# Global Incremental Flight Control for Agile Maneuvering of a Tailsitter Flying Wing

## Abstract

This paper proposes a novel control law for accurate tracking of agile trajectories using a tailsitter flying wing unmanned aerial vehicle that transitions between vertical takeoff and landing and forward flight. The global control formulation enables maneuvering throughout the flight envelope, including uncoordinated flight with sideslip. Differential flatness of the nonlinear tailsitter dynamics with a simplified aerodynamics model is shown. Using the flatness transform, the proposed controller incorporates tracking of the position reference along with its derivatives velocity, acceleration, and jerk, as well as the yaw reference and yaw rate. The inclusion of jerk and yaw rate references through an angular velocity feedforward term improves tracking of trajectories with fast-changing accelerations. The controller does not depend on extensive aerodynamic modeling but instead uses incremental nonlinear dynamic inversion to compute control updates based on only a local input–output relation, resulting in robustness against discrepancies in the simplified aerodynamics equations. Exact inversion of the nonlinear input–output relation is achieved through the derived flatness transform. The resulting control algorithm is extensively evaluated in flight tests, where it demonstrates accurate trajectory tracking and challenging agile maneuvers, such as sideways flight and aggressive transitions while turning.

Nomenclature | |
---|---|

$\mathit{a}$ | linear coordinate acceleration (in world-fixed frame, unless noted otherwise), ${\mathrm{m}/\mathrm{s}}^{2}$ |

${\mathit{b}}_{x}$, ${\mathit{b}}_{y}$, ${\mathit{b}}_{z}$ | basis vectors of body-fixed frame |

$c$ | propulsion and $\phi $-theory aerodynamic coefficients |

$\overline{c}$ | Buckingham $\pi $ aerodynamic coefficients |

$\mathit{f}$ | force vector (in world-fixed frame, unless noted otherwise), N |

$g$ | gravitational acceleration, ${\mathrm{m}/\mathrm{s}}^{2}$ |

${\mathit{i}}_{x}$, ${\mathit{i}}_{y}$, ${\mathit{i}}_{z}$ | standard basis vectors |

$\mathit{J}$ | vehicle moment of inertia tensor, $\mathrm{kg}\cdot {\mathrm{m}}^{2}$ |

$\mathit{j}$ | jerk in world-fixed frame, ${\mathrm{m}/\mathrm{s}}^{3}$ |

$\mathit{K}$ | diagonal control gain matrix |

$l$ | flap/rotor location |

$m$ | vehicle mass, kg |

$\mathit{m}$ | moment vector in body-fixed frame, $\mathrm{N}\cdot \mathrm{m}$ |

$\mathit{q}$ | throttle input |

${\mathit{R}}_{b}^{i}$ | rotation matrix from frame $b$ to frame $i$ |

$T$ | total thrust, N |

${T}_{i}$ | thrust by rotor $i$, N |

$\mathit{v}$ | velocity (in world-fixed frame, unless noted otherwise), $\mathrm{m}/\mathrm{s}$ |

$\mathit{x}$ | position in world-fixed frame, m |

${\alpha}_{T}$ | thrust angle, rad |

${\alpha}_{0}$ | zero-lift angle of attack, rad |

$\overline{\alpha}$ | sum of zero-lift angle of attack and thrust angle, rad |

${\mathit{\alpha}}_{x}$, ${\mathit{\alpha}}_{y}$, ${\mathit{\alpha}}_{z}$ | basis vectors of zero-lift frame |

$\mathrm{\Delta}T$ | differential thrust, N |

$\delta $ | sum of flap deflections, rad |

${\delta}_{i}$ | deflection of flap $i$, rad |

$\theta $ | vehicle pitch angle, rad |

$\overline{\theta}$ | zero-lift reference frame pitch angle, rad |

${\mu}_{i}$ | torque by rotor $i$, $\mathrm{N}\cdot \mathrm{m}$ |

$\mathit{\xi}$ | normed quaternion attitude vector |

${\mathit{\sigma}}_{\mathrm{ref}}$ | reference trajectory |

$\varphi $ | vehicle roll angle, rad |

$\psi $ | vehicle yaw angle, rad |

${\omega}_{i}$ | angular speed of rotor $i$, $\mathrm{rad}/\mathrm{s}$ |

$\mathbf{\Omega}$ | vehicle angular velocity in body-fixed frame, $\mathrm{rad}/\mathrm{s}$ |

## I. Introduction

A tailsitter flying wing is a tailsitter aircraft without fuselage, tail, and vertical stabilizers or control surfaces. Forgoing these structures simplifies the aerodynamic and mechanical design of the aircraft and potentially improves performance by lowering mass and aerodynamic drag. Due to the lack of vertical aerodynamic surfaces, flying wing aircraft often require active directional stabilization. The fast and relatively powerful brushless motors found on many small UAVs are particularly suitable to fulfill this task through differential thrust. By placing flaps that act as elevator (deflecting collectively) and as aileron (deflecting differentially), i.e., *elevons*, in the rotor wash, the aircraft remains controllable throughout its flight envelope, including static hover conditions. The reduced stability of flying wing aircraft may also result in increased maneuverability. Specifically, the lack of vertical surfaces enables maneuvers such as fast skidding turns and knife edge flight where the wing points in the direction of travel. In general, it permits uncoordinated flight, where the vehicle incurs nonzero lateral velocity.

