184:
de-spin antennas or optical instruments that must be pointed at targets for science observations or communications with Earth. Three-axis controlled craft can point optical instruments and antennas without having to de-spin them, but they may have to carry out special rotating maneuvers to best utilize their fields and particle instruments. If thrusters are used for routine stabilization, optical observations such as imaging must be designed knowing that the spacecraft is always slowly rocking back and forth, and not always exactly predictably. Reaction wheels provide a much steadier spacecraft from which to make observations, but they add mass to the spacecraft, they have a limited mechanical lifetime, and they require frequent momentum desaturation maneuvers, which can perturb navigation solutions because of accelerations imparted by the use of thrusters.
353:
which allows for the unity constraint on the quaternion to be better handled. It is also common to use a technique known as dynamic model replacement, where the angular rate is not estimated directly, but rather the measured angular rate from the gyro is used directly to propagate the rotational dynamics forward in time. This is valid for most applications as gyros are typically far more precise than one's knowledge of disturbance torques acting on the system (which is required for precise estimation of the angular rate).
1287:
259:
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255:. A rotation matrix, on the other hand, provides a full description of the attitude at the expense of requiring nine values instead of three. The use of a rotation matrix can lead to increased computational expense and they can be more difficult to work with. Quaternions offer a decent compromise in that they do not suffer from gimbal lock and only require four values to fully describe the attitude.
169:, must be occasionally removed from the system by applying controlled torque to the spacecraft to allowing the wheels to return to a desired speed under computer control. This is done during maneuvers called momentum desaturation or momentum unload maneuvers. Most spacecraft use a system of thrusters to apply the torque for desaturation maneuvers. A different approach was used by the
1533:. Conversely, by inducing a counter-current, using solar cell power, the orbit may be raised. Due to massive variability in Earth's magnetic field from an ideal radial field, control laws based on torques coupling to this field will be highly non-linear. Moreover, only two-axis control is available at any given time meaning that a vehicle reorient may be necessary to null all rates.
1372:(proportional to exhaust velocity) and the smallest torque impulse it can provide (which determines how often the thrusters must fire to provide precise control). Thrusters must be fired in one direction to start rotation, and again in the opposing direction if a new orientation is to be held. Thruster systems have been used on most crewed space vehicles, including
213:
can compute the proper direction to point the appendages. It logically falls to the same subsystem – the
Attitude and Articulation Control Subsystem (AACS), then, to manage both attitude and articulation. The name AACS may even be carried over to a spacecraft even if it has no appendages to articulate.
1104:
where a crystal cup shaped like a wine glass can be driven into oscillation just as a wine glass "sings" as a finger is rubbed around its rim. The orientation of the oscillation is fixed in inertial space, so measuring the orientation of the oscillation relative to the spacecraft can be used to sense
164:
back and forth between spacecraft and wheels. To rotate the vehicle on a given axis, the reaction wheel on that axis is accelerated in the opposite direction. To rotate the vehicle back, the wheel is slowed. Excess momentum that builds up in the system due to external torques from, for example, solar
1553:
technology demonstration. In low Earth orbit, the force due to drag is many orders of magnitude more dominant than the force imparted due to gravity gradients. When a satellite is utilizing aerodynamic passive attitude control, air molecules from the Earth's upper atmosphere strike the satellite in
212:
main engine nozzles were steerable. Knowing where to point a solar panel, or scan platform, or a nozzle — that is, how to articulate it — requires knowledge of the spacecraft's attitude. Because a single subsystem keeps track of the spacecraft's attitude, the Sun's location, and Earth's location, it
183:
There are advantages and disadvantages to both spin stabilization and three-axis stabilization. Spin-stabilized craft provide a continuous sweeping motion that is desirable for fields and particles instruments, as well as some optical scanning instruments, but they may require complicated systems to
77:
may be accurately pointed to Earth for communications, so that onboard experiments may accomplish precise pointing for accurate collection and subsequent interpretation of data, so that the heating and cooling effects of sunlight and shadow may be used intelligently for thermal control, and also for
1546:
hysteretic materials or a viscous damper. The viscous damper is a small can or tank of fluid mounted in the spacecraft, possibly with internal baffles to increase internal friction. Friction within the damper will gradually convert oscillation energy into heat dissipated within the viscous damper.
