Optimization-Based Control for a Large-Scale Electrical Vertical Take-Off and Landing during an Aircraft’s Vertical Take-Off and Landing Phase with Variable-Pitch Propellers
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1. Introduction
UAV platforms have been paid more and more attention by researchers and companies delighted with their characteristics of having a low cost, high efficiency, and minimal risk. With the intention of accomplishing long-distance flying scenarios as well as utilizing the advantage of vertically taking off and landing without an airport at the same time, the demand for UAVs with vertical take-off and landing (VTOL) capabilities has sufficiently increased. At present, VTOL UAVs have been developed in various layouts, such as a dual-system UAV [1,2], tilt-rotor UAV [3,4], tilt-wing UAV [5,6], and tail-sitter UAV [7,8], mostly using blended wings in order to adapt both the VTOL phase and the horizontally cruising phase simultaneously.
To satisfy certain needs of executing under two totally different flight modes, a number of propulsion system design schemes were proposed since normal propellers remain insufficient to provide thrust for the cruising operations of VTOLs. Some aircraft such as Pegasus PAV and VoloRegion rely on a propeller system consisting of two different sets of propellers, one set operating during a phase, which is redundant with negative effects because the aircraft only operates one flight mode at a time. Specifically, the other set of propellers usually does harm to perfect operations under the current flying phase, such as preventing the cruising flight from being faster.
In order to overcome the relatively high rotor advance ratios of normal propellers during the cruise phase, variable-pitch propellers have been designed as the propulsion system of various kinds of VTOL aircraft such as the V-22 Osprey and V-280 Valor aircraft. During the VTOL phase, the inflow angle is small, so the propellers are supposed to be equipped with small pitch angles for high-efficiency operations. Conversely, the propellers require large pitch angles to match increasingly bigger inflow angles with the aim of achieving higher efficiency during the cruise phase. Compared to regular propellers, variable-pitch propellers are helpful as they improve the efficiency of VTOL UAVs with their flexible pitch angles during both the cruise phase and VTOL phase.
Nowadays, delighted with the rapid development of electric power technology, especially the increasing power of electric motors and the growing ability of energy storage, electric VTOL (eVTOL) has become an emerging aircraft platform considering the pollution and noise produced by traditional aircraft. Different from traditional oil-driven VTOLs which rely on large fuel pipelines, 100 kg level eVTOLs succeed in making use of a distributed electric propulsion system due to the convenience and small size of electric cables. For instance, a typical eVTOL aircraft has been built by the Joby company and utilizes six electric propellers as its propulsion system. Therefore, designing and controlling distributed variable-pitch propeller systems on eVTOL aircraft are becoming meaningful and urgent research topics.
Unfortunately, current control strategies lack suitable solutions concerning satisfying flight in both phases, especially for large-scale eVTOL aircraft. Since control issues with regard to the VTOL phase of the flight envelope are mainly discussed in this paper, previous related research work considering the control of variable-pitch propellers during the VTOL phase is particularly discussed. Meanwhile, some of the scenarios where quadrotors are equipped with variable-pitch propellers are also worth focusing on as there is a lot in common between quadrotor flight and the VTOL phase flight.
Keran et al. [9] introduced methods concerning the design and analysis of passive variable-pitch propellers for VTOL UAVs. Although the passive structure decreases mechanical complexity, it sacrifices the controllability of many degrees of freedom and thus reduces the robustness and flexibility of the whole system, especially during the VTOL phase. Cutler et al. [10,11] proposed control methods based on quaternions, which offer an exquisite framework of control and trajectory generation concerning variable-pitch propellers. However, the authors concentrate on small quadrotors and their agile flights, thus maintaining four rotation speeds as high as possible in order to perform aggressive flights, leading to a waste of control resources on these four variables to a certain extent. Meanwhile, large-scale aircraft pay more attention to subjects like minimizing energy consumption and stabilizing the overall flight process, where the above control framework is not suitable to be directly applied. Xu et al. [12] introduced an aggressive trajectory tracker based on MPC and a nonlinear attitude controller for quadrotors, which is utilized on quadrotor platforms without limiting energy consumption. Unlike tiny quadrotors which are able to sacrifice some targets in order to elegantly track some aggressive trajectories, large eVTOLs should consider many more aspects of the flight, including the robustness against external disturbances and measurement noise, and energy consumption analysis. Portillo et al. [13] discussed a method to eliminate unmodeled dynamics or external disturbances at the control level, which provides a feasible solution to overcome these problems. Meanwhile, Bianchi et al. [14] addressed the energy consumption of quadrotors when tracking a trajectory in an analytical way. Though the proposed energy consumption function cannot be directly utilized in our work, the calculating ideas of the authors are still worth learning from.
