4. Attitude laws

72

4. Attitude laws

This chapter “zooms” inside the satellite, considering it as a 3D rigid body, and explores some common attitude laws that allow the satellite to satisfy simple pointing requirements. At the end of the chapter, all the instruments for a global environmental model will be available for next analyses.

Notice that some symbols defined in this context only stand for this chapter (e.g., rotational angles in Par. 4.1).

4.1 Attitude determination

To describe the attitude of a satellite, it is useful to introduce two further reference frames, both centered in the satellite’s center-of-gravity:

 Body Frame (BF), whose axes x, y, z move in time due to a prescribed attitude law.

 Satellite Fixed Frame (SFF), whose axes x’, y’, z’ are always parallel to those of GRF;

The attitude of a generic spacecraft is specified by the orientation of BF, which should be aligned with some desired frame according to the scope of the mission.

Spacecraft attitude determination is to provide the information of the distance between the spacecraft BF and the desired frame, [30]. Anytime the spacecraft attitude is not in the perfect position, the attitude information will be compared automatically with the desired attitude, and the error information is then used to calculate how much action is needed for each actuator to bring the spacecraft to the desired attitude. This is an active way to perform attitude control. Assuming a perfect attitude control, the spacecraft would exactly follow the prescribed attitude law and BF would coincide with the desired frame. This hypothesis in retained throughout the chapter.

At any time, the distance between BF and SFF can be represented by a single

rotation if quaternion is used or a series of rotations if Euler angles are used. In the

latter case, the sequence of the rotations is very important. Every rotation is about a

73 certain rotational axis for some angle, [30]. There are 12 possible Euler angle sequences and each of them has a Direct Cosine Matrix (DCM), indicated as Q, which is the product of three elementary rotation matrices R. A commonly used set of Euler angles for rotating BF into alignment with SFF is the “yaw-pitch-roll”

sequence, [31]:

𝑸 = 𝑹 _𝑥 (𝛾)𝑹 _𝑦 (𝛽)𝑹 _𝑧 (𝛼) (4.1) where subscripts indicate the axis about which the rotation takes place (the order of rotations is from right to left, see the figure below) and:

 α is the yaw angle (0° ≤ α < 360°),

 β is the pitch angle (-90° < β < 90°),

 γ is the roll angle (0° ≤ γ < 360°).

This means that, at any instant of the motion, SFF axes are obtained by rotating the BF axes through the sequence of yaw, pitch and roll angles (exactly in this order).

Figure 4.1 – Yaw-pitch-roll sequence transforming BF into SFF, [31]

74 The expression for DCM as a function of yaw-pitch-roll angles is:

𝑸 = [

cos 𝛼 cos 𝛽 sin 𝛼 cos 𝛽 − sin 𝛽

cos 𝛼 sin 𝛽 sin 𝛾 −sin 𝛼 cos 𝛾 sin 𝛼 sin 𝛽 sin 𝛾 −cos 𝛼 cos 𝛾 cos 𝛽 sin 𝛾 cos 𝛼 sin 𝛽 cos 𝛾 − sin 𝛼 sin 𝛾 sin 𝛼 sin 𝛽 cos 𝛾 − cos 𝛼 sin 𝛾 cos 𝛽 cos 𝛾

] (4.2)

To determine BF, the spacecraft current position and the time must be known. The mostly used time in aerospace engineering is the Universal Time (UT) and a way to express the current date and time (e.g. to identify the beginning of a mission, the so- called “time t 0 ”) is the Julian Date jd, [day], a positive number whose zero is midnight of 1 ^st January 4713 B.C., [21]. As a reference date, one can take 1 ^st January 2000, 12:00, for which jd 2000 = 2451545.

Position and time can be used to obtain the ephemeris astronomical direction information, such as the Sun direction. In the previous chapter, a simplified approach to the problem has been shown. Here, because of the strong pointing requirements of some kinds of missions, a more detailed model is proposed (Par. 4.3.1).

Starting from the choice of a target, the attitude of an Earth-orbiting satellite is described by the three angles α(t), β(t), γ(t) throughout the mission, after having specified the satellite’s orbital elements and the starting date. A Matlab ^® script called spacecraft_attitude.m has been created to perform this task, whose structure is highlighted below.

Figure 4.2 - Structure of spacecraft_attitude.m

If required, also the angular velocity vector ω(t), [rad/s], of BF w.r.t. SFF can be

extracted as output, in order to better comprehend the results.

