Attitude ControlWeek-13

Fundamentals of Spacecraft Attitude Determination

and Control

F. Landis Markley • John L. Crassidis

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Introduction• Spacecraft attitude control is essential to

meet mission pointing requirements.

• There are two types of stabilization methodfor S/C.• Three-axis stabilization• Spin stabilization

– In three-axis stabilization method reactionwheels or control moment gyros spun upto maintain a certain level of gyroscopicstiffness and the wheel is aligned along thepitch axis.

• Early spacecraft mission designs used passive spin stabilization to hold one axisrelatively fixed by spinning the spacecraft around that axis, usually the axis ofmaximum moment of inertia.

• Spin stabilization was mostly used due to:• The limited control actuation and,• Lack of sophisticated computer technology to

implement complex control laws.

https://celestrak.com/columns/v04n09/

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• Spin-stabilized spacecraft are very stable, but they have to be sensitively balanced; every component has to be designed and located with spacecraft balance in mind.

• This can be extremely difficult to accomplish to the required accuracy.

• In most cases the last few weights are added and adjusted only after actual flight hardware is delivered and installed, and the spacecraft is experimentally spin tested.

• Allowances must also be made for everything onboard that can move during flight.

Introduction

SlideShare

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Control of Spinning S/C• S/C spinning about its minor axis is

unstable in the presence of internalenergy dissipation.

• But S/C often required to spin abouttheir minör axis for several reason.• Because of the fairing constrains require

that minor axis of of payload S/C be aligned with longitudinal axis of launchvehicle.

• Most launch vehicles spin about theirlongitidunal axis prior to payloadseperation, resulting in a minor axis spinof S/C after seperation.

• Some launch vehicles or upper stagesdo not have spin-up capability forpayload S/C and spin up of the S/C is achived after seperation.

• Because of initial angular rates at seperation, a typical spin up maneuverusually results in a residual nutationangle and spin-axis precession fromseperation attitude.

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Deagel.com

Spin movie

• S/C spinning about their minor axis are oftenstabilized using an active nutation control sytemconsisting of thrusters and accelerometers.

• Spinning S/C are also required to spin down orreorient their spin axis.

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Control of Spinning S/C

Spin-up Maneuver• An axisymmetric, spin stabilized S/C equipped with

axial and spin thrusters.

• The axial thrusters are used during spin-axisreorientation.

• Spin thrusters are used during spin-up or spin-down maneuvers as well as spin rate controls.

• Reference frame B is coincides with principal axes.

• The first and second axes are called transverseaxes.

• Third is axis of symmetry.

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• Euler’s rotational equations of motion of an axisymmetric S/C (J1=J2=J) and (M1=M2=0) is:

• 𝐽 ሶ𝜔1 − 𝐽 − 𝐽3 𝜔3𝜔2 = 0

• 𝐽 ሶ𝜔2 − 𝐽 − 𝐽3 𝜔3𝜔1 = 0

• 𝐽3 ሶ𝜔3 = 𝑀3

• M3 is constant spin-up moment.

• Kinematic equation for 𝐶1 𝜃1 − 𝐶2 𝜃2 − 𝐶3 𝜃3 sequence:

• Example 1:

• ሶ𝜃1 =1

𝑐𝑜𝑠𝜃2𝑐𝑜𝑠𝜃3𝜔1 − 𝑠𝑖𝑛𝜃3𝜔2

• ሶ𝜃2 = 𝑠𝑖𝑛𝜃3𝜔1 + 𝑐𝑜𝑠𝜃3𝜔2

• ሶ𝜃3 = 𝑡𝑎𝑛𝜃2 −𝑐𝑜𝑠𝜃3𝜔1 + 𝑠𝑖𝑛𝜃3𝜔2 +𝜔3

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Control of Spinning S/C

• Steady-State Precession Angle Equations foraxisymmetric body with 𝑱𝟏 = 𝑱𝟐 = 𝑱 ≠ 𝑱𝟑 given by:

• 𝜃1 ∞ = 𝜔1 0𝐽

2𝑀3

𝜋

2− 𝜔2 0

𝐽

2𝑀3

𝜋

2

• 𝜃2 ∞ = 𝜔1 0𝐽

2𝑀3

𝜋

2+ 𝜔2 0

𝐽

2𝑀3

𝜋

2

Example 2:

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Control of Spinning S/C

Flat Spin Transition Maneuver

• One of the simplest rotational maneuvers is thereorientation of the spin axis of a S/C usinginternal energy dissipation.

