Attitude Determination and Control
(ADCS)
Olivier L. de Weck
Department of Aeronautics and Astronautics
Massachusetts Institute of Technology
16.684 Space Systems Product Development
Spring 2001
ADCS Motivation
G129 Motivation
— In order to point and slew optical
systems, spacecraft attitude control
provides coarse pointing while
optics control provides fine
pointing
G129 Spacecraft Control
— Spacecraft Stabilization
— Spin Stabilization
— Gravity Gradient
— Three-Axis Control
— Formation Flight
— Actuators
— Reaction Wheel Assemblies
(RWAs)
— Control Moment Gyros
(CMGs)
— Magnetic Torque Rods
— Thrusters
— Sensors: GPS, star trackers, limb
sensors, rate gyros, inertial
measurement units
— Control Laws
G129 Spacecraft Slew Maneuvers
— Euler Angles
— Quaternions
Key Question:
What are the pointing
requirements for satellite ?
NEED expendable propellant:
On-board fuel often determines life
Failing gyros are critical (e.g. HST)
Outline
G129 Definitions and Terminology
G129 Coordinate Systems and Mathematical Attitude Representations
G129 Rigid Body Dynamics
G129 Disturbance Torques in Space
G129 Passive Attitude Control Schemes
G129 Actuators
G129 Sensors
G129 Active Attitude Control Concepts
G129 ADCS Performance and Stability Measures
G129 Estimation and Filtering in Attitude Determination
G129 Maneuvers
G129 Other System Consideration, Control/Structure interaction
G129 Technological Trends and Advanced Concepts
Opening Remarks
G129 Nearly all ADCS Design and Performance can be viewed in
terms of RIGID BODY dynamics
G129 Typically a Major spacecraft system
G129 For large, light-weight structures with low fundamental
frequencies the flexibility needs to be taken into account
G129 ADCS requirements often drive overall S/C design
G129 Components are cumbersome, massive and power-consuming
G129 Field-of-View requirements and specific orientation are key
G129 Design, analysis and testing are typically the most
challenging of all subsystems with the exception of payload
design
G129 Need a true “systems orientation” to be successful at
designing and implementing an ADCS
Terminology
ATTITUDE : Orientation of a defined spacecraft body coordinate
system with respect to a defined external frame (GCI,HCI)
ATTITUDE DETERMINATION: Real-Time or Post-Facto knowledge,
within a given tolerance, of the spacecraft attitude
ATTITUDE CONTROL: Maintenance of a desired, specified attitude
within a given tolerance
ATTITUDE ERROR: “Low Frequency” spacecraft misalignment;
usually the intended topic of attitude control
ATTITUDE JITTER: “High Frequency” spacecraft misalignment;
usually ignored by ADCS; reduced by good design or fine
pointing/optical control.
Pointing Control Definitions
target desired pointing direction
true actual pointing direction (mean)
estimate estimate of true (instantaneous)
a pointing accuracy (long-term)
s stability (peak-peak motion)
k knowledge error
c control error
target
estimate
true
c
k
a
s
Source:
G. Mosier
NASA GSFC
a = pointing accuracy = attitude error
s = stability = attitude jitter
Attitude Coordinate Systems
X
Z
Y
^
^
^
Y = Z x X
Cross product
^^^
Geometry: Celestial Sphere
G68G3: Right Ascension
G71 : Declination
(North Celestial Pole)
G68
G71
A
r
c
le
n
g
t
h
dihedral
Inertial Coordinate
System
GCI: Geocentric Inertial Coordinates
VERNAL
EQUINOX
X and Y are
in the plane of the ecliptic
Attitude Description Notations
Describe the orientation of a body:
(1) Attach a coordinate system to the body
(2) Describe a coordinate system relative to an
inertial reference frame
A
Z
?
A
X
?
A
Y
?
}{ w.r.t.vector Position
Vector
system Coordinate }{
AP
P
A
=
=
=?
G42
G42
P
A
G42
y
P
x
P
z
P
?
?
?
?
?
?
?
?
?
?
=
z
y
x
A
P
P
P
P
G42
[]
?
?
?
?
?
?
?
?
?
?
==
1 0 0
0 1 0
0 0 1
}{ of vectorsUnit
AAA
ZYXA
???
Rotation Matrix
Rotation matrix from {B} to {A}
Jefferson Memorial
A
Z
?
A
X
?
A
Y
?
system coordinate Reference }{ =A
B
X
?
B
Y
?
B
Z
?
system coordinate Body }{ =B
[ ]
BBB
AA
B
ZYXR
???
AA
=
Special properties of rotation matrices:
1
,
?
== RRIRR
TT
1=R
(1) Orthogonal:
RRRR
AB
C
B
C
A
B B
≠
Jefferson Memorial
A
Z
?
A
X
?
A
Y
?
B
X
?
B
Y
?
B
Z
?
θ
θ
?
?
?
?
?
?
?
?
?
?
=R
A
B
cos sin 0
sin- cos 0
0 0 1
(2) Orthonormal:
(3) Not commutative
Euler Angles (1)
Euler angles describe a sequence of three rotations about different
axes in order to align one coord. system with a second coord. system.
?
?
?
?
?
?
?
?
?
?
=
1 0 0
0 cos sin
0 sin- cos
αα
αα
R
A
B
α by about Rotate
A
Z
?