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to provide attitude control. Although a CMG provides control about the two axes orthogonal to the gyro spin axis, triaxial control still requires two units. A CMG is a bit more expensive in terms of cost and mass, because gimbals and their drive motors must be provided. The maximum torque (but not
352:
could be used, and can provide benefits in cases where the initial estimate is poor). Multiple methods have been proposed, however the
Multiplicative Extended Kalman Filter (MEKF) is by far the most common approach. This approach utilizes the multiplicative formulation of the error quaternion,
103:
Spin stabilization is accomplished by setting the spacecraft spinning, using the gyroscopic action of the rotating spacecraft mass as the stabilizing mechanism. Propulsion system thrusters are fired only occasionally to make desired changes in spin rate, or in the spin-stabilized attitude. If
1541:
Three main types of passive attitude control exist for satellites. The first one uses gravity gradient, and it leads to four stable states with the long axis (axis with smallest moment of inertia) pointing towards Earth. As this system has four stable states, if the satellite has a preferred
1500:. The upper end of the vehicle feels less gravitational pull than the lower end. This provides a restoring torque whenever the long axis is not co-linear with the direction of gravity. Unless some means of damping is provided, the spacecraft will oscillate about the local vertical. Sometimes
1545:
The second passive system orients the satellite along Earth's magnetic field thanks to a magnet. These purely passive attitude control systems have limited pointing accuracy, because the spacecraft will oscillate around energy minima. This drawback is overcome by adding damper, which can be
1474:
Small solar sails (devices that produce thrust as a reaction force induced by reflecting incident light) may be used to make small attitude control and velocity adjustments. This application can save large amounts of fuel on a long-duration mission by producing control moments without fuel
1080:
Many sensors generate outputs that reflect the rate of change in attitude. These require a known initial attitude, or external information to use them to determine attitude. Many of this class of sensor have some noise, leading to inaccuracies if not corrected by absolute attitude sensors.
1554:
such a way that the center of pressure remains behind the center of mass, similar to how the feathers on an arrow stabilize the arrow. GASPACS utilized a 1 m inflatable 'AeroBoom', which extended behind the satellite, creating a stabilizing torque along the satellite's velocity vector.
202:, for example, were designed with scan platforms for pointing optical instruments at their targets largely independently of spacecraft orientation. Many spacecraft, such as Mars orbiters, have solar panels that must track the Sun so they can provide electrical power to the spacecraft.
934:
Another important and common control algorithm involves the concept of detumbling, which is attenuating the angular momentum of the spacecraft. The need to detumble the spacecraft arises from the uncontrollable state after release from the launch vehicle. Most spacecraft in
1421:
driven rotors made to spin in the direction opposite to that required to re-orient the vehicle. Because momentum wheels make up a small fraction of the spacecraft's mass and are computer controlled, they give precise control. Momentum wheels are generally suspended on
1495:
In orbit, a spacecraft with one axis much longer than the other two will spontaneously orient so that its long axis points at the planet's center of mass. This system has the virtue of needing no active control system or expenditure of fuel. The effect is caused by a
372:, to complex nonlinear estimators or many in-between types, depending on mission requirements. Typically, the attitude control algorithms are part of the software running on the computer hardware, which receives commands from the ground and formats vehicle data
555:
1456:
the maximum angular momentum change) exerted by a CMG is greater than for a momentum wheel, making it better suited to large spacecraft. A major drawback is the additional complexity, which increases the number of failure points. For this reason, the
1359:
are the most common actuators, as they may be used for station keeping as well. Thrusters must be organized as a system to provide stabilization about all three axes, and at least two thrusters are generally used in each axis to provide torque as a
231:
it occupies. Attitude and position fully describe how an object is placed in space. (For some applications such as in robotics and computer vision, it is customary to combine position and attitude together into a single description known as
308:
systems allows for precise position knowledge to be obtained easily. This problem becomes more complicated for deep space vehicles, or terrestrial vehicles operating in Global
