COLORES DE SEGURIDAD

In general, in this research a formation of satellites with a single leader and n followers is considered. Three coordinate frames are used, namely: (1) a reference frame F0 is used as an inertial frame whose origin is at the center of the Earth,

(2) a reference frame FL embedded at the center of mass of the leader as a body

frame which rotates with the spacecraft and represents its orientation, and (3) a local vertical local horizontal (LVLH) reference frame Fi attached to the orbit of

each satellite. The RAC-reference frame is used as a spacecraft-centered frame (as opposed to the roll, pitch, yaw (RPY) frame) in which x, y, z coordinates correspond to radial, along-track (direction of motion), and cross-track (perpendicular to orbit plane) directions, respectively.

Attitude of a satellite can be represented in different ways with sets of variables such as the Euler angles, direction cosine matrix (DCM), Euler parameters (some- times called quaternion), etc. Euler angles consist of three successive rotation angles (φ, θ, ψ) around the x, y, z axes of satellite’s body frame. The main advantage of

the Euler angle representation is that it provides easy visualization of the rotations and orientation of the satellite. Despite its advantages, the presence of singularities at specific angles limits applicability of the Euler angle representation to small rota- tions. For this reason, in this thesis, quaternion representation [133, 134] is utilized for spacecraft attitude. An angular displacement can be specified by the Euler pa- rameters or unit quaternion ¯q = (~q, q0) which is defined as: ~q = [q1, q2, q3]T , ~a. sinφ₂

and the auxiliary parameter q0 , cosφ₂. In this notation, ~a is a unit vector in the

direction of rotation with coordinate representation [a1, a2, a3] T

, called eigenaxis, and φ is the rotation angle about ~a. By definition, unit quaternion is subject to the constraint that ~qT_{~q + q}2

0 = 1. Note that a unit quaternion is not unique since ¯q and

−¯q represent the same attitude. However, uniqueness can be achieved by restricting φ to 0 ≤ φ ≤ π so that q0 ≥ 0.

Furthermore, given a vector ~v = [v1, v2, v3]T, the cross product operator is

denoted by: ~v×=       0 −v3 v2 v3 0 −v1 −v2 v1 0       (3.1)

which represents the fact that ~v × ~ω = ~v×~ω. The product of two unit quaternion ¯p and ¯q is defined as:

¯ p¯q =   q0~p + p0~q + ~q × ~p q0p0− ~qT~p   (3.2)

which is a unit quaternion as well. The conjugate of ¯q is defined as q? =−~qT, q0

T . The multiplicative identity quaternion is denoted by 1 = [0, 0, 0, 1]T, where, ¯q?_{q =¯}

¯

q ¯q? _{= 1 and ¯q1 = 1¯q = ¯q. If ¯q}d _{and ¯q represent desired and actual attitude}

respectively, then the attitude error is given by ¯qe = ¯qd?q, which represents the¯

rotational dynamics of a spacecraft relative to the inertial frame F0 are as follows: ˙ ~ q = −1 2~ω × ~q + 1 2q0~ω ˙ q0 = − 1 2~ω.~q (3.3) J ˙~ω = −~ω × (J ~ω) + ~τ

where, ~ω = [ωx, ωy, ωz]T represents the angular velocity of the satellite, ~q is the vector

and q0is the scalar part of the quaternion, J is the inertia matrix and τ = [τx, τy, τz]T

are the torques associated with the spacecraft.

3.2.1

Leader-Follower Formation Attitude Control

It is assumed in this thesis that the 5 (five) satellites in the formation under the assumed mission scenario (as discussed in Section 3.1) would perform syn- chronous/collaborative attitude maneuvers. Since the follower satellites circle around the Leader at the orbital rate, in order to ensure contact or line-of-sight with the followers, the leader rotates around the X axis at the same rate. In addition, it also rotates around the Z axis for the Earth pointing. Now, according to the leader- follower formation control architecture, the followers are required to maintain their position as well as the attitude with respect to the leader. As mentioned in Section 3.1, a fixed formation configuration is assumed. Therefore, in this case the follow- ers are required to maintain the same attitude as that of the leader relative to the inertial frame F0.

