Results and Discussion
6.1 General discussion
6.4.4 Discussion for the path-following using an extra quater- quater-nion in the cost functionquater-nion in the cost function
The initial guess for the L0-controller were the output from the L1-controller for a single axis maneuver from (0◦,0◦,0◦) to (90◦,0◦,0◦). This initial guess yielded better results than using an initial guess from the L1-controller for a path-following maneuver from (0◦,0◦,0◦) through (30◦,45◦,15◦) to (0◦,0◦,0◦), similarly to the observations made for the path-following maneuver in Sec-tion 6.4.1. As menSec-tioned previously, a possible explanaSec-tion to why the initial guess from the path-following maneuver did not yield satisfactory results may be that the optimization solver could have become stuck in a local minimum, and is therefore not able to find the optimal solution.
For the moving L0-controller, the vector hN was chosen such that it would cost less for the control input to occur between t = 10 s and t = 20 s, and t = 49s andt= 59s. Fig. 6.22 and Fig. 6.23 show that no control inputs occur within these time intervals. A reason for this could be that even though it is cheaper to provide control inputs within these two intervals, doing so may not minimize the cost function in (5.9b) while satisfying the constraints. In other words, to minimize the cost function, it is more important to minimize the terms in the cost function which relate to attitude and angular velocity, than it is to move the control inputs to inside the predefined intervals. The control inputs occur outside t = 10 s and t = 20 s, andt = 49 s andt = 59s, which is acceptable behaviour, since there are no terms in the movingL0-optimal control problem that prevents the control torques from occurring outside the cheapest time intervals. Thus, this result does not indicate that the movingL0-controller does not work, but it illustrates that the control inputs are only moved to the predefined interval if it contributes to minimizing the objective function while satisfying the constraints.
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Figure 6.20:Euler angles for path-following using an extra quaternion in the cost func-tion, from(0◦,0◦,0◦)through(30◦,45◦,15◦)to(0◦,0◦,0◦).
The saturation limits for the reaction wheel torque areτlimit =±3·10−3N·m, given in Table 5.2. Fig. 6.23 shows that, for all three controllers, the reaction wheel related to τw,2 saturates at the time instant where the second control torque occur, at aboutt = 34s. The first control input, which occurs aroundt = 2s, pushes the spacecraft towards the desired intermediate attitude. The second control input, which occurs aroundt= 34 s, pushes the spacecraft towards the final attitude, but in doing so it also has to oppose the motion of the spacecraft resulting from the first control input. At the end of the optimization, the last control input stops the rotation of the spacecraft. Therefore, it makes sense that the second control input, which occurs aroundt = 34 s, is larger than the other two, and could explain why it saturates.
The path-following maneuver which uses an extra quaternion yielded a
con-104 Chapter 6. Results and Discussion
Figure 6.21:Angular velocity,ωbob, for path-following using an extra quaternion in the cost function, from(0◦,0◦,0◦)through(30◦,45◦,15◦)to(0◦,0◦,0◦).
trol signal which was more sparse than the control signal resulting from the path-following maneuver which uses multiple optimizations. The difference in sparsity can be seen by comparing Table 6.8 to Table 6.6. These results make sense because in the case for the path-following maneuver which uses an ex-tra quaternion, the optimization of the maneuver is performed over the whole interval. For this maneuver, one control torque is required to push the space-craft towards the intermediate orientation, one torque is required to turn, and one torque is required to stop the spacecraft at the end of the optimization in-terval. In the case for path-following which uses multiple optimizations, two separate optimizations are performed for two separate maneuvers which are put together to form the total path. For each of these separate maneuvers, the angular velocity ωbob goes to zero at the end of the optimization, due to the
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Figure 6.22:Control input body frame for path-following using an extra quaternion in the cost function, from(0◦,0◦,0◦)through(30◦,45◦,15◦)to(0◦,0◦,0◦).
formulation of the cost functions in (5.3a), (5.5a), and (5.7a). Therefore, one torque is applied at the beginning of each maneuver to push the spacecraft to-wards the desired attitude, and a second torque is applied at the end of the optimization to stop the spacecraft, which yields a sparsity of 4 for theL1- and L0-control signals. A control signal having a sparsity of 4 is the sparsest possible control signal for a path-following maneuver split into two optimizations, due to the start and stop torques which has to be applied for each optimization.
Similarly, a sparsity of 3 provides the sparsest possible control signal for the path-following which uses an intermediate quaternion, when the cost functions are given as in (5.9a) to (5.9c) and hN is chosen such that it would cost less for the control input to occur betweent = 10 s and t = 20 s, and t = 49 s and t= 59s.
106 Chapter 6. Results and Discussion
Figure 6.23:Control input wheel frame for path-following using an extra quaternion in the cost function, from(0◦,0◦,0◦)through(30◦,45◦,15◦)to(0◦,0◦,0◦).
The response in the spacecraft’s states and the control signals are identi-cal for all three controllers, as shown in Fig. 6.20, Fig. 6.21, Fig. 6.22, and Fig. 6.23. As discussed before, a control signal having the sparsity of 3 is the sparsest possible control signal for a path-following maneuver which uses an intermediate quaternion in the cost function. These findings suggest that all controllers have been able to find optimal control signals for the path-following maneuver which uses an intermediate quaternion in the cost function from (0◦,0◦,0◦)through(30◦,45◦,15◦)to(0◦,0◦,0◦), since all the control signals have a sparsity of 3 and the spacecraft reaches its desired states. A possible expla-nation to why the three optimal solutions are identical could be that the initial guesses for each of the optimization procedures provide a starting point close to an identical local minimum.
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