Infusion Pump Control System

Control of a Rigid Wing Pumping Airborne Wind Energy System in

All Operational Phases

Davide Todeschini and Lorenzo Fagiano and Claudio Micheli and Aldo Cattano ∗

Abstract

The control design of an airborne wind energy system with rigid aircraft, vertical take-off and landing, and pumping operation is described. A hierarchical control structure is implemented, in order to address all operational phases: take-off, transition to power generation, pumping energy generation cycles, transition to hovering, and landing. Control design at all hierarchical levels is described. The design approach is conceived and developed with real-world applicability as main driver. Aircraft design considerations in light of system maneuverability are presented, too, as well as three possible alternative strategies for the retraction phase of the pumping cycle. The automatic control approach is assessed in simulation with a realistic model of the overall system, and the results yield a comparison among the three retraction strategies, clearly indicating the most efficient one. The presented results allow one to simulate the dynamical behavior of an AWE system in all operational phases, enabling further studies on all-round system automation, towards fully autonomous and reliable operation.

Introduction

An Airborne Wind Energy (AWE) system converts wind energy into electricity with an autonomous aircraft that carries out periodic trajectories in the wind flow [1, 2] and is tethered to a ground station. Deemed a potential game-changing solution , AWE is attracting the attention of researchers, entrepreneurs, and policy makers alike [4, 5], with the promise of producing large amounts of cost-competitive electricity and with wide applicability worldwide [6, 7, 8, 9]. Today, AWE is the umbrella name for a series of technologies under development, which can be classified according to different criteria, such as the operating principle (drag power or pumping operation), aircraft type (rigid or flexible), or take-off and landing method (vertical or linear), see, e.g., [2, 4] for an overview and [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] for contributions on specific aspects.

No fully autonomous AWE system has been commercialized so far: many groups in industry and academia are working to develop the technologies. Fully autonomous and reliable operation is currently one of the major research and development priorities . An essential component to reach this goal is the automatic control system that shall operate the aircraft and the ground station in all the different system and environment conditions, and in all phases: take-off, power generation, and landing. Such a control system features a hierarchical topology, with control functions at different layers distributed across the various subsystems. In the scientific literature, most contributions focus on control design for the power generation phase, mainly with flexible wings (see, e.g., [12, 13, 16]) but also with rigid ones [25, 15, 24]. A few works deal with the control aspects of take-off and/or landing phases [26, 27, 20]. To the best of the authors’ knowledge, no contribution so far has described the control of an AWES in all operational phases, allowing one to simulate the dynamical system behavior from take-off, to power generation, to landing, including all transition phases, and eventually to implement the control approach on real-world systems.

This paper provides its main contribution in this direction, by describing a design approach for an automatic control logic able to drive an AWE system in all operational phases. The considered concept employs a rigid aircraft with vertical take- off and landing (VTOL) capability, single tether, and pumping operation, in particular the one with boxed-wing design developed by Skypull SA . We adopt a realistic model of the whole system, and design the feedback controllers at all hierarchical levels, from low-level attitude control up to the supervisory state machine that defines the phase transitions, to realize fully autonomous operation in normal (i.e., non-faulty) conditions. In the same spirit as , the control system design is approached with simplicity and effectiveness in mind, making it highly suited for implementation on a real prototype. Moreover, this article delivers two additional contributions: 1) a study on the links between aircraft/control design and its maneuverability, exploring the minimum steering radius for a given design as a function of the wind speed, and 2) three possible strategies for the retraction (or reentry) phase of the pumping cycle: a “free-flight” one, where the aircraft glides upwind with very low force on the tether, and two alternatives with taut tether and different flight ∗D. Todeschini and L. Fagiano are with the Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Claudio Micheli is now with Auterion Inc., Giesshübelstrasse 40, 8045 Zürich, Switzerland. Aldo Cattano is with Skypull This is a preprint of a paper submitted to Control Engineering Practice.