75

4.2 Kinematic model

The simplest model to describe attitude is to schematize a spacecraft with a flat plate and consider its normal n as the vector pointing to the target. With no loss of generality, one can set x to coincide with n. If no other condition were assigned, the flat plate would be free to rotate about x, thus creating an indetermination. This is consistent with the fact that the attitude of a rigid body may be defined relative to a given reference frame by means of a pair of right-hand orthonormal triplets of vectors with a common origin, [32]. To eliminate this degree of freedom, at initial time y is set to lie on the x’y’ plane in such a way that the angle between z and z’ is ≤ 90°.

Figure 4.3 - Flat plate attitude at initial time

This approach is acceptable at didactic level only: a real spacecraft will surely have

to satisfy more conditions with a given attitude (e.g., solar panels to the Sun, sensors

or camera toward a celestial body, antennas to the ground station…).

76 The positions of x, y, and z in SFF at t 0 are known once the target is engaged and are indicated by column-vectors x 0 , y 0 , z 0 respectively. The DCM between BF and SFF at t 0 is:

𝑸 ₀ = [𝒙 0 𝒚 ₀ 𝒛 ₀ ] = [

𝑄 ₁₁ 𝑄 ₁₂ 𝑄 ₁₃ 𝑄 ₂₁ 𝑄 ₂₂ 𝑄 ₂₃ 𝑄 ₃₁ 𝑄 ₃₂ 𝑄 ₃₃

] (4.3)

By comparison of Eq. (4.2) and (4.3), yaw-pitch-roll angles at t 0 can be determined:

tan 𝛼 = ^{𝑄 12}

𝑄 ₁₁ (4.4)

sin 𝛽 = −𝑄 ₁₃ (4.5)

tan 𝛾 = ^{𝑄 23}

𝑄 ₃₃ (4.6) There is no quadrant uncertainty in computing β because the principal values of the arcsine function coincide with the range of the pitch angle. Finding α and γ involves computing the inverse tangent, so we must once again be careful to place the results of these calculations in the appropriate range, [31]. This is performed by a Matlab ^® function called atan2d_0_360.m.

In order to compute attitude at any subsequent time, the rate of change of DCM is required. A discrete approach to the problem has been implemented: after setting an appropriate time-step Δt, the change of DCM in time is given by:

𝑸(𝑡 + ∆𝑡) = 𝑪𝑸(𝑡) (4.7) Eq. (4.7) is the heart of the kinematic model. C is a general rotational matrix that brings x(t) into x(t+∆t), thus allowing the satellite to follow its target. It is characterized by a rotational axis e and a rotational angle θ. Actually, there are infinitely many combinations of e and θ to perform this rotation: if θ is chosen as the angle between x(t) and x(t+∆t), then the transformation is called a “minimum-angle” rotation, [30].

Intuitively, this is the most efficient (i.e., the fastest) way to satisfy pointing

requirements, and both e and θ are known from information on target.

77 Figure 4.4 - Minimum-angle rotation (adapted from [30])

The operative definition of C is the following:

𝑪 = cos(𝜃)𝑰 + (1 − cos(𝜃))𝒆𝒆 ^𝑇 − sin(𝜃)𝑬 (4.8) where I is a 3x3 identity matrix, e is a column unit-vector having components e 1 , e 2 , e 3 in SFF and:

𝑬 = [

0 −𝑒 ₃ 𝑒 ₂

𝑒 ₃ 0 −𝑒 ₁

−𝑒 ₂ 𝑒 ₁ 0

] (4.9)

In the end, from the definition of θ the angular velocity vector is computed at any time-step:

𝝎 = ^𝜃

∆𝑡 𝒆 (4.10)

4.3 Target selection and modelling

Four different kinds of target are proposed, each of which has its own peculiarities.

In order to have an effective pointing, some hypotheses made in the previous

chapters must be released, as discussed below. Since the pointing law is specified

only for n, these models are sometimes called “one pointing vector laws”, [33].

78 4.3.1 Fixed point in space

This kind of target is typical of space observation missions. The present day, the most popular mission is maybe the one of Hubble Space Telescope, which is next to its End-Of-Life time and whose complete decay is planned in the decade 2030- 2040. Targets like stars, which are several light-years far away from the Sun, can be considered essentially fixed in space, since their proper motion is very much slower than, for example, the motion of Earth around the Sun.

