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BeacSon and Momentum Wheels

0.4 to 400 Nam*s for momentum wheels at 1,200 to 5,000 rpm; max torques from 0.01 to 1 N-m

2 to 20

10 to 110

Control Moment Gyros (CMG)

25 to 500 N-m of torque

>10

90 to 150

Magnetic Torquers

1 to 4,000 A-m2*

0.4 to 50

0.6 to 16

* Multiply by moment arm (typically 1 to 2 m) to get torque, t Chap. 17 discusses weight and power for thruster systems In more detail.

i For 700-km orbit and maximum Earth field of 0.4 gauss, the maximum torques would be 4.5 x 10-5 N-m to 0.18 N-m (see Table 11-9B).

* Multiply by moment arm (typically 1 to 2 m) to get torque, t Chap. 17 discusses weight and power for thruster systems In more detail.

i For 700-km orbit and maximum Earth field of 0.4 gauss, the maximum torques would be 4.5 x 10-5 N-m to 0.18 N-m (see Table 11-9B).

fails. If the wheels are not orthogonal (and the redundant one never is), additional torque and momentum authority may be necessary to compensate for the unfavorable geometry. It is also common to use wheels larger than the minimum required in order to use a standard component

For spin-stabilized or momentum-bias systems, the cyclic torques will cause cyclic rates, while the secular torques cause gradual divergence. We typically design the stored angular momentum, determined by spin rate and inertia of the spinning body, to be large enough to keep the cyclic motion within our pointing specification without active control during an orbit Periodic torquing will still be required to counteract the secular disturbances. The more angular momentum in the body, the more resistant it is to external torques. An upper limit on the stored momentum, if one exists, may be defined by the fuel cost to precess this angular momentum.

For high-torque applications, control-moment gyros may be used instead of reaction wheels. These are single- or double-gimbaled wheels spinning at constant speed. By turning the gimbal axis, we can obtain a high-output torque whose size depends on the speed of the rotor and the gimbal rate of rotation. Control systems with control moment gyros can produce large torques about all three of the spacecraft's orthogonal axes, so we most often use them for agile (high-rate) maneuvers. They require a complex control law and momentum exchange for desaturation. Other disadvantages are high cost and weight

Spacecraft also use magnetic torquers as actuation devices. These torquers use magnetic coils or electromagnets to generate magnetic dipole moments. Magnetic torquers can compensate for the spacecraft's residual magnetic fields or attitude drift from minor disturbance torques. They also can desaturate momentum-exchange systems but usually require much more time than thrusters. A magnetic torquer produces torque proportional (and perpendicular) to the Earth's varying magnetic field. Electromagnets have the advantage of no moving parts, requiring only a magnetometer for field sensing and a wire-wound, electromagnetic rod in each axis. Because they use the Earth's natural magnetic fields, they are less effective at higher orbits. We can easily specify the rod's field strength in amp tum-m2 and tailor it to any application. Table 11-12 describes sizing rules of thumb for wheels and magnetic torquers.

TABLE 11-12. Simplified Equations for Sizing Reaction Wheels, Momentum Wheels, ar Magnetic Torquers. FlreSat momentum wheels are sized tor the baselft requirements. Reaction wheels are sized for the optional design with 30-dc slew requirement

Parameter

Simplified Equations

Application to FlreSat Example

Torque from Reaction Wheel for Disturbance Rejection

Reaction-wheel torque must equal worst-case anticipated disturbance torque plus some margin: Tffn = ( 1i) ) (Margin Factor)

For the example spacecraft, TD = 4.5x 10"5 N-m (Table 11-9). This is below almost all candidate reaction wheels. We win select a wheel based on storage requirements or slew torque, not disturbance rejection. See below.

Slew Torque tor Reaction Wheels

For max-acceleration slews (1/2 distance in 1/2 time):

2 2/UJ

For the 30-deg slews of the 90 kg-nn? spacecraft (Fig. 11-1) in 10 mln, this becomes:

= 5.2xl0~Vm This is also a small value.

Momentum Storage in Reaction Wheel

One approach to estimating wheel momentum, K is to integrate the worst-case disturbance torque, T0, over a tun orbit It the disturbance is gravity gradient, the maximum disturbance accumulates In 1/4 of an orbit A simplified expression tor such a sinusoidal disturbance Is:

where 0.707 Is the rms average ot a sinusoidal function.

