As shown in Fig. 19-12(c), after the trajectory intersects the switching line, it will proceed directly toward the origin. For example, an initial state at A will intersect the switching line at B, change directions, and move along the switching line to the origin where the control is set to zero. Similarly, an initial state at C will intersect the switching line at D and proceed to the origin. This case is discussed in more detail by Hsu and Meyer . In practical applications, modifications are required to take into account any instrument or system rate limits which may exist Also, the wheel torque (and acceleration) are neither perfectly. known nor necessarily constant over the entire wheel speed range. The minimum expected wheel torque should be used in designing the control law to avoid relay chattering. A modified version of this optimally damped bang-bang control law has been implemented in the IUE and SMM attitude control system computers.
When a system has been shown to be stable and optimal (in some sense) when operating continuously, it does not necessarily follow that the discrete or pulsed version is optimal or controllable except in the limit of very high pulse rates. A discrete system periodically samples attitude sensors and controls the actuator. The time step is the time between consecutive samples and may be limited by the time constants of an analog control system or the capacity of an onboard computer. As the length of the time step is increased, control may be lost at periods corresponding to the resonant frequencies of the system. For example, if the time step is an integral multiple of the nutation period, the oscillations will not be damped. An example of this problem is given by Schmidtbauer, et al., . In such cases, further analysis, including digital simulations, will be required to analyze the system performance.
For an orthogonal set of reaction wheels aligned with the body axes, we can reorient the spacecraft from one celestial target to another by a sequence of single-axis slews. The required direction cosine matrix is
C= Rj(a2)Rj(- 82)R?( &)/?,( /¡,)/?2(- «,)/?,(«,) (19-69)
where /?,(#) is a matrix representing the rotation about axis / through an angle 9, and the target coordinates (a,j,/3) are the right ascension and declination of the target and an azimuthal rotation angle about the target. The matrix C is uniquely specified by the initial and final target coordinates. This does not, however, completely specify the slew sequence.
A sequence of three slews is sufficient to accomplish any maneuver in the . general case where the rdtation angles, denoted for convenience as roll, pitch, and yaw, can take any value between 180 deg and —180 deg. There are 12 possible combinations of the form roll-pitch-yaw, where consecutive rotations about the same axis, such as roll-roll-yaw are excluded but nonconsecutive rotations about the same axis, such as roll-pitch-roll, are allowed. There are 3x(3- l)x(3-1)= 12 such combinations.
As an example, if we choose to perform a roll-pitch-yaw maneuver, we could calculate the required angles by setting (see Appendix E)
cos if cos 0 cosi^sin0sin<()+ sini^cos<() -cosifsin0cos<f> + sin^sin<f> — sin^cos0 -sin^sin0sin<f> + cosi^cos<f> sin^sin0eos<f> + cosi/»sin<> sin0 -cos0sin<|> cos0cos<f>
Because the C^ matrix elements are known, the angles can be calculated as follows:
- C2i/cos0 Cn/cos0
This sequence is ambiguous because either sign of the radical can be taken, leading to separate roll-pitch-yaw slew sequences where the rotation angles are supplements of each other.* This same ambiguity exists for all cases, yielding a total of 24 possible slew sequences to perform a maneuver.
Because the computer operations involved are rapid, all 24 possibilities can be computed and ordered according to a suitable criterion. Typically, the minimum total path or minimum slewing time is chosen, but other criteria may be used. Oh OAO it was found that the performance error due to gyro misalignments grew linearly with slew angle, and the sequence was chosen which had the shortest maximum slew leg.
Although errors may arise from gyro misalignments, no error is caused if a wheel is misaligned because any undesired momentum components will be sensed by the gyros and controlled to zero. The design for the IUE spacecraft takes advantage of this feature by mounting a redundant fourth wheel along the diagonal of the orthogonal cube formed by the three primary wheels. If one of the three orthogonal wheels fails, relays will be set in the control electronics to send its commands to the skewed wheel. The rate component along the desired axis will be executed at 1 /V3~ times the normal rate due to the projection of the commanded axis on the skewed axis, while the undesired components are temporarily absorbed by the reaction wheels on the other two axes.
A major advantage of using a sequence of single-axis slews is that the other two axes are controlled to zero, which minimizes the coupling of the axes in Euler's equations. This allows the three attitude angles to be estimated separately according to the accumulated gyro angles. Another advantage of this single-axis sequence lies in the ease of checking constraints. On IUE, for example, maneuvering ts severely restricted by constraints on the position of the body axes with respec t to the Sun, the Earth, and the Moon both while the spacecraft is pointed and whiles
•The ambiguity may be resolved, however, by limiting the intermediate rotation to the range 0 to 180 deg. In the above example, we would then have 0=arccos(+^l - Cf, ). This convention is generally adopted for Euler angle sequences. See, for example Goldstein  and Appendix E.
moving. Constraint checking is simplified by first transforming these inertially known vectors into the body system using the current attitude direction cosine matrix. By the geometry of the single-axis slews it is possible to perform even the dynamic constraint checks geometrically without simulating the maneuver.
