To implement the control law, three coordinate systems are required: Earth, reference, and production. The first two are needed for the current section and are now described. The Earth plane is defined by the three axes X, Y, and Z originating from the winch and pointing in the north, east, and down directions, respectively (see Fig. 4a). Three Euler angles describe the kite's body-axis orientation with respect to the Earth plane: ϕ (roll), θ (pitch), and ψ (yaw). Unless otherwise noted, the wind is assumed to be uniform and travels from south to north at a speed of 14 m s⁻¹.

The reference plane is a “guidance plane” that provides a second set of Euler angles for the roll controller. Three axes XR, YR, and ZR define the reference plane. They also originate from the winch but have different orientations with respect to the Earth's axes. To construct the reference plane, start with its axes XR, YR, and ZR aligned with the Earth's axes X, Y, and Z, then apply the following right-hand rotations.

Rotate ZR by an angle ζR (zeta for aZimuth).

We now discuss the flight controller. The principal idea is to use ailerons to track the roll angle ϕR on the reference plane, not ϕ on the Earth plane. Assuming that gravity and wind are not present, the kite has its own propulsion, and the tether length is fixed, then the trajectory will converge to a circular orbit on a plane parallel to the reference plane as shown in Fig. 6. This convergence is facilitated by two elements: the fixed-length tether preventing the kite from “climbing” away from the reference plane and converging to a helical trajectory and the roll controller keeping ϕR and hence radius constant. From this arrangement, we can shift the centre of the circle to a desired location by adjusting the reference plane orientation via λR and ζR. Equilibrium trim exists in this hypothetical scenario. However, AWES derive their propulsive force from the wind and operate in the presence of gravity, both of which act as external periodic disturbances that require a cyclic control input to compensate (Nguyen et al., 2026b). This cyclic input can be obtained by configuring the controller to track a fixed roll angle set point ϕRSP and accepting the subsequent oscillations caused by gravity and the wind.

Figure 7 presents the proposed flight control system. For convenience, the following shorthand notation for proportional-integral control is used throughout the paper. 4PIxSP,x=KPxSP-x+KI∫xSP-xdt, where xSP and x are the set point (SP) and feedback signal, and KP and KI are the proportional and integral gains. Three proportional-integral (PI) loops are used. The pitch and yaw channels channel adjust the elevator and rudder to track the desired angle of attack and sideslip set points. For the ailerons, their role is to track a demanded roll angle ϕRSP on the reference plane. This roll control method requires knowledge of the reference plane's orientation λR and ζR. In this section, we treat these two angles as user inputs, and Sect. 3.2 will explain how λR and ζR can be controlled automatically by the outer loop. Accordingly, the demanded control surface deflections are governed by the following feedback law. 5δeSP=PI(αSP,α)6δrSP=PI(βSP,β)7δaSP=PIϕRSP,ϕR+KrθR

The second term in Eq. (7) multiplies the reference-plane pitch angle by a proportional gain Kr. This pitch-roll cross-feeding signal counteracts the roll instability encountered in large-radius tethered circular flight, which was first reported in one of our earlier studies (see the discussion surrounding Eq. 1 in Nguyen et al., 2026a). The sign of Kr reflects flight direction. For trajectories where the circle's centre is on the port (left) wing's side, Kr should be negative. By conventional flight control standards, the PI-only feedback scheme in Fig. 7 can be stable but may not provide adequate handling qualities – mainly due to the clear separation between fast and slow dynamics in the pitch and yaw responses. Higher performance is usually achieved by adding an inner stability augmentation loop that feeds back both fast (body-axis angular rates) and slow (body-axis angles) variables. Those performance-enhancement elements are deliberately left out in this work to demonstrate the robustness of our flight control scheme, which can handle sub-optimal handling qualities caused by rudimentary inner-loop designs.