Arxiv:2006.11141V1 [Eess.Sy] 19 Jun 2020

trajectories. The automatic control system is assessed via numerical simulations, showing the good performance of the approach and allowing us to compare the three reentry alternatives. The comparison clearly confirms that the free-flight strategy achieves the best conversion efficiency of the pumping cycle, with about 70% ratio between cycle power and traction phase power. A very preliminary study connected to this paper is , where we considered the control in all phases of an untethered aircraft, hence with a simpler model, without the system and control design aspects for tethered flight and power generation, and without the study on different reentry strategies.

The paper is structured as follows. In Section 2, a concise description of the system under study and its operational phases is given. Section 3 presents the analysis of aircraft design vs. maneuverability. Section 4 describes the dynamical system model, divided into its main subsystems: the drone, the tether, and the ground station. Section 5 presents the automatic control strategy with the different reentry alternatives. Section 6 describes simulation results and the comparison among the reentry strategies. Finally, Section 7 contains concluding remarks and points out the next research and development steps.

System Description

A sketch of the system considered in this paper is presented in Figure 1. It employs a tethered rigid wing aircraft (also referred to as “drone” in the remainder) with a single tether and ground-based power conversion via pumping cycle operation. The drone has VTOL capability thanks to onboard propellers. One of the main peculiarities of this system with

Weight Polyethylene

Figure 1: Conceptual sketch of the considered AWE system with its main components and of pumping operation. respect to the other concepts currently under development is the boxed-wing design of the drone with a propeller at each corner, see Figure 2. Due to the distribution of masses and the positioning of batteries in the lower corners of the frame, the geometric center and the center of gravity are different (see Figure 2). This allows one to identify a lower wing, the one that is closer to the center of gravity, and an upper one (i.e. the opposite one), and consequently a left and right wing by looking at the drone from behind.

On each lateral wing there are two discrete control surfaces, while the upper and lower wings feature three discrete control surfaces each. These surfaces can be actuated either symmetrically, on a single wing or on two opposite wings, leading to a symmetric change of aerodynamic coefficients (hence of lift and drag forces), or in opposite directions. In the second case, the effect is to generate turning moments. All the possible combinations of movements of control surfaces lead to a rather high control authority when operating in dynamic flight mode. In addition, the four propellers act as in standard quad-copters in hovering mode, and they can further contribute to turning moments also in dynamic flight mode, besides providing the forward thrust to keep cruise velocity. In this report, we assume that the control system can inject turning moments around the axes of the body coordinate system B = (xB, yB, zB) depicted in Figure 2, plus a thrust force in the zB direction. Therefore, we have four control inputs (three turning moments and the thrust force). The translation from turning moments to individual surface/propeller commands is managed by a low-level logic that aims to provide the wanted action with minimal energy consumption. When operating in tethered mode in presence of strong-enough wind, the propellers are not used to generate thrust force, rather the forward apparent speed is provided by the well-known phenomena of crosswind kite power ,.

𝑦𝑦𝐵𝐵

Figure 2: Position of the center of gravity and geometric center of the drone, and adopted body reference frame B. Regarding the tether, this is made of Ultra-High Molecular Weight Polyethylene (UHMWPE), as in most pumping AWE systems. Finally, the ground station features a winch connected to an electric machine that serves both as generator and as motor during pumping operation, together with all the required subsystems (power electronics, energy storage, mechanical frame, tether spooling system, ground control system, ground sensors, etc.).

Operating Principle

In normal (i.e. non-faulty) conditions, the working principle of this class of system features the following phases: • Vertical take-off. When the wind conditions at the target operating altitude are suitable to generate energy (cut-in wind speed), the system takes-off vertically using the on-board propellers, in hovering mode. This is the typical flight condition of quad-rotors, with the addition of the tether. The propellers sustain the weight of the vehicle and control the attitude.

• Transition from hovering to power generation. In this phase, the drone must quickly speed up and rotate to achieve the attitude and reference speed that can sustain its weight during the power generation phase. • Power generation. The system transitions from multicopter mode to dynamic flight and enters into power gen- eration mode. This exploits the so-called “pumping operation”, composed of two phases (see Figure 1): in the traction phase the drone flies fast in crosswind patterns (i.e. roughly perpendicular to the wind flow), like a steer- able kite, and the tether is reeled-out under high force, generating energy, while in the retraction phase the drone glides towards the ground station and the tether is quickly reeled-in under very low force, spending about 10% of the energy previously generated. Two transition phases link the traction and retraction ones to achieve a repetitive power generation cycle. During power generation, the aircraft is kept airborne by large aerodynamic forces, the onboard propellers are therefore not employed, or they are used as generators to recharge the batteries and supply power to the onboard electronics and actuators. This is the typical flight condition of an airplane, with the addition of the tether.

