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Sun Pointing

May need for power generation or thermal control

Pointing During Thrusting

May need for guidance corrections

Communications Antenna Pointing

Toward a ground station or relay satellite

Pointing in a particular direction requires control of angular orientation about each of the 2 axes perpendicular to the pointing axis. If, for example, a payload or antenna must point toward the Earth, we need to control its attitude about 2 horizontal axes. If the payload is fixed to the spacecraft's body, these 2 axes are 2 of the 3 axes. Thus, we can use the third axis—rotation about the pointing axis—to satisfy a second pointing requirement, such as pointing one axis in the direction of flight. Table 10-14 lists types of pointing systems.

Either spin stabilization or 3-axis control using sensors and torquers can be used to control the spacecraft's attitude. Spin stabilization divides into passive spin, spin with precession control, or dual spin (spin with a despun platform). Classes of 3-axis control depend on the sensor type or torquing method. Possible sensors include Earth, Sun and star sensors, gyroscopes, magnetometers, and directional antennas. Torquers include gravity gradient, magnetic, thrusters, and wheels. Wheels include variable speed reaction wheels; momentum wheels, which have a nominal nonzero speed and therefore provide angular momentum to the spacecraft; and control moment gyros.

which are fixed-speed gimballed wheels. Table 10-15 summarizes these methods of control during thrusting and nonthrusting.

TABLE 10-14; Design Approaches for Selected Pointing Requirements.

Requirement

System

Nadir-Pointed Payload

Body-fixed payload using 2 axes of body attitude control to meet the Earth-pointing requirement The third axis is used to point a horizontal axis in the direction of flight

Can also use spin-stabilized spacecraft with spin aids normal to the orbit plane and payload mounted on despun platform.

Payload Pointed In a Flxed-lnertial Direction

Body-fixed payload using 2 axes of body attitude for payload-pointing direction in inertia) space. Third aids is used to keep one side toward the Sun.

Sun-Oriented Solar Array

Planar array requires 2 axes of control. May be achieved by

1 axis of body attitude and 1 rotation axis.

Cylindrical array with array axis perpendicular to Sun line.

Communications Antenna

2-axis mechanism.

TABLE 10-15. Types of Attitude Control.

Control Mode

Type of Control

Control During Thrusting: Spin Stabilization with Axial Thrust

Spin Stabilization with Radial Thrust

3-Axis Control

Passive spin in a fixed direction with thrust applied parallel to the spin axis.

Passive spin in a fixed direction with thrust applied perpendicular to the spin axis in short pulses.

Attitude is sensed with sensors whose output is used to control torquers. Torquers include thrusters operated off-on or swlveled to control thrust direction.

Control While Not Thrusting:

Spin Stabilization with Precession Control

Dual Spin

3-Axis Control

Spin direction is controlled by applying precession torque with an off-axis thruster.

Spin-stabilized with a despun platform.

Control using attitude sensors and torquers.

We use spin stabilization extensively for attitude control during kick-stage firing and for small spacecraft Spin-stabilized satellites with a despun platform, called a dual spin system, frequently support communications payloads. In this case, we mount the payload antenna on the despun platform so we can control its pointing. If the spin axis is roughly perpendicular to the Sun line, we can mount solar cells on the spacecraft's cylindrical skin to produce electric power.

Three-axis approaches range from passive control using gravity gradient or magnetics to full active control with propulsion thrusters and wheels. Passive techniques can provide coarse control to support low-accuracy pointing requirements and simple spacecraft. The active 3-axis method gives us highly accurate pointing control, more efficient solar arrays (by allowing oriented planar arrays), and pointing of several payloads or spacecraft appendages. But active 3-axis systems are more complex and usually heavier than spin-stabilized ones, and we may need to consider both approaches carefully before deciding between them. Table 10-16 summarizes how various requirements affect this decision.

TABLE 10-16. Implication of Pointing Requirements on Attitude Control Approach.

