N

Power • Solar Array Pointing Required?

Communications

• Antenna Pointing Accuracy

Structures • Center ot

Constraints

• Inertia Constraints

• Flexibility Constraints

• Thruster Location

• Sensor Mounting

Fig. 11-2. The Impact of Mission Requirements and Other Subsystems on the ADCS Subsystem. Direction of arrows shows requirements flow from one'subsystem to another.

In most cases, we do not need to rotate the spacecraft quickly. But retargeting time may be critical for some applications. In either case, slewing mainly influences the choice and size of actuators. For example, the vehicle's maximum slew rate determines the thrusters' size or the reaction wheel's maximum torque. High-rate maneuvers may require other actuation systems, such as a second set of high-thrust reaction jets or perhaps control moment gyros.

For FireSat, we assume that the launch vehicle places us in our final orbit, with no need for ADCS control during orbit insertion. The normal pointing requirement is 0.1 deg, nadir-oriented. Attitude determination must be autonomous, providing Earth-relative knowledge better than 0.1 deg (to support the pointing requirement) while the vehicle is within 30 deg of nadir. In addition to these basic requirements, we will consider an optional requirement for occasional repointing of the spacecraft to a region of interest We want to examine how such a requirement would influence the design, increasing the complexity and capability of the ADCS. For this option, we will assume the requirement to repoint the vehicle once every 30 days. It must repoint, or slew, up to 30 deg in under 10 min, and hold the relative nadir orientation for 90 min.

11.1.2 Selection of Spacecraft Control Type

Once we have defined the subsystem requirements, we are ready to select a method of controlling the spacecraft. Table 11-4 lists several different methods of control, along with typical characteristics of each.

TABLE 11-4. Attitude Control Methods and Their Capabilities. As requirements become tighter, more complex control systems become necessary.

Type

Pointing Options

Attitude Maneuverability

Typical Accuracy

Lifetime Limits

Gravity-gradient

Earth local vertical only

Very limited

±5 deg (2 axes)

None

Gravity-gradient and Momentum Bias Wheel

Earth local vertical only

Very limited

±5 deg (3 axes)

Life of wheel bearings

Passive Magnetic

North/south only

Very limited

±5 deg (2 axes)

None

Pure Spin Stabilization

InertiaUy fixed any direction Repolntwith precession maneuvers

High propellant usage to move stiff momentum vector

±0.1 deg to ±1 deg in 2 axes (proportional to spin rate)

Thrusterpropeliant (if applies)*

Dual-Spin Stabilization

Limited only by articulation on despun platform

Momentum vector same as above Despun platform constrained by its own geometry

Same as above for spin section

Despun dictated by -payload reference and pointing

Thruster propellant (if applies)* Despin bearings

Bias Momentum (1 wheel)

Best suited for local vertical pointing

Momentum vector of the bias wheel prefers to stay normal to orbit plane, constraining yaw maneuver

±0.1 deg to ±1 deg

Propellant (if applies)* Life of sensor and wheel bearings

Zero Momentum (thwsteronly)

No constraints

No constraints High rates possible

±0.1 deg to ±5 deg

Propellant

Zero Momentum (3 wheels)

No constraints

No constraints

±0.001 deg to ±1 deg

Propellant (if applies)* Life of sensor and wheel bearings

Zero Momentum CMG

No constraints

No constraints High rates possible

±0.001 deg to ±1 deg

Propellant (if applies)* Life of sensor and wheel bearings

'Thrusters may be used for slewing and momentum dumping at ail altitudes. Magnetic torquers may be used from LEO to GEO.

'Thrusters may be used for slewing and momentum dumping at ail altitudes. Magnetic torquers may be used from LEO to GEO.

Passive Control Techniques. Gravity-gradient control uses the inertial properties of a vehicle to keep it pointed toward the Earth. This relies on the fact that an elongated object in a gravity field tends to align its longitudinal axis through the Earth's center. The torques which cause this alignment decrease with the cube of the orbit radius, and are symmetric around the nadir vector, thus not influencing the yaw of a spacecraft around the nadir vector. This tendency is used on simple spacecraft in near-Earth orbits without yaw orientation requirements, often with deployed booms to achieve the desired inertias.