Figure 4.5 - Fixed point in space (adapted from [33])

To specify such targets, it is convenient to introduce the Heliocentric Ecliptic Frame (HEF), [21], whose origin is at the centre of the Sun and:

 X-axis (unit vector i e ) is the same as the one of GRF and is given by the intersection between the equatorial and ecliptic planes;

 Z-axis (unit vector k e ) is perpendicular to the ecliptic plane, in the northerly direction, so that k e and k lie in the same hemi-space;

 Y-axis (unit vector j e ) makes up a right-handed orthogonal set with the

previous two.

79 Figure 4.6 - Elliptic Earth orbit in HEF

Both GRF and HEF have an axis in common and they only differ in the angle ϵ between ecliptic and equatorial plane, whose mean value is about 23.5°.

Because of nutation and precession phenomena, the terrestrial axis slightly moves, therefore ϵ is not constant but tends to decrease in time, [21]. Moreover, the assumption that Earth’s orbit around the Sun is circular cannot satisfy so strong pointing requirements. The definition of the Earth’s ellipse in HEF must be considered in order to compute the correct Earth-Sun relative position (which can be either Earth’s position in HEF or Sun’s position in GRF). The following semi-analytic polynomial formulation, taken from [21], allows evaluating a more refined position of the Sun in GRF, in which also the target coordinates can be rewritten after coordinate transformation from HEF to GRF.

1. Evaluate the number of centuries C from jd 2000 to jd of observation:

𝐶 = ^{𝑗𝑑−𝑗𝑑 2000}

36525 (4.11) 2. Compute the Sun’s mean longitude L m,Sun , [deg], with a first-order polynomial

expression:

𝐿 _{𝑚,𝑆𝑢𝑛} = 280.4606184° + 36000.77005361 ∙ 𝐶 (4.12) 3. Compute the Sun’s mean anomaly M Sun , [deg], with a first-order polynomial

expression:

𝑀 _𝑆𝑢𝑛 = 357.5277233° + 35999.05034 ∙ 𝐶 (4.13)

4. Compute the Sun’s ecliptic longitude L Sun , [deg] (already defined in Eq. (3.4)):

80 𝐿 _𝑆𝑢𝑛 = 𝐿 _{𝑚,𝑆𝑢𝑛} + 1.91466471° ∙ sin(𝑀 _𝑆𝑢𝑛 ) + 0.019994643° ∙ sin(2𝑀 _𝑆𝑢𝑛 ) (4.14) 5. The Earth-Sun distance in AU (indicated as r Sun in GRF) is given by:

𝑟 _𝑆𝑢𝑛 = 1.000140612 − 0.016708617 ∙ cos(𝑀 _𝑆𝑢𝑛 ) − 0.000139589 ∙ cos(2𝑀 _𝑆𝑢𝑛 ) (4.15) 6. The variation of ϵ in time is obtained from:

𝜖 = 23.439291° − 0.0130042 ∙ 𝐶 (4.16) Finally, the Sun position-vector in GRF is given by Eq. (3.3) with the quantities evaluated as above and multiplied by r Sun .

At any time, pointing vector n is the norm of satellite-target vector in GRF, which is determined from the knowledge of Sun, satellite and target’s position.

As an example, consider the following case: from astronomical calculations, it is known that, at midday of 1 ^st January 2000, Saturn and its moons transit at the following point in HEF (data taken from [21]):

𝒓 _{𝑡𝑎𝑟𝑔𝑒𝑡} = −8.9625𝒊 _𝒆 + 2.6322𝒋 _𝒆 + 0.3101𝒌 _𝒆 [AU] (4.17)

Figure 4.7 - Sketch of Earth’s orbit and target position

81 A GEO satellite has the task to make surveillance on this point over one year, beginning on 1 ^st July 1999, 12:00. Satellite’s attitude is visualized below in terms of yaw-pitch-roll angles.

Figure 4.8 - Yaw-pitch-roll angles over one year

In fact, one appreciates considerable variations of attitude angles only along one

Earth’s orbit around the Sun, due to the enormous distances separating target and

satellite if compared to r Sun . The behavior of α(t), β(t), γ(t) is better understood after

plotting the components of ω, which are expected to be very low in this case and, in

fact, are on the order of 10 ^-8 rad/s.

82 Figure 4.9 – Components of ω over one year

Since the inclination of Saturn’s orbit is quite small (≈2.48°), the target is “almost” on the HEF X-Y plane ⁵ . This explains why global maximum excursion (≈15°) is reached by α, which describes a rotation about z, while maximum excursion for β is smaller (≈5°). Moreover, n ≡ x, therefore rotations about x (γ angle, and so ω x ) are very small just from the beginning, when the target is acquired.