Requirement

Implication

Control during kick-stage firing Coarse control (>10 deg) Low-accuracy pointing (> 0.1 deg) Low power requirement (< 1 kW) High-accuracy pointing (< 0.1 deg) High power requirement (> 1 kW) Multiple pointing requirement Attitude slewing requirement

Spin stabilization preferred in most cases

Spin stabilization or passive control using gravity gradient

Either 3-axis control or dual spin

Either oriented planar array or spinning cylindrical array

3-axis control

Oriented planar array

3-axis control

Dual spin with articulation mechanism or 3 axis with wheels

The guidance and navigation function on most spacecraft is a basic form of radio guidance. It uses ground tracking to measure the flight path (or spacecraft ephemeris), ground computing of desired velocity corrections, and command of the correction through the communications and command subsystems. The direction of the velocity correction is governed by the attitude control of the spacecraft body and the magnitude is controlled by engine firing time. Two elements limit performance: ground-tracking and spacecraft attitude-control during thrusting. The Global Positioning System (GPS) may provide another way to measure the flight path. Coupled with an appropriate guidance computer, a GPS receiver should be able to guide die boost phase and allow the spacecraft to navigate autonomously in orbit, at least for low altitudes. Other autonomous navigation methods are also available (see Sec. 11.7). Guiding spacecraft to intercept or rendezvous usually requires a guidance radar, a gyroscope reference assembly, and often accelerometers.

Accurate attitude control depends on the attitude sensors. Table 10-17 summarizes what present sensors can do. Each sensor class is available in either a 1- or 2-axis version. Magnetometers, Earth sensors, and Sun sensors are available in forms which use the spin motion of a spinning spacecraft to scan the sensor's field of view. Some magnetometers and Earth and Sun sensors do not require scanning, but some highly accurate Earth sensors use scanning detectors. Magnetometers apply only to altitudes below about 6,000 km because the Earth's magnetic field falls off rapidly with altitude. Uncertainty in the Earth's magnetic field and its variability with time limits the accuracy of a magnetometer. In the same way, horizon uncertainty limits an Earth sensor's accuracy. Star sensors, however, allow us to measure attitude very accurately. Most star sensors are too slow (typically several seconds) to control a spacecraft's attitude directly, so we normally use them with gyroscopes for high accuracy and rapid response. Gyroscope accuracy is limited by instrument drift, so most gyroscope systems are used in conjunction with an absolute reference such as a star sensor or a directional antenna.

After-the-fact processing can improve our knowledge of attitude. For example, we can monitor the Earth's magnetic field continuously and therefore partially correct the variable effects of magnetic sensing. Variations in the Earth's horizon tend to follow a daily cycle, so we can apply some filtering correction.

TABLE 10-17. Ranges of Sensor Accuracy. Microprocessor-based sensors are likely to improve accuracies in the future.

Sensor

Accuracy

Characteristics and Applicability

Magnetometers

Earth Sensors

Sun Sensors Star Sensors Gyroscopes

Directional Antennas

0.05 deg (GEO) 0.1 deg (low altitude) 0.01 deg 2 arc sec 0.001 deg/hr

0.01 deg to 0.5 deg

Attitude measured relative to Earth's local magnetic field. Magnetic field uncertainties and variability dominate accuracy. Usable only below -6,000 km.

Horizon uncertainties dominate accuracy. Highly accurate units use scanning.

Typical field of view ±130 deg.

Typical field of view ±16 deg.

Normal use Involves periodically resetting the reference position.

Typically 1% of the antenna beamwidth.

The attitude-control system can also produce attitude motions which combine with the attitude sensor's accuracy to affect the total control accuracy. Control systems that use thrusters alone require a dead zone to avoid continuous firing of the thrusters. The control system's accuracy is limited to half the dead-zone value plus the sensor accuracy. Systems which use wheels (either speed-controlled reaction wheels or gimballed constant-speed wheels) can avoid dead-zone attitude errors, so they usually can operate close to die sensor accuracy.