In this paper, we propose a novel flight control algorithm that is specifically designed for tracking of agile trajectories using the tailsitter flying wing aircraft shown in Fig. 1. The proposed controller uses differential flatness to track the reference position, velocity, acceleration, and jerk (the third derivative of position), as well as yaw angle and yaw rate. It is based on a global formulation, without mode switching or blending, and able to exploit the entire flight envelope, including uncoordinated flight conditions, for agile maneuvering. We derive the controller based on a simplified aerodynamics model and apply incremental nonlinear dynamic inversion (INDI) to achieve accurate trajectory tracking despite model discrepancies.

We use $\phi $ theory to model the aerodynamic force and moment [1]. The method captures dominant contributions over the entire flight envelope, including poststall and uncoordinated flight conditions, in a single, global formulation. The $\phi $-theory model does not suffer from singularities that methods based on the angle of attack and sideslip angle may incur around hover, where these angles are not defined.

Incremental, or sensor-based, nonlinear dynamic inversion is a version of nonlinear dynamic inversion (NDI) control [2] that alleviates the lack of robustness associated with NDI [3

Differential flatness is a property of nonlinear dynamics systems that guarantees the existence of an equivalent controllable linear system [10

Existing flight control designs for tailsitter aircraft are based on various approaches. Blending of separate controllers [21,22], gain scheduling [23,24], or preplanned transition maneuvers [25] can be used to handle the change of dynamics between hover and forward flight. However, when performing agile maneuvering at large angle of attack, the aircraft continuously enters and exits the transition regime, and it is preferable to utilize a controller without blending or switching [26]. A global formulation for trajectory tracking in coordinated flight is proposed by [27]. The controller is based on numerical inversion of a global first-principles model, but does not account for model discrepancies, leading to a systematic pitch tracking error. Wind tunnel testing can be used to improve accuracy of the dynamics model [28,29]. However, building an accurate model from measurements can be a time-consuming process that may need to be repeated if the controller is transferred to a different vehicle.

Robustification can be used to design a performant controller that does not rely on an accurate model of the vehicle dynamics, e.g., by using model-free control [30]. INDI has also been leveraged for robust control of various types of transitioning aircraft, such as tiltrotor aircraft [31,32] and aircraft with dedicated lift rotors [33

The trajectory generation algorithm by [39] utilizes differential flatness of a simplified longitudinal dynamics model to design transitions for a quadrotor biplane. The resulting trajectories are limited to forward motion and consider acceleration but no higher-order derivatives. The algorithm by [29] employs a predesigned constant angular velocity feedforward input to improve transition. Theoretically, this feedforward signal corresponds to the acceleration rate of change, i.e., jerk. However, it is not applied beyond the predesigned transition maneuver.

The main contribution of this paper is a global control design for tracking agile trajectories using a flying wing tailsitter. Our proposed control design is novel in several ways. Firstly, we derive a differential flatness transform for the tailsitter flight dynamics with simplified $\phi $-theory aerodynamics model. Secondly, we present a method to incorporate jerk tracking as an angular velocity feedforward input in tailsitter control design. As far as we are aware, this is the first tailsitter controller that achieves jerk tracking, making it suitable to fly agile trajectories with fast-changing acceleration references. Thirdly, we apply INDI to control a tailsitter aircraft in agile maneuvers that include large flight path angles and uncoordinated flight conditions. Our control design is based on direct nonlinear inversion and, contrary to existing INDI implementations, does not rely on local linearization of the dynamics for inversion. Fourthly, we detail our methodology for analytical and experimental estimation of the $\phi $-theory aerodynamic parameters used by the controller. Finally, we demonstrate the proposed controller in extensive flight experiments reaching up to $8\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$ in an indoor flight space measuring $18\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}\times 8\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}$. The flight experiments include agile maneuvers, such as aggressive transitions while turning, differential thrust turning, and uncoordinated flight.

The paper is structured as follows: In Sec. II, we provide an overview of the flight dynamics and aerodynamics model. We derive the corresponding differential flatness transform in Sec. III. The design of the trajectory-tracking controller is presented in Sec. IV. Section V details our methodology for analytical and experimental estimation of the aerodynamic parameters used by the controller. Extensive experimental flight results are presented in Sec. VI. An initial version of this paper, not including Sec. V, was presented at AIAA Aviation Forum 2021 [40]. Additional aerobatics flight experiments and an experimental comparison that explicitly shows the advantages of incremental control and feedforward references can be found in [41].

## II. Flight Dynamics Model

This section provides a detailed overview of the flight dynamics model employed in our proposed control algorithm. The algorithm, described in Sec. IV, is based on the notion of incremental control action and therefore utilizes the dynamics model solely as a local approximation of the flight dynamics. Unlike conventional inversion-based controllers, it does not require a globally accurate dynamics model.

The model is employed by the incremental controller to predict i) the change in linear acceleration due to increments in attitude and collective thrust, and ii) the change in angular acceleration due to increments in differential thrust and flap deflections. By inversion of these relationships, the control algorithm computes the increments required to attain the commanded changes in linear and angular acceleration. To maintain analytical invertibility and avoid undue complexity, the dynamics model omits any contributions that do not directly affect the aforementioned incremental relationships. For example, the velocity of the aircraft relative to the atmosphere may result in a significant aerodynamic moment. However, when compared to the fast dynamics of the motors and servos controlling the propellers and flaps, this moment contribution is relatively slow-changing, as it relates to the orientation and velocity of the entire vehicle. Consequently, it is assumed to be constant between control updates and does not need to be included in the incremental dynamics model. It is nonetheless accounted for in the control algorithm together with other unmodeled contributions to linear and angular acceleration through inertial measurement feedback, as described in Sec. IV.