• Transition from flight to hovering. In this phase, the drone must slow down and rotate with propellers pointing up, to achieve a stationary hovering condition. • Vertical landing. when the wind is too weak to generate energy, or too strong to operate the system safely, the drone carries out a landing manoeuvre composed of a first gliding part up to about 50 m of altitude, followed by a controlled vertical landing, after transitioning from dynamic flight back into multicopter mode.

The main overall control objective is to obtain a fully autonomous flight cycle from take-off to landing, going through all the phases described above. The model and control algorithms presented in this paper allow one to realize such an all-round operation.

Alternative Reentry Strategies

In order to maximize the efficiency of the pumping cycle, it is important analyze in detail the reentry phase and select the best strategy among different possible ones. One of the contributions of this paper is indeed to compare three different alternatives and their advantages and disadvantages. It is appropriate to describe qualitatively these three alternative strategies at this point, while in the next chapters we will point out the corresponding specific control solutions. In the remainder, we adopt the term “free-flight” to indicate a flight mode with slack tether, as opposed to “taut-tether” flight.

Moreover, it is assumed that the traction phase is carried out with figure-of-eight flight patterns in crosswind conditions.

Free-Flight Reentry

With this strategy, the reentry starts with the drone transitioning from taut-tether flight into free-flight. The ground station controller regulates the tether speed in order to limit the pulling force at very small values, thus limiting interference with the drone control system that aims to stabilize the drone in a free-flight condition. Then, the drone starts to glide upwind towards the ground station at controlled speed. To facilitate the restart of the next traction phase, in which the drone enters a figure of eight path, the gliding trajectory is carried out pointing to one side of the ground station with respect to the wind direction. When the drone is sufficiently close to the ground station, its control systems has to cooperate with the ground station’s one in order to transition again from free-flight to tethered flight. This is the most critical part of the trajectory and requires a very accurate coordination of on-board and on-ground controllers.

Complete Rotation Around The Ground Station

The second reentry strategy is a lateral reentry with a complete rotation around the ground station. In this case, the tether is maintained taut at all times, and the ground station controller has to reel-in at constant speed. Note that the drone makes a spiraling reentry, rotating around the ground station. Furthermore, to maximize the cycle efficiency, the reentry controller reference velocity has to be adequately calibrated, in order to have no high peaks of required power and at the same time guarantee a sufficiently fast tether reel in.

Climb And Descend Reentry

In the third reentry strategy, the drone continues to follow a figure of eight path, but at an higher altitude than the one used in the traction phase. Doing this maneuver, the elevation angle increases and the relative wind seen by the drone decreases. As a consequence, the force needed to reel in the tether can be reduced. Also in this case, the reference tether velocity has to be suitably calibrated to optimize the cycle performance.

3

Steering authority analysis and drone design considerations Steering authority during tethered flight is a crucial aspect for an AWE system, because of the particular trajectories to be achieved. We term steering authority of the tethered system the minimum turning radius that it can achieve while flying with taut tether under high lift force and apparent speed. The smaller such a steering radius, the higher the steering authority. The latter is affected by different aspects, including the steering strategy which, for the considered box-wing design, can be either “roll-based” or “yaw-based”.

In our research, we went through an initial design phase in which we analyzed these two options, and more in general the link between system design and steering authority, until eventually opting for a yaw-based strategy. In this section, we present the key points of our analysis, since they motivate subsequent control design choices and they can be useful also for other AWE systems’ developments. In particular, we analyze the steering authority in two different conditions: 1) trajectories parallel to ground, and 2) trajectories perpendicular to ground. These are indeed two extreme cases of all the situations that can occur in tethered crosswind flight. In the analysis, we assume for the sake of simplicity that the apparent speed magnitude coincides with the drone’s velocity vector magnitude (i.e. no absolute wind speed is present).