For T„ = 45 x 10-5 N • m (Table 11-9B) and a 700-kn orbital period of 9S.8 mln

. ,, _ -5 / 98.8 mlnY 60 secY. __ h =<4.5X10 N-m( 4 | ^ J(0.707)

A small reaction wheel which gives us storage ol 0.4 N-m-s would be sufficient It provides a margin of > 9 in storage for the worst-case torques.

Momentum Storage in Momentum Wheel

Roll and yaw accuracy depend on the wheel's momentum and the external disturbance torque. A simplified expression for the required momentum storage te

To torque P= orbit period h = angular 0g ~ allowable momentum motion

The value of h for a 0.1 deg yaw accuracy would be

For a 1 deg accuracy, we would need only 3.8 N-m-s

Momentum Storage In Spinner

Same as for a momentum wheel, but with the spin rate:

For the 0.1 deg accuracy, the spin rate Is:

_ (37.3)N-m-s =0 42rad/sec_ 41 ^ 90 kg-m

Torque from

Magnetic

Torquers

Magnetic torquers use the Earth's magnetic field, B, and electrical current through the torquer to create a magnetic dipole (D) that results In torque (7) on the vehicle:

Magnets used for momentum dumping must equal the peak disturbance + margin to compensate tor the lack ol complete directional control.

Table 11-9B estimates the worst-case Earth field, 0, to be 4.5 x 1 o-s tesla. We calculate the torque rod's magnetic torquing ability (dipole) to counteract the worst-case gravity gradient disturbance, TD, of 4.5x10-8 N-m as

B 4.5x10 tesla which is a »nail actuator. The Earth's field is cyclic at twice orbital frequency, thus, maximum torque is available only twice per orbit A torquer of 3 to 10 A- nfi capacity should provide sufficient margin.

Nate: For actuator sizing, the magnitude and direction ot the disturbance torques must be considered. In particular, momentum accumulation in inertia! coordinates must be mapped to body-fixed wheel axes, where necessary.

Nate: For actuator sizing, the magnitude and direction ot the disturbance torques must be considered. In particular, momentum accumulation in inertia! coordinates must be mapped to body-fixed wheel axes, where necessary.

Gas jets or thrusters produce torque by expelling mass, and are not governed by the same concerns as momentum storage devices. We consider them to be a hot-gas system, either bipropellant or monopropellant, when a chemical reaction produces the energy. They are a cold-gas system when energy comes from the latent heat of a phase change or from the work of compression without a phase change. Cold-gas systems usually apply to small spacecraft and low-impulse requirements.

Thrusters produce torques and forces that:

• Control attitude • Adjust orbits

• Control nutation • Control the spin rate

• Maneuver spacecraft • Dump extra momentum from a momentum over large angles wheel, reaction wheel, or control moment gyro

Unfortunately, their plumes may impinge on the spacecraft, contaminating surfaces, and they require expendable propellant dictating spacecraft life. An advantage is that they can provide large, instantaneous torques at any point in the orbit

We must decide whether we need thrusters, how many we need, and where to locate them. For applications that demand fine control from the thrusters, we may have to specify the minimum impulse from a single thruster pulse—usually 20 ms or greater. Single thrust levels are usually used, unless the complication of dual or variable thrust is required.

Although the baseline FireSat spacecraft will use magnetic torquers, we illustrate the thruster sizing calculations for momentum dumping and the optional slew requirement We will assume the thruster's moment arm is 0.5 m. Table 11-13 gives procedures and simplified equations, where applicable, for sizing thrusters and estimating propellant. Refer to Chap. 17 for a thorough discussion of propulsion subsystems.

The size of the thrusters and required propellent are small for this example. For the optional system with reaction wheels, slewing can be accomplished with the wheels, avoiding use of propellent For the baseline momentum bias system, we would use thrusters for the optional slews, though large electromagnets could be used if thrusters were not available and maneuver time were not important.

Sensors. We complete this hardware unit by selecting the sensors needed for control. Consult Table 11-14 for a summary of typical devices, as well as their performance and physical characteristics. Note, however, that sensor technology is evolving rapidly, promising more accurate, lighter-weight sensors for future mission.