A number of articles have concentrated on control algorithms directly designed for variable-pitch quadrotors and their flying strategies. Sheng et al. [15] suggested that variable-pitch quadrotors should be controlled based on the minimum power consumption principle, focusing less on the smoothness of control variables such as motor speeds and sometimes leading quadrotors to unhealthy flight behaviors. Jan et al. [16] proposed a model-based controller designed for variable-pitch propellers, whose pitch angles minimized power consumption within the electric propulsion system for the given thrust value instead of optimizing rotation speeds and pitch angles at the same time. Gupta et al. [17] designed a model-based nonlinear controller for variable-pitch quadrotors, which might request larger computation resources, causing immediate changes in control variables. Moreover, the flexibility of pitch angle control was not utilized because the authors still implemented the control allocation module with only four variables. Similarly, Bo et al. [18] designed a nonlinear robust attitude controller using variable-pitch propellers as the propulsion system, but the control variables including rotation speeds and pitch angles were computed by forces with one-to-one correspondence, thus failing to utilize these variables in a satisfying manner. Additionally, the control system was designed based on a linear model with transfer-function-based controller design methods, causing negative effects to the robustness of the aircraft. In conclusion, though plenty of research has been conducted regarding variable-pitch propellers on various kinds of platforms, problems related to the control issues during the VTOL phase of large-scale eVTOLs still exist. Most controllers failed to sufficiently exploit the control potential of all eight control variables, and normally could not meet the specific application scenarios of large-scale VTOLs.
The purpose of this study is, therefore, to design and control a set of variable-pitch propellers for a typical electric VTOL (eVTOL) platform in order to track various kinds of trajectories under considerable external disturbances and measurement noise during the VTOL phase. Meanwhile, the power consumption of all propellers should be considered in case of exceeding the rated power. Detailed contributions are listed below.
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A typical tail-sitter eVTOL platform with distributed variable-pitch propellers is designed, which is able to achieve excellent flying performances during both the VTOL and cruise phase. Simulation data of the platform are then utilized as the objective for the designed controller with nonlinear flight dynamics. In addition, an actuator system consisting of four variable-pitch propellers is designed and its thrust and torque models are accurately analyzed.
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A specific optimization-based control allocation module is achieved to fully excavate the control potential of the variable-pitch propellers with four extra control variables, and the control allocation solution can be generated with low-dimensional quadratic programming solvers.
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This module is then mounted in a complete control system including position and attitude controllers to track the given trajectory. Constraints such as power consumption and the maximum rate of the control variables are considered to maintain the stability of the system.
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A series of simulation experiments are accomplished in order to validate the effectiveness of the designed controller under different circumstances, such as set-point arrival and aggressive trajectory tracking under measurement noise and external disturbances.
The structure of this paper is as follows: First, the platform design and analysis work for both the eVTOL and the variable-pitch propellers system, including flight dynamics interpretation, are addressed in Section 2. Then, the control and optimization strategies for the eVTOL utilizing the variable-pitch propeller system as the actuator are developed in Section 3, followed by simulation experiments and the results in Section 4. Finally, conclusions are drawn in Section 5.
4. Simulation Experiments
This section presents three simulation scenarios to verify the effectiveness of the designed control law. The simulations are performed using Simulink in MATLAB, where the SDQP solver is able to solve the problem in less than 1 ms. Thanks to the conversion from the original high-level optimization problem to a low-dimensional structured QP problem, the efficiency of solving it improves dramatically. Therefore, the frequency of the positional controller is set as 50 Hz and the attitude controller as well as the control allocation module are set as 500 Hz. Additionally, all of these scenarios are conducted with the eVTOL model introduced in Section 2, and without loss of generality. When bringing up the comparison between variable-pitch propellers and regular ones, the simulation sets the motor parameters of the latter under the condition that pitch angle remains constant, normally at 5° or 10° since the force efficiency is too low for the aircraft to take off if gets any higher.
The first scenario suggests the effectiveness of the controller-given set-point control scene, the second scenario indicates its ability to track more aggressive trajectories, and the last scenario proves its robustness under noisy measurements and external disturbances.
The performance of the controller is validated in this section under different scenarios. The first scenario is about the set-point-arriving scene in Section 4.1, while the second scenario provides a much more complicated and aggressive trajectory in Section 4.2. Moreover, the third scenario concentrates more on measurement noises and external disturbances, as shown in Section 4.3. Finally, the numerical error analysis results are shown in Section 4.4. The parameter table is depicted in Table 1.
4.1. Set-Point Control
One of the basic missions of any successfully designed controller is the set-point controlling task. It is worth mentioning that, throughout this section, the initial condition of the four variable-pitch propellers is set as
For regular propeller conditions, the virtual “pitch angle” is kept at 5° or 10° constantly in order to verify the effectiveness of the designed controller. Consequently, the initial condition is set as
The coordinates of the target point are denoted as
Even though the target point seems fairly close to the original, the relative distance is reasonable because consecutive intermediate points are calculated from the trajectory generation module instead of an extremely far target. The performance of the proposed controller is depicted in Figure 9. As is described in these results, the eVTOL aircraft arrives at the designated target point in about 6 s, with approximately 6.5 kW the highest power and a power of 3.8 kW, ultimately, for each propeller. Therefore, the overall controlling performance of the variable-pitch propeller system is acceptable.