β by about Rotate
B
Y
?
γ by about Rotate
C
X
?
A
Z
?
A
X
?
A
Y
?
B
X
?
B
Y
?
B
Z
?
α
α
B
Z
?
B
X
?
B
Y
?
C
X
?
C
Y
?
C
Z
?
β
β
C
Z
?
C
X
?
D
Y
?
D
X
?
C
Y
?
D
Z
?
γ
γ
?
?
?
?
?
?
?
?
?
?
=
ββ
ββ
cos 0 sin-
0 1 0
sin 0 cos
R
B
C
?
?
?
?
?
?
?
?
?
?
=
γγ
γγ
cos sin 0
sin- cos 0
0 0 1
R
C
D
RRRR
C
D
B
C
A
B
A
D
=
Euler Angles (2)
G129 Concept used in rotational
kinematics to describe body
orientation w.r.t. inertial frame
G129 Sequence of three angles and
prescription for rotating one
reference frame into another
G129 Can be defined as a transformation
matrix body/inertial as shown: TB/I
G129 Euler angles are non-unique and
exact sequence is critical
Zi (parallel to r)
Yaw
Pitch
Roll
Xi
(parallel
to v)
(r x v direction)
Body
CM
Goal: Describe kinematics of body-fixed
frame with respect to rotating local vertical
Yi
nadir
r
/
YAW ROLL PITCH
cos sin 0 1 0 0 cos 0 -sin
-sin cos 0 0 cos sin 0 1 0
0 0 1 0 -sin cos sin 0 cos
BI
T
ψ ψθθ
ψψ φφ
φ φθ θ
??????
??????
=??
??????
G9G12G12G12G10G12G12G12G11G9G12G12G12G10G12G12G12G11G9G12G12G12G10G12G12G12G11
Note:
about Yi
about X’
about Zb
θ
φ
ψ
1
// /
T
BI IB BI
TTT
?
==
Transformation
from Body to
“Inertial” frame:
(Pitch, Roll, Yaw) = (G84G15G73G15G92)EulerAngles
Quaternions
G129 Main problem computationally is
the existence of a singularity
G129 Problem can be avoided by an
application of Euler’s theorem:
The Orientation of a body is uniquely
specified by a vector giving the direction
of a body axis and a scalar specifying a
rotation angle about the axis.
EULER’S THEOREM
G129 Definition introduces a redundant
fourth element, which eliminates
the singularity.
G129 This is the “quaternion” concept
G129 Quaternions have no intuitively
interpretable meaning to the human
mind, but are computationally
convenient
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
=
4
4
3
2
1
q
q
q
q
q
q
Q
G42
Jefferson Memorial
A
Z
?
A
X
?
A
Y
?
B
X
?
B
Y
?
B
Z
?
θ
K
A
?
?
?
?
?
?
?
?
?
?
?
=
z
y
x
A
k
k
k
K
?
?
?
?
?
?
?
=
?
?
?
?
?
?
=
?
?
?
?
?
?
=
?
?
?
?
?
?
=
2
cos
2
sin
2
sin
2
sin
4
3
2
1
θ
θ
θ
θ
q
kq
kq
kq
z
y
x
rotation. of axis
the describesvector A=q
G42
rotation. ofamount
the describesscalar A=
4
q
A: Inertial
B: Body
Quaternion Demo (MATLAB)
Comparison of Attitude Descriptions
Method Euler
Angles
Direction
Cosines
Angular
Velocity G90
Quaternions
Pluses
If given φ,ψ,θ
then a unique
orientation is
defined
Orientation
defines a
unique dir-cos
matrix R
Vector
properties,
commutes w.r.t
addition
Computationally
robust
Ideal for digital
control implement
Minuses Given orient
then Euler
non-unique
Singularity
6 constraints
must be met,
non-intuitive
Integration w.r.t
time does not
give orientation
Needs transform
Not Intuitive
Need transforms
Best for
analytical and
ACS design work
Best for
digital control
implementation
Must store
initial condition
Rigid Body Kinematics
Inertial
Frame
Time Derivatives:
(non-inertial)
X
Y
Z
Body
CM
Rotating
Body Frame
i
J
K
^
^
^
^
^
^
j
k
I
r
R
G85
G90 = Angular velocity
of Body Frame
BASIC RULE:
INERTIAL BODY
ρρωρ=+×
G6G6
Applied to
position vector r:
()
BODY
BODY BODY
2
rR
rR
rR
ρ
ρωρ
ρ ωρ ωρωωρ
=+
=+ +×
=+ + × +×+××
G6
G6G6
G6G6
G6G6 G6G6G6 G6
Position
Rate
Acceleration
Inertial
accel of CM
relative accel
w.r.t. CM
centripetal
coriolis
angular
accel
Expressed in
the Inertial Frame
Angular Momentum (I)
Angular Momentum
total
1
n
iii
i
Hrmr
=
=×
∑
G6
m1
mn
mi
X
Y
Z
Collection of point
masses mi at ri
ri
r1
rn
rn
ri
r1
.
.
.