Navigation Satellite System (GNSS) denied environments (see
287:) from a set of measurements (often using different sensors). This can be done either statically (calculating the attitude using only the measurements currently available), or through the use of a statistical filter (most commonly, the
1223:
uses a horizon sensor to sense the direction to Earth's center, and a gyro to sense rotation about an axis normal to the orbit plane. Thus, the horizon sensor provides pitch and roll measurements, and the gyro provides yaw. See
299:
For some sensors and applications (such as spacecraft using magnetometers) the precise location must also be known. While pose estimation can be employed, for spacecraft it is usually sufficient to estimate the position (via
60:
to command the actuators based on (1) sensor measurements of the current attitude and (2) specification of a desired attitude. The integrated field that studies the combination of sensors, actuators and algorithms is called
383:, most spacecraft make use of active control which exhibits a typical attitude control loop. The design of the control algorithm depends on the actuator to be used for the specific attitude maneuver although using a simple
926:
839:
752:
1203:. This sensor provides orientation with respect to Earth about two orthogonal axes. It tends to be less precise than sensors based on stellar observation. Sometimes referred to as an Earth sensor.
658:
397:
The appropriate commands to the actuators are obtained based on error signals described as the difference between the measured and desired attitude. The error signals are commonly measured as
379:
The attitude control algorithms are written and implemented based on requirement for a particular attitude maneuver. Asides the implementation of passive attitude control such as the
368:
that receive data from vehicle sensors and derive the appropriate commands to the actuators to rotate the vehicle to the desired attitude. The algorithms range from very simple, e.g.
1525:
exert a moment against the local magnetic field. This method works only where there is a magnetic field against which to react. One classic field "coil" is actually in the form of a
415:
251:. While Euler angles are oftentimes the most straightforward representation to visualize, they can cause problems for highly-maneuverable systems because of a phenomenon known as
1798:
283:
Before attitude control can be performed, the current attitude must be determined. Attitude cannot be measured directly by any single measurement, and so must be calculated (or
1998:
993:
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are used to connect two parts of a satellite, to increase the stabilizing torque. A problem with such tethers is that meteoroids as small as a grain of sand can part them.
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to the vehicle. Their limitations are fuel usage, engine wear, and cycles of the control valves. The fuel efficiency of an attitude control system is determined by its
1330:
strength and, when used in a three-axis triad, magnetic field direction. As a spacecraft navigational aid, sensed field strength and direction is compared to a map of
585:
1433:
To maintain orientation in three dimensional space a minimum of three reaction wheels must be used, with additional units providing single failure protection. See
667:
making use of either momentum or reaction wheels as actuators. Based on the change in momentum of the wheels, the control law can be defined in 3-axes x, y, z as
1036:
1016:
605:
1399:
To minimize the fuel limitation on mission duration, auxiliary attitude control systems may be used to reduce vehicle rotation to lower levels, such as small
291:) that statistically combine previous attitude estimates with current sensor measurements to obtain an optimal estimate of the current attitude.
130:
is an alternative method of spacecraft attitude control in which the spacecraft is held fixed in the desired orientation without any rotation.
1312:(s) or a camera. It uses magnitude of brightness and spectral type to identify and then calculate the relative position of stars around it.
2121:
340:
Kalman filtering can be used to sequentially estimate the attitude, as well as the angular rate. Because attitude dynamics (combination of
344:
and attitude kinematics) are non-linear, a linear Kalman filter is not sufficient. Because attitude dynamics is not very non-linear, the
1802:
73:
A spacecraft's attitude must typically be stabilized and controlled for a variety of reasons. It is often needed so that the spacecraft
341:
1549:
A third form of passive attitude control is aerodynamic stabilization. This is achieved using a drag gradient, as demonstrated on the
1334:
stored in the memory of an on-board or ground-based guidance computer. If spacecraft position is known then attitude can be inferred.
