1. Introduction

2. Materials and Methods

3. Results

3.1. Identify Dynamic Models

When experimental data was collected, it was possible to identify dynamic models for every flight state. The models were identified according to a section of gathered data corresponding to the same state in actual flight. Then, to verify it, dynamic models were compared to a suitable part of data from other flights. At last, the entire flight was simulated using identified models mimicking the real-life test, and it allowed us to compare the total energy consumption and scale of errors.

Identification was performed in Matlab. Input data was a control signal from the flight controller, and output data was registered flight parameters. Identified parameters were a change of Power over time, vertical displacements, and horizontal velocities over time during every considered flight state. The dynamic models were presented as a continuous time transfer function.

The only difference was hover, in which the power was calculated as the average power required to hover at a given altitude. Then, knowing the result for a few different altitudes, the dependence of power on time was approximated with function. According to Rotratu [9], the power required to hover is given by:

$$P=\frac{T^{\frac{2}{3}}}{\sqrt{2\rho A}}$$

(3)

where: T – total thrust force,

A – total area swept by rotors,

ρ – air density.

Assuming that during hover, T and A are constant, power depends on air density, which in International Standard Atmosphere [10] can be approximated as revered exponential proportional to flight altitude H. In this case, the power required to hover can be written as:

$$P=K·H^z$$

(4)

where: $K=\frac{T^{\frac{2}{3}}}{\sqrt{2A}}=const$

z – proportion exponent.

Using collected data, the equation describing power required to hover on flight altitude H was approximated with:

$$P\left(H\right)={119.49H}^{0.02}\left[W\right]$$

(5)

3.1.1. Vertical Movement Models

The first dynamic models to identify were vertical movements: ascending and descending. The data for the process was four subsequent climbs by 100 meters from 0 to 400m for ascending. Analogically, there were four subsequent decreases by 100m from 400m to 0 for the descent. Models for the altitude changes were identified together for both states, and models of power were created separately. A comparison of experimental data and identified models for altitude change was shown in Figure 5.

As seen in Figure 5, the simulated characteristic matches the experimental data; hence, the simulation can be assumed to mimic the vertical movement of the drone with sufficient precision. The following transfer function described the identified model:

$$G_1\left(s\right)=e^{-2.9s}\frac{0.03071}{s^2+0.2923s+0.03071}$$

(6)

The second part of the identification was made for the power required to climb and descend on a 100m difference in altitude. As the system was assumed to be linear, offsets were brought to 0 not to break homogeneity and superposition principles. Then, it was added to the stable state power required for hover. The power characteristics for climb and descent were shown in Figure 6 and Figure 7.

As can be seen, at the beginning of the climb, the power rapidly increased as the drone had to accelerate upwards. At some point, the UAV reached sufficient speed to reach the desired altitude, requiring less and less thrust. Hence, the power started to decrease. At the climb’s final stage, the drone had to lose gathered speed using a force of gravity. At that point, the power dropped below a stable state shortly after stabilizing to the value required to hover. The identified model for power during the climb was presented below:

$$G_2\left(s\right)=e^{-0.2s}\frac{0.07023s+\mathrm{0,02383}}{s^2+0.006887s+0.06957}$$

(7)

When the drone needed to descend, power first dropped, and reduced thrust became lower than gravity force, making the drone accelerate downwards. From some point, the power rose to slow the rate of descent. Subsequently, the thrust had to grow over the force of gravity for some time to lose downward velocity and stabilize the flight to hover at the desired altitude. The transfer function for power during descent was presented below:

$$G_3\left(s\right)=e^{-0.5s}\frac{0.09489s-\mathrm{0,00001339}}{s^2+0.228s+0.05344}$$

(8)

In both cases, the experimental data was distorted by a noise that could not be seen in the simulated response. Optical evaluation allowed us to conclude that the signal trends were similar in both cases, yet further validation of the models had to be done.

3.1.2. Horizontal Movement Identification

Analogical identification was performed for horizontal movement. The flight states were accelerating, moving with constant velocity, and decelerating (stopping). The energy consumption had linear dependence on acceleration and flying with constant velocity. While accelerating, the temporal power grew proportionally to velocity, and during the constant speed movement, the energy consumption was settled around a constant value. This value was calculated as an average take from the segment where the drone flew with constant velocity. The identified model for horizontal flight was presented below:

$$G_4\left(s\right)=e^{-0.6s}\frac{\mathrm{0,03569}}{s^2+0.935s+0.3588}$$

(9)

When the drone was commanded to stop, the desired velocity was instantly changed to 0, so the UAV not only aborted the flying forward but also tried to stop in the shortest time. Hence, it rotated backward and created counter thrust to its movement direction, creating backward force. The power change for deacceleration was shown in Figure 9.

Figure 8. The control signal and characteristic comparison of experimental data and simulation for airspeed during horizontal flight.

Figure 8. The control signal and characteristic comparison of experimental data and simulation for airspeed during horizontal flight.

Figure 9. The control signal and characteristic comparison of experimental data and simulation for power during deacceleration.

Figure 9. The control signal and characteristic comparison of experimental data and simulation for power during deacceleration.

As can be seen after being commanded to stop, the power consumption rapidly increased as the drone started the backward rotation and increased the thrust. Then, it rotated forward, decreasing the power to settle in the hover. Below, the identified transfer function for power change during deacceleration was presented:

$$G_5\left(s\right)=e^{-0.6s}\frac{\mathrm{0,02583}s-\mathrm{0,3294}}{s^2+1.905·{10}^{-10}s+0.2492}$$

(10)

4. Discussion

References

1. Sanchez-Lopez, Jose Luis, et al. "A reliable open-source system architecture for the fast designing and prototyping of autonomous multi-uav systems: Simulation and experimentation." Journal of Intelligent & Robotic Systems 84 (2016): 779-797. [CrossRef]
2. Soria, Enrica, Fabrizio Schiano, and Dario Floreano. "SwarmLab: A MATLAB drone swarm simulator." 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2020. [CrossRef]
3. de Winter J.C.F., van Leeuwen P.M., Happee R. “Advantages and Disadvantages of Driving Simulators: A Discussion” Proceedings of Measuring Behavior 2012 8 th International Conference on Methods and Techniques in Behavioral Research Utrecht, August 28-31, 2012, p. 47-49.
4. Higgins, Fiona, Allan Tomlinson, and Keith M. Martin. "Threats to the swarm: Security considerations for swarm robotics." International Journal on Advances in Security 2.2&3 (2009), p. 288-297.
5. Popov, Luben Nikolaev. Drone swarm simulation. 2020. PhD Thesis. University of Houston, page 8.
6. Li, Jing, et al. "A path planning method for sweep coverage with multiple UAVs." IEEE Internet of Things Journal 7.9 (2020): 8967-8978. [CrossRef]
7. Dudek, Magdalena, et al. "Hybrid fuel cell–battery system as a main power unit for small unmanned aerial vehicles (UAV)." Int. J. Electrochem. Sci 8.6 (2013): 8442-8463.
8. Cesare, Kyle, et al. "Multi-UAV exploration with limited communication and battery." 2015 IEEE international conference on robotics and automation (ICRA). IEEE, 2015. [CrossRef]
9. Rotaru, C., Todorov, M.(2018), Helicopter Flight Physics. In Flight Physics Models, Techniques and Technologies; Volkov, K., Ed.; IntechOpen: London, UK; ISBN 978-953-51-3807-5.
10. Cavcar M. (2000) The International Standard Atmosphere (ISA), Import from: https://scholar.google.com/citations?user=WpEp_wcAAAAJ&hl=en , date of import: 30.08.2023.