System in
motion relative
to Inertial Frame
If we assume that
(a) Origin of Rotating Frame in Body CM
(b) Fixed Position Vectors ri in Body Frame
(Rigid Body)
Then :
BODY
total
11
ANGULAR MOMENTUM
OF TOTAL MASS W.R.T BODY ANGULAR
INERTIAL ORIGIN MOMENTUM ABOUT
CENTER OFMASS
nn
ii
ii
H
HmR mρρ
==
??
=×+×
??
??
∑∑
G0
G6
G6
G9G12G12G10G12G12G11
G9G12G12G10G12G12G11
Note that G85
i
is
measured in the
inertial frame
Angular Momentum Decomposition
Angular Momentum (II)
For a RIGID BODY
we can write:
,BODY
RELATIVE
MOTION IN BODY
ii i i
ρ ρωρωρ=+×=×
G6G6
G9G12G10G12G11
And we are able to write:
HIω=
“The vector of angular momentum in the body frame is the product
of the 3x3 Inertia matrix and the 3x1 vector of angular velocities.”
RIIGID BODY, CM COORDINATES
H and G90 are resolved in BODY FRAME
Inertia Matrix
Properties:
11 12 13
21 22 23
31 32 33
III
II I I
III
??
??
=
??
Real Symmetric ; 3x3 Tensor ; coordinate dependent
()
()
()
22
11 2 3 12 21 2 1
22
22 1 3 13 31 1 3
22
33 1 2 23 32 2 3
11
nn
ii i iii
ii i iii
nn
ii i iii
Im II m
Im II m
Im II m
ρ ρρρ
ρ
ρ ρρρ
==
==
=+ ==?
=+ ==?
=+ ==?
∑∑
∑∑
Kinetic Energy and Euler Equations
22
total
11
E-ROT
E-TRANS
22
nn
iii
EmRmρ
==
??
=+
??
??
∑∑
G6
G6
G9G12G10G12G11
G9G12G12G10G12G12G11
Kinetic
Energy
For a RIGID BODY, CM Coordinates
with G90 resolved in body axis frame
ROT
11
22
T
EHIωωω=?=
HT Iωω
??
=?×
??
G6
Sum of external and internal torques
In a BODY-FIXED, PRINCIPAL AXES CM FRAME:
1111 2 323
2 2 2 2 33 11 3 1
3333 1 212
()
()
()
HI TI I
HI TI I
HI TII
ωωω
ωωω
==+?
==+?
==+?
G6
G6
G6
G6
G6
G6
Euler Equations
No general solution exists.
Particular solutions exist for
simple torques. Computer
simulation usually required.
Torque Free Solutions of Euler’s Eq.
TORQUE-FREE
CASE:
An important special case is the torque-free motion of a (nearly)
symmetric body spinning primarily about its symmetry axis
By these assumptions:
,
xy z
ωω ω<< =? xx yy
II?
And the Euler equations become:
0
x
y
zz yy
xy
xx
K
zz xx
yy
yy
K
z
II
I
II
I
ωω
ωω
ω
?
=? ?
?
=?
=
G6
G9G12G10G12G11
G6
G9G12G10G12G11
G6
The components of angular velocity
then become:
() cos
() cos
xxn
yyn
ttωωω
ωωω
=
=
The G90
n
is defined as the “natural”
or “nutation” frequency of the body:
22
nxy
KKω =?
B
o
d
y
C
o
n
e
S
p
a
c
e
C
o
n
e
Z
H
G90
zxy
III<=
H
Z
G90
B
od
y
C
on
e
Space
Cone
zxy
III>=
G81 : nutation
angle
G81
G81
H and G90 never align
unless spun about
a principal axis !
Spin Stabilized Spacecraft
UTILIZED TO STABILIZE SPINNERS
G58
Xb
Yb
Zb
G129 Two bodies rotating at different rates
about a common axis
G129 Behaves like simple spinner, but part
is despun (antennas, sensors)
G129 requires torquers (jets, magnets) for
momentum control and nutation
dampers for stability
G129 allows relaxation of major axis rule
DUAL SPIN
Perfect Cylinder
BODY
G58
Antenna
despun at
1 RPO
2
2
2
43
2
xx yy
zz
mL
II R
mR
I
??
== +
??
??
=
Disturbance Torques
Assessment of expected disturbance torques is an essential part
of rigorous spacecraft attitude control design
G129 Gravity Gradient: “Tidal” Force due to 1/r2 gravitational field variation
for long, extended bodies (e.g. Space Shuttle, Tethered vehicles)
G129 Aerodynamic Drag: “Weathervane” Effect due to an offset between the
CM and the drag center of Pressure (CP). Only a factor in LEO.
G129 Magnetic Torques: Induced by residual magnetic moment. Model the
spacecraft as a magnetic dipole. Only within magnetosphere.
G129 Solar Radiation: Torques induced by CM and solar CP offset. Can
compensate with differential reflectivity or reaction wheels.
G129 Mass Expulsion: Torques induced by leaks or jettisoned objects
G129 Internal: On-board Equipment (machinery, wheels, cryocoolers, pumps
etc…). No net effect, but internal momentum exchange affects attitude.
Typical Disturbances
Gravity Gradient
Gravity Gradient:
1) ⊥ Local vertical
2) 0 for symmetric spacecraft
3) proportional to ∝ 1/r
3
Earth
r
^
-sinG84
Zb
Xb
G84
3
/ ORBITAL RATEnaμ==
2
??