An alternative to the single-axis slew sequence, called the eigenaxis method, [TRW, 1976] has been designed for HEAO-B, where a quaternion attitude estimate is available in the onboard processor. This method uses quaternion multiplication to compute the unique rotation axis, e, and angle, <f>, which can achieve the desired three-axis reorientation. The error quaternion, qE, is given by fe-i'fc (>9-72)
where q is the current attitude state and qD is the desired attitude (see Section 12.1 and Appendix D). The eigenaxis, e, is identified by expressing the four components of the error quaternion as esiny ,cosy
and performing the maneuver by rotating about k through the angle In theory, this yields a minimum path maneuver, but in the actual design, large deviations are expected from the eigenaxis due to system nonlinearities such as torque motor response, torque or wheel saturation, and nonorthogonal reaction wheels. Compensation is made for these effects by continually updating the eigenaxis and recomputing the motor torques.
19.5 Attitude Acquisition
Attitude acquisition consists of the series of attitude maneuvers, commands, and procedures necessary to reorient and reconfigure the spacecraft from the attitude state at separation from the launch vehicle to an attitude state suitable for the initiation of normal mission operations. The latter configuration is referred to as mission mode. This section describes the problems and procedures unique to attitude acquisition, including the deployment of extendable booms, antennas, and solar panels and the inflight checkout of both hardware and software.
Most missions require some period of attitude acquisition. The simplest acquisition sequences require maneuvers such as despin, deployment of solar panels, activation of the onboard sensors and experimental hardware, and a maneuver to the first mission attitude. This sequence is similar to that used by SAS-3 and most stellar-oriented missions. Slightly more complex sequences are required for geosynchronous spacecraft which employ a transfer orbit such as SMS/GOES or CTS. For these missions, a prolonged sequence of interspersed attitude and orbit maneuvers, lasting a week or more, is required to attain the proper position, orbit, and attitude. (See Section 1.1 for an example.) A sequence of maneuvers lasting 5 months was employed by RAE-2, initially to achieve a circular orbit about the Moon and finally to deploy four extendable antennas in a timed sequence to a total length of 450 m to achieve a three-axis, gravity-gradient stabilized attitude [Werking, el al., 1974].
Some missions may require periods of attitude reacquisition to reacquire mission mode in the event of hardware or software failure or operator error. As an example, reacquisition is required for autonomous missions, such as the IUE, if the attitude error after a commanded maneuver exceeds the Fine Error Sensor field of view [Blaylock and Berg, 1976).
Clearly, specific details of attitude acquisition are very mission dependent—a function not only of attitude requirements, but of onboard hardware, ground support hardware (e.g., the availability of telemetry and command stations), ground support software, and power and thermal constraints. Despite the numerous constraints placed on this phase of a mission, some of which may be quite severe, considerable flexibility is available to the mission planner and the opportunities for innovative solutions are great. Although the implemented procedures for GEOS-3 (see Section 19.5.3) and CTS [Basset, 1976] were both specific and intricate, numerous alternatives were considered [Repass, et al., 1975; Lerner, et al., 1976; Kjosness. 1976] and discarded. Frequent improvements to the baseline procedures were made in the days preceding launch and probably could have continued. Although the end points of attitude acquisition sequences are fixed, the possible paths are distinctly nonunique; many must be traveled and pitfalls mapped in prelaunch planning before the best can be selected.
Attitude acquisition may be categorized by the degree of autonomy of the spacecraft hardware or, conversely, by the amount of ground support required. The spacecraft may be (I)fully autonomous; (2) semi autonomous, i.e.. using a mixture of onboard and ground support; or (3) ground controlled. Fully autonomous attitude acquisition is accomplished either through the use of analog, preprogrammed electronics or a digital onboard computer, or OBC (see Section 6.9). Sensor data is used in a control law which is implemented via the analog electronics or OBC to command torquing devices such as electromagnets, wheels, and thrusters. For example, the German Aeronomy satellite, AEROS, used error signals from an analog Sun sensor and a magnetometer to control an electromagnet and torque the spin axis to the Sun. HCMM uses magnetometers and a wheel-mounted horizon scanner to control a magnetic torquing system to achieve a stable, three-axis Earth-pointing attitude [Stickler and Alfriend, 1974].
For semiautonomous spacecraft, a mixture of onboard and ground support is used to achieve acquisition. HEAO-I used an onboard analog control system to place the spin axis within several degrees of the Sun and ground software using star tracker data to determine a precise three-axis attitude and calibrate the gyro-based control system. After calibration, the control system maneuvered the spacecraft to a target attitude and maintained it there using hydrazine thrusters to null the difference between the target and the gyro-propagated onboard attitude (see Section 19.4).
Ground-based attitude control may be either open-loop or closed-loop. Closed-loop control is similar to that provided on board: sensor data is telemetered in real time to the ground support computer; the data are processed and torquing commands are computed; and, finally, the software uplinks the requisite commands. The main advantage of closed-loop control is the flexibility and power of large ground-based computers. Continuous, rapid-response commanding capability is provided without requiring increased onboard weight and attendant complexity. The main disadvantages are the requirement For continuous uplink and downlink contact during operations and increased opportunities, for hardware or software failure or operator error due to the extended communication lines. Attitude acquisition for the CTS spacecraft [Basset, 1976] used closed-loop control with a Hewlett Packard 2I00A minicomputer.
