Analysis and Enhancement of Steady Climb Performance with Control Input Redundancy for a Dual-Propulsion VTOL UAV

Abstract

Dual-propulsion UAVs employ separate rotors for rotary-wing and fixed-wing modes to achieve VTOL (vertical take-off and landing) and high-speed cruise. This paper analyzes steady climb in high-speed flight by utilizing the redundant rotary-wing rotors. We develop the models of aerodynamic forces and thrust forces of a dual-propulsion UAV to obtain its longitudinal dynamic model. The maneuverability of the UAV is analyzed based on the dynamic model to reveal whether a steady climb at a given climb angle is possible within allowable thrust forces. The analytical results show that the climb flight performance of the UAV can be enhanced by utilizing the redundant control inputs during high-speed flights. Flight experiments not only demonstrate that several climb flight states predicted by the analysis are successfully realized, but also that steady climb at a higher climb angle, unattainable in conventional fixed-wing mode, is made possible by simultaneously using the rotors for rotary-wing mode. The enhanced flight performance would increase the number of missions that the UAV can accomplish.

1. Introduction

Vertical take-off and landing (VTOL) UAVs (Unmanned Aerial Vehicles) can switch the flight mode between rotary-wing mode and fixed-wing mode. They can travel long distances at high speed in fixed-wing mode, and they can take off and land vertically in rotary-wing mode. This flight capability allows VTOL UAVs to achieve tasks that require long-range and high-speed flight without limiting takeoff/landing locations, such as inspections and deliveries in extensive mountainous areas [1,2].

The flight performance of VTOL UAVs in high-speed cruise would conform to that of fixed-wing aircraft such as turning radius and climb rate. However, the actuators mounted for rotary-wing mode or flight mode transition are not used in the conventional fixed-wing mode of VTOL UAVs. The flight performance of VTOL UAVs would be degraded due to the weight of the actuators compared to fixed-wing aircraft.

Conversely, the actuators for rotary-wing mode or flight mode transition provide redundancy in control inputs to VTOL UAVs during high-speed cruise. For example, dual-propulsion VTOL UAVs are equipped with rotors for rotary-wing mode and rotors for fixed-wing mode independently. In high-speed flight, the rotors for rotary-wing mode do not need to be used but can be used as redundant control inputs. Tilt-rotor UAVs can switch their flight mode by tilting actively some rotors mounted on them. In the conventional fixed-wing mode, the tilt angle is fixed at a predetermined angle, but it can be chosen within a certain range.

The redundancy in control inputs during high-speed flight can enhance the flight performance of VTOL UAVs. Recently, unified control approaches have become popular, in which a controller is built without dividing the flight envelope into rotary-wing mode and fixed-wing mode to utilize the redundant control inputs to their full extent [3]. In [4,5,6], flight controllers for tilt-rotor and tilt-wing UAVs have been designed based on dynamic inversion and model predictive control. However, the flight performance of VTOL UAVs with control input redundancy has not been investigated sufficiently. A comprehensive understanding of flight performance would be necessary to clearly identify applicable missions for VTOL UAVs.

The possible acceleration of a UAV can be estimated from its dynamic model and actuator limitations. The set of possible acceleration is obtained from the maximum and minimum values of thrust forces and control surface angles, and it is called maneuverability in this paper. For multicopter UAVs, their maneuverability has been investigated in the literature, where the aerodynamic forces acting on the UAVs other than thrust forces are ignored because they are small enough [7,8]. However, the maneuverability of VTOL UAVs depends largely on the aerodynamic forces, and its analysis is more challenging than in the case of multicopter UAVs. We have analyzed the maneuverability of a dual-propulsion UAV to achieve a back-transition with large deceleration in [9]. In particular, steep turns during high-speed flight can be performed due to the maneuverability enhanced by redundant control inputs, which has been shown for a dual-propulsion UAV in [10].