### A. Reference Frame Conventions

We define the basis of the body-fixed reference frame, shown in Fig. 2a, as the vectors ${\mathit{b}}_{x}$, which coincides with the chord line and the wing symmetry plane; ${\mathit{b}}_{y}$, which is perpendicular to this symmetry plane; and ${\mathit{b}}_{z}$, which is defined to satisfy the right-hand rule. These vectors form the rotation matrix ${\mathit{R}}_{b}^{i}=[\begin{array}{ccc}{\mathit{b}}_{x}& {\mathit{b}}_{y}& {\mathit{b}}_{z}\end{array}]\in SO(3)$, which gives the transformation from the body-fixed reference frame (indicated by the subscript $b$) to the world-fixed north-east-down (NED) reference frame (indicated by the superscript $i$) consisting of the columns of the identity matrix $[\begin{array}{ccc}{\mathit{i}}_{x}& {\mathit{i}}_{y}& {\mathit{i}}_{z}\end{array}]$. The zero-lift axis system, depicted in Fig. 2b, is obtained by rotating the body-fixed axis system around its negative ${\mathit{b}}_{y}$ axis by the zero-lift angle of attack ${\alpha}_{0}$, which is defined as the angle of attack for which the aircraft produces zero lift. For symmetric airfoils ${\alpha}_{0}=0$, while most cambered airfoils have ${\alpha}_{0}<0$. Finally, the thrust angle ${\alpha}_{T}$ is defined as the angle of the thrust line with regard to the ${\mathit{b}}_{x}\text{-}{\mathit{b}}_{y}$ plane. The motors may be slightly tilted down to obtain a horizontal thrust vector in forward flight with positive angle of attack, leading to ${\alpha}_{T}<0$.

### B. Vehicle Equations of Motion

We apply the Newton–Euler equations to describe the vehicle translational and rotational dynamics. Their translational component is given by

The rotational dynamics are given by

*external*with regard to ${\mathit{f}}_{\mathrm{ext}}$ and ${\mathit{m}}_{\mathrm{ext}}$ refers to unmodeled force and moment contributions, i.e., external to the model, but not necessarily due to physically external influences, such as gusts.

### C. Aerodynamic and Thrust Force and Moment

We employ $\phi $-theory parameterization to model the aerodynamic force and moment [1]. This parameterization has several advantages over standard expressions for aerodynamic coefficients. Firstly, it provides a simple global model that includes dominant contributions over the entire flight envelope, including poststall conditions. Our simplified model relies on only nine scalar aerodynamic coefficients: two for the wing, two for the flaps, two for the propellers, and three for propeller–wing interaction. We use experimental flight data to determine these coefficients, leading to improved model accuracy, as described in Sec. V. Secondly, $\phi $-theory parameterization avoids the singularities that methods based on angle of attack and sideslip angle incur near hover conditions, where these angles are undefined.

We obtain the force in the zero-lift axis system by summing contributions due to thrust, flaps, and wings. The thrust force is given by

The moment is obtained by summation of contributions due to motor thrust and torque, and flap deflections. We ignore the wing moment due to velocity, attitude, and rotation rates, as these state variables are relatively slow-changing compared to the motor speeds and flap deflections. The corresponding contributions are incorporated in the unmodeled moment ${\mathit{m}}_{\mathrm{ext}}$ and accounted for through angular acceleration feedback. The moment due to motor thrust is given by

## III. Differential Flatness

The purpose of our control design is to accurately track the trajectory reference

Differential flatness of a nonlinear dynamics system entails the existence of an equivalent controllable linear system. An important property of flat systems is that their state and input variables can be directly expressed as a function of the flat output and a finite number of its derivatives. This property is of major importance when developing trajectory generation and tracking algorithms, as it allows one to readily obtain state and input trajectories corresponding to an output trajectory, effectively transforming the output tracking problem into a state tracking problem. In practice, the state trajectory can serve as a feedforward control input that enables tracking of higher-order derivatives of the flat output. Inclusion of these feedforward inputs improves trajectory tracking performance by reducing the phase lag in response to rapid changes in the flat output. For details on differential flatness and its applications in general, we refer the reader to [10

In this section, we show differential flatness of the dynamics system described in Sec. II—with some simplifications—by deriving expressions of the state and control inputs as a function of the flat output defined by Eq. (15). The expression for angular velocity is used in our trajectory-tracking controller to obtain a feedforward input based on the reference jerk and yaw rate, and the expressions for attitude and the control inputs are used for linear and angular acceleration control, respectively.

### A. Attitude and Collective Thrust

The position and velocity states are trivially obtained from Eq. (15). We arrive at expressions for the attitude and collective thrust by rewriting Eq. (2) as

- The yaw angle reference $\psi $
- The fact that ${\mathit{i}}_{y}^{\top}{\mathit{f}}^{\alpha}=0$ according to Eq. (5)
- The forces in the vehicle symmetry plane, i.e., ${\mathit{i}}_{x}^{\top}{\mathit{f}}^{\alpha}$ and ${\mathit{i}}_{z}^{\top}{\mathit{f}}^{\alpha}$

In this section, we express the attitude using the $ZXY$, or 3-1-2, Euler angles $\psi $, $\varphi $, and $\theta $. The angle symbols are also used to refer to rotation matrices between intermediate frames; e.g., the rotation matrix ${\mathit{R}}_{i}^{\varphi}$ represents the rotations by $\psi $ and $\varphi $. The $ZXY$ Euler angles form a valid and universal attitude representation that is suitable for the flat transform because each of the angles is uniquely defined by one of the constraints, as shown in Fig. 3. To avoid the well-known issues with Euler angles, we convert the obtained attitude to quaternion format before it is used by the flight controller.