Torquing methods for 3-axis-controlled spacecraft include gravity gradient, magnetic, thrusters, and wheels. Spacecraft using gravity-gradient and magnetic torquing are clean and simple but do not provide high levels of control torque. We use thrusters on most spacecraft because they produce large torque and can control the spacecraft's translational velocity as well as attitude. If the spacecraft must maneuver or suffers cyclical disturbances, such as those at orbit rate or twice orbit rate (see Chap. 11), we need to use wheels. A wheel can cyclically speed up or slow down, thus producing maneuvering torque or counteracting disturbance torques. A wheel system consumes less propellant than a thruster-only system because periodic effects do not require propellant use. On the other hand, wheel systems are heavier and more complex than those without wheels. Variable-speed reaction wheels produce only limited control torque (less than 1 N-m). To obtain large values of cyclic torque, we use control moment gyros—constant-speed wheels gimballed about an axis perpendicular to the spin axis. These gyros can develop torques up to several thousand N * m. When we need more degrees of freedom or better pointing accuracy, we use mechanisms to point spacecraft appendages, such as solar arrays or directional antennas.

Estimating Torque Requirements

One important sizing parameter for the control subsystem is its torque capability. This capability, often called the control authority, must be large enough to counterbalance disturbance torques and control the attitude during maneuvers and following transient events such as spacecraft separation, deployment, and failure recovery. These latter events usually size the torque requirement The separation transient is usually specified in terms of tip-off rate—the angular velocity, typically 0.1 to 1 deg/s, imparted to the spacecraft at release from the booster. We size the attitude control thrusters to capture or stabilize the spacecraft attitude before it has exceeded a sped-

fied value, as shown in Fig. 10-2A. The relation between required torque T, the tip-off rate, a>, (in rad/s), spacecraft moment of inertia, Is, and the maximum attitude excursion, Own (in rad), is:

The torquing ability of a thruster system, a reaction wheel, a control moment gyro, or a pointing mechanism may be set by an acceleration requirement such as that arising from an attitude slew maneuver, shown in Fig 10-2B. The torque is simply la where / is the moment of inertia and a is the acceleration. Sometimes the attitude maneuver is specified as a change in angle of 6 in a time t^. The torque in this case is:

As Fig. 10-2B shows, this is based on applying full accelerating torque for t^ / 2 and full decelerating torque for the remaining time.

Fig. 10-2. Estimating Torque for Attitude Capture and Maneuvering.

A Attitude Capture Following Separation B. Attitude Maneuver in Time, tdur

Fig. 10-2. Estimating Torque for Attitude Capture and Maneuvering.

The control torque required to stabilize a spacecraft during velocity-correction firing of a rocket motor is the product of the rocket's thrust level and the distance that its line-of-action is offset from the spacecraft center-of-mass. This torque can be due either to thruster misalignment or eg offset (see Table 10-18).

Estimating Angular Impulse for 3-Axis Control

Another major sizing parameter for the attitude control subsystem is the angular impulse capability of its torquers. Angular impulse is the time integral of torque. For thruster-produced torque, the angular impulse is related to the propellant mass expended. For reaction wheels and control moment gyros, the angular impulse is related to wheel moment of inertia and speed. In all cases, angular impulse is related to control system weight.

For 3-axis control systems, we calculate angular impulse by evaluating that needed for attitude maneuvering, for counteracting the effects of disturbance torques, and for oscillation or limit cycling. We determine the angular impulse required for maneuvering from spacecraft moment of inertia and maneuver angular rate. The angular impulse required to start an attitude maneuver Lstart is:

where Is is spacecraft moment of inertia and a),^ is the angular rate (rad/s) of the

TABLE 10-18. Disturbance Torques. These are vector equations where V denotes vector cross product and denotes vector dot product See Sec. 11.1 for simplified _equations._______

Disturbance

Equation

Definition of Terms

AV Thruster Misalignment

s x T

s vector distance from center of mass to thrust application point T vector thrust

Aerodynamic Torque

±pVzCdA(i^xscp)

p atmospheric density Cd drag coefficient (typically 235) A area perpendicular to Uy V velocity

Uy unit vector in velocity direction scp vector distance from center of mass to center of pressure