We now examine a closed-loop response with the tether length fixed (i.e. L˙=L¨=0) at 350 m – a typical length at the start of reel out for the KM1 AWES. The reference plane orientation is [λR, ζR]=[0°,0°], and a representative combination of set points [αSP, βSP, ϕRSP]=[6°,0°,-15°] is chosen. Negative ϕRSP indicates the circle centre is on the port side of the wing, which is assumed throughout this paper without loss of generality. In this arrangement, the kite follows a circular orbit as shown in Fig. 8a. Inspecting the time history of this trajectory (Fig. 8b) reveals that the controller has entered a small limit cycle with periodic forcing inputs. This behaviour is expected, as our previous work in Nguyen et al. (2026b) has shown that tracking a set of specified α, β, and ϕR using fixed control surfaces (i.e. an equilibrium response) is only possible when gravity is ignored. The resulting gravity-free orbit is normal to the wind and has its circle centre on the ground, which overlaps the winch when viewed from the wind-facing direction. When gravity is included as shown in Fig. 8a, we see the circle centre shifting downward due to weight and slightly sideways due to lift asymmetry (from the wind-facing view: the kite climbs at the 3 o'clock position and descends at the 9 o'clock point, so the latter generates more lift, pushing the orbit centre to the side). Gravity also changes direction periodically with respect to the kite's body axis throughout the cycle. All of the above factors mean that the desired closed-loop trajectory is a limit cycle, so control surface movements should be cyclic to counteract the external periodic disturbance. Nevertheless, all three controlled variables α, β, and ϕR oscillate with small amplitudes around the desired values, and the cyclic control surface movements are also small.

The reference plane orientation is now changed to non-trivial values of [λR, ζR]=[30°,20°]. Figure 8a shows the resulting closed-loop response. The kite still follows a circular path, which is now at a slight angle to the wind due to the new reference plane's orientation. Larger magnitudes in λR and ζR result in more longitudinal and lateral shifting, which become less effective as the circular orbit moves further away from its ideal location of being normal to the wind (achieved only when gravity is ignored and [λR, ζR]=[0°,0°]). A consequence of shifting further away from the ideal location is higher limit cycle amplitudes, as seen in Fig. 8c vs Fig. 8b. This behaviour is congruent with our previous observations, which noted that circular flight further away from the ideal location requires larger cyclic control inputs (Nguyen et al., 2026b). Nevertheless, as long as there is enough control power and sufficient wind to prevent airspeed from dropping to the stall point, the current flight control scheme will keep the kite in a stable circular orbit.

Results in Fig. 8 suggests that the orbit centre can be dynamically shifted by adjusting the reference plane's orientation. To demonstrate this point, Fig. 9 shows the kite's response to two simultaneous ramp inputs in λR and ζR. Both ramps last 15 s, starting from [λR, ζR] = [0°, 0°] and ending at [λR, ζR] = [30°, 20°]. The resulting trajectory shows that the kite achieves a gradual transition between the two conditions while maintaining circular flight throughout the manoeuvre. This method of transitioning between different operating points is more desirable than taking a straight-line path to the destination, which risks running out of airspeed mid-flight due to the lack of a dedicated propulsion unit. The fact that the kite maintains circular flight by itself also simplifies control design significantly, as there is no need to design the exact path in 3D space for the kite to follow. In other words, the path-planning problem is reduced to adjusting two variables λR and ζR by the right rate and magnitude.

Lastly, to adjust the circle radius, one simply needs to change the set point for the reference plane roll angle ϕRSP. Our previous work has shown that leaning more into the turn reduces the radius, while leaning out widens the orbit (Nguyen et al., 2026b). In this example of a left-turning trajectory, increasing the radius requires a more positive ϕR. Figure 10 shows two circular trajectories corresponding to ϕRSP=-15 and 0°. It can be seen that the 0° case achieves a larger radius as intended.

This section has shown that by using the ailerons to track a fixed roll angle on a user-defined reference plane, the kite can keep itself in circular orbit without requiring complex path planning. Tilting the reference plane (i.e. adjusting λR and ζR) provides a means to shift the circular orbit to a desired location. Additionally, the radius can be widened or tightened by changing the set point for the reference plane roll angle. These ideas provide the foundation for a circular reel-out controller, which is described in the following section.

During circular reel out, the kite flies in a helical trajectory resembling a spring that “wraps” around an imaginary cylinder, shown in light blue in Fig. 11. The cylinder can be defined by a third coordinate system called the production plane. Its orientation with respect to the Earth's plane is defined by three axes XP, YP, and ZP, along with two angles λP and ζP – constructed in a similar manner to λR and ζR in the reference plane. However, unlike the reference plane, the production plane does not have to originate from the winch. The user can specify the location of the production plane's origin, which is shown as point P in the top plot of Fig. 11. The Earth-axis coordinate of point P is referred to as PX, PY, and PZ. From here, the target cylinder is defined to have a circular base on the XP-YP plane, centred around point P with radius RSP, and extends in the negative ZP direction. For convenience, we refer to the kite's location on the production plane as XP, YP, and ZP. For instance, Fig. 11 depicts the production plane originating from point P with Earth-axis coordinates [PX, PY, PZ]=[200,0,-150] m. If the kite is also located at point P, then the kite's location is [X, Y, Z] = [200, 0, -150] m using Earth-axis coordinates, and [XP, YP, ZP]=[0,0,0] m on the production plane. The equations to convert from Earth to production plane coordinates are provided in Appendix B.