We define the yaw angle $\psi $ as the angle between the world-fixed ${\mathit{i}}_{y}$ axis and the projection of the body-fixed ${\mathit{b}}_{y}$ axis onto the horizontal plane, i.e., the plane perpendicular to ${\mathit{i}}_{z}$. While this angle is undefined if the wingtips are pointing straight up/down (i.e., ${\mathit{i}}_{z}^{\top}{\mathit{b}}_{y}=\pm 1$), we avoid ambiguity by performing the yaw rotation $\psi {\mathit{i}}_{z}$ from the identity rotation (i.e., from ${\mathit{R}}_{i}^{b}=\mathit{I}$).

Next, we satisfy constraint (ii) by rotation around the yawed $x$-axis ${\mathit{R}}_{\psi}^{i}{\mathit{i}}_{x}$ by

To satisfy constraint (iii), we solve Eq. (9) for the collective thrust $T={T}_{1}+{T}_{2}$ and for the rotation angle $\overline{\theta}$ around the vehicle $y$ axis. To find these expressions, we assume that the flap angles are constant and known. We can make this assumption without consequence because of a limitation of the INDI acceleration controller. As described in Sec. IV, we only consider the low-frequency component of the flap deflection when controlling the linear acceleration. This slow-changing component is virtually constant between control updates.

Since the individual thrust values are still undetermined, we assume that the difference between the steady-state flap deflections is negligible so that

Note that we purposely selected the $ZXY$ rotation sequence and the definition of yaw, such that $\varphi $ and $\theta $ do not affect the satisfaction of constraint (i), and $\theta $ does not affect the satisfaction of constraint (ii). Given that the Euler angles are uniquely defined (up to addition of $\pi $) by the yaw reference, Eqs. (17), and (21), this implies that these expressions give the attitude as a function of ${\mathit{\sigma}}_{\mathrm{ref}}$.

### B. Angular Velocity

By taking the derivative of Eq. (17), we obtain

### C. Control Inputs

At this point, we have expressed the state variables as a function of the flat output (15). To obtain an expression for the control inputs, an expression for angular acceleration is obtained as the derivative of Eq. (35) and substituted into Eq. (4) to obtain an expression for $\mathit{m}$ [41]. The angular acceleration can be utilized as a feedforward input corresponding to snap, the fourth derivative of position, and yaw acceleration [18]. However, calculation of this feedforward input significantly complicates the controller expressions, and its benefit may be marginal given how challenging it is to perform accurate feedforward control of the angular acceleration of a fixed-wing aircraft. Hence, we do not incorporate the angular acceleration feedforward input in our control design.

As described in Sec. IV.D, our control design obtains a moment command using INDI. To find the corresponding control inputs, i.e., flap deflections and differential thrust $\mathrm{\Delta}T={T}_{1}-{T}_{2}$, we solve Eq. (14) for these inputs. We find an expression for the differential thrust $\mathrm{\Delta}T$ by equating

## IV. Trajectory-Tracking Control

Our proposed controller is designed to accurately track the dynamic position reference ${\mathit{\sigma}}_{\mathrm{ref}}$. It consists of several components based on various control methodologies, as shown in Fig. 4. Each component employs a global formulation that enables seamless maneuvering throughout the flight envelope. By separating kinematics and dynamics, we are able to employ proportional-derivative (PD) control on the translational and rotational kinematics. Application of the resulting linear and angular acceleration commands is performed using INDI control. INDI enables accurate control by incremental adjustment of control inputs, based on the inverted dynamics model derived in Sec. III. Due to its incremental formulation, the controller only depends on local accuracy of the input–output relation, resulting in favorable robustness against modeling errors and external disturbances. As we will detail in this section, these errors and disturbances (i.e., ${\mathit{f}}_{\mathrm{ext}}$ and ${\mathit{m}}_{\mathrm{ext}}$) are implicitly estimated and corrected for based on the difference between sensor-based and model-based force and moment estimates. By directly incorporating linear and angular acceleration measurements to obtain the sensor-based estimates, the controller is able to quickly and wholly counteract errors and disturbances, without relying on integral action.

Our proposed control design uses a state estimate consisting of position $\mathit{x}$, velocity $\mathit{v}$, and attitude $\mathit{\xi}$. Additionally, linear acceleration ${\mathit{a}}^{b}$ and angular velocity $\mathbf{\Omega}$ measurements in the body-fixed reference frame are obtained from the inertial measurement unit (IMU). Motor speeds $\mathit{\omega}$ and flap deflections $\mathit{\delta}$ are measured and utilized as well.

### A. PD Position and Velocity Control

We use cascaded proportional-derivative (PD) controllers for position and velocity control, resulting in the following expression for the acceleration command:

The gravity-corrected linear acceleration in the world-fixed reference frame is obtained as

### B. INDI Linear Acceleration Control

INDI control incrementally updates the attitude and collective thrust to track the acceleration command ${\mathit{a}}_{c}$. The controller estimates the unmodeled force ${\mathit{f}}_{\mathrm{ext}}$ by comparing the measured acceleration to the expected acceleration according to the vehicle aerodynamics model and motor speed measurements. By rewriting Eq. (2), we obtain

### C. PD Attitude and Angular Rate Control

Given the extensive attitude envelope of the tailsitter vehicle, our attitude controller employs quaternion representation to avoid kinematic singularities. The attitude error quaternion is obtained as

### D. INDI Angular Acceleration Control

The angular acceleration controller has a similar construction as its linear acceleration counterpart described in Sec. IV.B. By rewriting Eq. (4), we obtain the following expression for the unmodeled moment:

### E. Integrative Motor Speed Control

While the flaps are controlled by servos equipped with closed-loop position control, the propellers are driven by brushless motors that cannot directly track motor speed commands. Instead, we use the second-order polynomial $p$ to find the corresponding throttle input. This function was obtained from regression analysis of static test data relating motor speed to throttle input. We add integral action to account for changes due to the fluctuating battery voltage, so that the throttle command that is sent to the motor electronic speed controller (ESC) is obtained as

## V. Estimation of Aerodynamic Parameters

The controller utilizes several mass, geometric, and aerodynamic properties of the vehicle. The vehicle mass $m$, moment of inertia $\mathit{J}$, motor position ${l}_{{T}_{y}}$, motor incidence angle ${\alpha}_{T}$, and flap position ${l}_{{\delta}_{y}}$ can be measured using standard methods or determined based on the design. The propeller thrust coefficient ${c}_{T}$, torque coefficient ${c}_{\mu}$, and throttle response curve $p$ are obtained using static bench tests. For a simple wing without twist, the zero-lift angle of attack ${\alpha}_{0}$ is determined by the airfoil and can thus be obtained from literature or two-dimensional analysis.

What remains are the $\phi $-theory aerodynamic coefficients ${c}_{{L}_{V}}$, ${c}_{{D}_{V}}$, ${c}_{{L}_{T}}$, ${c}_{{D}_{T}}$, ${c}_{{L}_{V}}^{\delta}$, ${c}_{{L}_{T}}^{\delta}$, and ${c}_{{\mu}_{T}}$, as well as the position of the aerodynamic center of the flaps ${l}_{{\delta}_{x}}$. Instead of relying on extensive analysis (e.g., through computational fluid dynamics [CFD] or wind tunnel tests), we initially estimate these parameters using back-of-the-envelope calculations and then refine them using flight test data.

### A. Analytical Model

We approximate the $\phi $-theory coefficients based on conventional Buckingham $\pi $ dimensionless aerodynamic coefficients obtained using Prandtl’s classical lifting-line theory. To avoid confusion, we use $\overline{c}$ to denote the conventional Buckingham $\pi $ coefficients, whereas the $\phi $-theory and propulsion coefficients used by the model described in Sec. II are written without an overbar. The lift slope of a finite wing without twist or sweep is given by

From momentum disc theory, we obtain

Flap deflection changes the lift coefficient in two major ways: change in angle of attack and change in airfoil camber [46]. For simplicity, we only consider the change in angle of attack due to flap deflection, so that for small angles

Wing properties corresponding to the tailsitter aircraft shown in Fig. 1 are given in Table 1. Using these values and $\rho =1.225\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kg}/{\mathrm{m}}^{3}$ for standard sea-level conditions, we obtain the aerodynamic parameters given in the middle column of Table 2. We set the thrust pitch moment coefficient ${c}_{{\mu}_{T}}$ to zero for now, because its analytical estimation is complicated by the propeller–wing interaction. The flap pitch moment effectiveness ${l}_{{\delta}_{x}}$ is set to 0.075 m based on the location of the flap quarter-chord point.

### B. Experimental Data

We found that our control algorithm is able to stabilize the tailsitter in flight when using the analytically obtained aerodynamic parameters from Table 2. The incremental nature of the algorithm enables the controller to compensate for discrepancies in the dynamics model, such as inaccurate aerodynamics parameters. Despite the fact that the parameters were obtained using small-angle-of-attack assumptions, we found that they also enable stable flight at large angles of attack and in static hover. A description of the experimental setup is given in Sec. VI.A.

We use experimental data to improve our estimate of the aerodynamic parameters. Specifically, we use the fact that the force contributions in the zero-lift reference frame ${\mathit{f}}_{T}^{\alpha}$, ${\mathit{f}}_{\delta}^{\alpha}$, and ${\mathit{f}}_{w}^{\alpha}$ are linear in the parameters. This allows us to formulate the parameter estimation problem as a multiple linear regression. For the drag coefficients we have

The force measurements are shown in Fig. 5, along with estimates based on analytical and experimental aerodynamic parameters. Since we set both ${c}_{{D}_{V}}$ and ${c}_{{D}_{T}}$ to zero, the analytical force estimate along ${\mathit{\alpha}}_{x}$, shown in Fig. 5a, consists solely of the direct thrust contribution. At the beginning of the trajectory, where the vehicle is in static hover, this analytical estimate closely matches the measured force, meaning that the presence of the wing has no significant effect on the thrust magnitude and ${c}_{{D}_{T}}$ is indeed close to zero. As the speed increases, the measured forward force increases beyond the analytical thrust force estimate. This may be due to (a combination of) various reasons. The actual thrust may be underestimated because of increasing efficiency of the high-pitch propellers as the blade angle of attack is reduced in forward flight. Another reason may be the lift-induced forward force given by Eq. (58). Possibly, this forward force eclipses the parasitic drag force acting in the opposite direction at the speeds we achieve in the indoor flight space. Regardless of its exact cause, the increasing forward force drives us to set ${c}_{{D}_{V}}$ to zero. When flying at higher speeds, it may be possible to obtain an accurate nonzero estimate of ${c}_{{D}_{V}}$ using the regression equation. We found that, in practice, the discrepancy in the forward force estimate has little influence on controller performance. It is closely aligned with the thrust force and can thus be counteracted quickly and accurately by the incremental controller. The lateral force component, shown in Fig. 5b, is close to zero, as expected. The small bias in its measurement may be due to an imbalance or misalignment. The measured and estimated force component along ${\mathit{\alpha}}_{z}$ is shown in Fig. 5c. It can be seen that the estimate based on the analytical parameters has a significant deviation at increased speeds. The estimate based on the experimental regression parameters closely matches the measurements throughout the trajectory with a coefficient of determination, i.e., ${R}^{2}$ value, of 0.97. The resulting parameters are given in the final column of Table 2 and have similar order of magnitude as the corresponding analytical parameters. The analytical underestimation of ${c}_{{L}_{V}}^{\delta}$ may be because the change of camber due to flap deflection is not considered.

