The most important goal of an EO satellite’s ADCS is to provide a stable and accurate platform for imaging manoeuvres. The selection of ADCS hardware and software to be used during imaging greatly determines the image quality. The actuators, controllers, sensors, and estimators that will be used during an imaging manoeuvre must thus be chosen carefully.
An FMC imaging manoeuvre (as depicted in Section 1.3) consists of several stages. The manoeuvre is initialised by pointing in the nadir direction. The satellite will then slew to the roll offset angle and the initial pitch angle, after which the rate about YO (ωY) will be ramped down to the desired FMC
angular rate (ωf mc). Once the angular rate is constant, the imaging of the target area can start. As soon as the imaging has finished, the angular rate will be ramped back up, after which the satellite
will stabilise itself and return to a nadir-pointing attitude. A basic flow diagram of the FMC Imaging Mode can be seen in Figure 7.11.
Figure 7.11 – A flow diagram of the ADCS FMC Imaging Mode
7.6.1 FMC Schedule
The scheduling of an imaging manoeuvre is a critical part of any EO satellite mission. The high velocity at which a LEO satellite travels brings forth a tight timing requirement when imaging a target area. An FMC manoeuvre contains added complexity as a result of the rotation compensation, which means that the schedule of this manoeuvre must be planned with precision.
The FMC Imaging Mode will most likely be activated as a result of a telecommand. The telecommand will also contain a specific Unix time t0 at which the satellite must be at the centre of the imaging manoeuvre (see Section 1.3). The first phase of the FMC Imaging Mode is to slew to nadir-pointing
Tnadir before t0 (both in seconds), thus at t = t0− Tnadir.
The next FMC phase, during which the satellite will be slewed to RPY = hφf mc θinitial 0 iT
, will be executed at t = t0− Tslew. The roll offset angle φf mcwill depend on the location of the image target
and the initial pitch angle θinitial will be determined by the duration of the pitch rate ramp Tramp and
the pitch angle before imaging θf mc (see Equation 1.3.32). Tf mc, the duration for which the on-board image sensor will be collecting data, can be calculated from Limage and nf mc using Equation 1.3.31.
CHAPTER 7. AN APPLICATION SPECIFIC ADCS 96
At t = t0−1
2Tf mc+ Tbuf + Tramp
the angular rate about the orbit Y-axis ωY should start being ramped down to a constant rate of −ωf mc. The satellite will be rotating at a rate of −ωf mc for a
period of Tf mc+ Tbuf, where Tbuf is an added initial buffer period where the inner reaction wheel speed control dynamics will be allowed to stabilise. A special open loop FMC wheel controller will be used to perform the rate ramp. Once −ωf mc has been reached at t = t0 −
1
2Tf mc+ Tbuf
, the buffer period will start. The satellite will be ready to start imaging once the buffer period has passed at t = t0− 12Tf mc.
The imaging payload will stop collecting data at t = t0+12Tf mc, after which the angular rate ωY will
be ramped back up to approximately zero over a period of Tramp. The satellite dynamics will then be allowed to stabilise for a period of Tstable, during which the Quaternion Feedback wheel controller will
once again be activated.
The FMC imaging manoeuvre will end with a slew to a nadir-pointing attitude at t = t0+12Tf mc+ Tramp+ Tstable, after which the FMC Imaging Mode will conclude at t = t0+ Tend. The timeline of an
FMC imaging manoeuvre (as explained in this section) can be seen in Figure 7.12.
Figure 7.12 – The schedule of an FMC imaging manoeuvre
7.6.2 FMC Wheel Controller
A special wheel controller is required for the FMC imaging manoeuvre. Although the Quaternion Feedback controller offers accurate pointing, its closed loop nature will cause unwanted jitter during imaging. The FMC wheel controller will therefore be an open loop controller.
As a result of the roll offset angle, both the Y-wheel and the Z-wheel will need to be employed to reach the desired Y-axis orbit rate of ωY = ωf mc. The X-wheel will however still be controlled by
the Quaternion Feedback controller. The Y-axis and Z-axis open loop control law for the FMC wheel controller is given by
Nwy,req = IyyNf mccos (φf mc) − Ngyro,y (7.6.1)
and Nwz,req = −IzzNf mcsin (φf mc) − Ngyro,z , (7.6.2)
where Nf mc is the torque required to ramp the satellite’s rate about YO to and from −ωf mc. Nf mc is furthermore dependent on Tramp and ωf mc through
Nf mc = ωf mc
Tramp . (7.6.3)
Equation 7.6.3 thus implies that a wheel torque of Nf mc is required to ensure a change in angular rate of ωf mc over a period of Tramp. Once the rate ramping process is done, Nf mc should be set to zero to
ensure a constant FMC rate.