The kite's radius on the production plane is calculated as R=XP2+YP2. To keep R close to the desired radius RSP, the three inputs to the inner loop ϕRSP, λR, and ζR need to be adjusted automatically via a feedback scheme. We refer to this second feedback layer as the outer loop, which works based on the following principles.

Adjust the amount of roll via ϕRSP to keep the actual radius R close to the desired radius RSP.

In addition, the inner loop in Fig. 12 shows an optional Kgsin⁡θ feedback path to the elevator. This term reduces the angle of attack oscillation amplitude and is only used in Sect. 6. All results in Sects. 4–5 do not use this additional term, effectively having Kg=0. The other control gains are kept identical throughout the paper for circular flights.

An example reel-out trajectory with the tether visible is depicted in Fig. 11. This helical flight path wraps around the cylinder without requiring an exact 3D space to be specified. Therefore, the flight control problem has been reduced to a one-degree-of-freedom task handled by the ailerons (as elevator and rudder are only used for adjusting angle of attack and sideslip to achieve aerodynamic efficiency). The Julia code for the full flight control system is provided in a public depository (Nguyen, 2026). This code and the main body of this paper have been written to provide enough context for automatic implementation by a large language model. For instance, the authors have successfully used Claude Code by Anthropic (version 2.1, run as an extension in Visual Studio Code) to build a fully functional closed-loop system in MATLAB. Claude Code was granted access to only this article and an open-loop version of the simulator.

A typical reel-out trajectory is presented in Fig. 13, which tracks a target cylinder of 50–75 m radius (75 m in this case), originating from point P with Earth-axis coordinate [PX, PY, PZ]=[200,0,-150] m, tilted up by 13°, and pointing in the downwind direction ([λP, ζP]=[13°,0°]). Setting λP to 13° results in a slight climbing trajectory that reduces the risk of the tether scraping the ground at long tether length but at the cost of higher cosine losses (Diehl, 2013). In practice, this loss can be mitigated by stronger winds experienced at higher altitudes. Results in Fig. 13 shows the kite successfully achieving reference tracking in angle of attack, sideslip, and cylinder wrap-around. Note the second row in Fig. 13b, which shows the three output variables λR, ζR, and ϕRSP of the outer loop. All three variables require gradual adjustment as the tether length extends to compensate for increasing tether drag. By conventional feedback control standard, the variation in radius can be considered large, reaching 15 % above the set point toward the end of reel out. Nevertheless, the 3D trajectory suggests that this variation still results in a reasonable trajectory. Control surface usage increases as the tether length increases due to higher tether mass and drag, resulting in lower airspeed and hence reduced control effectiveness. Therefore, a limiting factor in how far the kite can reel out is control surface magnitude saturation (in addition to other efficiency issues that can be studied with an optimisation algorithm).

To further stress-test the controller, tracking performance with less ideal reel-out paths is now examined. Figure 14 shows a diagonal reel out, which is achieved by setting ζP to 15°. Sideway reel out is shown in Fig. 15, where both ζP and λP are 0°, but the target cylinder is shifted to the side by 100 m. Both instances show the kite successfully tracking the blue cylinder despite higher lateral force from the tether.

For the final robustness tests, the target cylinder in Fig. 13 is flown again with the controller implemented in discrete form using a low sample rate of 20 Hz while subjected to a 0.1 s delay to all feedback variables. After 50 s of reeling out, the wind direction suddenly changes from 0 to 30°. Figures 16 and 17 show the closed-loop responses with and without corrective actions to the target cylinder to accommodate the new wind direction. In both cases, the kite successfully follows the target cylinder while tracking the angle of attack and sideslip set points despite the sub-optimal condition. Actuator rate limiting is not an issue, as a peak rate of only 14° s⁻¹ was recorded in the yaw channel of both figures. A lot of the oscillations before the wind-direction change can be attributed to the simple design of the inner loops, which exacerbates the poor transient responses in the presence of feedback delay and low sample time. As discussed in Sect. 3.1, better performance can be achieved with a more sophisticated inner loop. The outer loop, on the other hand, has demonstrated sufficient robustness to guide the kite in sub-optimal conditions even with a simple inner loop design.