Next, the same control algorithms are simulated under conditions and with the pitch angles of the four propellers remaining the same. The results for conditions and are shown in Figure 10 and Figure 11, respectively. As for case , which is higher than most angles shown in Figure 9, the controller fails to arrive at the given set-point despite the fact that the propellers are able to constantly offer sufficient force and torque for the aircraft platform. The main reason why the controller fails at this scenario is that the four pitch angles are no longer controllable, which causes the robustness of the control allocation module to decrease dramatically. Then, the virtual pitch angle is continually increased to against the proposal, meaning that the propeller system can finally arrive at the target stably. The position curve under this virtual situation is very close to that of the variable-pitch propellers in Figure 9, but the attitude curve develops differently. Nevertheless, since the force provided by this kind of propeller is larger than the proposed variable-pitch propellers, the ultimate mechanical power for each propeller is 4.5 kW, which is higher.
4.2. Aggressive Trajectory Tracking
A typical method to validate the effectiveness of a flight controller is to simulate whether it can track some aggressive trajectories. Different from quadrotors which are able to dance along some fascinating trajectories with extremely high speeds, large-scale eVTOLs pay more attention to the stability, robustness, and energy consumption of the system. As a result, a spiral ascent trajectory is defined as in Equation (31), which is indeed an exceedingly aggressive trajectory for large-scale eVTOLs.
where
The behavior of the designed control system is shown in Figure 12. Generally speaking, the proposed control system successfully tracks the given trajectory. From the results, it can be summarized that the controller utilizes all of the eight control variables to track the reference trajectory with an approximately 0.5 s time delay, which is common using a nonforward model-free controller structure such as cascade PID. The power curve is also constrained under 10 kW, which prevents the eVTOL from accidents due to there being insufficient power. Additionally, the dramatic changes in the four pitch angles demonstrate that the role of the variable-pitch controller is irreplaceable.
In contrast, the controller fails to track the given aggressive trajectory given motor parameters under the condition, which explains that under a bunch of constraints such as energy consumption and the maximum change rate of rotation speed, it is not suitable to control such a large eVTOL aircraft by just using four motor speeds during the VTOL phase. The control results of this scenario are shown in Figure 13.
4.3. Flying under Measurement Noise and External Disturbances
In this subsection, the efficiency of the proposed control algorithms is validated under significant measurement noise and external disturbances. The places where these noise and disturbances are introduced are depicted in Figure 14. The noisy measurement situation is introduced in advance since it is normally a common situation in real flight, followed by dramatic external disturbing forces. The amplitudes of white-noise error signals are listed in Table 2.
The results of adding the measurement noise situation are depicted in Figure 15, and the noises are provided from the Band-Limited White Noise block in MATLAB as shown in Figure 16. It can be concluded from the results that the controller is able to track the given aggressive trajectory with a similar time delay after adding relatively large measurement noise; thus, the robustness of the given controller can be proven. In addition, the mechanical power of each variable-pitch propeller is constrained under 10 kW, which guarantees the security of the flight even with some unexpected accidents occurring. Moreover, conclusions can be drawn from the pitch angle figure that the potential of pitch angle control has been fully motivated toward, which prevents the rotation speeds from changing so dramatically and causing harm that could have been avoided.
Finally, dramatic external forces are imposed on the aircraft as shown in Equation (33) with measurement noise added simultaneously. The behaviors are drawn in Figure 17 and noise figures of the position and attitude are drawn in Figure 18. From the position curve, a dramatic change can be observed when external forces are imposed at 6–10 and 25–30 s, causing suddenly enormous shifting trends from the original position. Nevertheless, these positions quickly bounce back to the places where they should be, which illustrates the ability and robustness of the designed controller. Meanwhile, despite these disturbances, the mechanical powers of all four propellers are constrained to less than 10 kW, showing that it is possible to reduce energy consumption while tracking aggressive trajectories with acceptable accuracy at the same time with variable-pitch propellers as the actuator system.
4.4. Numerical Analysis
4.4.1. Set-Point Control
Power indexes are established to compare the control performances between variable-pitch propellers and regular propellers with a virtual pitch of . The numerical results are listed in Table 3.
From Table 3, conclusions can be drawn that variable-pitch propellers require much less power than regular propellers when executing the set-point control mission using the eVTOL prototype. This is because the actual pitch angle of variable-pitch propellers is maintained at around 5 degrees, which is much less than the virtual pitch of 10 degrees, while still being able to control the aircraft.
4.4.2. Aggressive Trajectory Control
Under this scenario, a trajectory tracking error should be introduced to describe the validity of the designed control algorithms. The MSE is calculated to depict the tracking error during the flying process, which is shown in Equation (34).
However, the MSE is only able to describe the absolute trajectory tracking error. Therefore, the target trajectory length is used to regularize the tracking error index. The length can be represented as
Consequently, the new numerical index AVGMSE describing the tracking error can be defined as
The AVGMSE, maximum power, and average power are recognized as numerical indexes to describe the abilities of the control algorithm under the aggressive trajectory control scenario. Since the scenarios in Section 4.2 and Section 4.3 are similar, the numerical results are listed together in Table 4.
From Table 4 conclusions can be drawn that the trajectory tracking error is acceptable even under measurement noise and external disturbances, proving the validity of the designed controller under the aggressive trajectory tracking scenario.
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