We also attempted to use multiple linear regression on the pitch moment ${\mathit{i}}_{y}^{\top}\mathit{m}$ to obtain experimental estimates for ${c}_{{\mu}_{T}}$ and ${l}_{{\delta}_{x}}$. This approach did not result in consistent parameters and good moment predictions, most likely due to significant moment contributions that are not captured by the simplified aerodynamic model, e.g., due to airspeed, angle of attack, and angular rates. Modeling these contributions is not required for our control design, which relies only on an incremental expression that relates the change in moment to changes in differential thrust and flap deflections. However, their absence in the aerodynamic model makes accurate estimation of ${c}_{{\mu}_{T}}$ and ${l}_{{\delta}_{x}}$ more difficult. We found that the initial estimate of ${l}_{{\delta}_{x}}=0.075\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}$ results in satisfactory pitch control, so we leave this value unchanged. Finally, we set ${c}_{{\mu}_{T}}$ based on the trim flap deflections in static hover, as follows:

The flight test results presented in Sec. VI were obtained using the experimental parameter estimates. We found that these parameters result in improved trajectory-tracking performance when compared to the analytical parameters. As expected, the difference is most significant at increased speeds where the discrepancy between the force estimates is largest. Specifically, we found that altitude oscillations may occur due to the underestimated ${c}_{{L}_{V}}$ and ${c}_{{L}_{V}}^{\delta}$ analytical parameters.

## VI. Experimental Results

In this section, we evaluate controller performance on various trajectories that include challenging flight conditions, such as large accelerations, transition on curved trajectories, and uncoordinated flight. An overview of the trajectories is given in Table 3. Video of the experiments can be found at https://youtu.be/OiD0rQMQ0r0.

### A. Experimental Setup

Experiments were performed in an $18\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}\times 8\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}$ indoor flight space using the tailsitter vehicle shown in Fig. 1 and described in [47]. The vehicle is 3D-printed using Onyx filament with carbon fiber reinforcement. It weighs 0.7 kg and has a wingspan of 55 cm from tip to tip. It is equipped with two T-Motor F40 2400 KV motors and Gemfan Hulkie 5055 propellers with 5.0 inch tip diameter and 5.5 inch advance pitch. The motor speeds are measured using optical encoders at one measurement per rotation, i.e., at about 200 Hz in hover. MKS HV93i micro-servos are used to control the flaps. We obtain the flap deflection measurement as an analog signal by connecting a wire to the potentiometer in the servo.

The flight control algorithm runs onboard on an STM32 microcontroller using custom firmware. The microcontroller has a clock speed of 400 MHz and takes $25\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{s}$ to compute a control update at 32-bit floating point precision. Accelerometer and gyroscope measurements are obtained from an Analog Devices ADIS16475-3 IMU at 2000 Hz, and control updates are performed at the same frequency. Position and attitude measurements are provided by a motion capture tracking system at 360 Hz and sent to the vehicle over Wi-Fi with an average latency of 18 ms. The motion capture data are propagated using IMU measurements to correct for the latency.

We used time-domain flight data to find control gains that attain the fast rise time required for aggressive maneuvering while maintaining small overshoot. Second-order Butterworth filters are applied for signal processing, because this type of filter provides the most uniform sensitivity in the passband. To achieve high-bandwidth disturbance rejection, we set the cutoff frequency of the low-pass filters to the maximum frequency where controller performance is not affected by inertial measurement noise; in our case, 15 Hz. While the motor speed and flap deflection measurements contain relatively much less noise, we apply identical low-pass filters to these signals to avoid phase difference. The transient flap deflections are obtained by subsequent filtering using a second-order Butterworth high-pass filter with cutoff frequency of 1 Hz. We raised this cutoff to increase linear acceleration control authority, until we observed pitch oscillations due to the nonminimum phase acceleration response.

### B. Lemniscate Trajectory

Figure 6 shows experimental results for tracking of a Lemniscate of Bernouilli with a constant speed of $6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$. Throughout the trajectory, ${\psi}_{\mathrm{ref}}$ is set perpendicular to the velocity, leading to coordinated flight. The reference and flown trajectories over eight consecutive laps (of 7 s each) starting at the $y$ extreme are shown in Fig. 6a. It can be seen that the tracking performance is very consistent between laps. Figure 6b shows that the largest position deviation occurs at the end of the circular parts where the vehicle does not fully maintain the reference acceleration of almost 2$g$, as can be seen in Fig. 6d. Over the middle part of the trajectory, the vehicle increases speed to catch up (see Fig. 6c), and the position error is reduced again. Overall, the controller achieves a position tracking error of 17 cm root mean square (RMS) with a maximum error of 33 cm.

Figure 6e shows the commanded and flown attitude. The attitude controller uses quaternion representation, but to ease interpretation the figure uses the $ZXY$ Euler angles described in Sec. III. For this trajectory, $\varphi $ corresponds to the bank angle and reaches almost 1 rad, which matches the acceleration nearing 2$g$ in Fig. 6d. The maximum attitude error occurs during a small period in the turn, where the vehicle incurs a pitch error of 6 deg. Controlling the pitch angle of a flying wing during aggressive maneuvers is generally challenging due to the lack of an elevator, and the pitch deviation is likely the cause of the relatively large position deviation at the exit of the turn. Overall, the controller is able to track the dynamic attitude command well, and it maintains an attitude error of less than 2 deg on each axis during the rest of the trajectory.