Open-loop ground-based attitude control uses ground software to process and display sensor data and to compute and evaluate (often via simulation) command sequences. Analysts then select appropriate commands which are uplinked for execution on board. Open-loop control requires a time delay from 30 sec to several hours between receipt of sensor data and command execution whereas closed-loop ground-based control delays are of the order of several seconds. The advantage of open-loop control is the software simplicity and reliability afforded by relaxing the severe time constraints to permit analysts to evaluate and verify computed commands while retaining the power and flexibility of the ground-based computer facilities. The analyst can also respond to contingencies not foreseen in prelaunch analysis and rely on his judgment and experience in evaluating commands. The disadvantages are the limited control afforded by the slow response time,* and the increased possibility of operator error when many individual decisions and actions are required from computation to uplink of commands. Open-loop control has been the most widely used to date. Examples include the generation of command sequences for attitude maneuvers or maintenance for AE, SMS/GOES, and CTS and the GEOS-3 acquisition sequence described in Section 19.5.2.
This subsection describes attitude maneuvers that are unique to attitude acquisition. Table 19-1 (page 666) illustrates the types of maneuvers and constraints required for representative acquisition sequences. The initial state is determined largely by the configuration of the spacecraft within the last rocket stage and whether or not that stage is spin stabilized. The detailed release mechanism for spacecraft separation from the last stage and the performance of the yo-yo despin mechanism, described below, are also important. The final state includes the attitude, attitude rate, and spacecraft configuration (e.g., solar panel and antenna deployment, momentum wheel spinup, and attitude sensor and experiment turn-on).
In the event that the spacecraft cannot be commanded due to an onboard or ground support failure, intermediate attitudes between the initial and final state should be "safe harbors," viz capable of being maintained for prolonged periods without endangering the success of the mission. As an alternative, opportunities for easy access to safe harbors should be mapped and exploited as necessary by, for example, loading backup commands to be executed automatically on board the spacecraft at some later time.
Yo-Yo Despin. This maneuver is frequently employed for reducing the spacecraft's spin rate shortly after separation from the last stage of a booster
•Open-loop control employing several analysts and telephone lines »<ts employed for RAE-I to minimize time delays [IBM, 1968].
rocket. The last stage of rockets such as the Scout and Delta is often spin stabilized at a high angular velocity, e.g., 150 rpm, and this spin rate is maintained by the spacecraft at separation.
As illustrated in Fig. 19-13, we assume that a cylindrical spacecraft with axial moment of inertia / and radius R is rotating without nutation about its longitudinal axis with angular velocity 8. Two equal masses, /w, and m2, are attached to separate cables of length / wrapped around the spacecraft perimeter opposite the
direction of rotation. At time f=0, the masses are released and travel tangentially away from the spacecraft. As the cables unwind, they increase the moment of inertia of the system about the z axis and decrease the spacecraft's angular velocity. When completely unwound, the cables and attached masses are jettisoned, carrying off a substantia] fraction of the system angular momentum. The relationship between the final spin rate, the spacecraft size and inertia, the cable length, and the yo-yo masses is derived as follows.
We define the body coordinate frame, x, y, and z, to be fixed in the spacecraft which is routing about z at angular velocity B relative to inertial space (see Fig. 19-14). The masses are initially at ±x. Similarly, the cable frame, i, j, and k, is rotating in the body such that the cables are tangent to the spacecraft perimeter along the ±i axis; i.e., the coordinates of the points where the cables are tangent to the body are fixed in the cable frame at (± £,0,0).
Assuming no energy loss during the despin, we can compute 8(0 fr°m conservation of energy and angular momentum. By symmetry, the angular momentum and kinetic energy of both masses are equal and it suffices to consider only m,. The position and velocity of m, in the cable frame (see Fig. 19-14) are
where is the angular separation between the cable and body frames in radians. The angular velocity of the cable frame, relative to inertial space, is
From Section 16.1, the velocity, v„ of m„ in inertial space, expressed in the cable frame, can be written in terms of <o, r„ and r, as, v,=rl+<oXr1 = /ty(S+^)i + /?8j (19-76)
The angular momentum of mt in the cable frame is h, =r, X m,v, = m,/?2[0+<>2(0 + <>)]k (19-77)
Because k is fixed in inertial space in the direction of total system angular momentum, we may use the conservation of angular momentum to obtain
(/ + 2mI«2)80=/n(i) + 2mIrt2[R(f)-lV(8 + <>)] (19-78)
where R0=R(f=0) and the total moment of inertia is the sum of the inertia of the spacecraft body (/) and the two masses (2m,/?2). The kinetic energy of m, is r. = ^'"iVv1 = ^m1/*2[*2(0+^)2+82] (19-79)
From the conservation of energy we obtain
Equations (19-78) and (19-80) may be solved simultaneously for 4> and Q to obtain
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