Frequently, we add dampers to gravity-gradient spacecraft to reduce librarian —small oscillations around die nadir vector caused by disturbances. Gravity-gradient spacecraft are particularly sensitive to thermal shocks on long deployed booms when entering or leaving eclipses. They also need a method of ensuring attitude capture with the correct end of the spacecraft pointed at nadir—the gravity-gradient torques make either end along the minimum inertia axis equally stable.

In the simplest gravity-gradient spacecraft, only two orientation axes are controlled. The orientation around the nadir vector is unconstrained. To control this third axis, a small, constant-speed momentum wheel is sometimes added along the intended pitch axis (i.e., an axis peipendicular to the nadir and velocity vectors). This "yaw" wheel is stable when it aligns with the orbit normal, and small energy dissipation mechanisms on board cause the spacecraft to seek this minimum energy, stable orientation without active control.

A third type of purely passive control uses permanent magnets on board the spacecraft to force alignment along the Earth's magnetic field. This is most effective in near-equatorial orbits where the field orientation stays almost constant for an Earth-pointing vehicle.

Spin Control Techniques. Spin stabilization is a passive control technique in which the entire spacecraft rotates so that its angular momentum vector remains approximately fixed in inertial space. Spin-stabilized spacecraft (or spinners), employ the gyroscopic stability discussed earlier to passively resist disturbance torques about two axes. The spinning motion is stable (in its minimum energy state) if the vehicle is spinning about die axis having the largest moment of inertia. Energy dissipation mechanisms on board, such as fuel slosh and structural damping) will cause any vehicle to head toward this state if uncontrolled. Thus disk-shaped spinners are passively stable while pencil-shaped vehicles are not. Spinners can be simple, survive for long periods without attention, provide a thermally benign environment for components, and provide a scanning motion for sensors. The principal disadvantages of spin stabilization are (1) that the vehicle mass properties must be controlled to ensure the desired spin direction and stability and (2) that the angular momentum vector requires more fuel to reorient than a vehicle with no net angular momentum, reducing the usefulness of this technique for payloads that must be repointed frequently.

It takes extra fuel to reorient a spinner because of the gyroscopic stiffness which also helps it resist disturbances. In reorienting a spinning body with angular momentum, h, a constant torque, T, will produce an angular velocity, <a, perpendicular to the applied torque and angular momentum vector, of magnitude co = T/h. Thus, the higher the stored momentum is, the more torque must be applied for a given co. For a maneuver through an angle 6, the torque-time product—an indication of fuel required for the maneuver—is a constant equal to hQ. Conversely, for a nonspinning vehicle with no initial angular velocity, a small torque can be used to start it rotating, with an opposite torque to stop it The fuel used for any angle maneuver can be infinitesimally small if a slow maneuver is acceptable.

A useful variation of spin control is called dual-spin stabilization, where the spacecraft has two sections spinning at different rates about the same axis. Normally, one section, the rotor, spins rapidly to provide angular momentum, while the second section, the stator or platform, is despun to keep one axis pointed toward the Earth or Sun. By combining inertially fixed and rotating sections on the same vehicle, dual spinners can accommodate a variety of payloads m a simple vehicle. Also, by adding energy dissipation devices to the platform, a dual spinner can be passively stable spinning about the axis with the smallest moment of inertia. This permits more pencil-shaped spacecraft, which fit better in launch vehicle fairings. The disadvantage of dual-spin stabilization is the added complexity of the platform bearing and slip rings between the sections. This complexity can increase cost and reduce reliability compared to simple spin stabilization.

Spinning spacecraft, both simple and dual, exhibit several distinct types of motion which often are confused. Precession is the motion of the angular momentum vector caused by external torques such as thruster firings. Wobble is the apparent motion of the body when it is spinning with the angular momentum vector aligned along a principal axis of inertia which is offset from a body reference axis—for example, the intended spin axis. This looks like motion of the intended spin axis around the angular momentum vector at the spin rate.

Nutation is the torque-free motion of the spacecraft body when the angular momentum vector is not perfectly aligned along a principal axis of inertia. For rod-shaped objects, this motion is a slow rotation (compared to spin rate) of the spin axis around the angular momentum vector. For these objects, spinning about a minimum inertia axis, additional energy dissipation will cause increased nutation. For disk-shaped objects, spinning around a maximum inertia axis, nutation appears as a higher-than-spin-rate tumbling. Energy dissipation for these objects (e.g., with a passive nutation damper) reduces nutation, resulting in a clean spin.

