Numerical Simulation Problems Part 2

Continuing from the previous post...

The next plots show the roll and yaw angle history for the constrained model only. For the unconstrained model, these angles remained absolutely zero, but a numerical error appeared in the values for the constrained model:

Figure 13: Roll angle (left) and yaw angle (right) at 15 degree ref pitch for constrained model

Figure 14: Roll angle (left) and yaw angle (right) at 30 degree ref pitch for constrained model

Figure 15: Roll angle (left) and yaw angle (right) at 45 degree ref pitch for constrained model

Figure 16: Roll angle (left) and yaw angle (right) at 60 degree ref pitch for constrained model

Figure 17: Roll angle (left) and yaw angle (right) at 75 degree ref pitch for constrained model

Figure 18: Roll angle (left) and yaw angle (right) at 90 degree ref pitch for constrained model

As can be seen from the above figures, a small numerical error is present for the roll and yaw angles in the cases of 15, 30, 45, and 60 degree reference pitch angles. However, for the 75 degree and 90 degree reference pitch angles, the roll and yaw angles go completely unstable and generate a massive error.

To investigate what is actually happening when the system goes unstable, a simulation for the 75 degree reference pitch angle case was run for 6.2 seconds, which is the point immediately before the constrained system goes unstable. Figure 19 below shows the roll and yaw angles for this case:

Figure 19: Roll angle (left) and yaw angle (right) at 75 degree ref pitch for constrained model

The behavior of the plots in figure 19 is perplexing. The error seems to spike at around 5.5 seconds. I am not sure what causes this, and need to investigate further. I hypothesize that there may be a numerical instability issue with the integration technique, but I don't understand why the controller would not stabilize the system when the roll and pitch angles deviate from the zero degree reference.

One thing to note: When reducing the time step from 1 millisecond to 0.5 milliseconds, the constrained model is stable for the reference pitch angle of 75 degrees, but still not for the 90 degree reference. It is not feasible for me to reduce the  time step much beyond this point.

Another thing to note: the behavior of the error in figures 13 through 16 may seem random and one might think they would differ from run to run, but they don't. I am getting repeatable graphs.