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Optimal-Time Quadcopter Descent Trajectories Avoiding the Vortex Ring and Autorotation States

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Optimal-Time Quadcopter Descent Trajectories Avoiding the Vortex Ring and Autorotation States 1 Optimal-time Quadcopter Descent Trajectories Avoiding the Vortex Ring and Autorotation States Amin Talaeizadeha, Duarte Antunesb;∗, Hossein Nejat Pishkenaria;∗ and Aria Alastya Abstract—It is well-known that helicopters descending fast attention in the context of quadcopters, despite the fact that may enter the so-called Vortex Ring State (VRS), a region they can play a very important role due to their agility. Some in the velocity space where the blades lift differs significantly exceptions are [8]–[16]. from regular regions. This may lead to instability and therefore this region is avoided, typically by increasing the horizontal In [8] it is reported that for descending trajectories and for speed. This paper researches this phenomenon in the context of some velocity state space regions a quadcopter can become small-scale quadcopters. The region corresponding to the VRS unstable. Other studies show that these instabilities are due is identified by combining first-principles modeling and wind- to the VRS effects, see [9]–[11], where simple models are tunnel experiments. Moreover, we propose that the so called defined for explaining the boundary of these regions. These Windmill-Brake State (WBS) or autorotation region should also be avoided for quadcopters, which is not necessarily the case simple models are similar to helicopter models and are used for helicopters. A model is proposed for the velocity constraints in some works to design controllers that avoid entering the that the quadcopter must meet to avoid these regions. Then, VRS [12]–[16]. In fact, these studies are similar to helicopter the problem of designing optimal time descend trajectories that VRS avoiding controllers introduced in [17], [18]. However, avoid the VRS and WBS regions is tackled. Finally, the optimal the results of these models and control strategies have only trajectories are implemented on a quadcopter. The flight tests show that by following the designed trajectories, the quadcopter been shown through simulation and have not been validated is able to descend considerably faster than purely vertical through real measurements. Moreover, due to differences in trajectories that also avoid the VRS and WBS. the blades mechanisms between helicopters and quadcopters1, Index Terms—Vortex ring state, Quadcopter, Windmill brake high fluctuation regions are different. Furthermore, such con- state, Optimal trajectory design, Vortex ring avoidance trajectory, trollers are not time-optimal, which is often required in the Quadcopter fast descent. context of aggressive maneuvers. (see [19]–[21] which tackle the problem of finding time-optimal trajectories without con- sidering aerodynamic constraints.) I. INTRODUCTION In this paper, we conduct static experimental tests on a UADCOPTERS have been introduced in recent years wind tunnel to identify the regions where fluctuations occur. Q and are expected to have a large impact on many Combining this information with the VRS and TWS regions applications, such as agriculture monitoring, industrial inspec- predicted by a theoretical model, we provide a simple model tion, entertainment industry, among others [1]. One of the for the region in the velocity space to be avoided by the interesting features of these aerial robots is that they are able quadcopter. As we will discuss, we claim that the quadcopter, to perform aggressive maneuvers due to their agility. In order contrarily to helicopters, should also avoid the WBS region. to safely conduct these aggressive maneuvers, it is important According to the model for constraints and state bounds, to model and understand the dynamic behavior at high speeds, different optimal trajectories are designed, considering planar where non-trivial aerodynamic effects come into play. 2D and full 3D models for the motion of the quadcopter. Such aerodynamic effects have been extensively studied Due to the complexity of this nonlinear optimal control in the context of helicopters, where one can find several arXiv:1909.09069v1 [math.DS] 19 Sep 2019 problem, we resort to the optimal control toolbox, GPOPS- seminal papers [2]–[7]. As explained in these papers, there are II [22]. The designed trajectories to descend as fast as possible three regions in the velocity space called Vortex Ring State boil down to intuitive maneuvers, such as oblique and flips (VRS), Windmill-Brake State (WBS) and Turbulence Wake maneuvers in 2D, and helix type maneuvers in 3D. Finally, State (TWS) in which the behavior of the helicopter and the the optimal trajectories are implemented on a quadcopter. The effect of the environment is different from regular regions. flight tests show that by following the designed trajectories, the The VRS and TWS regions must be avoided, but it is still quadcopter is able to descend considerably faster than