847:
760:
673:
2005:
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without reliance on the observation of external objects. Classically, a gyroscope consists of a spinning mass, but there are also "
1899:
160:, also called momentum wheels, which are mounted on three orthogonal axes aboard the spacecraft. They provide a means to trade
1200:
1118:
610:
2137:
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304:) separate from the attitude estimation. For terrestrial vehicles and spacecraft operating near the Earth, the advent of
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orientation, e.g. a camera pointed at the planet, some way to flip the satellite and its tether end-for-end is needed.
1183:
This class of sensors sense the position or orientation of fields, objects or other phenomena outside the spacecraft.
1731:
1101:
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or torque rods as control actuators. The control law is based on the measurement of the rate of change of body-fixed
62:
1746:
Crassidis, John L.; Markley, F. Landis (May 23, 2012). "Unscented
Filtering for Spacecraft Attitude Estimation".
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to sense local gravity and force its gyro into alignment with Earth's spin vector, and therefore point north, an
2142:
1490:
380:
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in a planetary magnetic field. Such a conductive tether can also generate electrical power, at the expense of
550:{\displaystyle T_{c}(t)=K_{\text{p}}e(t)+K_{\text{i}}\int _{0}^{t}e(\tau )\,d\tau +K_{\text{d}}{\dot {e}}(t),}
2157:
2152:
1593:
1165:
409:. The PID controller which is most common reacts to an error signal (deviation) based on attitude as follows
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employ this method, and have used up about three quarters of their 100 kg of propellant as of July 2015.
1827:
326:
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sensing is often used, which senses the comparative warmth of the atmosphere, compared to the much colder
1457:
1195:
is an optical instrument that detects light from the 'limb' of Earth's atmosphere, i.e., at the horizon.
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of allowed attitude error. Thrusters may also be referred to as mass-expulsion control (MEC) systems, or
38:
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One method is to use small thrusters to continually nudge the spacecraft back and forth within a
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that accelerate ionized gases electrically to extreme velocities, using power from solar cells.
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2012, a high-altitude balloon-borne cosmology experiment launched from
Antarctica on 2012-12-29
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173:, which had sensitive optics that could be contaminated by thruster exhaust, and instead used
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1278:, but Earth sensors are still integrated in satellites for their low cost and reliability.
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Markley, F. Landis; Crassidis, John L. (2014), "Static
Attitude Determination Methods",
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1100:" utilizing coherent light reflected around a closed path. Another type of "gyro" is a
1021:
1001:
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A simple implementation of this can be the application of the proportional control for
590:
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99:
1907:
325:. Many solutions have been proposed, notably Davenport's q-method, QUEST, TRIAD, and
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Another method for achieving three-axis stabilization is to use electrically powered
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Attitude control of spacecraft is maintained using one of two principal approaches:
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Attitude can be described using a variety of methods; however, the most common are
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2044:"Studying the Effects of Disturbance Torques on a 2U CubeSat in Low Earth Orbits"
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939:(LEO) makes use of magnetic detumbling concept which utilizes the effect of the
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adjusted its attitude using its solar cells and antennas as small solar sails.
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guidance: short propulsive maneuvers must be executed in the right direction.
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to apply the torques needed to orient the vehicle to a desired attitude, and
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probes in the outer Solar System are examples of spin-stabilized spacecraft.
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1300:
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is usually sufficient (however
Crassidis and Markely demonstrated that the
248:
1999:"Investigation of Pulsed Plasma Thrusters for Spacecraft Attitude Control"
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1212:
401:(Φ, θ, Ψ), however an alternative to this could be described in terms of
252:
1196:
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desired, the spinning may be stopped through the use of thrusters or by
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Attitude and
Determination Control Systems for the OUFTI nanosatellites
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Crassidis, John L., and John L. Junkins.. Chapman and Hall/CRC, 2004.