For control and especially for diagnosis purpose, it is reasonable to utilize Eu- ler angles. The reason for utilizing Euler angles is that attitude control performance specifications are usually available in terms of error in the Euler angles and the ground support personnel responsible for fault diagnosis are more comfortable deal- ing with Euler angles. Since the spacecraft attitude dynamics is specified in inertial

frame, PID controllers [135, 136] (with ±5V limit or saturation for the control sig- nal) are implemented for each axis of the leader based on “errors in terms of Euler angles” w.r.t. inertial frame obtained by performing the quaternion to Euler angle transformation (as discussed in Section 3.2). Followers’ PID controllers (with ±5V limit or saturation for the control signal) have been designed based on “errors in term of the Euler angles” w.r.t. Leaders body-fixed frame obtained by performing the quaternion to Euler angle transformations.

3.2.2

Subsystem Component Model

In the model for synthetic data generation, the high fidelity mathematical model of the Ithaco Type-A reaction wheel that is available in [123] is utilized. As shown with a simplified schematic diagram in Figure 3.2, the model consists of detailed mathematical representation of a reaction wheel containing relationships to repre- sent motor driver/torque control mechanism, motor disturbances such as cogging and ripple torques, Coulomb and viscous friction, torque noise that results due to lubricant dynamics, the “EMF torque limiting” phenomenon at high speed, safety mechanism for limiting speed, and torque bias discontinuity. Detailed discussion on the model and the reaction wheel parameter values are available in [123] and is not presented here. Under fault-free conditions, the reaction wheel is capable of deliver- ing a maximum of ±40mN-m torque in response to a ±5V torque command input voltage and operates within the speed range of ±5100rpm. For this research, the reaction wheel model has been modified to accommodate and support fault injection capabilities.

Ideal dynamics for attitude sensors are assumed; i.e., signals from sun sensors, horizon scanners, magnetometers, etc. have been assumed to be fed back to the attitude controller without any error or time delay.

Figure 3.2: A simplified schematic diagram of a reaction wheel (adopted from [123]).

3.2.3

Environmental Disturbance Models

Satellites in a LEO orbit are subjected to different types of environmental distur- bance torques such as magnetic, gravity gradient, aerodynamic, and solar radiation. The following disturbance models and parameters have been selected based on the information that is available in [137] and the mission specifications presented in Section 3.1.

Magnetic: The maximum magnetic torque that occurs in a polar orbit is given by: Tmag = DB; where D is the residual dipole of the spacecraft and B is the Earth

magnetic field. B can be approximated as B = 2M_R3; where, M is the magnetic

moment of the Earth (M = 7.96 × 1015 tesla-m2) and R is the radius of the orbit (in our case, R = 6928 × 103 m). Note that the magnetic torque is cyclic over the spacecraft orbit (as selected in Section 3.1) with peak value at the poles and half of the peak value at the equator. Consequently, the frequency of this cycle is twice that of the orbit. In this thesis the following is assumed, i.e.: D = 0.8 A-m2

Gravity Gradient: The maximum gravity torque is given by: Tgrg = _2R3µ3|Ixx −

Izz| sin(2θ); where µ is the Earth’s gravity constant (3.986 × 1014 m3/sec2), θ is

the maximum deviation of x-axis (radial direction) from local vertical in radians, and Ixx and Izz are the spacecraft inertia as provided in Section 3.1. In this thesis

θ = 0.1 degree is assumed. Therefore, in this case, Tgrg = 6.28 × 10−9 N -m.

Aerodynamic: The maximum aerodynamic torque is given by: Taero = F (cpa−cg);

where F is the force. F is approximated by F = 0.5ρCdAV2; where cpa is the center

of aerodynamic pressure, cg is the center of gravity, ρ is the atmospheric density,

Cd is the drag coefficient (2 − 2.5), A is the surface area, and V is the spacecraft

velocity. In the selected 550 km orbit, V = 7585.2 m/s. In this thesis, the following values are assumed: (cpa− cg) = 0.3 m, ρ = 1.04 × 10−13 kg/m3, Cd = 2.3, A = 1

m2_{. Therefore, in this case, T}

aero = 2.06 × 10−6 N -m.