Nutation is caused by disturbances such as thruster impulses, and can be seen as varying signals in body-mounted inertial and external sensors. Wobble is caused by imbalance and appears as constant offsets in body-mounted sensors. Such constant offsets are rarely discernible unless multiple sensors are available.

Spin stability normally requires active control, such as mass expulsion or magnetic coils, to periodically adjust the spacecraft's attitude and spin rate to counteract disturbance torques. In addition, we may need to damp the nutation caused by disturbances, precession commands, or fuel slosh. Aggravating this nutation is the effect of structural flexure and fuel slosh, which is present in any space vehicle to one degree or another. Once the excitation stops, nutation decreases as these same factors dissipate the energy. But this natural damping can take hours. We can neutralize this source of error in minutes with nutation dampers (see Sec. 11.1.2). We can also reduce the amount of nutation from these sources by increasing the spin rate, thus increasing the stiffness of the spinning vehicle. If the spin rate is 20 rpm, and the nutation angle is 3 deg, then at 60 rpm the nutation angle would decrease by a factor of three. We seldom use spin rates above 90 rpm because of the large centrifugal forces and their effect on structural design and weight. In thrusting and pointing applications, spin rates under 20 rpm may allow excessive nutation and are not used. However, noncritical applications, such as thermal control, are frequently insensitive to nutation and may employ very low spin rates.

Three-axis Control Techniques. Spacecraft stabilized in 3 axes are more common today than those using spin or gravity gradient They maneuver and can be stable and accurate, depending on their sensors and actuators. But they are also more expensive and more complex. The control torques about the axes of 3-axis systems come from combinations of momentum wheels, reaction wheels, control moment gyros, thrusters, or magnetic torquers. Broadly, however, these systems take two forms: one uses momentum bias by placing a momentum wheel along the pitch axis; the other is called zero momentum with a reaction wheel on each axis. Either option usually needs thrusters or magnetic torquers as well as the wheels.

In a zero-momentum system, reaction wheels respond to disturbances on the vehicle. For example, a vehicle-pointing error creates a signal which speeds up the wheel, initially at zero. This torque corrects the vehicle and leaves the wheel spinning at low speed, until another pointing error speeds the wheel further or slows it down again. If the disturbance is cyclic during each orbit, the wheel may not approach saturation speed for several orbits. Secular disturbances, however, cause the wheel to drift toward saturation. We then must apply an external torque, usually with a thruster or magnetic torquer, to force the wheel speed back to zero. This process, called desaturation, momentum unloading, or momentum dumping, can be done automatically or by command from the ground.

When high torque is required for large vehicles or fast slews, a variation of 3-axis control is possible using control moment gyros, or CMGs. These devices work like momentum wheels on gimbals. (See Sec. 11.1.4 for a further discussion of CMGs.) The control of CMGs is complex, but their available torque for a given weight and power can make them attractive.

As a final type of zero momentum 3-axis control, simple all-thruster systems are used for short durations when high torque is needed, such as orbit insertion or during A V burns from large motors. These thrusters then may be used for different purposes such as momentum dumping during other mission modes.

Momentum bias systems often have just one wheel with its spin axis mounted along the pitch axis, normal to the orbit plane. The wheel is run at a nearly constant, high speed to provide gyroscopic stiffness to the vehicle, just as in spin stabilization, with similar nutation dynamics. Around the pitch axis, however, the spacecraft can control attitude by torquing the wheel, slightly increasing or decreasing its speed. Periodically, the pitch wheel must be desaturated (brought back to its nominal speed), as in zero-momentum systems, using thrusters or magnets.

The dynamics of nadir-oriented momentum-bias vehicles exhibit a phenomenon known as roll-yaw coupling. To see this coupling, consider an inertially-fixed angular momentum vector at some angle with respect to the orbit plane. If the angle is initially a positive roll error, then 1/4 orbit later it appears purely about the yaw axis as a negative yaw error. As the vehicle continues around the orbit, the angle goes through negative roll and positive yaw before realigning as positive roll. This coupling, which is due to the apparent motion of the Earth and, therefore, the Earth-fixed coordinate frame as seen from the spacecraft, can be exploited to control roll and yaw over a quarter orbit using only a roll sensor.