In this paper, we examine the steady climb performance of a dual-propulsion VTOL UAV based on its maneuverability. To analyze the maneuverability with control input redundancy, we develop a longitudinal dynamic model of the UAV combining an aerodynamic force model and a thrust force model. The maneuverability obtained from the dynamic model can reveal whether steady flight at a given climb angle is possible within the thrust force limitations. Utilizing the rotors mounted for rotary-wing mode even during high-speed flights allows for various flight states to achieve a given climb angle, as well as enhancing the steady climb performance. The results obtained from the maneuverability analysis are verified by flight experiments. The main contributions of this paper are as follows: (1) We present a method to analyze the steady climb performance of a dual-propulsion VTOL UAV based on its dynamic model. (2) We demonstrate through numerical analysis and flight experiments that the steady climb state during high-speed flight can be changed by using the rotors mounted for rotary-wing mode. (3) We show experimentally that the dual-propulsion VTOL UAV with redundant control inputs can climb steadily at higher climb angles than in the conventional fixed-wing mode flight.

The remainder of this paper is organized as follows. Section 2 presents the mathematical model of the dual-propulsion UAV. Section 3 analyzes the maneuverability to clarify the steady climb states that can be achieved. Section 4 presents the results of the flight experiments and discusses them in relation to the maneuverability analysis. Section 5 concludes the paper.

2. Mathematical Model of Dual-Propulsion VTOL

2.1. Dual-Propulsion VTOL

This paper deals with a dual-propulsion VTOL UAV, which is developed by Aerosense Inc. in 2020 and called “Aerobo Wing” (Figure 1). The UAV can fly for 40 min and cover a distance of $50\mathrm{km}$ with payloads such as cameras for surveying. It weighs $9.45\mathrm{kg}$, has a wingspan of $2.15\mathrm{m}$, and a total length of $1.24\mathrm{m}$.

The UAV is equipped with two types of rotors: four rotors for rotary-wing mode and one rotor for fixed-wing mode. The four rotors for rotary-wing mode are placed symmetrically around the origin of the fuselage, and their thrust axes are inclined slightly from the upward direction of fuselage to gain yaw torque. The UAV can hover like a multicopter UAV when in rotary-wing mode. We refer to the four rotors as MC rotors in this paper and each individual rotor as rotors 1, 2, 3, and 4, as shown in Figure 2. The one rotor for fixed-wing mode is located at the aft end of fuselage, and its thrust axis passes through the mass center of the UAV. In the conventional fixed-wing mode, the UAV is propelled by the force from the rotor without using the MC rotors. We call the rotor FW rotor in this paper, and we also refer to it as rotor 5. The left and right wings have control surfaces called elevons at their trailing edges. They are used to control the attitude of the UAV in fixed-wing mode.

In typical operations, the UAV performs a vertical take-off in rotary-wing mode, followed by a transition to fixed-wing flight at altitude. Upon mission completion, the aircraft executes a back-transition to rotary-wing mode for a vertical landing. During the transitions between rotary-wing mode and fixed-wing mode, all actuators are utilized to control the UAV’s position and attitude while changing the airspeed. In this paper, we investigate the improvement in climb performance achieved by employing the MC rotors even during high-speed flight.

2.2. Equation of Motion

In this paper, we derive a longitudinal dynamic model under the following assumption.

Assumption 1.

The wind is constant and horizontal.

Denoting the wind vector by ${\mathit{V}}_w$, the airspeed vector of the UAV ${\mathit{V}}_a$ can be represented as

$$\begin{array}{cc}\hfill {\mathit{V}}_a& ={\mathit{V}}_g-{\mathit{V}}_w,\hfill \end{array}$$

(1)

where ${\mathit{V}}_g$ is the ground speed vector of the UAV. As shown in Figure 3, we introduce three coordinate frames: the inertial frame composed of the $x_e$ and $z_e$ axes, the body frame composed of the $x_b$ and $z_b$ axes, and the stability frame composed of the $x_s$ and $z_s$ axes where the $x_s$ axis is aligned with the airspeed vector ${\mathit{V}}_a$. The origin of each coordinate frame is located at the center of gravity. The mounting angle of rotor i relative to the $x_b$ axis is denoted as ${\theta }_{Ti}$, and the elevon angle is expressed as ${\delta }_e$.