Sun sensors are visible-light detectors which measure one or two angles between their mounting base and incident sunlight They are popular, accurate and reliable, but require clear fields of view. They can be used as part of the normal attitude determination system, part of the initial acquisition or failure recovery system, or part of an independent solar array orientation system. Since most low-Earth orbits include eclipse periods, Sun-sensor-based attitude determination systems must provide some way of tolerating the regular loss of this data without violating pointing constraints.

Sun sensors can be quite accurate (< 0.01 deg) but it is not always possible to take advantage of that feature. We usually mount Sun sensors near the ends of the vehicle to obtain an unobstructed field of view. Sun sensor accuracy can be limited by structural bending on large spacecraft Spinning satellites use specially designed Sun sensors that measure the angle of the Sun with respect to the spin axis of the vehicle. The data may be sent to the ground for processing or used in a closed-loop control system on board the vehicle.

TABLE 11-13. Simplified Equations for Preliminary Sizing of Thruster Systems. FlreSat thruster requirements are small for this low-disturbance, minimal slew application.

Simplified Equations

Application to FlreSat Example

Thruster force level sizing for external disturbances:

f=td/l f is thruster force, td Is worst-case disturbance torques, and L is the thruster's moment arm

For the worst case of 4.5x10"5 N-m (Table 11-7) and a thruster moment arm of 0.5 m

This small value indicates slewing rate,'not disturbances, will more likely determine size. Also, using thrusters to fight cyclic disturbances uses much fuel.

Sizing force level to meet slew rates (optional zero momentum system):

Determine highest slew rate required In the mission profile.

Develop a profile that accelerates the vehicle to that rate, coasts, then decelerates. We calculate the thruster force from the acceleration value using the following relationships:

T = F L = lS Solve for F

Assume a 30-deg slew in less than 1 min (60 sec), accelerating for 5% of that time, coasting for 90%, and decelerating for 5%. .

To reach 0.5 deg/s In 5% of 1 min, which is 3 sec, requires an acceleration s 0 _ 0.5 deg/sec _ 0.167 deg/sec2 -2.91x10-3rad/sec2 t 3 sec

L 0.5 m This is small but feasible.

Sizing force level for slewing a momentum-bias vehicle:

The applied torque T Is

F= average thruster force L = moment arm d= thruster duty cycle

(fraction of spin period) h = angular momentum (0 = slew rate

For FlreSat, allowing 10 min for a 30-deg slew, with 10% duty cycle i^N-m-i^K *

hat \ 600 sec J 180 deg Ld (0.5m)(0.1) = 0.67 N

Thruster pulse life:

Develop detailed maneuver profile from mission sequence of events and determine pulse number and length for each segment

Assume example spacecraft uses thrusters only for large mission maneuvers and momentum dumping for 1 sec each wheel once a day. The large maneuver, 30 deg, in 2 axes each week includes a 3-sec acceleration pulse and a 3-sec deceleration pulse.

Total Pulses = 2 pulses (start & stop) x 2 axes x 12/yr x 5 yr (maneuver)

+ 1 pulse x 3 wheels x 365 days/yr x 5 yr (momentum dump)

This is below the typical 20,000 to 50,000 pulse ratings for small thrusters.

TABLE 11-13. Simplified Equations for Preliminary Sizing of Thruster Systems.

(Continued) FireSat thruster requirements are small for this low-disturbance, minimal slew application.

TABLE 11-13. Simplified Equations for Preliminary Sizing of Thruster Systems.

(Continued) FireSat thruster requirements are small for this low-disturbance, minimal slew application.

Simplified Equations

Application to FireSat Example

Sizing force level for momentum dumping:

where h = stored momentum (from wheel capacity or disturbance torque X time) L = moment arm /= bum time

For FireSat with 0.4 N-m-s wheels and 1-sec bums,

F = 0A"-m'a =0.8N (0.5 m)x(1sec)

Propellent

Estimate propellant mass (Mp) by determining the total pulse length, t, for the pulses counted above, multiplying by thruster force (F), and dividing by specific impulse (1^, and g as follows:

m R P 'sp9

To derive the propellant weight from the pulses above, use 3 sec for on-time for each large maneuver pulse, and 1 sec for each momentum-dump pulse at the computed force levels (actual times will change when a thruster is chosen, but the total impulse will be the same). Total Impulse = /=

240 pulses 3 sec/pulse 0.52 N + 5,475 pulses x 1 sec/pulse x 0.8 N = 4,754 N-s then

/ = 4,754 N-s =2.43 kg 'sp9 200 sec x 9.8 m /sec where an of 200 sec for hydrazine is a conservative estimate.