Effects of Requirements on Control Type. With the above knowledge of control types, we can proceed to select a type which best meets mission requirements. Tables 11-5 through 11-7 describe the effects of orbit insertion, payload pointing, and payload slew requirements on the selection process.

A common control approach during orbit insertion is to use the shoTt-term spin stability of the spacecraft-orbit-insertion motor combination. Once on station, the motor may be jettisoned, the spacecraft despun using jets or a yo-yo device, and a different control technique used.

Payload pointing will influence the ADCS control method, the class of sensors, and the number and kind of actuation devices. Occasionally, pointing accuracies are so stringent that a separate, articulated platform is necessary. An articulated platform can perform scanning operations much easier than the host vehicle, with better accuracy and stability.

TABLE 11-5. Orbit Transition Maneuvers and Their Effect Using thrusters to change orbits creates special challenges for the ADCS.

Requirement

Effect on Spacecraft

Effect on ADCS

"Large impulse to complete orbit Insertion (thousands of nrt/s)

Solid motor or large blpropellant stage Large thrusters or a glmbaled engine or spin stabilization for attitude control during bums

Inertia! measurement unit for accurate reference and velocity measurement Different actuators, sensors, and control laws for bum vs. coasting phases Need for navigation or guidancev

On-orblt plane changes to meet payload needs or vehicle operations (hundreds of m/s)

More thrusters, but may to enough If coasting phase uses thrusters

Separate control law for thrusting Actuators sized for thrusting disturbances Onboard attitude reference for thrusting phase

Orbit maintenance trim maneuvers (<100 m/s)

One set of thrusters

Thrusting control law Onboard attitude reference

TABLE 11-6. Effect of Payload Pointing Directions on ADCS Design. The payload pointing requirements are usually the most Important factors for determining the type of actuators and sensors.

Requirement

Effect on Spacecraft

Effect on ADCS

Bum-pointing

• Scanning •Off-nadir pointing

• Gravity-gradient fine for low accuracies (>1 deg) only

• 3-axis stabilization acceptable with Earth local vertical reference

If gravity-gradient

• Booms, dampers, Sun sensors, magnetometer or horizon sensors for attitude determination

• Momentum wheel for yaw control tf&ixfs

• Horizon sensor for local vertical reference (pitch and roll)

• Sun or star sensor for third-axis reference and attitude determination

• Reaction wheels, momentum wheels, or control moment gyros for accurate pointing and propetlant conservation

• Reaction control system for coarse control and momentum dumping

• Magnetic torquers can also dump momentum

• Inertlal measurement unit for maneuvers and attitude determination

Inertlal pointing •Sun •Celestial targets • Payload targets of opportunity

• Spin stabilization fine for medium accuracies with few attitude maneuvers

• Gravity gradient does not apply

•3-axis control is most versatile for frequent reorientations

If spin

• Payload pointing and attitude sensor operations limited without despun platform

• Needs thrusters to reorient momentum vector

• Requires nutation damping

• Typically, sensors Include Sun sensors, star tracker, and inertia! measurement unit

• Reaction wheels and thrusters are typical actuators

• May require articulated payload (e.g., scan platform)

TABLE 11-7. Slewing "Requirements That Affect Control Actuator Selection. Spacecraft slew agility can demand larger actuators for Intermittent use.

Slewing

Effect on Spacecraft

Effect on ADCS

Nora

Spacecraft constrained to one attitude—highly Improbable

• Reaction wheels, if planned, can be smaller

• If magnetic torque can dump momentum, may not need thrusters

Nominal rates— 0.05 deg/s (maintain local vertical) to 0.5 deg/s

• Reaction wheels adequate by themselves only for a few special cases

High rates— > 0.5 deg/s

• Structural Impact on appendages

• Weight and cost increase

• Control moment gyros very likely or two thruster force levels—one for stationkeeplng and one for high-rate maneuvers

Trade studies on pointing requirements must consider accuracy in determining attitude and controlling vehicle pointing. We must identify the most stringent requirements. Table 11-8 summarizes effects of accuracy requirements on the spacecraft's ADCS subsystem approach. Section 5.4 discusses how to develop pointing budgets.

