Detumbling Simulation
Figures 5.2, 5.3 and 5.4 show the results of a detumbling simulation. The simulation has been performed to verify the working of the controller. For this reason, neither measurement noise nor estimators were simulated. The initial conditions and orbit characteristics in the simulation are shown in Table 5.2. Figure 5.1 shows the orbit used in the simulation.
Figure 5.1: Spacecraft orbit used in the detumbling simulation
Orbit characteristics Circular orbit, Altitude = 470 km, Inclination = 51.6 Initial attitude rΦ, Θ, Ψs r5, 5, 5s rdegs
Initial angular rates rωx, ωy, ωzs r10, 10, 10s rdeg{ss Rates Control rate = 5 s, Sampling rate = 0.5 s Constraints Maximum dipole moment = 0.5 Am2
Table 5.2: Detumbling simulation setup
Figure 5.2: Orbit referenced angular velocities of the spacecraft during the detum-bling simulation. A focus of the third orbit is shown in the figure to show the detail of the angular velocity value
Figure 5.3: Norm of angular velocities
0.5 0.52 0.54
-0.2 -0.1 0 0.1 0.2
Figure 5.4: Dipole moment
From Figure 5.2, it can be seen that in the ideal case the spacecraft is detumbled in 2 orbits since the angular velocities are already below the imposed requirement given in Table 3.8. However, this is the ideal case, in the non-ideal case with disturbances three orbits will be needed to fulfil the requirement. A focus of the control signal was included in Figure 5.4 to highlight the stepwise nature due to ZOH.
5.1.2 Momentum Dumping
As mentioned earlier, another important use of magnetic torquers for low-Earth orbiting space-craft is momentum dumping. In Section 3.2.5, the difference between cyclic and secular torques was discussed. While cyclic torques result in a cyclic variation of reaction wheels speed, secular torques result in a linear increase because the wheel constantly absorbs angular momentum. Even-tually saturation of the wheels will occur due to the excess momentum, which can only be dumped through external torques.
A common approach to design a magnetic torquer control law for momentum dumping is to com-mand a magnetic dipole moment [18, 31]
¯
m k
∥ ¯B∥
¯h ˆb (5.6)
which is similar to Equation 5.2 but with the angular velocity vector replaced by the wheel angular momentum.
Momentum Dumping Simulation
Here again, an ideal simulation was carried out to verify the correct desaturation of the reaction wheels. The simulation starts from a nadir pointing condition with unit quaternion and zeros an-gular velocities, with a certain anan-gular momentum stored in the reaction wheels defined by the random choice of the initial reaction wheel velocities. The momentum dumping magnetic con-troller is implemented simultaneously with the nadir pointing concon-troller to desaturate the wheels almost entirely in one orbit. The orbit characteristics are the same as those shown in Table 5.2.
The simulation setup is summarised in Table 5.3.
The controller gain indicated by k in Equation 5.6 determines how fast the wheels will be desatu-rated. However, as discussed in Section 3.2.5, this desaturation speed is limited by the maximum dipole moment deliverable by the magnetic torquers. The gain chosen is such that the momentum dumping is spread within an orbit.
Initial attitude rΦ, Θ, Ψs r0, 0, 0s rdegs Initial angular rates rωx, ωy, ωzs r0, 0, 0s rdeg{ss
Initial RWsangular rates rω1, ω2, ω3, ω4s r1000, 1500, 500 500s rrpms Rates Control rate = 1 s, Sampling rate = 1 s
Constraints Maximum dipole moment = 0.5 Am2 Controller gain k diagr50, 50, 50s
Reaction wheels
Table 5.3: Momentum dumping simulation setup
Results are shown in diagrams 5.5, 5.6, 5.7, 5.8, 5.9 and 5.10. As represented in Figures 5.5 and 5.6, during momentum dumping the spacecraft maintains nadir pointing, which means that the torques applied by the magnetic torquers are opposed by the reaction wheels and are such that the angular momentum of the wheels decreases over time. This can be seen in Figure 5.9 which shows how the wheel momenta approach zero thanks to the execution of the momentum dumping control law.
Figure 5.5: Quaternions resulting from the reaction wheels momentum dumping sim-ulation
Figure 5.6: Spacecraft angular velocities resulting from the reaction wheels momen-tum dumping simulation
From Figure 5.7, it can be seen that the angular momentum is almost totally damped in one orbit as intended. The chosen gain results in no costly desaturation manoeuvres in terms of applied control torques. Indeed, Figure 5.8 shows that the dipole moment assumes small values during the simulation without ever reaching the maximum value indicated in Table 5.3.
Figure 5.7: Angular momentum magnitude Figure 5.8: Dipole moment
Although the reaction wheels model is discussed in Section 6.1.2, diagrams showing their angular velocities and accelerations are also shown for completeness. There are four reaction wheels and they are arranged in the pyramidal configuration previously discussed. The angular velocity and inertia of the wheels determine the angular momentum they store. The technological limit of maximum permissible wheel speed defines a maximal storable angular momentum. Once the maximum speed limit is reached, desaturation is necessary to continue operating the wheel. In practice, efforts are made to avoid approaching this limit and not to reduce the angular momentum to zero to prevent problems with the motors that drive the wheels.
Figure 5.9: Angular momentum and magnetic torques resulting from the reaction wheels momentum dumping simulation. On the left angular momentum, on the right magnetic torques
From Figure 5.10, it can be seen that during the simulation, desaturation occurred correctly as the angular velocity of the wheels, and therefore the angular momentum decreases over time.
The wheels acceleration shown in Figure 5.10 is closely related to the control torques applied by the nadir pointing controller. This torque is not shown here, but it is equal to and opposite to that shown in Figure 5.9.
Figure 5.10: Reaction wheels angular rates and acceleration resulting from the reac-tion wheels momentum dumping simulareac-tion