TABLE 11-14. Typical ADOS Sensors. Sensors have continued to improve in performance while getting smaller and less expensive.

Sensor

Typical Performance Range

WtRange (kg)

Measurement Unit (Gyros & Accelerometers)

Gyro drift rate=0.003 deg/hr to 1 deg/hr, accel.

Linearity = 1 to 5 x 10-8 g/g2 over range of 20 to 60 g

1 to 15

10 to 200

Sun Sensors

Accuracy = 0.005 deg to 3 deg

0.1 to 2

0to3

Star Sensors (Scanners & Mappers)

Attitude accuracy = 1 arc sec to 1 arc min 0.0003 deg to 0.01 deg

2to5

5 to 20

Horizon Sensors

• Scanner/Pipper

• Fixed Head (Static)

Attitude accuracy: 0.1 deg to 1 deg (LEO) < 0.1 deg to 0.25 deg

1 to 4 0.5 to 3.5

5 to 10 0.3 to 5

Magnetometer

Attitude accuracy = 0.5 deg to 3 deg

0.3 to 1.2

< 1

Star sensors have evolved rapidly in the past few years, and represent the most common sensor for high-accuracy missions. Star sensors can be scanners or trackers. Scanners are used on spinning spacecraft. Stars pass through multiple slits in a scan ner's field of view. After several star crossings, we can derive the vehicle's attitude. We use trackers on 3-axis attitude stabilized spacecraft to track one or more stars to derive 2- or 3-axis attitude information. The most sophisticated units not only track the stars as bright spots, but identify which star pattern they are viewing, and output the sensor's orientation compared to an inertia] reference. Putting this software inside the sensor simplifies processing requirements of the remaining attitude control software.

While star sensors excel in accuracy, care is required in their specfication and use. For example, the vehicle must be stabilized to some extent before the trackers can determine where they point This stabilization may require alternate sensors, which can increase total system cost. Also, star sensors are susceptible to being blinded by the Sun, Moon, or even planets, which must be accommodated in their application. Where the mission requires the highest accuracy and justifies a high cost we use a combination of star trackers and gyros. These two sensors complement each other nicely: the gyros can be used for initial stabilization, and during periods of sun or moon interference in the trackers, while the trackers can be used to provide a high-accuracy, low frequency, external reference unavailable to the gyros. Work continues to improve the sample rate of star trackers and to reduce their radiation sensitivity.

Horizon sensors are infrared devices that detect the contrast between the cold of deep space and the heat of the Earth's atmosphere (about 40 km above the surface in the sensed band). Simple narrow field-of-view fixed-head types (called pippers or horizon crossing indicators) are used on spinning spacecraft to measure Earth phase and chord angles which, together with orbit and mounting geometry, define two angles to the Earth (nadir) vector. Scanning horizon sensors use a rotating mirror or lens to replace (or augment) the spinning spacecraft body. They are often used in pairs for improved performance and redundancy. Some nadir-pointing spacecraft use staring sensors which view the entire Earth disk" (from GEO) or a portion of the limb (from LEO). The sensor fields of view stay fixed with respect to the spacecraft This type works best for circular orbits.

Horizon sensors provide Earth-relative information directly for Earth-pointing spacecraft, which may simplify onboard processing. The scanning types require clear fields of view for their scan cones (typically 45, 60, or 90 deg, half-angle). Typical accuracies for systems using horizon sensors are 0.1 to 0.25 deg, with some applications approaching 0.03 deg. For the highest accuracy in low-Earth orbit it is necessary to correct the data for Earth oblateness and seasonal horizon variations.