The results hold qualitatively also in presence of wind.

Trajectories Parallel To Ground

The centrifugal force during a turn at constant radius r can be expressed as:

(1)

where m is the drone mass and v is its speed. The centrifugal force has to be counterbalanced by a centripetal one in order to turn at constant radius r. The generation mechanism of the centripetal force depends on the considered steering strategy, as described below.

Roll-Based Steering Strategy

With roll-based steering, the centripetal force is the projection of the lift force on the plane where the drone’s trajectory

(2)

where FL is the total lift force (generated by upper and lower aerodynamic surfaces) and ϕ is the roll angle. The lift force

Fd2

Figure 3: Sketch of steering using (a) roll-based or (b) yaw-based strategies.

(3)

where ρ the air density, S is the effective wing surface, cL the lift coefficient, and α, β the angle of attack and side-slip angle, respectively. The aerodynamic coefficients employed in our study are reported in Section 6 (Figure 13). Equating (1) and (2) and inserting (3) into (2) we have

(5)

From this equation it can be noted that, to increase the steering authority using a roll-based strategy, one of the following measures can be adopted, compatibly with other design and control requirements:

• Reduce The Drone Mass M;

• Increase the upper and lower wings surfaces Sup, Sdown; • Increase the lift coefficient cL (e.g. higher angle of attack α); • Increase the roll angle ϕ.

Yaw-Based Steering Strategy

In yaw-based steering (see Figure 3-(b)), the centripetal force is provided by the total lift force FLlateral of the lateral

(6)

where Sright, Sleft are, respectively, the surface of the left and right wings, and cLlateral(β) their lift coefficient. The latter is a function of the side-slip angle β, which for the lateral wings corresponds to the angle of attack. The turning radius is thus given by: solving for turning radius

(7)

Thus, to increase the steering authority with a yaw-based strategy, one of the following measures can be adopted:

• Reduce The Drone Mass M;

• Increase the lateral wings’ surfaces Sright, Sleft; • Increase the lift coefficient of the lateral wings, cLlateral; • Increase the side slip angle β (without exceeding its limits for a stable flight).

Trajectories Perpendicular To Ground

In the previous section, results for trajectories parallel to ground have been presented. However, in case of up-loops or down-loops, the steering ability is affected also by the gravity force Fg and its orientation during the path. In a part of the path, Fg has a centripetal component, which helps during the turns. In the opposite part of the path, Fg has a centrifugal component, which makes the turns more difficult. Thus, during trajectories not parallel to ground, steering ability (and minimum turning radius) is bounded between 2 extreme values, as described below.

(9)

where Fcentripetal is provided by either one of the two strategies presented in the previous section. From eq. (9) it can be noted that, in the upper part of the trajectory, the gravity force is a centripetal one, helping the drone to turn. Moreover, the turning radius increases with larger speed values.

(11)

from which it can be noted that, in the bottom part of the trajectory gravity provides a centrifugal contribution, making the turn more difficult. Equation (11) has a singular point for a value of speed vlim: below this speed, the drone is not able to steer in the lower part of trajectory, because the available centripetal force is not large enough to compensate weight. In this case though, the turning radius decreases with larger speed values.

(B)

Figure 4: Sketch of turning maneuver with path perpendicular to ground: top of trajectory (a) and bottom of trajectory (b). Table 1: Parameters used in for steering authority analysis

Turning Radius [M]

Figure 5: Example of relation between turning radius and front speed for a path perpendicular to ground, for the Skypull system considered in this study. In blue, the turning radius for the upper part of the trajectory, in red for the bottom part. The green area represents the values of the turning radius during the whole maneuver, for each front speed value. Drone parameters reported in Table 1.

Conclusions

From the previous analysis, the following considerations emerge: • During trajectories perpendicular to ground, the turning radius is bounded between extreme values: rtop < r <

Rbottom;

• For trajectories parallel to ground, the speed of the drone does not affect the steering authority. Instead, for tra- jectories perpendicular to ground, the steering authority is highly affected by the drone speed and it approaches a unique asymptotic value with larger speed, either from above (in the top part of the trajectory) or from below (in the bottom part). This effect is presented in Figure 5.