FireSat Control Selection. For FireSat, we consider two options for orbit insertion control. First, the launch vehicle may directly inject the spacecraft into its mission oibiL This common option simplifies the spacecraft design, since no special insertion mode is needed. An alternate approach, useful for small spacecraft such as FireSat, is to use a monopropellant system on board the spacecraft to fly itself up from a low parking orbit to its final altitude. For small insertion motors, reaction wheel torque or momentum bias stabilization may be sufficient to control the vehicle during this bum. For larger motors, AV thruster modulation or dedicated ADCS thrusters become attractive.

Once on-station, the spacecraft must point its sensors at nadir most of the time and slightly off-nadir for brief periods. Since the payload needs to be despun and the spacecraft frequently reoriented, spin stabilization is not the best choice. Gravity-gradient and passive magnetic control cannot meet the 0.1 deg pointing requirement or the 30 deg slews. This leaves 3-axis control and momentum-bias stabilization as viable options for the on-station control as well.

Depending on other factors, either approach might work, and we will baseline momentum bias control with its simpler hardware requirements. In this case, we will use a single pitch wheel for momentum and electromagnets for momentum dumping and roll and yaw control.

For the optional off-nadir {minting requirement, 3-axis control with reaction wheels might be more appropriate. Also, 3-axis control often can be exploited to simplify the solar array design, by using one of the unconstrained payload axes (yaw, in this case) to replace a solar array drive axis. Thus, the reduced array size possible with 2 deg of freedom can be achieved with one array axis drive and one spacecraft rotation.

11.1.3 Quantify the Disturbance Environment

In this step, we determine the size of the external torques the ADCS must tolerate. Only three or four sources of torque matter for the typical Earth-orbiting spacecraft They are gravity-gradient effects, magnetic-field torques on the vehicle, impingement

TABLE 11-8. Effect of Control Accuracy on Sensor Selection and ADCS Design. Accurate pointing requires better, higher cost, sensors, and actuators.

Required Accuracy (3o)

Effect on Spacecraft

Effect on ADCS

>5 deg

• Permits major cost savings

• Permits gravity-gradient (GG) stabilization

Without attitude determination

• No sensors required for GG stabilization

• Boom motor, GG damper, and a bias momentum wheel are only required actuators

With attitude determination

• Sun sensors & magnetometer adequate for attitude determination at z 2 deg

• Higher accuracies may require star trackers or horizon sensors

• Spin stabilization feasible If stiff, inertially fixed attitude is acceptable

• Payload needs may require despun platform on spinner

• 3-axis stabilization will work

• Sun sensors and horizon sensors may be adequate for sensors, especially a spinner

• Accuracy for 3-axis stabilization can be met with RCS deadband control but reaction wheels will save propellant for long missions

• Thrusters and damper adequate for spinner' actuators

• Magnetic torquers (and magnetometer) useful

0.1 deg to 1 deg

• 3-axis and momentum-bias stabilization feasible

• Dual-spin stabilization also feasible

• Need for accurate attitude reference leads to star tracker or horizon sensors & possibly gyros

• Reaction wheels typical with thrusters for momentum unloading and coarse control

• Magnetic torquers feasible on light vehicles (magnetometer also required)

< 0.1 deg

• 3-axis stabilization is necessary

• May require articulated & vibration-isolated payload platform with separate sensors

• Same as above for 0.1 deg to 1 deg but needs star sensor and better class of gyros

• Control laws and computational needs are more complex

• Flexible body performance very Important

by solar-radiation, and, for low-altitude orbits, aerodynamic torques. Section 8.1 discusses the Earth environment in detail, and Chap. 10 and Singer [1964] provide a discussion of disturbances. Tables 11-9A and 11-9B summarize the four major disturbances, provide equations to estimate their size for the worst case, and calculate values for the FireSat example.

Disturbances can be affected by the spacecraft orientation, mass properties, and design symmetry. For the normal FireSat orientation, the largest torque is due to the residual magnetism in the spacecraft If, however, we use the optional 30-deg off-nadir pointing, the gravity-gradient torque increases over an order of magnitude, to become as large as the magnetic torque. Note that we use 1 deg in the gravity-gradient calculations, rather than the 0.1 deg pointing accuracy. This is to account for our uncertain knowledge of the principal axes. If the principal axes are off by several degrees, that angle may dominate in the disturbance calculations. We also note that a less symmetric solar array arrangement would have increased both the aerodynamic and solar torques, making them closer to the magnetic torque in this example.