3TnrIr
??
=?×
??
G0
Gravity Gradient
Torques
In Body Frame
[]
22
?
sin sin 1 sin sin 1
T
T
r θφ θ φ θφ
??
=? ? ? ??
??
Small
angle
approximation
Typical Values:
I=1000 kgm
2
n=0.001 s
-1
T= 6.7 x 10
-5
Nm/deg
Resulting torque in BODY FRAME:
2
()
3( )
0
zz yy
zz xx
II
TnII
φ
θ
?
??
??
∴? ?
??
??
()
3
xx zz
lib
yy
II
n
I
ω
?
=
Pitch Libration freq.:
Aerodynamic Torque
a
TrF=×
r = Vector from body CM
to Aerodynamic CP
Fa = Aerodynamic Drag Vector
in Body coordinates
2
1
2
aD
FVSCρ=
12
D
C≤≤
Aerodynamic
Drag Coefficient
Typically in this Range for
Free Molecular Flow
S = Frontal projected Area
V = Orbital Velocity
G85 = Atmospheric Density
Exponential Density Model
2 x 10
-9
kg/m
3
(150 km)
3 x 10
-10
kg/m
3
(200 km)
7 x 10
-11
kg/m
3
(250 km)
4 x 10
-12
kg/m
3
(400 km)
Typical Values:
Cd = 2.0
S = 5 m
2
r = 0.1 m
r = 4 x 10
-12
kg/m
3
T = 1.2 x 10
-4
Nm
Notes
(1) r varies with Attitude
(2) G85 varies by factor of 5-10 at
a given altitude
(3) C
D
is uncertain by 50 %
Magnetic Torque
TMB=×
B varies as 1/r3, with its direction
along local magnetic field lines.
B = Earth magnetic field vector in
spacecraft coordinates (BODY FRAME)
in TESLA (SI) or Gauss (CGS) units.
M = Spacecraft residual dipole
in AMPERE-TURN-m2 (SI)
or POLE-CM (CGS)
M = is due to current loops and
residual magnetization, and will
be on the order of 100 POLE-CM
or more for small spacecraft.
Typical Values:
B= 3 x 10
-5
TESLA
M = 0.1 Atm
2
T = 3 x 10
-6
Nm
Conversions:
1 Atm2 = 1000 POLE-CM , 1 TESLA = 104 Gauss
B ~ 0.3 Gauss
at 200 km orbit
Solar Radiation Torque
s
TrF=×
r = Vector from Body CM
to optical Center-of-Pressure (CP)
Fs = Solar Radiation pressure in
BODY FRAME coordinates
()
1
ss
FKPS=+
K = Reflectivity , 0 < K <1
S = Frontal Area
/
ss
PIc=
I
s
= Solar constant, depends on
heliocentric altitude
2
1400 W/m @ 1 A.U.
s
I =
Significant for
spacecraft
with large
frontal area
(e.g. NGST)
SUN
Typical Values:
K = 0.5
S =5 m
2
r =0.1 m
T = 3.5 x 10
-6
Nm
Notes:
(a) Torque is always ⊥ to sun line
(b) Independent of position or
velocity as long as in sunlight
Mass Expulsion and Internal Torques
Mass Expulsion Torque:
TrF=×
Notes:
(1) May be deliberate (Jets, Gas venting) or accidental (Leaks)
(2) Wide Range of r, F possible; torques can dominate others
(3) Also due to jettisoning of parts (covers, cannisters)
Internal Torque:
Notes:
(1) Momentum exchange between moving parts
has no effect on System H, but will affect
attitude control loops
(2) Typically due to antenna, solar array, scanner
motion or to deployable booms and appendages
Disturbance Torque for CDIO
ground
Air
Bearing
Body
CM
Pivot Point
Air Bearing
G39 offset
Expect residual
gravity torque to be
largest disturbance
Initial Assumption:
0.001 100 9.81 1 [Nm]Trmg=× ? ? ? ?
r
mg
Important
to balance
precisely !
Passive Attitude Control (1)
G129 Requires Stable Inertia Ratio: Iz > Iy =Ix
G129 Requires Nutation damper: Eddy Current, Ball-in-
Tube, Viscous Ring, Active Damping
G129 Requires Torquers to control precession (spin axis
drift) magnetically or with jets
G129 Inertially oriented
Passive control techniques take advantage of basic physical
principles and/or naturally occurring forces by designing
the spacecraft so as to enhance the effect of one force,
while reducing the effect of others.
G90
Precession:
G39H
H
G39G84
r
TrF=×
F into page
HTrF==
G6
SPIN STABILIZED
dH H
H
dt t
?
=?
?
G6
HrFt∴? ? ?
2sin
2
HH H I
θ
θωθ
?
?= ??= ??
rF t rF
t
HI
θ
ω
?
?? = ?
Large G90
=
gyroscopic
stability F
Passive Attitude Control (2)
GRAVITY GRADIENT
G129 Requires stable Inertias: I
z
<< I
x
, I
y
G129 Requires Libration Damper: Eddy Current,
Hysteresis Rods
G129 Requires no Torquers
G129 Earth oriented
G129 No Yaw Stability (can add momentum wheel)
Gravity Gradient with Momentum wheel:
nadir
d
o
w
n
f
o
r
w
a
r
d
Wheel spins
at rate G58
BODY rotates at
one RPO (rev per orbit)
O.N.