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Attitude is part of the description of how an object is placed in the
1309:
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The STARS real-time star tracking software operates on an image from
1251:
1089:
943:. The control algorithm is called the B-Dot controller and relies on
373:
149:
143:
53:
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is the commanded magnetic dipole moment of the magnetic torquer and
1977:
Magnetically suspended momentum wheels for spacecraft stabilization
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1649:
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Attitude control can be obtained by several mechanisms, including:
1216:
134:
1125:. They have applications outside the aeronautical field, such as:
19:"Attitude control" redirects here. For the use in psychology, see
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258:
49:
921:{\displaystyle T_{c}z=-K_{\text{q3}}q_{3}+K_{\text{w3}}{w_{z}},}
834:{\displaystyle T_{c}y=-K_{\text{q2}}q_{2}+K_{\text{w2}}{w_{y}},}
747:{\displaystyle T_{c}x=-K_{\text{q1}}q_{1}+K_{\text{w1}}{w_{x}},}
2116:
1452:
2092:"GASPACS Get Away Special Passive Attitude Control Satellite"
1716:
Fundamentals of
Spacecraft Attitude Determination and Control
1551:
Get Away
Special Passive Attitude Control Satellite (GASPACS)
1426:
to avoid bearing friction and breakdown problems. Spacecraft
1267:
1105:
the motion of the spacecraft with respect to inertial space.
664:
1975:
Henrikson, C.H.; Lyman, J.; Studer, P.A. (January 1, 1974).
1903:
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uses a set of four CMGs to provide dual failure tolerance.
1305:
192:
Many spacecraft have components that require articulation.
1646:"Basics of Space Flight Section II. Space Flight Projects"
16:
Process of controlling orientation of an aerospace vehicle
1243:
1117:
with single- or multi-axis motion sensors. They utilize
1304:
is an optical device that measures the position(s) of
931:
This control algorithm also affects momentum dumping.
653:{\displaystyle K_{\text{p}},K_{\text{i}},K_{\text{d}}}
1695:. Basics of Spaceflight Section II (Report). NASA JPL
1274:; nowadays the main method to detect attitude is the
1067:
is the rate of change of the Earth's magnetic field.
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These are rotors spun at constant speed, mounted on
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Static attitude estimation methods are solutions to
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High speed craft motion control and damping systems
33:is the process of controlling the orientation of a
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987:
920:
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1121:. Some multi-axis MRUs are capable of measuring
335:
1822:
1820:
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1242:is a device that senses the direction to the
1817:
1178:
1075:
1536:
1406:
386:proportional–integral–derivative controller
45:, certain fields, and nearby objects, etc.
37:(vehicle or satellite) with respect to an
2067:
1748:Journal of Guidance, Control and Dynamics
1266:is a device that senses the direction to
1250:and shades, or as complex as a steerable
1164:Orientation and attitude measurements on
1108:
503:
278:
81:
1854:"Gyrocompass for Orbital Space Vehicles"
1851:
1285:
257:
23:. For attitude control of aircraft, see
1801:. Kongsberg Maritime AS. Archived from
1718:, Springer New York, pp. 183–233,
1440:
607:is the attitude deviation signal, and
2130:
1211:Similar to the way that a terrestrial
1206:
376:for transmission to a ground station.
48:Controlling vehicle attitude requires
2048:Journal of Physics: Conference Series
2032:. Vincent Francois-Lavet (2010-05-31)
2004:. Erps.spacegrant.org. Archived from
1928:
1693:"Chapter 11. Typical Onboard Systems"
1430:often use mechanical ball bearings.
1254:, depending on mission requirements.
1170:Remotely operated underwater vehicles
1132:motion compensation and stabilization
1113:Motion reference units are a kind of
1507:
1161:Offshore structure motion monitoring
294:
2042:Mohammad Nusrat Aman, Asma (2019).
1092:are devices that sense rotation in
660:are the PID controller parameters.
13:
1152:Motion compensation of single and
317:Static attitude estimation methods
14:
2169:
2109:
1929:Acuña, Mario H. (November 2002).