TABLE 11-9A. Simplified Equations for Estimating Worst-Case Disturbance Torques

Disturbance torques affect actuator size and momentum storage requirements.

TABLE 11-9A. Simplified Equations for Estimating Worst-Case Disturbance Torques

Disturbance torques affect actuator size and momentum storage requirements.

Disturbance

Type

Influenced Primarily by

Formula

Gravity-gradient

Constant torque for Earth-oriented vehicle, cyclic for inertiaiiy oriented vehicle

• Spacecraft inertias

where Tg Is the max gravity torque; n is the Earth's gravity constant (3.986 x 1014 rr^/s2); Ris orbit radius (m), 0is the maximum deviation of the Z-axis from local vertical in radians, and lz and ly are moments of inertia about z and / (or x, if smaller) axes in kg-m2.

Earth-oriented vehicle, constant for solar-oriented vehicle or platform

• Spacecraft geometry

• Spacecraft surface reflectivity

•Spacecraft geometry and eg location

Solar radiation pressure, 7^,, is highly dependent on the type of surface being illuminated. A surface is either transparent, absorbent, or a reflector, but most surfaces are a combination of the three. Reflectors are classed as diffuse or specular. In general, solar arrays are absorbers and the spacecraft body is a reflector. The worst case solar radiation torque is

and Fs is the solar constant, 1,367 W/m2, c is the speed of light, 3 X108 m/s, As is the surface area, c^ is the location of the center of solar pressure, eg is the center of gravity, q Is the reflectance factor (ranging from 0 to 1, we use 0.6), and / is the angle of incidence of the Sun.

Magnetic Field

• Residual spacecraft magnetic dlpoie

where Tm Is the magnetic torque on the spacecraft; D is the residual dipole of the vehicle in amp -turn -m2 (A-m2), and B Is the Earth's magnetic field in tesla. B can be approximated as 2MIR3 for a polar orbit to half that at the equator. Mis the magnetic moment of the Earth, 7.98 x 1015 tesla-m3, and R Is the radius from dipole (Earth) center to spacecraft in m.

Aerodynamic

Constant for Earth-oriented vehicles, variable for Inertiaiiy oriented vehicle

• Orbit altitude •Spacecraft geometry and eg location

Atmospheric density for low orbits varies significantly with solar activity.

where F= 0.5 [p CdAV2]; F being the force; Cd the drag coefficient (usually between 2 and 2.5); p the atmospheric density; A, the surface area; V, the spacecraft velocity; c^ the center of aerodynamic pressure; and eg the center of gravity.

TABLE 11-9B. Example of Worst Case Disturbance Torque Estimates for FlreSaL Magnetic and aerodynamic disturbances are the largest for this small spacecraft

Disturbance

FlreSat Example

Gravity-gradient

For R = (6,378 + 700) km = 7,078 km; lz = 30 kg-m2, ly = 60 kg-m2 and

0=1 deg (normal mode) or 30 deg (optional target-of-opportunity mode): normal:

(3)(3.986 x 1014 m3/ s2)(30 kg- mz)sin(2 deg) 9 ~ (2)(7.078x106m)3

= 1.8x10-8 N-m optional target-of-opportunity. Tg = 4.4 x 10-5 N-m

Solar Radiation

For a 2 m by 1.5 m spacecraft cross-section, a center-of-solar-pressure to center-of-mass difference of 0.3 m, incidence angle of 0 deg and coefficient of reflectivity of 0.6.

Tsp = (1,367W/m2)(2mx 1,5m) (0.3m)(1 + 0.6) (cos0deg)/(3x108m/s) = 6.6X10-8 N-m

Magnetic Field

For R=7,078 km, a spacecraft magnetic dipole of 1 A-m2* and the worst-case polar magnetic field, M= 2 (7.96 x1015 tesIa-m3)/(7.078 x 108 m)3 = 4.5x10-5 tesla (= 0.45 gauss)

Tm =1 x4.5x 10-5 = 4.5 x 10"5 N-m

Aerodynamics

For Illustration purposes we assume a 3 m2 surface, offset from the center of mass by 0.2 m. In a 700-km orbit the velocity Is = 7,504 m/s, the atmospheric density (p) Is = 10~13 kg/m3. For Cd, the drag coefficient, use 2.0.