Gravity Gradient Configuration
with momentum wheel for
yaw stability
“DUAL SPIN” with GG
torque providing
momentum control
Active Attitude Control
G129 Reaction Wheels most common actuator
G129 Fast; continuous feedback control
G129 Moving Parts
G129 Internal Torque only; external still
required for “momentum dumping”
G129 Relatively high power, weight, cost
G129 Control logic simple for independent axes
(can get complicated with redundancy)
Active Control Systems directly sense spacecraft attitude
and supply a torque command to alter it as required. This
is the basic concept of feedback control.
Typical Reaction (Momentum) Wheel Data:
Operating Range: 0 +/- 6000 RPM
Angular Momentum @ 2000 RPM:
1.3 Nms
Angular Momentum @ 6000 RPM:
4.0 Nms
Reaction Torque: 0.020 - 0.3 Nm
Actuators: Reaction Wheels
G129 One creates torques on a spacecraft by creating equal but opposite
torques on Reaction Wheels (flywheels on motors).
— For three-axes of torque, three wheels are necessary. Usually use four
wheels for redundancy (use wheel speed biasing equation)
— If external torques exist, wheels will angularly accelerate to counteract
these torques. They will eventually reach an RPM limit (~3000-6000
RPM) at which time they must be desaturated.
— Static & dynamic imbalances can induce vibrations (mount on isolators)
— Usually operate around some nominal spin rate to avoid stiction effects.
Needs to be carefully balanced !
Ithaco RWA’s
(www.ithaco.com
/products.html)
Waterfall plot:
Actuators: Magnetic Torquers
G129 Often used for Low Earth Orbit
(LEO) satellites
G129 Useful for initial acquisition
maneuvers
G129 Commonly use for momentum
desaturation (“dumping”) in
reaction wheel systems
G129 May cause harmful influence on
star trackers
Magnetic Torquers
G129 Can be used
— for attitude control
— to de-saturate reaction wheels
G129 Torque Rods and Coils
— Torque rods are long helical coils
— Use current to generate magnetic
field
— This field will try to align with the
Earth’s magnetic field, thereby
creating a torque on the spacecraft
— Can also be used to sense attitude
as well as orbital location
ACS Actuators: Jets / Thrusters
G129 Thrusters / Jets
— Thrust can be used to control
attitude but at the cost of
consuming fuel
— Calculate required fuel using
“Rocket Equation”
— Advances in micro-propulsion
make this approach more feasible.
Typically want I
sp
> 1000 sec
G129 Use consumables such as Cold Gas
(Freon, N2) or Hydrazine (N2H4)
G129 Must be ON/OFF operated;
proportional control usually not
feasible: pulse width modulation
(PWM)
G129 Redundancy usually required, makes
the system more complex and
expensive
G129 Fast, powerful
G129 Often introduces attitude/translation
coupling
G129 Standard equipment on manned
spacecraft
G129 May be used to “unload” accumulated
angular momentum on reaction-wheel
controlled spacecraft.
ACS Sensors: GPS and Magnetometers
G129 Global Positioning System (GPS)
— Currently 27 Satellites
— 12hr Orbits
— Accurate Ephemeris
— Accurate Timing
— Stand-Alone 100m
— DGPS 5m
— Carrier-smoothed DGPS 1-2m
G129 Magnetometers
— Measure components Bx, By, Bz of
ambient magnetic field B
— Sensitive to field from spacecraft
(electronics), mounted on boom
— Get attitude information by
comparing measured B to modeled B
— Tilted dipole model of earth’s field:
3
29900
6378
0190
22 2 530
north
east
km
down
BCSCS
BSC
r
CS
??λ?λ
λλ
??λ?λ
??
??
?? ? ?
??
??
?? ? ?
=??
??
?? ? ?
??
?? ? ?
?? ?
?? ? ?
??
Where: C=cos , S=sin, φ=latitude, λ=longitude
Units: nTesla
+Y
+Z
flux
lines
+X
Me
ACS Sensors: Rate Gyros and IMUs
G129 Rate Gyros (Gyroscopes)
— Measure the angular rate of a
spacecraft relative to inertial space
— Need at least three. Usually use
more for redundancy.
— Can integrate to get angle.
However,
— DC bias errors in electronics
will cause the output of the
integrator to ramp and
eventually saturate (drift)
— Thus, need inertial update
G129 Inertial Measurement Unit (IMU)
— Integrated unit with sensors,
mounting hardware,electronics and
software
— measure rotation of spacecraft with
rate gyros
— measure translation of spacecraft
with accelerometers
— often mounted on gimbaled
platform (fixed in inertial space)
— Performance 1: gyro drift rate
(range: 0 .003 deg/hr to 1 deg/hr)
— Performance 2: linearity (range: 1
to 5E-06 g/g^2 over range 20-60 g
— Typically frequently updated with
external measurement (Star
Trackers, Sun sensors) via a
Kalman Filter
G129 Mechanical gyros
(accurate, heavy)
G129 Ring Laser (RLG)
G129 MEMS-gyros
Courtesy of Silicon Sensing Systems, Ltd. Used with permission.