We consider the airspeed $V_a$ and the climb angle ${\gamma }_a$ relative to the air as follows:

$$\begin{array}{cc}\hfill V_a& =\parallel {\mathit{V}}_a\parallel ,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\gamma }_a& =-{sin}^{-1}({\mathit{e}}_z·{\mathit{V}}_a/V_a),\hfill \end{array}$$

(3)

where $e_z$ is a unit vector along the $z_e$ axis. The acceleration of the UAV is expressed in the stability frame as $\mathit{a}={[a_{x_s},a_{z_s}]}^{\top }$, where the components parallel to and perpendicular to the airspeed vector ${\mathit{V}}_a$ are denoted as $a_{x_s}$ and $a_{z_s}$, respectively. Using $V_a$ and ${\gamma }_a$, the acceleration of the UAV at stability frame can be represented as

$$\begin{array}{cc}\hfill a_{x_s}& ={\dot{V}}_a,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill a_{z_s}& =-V_a{\dot{\gamma }}_a.\hfill \end{array}$$

(5)

The force vector composed of the thrust forces of rotors and the aerodynamic forces acting on the wings is also expressed in the stability frame as $\mathit{F}={[F_{x_s},F_{z_s}]}^{\top }$. Then, the equation of motion for the UAV can be written as

$$\begin{array}{cc}\hfill m{\dot{V}}_a& =F_{x_s}-mgsin{\gamma }_a,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill m{V}_a{\dot{\gamma }}_a& =-F_{z_s}-mgcos{\gamma }_a,\hfill \end{array}$$

(7)

where m is the mass of the UAV and g is the gravitational acceleration.

We assume that the force $\mathit{F}$ can be divided into the force from the rotor thrusts ${\mathit{F}}_t$ and the aerodynamic force ${\mathit{F}}_a$ as follows:

$$\mathit{F}(V_a,\alpha ,\mathit{T})={\mathit{F}}_t(\alpha ,\mathit{T})+{\mathit{F}}_a(V_a,\alpha ,\mathit{T}).$$

(8)

Denoting the thrust force of rotor i by $T_i$ and summarizing them in a vector form as $\mathit{T}={[T_1,T_2,T_3,T_4,T_5]}^{\top }$, ${\mathit{F}}_t$ can be calculated as

$${\mathit{F}}_t=\mathit{H}\mathit{T},$$

(9)

$$\begin{array}{cc}\hfill \mathit{H}\left(\alpha \right)& =\left[\begin{array}{cccc}cos(\alpha +{\theta }_{T_1})& cos(\alpha +{\theta }_{T_2})& \cdots & cos(\alpha +{\theta }_{T_5})\\ -sin(\alpha +{\theta }_{T_1})& -sin(\alpha +{\theta }_{T_2})& \cdots & -sin(\alpha +{\theta }_{T_5})\end{array}\right],\hfill \end{array}$$

(10)

where $\alpha$ is the angle of attack (AoA) that is defined as the angle between the $x_b$ axis and the airspeed vector ${\mathit{V}}_a$ as shown in Figure 3. The aerodynamic force ${\mathit{F}}_a$ is represented using lift L and drag D as follows:

$$\begin{array}{cc}\hfill {\mathit{F}}_a& ={[-D,-L]}^{\top }.\hfill \end{array}$$

(11)

It should be noted that the equation of rotational motion about the pitch axis is omitted because it is not used in this paper under the following assumption.

Assumption 2.

The pitch moment of the UAV can be easily balanced using the elevon angle ${\delta }_e$ without changing the lift L and drag D.