• In a drone with lower mass, higher lift, or larger effective surface, the singularity point vlim is shifted to lower values, with an increase of the steering authority. Figure 6 shows the sensitivity of the turning radius with respect to the effective lateral wings’ surface, keeping constant all other parameters. A larger surface decreases the turning radius in all conditions and reduces the minimum front speed required to steer at bottom of the path. From this analysis, it can be noted that doubling the lateral surfaces has a large impact on the steering authority, while further doubling it has a relatively lower effect, since the relationship is hyperbolic (see, e.g., (7)). Moreover, this simplified analysis is not valid anymore for small radius values (as compared with the drone’s wingspan), since it does not take into account that when the turn is too narrow the lift distribution along the wing becomes highly uneven, and the total lift decreases significantly.

Based on the presented analysis, in our research we eventually decided to adopt a yaw-based strategy and to double the lateral wing surfaces with respect to the original drone design, in order to improve maneuverability.

System Model

In this section, we describe the models of the drone and of the ground station that we use to simulate the system. For control design, described in Section 5, we used different simplified models, hence introducing a mismatch between the latter and the plant model used to evaluate the control system behavior.

Turning Radius [M]

Figure 6: Turning radii of the Skypull system considered in this study as a function of front speed, for the baseline total lateral surface (dotted line), and for increased surface values: twice the baseline (dash-dotted), and four times the baseline (solid). Drone parameters reported in Table 1.

Reference Frames

The considered model is derived from established 6-dof aircraft model equations , with modifications due to the box- wing architecture of the drone and the presence of the four propellers. In the remainder, · T denotes the matrix transpose operation.

We start by introducing the drone’s position⃗p(F ) = [xF , yF , zF ]T in the inertial reference frame F, which is fixed to a point on ground, with component zF pointing up and xF along a chosen direction (usually the prevalent wind direction at the considered location). For taut-tether control purposes, it is useful to convert the coordinates of the inertial system into

(12)

where d is the drone distance from ground station, θel is the elevation angle and ϕaz is the azimuth angle (Figure 7, left). For convenience when describing the control approach, we introduce three additional right-handed reference frames: • B (xB, yB, zB) is the body reference frame, fixed to the UAV and with origin in its center of gravity (Figure 2).

The zB axis is aligned with propellers’ axes, while xB points towards the upper wing. This reference system is aligned with the inertial one, when the UAV is stationary in hovering mode. • Aw (xAw, yAw, zAw) is the apparent wind reference system, with xAw aligned with the apparent wind direction, yAw parallel to the upper wing, and zAw pointing up.

• L (LT , LN, LW ) is the local (or tether) reference frame, fixed to the UAV and with LT aligned with tether, pointing opposite to the ground station, LN is pointing to the local North while LW is pointing to local West (see Figure 7, left).

The relative orientation between frames F and B can be expressed by a quaternion⃗q(t) =

Q4(T)T ,

where t ∈R is the continuous time variable. For the sake of notational simplicity, in the remainder we omit the explicit dependency of the various quantities on t when this is clear from the context, and we point out the constant (i.e. not time-dependent) quantities and parameters.

𝑧𝑧𝐵𝐵

Figure 7: Left: spherical coordinates. The drone position can be expressed in terms of distance from ground station, elevation angle θel, and azimuth angle ϕaz. A sketch of the local reference system L is presented as well: Lt is directed along the tether, Ln points to the local North (zenith) and Lw points the local West. Right: close-up view of the local reference system, and Euler angle ψL.

A vector in the F system can be expressed in the B system by means of the following rotation matrix:

(13)

and a vector expressed in Aw reference can be expressed in B system by the following rotation matrix :

(14)

where, as already introduced in Section 3, α is the drone’s angle of attack and β is the side-slip angle. Finally, the rotation

Drone Model

Due to the unbalance in mass distribution mentioned in Section 2, the constant inertia matrix computed with respect to the B frame has non-zero terms out of the diagonal:

(16)

The external forces and moments acting on the UAV considered in the model are:

• Aerodynamic Force And Moments;

• Tether force and moments.

