To help characterize the performance of the quaternion based PD
controller, responses were recorded for steps about the roll, pitch,
and yaw axes for both low, and high proportional gain settings. For
large steps, the error dynamics are nonlinear, so these step
responses may not be analyzed using the same methods as used in
linear control theory. However it is informative to apply several of
the standard step response metrics to these signals to help
characterize their performance.
Figure 5-7 45° roll step response (high gain: =68, =1)
Figure 5-8 45° roll step response (low gain: =18, =1)
Figure 5-9 45° pitch step response (high gain: =68, =1)
Figure 5-10 45° pitch step response (low gain: =18, =1)
Figure 5-11 45° yaw step response (high gain: =68, =1)
Figure 5-12 45° yaw step response (low gain: =18, =1)
The major characteristics of these responses are compared in
Figure 5-13 through Figure 5-15.
Figure 5-13 High and low gain roll step characteristics
Figure 5-14 High and low gain pitch step characteristics
Figure 5-15 High and low gain yaw step characteristics
All of the step responses, both high and low gain, immediately
request full thruster output. This causes the very beginning of both
the high and low gain responses to look very similar. Saturation does
not persist as long in the low gain responses, and the lower thrust
values reduce overshoot. Response about the roll axis is faster,
because inertia and drag about that axis is lower, while thrust
capability is approximately the same.
The final steady state offset is much greater for the low gain controller. This offset is produced by buoyancy moment, and the error indicates the level of angular at which the commanded torque produced by the proportional error matches the buoyancy moment. The balance in this test was not as accurate as in the tests described in the previous section. Because of the buoyancy offset, the high gain average steady state error is about 7° which is higher than 0.3° shown in Figure 2-2. Another effect of this offset can be seen in the low gain pitch response (Figure 5-10). The buoyancy offset was causing the vehicle nose to hang down about 9°. The commanded step in desired attitude was 45° pitch down, however the steady state error caused the difference between desired and actual attitudes to be only 36° immediately after the 45° was commanded. The vehicle was manually balanced for this test, and the inaccuracies are evident in the results. The automatic balance algorithm which was implemented after these tests were conducted significantly increased the repeatability and accuracy of rotational buoyancy compensation. The automatic balance results are described in Section 0.