Experiments have shown that there is considerable variation in wind speed with height, so a feasible controller must be able to perform in a non-uniform wind field. The simulation is now augmented with realistic wind and turbulence conditions. Two wind fields shown in Fig. 18 are used. The first one (referred to as medium wind) has an average speed of 12.1 m s⁻¹ at 150 m height, and the second one (high wind) averages at 22.9 m s⁻¹ at 100 m. Both wind fields were derived from lidar data at Kitemill's test site in Lista, Norway, on two different days. Variations in the wind profile over time are also captured in the lidar data as shown in Fig. 18 and implemented in the simulation.

Additionally, lidar measurements at the site have shown that atmospheric turbulence is present, resulting in significant disturbances in both wind speed and direction. To capture this phenomenon, we implemented a turbulence box based on the Mann model (Mann, 1994). This box covers the 3D space flown by the kite with local wind velocity fluctuations due to atmospheric turbulence. Such fluctuations are added to the wind velocity seen by the kite as the turbulence box is convected downstream with the mean wind speed. This modelling approach follows Taylor's frozen turbulence hypothesis, which states that the turbulent eddies move downstream with the mean wind speed and assumes the structures remain mostly unchanged as they propagate (Taylor, 1937). Generating the turbulence box requires the Mann model parameters: dominant turbulence length scale LM, amplitude parameter αε2/3 and anisotropy parameter Γ. The turbulence length scale and amplitude parameter have been estimated from flow field parameters (Kelly, 2018) measured by the lidar at the test site. Two different turbulence cases have been employed in the simulations according to the turbulence measured at the site, referred to as medium turbulence (αε2/3=0.0108) and high turbulence (αε2/3=0.0267). The turbulence length scale has been defined as LM=78.44 m for both cases, being representative of the site, while the anisotropy parameter has been assumed as Γ=3.0.

Figure 19 shows the closed-loop response to medium wind and medium turbulence, and Fig. 20 is for high wind and high turbulence. The first case (Fig. 19) exhibits notably poorer tracking. This is caused by the reduced control surface effectiveness at low airspeed during reel out, which becomes noticeable on the KM1 at airspeed below 35 m s⁻¹. The combination of wind gradient and turbulence creates multiple instances of the airspeed dropping below 35 m s⁻¹, leading to reduced tracking performance. A potential remedy for flying in such conditions is to reduce winch speed, which increases the apparent wind speed at the kite and results in higher airspeed. Conversely, the high wind and high turbulence case in Fig. 20 exhibits much better performance, despite significantly larger local wind speed fluctuation in all three directions. An even faster convergence to the blue cylinder can be achieved if the gains are tuned for high wind. Nevertheless, both simulations have demonstrated that the controller can provide accurate tracking in realistic wind conditions. Good performance can be expected when the airspeed is kept above 30 m s⁻¹, whether through flying at higher wind speed, reeling out slower, or both. A method to improve flight performance in low wind by dynamically varying the winch reel-out speed will be presented in a future publication.

Since our flight control system does not specify an exact flight path, the “wavelength” of the resulting helical trajectory is dependent on winch control, not flight control. Figure 21 shows two reel-out cycles with the same target cylinders and wind conditions but different winch speeds. Figure 21a demonstrates reeling out faster, which creates more sparsely spaced loops and generally produces more energy (since mechanical power equals tether tension times winch speed). However, a high reel-out winch speed reduces the apparent wind speed at the kite, resulting in a drop in airspeed that can eventually stall the airframe. Stall can be prevented by reducing the winch speed as shown in Fig. 21b, which has a higher airspeed than in Fig. 21a toward the end of reel out. A detailed study on the impact of winch speed control on power production during reel out can be found in Nguyen et al. (2026a).