10

The gravitational force is computed in the inertial frame as⃗Fg(F ) = m

GT , Where The Constants M And G Are,

respectively, the drone’s mass and the gravity acceleration. Each propeller generates a thrust force,⃗Fp(B)j, and a drag torque in zB direction, Tp(B)j, j = 1 . . . , 4, expressed in body

(17)

Where ωj is the rotational speed the jth propeller, and bj, cj are constant parameters. The propellers’ forces can be linearly combined to obtain the total thrust in zB direction, denoted by U1, and rotational moments around xB (∆Up2),

(18)

where dxBi, dyBi, i = 1, . . . , 4 are the constant position coordinates of each propeller with respect to the origin of B. When deriving the model of the drone’s aerodynamics, we took into account the various application points of the aero- dynamic force and moment vectors generated by the four wings. Moreover, the angle of attack of one wing corresponds to the side-slip angle for the perpendicular one, thus resulting in a rather complex computation of the overall forces and moments acting at each instant on the drone. As a matter of fact, the overall aerodynamic force and moment vectors are nonlinear mappings whose inputs are the drone’s angle of attack and side-slip, the apparent wind speed vector, and the actuators’ position. In the apparent wind reference system (Aw), the aerodynamic force vector can be expressed as:⃗

(19)

where FD, FS, FL are the drag component, lateral component, and lift component, respectively, computed as:

(20)

Where ρ is the air density, S is the wing area (both constant parameters), Wa is the magnitude of the apparent wind seen by the UAV, cL, cD and cS are respectively the lift, drag and side forces’ coefficients, which depend on (α, β). These lumped parameters take into account the specific geometry of the wing. Similarly, the overall aerodynamic moment is denoted as⃗Maero and it is computed in frame Aw as a quadratic function of Wa and a nonlinear function of (α, β), featuring suitable constant aerodynamic coefficients. The apparent wind vector⃗Wa can be computed in the inertial frame



where⃗W(F ) is the absolute wind vector. Regarding the tether, its connection point to the drone is located on a rigid frame attached below the box-wing aircraft (see Figure 8). So, the pulling force also produces a turning moment on the drone. Computing forces with respect to the

B

is the tether direction, computed as (assuming taut tether):⃗

Dtether(B) = Ht

BF⃗p(F ).

(23)

where⃗ptether(B) is the position of the connection point in body reference frame, relative to the center of gravity of the drone. The force magnitude |⃗Ftether| is computed considering non-zero load transfer only with taut tether:

Connection

Figure 8: Sketch of the tether connection. The pulling force generates both forces and moments on the drone’s center of gravity. where ks is the tether stiffness, d is the drone distance from the GS and Ltether is the unrolled tether length, i.e. (d − Ltether) is the tether elongation. Tether stiffness is a parameter that depends on the length of unrolled tether:

(25)

where Fmax is the maximum force that the tether can withstand, ∆Lmax is the corresponding maximum elongation, and ϵmax is the maximum elongation relative to the length. Being Fmax ϵmax a constant parameter for a given tether, the stiffness depends only on its length, and decreases with it according to the hyperbolic equation (25).

To avoid numerical solution problems, in numerical integration routines it is advisable to implement an approximated

(26)

This formulation is actually identical to (25) for positive elongation, and uses a decreasing exponential approximation for negative elongation. This allows one to avoid the non-differentiable point at zero elongation, which is instead present in (24). The resulting curve is shown in Figure 9. Finally, tether drag is accounted for as an incremental term that increases

(27)

where cD is the drone drag coefficient, cD,l is the tether drag coefficient, Sup is the total aerodynamic upper surface, dt and Lt are respectively the tether diameter and the unrolled length. The equation is derived from a momentum balance considering a linear dependency of the apparent speed seen by each infinitesimal segment of tether, and the distance of such a segment from the ground station, see e.g. .

Considering all of the described external forces and moments, the resulting drone model consists of 13 ordinary differential

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