor creates a magnetic
moment m, which in turn causes a torque g = m × B
moment m, which in turn causes a torque g = m × B
•
• Companies make magnetic torquer rods and coils specifically for
Companies make magnetic torquer rods and coils specifically for
this ACS application
this ACS application
•
• There’s a simple controller called the B-dot controller that can
There’s a simple controller called the B-dot controller that can
spin up or despin a satellite using this torque
spin up or despin a satellite using this torque
Rotational Maneuvers
• Many systems require reorienting the
• Many systems require reorienting the
spacecraft from one attitude to another
spacecraft from one attitude to another
• Similar to three-axis stabilization, but with
• Similar to three-axis stabilization, but with
additional capability
additional capability
• Uses thrusters, momentum wheels, reaction
• Uses thrusters, momentum wheels, reaction
wheels, or control moment gyros
wheels, or control moment gyros
• Example: Hubble Space Telescope uses
• Example: Hubble Space Telescope uses
momentum wheels, and turns at about the
momentum wheels, and turns at about the
same speed as a minute hand on a clock
same speed as a minute hand on a clock
Hubble Pointing
Hubble is the most precisely pointed machine ever devised for
Hubble is the most precisely pointed machine ever devised for
astronomy.
astronomy.
Requirement: The telescope must be able to maintain lock on
Requirement: The telescope must be able to maintain lock on
a target for 24 hours without deviating more than 7/1,000ths
a target for 24 hours without deviating more than 7/1,000ths
(0.007) of an arc second (2 millionths of a degree) which is
(0.007) of an arc second (2 millionths of a degree) which is
about the width of a human hair seen at a distance of a mile.
about the width of a human hair seen at a distance of a mile.
A laser with the stability and precision of the Hubble, mounted
A laser with the stability and precision of the Hubble, mounted
on top of the United States Capitol could hold a steady beam
on top of the United States Capitol could hold a steady beam
on a dime suspended above New York City, over 200 miles
on a dime suspended above New York City, over 200 miles
distant. This level of stability and precision is comparable to
distant. This level of stability and precision is comparable to
sinking a hole-in-one on a Los Angeles golf course from a tee in
sinking a hole-in-one on a Los Angeles golf course from a tee in
Washington, DC, over 2,000 miles away, in 19 out of 20
Washington, DC, over 2,000 miles away, in 19 out of 20
attempts.
attempts.
Course Overview
• Some Mission Analysis concepts
• Some Mission Analysis concepts
• Kinematics: Vectors, Rotation matrices, Euler
• Kinematics: Vectors, Rotation matrices, Euler
angles, Euler parameters (aka quaternions)
angles, Euler parameters (aka quaternions)
• Attitude determination
• Attitude determination
• Rigid body dynamics (Euler’s equations)
• Rigid body dynamics (Euler’s equations)
• Satellite dynamics applications
• Satellite dynamics applications
• Attitude control
• Attitude control