To control the attitude of spacecraft, reliance is placed on sensors of various types. This is particularly the case for three-axis stabilized vehicles but also applies to spin-stabilized vehicles.

Sensors that are directed toward a star or toward the earth's horizon depend on sources external to the spacecraft. Others, such as gyroscopes, are internal. In all cases, an error signal is generated that then is used by the attitude control system to initiate the needed correction, for instance, by firing the appropriate attitude control thrusters.

Usually, the attitude of a spacecraft is defined relative to inertial space, that is, relative to axes that do not rotate. For orbiting spacecraft, it is sometimes preferred to relate the vehicle attitude to the type of reference frame familiar from aircraft attitude control by introducing the pitch (relative to the local horizontal), roll, and yaw angles. It should be noted, however, that as the spacecraft moves on its orbit, the local horizontal rotates. Hence allowance must be made for the fact that the aircraft-type reference is not an inertial reference.

In deep-space missions, as opposed to earth or planet orbiting missions, the spacecraft attitude is usually determined by observations of stars by small, optical telescopes. The attitude is then specified by the orientation of a set of spacecraft-fixed reference axes relative to the three mutually orthogonal axes that can be defined by the ecliptic plane and the vernal equinox line (Fig. 1.1). For this purpose, the declination and right ascension (Fig. 1.4) of selected guide stars are programmed into a star catalog contained in the memory of the flight computer.

To fix the attitude, a minimum of two bright stars ("guide stars") need to be sighted. For high accuracy, the angular separation between the two stars should be not much less than 45°. To avoid locking on the wrong star, the immediate vicinity of the selected guide stars should be void of other bright stars.

A frequently selected star is Capella. It satisfies these conditions and also has the advantage of a high declination. For deep-space missions close to the ecliptic plane, as most such missions are, a high declination avoids optical interference from the sun or from the planets.

In addition to the declination and right ascension of the guide stars, the flight computer should contain in its memory the positions of other bright stars. This greatly facilitates the reacquisition of the guide stars in case lock-on has temporarily been lost.

The attitude of the spacecraft is therefore determined by the lines of sight to two guide stars relative to spacecraft-fixed reference axes. The geometric relation between the selected astronomical reference (e.g., based on the ecliptic plane together with the vernal equinox line) and the spacecraft reference axes can then be specified either by the direction cosines between the two systems of axes or by their Euler angles.

Alternatively, a single guide star together with the sun may be used. In deep-space missions, the direction in the heliocentric reference of the spacecraft-to-sun line changes depending on the position of the spacecraft along its trajectory. Hence, in this case the position of the spacecraft at any given time has to be known.

Sun sensors are used to find the declination, say S, of the sensor axis relative to the heliocentric reference. The right ascension cannot be determined from a sun sensor. Therefore, to determine the spacecraft attitude completely, additional information, for instance from a star sensor or from a gyrocompass, is needed.

Most sun sensors are made to rotate continuously by an electric motor. Alternatively, an oscillating mirror can be used to scan the sun. In the case of spin-stabilized vehicles, the needed rotation can be provided by attaching the sensor directly to the spinning part.

The principle of operation is explained in Fig. 6.6. Figure 6.6a is a view parallel to the sensor axis of rotation. Figure 6.6b is the projection on the plane containing the sensor axis of rotation and the perpendicular to the ecliptic plane.

The functioning of the sensor can best be explained by basing it on the geometry of a sphere with center C on the axis of rotation. For this reason, the sensor is depicted in the figure as a sphere of unit radius. In practice, the sensor outer surface could just as well be a flat surface.

There are two slots, Si and S2. When the sun is in the plane C-Si, a portion of the light will strike the photocell Pi, which is located on the line C-Ai approximately through the midpoint of Si. Hence the photocell is

Figure 6.6 Sun sensor represented by unit sphere, (a) Top view; (b) elevation; (c) spherical triangle used for calculation.