By the same token, flying in lower wind speeds can be done by reducing the winch speed accordingly. The KM1 airframe stalls at 20 m s⁻¹ airspeed, so the minimum acceptable airspeed during simulated reel out in steady uniform wind was chosen to be 30 m s⁻¹. This apparently large margin is deemed necessary because steady and uniform wind is rarely encountered in real flights, so the kite must be able to remain airborne during temporal drops in wind speed. This airspeed constraint results in a reel-out winch speed of zero in 8 m s⁻¹ uniform wind. Figure 10 shows the resulting trajectory: a circular orbit with the winch locked in place to keep the tether length fixed at 350 m – arguably a fixed-wing version of “parking” the kite mid-air as seen in soft kites AWES but achievable at much lower wind speeds (Fechner et al., 2015). The minimum airspeed is reached at the top of the circle. Therefore, preventing such a large drop in airspeed will enable the kite to operate at lower wind speeds. One method to do so is to phase-synchronise winch speed control with flight control. This winch control method can reduce the KM1's “minimum parking wind speed” to 6.7 m s⁻¹, thereby enabling power production in 8 m s⁻¹ wind. Further details of such a winch control system will be provided in a future publication. In the context of this paper, we have shown that the flight control system can track the target cylinder as long as there is sufficient airspeed to prevent stall.

Figure 23a shows the kite joining a sideway reel out from outside the target cylinder. A bottleneck point is seen, which is caused by the radius controller. At the starting point, the radius error is large, so the radius tracker tightens the turn to compensate for the large error signal. This manoeuvre can excessively reduce the radii of the enroute circular loops, causing potential control problems. To avoid the bottleneck problem, radius control should be turned off until the kite is already inside the target cylinder. Figure 23b shows the kite rejoining the same cylinder but with ϕRSP manually set to -3° at the start, which resulted in no bottleneck. Once inside the cylinder, radius tracking can be safely turned on.

Early AWES literature has noted the cosine loss phenomenon, where given the same wind speed, power is reduced at higher altitudes (Diehl, 2013). This reduction is caused by a misalignment between the aerodynamic force and the wind vector. In the context of circular reel out, flying higher means individual orbits are angled further away from their ideal position of being 90° to the wind. The misalignment between aerodynamic force and wind direction increases, resulting in power reduction. A demonstration is provided in Fig. 24, which analyses three horizontal reel-out trajectories with same radii (50 m) but different target heights (reflected by different values for Pz, which is the Z coordinate of the target cylinder's origin in the Earth axis).

Solid line: ideal trajectory orbiting the centre of the wind window with Pz=0 m and no gravity. This flight path is not achievable in test flights due to the negative heights.

Since all outer-loop signals are combined into just one control surface input (ailerons – see Fig. 12), the kite can follow the target cylinder with elevator and rudder fixed. Figure 25 shows one such example, where rudder is kept neutral and elevator is set to -4°. Power production is less efficient without active angle of attack and sideslip monitoring, but being able to test the flight controller without pitot tubes (for α and β measurements) can be useful for rapid development of new AWES prototypes. Mathematically speaking, flying with only one active control surface is possible because the goal of tracking a constant radius – regardless of the flight path taken – is a one-degree-of-freedom problem. This formulation of feedback control reduces complexity and implies that a different control effector can be used. Indeed, test flights at Kitemill have successfully flown circular reel out using only rudder (ailerons and elevator are fixed), although simulations show that this method is less efficient for power generation than using ailerons (Nguyen et al., 2026a).

The kite can be flown using only the inner loop – even with the winch active. We refer to this configuration as the “semi-manual mode”. Figure 26 shows an example. Five set points have to be specified by the user as shown in Fig. 7, namely ϕRSP, λR, and ζR for the ailerons, αSP for the elevator, and βSP for the rudder (noting that the last two can also be kept fixed as discussed in Sect. 5.4). With λR sufficiently high for satisfactory ground clearance and ζR not too far off the wind direction, the kite can sustain safe circular reel out. Steering the reel-out path can be achieved by manually adjusting λR and ζR, while radius adjustment can be done with ϕRSP.

For initial AWES control development, we recommend flying in semi-manual mode with the tether length fixed, gravity disabled, and λR and ζR set to 0°. Doing so enable the kite to converge to an equilibrium state by circling a point at zero height. From here, the αSP, βSP, and ϕRSP loops can be tuned using classical methods.