ACS Sensor Performance Summary
Reference Typical
Accuracy
Remarks
Sun 1 min Simple, reliable, low
cost, not always visible
Earth 0.1 deg Orbit dependent;
usually requires scan;
relatively expensive
Magnetic Field 1 deg Economical; orbit
dependent; low altitude
only; low accuracy
Stars 0.001 deg Heavy, complex,
expensive, most
accurate
Inertial Space 0.01 deg/hour Rate only; good short
term reference; can be
heavy, power, cost
CDIO Attitude Sensing
Will not be able to
use/afford STAR TRACKERS !
From where do we get
an attitude estimate
for inertial updates ?
Potential Solution:
Electronic Compass,
Magnetometer and
Tilt Sensor Module
Problem: Accuracy insufficient to meet requirements alone,
will need FINE POINTING mode
Specifications:
Heading accuracy: +/- 1.0 deg RMS @ +/- 20 deg tilt
Resolution 0.1 deg, repeatability: +/- 0.3 deg
Tilt accuracy: +/- 0.4 deg, Resolution 0.3 deg
Sampling rate: 1-30 Hz
Spacecraft Attitude Schemes
G129 Spin Stabilized Satellites
— Spin the satellite to give it
gyroscopic stability in inertial
space
— Body mount the solar arrays to
guarantee partial illumination by
sun at all times
— EX: early communication
satellites, stabilization for orbit
changes
— Torques are applied to precess the
angular momentum vector
G129 De-Spun Stages
— Some sensor and antenna systems
require inertial or Earth referenced
pointing
— Place on de-spun stage
— EX: Galileo instrument platform
G129 Gravity Gradient Stabilization
— “Long” satellites will tend to point
towards Earth since closer portion
feels slightly more gravitational
force.
— Good for Earth-referenced pointing
— EX: Shuttle gravity gradient mode
minimizes ACS thruster firings
G129 Three-Axis Stabilization
— For inertial or Earth-referenced
pointing
— Requires active control
— EX: Modern communications
satellites, International Space
Station, MIR, Hubble Space
Telescope
ADCS Performance Comparison
Method Typical Accuracy Remarks
Spin Stabilized 0.1 deg Passive, simple; single axis
inertial, low cost, need slip
rings
Gravity Gradient 1-3 deg Passive, simple; central
body oriented; low cost
Jets 0.1 deg Consumables required, fast;
high cost
Magnetic 1 deg Near Earth; slow ; low
weight, low cost
Reaction Wheels 0.01 deg Internal torque; requires
other momentum control;
high power, cost
3-axis stabilized, active control most common choice for precision spacecraft
ACS Block Diagram (1)
Feedback Control Concept:
+
-
error
signal
gain
K
Spacecraft
Control
Actuators
Actual
Pointing
Direction
Attitude Measurement
c
TKθ=??
Correction
torque
= gain x error
desired
attitude
G84
G39G84
T
c G84
a
Force or torque is proportional to deflection. This
is the equation, which governs a simple linear
or rotational “spring” system. If the spacecraft
responds “quickly we can estimate the required
gain and system bandwidth.
Gain and Bandwidth
Assume control saturation half-width θ
sat
at torque command T
sat
, then
sat
sat
T
K
θ
?
hence
0
sat
K
I
θθ
??
+?
??
??
G6G6
Recall the oscillator frequency of a
simple linear, torsional spring:
[rad/sec]
K
I
ω=
I = moment
of inertia
This natural frequency is approximately
equal to the system bandwidth. Also,
12
[Hz] =
2f
f
ωπ
τ
πω
=?
Is approximately the system time constant G87.
Note: we can choose any two of the set:
,,
sat
θθ ω
G6G6
EXAMPLE:
2
10 [rad]
sat
θ
?
=
10 [Nm]
sat
T =
2
1000 [kgm ]I =
1000 [Nm/rad]K∴=
1 [rad/sec]ω=
0.16 [Hz]f =
6.3 [sec]τ=
Feedback Control Example
Pitch Control with a single reaction wheel
Rigid Body
Dynamics
B
O
D
Y
wext
ITT I Hθω=+==
G6G6 G6
G6
?
Wheel
Dynamics
()
w
JThθ?+ =? =
G6G6G6G6
Feedback
Law, Choose
G44
G44
wpr
TKKθθ=? ?
G6
Position
feedback
Rate
feedback
Then:
()
()
()
()
2
22
//0
//0
20
rp
rp
K I K I Laplace Transform
sKIsKI
ss
θθ
ζω ω
++=→
++=
++=
G6G6 G6
Characteristic Equation
r
/ =K / 2
pp
KI KIωζ=
Nat. frequency
damping
Stabilize
RIGID
BODY
Re
Im
Jet Control Example (1)
T
c
F
F
G84
l
l
Introduce control torque T
c
via
force couple from jet thrust:
c
ITθ=
G6G6
Only three possible values for T
c
:
0
c
Fl
T
Fl
?
?
=
?
?
?
?
Can stabilize (drive G84 to zero)
by feedback law:
On/Off
Control
only
()
sgn
c
TFlθτθ=? ? +
G6
prediction
term
Where
()
sgn
x
x
x
=
G87 = time constant
G84
G84
.
START
“PHASE PLANE”
SWITCH
LINE
“Chatter” due to minimum
on-time of jets.