F= 1/2 [(10-13 kg/m3) (2)(3 m2) (7,504 m/s)2] = 1.7 x 10"5 N

This is small. At a 100-km orbit, however, p = 10-9 kg/m?. This results In T= 3.3 x 10-2 N -m, which Is significant for our small spacecraft

* Residual magnetic dlpoles can range anywhere from 0.1 to > 20 A-m2 depending on the spacecraft's size and whether any onboard compensation Is provided. On a small-sized, uncompensated vehicle, 1 A-m? Is typical (1 A-m2 = 1,000 pole -cm).

* Residual magnetic dlpoles can range anywhere from 0.1 to > 20 A-m2 depending on the spacecraft's size and whether any onboard compensation Is provided. On a small-sized, uncompensated vehicle, 1 A-m? Is typical (1 A-m2 = 1,000 pole -cm).

The other disturbances on the control system are internal to the spacecraft Fortunately, we have some control over them. If we find that one is much larger than the rest we can respecify it to tighter values. This change would reduce its significance, but most likely add to its cost or weight Table 11-10 summarizes the common internal disturbances. Misalignments in the center of gravity and in thrusters will show up during thrusting only and are corrected in a closed-loop control system. The slosh and operating machinery torques are of greater concern but depend on specific hardware. If a spacecraft component has fluid tanks or rotating machinery, the system designer should investigate disturbance effects and ways to compensate for the disturbance, if required. Standard techniques include slosh baffles or counter-rotating elements.

TABLE 11-10. Principal Internal Disturbance Torques. Spacecraft designers can minimize internal disturbances through careful planning and precise manufacturing which may increase costs.

Disturbances

Effect on Vehicle

Typical Values

Uncertainty In Center of Gravity (eg)

Unbalanced torques during firing of coupled thrusters Unwanted torques during translation thrusting

1 to 3 cm

Thruster Misalignment

Same as eg uncertainty

0.1 to 0.5 deg

Mismatch of Thruster Outputs

Similar to eg uncertainty

±5%

Rotating Machinery (pumps, tape recorders)

Torques that perturb both stability arid accuracy

Dependent on spacecraft design; may be compensated by counter-rotating elemente.

Liquid Sloshing

Torques due to fluid motion arid variation in center-of-mass location

Dependent on specific design; may be controlled by bladders or baffles.

Dynamics of Flexible Bodies

Oscillatory resonance at bending frequencies, limiting control bandwidth

Depends on spacecraft structure.

Thermal Shocks on Flexible Appendages

Attitude disturbances when entering/leaving eclipse

Depends on spacecraft structure. Worst for gravity gradient systems with long Inertia booms.

11.1.4 Select and Size ADCS Hardware

We are now ready to evaluate and select the individual ADCS components.

Actuators. We first discuss the actuators, as summarized in Table 11-11, beginning with reaction and momentum wheels. Reaction wheels are essentially torque motors with high-inertia rotors. They can spin in either direction, and provide one axis of control for each wheel. Momentum wheels are reaction wheels with a nominal spin rate above zero to provide a nearly constant angular momentum. This momentum provides gyroscopic stiffness to two axes, while the motor torque may be controlled to precisely point around the third axis.

In sizing wheels, it is important to distinguish between cyclic and secular disturbances, and between angular momentum storage and torque authority. For 3-axis control systems, cyclic torques build up cyclic angular momentum in reaction wheels, as the wheels provide compensating torques to keep the vehicle from moving. We typically size the angular momentum capacity of a reaction wheel (limited by its saturation speed) to handle the cyclic storage during an orbit without the need for frequent momentum dumping. Thus, the average disturbance torque for 1/4 or 1/2 orbit determines the minimum storage capability. The secular torques and our total storage capacity then define how frequently angular momentum must be dumped.

The torque capability of the wheels usually is determined by slew requirements or the need for control authority above the peak disturbance torque in order for the wheels to maintain pointing accuracy.

For 3-axis control, at least three wheels are required with their spin axes not coplanar. Often, a fourth redundant wheel is carried in case one of the three primaries

TABLE 11-11. Typical ADCS Actuators. Actuator weight and power usually scale with performance.

Actuator

Typical Performance Range

Weight (kg)

Hot Gas (Hydrazine) Cold Gas

0.5 to 9,000 N* <5N*

Variablet Variablet

0 0

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