• A semirigid S/C with internal energy dissipation is stable only when spinning about its major axis.

• A S/C spinning about its minor axis in the presence of internal energy dissipation is unstable.

• S/C will eventually reorient to spin about its majoraxis.

• Such a passive reorientation maneuver is called a flat-spin transition maneuver.

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• S/C can end up with either a positive or negative spinabout the major axis.

• The rotational equations of motion are:𝐽1 − 𝐽 ሶ𝜔1 = 𝐽2 − 𝐽3 𝜔3𝜔2 + 𝜇𝜎1 +𝑀1𝐽2 − 𝐽 ሶ𝜔2 = 𝐽3 − 𝐽1 𝜔3𝜔1 + 𝜇𝜎2 +𝑀2𝐽3 − 𝐽 ሶ𝜔3 = 𝐽1 − 𝐽2 𝜔1𝜔2 + 𝜇𝜎3 +𝑀3ሶ𝜎1 = − ሶ𝜔1 − Τ𝜇 𝐽 𝜎1 − 𝜔2𝜎3 + 𝜔3𝜎2ሶ𝜎2 = − ሶ𝜔2 − Τ𝜇 𝐽 𝜎2 −𝜔3𝜎1 + 𝜔1𝜎3ሶ𝜎3 = − ሶ𝜔3 − Τ𝜇 𝐽 𝜎3 −𝜔1𝜎2 + 𝜔2𝜎1

• J1,J2,J3 are principal moments of inertia of S/C

• J spherical fuel slug of inertia

• 𝝈𝟏, 𝝈𝟐 , 𝝈𝟑 are relative rates between rigid body and fuel slug

• 𝝁 𝒊𝒔 viscous damping coefficient of the fuel slug

• M1, M2, M3 are control torques10

Flat Spin Transition Maneuver

Example 3:

• In the modern era advancements in sensors, actuators, and computer processors allow for three-axis stabilized spacecraft designs.

• Attitude control law theory also has been extensively studied and advanced, allowing for guaranteed control stability even with nonlinearattitude dynamics.

• The control of spacecraft for large angle slewing maneuvers poses a difficult problem, however.

• These difficulties include:• Highly nonlinear characteristics of

the governing equations, • Control rate, and saturation

constraints and limits, and• Incomplete state knowledge due to

sensor failure or omission.

Introduction

Nanosat Revolution

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• The control of spacecraft with large angle slews can be accomplished by either open-loop or closed-loop schemes.

• Open-loop schemes usually require a pre-determined pointing maneuver and are typically determined using optimal control techniques, which involve the solution of a two-point boundary value problem.

• Open-loop schemes are sensitive to spacecraft parameter uncertainties and unexpected disturbances.

• Closed-loop systems can account for parameter uncertainties and disturbances, and thus provide a more robust design methodology.

Introduction

SlideShare12

• The aforementioned techniques all utilize full state knowledge:• attitude and• rate feedback.

• The problem of controlling a spacecraft without full state feedback is more complex.

• The basic approaches used to solve this problem can be divided into methods which estimate the unmeasured states using a filter algorithm and methods which develop control laws directly from output feedback.

• Filtering methods such as the extended Kalman filter, have been successfullyapplied on numerous spacecraft systems without the use of rate-integrating gyro measurements.

Introduction

GitHub 13

• An advantage of these methods is that the attitude may be estimated by using only one set of vector attitude observations (such as magnetometer observations).

• However, these methods are usually much less accurate than methods that use gyro measurements.

• A more direct technique has been developed, which solves the attitude problem without rate knowledge.

• This method is based on a passivity approach, which replaces the rate feedback by a nonlinear filter of the quaternion.

• A model-based filter reconstructing the angular velocity is not needed in this case.

Introduction

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Quaternion-Feedback Reorientation Maneuvers

• Most three-axis stabilized S/C utilize a sequence of rotational maneuvers about each control axis.