1186:
2115:
1935:Review of Scientific Instruments
1828:Spacecraft Earth Horizon Sensors
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1564:
1364:in order to prevent imparting a
1246:. This can be as simple as some
63:guidance, navigation and control
52:to measure vehicle orientation,
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2069:10.1088/1742-6596/1152/1/012024
2035:
2023:
1991:
1968:
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1774:"Hemispherical Resonator Gyros"
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187:
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1521:or (on very small satellites)
1491:Gravity-gradient stabilization
1485:Gravity-gradient stabilization
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1166:Autonomous underwater vehicles
988:{\displaystyle m=-K{\dot {B}}}
541:
535:
500:
494:
460:
454:
435:
429:
394:satisfies most control needs.
381:gravity-gradient stabilization
41:or another entity such as the
1:
1833:(Report). NASA. December 1969
1609:
1594:Longitudinal static stability
1231:
1084:
1038:is the proportional gain and
336:Sequential estimation methods
1345:
1337:
1102:hemispherical resonator gyro
327:singular value decomposition
7:
2138:Spacecraft attitude control
2122:Spacecraft attitude control
1931:"Space-based magnetometers"
1724:10.1007/978-1-4939-0802-8_5
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1458:International Space Station
357:Attitude control algorithms
216:
177:for desaturation maneuvers.
68:
39:inertial frame of reference
31:Spacecraft attitude control
10:
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1906:. May 2004. Archived from
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1488:
1475:expenditure. For example,
1467:
1444:
1410:
1349:
1149:Hydro acoustic positioning
1123:roll, pitch, yaw and heave
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1060:{\displaystyle {\dot {B}}}
262:Changing orientation of a
220:
97:
18:
1878:10.1134/S0010952521030011
1179:Absolute attitude sensors
1115:inertial measurement unit
1076:Relative attitude sensors
1671:"Voyager Weekly Reports"
1537:Passive attitude control
1447:Control moment gyroscope
1407:Reaction/momentum wheels
1326:is a device that senses
141:(RCS). The space probes
139:reaction control systems
126:Three-axis stabilization
25:Aircraft flight dynamics
1852:Abezyaev, I.N. (2021).
1604:Reaction control system
1352:Reaction control system
1158:Ocean wave measurements
1094:three-dimensional space
587:is the control torque,
350:Unscented Kalman filter
1626:. NASA. March 26, 2007
1624:"The Pioneer Missions"
1332:Earth's magnetic field
1295:
1174:Ship motion monitoring
1154:multibeam echosounders
1109:Motion reference units
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989:
941:Earth's magnetic field
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346:Extended Kalman filter
279:Attitude determination
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171:Hubble Space Telescope
82:Types of stabilization
2143:Aerospace engineering
1779:. Northropgrumman.com
1599:Directional stability
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580:{\displaystyle T_{c}}
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2158:Dynamics (mechanics)
2153:Spaceflight concepts
2124:at Wikimedia Commons
1441:Control moment gyros
1417:Momentum wheels are
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370:proportional control
306:Satellite navigation
2060:2019JPhCS1152a2024N
1947:2002RScI...73.3717A
1870:2021CosRe..59..204A
1270:. It is usually an
1221:orbital gyrocompass
1207:Orbital gyrocompass
1136:Dynamic positioning
490:
342:rigid body dynamics
302:Orbit determination
223:Attitude (geometry)
165:photon pressure or
1799:"MRU Applications"
1586:Spaceflight portal
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1143:of offshore cranes
1141:Heave compensation
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362:Control algorithms
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100:Spin stabilization
93:Spin stabilization
2120:Media related to
2011:on April 22, 2014
1955:10.1063/1.1510570
1941:(11): 3717–3736.
1527:conductive tether
1523:permanent magnets
1514:Magnetic torquers
1508:Magnetic torquers
1424:magnetic bearings
1357:Vernier thrusters
1226:Tait-Bryan angles
1201:cosmic background
1054:
1031:{\displaystyle K}
1011:{\displaystyle m}
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703:
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634:
621:
600:{\displaystyle e}
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366:computer programs
295:Position/location
241:Rotation matrices
175:magnetic torquers
167:gravity gradients
75:high-gain antenna
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1197:Thermal infrared
1098:ring laser gyros
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1119:MEMS gyroscopes
1111:
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274:attached to it.