Gravity

Gradient

"0

(i Earth's gravitational coefficient

3.986 X 1014 rrrVs2 R0 Distance to Earth's center (m) 1 Spacecraft inertia tensor ue Unit vector toward nadir

Solar

Radiation

KsiUe-uJA^^jjxs,

Kg solar pressure constant

4.644 X 10-6 N/m2 sc vector from spacecraft center of mass to area A u„ unit vector normal to A us unit vector toward the Sun a surface absorptivity coefficient rs surface specular reflectance coefficient surface diffuse reflectance coefficient

(Note: a+ rs+ rd= 1)

maneuver. Stopping the maneuver requires an equal amount of angular impulse of opposite polarity.

To compute angular impulse to cancel disturbance torques, we examine the disturbances affecting the spacecraft as shown in Table 10-18. These disturbance torques are all vector quantities. They can be expressed in any convenient system of coordinates, although spacecraft body coordinates are generally used. For a given spacecraft configuration, orbit, and spacecraft attitude, these torques can be computed and integrated over the spacecraft lifetime. The result is the accumulated angular impulse which, if uncontrolled, will disturb the spacecraft attitude. The control subsystem must counteract these disturbance torques by applying control torque and the control subsystem angular impulse capability must be at least equal to the disturbance angular impulse.

In addition to the disturbance torques listed in Table 10-18, the control system may be sized by the requirement to interchange momentum between spacecraft body axes (sometimes referred to as Eider cross-coupling torque). Numerically this torque-like effect is:

and comes about when the spacecraft dynamic equations are written in rotating coordinates. TE is the torque, £2 is the angular velocity of the coordinate system, and Hj is the spacecraft angular momentum including that due to body rotation and internal moving parts (such as reaction wheels). For a circular orbit and the dynamic equations written in a coordinate system which rotates at orbit rate, the Euler cross-coupling torques are of the same form as gravity-gradient torques and these effects are often combined.

Note that, for any single axis, the disturbance torque may have cyclic terms that integrate to zero over an integer number of cycles and secular terms that are not periodic. Also note that angular impulse required for maneuvering is all cyclic. Reaction wheels and control moment gyros can counteract cyclic torques by changing speed or direction. If we use reaction wheels or control moment gyros (CMGs), we can size them for the cyclic terms and counteract only the secular terms with thrusters. But if we are designing a system that has no wheels or CMGs, we must expel propellant to counteract all disturbances, and the angular impulse requirement is the sum of the time integrals of the absolute value of disturbance torque computed about each axis. The process of computing control system angular impulse from disturbance torques and identifying cyclic and secular components is shown in Table 10-19.

Thruster control systems operate by pulsing a thruster when the attitude error exceeds a set value known as the dead-zone limit. The thruster's design determines the length of the pulse—typically from 0.02 sec to 0.1 sec. The propellant consumption of such a system is proportional to the size of the pulses and the rate of pulse firing. A well designed control system will fire a minimum length pulse each time the dead-zone limit is exceeded. The angular velocity change, Aft), produced by a minimum pulse Pmin is:

where P^ = Ts t^, T is thrust level of the thruster, s is the lever arm through which the thruster works to produce torque, t„^n is the minimum thruster firing time, and Is is the spacecraft's moment of inertia. The mean angular velocity of the spacecraft while in the dead-zone is Ato/2 which implies that the spacecraft transverses a dead-zone of 26d in 40^/Act) seconds. Since the pulse firing time is negligible relative to the time spent in traversing the dead-zone, the average impulse rate, IR, is one minimum pulse every 46^/Ato seconds, or:

The total angular impulse expended during the mission, L^, is IR times mission duration:

The torque produced by a thruster is equal to its thrust T times its lever arm s. The time integral of thrust is linear impulse and is related to the mass of propellant used by the rocket equation (Eq. 17-6). An appropriate expression relating angular impulse to propellant use is:

Lm = IR * mission duration

or equivalently:

TABLE 10-19. Computing Control System Angular Impulse Requirements.

Step

Operation

Comments

1. Calculate disturbance torques

Use equations in Table 10-18

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