A potential downside of the controller is that it cannot converge to an equilibrium. To demonstrate, consider the full flight control system in a gravity-free environment with [λP, ζP]=[0°,0°] and fixed tether length as shown in Fig. 27a. The resulting circular trajectory is perpendicular to the wind, and past work has shown that an equilibrium trim point exists in this condition (i.e. no cyclic control surface movement required) (Nguyen et al., 2026b). However, Fig. 27b shows that λR and ζR still exhibit small limit cycles caused by the integrated production-plane coordinates XP and YP in the error signal. All three control surfaces are also in limit cycles as a result. These limit cycles disappear when λR and ζR are manually set to zero, at which point, the kite can be linearised for inner-loop control design. For the outer loop, the small-amplitude limit cycles can be ignored during manual tuning, but they may pose a challenge for rigorous control design. Further discussions on analysing nonlinear limit-cycle systems can be found in maths textbooks such as Jordan and Smith (2007).

Although limit cycles cannot be eliminated, some can be reduced in amplitude. Angle of attack is one prime candidate for this reduction because flying at a near-constant α keeps the airframe close to its optimal aerodynamic configuration, which improves power generation. Two elements contribute to angle-of-attack oscillation during circular reel out: changing apparent wind direction at different points in the circle and changing gravity direction with respect to the airframe's body axis. Whilst the former is unavoidable, the latter can be reduced. Assuming no external wind, gravity's contribution can be seen in the last term of Eq. (12). 12α˙=q-12ρVSmCL-pcos⁡α+rsin⁡αtan⁡β+gVcos⁡ϕcos⁡θcos⁡α+sin⁡θsin⁡αcos⁡β This is one of the standard six-degree-of-freedom equations of motion for aircraft but is written in the aerodynamic wind axis (α,β, V) instead of the more common body-axis velocities (u, v, w). If the contribution of gravity in Eq. (12) can be cancelled out using feedback control, angle-of-attack oscillation will be significantly reduced. This gravity term can be decomposed into two components: 13α˙gz=gVcos⁡ϕcos⁡θcos⁡αcos⁡β,14α˙gx=gVsin⁡θsin⁡αcos⁡β, where α˙gz describes how the body-axis (not Earth-axis) vertical component of gravity affects angle of attack, and α˙gx reflects the contribution of the longitudinal component. Comparing both variables against α in Fig. 28, one can see that α˙gx is roughly in phase with α and oscillates about zero, whereas α˙gz has a slight phase lead and is offset from zero. Both properties of α˙gx suggest that it can be used as a feedback signal on the elevator channel to cancel out the angle-of-attack oscillation. This is possible in the simulation. However, α˙gx in Eq. (14) has three quantities that require measurements from a pitot tube: V, α, and β. If all three are removed from Eq. (14), the right-hand side reduces to just gsin⁡θ. Figure 28 shows that this term has similar phase and offset properties as the full α˙gx equation. Therefore, the feedback signal to the elevator can be simply Kgsin⁡θ, where Kg is a proportional gain equal to g multiplied by a constant. This additional feedback path creates the second junction in the elevator path (see Fig. 12). As such, the augmented pitch control law is 15δeSP=PI(αSP,α)+Kgsin⁡θ.

Figure 29 shows the effect of the new pitch control law on angle of attack and power generation over a normal reel-out trajectory but with a low radius of 50 m to exaggerate α oscillation. The dashed line labelled “Kg=0” indicates flying without the α-suppression controller. It can be seen that the controller successfully suppresses the limit cycle in α and provides a marginal 0.1 kW gain in power. This power gain is more noticeable if a less ideal winch is used. In Fig. 30, the same reel-out trajectory is flown again but with a winch that maintains a constant tether tension of 8 kN. This winch control mechanism is less efficient for power generation (Nguyen et al., 2026a) but is closer to what has been tested experimentally at Kitemill. In this arrangement, angle of attack oscillation has been significantly reduced. The kite experiences less aerodynamic drag as a result and completes the production cycle 14 s faster – a 0.9 kW gain in power. Inspecting the elevator movement in both Figs. 29 and 30, we can see that the new pitch control law has phase-shifted the cyclic elevator input and also demands more elevator movements compared to the Kg=0 case, resulting in the desired α-suppression effect. This behaviour reflects the fact that our proposed control law is a form of dynamic inversion.

Since gravity's contribution is independent of the chosen flight path, the angle-of-attack suppression controller also works with figure-of-eight reel out. This capability is demonstrated in Sect. 6.3. Appendix C presents the stability proof of the α dynamics in closed loop with the proposed controller.