272:reference frame
266:is the same as
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158:reaction wheels
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21:Attitude change
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2110:External links
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1990:
1979:(Report). NASA
1967:
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1864:(3): 204–211.
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1754:(4): 536–542.
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1413:Momentum wheel
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270:the axes of a
221:Main article:
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98:Main article:
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26:
22:
2095:. Retrieved
2086:
2051:
2047:
2037:
2025:
2015:September 9,
2013:. Retrieved
2006:the original
1993:
1983:December 30,
1981:. Retrieved
1970:
1960:December 30,
1958:. Retrieved
1938:
1934:
1924:
1912:. Retrieved
1908:the original
1894:
1861:
1857:
1847:
1835:. Retrieved
1807:. Retrieved
1803:the original
1793:
1783:September 9,
1781:. Retrieved
1768:
1751:
1747:
1741:
1715:
1709:
1697:. Retrieved
1687:
1675:. Retrieved
1665:
1653:. Retrieved
1640:
1628:. Retrieved
1618:
1548:
1544:
1540:
1517:
1494:
1476:
1473:
1450:
1435:Euler angles
1432:
1416:
1398:
1355:
1341:
1323:magnetometer
1321:
1319:
1316:Magnetometer
1301:star tracker
1299:
1297:
1282:Star tracker
1276:star tracker
1264:Earth sensor
1263:
1261:
1258:Earth sensor
1237:
1235:
1220:
1210:
1192:
1190:
1182:
1112:
1088:
1079:
997:
949:magnetometer
933:
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662:
559:
399:euler angles
396:
389:
385:
378:
360:
339:
331:
320:
298:
282:
249:Euler angles
238:
226:
204:
197:
191:
188:Articulation
182:
148:
142:
123:
115:
109:
90:
85:
72:
47:
30:
29:
2097:November 3,
1809:January 29,
1498:tidal force
1464:Solar sails
1366:translation
1248:solar cells
1213:gyrocompass
407:quaternions
253:Gimbal lock
245:Quaternions
2132:Categories
1837:January 1,
1699:January 1,
1673:. Nasa.gov
1630:January 1,
1610:References
1478:Mariner 10
1470:Solar sail
1392:, and the
1308:(s) using
1239:Sun sensor
1232:Sun sensor
1090:Gyroscopes
1085:Gyroscopes
311:Navigation
264:rigid body
117:Pioneer 11
111:Pioneer 10
58:algorithms
35:spacecraft
2078:127003967
1886:254423773
1346:Thrusters
1338:Actuators
1310:photocell
1252:telescope
1052:˙
980:˙
968:−
951:signals.
868:−
781:−
694:−
530:˙
508:τ
498:τ
478:∫
374:telemetry
285:estimated
150:Voyager 2
144:Voyager 1
54:actuators
1677:July 15,
1655:July 15,
1650:Nasa.gov
1558:See also
1217:pendulum
268:rotating
217:Geometry
135:deadband
69:Overview
2056:Bibcode
1943:Bibcode
1914:May 25,
1866:Bibcode
1502:tethers
1453:gimbals
1378:Mercury
1215:uses a
1130:Antenna
1071:Sensors
205:Cassini
199:Galileo
194:Voyager
50:sensors
2148:Orbits
2076:
1884:
1730:
1386:Apollo
1382:Gemini
1374:Vostok
1362:couple
998:where
560:where
247:, and
108:. The
2074:S2CID
2009:(PDF)
2002:(PDF)
1882:S2CID
1831:(PDF)
1777:(PDF)
1519:Coils
1390:Soyuz
1268:Earth
229:space
208:'
2099:2022
2052:1155
2017:2013
1985:2022
1962:2022
1916:2012
1904:NASA
1839:2023
1811:2015
1785:2013
1728:ISBN
1701:2023
1679:2015
1657:2015
1632:2023
1306:star
1292:EBEX
1168:and
364:are
234:Pose
196:and
147:and
114:and
2064:doi
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