5.2.2 PD Step Responses

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.