Problem
c
TFl=?
c
TFl=
Jet Control Example (2)
“Chatter” leads to a
“limit cycle”, quickly
wasting fuel
Solution:
Eliminate “Chatter” by “Dead Zone” ; with Hysteresis:
G84
G84
.
“PHASE PLANE”
c
TFl=?
c
TFl=
At Switch Line:
0θτθ+=
G6
SL
c
θ
C
T
2
1
Is
1 sτ+
ε θτθ=+
G6
+
-
E
1
E
2
ε?
Results in the following motion:
G84
G84
.
DEAD ZONE
1
ε?
2
ε?
1
ε
2
ε
max
θ
max
θ
G6
Low Frequency Limit Cycle
Mostly Coasting
Low Fuel Usage
G84 and G84 bounded
.
ACS Block Diagram (2)
Spacecraft
+
+
+
dynamic
disturbances
sensor noise,
misalignment
target
estimate
true
accuracy + stability
knowledge error
control
error
Controller
Estimator
Sensors
In the “REAL WORLD” things are somewhat more complicated:
G129 Spacecraft not a RIGID body, sensor , actuator & avionics dynamics
G129 Digital implementation: work in the z-domain
G129 Time delay (lag) introduced by digital controller
G129 A/D and D/A conversions take time and introduce errors: 8-bit, 12-bit,
16-bit electronics, sensor noise present (e.g rate gyro @ DC)
G129 Filtering and estimation of attitude, never get q directly
Attitude Determination
G129 Attitude Determination (AD) is the process of of deriving estimates
of spacecraft attitude from (sensor) measurement data. Exact
determination is NOT POSSIBLE, always have some error.
G129 Single Axis AD: Determine orientation of a single spacecraft axis
in space (usually spin axis)
G129 Three Axis AD: Complete Orientation; single axis (Euler axis,
when using Quaternions) plus rotation about that axis
2
filtered/corrected
rate
1
estimated
quaternion
Wc comp rates
Switch1
Switch
NOT
Logical
Kalman
Fixed
Gain
KALMAN
Constant
2
inertial
update
1
raw
gyro rate
Example:
Attitude
Estimator
for NEXUS
Single-Axis Attitude Determination
G129 Utilizes sensors that yield an arc-
length measurement between
sensor boresight and known
reference point (e.g. sun, nadir)
G129 Requires at least two independent
measurements and a scheme to
choose between the true and false
solution
G129 Total lack of a priori estimate
requires three measurements
G129 Cone angles only are measured, not
full 3-component vectors. The
reference (e.g. sun, earth) vectors
are known in the reference frame,
but only partially so in the body
frame.
X
Y
Z
^
^
^
true
solution
a priori
estimate
false
solution
Earth
nadir
sun
Locus of
possible S/C
attitude from
sun cone angle
measurement
with error band
Locus of
possible attitudes
from earth cone
with error band
Three-Axis Attitude Determination
G129 Need two vectors (u,v) measured in
the spacecraft frame and known in
reference frame (e.g. star position
on the celestial sphere)
G129 Generally there is redundant data
available; can extend the
calculations on this chart to include
a least-squares estimate for the
attitude
G129 Do generally not need to know
absolute values
()
?
/
/
?
??
iuu
juvuv
kij
=
=× ×
=×
Define:
Want Attitude Matrix T:
??
?? ??
BBB RRR
MN
ijk Tijk
????
=?
????
G9G12G12G10G12G12G11G9G12G12G10G12G12G11
So:
1
TMN
?
=
Note: N must be non-singular (= full rank)
,uv
Effects of Flexibility (Spinners)
The previous solutions for Euler’s equations were only valid for
a RIGID BODY. When flexibility exists, energy dissipation will occur.
HIω= CONSTANT
Conservation of
Angular Momentum
ROT
1
2
T
EIωω=
DECREASING
∴
Spin goes to maximum
I and minimum G90
CONCLUSION: Stable Spin is
only possible about the axis of
maximum inertia.
Classical Example: EXPLORER 1
initial
spin
axis
energy dissipation
Controls/Structure Interaction
G84
Spacecraft
Sensor
Flexibility
G129 Can’t always neglect flexible modes (solar
arrays, sunshield)
G129 Sensor on flexible structure, modes introduce
phase loss
G129 Feedback signal “corrupted” by flexible
deflections; can become unstable
G129 Increasingly more important as spacecraft
become larger and pointing goals become tighter
-2000 -1500 -1000 -500 0 500 1000
-200
0
200
NM axis 1 to NM axis 1
Gain
[dB]
Phase [deg]
Loop Gain Function: Nichols Plot (NGST)
Flexible
modes
Stable
no encirclements
of critical point
Other System Considerations (1)
G129 Need on-board COMPUTER
— Increasing need for on-board performance and autonomy
— Typical performance (somewhat outdated: early 1990’s)
— 35 pounds, 15 Watts, 200K words, 100 Kflops/sec, CMOS
— Rapidly expanding technology in real-time space-based computing
— Nowadays get smaller computers, rad-hard, more MIPS
— Software development and testing, e.g. SIMULINK Real Time Workshop,
compilation from development environment MATLAB C, C++ to target
processor is getting easier every year. Increased attention on software.
