Greenwich meridian
Spacecraft orbit
Lo
Figure 5.19: Overview of vectors involved in tracking control
where αG0 is given in Equation 2.6, the transpose of the transformation matrix in Equation 2.17 can be used to compute the vector in FECI
¯
xIT AIEx¯ET (5.78)
The spacecraft position vector expressed in FECI is denoted by ¯r as shown in Figure 5.19. It follows that the vector pointing from the satellite to the target in FECI can be calculated as
¯
xIS{T ¯xIT ¯r (5.79)
The same vector can be transformed in the local orbital frame FO through the transpose of the matrix defined in Equation 2.18
¯
xOS{T AOIx¯IS{T (5.80)
The unit direction vector of ¯xOS{T in FOis given by
¯
uOS{T x¯OS{T
∥¯xOS{T∥ (5.81)
The spacecraft axis representing the observation payload is denoted by the constant unit vector
¯ uBp
0 0 1
. Therefore, the vector ¯uOS{T can be seen as the desired orientation for ¯uBp during tracking. The desired quaternion can be defined using the following unit vector
¯
ud u¯Bp ¯uOS{T
∥¯uBp ¯uOS{T∥ (5.82)
and the angle between the time-dependent vector ¯uOS{T and the constant vector ¯uBp given by δ cos1 u¯Bp ¯uOS{T
(5.83) With these two elements, the desired quaternion to be used in Equation 5.74 can be calculated as shown in Equation 2.13
¯ qdes
cos δ{2
¯
ud sin δ{2
(5.84) To investigate the tracking accuracy it is necessary to compute ¯uBS{T as
¯
uBS{T ABOu¯OS{T (5.85)
where ABO is the attitude matrix defined in 2.16. Then, the tracking error, i.e. the angle between the vector ¯uBS{T and ¯uBp, is
θerr cos1 u¯Bp ¯uBS{T
(5.86) The procedure for calculating the desired angular velocity is more elaborate and is not presented here. The final result is given below, but the procedure is discussed in detail in [21]
ωOBdes x¯OS{T 9¯xOS{T
∥¯xOS{T∥2 (5.87)
where
9¯xOS{T 9AOI x¯IT ¯r
AOI A9IEx¯ET ¯v A9IE ωC
sinpαGq cospαGq 0 cospαGq sinpαGq 0
0 0 0
A9OI 9¯o1I 9¯o2I 9¯o3IT
9¯o1I 9¯o2I ¯o3I o¯2I 9¯o3I
9¯o2I I ¯o2Io¯T2I p¯r 9¯vq
∥¯r ¯v∥
9¯o3I I ¯o3Io¯T3I
¯ v
∥¯r∥
(5.88)
The desired time-varying quaternion and angular velocity can be used in control law 5.72 to per-form target tracking control. The tracking mode is only activated when the target is visible to the spacecraft, the boundary situation is the one discussed in Section 3.2.5 and shown in Figure 3.16.
The problem of large angle slew manoeuvering, which may be encountered if an instantaneous manoeuvre is performed from an attitude very different from that required for target tracking, was previously mentioned. It was anticipated that to account for the limited actuation capacity of reac-tion wheels it is necessary to perform a pre-manoeuvre, extended in time, before target tracking.
This is also discussed in [21] where it is suggested to define the following desired pre-manoeuvre quaternion as
¯
qdespt0q
cos δpt0q{2
¯
udpt0q sin δpt0q{2
(5.89) where t0 is the time when starting tracking control, and ¯udpt0q and δpt0q can be calculated from Equations 5.82 and 5.83 using the unit vector ¯uOS{Tpt0q calculated at t0.
As shown in Figure 3.16, it has been chosen to begin the pre-manoeuvre ahead of the starting tracking point by an angle ϕ. This makes it possible to begin the slew manoeuvre before arriving at the tracking start point and thus avoid large, sudden angular excursions.
Target Tracking Control Simulation
A simulation was carried out to validate the correct implementation of the algorithm, in which the target to be tracked lies in correspondence with the ground track of the satellite’s orbit as shown in Figure 5.20. In this way, it can be verified whether when the target is precisely below the spacecraft during tracking, the guidance provides the unit quaternion as desired. An ideal simulation is considered in which the sensors are noise-free. The spacecraft starts from an initial condition of nadir pointing.
Target Equator
Spacecraft orbit
Figure 5.20: Spacecraft orbit and target position in the target tracking control simu-lation. Target lies on the spacecraft ground track
Once it reaches the pre-manoeuvre start position in orbit, it begins to change its attitude to achieve the required attitude for tracking control. The simulation results are shown in Figures 5.21, 5.22, and 5.23. The diagrams show the guidance, hence the desired quaternion or angular velocity, and the actual output of the spacecraft, and also highlight the pre-manoeuvre and tracking manoeuvre regions. The results show that the spacecraft correctly tracks the desired output provided by the guidance function. In Figure 5.22, it can be seen that when the spacecraft is in correspondence with the target, the desired quaternion coincides with the unit quaternion, thus indicating the correct implementation of the algorithm. Figure 5.21 shows that the spacecraft body angular rate does not exactly track the desired angular velocity.
Figure 5.21: Angular velocities and torques output in the target tracking control sim-ulation. The legend is the same as in Figure 5.22
However, from Figure 5.23 it can be seen that the tracking error is very small. It reaches a maximum where the spacecraft is manoeuvring closest to the target. Due to the pre-and post-manoeuvre, the torques delivered during the simulation are very small, as represented in Figure 5.21. In this way, the technological actuation limit of the reaction wheels is not exceeded, and the risk of wheel saturation is minimised.
Figure 5.22: Quaternions output in the target tracking control simulation. Light gray indicates the pre-manoeuvre, while dark gray the tracking control
Figure 5.23: Tracking error output in the target tracking control simulation. The error angle is defined in Equation 5.86. Left diagram has the same legend as in Figure 5.22. Right diagram is a focus of the tracking control region
Chapter 6
Simulations and Results
This chapter presents the results from complete non-ideal simulations. In this regard, the sim-ulation setup is defined by introducing the models of the actuators and sensors and the possible combinations of the determination and control algorithms whose performance is studied.