Figure 6.6 Sun sensor represented by unit sphere, (a) Top view; (b) elevation; (c) spherical triangle used for calculation.

pulsed once on every revolution of the sensor. The slot S2 and the photocell P2 function analogously.

The two slots are arranged in the form of a letter V. Slot Si, but not S2, is in a plane that contains the axis of rotation.

As the sensor rotates, the path of the sun as seen by the unit sphere is incident on it on a plane parallel to the sphere's equator. The declination, S, of the sensor axis of rotation in the heliocentric reference is the angle in the diagram between the axis of rotation and the intersection of the plane of incidence with the unit sphere.

The period, say P, of the sun sensor, relative to inertial space, is equal to the time interval between two successive pulses of photocell P1. By averaging over many rotations, P is determined to high precision. If t\ is an instant of time when P] is triggered, and t2 the instant following t\ when P2 is triggered, then the azimuthal angle, traversed by the sensor during the interval t2 - t\ is

The angle <f> will be needed to find S. This is seen from considering the spherical triangle shown in Fig. 6.6c. The angles a and fi are geometrical properties of the sensor, hence are known. Also known, from (6.26), is 4>. The remaining side, a, of the spherical triangle can be eliminated as shown next:

From the two Napier formulas of spherical trigonometry,

The desired result for the declination S of the sensor axis follows from applying the inverse tangent function to both sides of each equation and adding, with the result that

-cotan a

The declination can therefore be found from the measured time intervals between photocell pulses and the application of software that is programmed to express Eqs. (6.26) and (6.27).

Orbiting spacecraft often use as an input to their attitude control system observations of the horizon of the orbited astronomical body. The required sensors usually operate in a scanning mode in which the sensors' lines of sight are swept repeatedly over the astronomical body such that they cross its horizon.

In the case of three-axis stabilized spacecraft, the scanning motion is achieved either by a continuous conical sweep of the sensors' lines of sight

\ Sensor Axis of Rotation

Figure 6.7 Horizon scanning sensor (only a single line-of-sight cone is shown). lA, start; fA, end of intersection with earth; 0, pitch angle; <j>, roll angle.

\ Sensor Axis of Rotation

Figure 6.7 Horizon scanning sensor (only a single line-of-sight cone is shown). lA, start; fA, end of intersection with earth; 0, pitch angle; <j>, roll angle.

or by motor-driven mirrors that oscillate back and forth through some finite angle. In spin-stabilized spacecraft the sensors can be rigidly attached to the spinning portion of the vehicle. In this case the vehicle itself provides the sweeping motion of the lines of sight.

The basic principle of operation of such sensors is illustrated in Fig. 6.7. In this figure the earth is taken as an example. The line of sight moves on a circular cone that intersects the earth during part of the time. As is illustrated in the figure, the intersection starts and terminates on the horizon as seen from the spacecraft. On a given sweep the intersection starts at some time and terminates at It is the difference between these two times that provides the information needed for the attitude control.

In the usual arrangement, illustrated in Fig. 6.8, there are two lines of sight, A and B, provided either by a single sensor or by two separate but synchronized sensors. The vehicle's pitch angle (relative to the local horizontal) is designated, in conformity with the usual aircraft terminology, by 0, the roll angle by <p, and the yaw angle (not shown) by f. It is evident that because of the spherical symmetry of the earth, xjr cannot be determined by horizon sensors. To obtain it requires information derived from gyroscopes. (More precisely, the outputs of two rate gyros, one for the roll rate and one for the yaw rate, are combined with the Kalman filtered roll angle obtained from the horizon sensors.)

The photodetectors and optics of horizon sensors are designed to operate in the near-infrared, usually in the wavelength range 10 to 20 /¿m. The advantage is that in this spectral range the earth's atmosphere is more sharply defined. Also, there is less variation of the signal's amplitude between a sunlit and a dark spacecraft horizon.