G129 Ground Processing
— Typical ground tasks: Data Formatting, control functions, data analysis
— Don’t neglect; can be a large program element (operations)
G129 Testing
— Design must be such that it can be tested
— Several levels of tests: (1) benchtop/component level, (2) environmental
testing (vibration,thermal, vacuum), (3) ACS tests: air bearing, hybrid
simulation with part hardware, part simulated
Other System Considerations (2)
G129 Maneuvers
— Typically: Attitude and Position Hold,Tracking/Slewing, SAFE mode
— Initial Acquisition maneuvers frequently required
— Impacts control logic, operations, software
— Sometimes constrains system design
— Maneuver design must consider other systems, I.e.: solar arrays pointed
towards sun, radiators pointed toward space, antennas toward Earth
G129 Attitude/Translation Coupling
— G11G20G12 G39v from thrusters can affect attitude
— (2) Attitude thrusters can perturb the orbit
G129 Simulation
— Numerical integration of dynamic equations of motion
— Very useful for predicting and verifying attitude performance
— Can also be used as “surrogate” data generator
— “Hybrid” simulation: use some or all of actual hardware, digitally simulate
the spacecraft dynamics (plant)
— can be expensive, but save money later in the program
CM
F
l
T
T
(1)
(2)
F
1
F
1
= F
2
G39F
H/W
A/D
D/A
sim
Future Trends in ACS Design
G129 Lower Cost
— Standardized Spacecraft, Modularity
— Smaller spacecraft, smaller Inertias
— Technological progress: laser gyros, MEMS, magnetic wheel bearings
— Greater on-board autonomy
— Simpler spacecraft design
G129 Integration of GPS (LEO)
— Allows spacecraft to perform on-board navigation; functions independently
from ground station control
— Potential use for attitude sensing (large spacecraft only)
G129 Very large, evolving systems
— Space station ACS requirements change with each added module/phase
— Large spacecraft up to 1km under study (e.g. TPF Able “kilotruss”)
— Attitude control increasingly dominated by controls/structure interaction
— Spacecraft shape sensing/distributed sensors and actuators
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
-1
-0.5
0
0.5
1
y/R
o
(ve locity ve ctor)
Circular Paraboloid
E llips e
Optimal Focus (p/R
o
=2.2076)
Projected Circle
z/R
o
(Cros s axis )
Hyperbola (Foci)
x/
R
o
(Z
e
n
i
t
h
N
a
d
i
r)
Advanced ACS concepts
G129 No ?V required for collector
spacecraft
G129 Only need ?V to hold combiner
spacecraft at paraboloid’s focus
Visible Earth Imager using
a Distributed Satellite System
Exploit natural orbital dynamics to
synthesize sparse aperture arrays
using formation flying
Hill’s equations exhibit closed “free-
orbit ellipse” solutions
2
x
y
2
z
x 2yn 3n x a
y 2xn a
z n z a
??=
+=
+=
G6G6 G6
G6G6 G6
G6G6
Formation Flying in Space
TPF
ACS Model of NGST (large, flexible S/C)
gyro
Wt true rate
WheelsStructural Filters
Qt true attitude
Qt prop
PID
Controllers
K
Estimated
Inertia
Tensor
KF Flag
Attitude
Determination
K
ACS Rate
Matrix
Command
Rate
Command
Position
72 DOF
72
4
3
3
3
4
4
3 63 3
3
6x1
Forces &
Torques
PID bandwidth is 0.025 Hz
3rd order LP elliptic filters for
flexible mode gain suppression
Kalman Filter blends 10 Hz IRU and
2 Hz ST data to provide optimal attitude
estimate; option exists to disable the KF
and inject white noise, with amplitude given
by steady-state KF covariance into the
controller position channel
Wheel model includes non-linearities
and imbalance disturbances
FEM
“Open” telescope (no
external baffling) OTA
allows passive
cooling to ~50K
Deployable
secondary
Mirror (SM)
Beryllium
Primary mirror (PM)
Spacecraft support module
SSM (attitude control,
communications, power,
data handling)
arm side
Science
Instruments
(ISIM)
Large (200m
2
) deployable
sunshield protects from sun,
earth and moon IR radiation(ISS)
Isolation truss
cold side
NGST
ACS
Design
Attitude Jitter and Image Stability
Guider
Camera
*
*
roll about boresight produces
image rotation (roll axis shown
to be the camera boresight)
“pure” LOS error from
uncompensated high-frequency
disturbances plus guider NEA
total LOS error at target
is the RSS of these terms
FSM rotation while guiding on a
star at one field point produces
image smear at all other field points
Target
Guide Star
Important to assess impact of attitude jitter (“stability”) on image
quality. Can compensate with fine pointing system. Use a
guider camera as sensor and a 2-axis FSM as actuator.
Source: G. Mosier
NASA GSFC
Rule of thumb:
Pointing Jitter
RMS LOS < FWHM/10
E.g. HST: RMS LOS = 0.007 arc-seconds
References
G129 James French: AIAA Short Course: “Spacecraft Systems Design and
Engineering”, Washington D.C.,1995
G129 Prof. Walter Hollister: 16.851 “Satellite Engineering” Course Notes,
Fall 1997
G129 James R. Wertz and Wiley J. Larson: “Space Mission Analysis and
Design”, Second Edition, Space Technology Series, Space Technology
Library, Microcosm Inc, Kluwer Academic Publishers