#### 4.2. Tasha III—Launch 1

Tasha III from

Figure 8b was launched from Kaitorete Spit on 22 July 2015. The aim of this launch was primarily to test the new avionics stack and fibreglass rear fins. In the previous flight of Smokey [

12], the fins were 3D printed. The reason for the fibreglass was to provide further strengthening suitable for supersonic flights in future research. In addition, to gain some useful data from the launch, a PD controller was implemented during flight in both the thrust and coast periods. Since the vehicle was finished only days before the launch, there was not sufficient time to do a full wind tunnel test, analysis of data and rigorous testing of gains. The only wind tunnel test performed was the stability test of

Section 3.2.

For the flight, the gains chosen were

${k}_{p}=1,\text{\hspace{0.17em}}{k}_{d}=0.1$, which were known to give a reasonable response from previous wind tunnel testing of Smokey with the gimbal frame [

12]. The reference was chosen to be a series of responses starting at 0° for a pre-determined amount after clearing the launch guide, followed by an alternation between −15° and 15° every second. As a precaution, since this was the first flight with the avionics, a maximum limit of 6° was enforced in each canard. Unfortunately, there was a significant fin offset in the airframe which was greater than the canards could compensate for with this maximum limit, so only about 1 s of oscillatory data was obtained. However, this data set was sufficient to identify parameters and thus analyse the rocket roll response.

The apogee for this flight was 522 m, the time to apogee was 10.2 s and the maximum velocity was 97 m·s

^{−1}. The wind speed was very low on the day, varying between 1 and 2 m·s

^{−1} on the ground. The flight was successful and the rocket safely recovered apart from a couple of broken canards that were easily replaced.

Figure 17 shows video stills from the ignition, take-off, onboard footage and parachute recovery.

Since there was only one step response from 0° to −15°, the control reference is modeled by a single Heaviside function and the proportional derivative (PD) control command is defined:

The Heaviside function in Equation (28) provides a delay of ${T}_{PD}$ after the launch detect to ensure the rocket is well clear of the launch guide before starting control. For launch detection to occur, the on board accelerometer on the rocket’s vertical axis needs to detect a consistent 2 g or higher acceleration over 0.2 s. This requirement prevents sensor errors or motor misfires from triggering launch control. Once the launch is detected, the rocket’s vertical axis acceleration is transformed to the earth reference frame and then double integrated to determine the rocket’s height above the launch guide. Using the two conditions of launch detection and launch guide clearance enables safe conditions for actuating the rocket’s canards.

The roll rate data including all the key points of the launch are given in

Figure 18, where

$t=0$ corresponds to 1.5 s before launch. The control was started when the rocket height was 2 m above the 4 m launch guide. However, due to the specified 0.2 s delay in the launch detect, the actual altitude when control started was 7 m which was 0.63 s after lift-off.

The commanded and encoder measured canard roll angle are defined as the average of all canard inputs which is standard in the literature [

19]:

The data is analyzed from just before control starts up to a couple of seconds before the parachute deployment. The roll rate and fin angle

${u}_{enc}$ which is computed from Equation (33) using the encoder outputs

$en{c}_{1},\dots ,en{c}_{4}$ for each canard, are plotted in

Figure 19, where the time is reset to 0. Note that for some of the thrust period and all of the coast period all the canards are set at their maximum values of 6°, yet the roll rate stays negative. The reason is there is a fin offset which is larger than the 6° that the canards can compensate for.

To identify the torque constant β, damping α and disturbance

${u}_{dist}(t)$ from the roll model of Equations (2)–(8), the first step is to specify the range in the β and α values as given in Equation (19). These ranges are defined:

The next step is to define the time values for the disturbance changes in Equation (7). These values are equally spaced in both the thrust period and the coast period of the data which are denoted in

Figure 19a. Let

${N}_{thrust}$ be the number of time points in the thrust period and

${N}_{coast}$ the number of time points in coast period. The values in Equation (7) are defined:

Since there are no dynamics in the canards during coast,

${N}_{coast}$ is set to the minimum possible value which obtains a reasonable match in the coast period. It was found empirically that higher values than 2 do not help identifiability due to the lack of dynamics during this period, and a value of 1 gave a consistently large error. Therefore

${N}_{coast}$ is set to 2 for all the analysis on this launch. The value of

${N}_{thrust}$ was also minimized and it was found that

${N}_{thrust}$ = 2 was also a good choice. Higher values gave a progressively better match to the data in the thrust period as would be expected, but were much slower computationally. Specifically, the values from

${N}_{thrust}$ = 2,⋯,6 gave virtually identical results with an improvement in the match to the roll angle by less than 0.1°. More importantly, the identified damping remained unchanged and the torque constant only varied by a maximum of 0.05. The results for

${N}_{thrust}$ = 2 and

${N}_{coast}$ = 2 are defined:

The model response is plotted against the measured values for both the roll rate and roll angle in

Figure 20. A zoomed in plot of

Figure 20a is given in

Figure 21a and the identified time-varying disturbance is plotted in

Figure 21b.

Figure 21a shows that although there is some error in the first controlled response, the overall trends are captured accurately. For example the data drops 27.2° from the local maximum at

t = 0.49 s to the local minimum at

t = 1.02 s where the model predicts a drop of 26.15° which corresponds to less than 4% error. The model also captures all the coast period, in both the roll angle and roll rate, as shown in

Figure 20. The disturbance is quite low initially in the first 0.5 s of the data. This behavior is caused due to low velocity and therefore the canard–fin disturbance dominates the dynamics, as the fin offset takes some time to take effect. After about 1 s the disturbance rapidly converges to a near constant value around −7° which remains for the rest of the flight with only a minor increase at the end. The results of

Figure 20 and

Figure 21 and Equations (38) and (39) show that quite a simple model with a relatively smooth disturbance function, is very effective in capturing the rocket roll response.

Note that the second PD controlled roll response in

Figure 21a has a very large steady state error, since the reference was −15°. The reason for this error is that the fin offset is having a major effect due to the increased velocity and the maximum and minimum canard constraints do not provide enough actuation to overcome this fin offset. However, the goal of this launch was not control, but to test the logistics of the new launch vehicle and avionics and provide an initial proof-of-concept of the model and methods.

The final validation of the model and methods for this launch is to match the closed-loop model of Equations (28)–(31) to the data. The results are:

These results are very close to the open-loop response with a mean error difference of 0.06 deg/s in the roll rate and 0.01° in the roll angle. However, there are some small differences in the thrust period in the roll angle as shown in

Figure 22a, but the overall behavior of the two responses are very similar. In addition, apart from a small period in the thrust, the identified closed and open-loop disturbances are virtually identical as shown in

Figure 22b.

Hence, in summary, there is no noticeable change in the output responses and identified parameters when using the closed-loop model over of the open-loop model, although the closed-loop model typically takes about 50% longer to simulate. The advantage of the open-loop model computationally is that the roll rate is decoupled from the roll angle which is simpler to handle numerically.