Although, strictly speaking, L and D would be affected by ${\delta }_e$, we assume that the variations in L and D due to ${\delta }_e$ is negligible. From Assumption 2, we will consider only the force balances in (6) and (7) to analyze the steady state in Section 3. In the following two subsections, we describe the details of modeling $T_i$, L, and D.

2.3. Rotor Thrust Model

In this subsection, we construct a thrust model for each rotor by combining a propeller aerodynamic model and a brushless motor electrical model. Although the following model is developed for each rotor i, we omit the subscript i for notational simplicity. The constructed model expresses the thrust force as a function of throttle input, battery voltage, and inflow velocity. The inflow velocity $V_{\mathrm{p}}$ is defined as the component of the airspeed vector along the thrust axis of each rotor:

$$\begin{array}{c}\hfill [V_{\mathrm{p}1}\cdots V_{\mathrm{p}5}]=\left[V_{a}0\right]\mathit{H}\left(\alpha \right).\end{array}$$

(12)

Since the propeller is directly attached to the motor shaft, the propeller and motor share the same rotational speed. The thrust model is defined as the thrust generated under steady-state conditions, where the rotational speed remains constant, implying that the propeller torque and the motor torque are balanced.

In the propeller model, the propeller thrust T and torque Q are determined by the rotational speed $\Omega$ and the inflow velocity $V_{\mathrm{p}}$ in the following forms:

$$\begin{array}{cc}\hfill T& =f_T(V_{\mathrm{p}},\Omega ),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill Q& =f_Q(V_{\mathrm{p}},\Omega ),\hfill \end{array}$$

(14)

where $f_T$ and $f_Q$ are constructed by building an aerodynamic model of a rotor blade, which is consistent with the propeller performance data reported in [11], using blade element theory. Thrust and torque values are computed at discrete points with a $1\mathrm{m}/\mathrm{s}$ step in inflow velocity $V_{\mathrm{p}}$ and a $1000\mathrm{RPM}$ step in rotational speed $\Omega$. Bilinear interpolation is applied to these computed data points to obtain continuous functions $f_T$ and $f_Q$.

In the brushless motor model, the throttle input, battery voltage, angular velocity, and torque are related. It is assumed that the motor’s electrical time constant $T_c$ is sufficiently smaller than the propeller’s response time, and that the motor torque is proportional to the average internal current $\overline{i}$ as follows:

$$\begin{array}{c}\hfill Q=K_t\overline{i},\end{array}$$

(15)

where $K_t$ is the motor torque constant. The average current $\overline{i}$ is determined in different ways depending on the type of ESC (electronic speed controller) used for each motor.

For ESCs employing complementary switching [12], the average motor current is given by

$$\begin{array}{c}\hfill \overline{i}(u,E_{\mathrm{B}},\Omega )={\displaystyle \frac{E_{\mathrm{B}}u-K_e\Omega }{R}},\end{array}$$

(16)

where u is the throttle input, $E_{\mathrm{B}}$ is the battery voltage, $K_e$ is the back EMF constant, and R is the motor resistance. If the ESC uses non-complementary switching, the motor current cannot become negative during the “OFF” phase of the PWM modulation inside the ESC, and the average current given by Equation (16) is no longer valid at low throttle inputs u. The condition under which Equation (16) breaks down is expressed as

$$\begin{array}{cc}\hfill T_P(1-u)& >T_0:=T_{c}log\left({\displaystyle \frac{E_{\mathrm{B}}}{K_e\Omega }}(1-e^{-{\displaystyle \frac{T_P}{T_c}}u})+e^{-{\displaystyle \frac{T_P}{T_c}}u}\right),\hfill \end{array}$$

(17)

where $T_P$ is the PWM period. The left-hand side of Equation (17) represents the PWM OFF duration, and the right-hand side indicates the time at which the current reaches zero during the OFF phase. When the condition (17) is satisfied, the average current is given by

$$\begin{array}{c}\hfill \overline{i}(u,E_{\mathrm{B}},\Omega )={\displaystyle \frac{E_{\mathrm{B}}-K_e\Omega }{R}}u-{\displaystyle \frac{T_0}{T_P}}{\displaystyle \frac{K_e\Omega }{R}}.\end{array}$$