• Many S/C also peforms rotational maneuvers aboutan inertially fixed axis during an acquisition mode, such as sun acquisition or Earth acquisition, so thata particular sensor will pick up a particular target.

• S/C are sometimes required to maneuver as fast as possible with in the physical limits of actuators andsensors.

• In this part, a feedback control logic for three-axis, large-angle reorientation maneuvers is introduced.

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Attitude Control: Regulation Case• Regulation control is defined as bringing the attitude to some fixed

location (usually the identity quaternion [0 0 0 1]T) and the angular velocity to zero [0 0 0]T.

• The quaternion attitude kinematics and Euler’s rotational equation of motion are given as:

ሶ𝐪 =

ሶ𝑞1ሶ𝑞2ሶ𝑞3ሶ𝑞4

=1

2𝚵 𝐪 𝛚 =

1

2

𝑞4 −𝑞3 𝑞2𝑞3 𝑞4 −𝑞1−𝑞2−𝑞1

𝑞1−𝑞2

𝑞4−𝑞3

𝜔1

𝜔2

𝜔3

ሶ𝐪 =

ሶ𝑞1ሶ𝑞2ሶ𝑞3ሶ𝑞4

=1

2𝛀 𝛚 𝐪 =

1

2

0 𝜔3 −𝜔2 𝜔1

−𝜔3 0 𝜔1 𝜔2

𝜔2 −𝜔1 0 𝜔3

−𝜔1 −𝜔2 −𝜔3 0

𝑞1𝑞2𝑞3𝑞4 Example 4:

Dynamic D.E. ሶ𝛚 =1

𝐉−𝛚 × 𝐉𝛚 + 𝐋

• Because quaternions are well suited for onboard real-time computation, S/C orientation is commonly described in terms of the quaternions.

• Linear state feedback controller of the following form can be considered for real-time implementation.

• 𝐋 = −𝑘𝑝𝐪𝐞 − 𝑘𝑑𝛚

• 𝐋 = −𝑘𝑝𝐬𝐢𝐠𝐧 𝑞4𝑒 𝐪𝐞 − 𝑘𝑑𝛚 Guarantee Shortest Path

• kp and kd are positive scalar gains.

• Attitude error quaternions 𝑞1𝑒 , 𝑞2𝑒 , 𝑞3𝑒, 𝑞4𝑒 arecomputed using the desired or commanded attitudequaternions 𝑞1𝑐 , 𝑞2𝑐 , 𝑞3𝑐 , 𝑞4𝑐 and the current attitudequaternions 𝑞1, 𝑞2, 𝑞3, 𝑞4 .

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Attitude Control: Regulation Case

•

𝑞1𝑒𝑞2𝑒𝑞3𝑒𝑞4𝑒

=

𝑞4𝑐 𝑞3𝑐 −𝑞2𝑐 −𝑞1𝑐−𝑞3𝑐 𝑞4𝑐 𝑞1𝑐 −𝑞2𝑐𝑞2𝑐 −𝑞1𝑐 𝑞4𝑐 −𝑞3𝑐𝑞1𝑐 𝑞2𝑐 𝑞3𝑐 𝑞4𝑐

𝑞1𝑞2𝑞3𝑞4

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Attitude Control: Regulation Case

Example 5:

• The aforementioned control laws can also be used with reaction wheels.

• For the reaction-wheel-only case expresses Euler’s rotational equation as:

• where J now includes the transverse inertia of the wheels, h is the wheel angular momentum, and ሶ𝐡 is the wheel torque.

• Note that the total angular momentum, Jω+h, is conserved.

• An equivalent but more useful form for control purposes is given by:

hhJJ

torquewheeleffective

L

Lhh

LJωωJ

Attitude Control: Regulation Case

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• Now define the following control laws for the wheel torques:

Attitude Control: Regulation Case

Example 6:

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Non-linear

Linear

Take Home Quiz-8

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• In Example-3 (Flat spin transition maneuverproblem) analyze:

• 𝜔1 𝑎𝑛𝑑 𝜔2

• 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒𝑠 𝑎𝑛𝑑 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠

• 𝑡𝑜 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝜔1 ∞ .