Local Horizontal

Local Horizontal

Let rg be the earth's radius (averaged for the orbit and including a correction for the effective height of the atmosphere in the near-infrared region) and r the distance of the spacecraft from the earth's center. The radius, rH, of the horizon seen from the spacecraft is therefore r„ = rgy 1 - r|/r2 (6.28)

Figure 6.8 shows the ground traces of the two lines of sight A and B for a given (here positive) roll angle <f>. Shown by dashed lines for reference are the ground traces for zero roll angle. These are symmetric with respect to the plane defined by the roll axis and the local vertical.

The angle, say 2/3, between the two sensor telescopes with the lines of sight A and B in principle can be chosen arbitrarily as long as A and B sometimes intersect the earth. For comparable sensitivity of the pitch and roll angle determinations it is, however, advantageous to choose fi such that for zero roll angle the four points of tangency with the earth form approximately a square. This is also shown in the figure. It then follows that the preferred choice for /3 is p = tan"1 (1 + 2r2 r^/r¡)~1/2 (6.29)

Let y (£) be the (positive) angle between the spacecraft roll axis (i.e., the reference axis for the pitch) and the plane containing the two lines of sight. This angle can be determined by a shaft encoder on the axis of rotation of the sensor.

As the plane is swept over the earth, there will be an instant, say tv, when this plane is vertical. This will occur at the time

The value of y at this time is sufficient to determine the pitch angle 9 of the spacecraft, since evidently

The period of the sweep is of the order of a few seconds, which is short compared with the time for appreciable changes of the spacecraft's pitch. The measurements of y at the midpoints between tA and £( and similarly between fB and averaged for greater precision as indicated in (6.30), in effect provide a quasi-continuous reading of the pitch angle.

The roll angle can be determined from the time intervals t^ — tA and t^- This is seen from Fig. 6.8 as follows: For roll angles 0/0, the lines of sight, when just touching the horizon, become displaced by comparison with the case 0 = 0, resulting in a lengthening or shortening of the ground traces, hence of the time intervals. If (Af)0 is the value of the time intervals for zero roll angle, it follows from the geometrical relations shown in the figure that t'-tA = (Af)0 +

2(1 + 2 r2/r|) 0 1 + 2r2r2/r4 ' v 2(1 + 2 r2/r2) 0 1 + 2r2r2/r| ' v valid for 1. The angular rate of the sensor axis of rotation (or of the rate of spin in the case of spin-stabilized vehicles with sensors fixed to the vehicle) is designated by v.

Subtracting the two equations and solving for 0 gives the final result

(l + 2r2rh/r^)v * = 4(1 + 2r2/r2) " * " " '")) (6-32)

Measuring the time intervals t'K - tA and t^ - iB and carrying out the calculations indicated in (6.28) and (6.32) therefore determines the roll angle. This supposes that the orbit radius, or more generally the instantaneous spacecraft-to-earth-center distance, r, is known, derived either from known orbital data or by a spacecraft position determination.

A particular design of a horizon scanning sensor is shown in Fig. 6.9. The (single) line of sight is through an infrared-transmitting germanium window, coated with an interference filter to define the passband. A wedge-shaped lens, rotated by an electric motor, directs the incident radiation toward a bolometer. In this way a conical scan of the earth is obtained. The angle of rotation, and from it the angular rate, is obtained from magnetic pickups. Infrared radiation from spacecraft components that may intrude into the scanning cone are electronically blanked out.

Two such instruments, working in tandem, are needed to complete the system illustrated in Fig. 6.8. Pitch and roll angle accuracies achieved are of the order of 0.1°.

Window Filter (fixed)

Line of Sight

Bolometer (fixed)

Figure 6.9 Schematic of a horizon scanning sensor. Courtesy of TELDIX GmbH, Germany.

Window Filter (fixed)

Lens/Wedge (rotates)

Bolometer (fixed)

Line of Sight

Figure 6.9 Schematic of a horizon scanning sensor. Courtesy of TELDIX GmbH, Germany.

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