### C. Knife Edge Transitioning Flight

Our proposed algorithm is able to control the vehicle in uncoordinated flight conditions where it has significant lateral velocity. In knife edge flight, the wingtip is pointing in the velocity direction, leading to instability due to the location of the center of gravity behind the quarter span point [38]. Figure 7 shows experimental results for a trajectory where ${\psi}_{\mathrm{ref}}$ is set to enforce knife edge flight on one side and coordinated flight on the other side. The results show that our controller is able to stabilize the knife edge flight condition, and that it is able to transition between knife edge and coordinated flight while at the same time performing a 1.6$g$ turn.

One lap of the oval trajectory takes 6.25 s to complete at a constant speed of $6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$. Referring to the view from above in Fig. 7a, the trajectory is flown in clockwise direction with the straight at the top in knife edge configuration and the straight at the bottom in coordinated flight. During each turn, the yaw reference is rotated by $\pi /2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{rad}$ to enforce the switch between configurations. Consequently, the coordinated flight segment is flown in inverted orientation every other lap. Figure 7a shows the reference and flown position over eight successive laps, and Figs. 7b and 7c show data corresponding to two laps starting and ending at the transition from knife edge flight to coordinated flight in inverted orientation. It can be seen that the reference is followed accurately during both regular and inverted coordinated flight. Even while performing the transition from knife edge to coordinated flight, the controller is able to track the turning trajectory. At the transition from coordinated flight to knife edge, we see that the trajectory is shifted during transitions from inverted orientation. This leads to a position tracking error of at most 48 cm at the start of the knife edge segment. During transition from regular coordinated flight to knife edge, a position error of at most 25 cm is incurred. Flight during the knife edge segment is consistent and stable, and the yaw reference is tracked within 3 deg. Over the whole trajectory, the controller achieves tracking of the position reference within 20 cm RMS and the yaw reference within 1.7 deg RMS.

### D. Circular Trajectory

Figure 8 shows experimental results for tracking of a circular trajectory reference with a 3.5 m radius at a speed of $8.1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$. The left column of figures corresponds to coordinated flight where the ${\mathit{b}}_{y}$ axis is perpendicular to the circle tangent, and the right column corresponds to knife edge flight where the wing tip points along the tangent of the circle. Position tracking performance is very similar between both flights. In both cases, the flown trajectory has a slightly smaller radius than the reference, reducing the flight speed to $7.8\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$. The RMS position tracking error is 15 cm for both coordinated and knife edge flight. The aircraft reaches a continuous acceleration of 2.1$g$. In coordinated flight, this requires a bank angle of 63 deg. In knife edge flight, the aircraft is rolled 14 deg toward the direction of travel to compensate for drag and pitched over by 152 deg to provide thrust toward the circle center. As shown in Fig. 9, maintaining this state requires flap deflections of 20 deg and rotor speeds of over $2000\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{rad}/\mathrm{s}$ in knife edge flight. In coordinated flight, the aircraft exploits the lift force to achieve circular flight more efficiently and requires rotor speeds of $1800\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{rad}/\mathrm{s}$ with nearly unchanged flap deflections. In contrast to the trajectory described in Sec. VI.C, the flap deflections are almost constant during the circular trajectory. Hence, there are no transient accelerations caused by the flaps that are not considered in the position controller, and very accurate trajectory tracking is achieved in knife edge flight. This shows that our controller is not only able to stabilize the knife edge flight condition, but also attains accurate trajectory tracking in knife edge flight at speeds close to $8\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$. Figure 9 also shows the commanded and achieved angular accelerations. To improve the readability of these figures, we have shifted the commanded angular acceleration curves by up to 70 ms to account for the response time of the servos and rotors. It can be seen that the controller is able to follow the commands with decent accuracy despite using only a simplistic model of the aerodynamic moment.

### E. Transitions

Figure 10 shows experimental results for transitions between static hover and coordinated flight at $8\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$ on a circular trajectory with 3.5 m radius. Each transition takes 3 s at a constant tangential acceleration of $2.7\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/{\mathrm{s}}^{2}$ and is completed in 12 m. The pitch angle varies over a range of 64 deg. While transitioning from and to hover, the controller tracks the circle trajectory with respectively 10 cm and 15 cm RMS, and 15 cm and 24 cm maximum position error. These maneuvers demonstrate that the controller is capable of performing aggressive transitions while simultaneously tracking turns with significant acceleration.

To evaluate the benefits of the feedforward input based on jerk and yaw rate, we also attempted to fly the same transitions without the angular velocity reference, i.e., with ${\mathbf{\Omega}}_{\mathrm{ref}}=\mathbf{0}$ in Eq. (49). We found that the controller is still able to perform 3 s transitions to speeds up to $3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$. However, if the target speed is increased and the corresponding tangential acceleration exceeds $1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/{\mathrm{s}}^{2}$, the absence of the feedforward term causes large deviations from the reference trajectory to the point where the vehicle cannot be stabilized anymore. This confirms the benefit of jerk and yaw rate tracking when flying aggressive maneuvers. Intuitively, the feedforward input enables the control to anticipate future accelerations by regulating not only the forces acting on the vehicle but also their temporal derivative. Additional experimental results that show the effect of feedforward control can be found in [41].

