GN/MAE155A 1

Introduction to Attitude Control Systems

MAE 155A

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Determination & Attitude Control Systems (DACS)

• Introduction

• DACS Basics

• Attitude Determination and Representation

• Basic Feedback Systems

• Stabilization Approaches

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Determination & Attitude Control Systems (DACS)

• Control of SC orientation: Yaw, Pitch, Roll• 3 Components to DACS:

– Sensor: Measure SC attitude– Control Law: Calculate Response– Actuator: Response (Torque)

• Example: Hubble reqts 2x10-6 deg pointing accuracy => equivalent to thickness of human hair about a mile away!

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Determination & Attitude Control Systems (DACS)

• Introduction

• DACS Basics

• Attitude Determination and Representation

• Basic Feedback Systems

• Stabilization Approaches

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DACS Basics

TorquersThrusters

Reaction WheelsMomentum Wheels

CMGs

S/C

SensorsGyros

Horizon Sensors

Sun Sensors

Correction Attitude Errors

ComputerControl

Law

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Determination & Attitude Control Systems (DACS)

•Spinning Spacecraftprovide simple pointingcontrol along single axis(low accuracy)

•Three axis stabilityprovides high accuracypointing control in anydirection

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DACS Design

Considerations:•Mission Reqts•Disturbance Calcs•DACS System Design

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DACS Reqts Definition• Summarize mission pointing reqts

– Earth (Nadir), Scanning, Inertial

• Mission & PL Pointing accuracy– Note that pointing accuracy is influenced by all 3 DACS components– Pointing accuracy can range from < 0.001 to 5 degrees

• Define Rotational and translational reqts for mission: Magnitude, rate and frequency

• Calculate expected torque disturbances• Select ACS type; Select HW & SW• Iterate/improve as necessary

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Torque Disturbances• External

– Gravity gradient: Variable g force on SC– Solar Pressure: Moment arm from cg to solar c.p.– Magnetic: Earth magnetic field effects– Aero. Drag: Moment arm from cg to aero center

• Internal– Appendage motion, pointing motors- misalign, slosh

• Cyclic and secular– Cyclic: varies in sinusoidal manner during orbit– Secular: Accumulates with time

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Determination & Attitude Control Systems (DACS)

• Introduction

• DACS Basics

• Attitude Determination and Representation

• Basic Feedback Systems

• Stabilization Approaches

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SC Attitude Determination Fundamentals

• Attitude determination involves estimating the orientation of the SC wrt a reference frame (usually inertial or geocentric), the process involves:– Determining SC body reference frame location from

sensor measurements

– Calculating instantaneous attitude wrt reference frame

– Using attitude measurement to correct SC pointing using actuators (or torquers)

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Basic SC Attitude Determination

Sensor Data

• Gyros• Star/Sun Sensor• Magnetometer

State Estimation

• Batch Estimators• Least Squares• Kalman Filtering

Attitude Calculation

• Euler Angles• DCM• Quaternions

Control LawUsed to Determine

Required Correction

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Attitude Sensors

• Performance requirements based on mission• Weight, power and performance trades performed to select optimal sensor• Multiple sensors may be used

Sensor Typical Performance (deg) Horizon 0.02 – 0.1 Magnetometer 0.5 – 1 Star Tracker 0.0002 – 0.08 Star Scanner 0.003 – 0.1 Gyro (Measures Rates) -

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State Estimation Approaches

• Estimate SC orientation using data measurements• Estimates typically improve as more data are

collected (assuming no ‘jerk motion’)• Estimation theory and statistical methods are used to

obtain best values– Least squares and Kalman filtering are most common

approaches

• Least squares minimizes square of error (assumes Gaussian error distribution)

• Kalman filter minimizes variance

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SC Attitude Representation

• SC frame of reference typically points SC Z axis anti-Nadir, and X axis in direction of velocity vector

• Relationship between SC and inertial reference frame can be defined by the 3 Euler angles (Yaw, Pitch and Roll)– Note that both magnitude and sequence of rotation

affect transformation between SC and inertial reference frame

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SC Attitude Representation Using Euler Rotation Angles

• The Direction Cosine Matrix (DCM) is the product of the 3 Euler rotations in the appropriate sequence (Yaw-Pitch-Roll)

DCM ~ R = R1 * R2 * R3

Ref: Brown, Elements of SC Design

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Direction Cosine Rotation Matrix

Ref: Brown, Elements of SC Design

R

cos cos

cos sin sin sin cos

sin sin cos sin cos

cos sin

cos cos sin sin sin

sin cos cos sin sin

sin

sin cos

cos cos

The DCM is given by:

Note that each transformation requires substantial arithmetic andtrigonometric operations, rendering it computationally intensive

An alternative, and less computationally intensive, approach to using DCM involves the use of Quaternions

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Quaternion Definition

• Euler’s theorem states that any series of rotation of a rigid body can be represented as a single rotation about a fixed axis– Orientation of a body axis can be defined by a

vector and a rotation about that vector– A quaternion, Q, defines the body axis vector

and the scalar rotation => 4 elements• Q = i.q1 + j.q2 + k.q3 + q4 , where

i2 = j2 = k2 = -1

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Basic Quaternion Properties• Given the quaternion Q, where