Figure 4.10 - Components of ω for the first 5 orbits

5

This statement is true if one refers to target coordinates in HEF. Just to give an idea of how “far” the

target is from the HEF X-Y plane, consider that 0.3101 AU ≈ 7273 DU, while the radius of the Sun is

about 109 DU.

83 The “confusing” plot of ω z can be clarified if one zooms inside the first 5 orbits to underline the short-period effects on the satellite orbiting around the Earth. As expected, all the three components of ω exhibit the same sine-like pattern along each orbit. Here, the maximum excursion for each ω component depends on satellite’s orbital elements.

In the end, plotting the modulus of the angular velocity ω 0 one can have an idea of how “fast” the change of attitude throughout the mission is. With reference to Fig.

4.7, ω 0 is smaller slightly before aphelion and perihelion of the Earth’s orbit (points P and A), where the satellite moves almost in the same direction as the Sun-target vector. On the contrary, it reaches a local maximum at the intersections of Earth’s orbit and Sun-target direction (points I 1 , I 2 ), where the satellite moves almost perpendicularly.

Figure 4.11 - ω 0 over one year

4.3.2 Satellite flying along another Earth orbit

This kind of mission is mainly concerned with telecommunication satellites, in particular for LEO constellations of satellites (e.g. Iridium, Globalstar) where mutual visibility is a necessary condition to establish intersatellite links by correctly pointing satellite’s antennas.

The target is defined by means of its orbit around the Earth, while the satellite is

supposed to follow the target even if it is shadowed by the Earth.

84 The following example considers two circular concentric orbits: the target (subscript t) is in the inner orbit, while the “follower” (subscript f) is in the outer one. Altitudes are appropriately sized such that P f = 2*P t , thus obtaining h f = 4500 km and h t ≈ 475 km.

Figure 4.12 - Orbits of target and follower

In this simple case, only yaw angle is always non-zero in time. It becomes again 180° at instants t 2 and t 3 , while ω = ω z *k becomes zero at times t 1 and t 4 when α reaches respectively maximum and minimum values.

Figure 4.13 - Yaw angle along one orbit

85 Figure 4.14 - ω z along one orbit

4.3.3 Fixed point on Earth surface

The observation of a fixed point of the Earth surface is of interest for both communication purposes (an antenna that points to a ground station) and surveillance purposes (a camera that makes phots over a certain region). Again, it is assumed that the satellite follows the target even if it is out of the satellite’s horizon.

The precision required for these missions makes unsatisfying the assumption of Earth as a sphere. Therefore, an ellipsoidal model of Earth surface is implemented.

It is obtained as a rotation of an ellipse of semi-axes a E and b E around the minor semi-axis b E . The following values have been adopted, [21]:

 a E = 6378.145 km is the Earth’s equatorial radius,

 b E = 6356.785 km is the Earth’s polar radius,

 e E = 0.08182 is the ellipse eccentricity.

The coordinates of the target are specified by two angles: the geodetic latitude φ in

the range [-90°, 90°] and the longitude λ in the range [0°, 360°). The geodetic altitude

is always 0. These coordinates can be converted in standard GRF Cartesian

coordinates by means of the following relations, [21]:

86

{

𝑋 = ( _√1−𝑒 ^{𝑎 𝐸}

𝐸 2 (sin 𝜑) ² ) cos 𝜑 cos 𝛼 _𝑟

𝑌 = ( ^{𝑎 𝐸}

√1−𝑒 _𝐸 ² (sin 𝜑) ² ) cos 𝜑 sin 𝛼 _𝑟 𝑍 = ( ^{𝑎 𝐸 (1−𝑒 𝐸 2 )}

√1−𝑒 _𝐸 ² (sin 𝜑) ² ) sin 𝜑

(4.18 - 4.20)

α r is the right ascension, the angle between GRF X-axis and the meridian that contains the target. It is defined in the range [0°, 360°) and is measured towards East along the equatorial plane. To exploit the previous set of equation, a relation between λ and α r must be established:

𝛼 _𝑟 = 𝜆 + 𝛼 _𝑟,𝐺 = {𝜆 + 𝛼 _𝑟,𝐺0 + 𝜔 _𝐸 (𝑡 − 𝑡 ₀ )}

𝑚𝑜𝑑 360° (4.21) where:

 α r,G is the right ascension of the Greenwich Meridian at generic time t,

 α r,G0 is the right ascension of the Greenwich Meridian at a fixed time t 0 , specified as parameter by jd,

 ω E ≈ 4.178*10 ^-3 deg/s is the mean value of Earth’s angular velocity about its polar axis; it defines the motion of the target.