### F. Differential Thrust Turning

Since the controller is not restricted to coordinated flight, it can perform turns without banking. The tailsitter aircraft is particularly suitable for quick turns using yaw, because of the absence of any vertical surfaces and the availability of relatively powerful motors. Figure 11 shows a fast turn that is executed using differential thrust. The reference trajectory, entered in coordinated flight at $7\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$, changes direction without deviating from a straight line. The controller responds with large differential thrust and flap deflections. At the onset of the turn, both flaps are almost fully deflected in opposite directions and the motors produce a differential thrust of 6.1 N. This causes the aircraft to turn at a maximum rate of $650\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}/\mathrm{s}$ and point in the opposite direction within half a second, while remaining within 1 m of the position reference.

## VII. Conclusions

This paper introduced a novel control design aimed at tracking agile trajectories using a tailsitter flying wing. It was shown that combining $\phi $ theory with robust incremental control enables effective global control that does not depend on extensive aerodynamic modeling. In fact, the proposed controller achieves accurate tracking of challenging maneuvers using only coarse aerodynamic parameters, estimated from limited flight test data. Hence, it has great potential for application to aircraft with (partially) unknown aerodynamic properties, allowing quick iteration of vehicle and control designs.

The current controller has some limitations that may be addressed to improve suitability for widespread deployment. Firstly, the aerodynamic model assumes that there are no wind or gusts. It only considers the vehicle attitude and velocity with regard to the world-fixed reference frame, not considering the aerodynamic angles with regard to the freestream velocity. While incremental control counteracts the resulting unmodeled force and moment to some extent, it would likely be beneficial to account for wind and gusts in the control design [48]. Additionally, the definition of the yaw reference in the world-fixed frame does not readily transform into a coordinated flight constraint in windy conditions. Thus, it may be practical to add an option to switch to a sideslip angle reference instead of the yaw reference. Secondly, the controller assumes that the reference trajectory is dynamically feasible. If this assumption is violated by the trajectory planner, saturation and clipping of the control inputs may occur, potentially resulting in loss of control. An extension of the proposed controller that optimizes control allocation in case of saturation may be used to prioritize controlled and stable flight, while temporarily sacrificing trajectory tracking accuracy.

Future work may include further exploration of the flight envelope in outdoor flights, including increased speeds that are unattainable in the constrained indoor space. This may require consideration of wind, as described above, as well as reconfiguration of the test platform to add satellite navigation and airspeed sensors. An extension to more conventional unmanned aircraft, i.e., with fuselage, tail, and vertical stabilizer, may also be considered. Feedforward jerk and yaw rate references could significantly improve tracking of agile maneuvers on these aircraft, similarly to our findings for the tailsitter flying wing.

## Acknowledgments

We thank Murat Bronz and John Aleman for the design, fabrication, and assembly of the aircraft, and we thank Lukas Lao Beyer and Nadya Balabanska for the implementation of a flight dynamics simulation tool. This work was supported by the Army Research Office through grant W911NF1910322.

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## Tables

Property | Symbol | Value |
---|---|---|

Airfoil lift slope, ${\mathrm{rad}}^{-1}$ | ${\overline{c}}_{{l}_{\alpha}}$ | 5.73 |

Wing surface area, ${\mathrm{m}}^{2}$ | $S$ | 0.070 |

Aspect ratio | $AR$ | 4.3 |

Taper ratio | $\lambda $ | 0.59 |

Flap chord ratio | ${c}_{f}/c$ | 0.5 |

Propeller diameter, m | $D$ | 0.13 |

Thrust angle, rad | ${\alpha}_{T}$ | $-5\frac{\pi}{180}$ |

Monoplane circulation coefficient | $\tau $ | 0.14 |

Span efficiency factor | $e$ | 0.97 |

Parameter | Analytical | Experimental |
---|---|---|

${c}_{{L}_{V}}$, $\mathrm{kg}/\mathrm{m}$ | 0.17 | 0.29 |

${c}_{{D}_{V}}$, $\mathrm{kg}/\mathrm{m}$ | 0 | 0 |

${c}_{{L}_{T}}$ | 3.4 | 2.23 |

${c}_{{D}_{T}}$ | 0 | 0 |

${c}_{{L}_{V}}^{\delta}$, $\mathrm{kg}/\mathrm{m}$ | 0.041 | 0.18 |

${c}_{{L}_{T}}^{\delta}$ | 1.7 | 1.25 |

Trajectory | max $\Vert \mathit{v}\Vert $, $\mathrm{m}/\mathrm{s}$ | max $\Vert \mathit{a}-{\mathit{i}}_{z}g\Vert $, $g$ | max $\Vert \mathbf{\Omega}\Vert $, $\mathrm{deg}/\mathrm{s}$ | max $\Vert {\mathit{x}}_{\mathrm{ref}}-\mathit{x}\Vert $, m | RMS $\Vert {\mathit{x}}_{\mathrm{ref}}-\mathit{x}\Vert $, m |
---|---|---|---|---|---|

Lemniscate | 6.2 | 1.7 | 133 | 0.17 | 0.33 |

Knife edge transitioning | 6.6 | 2.1 | 277 | 0.20 | 0.48 |

Circular (coordinated) | 7.8 | 2.2 | 154 | 0.15 | 0.18 |

Circular (knife edge) | 7.9 | 2.1 | 140 | 0.15 | 0.17 |

Transition from hover | 8.3 | 2.1 | 146 | 0.10 | 0.15 |

Transition to hover | 8.0 | 2.2 | 164 | 0.15 | 0.24 |

Differential thrust turn | 7.3 | 2.2 | 661 | 0.63 | 0.96 |