Q = i.q1 + j.q2 + k.q3 + q4 ; we have

ij = - ji = k; jk = -kj = i; ki = -ik = j• Two quaternions, Q and P are equal iff all their

elements are equal, i.e., q1 = p1 ; q2 = p2 ; q3 = p3

• Quaternion multiplication is order dependent, R=Q*P is given by: R = (i.q1 + j.q2 + k.q3 + q4)*(i.p1 + j.p2 + k.p3 + p4)

• The conjugate of Q is given by Q*, whereQ* = -i.q1 - j.q2 - k.q3 + q4

• The inverse of Q, Q-1 = Q* when Q is normalized

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Basic Quaternion Properties

DCM

2 q4 2 2 q

1 2 1

2 q1

q2

2 q4

q3

2 q1

q3

2 q4

q2

2 q1

q2

2 q4

q3

2 q4 2 2 q

2 2 1

2 q2

q3

2 q4

q1

2 q1

q3

2q4

q2

2 q2

q3

2 q4

q1

2 q4 2 2 q

3 2 1

The DCM can be expressed in terms of quaternion elements as:

• The quaternion transforming frame A into frame B is given by: VB = Qa

b VA (Qab)*

• Quaternions can also be combined as: Qa

c = Qbc Qa

b

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Comparison of 3-Axis Attitude Representation

Ref: Brown, Elements of SC Design

Method Advantages Disadvantages Euler Angles -No redundant

parameters -Clear physical interpretation

-Singularities -Trig functions -No convenient product rule

Direction Cosine -No singularities -Good physical representation -Convenient product rule

-Six redundant parameters -Trig functions

Quaternions -No singularities -No trig functions -Convenient product rule

-One redundant parameter -No physical meaning

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Determination & Attitude Control Systems (DACS)

• Introduction

• DACS Basics

• Attitude Determination and Representation

• Basic Feedback Systems

• Stabilization Approaches

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Feedback Loop Systems

• The control loop can use either an open or closed system. Open loop is used when low accuracy is sufficient, e.g., pointing of solar arrays.

• Generic closed-loop system:

Ref: Brown, Elements of SC Design

Control Law,Actuators

Spacecraft Dynamics

OutputReference

Disturbance

Error

aa

r

e = r + a

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Basic Rotation Equations Review

Angular displacement: = 1/2 t2 = d /dt(note ‘burn’ vs. maneuvering time)

Angular speed: = t

Angular acceleration: = T/Iv Angular Momentum: H = Iv => H = T t

Where,

~ rotation angle; ~ angular acceleration T ~ torque; Iv ~ SC moment of Inertia H ~ Angular Momentum; ~angular speed

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Basic Rotation Equations Review Torque equations: T = dH/dt = Iv d /dt = Iv

d2 / dt2 (Iv assumed constant)Note that the above equations are scalar representations of their vector forms (3D)

Hy

Hx

Spin axis precession

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Determination & Attitude Control Systems (DACS)

• Introduction

• DACS Basics

• Attitude Determination and Representation

• Basic Feedback Systems

• Stabilization Approaches

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Spin Stabilized Systems

• Spinning SC (spinner): resists disturbance toques (gyroscopic effect)– Disturbance along H vector affects spin rate

– Disturbance perp. to H => Precession

– Adv: Low cost, simple, no propel mgmt

– Disadv:- Low pointing accuracy (> 0.3 deg) - I about spinning axis >> other I - Limited maneuvering, pointing

• Dual spin systems: major part of SC spins while a platform (instruments) is despun

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SC Stabilization Systems

r1

r2

- Gravity Gradient (G2) Systems (passive):Takes adv of SC tendency to align

its long axis along g vector, g = GM/r; r1<r2 => F1>F2 => Restoring Torque

-Momentum Bias: Use momentum wheel to provide inertial stiffness in 2 axes, wheel speed provides control in 3rd axis

F2

F1

SCwheel Pitch axis (y)

NadirV

Stability condition: Ir r > (Ixx-Iyy) y

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SC Stabilization Systems

• Reaction Wheels (RW)– Motor spins a small free rotating wheel aligned

w. vehicle control axis (~low RPM)– One wheel per axis needed, however, additional

wheels are used for redundancy– RW only store, not remove torques

• Counteracting external torque is needed to unload the stored torque, e.g., magnetic or rxn jets (momentum dumping)

– Speed of wheel is adjusted to counter torque

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SC Stabilization Systems• RW at high RPM are termed momentum

wheels.– Also provides gyroscopic stability– Magnetic (torque) coils can be used to

continuously unload wheel• Wheels provide stability during

periods of high torque disturbances

– Control Moment Gyro• Gimbaled momentum wheel

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SC Stabilization Systems (External):

• Thrusters: Used to provide torque (external) on SC

• Magnetic torque rods– Can be used to provide a controlled external

torque on SC• Need to minimize potential residual disturbance

torqueT = M x B where M~dipole w. magnetic moment M B~Local Flux density

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Reaction Wheels

Magnetic Torquers

Magnetometers

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DACS Summary

Reqt GG Spin Dual Spin 3 Axis Momentum Bias

Nadir Pointing

Yes No Poor OK OK

Geo. NO OK OK OK OK Planetary No OK Ok Ok OK TVC No Good Good OK NO Maneuver. No Limited Limited Good Poor Pointing Acc. (deg)

5 1 0.1 0.001 0.1 to 3

Relative Cost

Small 1 1.2 2.1 1.5

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Conclusions

• Examples

• References

• Discussion & Questions