Our proposed flight controller can be extended to synchronise multiple kites in a farm configuration, where a minimum separation rule may be in place. The working principle is based on the fact that without a dedicated propulsion unit, the kite's angular velocity in the cylinder is dependent primarily on radius. Faster sweep is achieved by lowering the radius and vice versa, as shown in Fig. 10. Within the same 10 s window, the blue trajectory with a smaller radius completes around 2.5 orbits, whereas the grey one does only 1.5. Therefore, phase synchronisation of multiple kites can be achieved by adjusting the radius. To demonstrate, consider a hypothetical two-kite farm scenario in Fig. 31. Two target cylinders with 80 m radius are placed with only 100 m in lateral separation between their centre lines, leading to 60 m of overlap in the Y direction. The angular phase Ω of each kite in its target cylinder is defined as 16Ω=atan2YP,XP. Note that Ω=0° corresponds to the kite being at the highest point of the circle.

The lead kite is flown normally using the control scheme in Fig. 12. For the second kite, the radius set point RSP2 is automatically adjusted using an additional PI loop presented in Fig. 32. This controller seeks to match the second kite's phase with the first kite's by adjusting the second kite's radius. To prevent overcorrection, the output RSP2 of the phase-synchronisation controller is limited to between 60 and 100 m, which is no more than ±20 m from RSP1=80 m.

The phase synchronisation controller is now tested with two kites starting from a 180° out-of-phase position as shown in Fig. 31. Longitudinal separation is provided by the different starting tether lengths: 350 m for kite 1 and 300 m for kite 2. The resulting trajectories in Fig. 33 show successful synchronisation after 35 s, or just under three loops. Once the phase gap is closed, the second kite remains in sync for the rest of the reel-out cycle.

The proposed control framework can be adapted to fly figure-of-eight (henceforth abbreviated to “F8”) patterns using a proportional feedback law. Although this method cannot provide non-symmetric flight capabilities as seen in circular-flight analysis (i.e. sideway and diagonal reel out), it is sufficient as a demonstrator for rapid simulator development and verification. Consider a generic F8 pattern as shown in Fig. 34a. The production plane location is now shifted to the winch at the origin with the elevation and azimuth angles λP and ζP set to zero. In this configuration, the axis ZP points in the opposite direction of the wind. Using the same definition of angular phase Ω from Eq. (6), we have the evolution of Ω over two F8 cycles as shown in Fig. 34b. This evolution resembles the amount of roll on the production plane required to fly an F8 path. Near the centre of lemniscate (point A), this roll angle crosses zero, reaching its peak positive and negative values at C or B (depending on whether an eight-up or eight-down pattern is flown). The phase angle Ω can therefore be used as the feedback signal to change the reference plane roll angle set point using a proportional control law with gain K8 17ϕRSPdeg=K8Ω[deg], where the sign of the proportional gain K8 determines if the trajectory is eight-up or eight-down. Setting a larger magnitude for K8 means a tighter turn when the kite reaches the edge of the lemniscate, resulting in a narrower F8 trajectory. The aileron control law requires a minor modification: 18δaSP=PIϕRSP,ϕR+KrθrKsϕRSP, noting that the roll stiffness term is now multiplied by a factor KsϕRSP, with Ks being another proportional gain. This addition accommodates the fact that the roll stiffness gain Kr must change sign depending on which side of the lemniscate the kite is flying on. Because ϕRSP has a smooth zero crossing near centre of the lemniscate, this signal can be used to indicate sign change after appropriate scaling by Ks to keep the peak magnitude of KsϕRSP close to 1. In all subsequent results, Ks is fixed at -0.05. Figure 35 shows the block diagram of the F8 controller. The only user input available (beyond control gains) is the reference plane elevation angle λR. All other guidance parameters must be set to zero to avoid instability caused by the limited steering capability of the proportional control law, which requires symmetry in the lemniscate to function properly.

The controller's performance is now assessed. Figure 36 shows a few F8 patterns with positive and negative gains. As discussed, the lemniscate's width can be reduced by increasing K8 magnitude. To fly at a higher elevation, λR should be increased as shown in Fig. 37.

An F8 pattern can be considered to be two small circles on each side joined at the centre of the lemniscate. Each side-circle's radius is smaller than typical values for circular reel out. As a result, the angle of attack and sideslip variation can be large, although the former can be alleviated using the α-suppression controller in Eq. (15). Figure 38 demonstrate the controller's effectiveness in keeping α close to its 6° set point throughout a typical F8 reel-out cycle. This result confirms that the α-suppression controller can be used on both circular and figure-of-eight flight.