The variation in time of α r,G0 is a consequence of the slow variation of the ecliptic

plane in time. This is because of perturbative effects on Earth’s orbit like the

precession of equinoxes and the influence of the Moon. Values of α r,G0 are tabulated

on astronomical almanacs for every day of the year but, again, there are semi-

analytic approaches based on polynomial approximation of complex astronomical

calculations. The one implemented here is taken and adapted from the U.S. Naval

Observatory website. At the basis of the algorithm is the evaluation of the

discrepancy between Greenwich Mean Sidereal Time (GMST) and Greenwich

Apparent Sidereal Time (GAST), based on the “equation of the equinoxes” and the

perturbation of the Moon. Both GMST and GAST give a measure of the hour angle

of the vernal equinox (i.e., the position of GRF X-axis), but the former measures it

from the “mean equinox”, the latter from the “true equinox”. Therefore, GAST is

87 required to evaluate α r,G0 at the beginning of the mission jd. A way to compute it is expressed in the following procedure:

1. Evaluate GMST, [hour], at initial time:

𝐺𝑀𝑆𝑇 = {18.697374558 + 24.06570982441908 ∙ 𝑗𝑑} 𝑚𝑜𝑑 24−ℎ𝑜𝑢𝑟 (4.22) 2. Evaluate the longitude of the ascending node of the Moon L Moon , [deg]:

𝐿 _{𝑀𝑜𝑜𝑛} = 125.04 − 0.052954 ∙ 𝑗𝑑 (4.23) 3. Evaluate L m,Sun and ϵ (compare with Eq. (4.12 – 4.16)):

{ 𝐿 _{𝑚,𝑆𝑢𝑛} = 280.47 + 0.98565 ∙ 𝑗𝑑

𝜖 = 23.4393 − 0.0000004 ∙ 𝑗𝑑 (4.24 – 4.25) 4. The “equation of the equinoxes” is indicated as eq_eq = ∆ψ*cos(ϵ) where ∆ψ,

[hour], is the nutation in longitude and is approximated by:

∆𝜓 = −0.000319 ∙ sin(𝐿 _{𝑀𝑜𝑜𝑛} ) − 0.000024 ∙ sin(2𝐿 _{𝑚,𝑆𝑢𝑛} ) (4.26) 5. GAST is evaluated applying the correction eq_eq to GMST:

𝐺𝐴𝑆𝑇 = 𝐺𝑀𝑆𝑇 + 𝑒𝑞_𝑒𝑞 (4.27) 6. Obtain α r,G0 as:

𝛼 _{𝑟,𝐺𝑂} = 𝐺𝐴𝑆𝑇 ∙ ^360°

24 (4.28) As an example, consider a GEO satellite that points to the equator on 21 ^st March 2015, 00:00.

Figure 4.15 - GEO orbit and target at time t 0

88 It is expected that n orients itself in the negative direction of GRF X-axis in order to engage the target. Then, since both the satellite and the target are coplanar, only α should change and cover one turn after one orbit. This is confirmed by the figure below: α varies linearly with time, since angular velocity is constant.

Figure 4.16 - Plot of yaw angle and ω z

4.3.4 Earth centre

This is the simplest attitude law examined and is typical of Earth-observation satellites, whose camera is constantly pointing to the Earth centre in order to cover strips of ground surface during the motion of the satellite. In this case, BF has two axes coincident with the so-called Orbital Frame (OF) as defined in [21], Ch. 12, Par.

12.9, while the positive direction of the third axis changes (see next figure). For

89 circular orbits of radius r, it has been verified that the value of ω 0 coincides with the projection of the angular velocity of OF upon SFF, whose analytic formula is:

𝜔 _𝑂𝐹 = √ ^{𝜇 𝐸𝑎𝑟𝑡ℎ} _{𝑟 3} (4.29) where μ Earth ≈ 3.986*10 ¹⁴ m ³ /s ² is the Earth’s gravitational parameter.

Figure 4.17 - Comparison between OF and BF

The algorithm for evaluating the position of the Sun according to jd is rather precise,

so it is adopted in the following, when boundary conditions are integrated in the

thermal code.