Figure 7.13 illustrates the satellite’s orientation during an FMC manoeuvre. It is clear from Figure 7.13 that ωY will have components about YB and ZB that will be scaled by cos(φf mc) and sin(φf mc) respectively, which explains the use of these scalars in Equations 7.6.1 and 7.6.2.
Figure 7.13 – The satellite orientation during an FMC imaging manoeuvre
7.6.3 FMC Implementation
Although the explanation in Chapter 1 builds a basic understanding of an FMC imaging manoeuvre, several factors need to be considered when attempting practical implementation. The rate ramp duration Tramp should firstly be such that the wheel speed commands are not too large, as this will cause large torques and unwanted dynamics. As a rule of thumb for this project, Tramp should be
chosen in such a way that Nf mc ≈ 0.1Nw,max, where Nw,max is the maximum wheel torque. The
duration of Tramp and Tbuf also influences the required initial pitch angle θinitial, as illustrated by Figure 7.14.
θinitial can thus be calculated by integrating the area under the curve shown in Figure 7.14, thus
θinitial = 1 2ωf mcTramp+ ωf mcTbuf + 1 2ωf mcTf mc | {z } θf mc . (7.6.4)
The buffer period Tbuf should furthermore be greater or equal to the wheel speed control loop settling time.
CHAPTER 7. AN APPLICATION SPECIFIC ADCS 98
Figure 7.14 – The initial pitch angle for an FMC imaging manoeuvre
7.6.4 FMC Imaging Mode Simulation Results
The correct functioning of the FMC Imaging Mode was verified by simulating the 20 kg EO satellite during an FMC manoeuvre with the following specifications:
• FMC factor (nf mc): 4
• Image length (Limage): 40 km
• Altitude (h): ±500 km • Roll offset angle (φf mc): 30◦
• Buffer period (Tbuf): 5 s • Ramp duration (Tramp): 20 s • Start offset time (Tnadir): 300 s
• Initial attitude offset time (Tslew): 150 s • Stabilisation period (Tstable): 30 s
• Manoeuvre end offset time (Tend): 150 s
The FMC imaging duration Tf mcand desired angular rate ωf mcwere then calculated as approximately
22.6 s and 0.6095◦/s (using Equations 1.3.31 and 1.3.30) respectively. Using Equation 7.6.4, the initial
pitch angle θinitial was determined to be 14.53◦. The pitch angle at the start of the imaging process was calculated as θf mc= 12ωf mcTf mc= 6.90◦.
A star tracker was used to provide measurements to the EKF during the FMC simulation (except during the initial and final slew manoeuvres, where is angular rate will be too high for the star tracker
to produce solutions, and the FSS must be employed). Figure 7.15 shows the Euler angles as produced by the simulation and the resulting orbit-referenced angular rates can be seen in Figure 7.16.
−300 −150 −36 −11.3 0 11.3 32 62 150 −20 −10 0 10 20 30 40 Time (s) Angle (deg) Roll (φ) Pitch (θ) Yaw (ψ)
Figure 7.15 – The Euler angles during an FMC imaging manoeuvre
−36 −16 −11.3 0 11.3 32 −0.4 −0.2 0 0.2 0.4 0.6 Time (s)
Angular Rate (deg/s)
ωxo ωyo ωzo
Figure 7.16 – The angular rates during an FMC imaging manoeuvre
The 1-σ roll angle error in Figure 7.15 was calculated as 28.4 arcsec during imaging and the angular rates in Figure 7.16 followed the expected behaviour. The angular rate about YO during imaging (which was calculated as ωY = −qωyo2+ ωzo2) for the same simulation can be seen Figure 7.17. The
near-constant offset of roughly 0.007◦/s (or 25 arcsec/s) from the desired rate of ωf mc = −0.6097◦/s
can be attributed to an initial offset in angular rate (i.e. before the rate was ramped down). The stability of ωY, which is defined as the maximum deviation from the mean angular rate of −0.6031◦/s,
was calculated as 6.6 arcsec/s.