purely possible, although undesired, to control the helicopter in the vertical trajectories that also avoid the VRS. WBS region. However, these effects have received only limited The remainder of the paper is organized as follows. In Section II, the aerodynamic effects in descent maneuvers that a Mechanical Engineering Department, Sharif University of Technology, Tehran, I.R.Iran b Control Systems Technology Group, Department of Mechanical En- 1The helicopters blades have a constant rotational speed and their pitch gineering, Eindhoven University of Technology, 5600MB Eindhoven, The angle is variable with a swashplate mechanism for changing the direction of Netherlands the blade disk thrust. In contrast, quadcopters motor speed is variable, and their Email addresses: [email protected] (A. Talaeizadeh), blades pitch angle is usually fixed. Moreover, the hings of the helicopter blades [email protected] (D. Antunes), [email protected] (H. Nejat Pishkenari), provide a flapping degree of freedom, whereas in contrast to the quadcopter [email protected] (A. Alasty). where the blades are rigid. 2 H v vH jbd V kbd vz Vortex Ring State (VRS) V + S VS vz vi vH z Turbulence Wake State Vtv Tip Vortex v (TWS) Fig. 2: Tip vortexes velocities in oblique descent, used in Windmill Brake State (WBS) Onera’s criterion [13]. VS in this figure is the velocity of air stream in the blade disk. also considered as VRS in this paper because this region Fig. 1: The VRS, TWS and WBS regions in the velocity state is encompassed by the VRS region in the velocity state space for helicopters are shown in the figure. Both vertical, space. v , and horizontal v speeds are normalized by the induced z H • WBS:(or autorotation) In this region, while the helicopter velocity at hover, where vH and vz are horizontal and vertical still suffers high amplitude fluctuations, there is enough speeds in the blade disk frame, respectively [2]. lift created by the airflow (contrarily to the VRS) such that the helicopter can still be controlled.2 This can be achieved by switching off the motors and changing the may cause instability are described, and two major regions in pitch angle of the blades to simply control the angle the descent of quadcopters are introduced, based on a theoret- of attack (the pilot can, e.g., change a positive angle ical model borrowed from helicopter aerodynamic theory. In of attack to a negative one to produce the thrust and Section III, the wind tunnel test results are described and a change the thrust vector direction with the swashplate to model is suggested for the boundaries of the unstable regions control the orientation of the helicopter [7]). The blades in the velocity space. Section IV provides optimal minimum lift (and rotation) will now be caused by the air moving up time trajectories in 2D and 3D spaces that are designed through the blade disk and not by the motors. This effect considering the VRS constraints, a model of the quadcopter, is equivalent to the gliding flight in fixed-wing planes. and GPOPS-II. Section V presents the flight test experiments In Section II-2, the differences between helicopter and by which the quadcopter follows the optimal time trajectories quadcopter, and the behavior of the quadcopter in this while avoiding the VRS. Section VI provides a brief discussion region are explained. and research lines for future work. To determine the behavior of the quadcopter in the descent II. MODEL OF CRITICAL REGIONS DURING DESCENT phases, firstly, we will provide a model for the VRS and TWS regions. For the WBS region, we will discuss that due In this section, the behavior of the quadcopter in descent to the differences between the helicopter’s and quadcopter’s maneuvers is described based on the theory available for blade disks, the WBS models for helicopters are not valid for helicopters. When a helicopter is descending, in some regions quadcopters, and we will find the behavior of the quadcopter’s in the velocity state space, intensive vibrations and fluctuations blade disks by wind tunnel tests which are provided in in thrust occur. By increasing the horizontal speed of the Section III. helicopter, these effects can be reduced. Depending on the vertical and horizontal speeds, the amplitude of fluctuations 1) Vortex Ring State Model in Quadcopter: The momentum in thrust can vary. The low and high amplitude fluctuations theory is not valid in the VRS and TWS regions. However, it are known as VRS and TWS, respectively. In Figure 1, the is possible to find the boundary of these regions to identify VRS and TWS regions are illustrated in the velocity space if the quadcopter is in these regions or not. For modeling normalized by the induced velocity at hover. The induced the VRS and TWS boundaries, there are many suggestions velocity corresponds to the flow at the tip of the blade from the in the literature [23]–[25]. One of the closest models to the high-pressure air at the bottom of the blade to the low-pressure experiments is Oneras model [26]. Let kVtvk be the speed air at the top of the blade.
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