Magnetometers are simple, reliable, lightweight sensors that measure both the direction and size of the Earth's magnetic field. When compared to the Earth's known field, their output helps us establish the spacecraft's attitude. But their accuracy is not as good as that of star or horizon references. The Earth's field can shift with time and is not known precisely in the first place. To improve accuracy, we often combine their data with data from Sun or horizon sensors. When a vehicle using magnetic torquers passes through magnetic-field reversals during each orbit we use a magnetometer to control the polarity of the torquer output The torquers usually must be turned off while the magnetometer is sampled to avoid corrupting the measurement

GPS receivers are commonly known as high-accuracy navigation devices. Recently, GPS receivers have been used for attitude determination by employing the differential signals from separate antennas on a spacecraft Such sensors offer the promise of low cost and weight for LEO missions, and are being used in low accuracy applications or as back-up sensors. Development continues to improve their accuracy, which is limited by the separation of the antennas, the ability to resolve small phase differences, the relatively long wavelength, and multipath effects due to reflections off spacecraft components.

Gyroscopes are inertia] sensors which measure the speed or angle of rotation from an initial reference, but without any knowledge of an external, absolute reference. We use them in spacecraft for precision attitude sensing when combined with external references such as star or sun sensors, or, for brief periods, for nutation damping or attitude control during thruster firing. Manufacturers use a variety of physical phenomena, from simple spinning wheels (iron gyros using ball or gas bearings) to ring lasers, hemispherical resonating surfaces, and laser fiber optic bundles. The gyro manufacturers, driven by aircraft markets, steadily improve accuracy while reducing size and mass.

Error models for gyroscopes vary with the technology, but characterize the deterioration of attitude knowledge with time (degrees per hour or per square-root of time). When used with an accurate external reference, such as star trackers, gyros can provide smoothing (filling in the measurement gaps between star tracker samples) and higher frequency information (tens to hundreds of hertz), while the star trackers provide the low frequency, absolute orientation information that the gyros lack. Individual gyros provide one or two axes of information, and are often grouped together as an Inertial Reference Unit, IRU, for three full axes. IRUs with accelerometers added for position/velocity sensing are called Inertial Measurement Units, IMUs.

Sensor selection. Sensor selection is most directly influenced by the required orientation of the spacecraft (e.g., Earth- or inertial-pointing) and its accuracy. Other influences include redundancy, fault tolerance, field of view requirements, and available data rates. Typically, we identify candidate sensor suites and conduct a trade study to determine the best, most cost-effective approach. In such studies, the existence of off-the-shelf components and software can strongly influence the outcome. In this section we will only briefly describe some selection guidelines.

Full 3-axis knowledge requires at least two external vector measurements, although we use inertial platforms or spacecraft angular momentum (from spinning or momentum wheels) to hold the attitude between external measurements. In some cases, if attitude knowledge can be held for a fraction of an orbit, the external vectors (e.g., Earth or magnetic) will have moved enough to provide the necessary information.

For Earth-pointed spacecraft, horizon sensors provide a direct measurement of pitch and roll axes, but require augmentation for yaw measurements. Depending on the accuracy required, we use Sun sensors, magnetometers, or momentum-bias control relying on roll-yaw coupling for the third degree of freedom. For inertially-pointing spacecraft, star and Sun sensors provide the most direct measurements, and inertial platforms are ideally suited. Frequently, only one measurement is made in the ideal coordinate frame (Earth or inertial), and the spacecraft orbit parameters are required in order to convert a second measurement or as an input to a magnetic field model. The parameters are usually uplinked to the spacecraft from ground tracking, but autonomous navigation systems using GPS are also in use (see Sec. 11.7).

FireSat sensors. The external sensors for FireSat could consist of any of the types identified. For the 0.1 deg Earth-relative pointing requirement, however, horizon sensors are the most obvious choice since they directly measure two axes we need to control. The accuracy requirement makes a star sensor a strong candidate as well, although its information needs to be transformed to Earth-relative pointing for our use. The 0.1 deg accuracy is at the low end of horizon sensors' typical performance, and we need to be careful to get the most out of their data.

TABLE 11-15. FlreSat Spacecraft Control Components Selection. A simple, low-cost suite of components fits FlreSafs needs.

Type

Components

Rationale

Actuation

(1) Momentum Wheel

• Pitch axis torqulng

Devices

0 0

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