(18)

To satisfy the torque balance condition (15) between the motor and the propeller, the propeller angular velocity $\Omega$ is determined numerically for each operating point $(V_{\mathrm{p}},u)$ at a given $E_{\mathrm{B}}$. For a cost function $J(\Omega ;V_{\mathrm{p}},u):=|f_Q(V_{\mathrm{p}},\Omega )-K_t\overline{i}(u,E_{\mathrm{B}},\Omega )|$, its minimization is performed using a one-dimensional optimization algorithm. The final rotor thrust model $T(V_{\mathrm{p}},u,E_{\mathrm{B}})$ is then obtained by substituting the resulting $\Omega$ into the propeller thrust model $f_T(V_{\mathrm{p}},\Omega )$:

$$\begin{array}{c}\hfill T=T(V_{\mathrm{p}},u;E_{\mathrm{B}})=f_T(V_{\mathrm{p}},{\Omega }^{*}(V_{\mathrm{p}},u;E_{\mathrm{B}})),\end{array}$$

(19)

where ${\Omega }^{*}$ denotes the angular velocity that satisfies the torque balance.

The ESCs for MC rotors are assumed to employ complementary switching, while that for FW rotor utilizes non-complementary switching. To develop the rotor model $T(V_{\mathrm{p}},u;E_{\mathrm{B}})$ in Equation (19) numerically for both MC and FW rotors, the battery voltage $E_{\mathrm{B}}$ is set to $23.0\mathrm{V}$. The throttle command u is varied from 0 to 1 in $0.01$ increments. The inflow velocity $V_{\mathrm{p}}$ varies from $-8\mathrm{m}/\mathrm{s}$ to $8\mathrm{m}/\mathrm{s}$ for the MC rotors and from $0\mathrm{m}/\mathrm{s}$ to $30\mathrm{m}/\mathrm{s}$ for the FW rotor, both in $1\mathrm{m}/\mathrm{s}$ increments. At each operating point $(V_{\mathrm{p}},u)$, the thrust is computed, and a rotor thrust model is constructed via bilinear interpolation with respect to u and $V_{\mathrm{p}}$. Figure 4 shows the rotor thrust surfaces constructed by the above procedure: (a) for the fixed-wing rotor, and (b) for the rotary-wing rotor. The surfaces are projected onto the $(V_{\mathrm{p}},T)$ plane, and the contours with respect to u are plotted at increments of 0.1. Furthermore, the maximum and minimum thrusts, denoted by $T_{max}$ and $T_{min}$, are computed as

$$\begin{array}{cc}\hfill T_{max}(V_{\mathrm{p}};E_{\mathrm{B}})& =\underset{u}{max}T(V_{\mathrm{p}},u;E_{\mathrm{B}});\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill T_{min}(V_{\mathrm{p}};E_{\mathrm{B}})& =\underset{u}{min}T(V_{\mathrm{p}},u;E_{\mathrm{B}}).\hfill \end{array}$$

(21)

These values represent the upper and lower bounds of achievable thrust at a given inflow speed $V_{\mathrm{p}}$, and are illustrated by solid curves in Figure 4.

2.4. Aerodynamic Force Model

In this paper, we represent the lift L and drag D as follows:

$$\begin{array}{cc}\hfill L(V_a,\alpha )& ={\displaystyle \frac{1}{2}}\rho V_a^{2}S{C}_L\left(\alpha \right),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill D(V_a,\alpha ,\mathit{T})& ={\displaystyle \frac{1}{2}}\rho V_a^{2}S{C}_D\left(\alpha \right)+\sqrt{\sum _{i=1}^4\left|T_i\right|}·V_a{C}_{DT},\hfill \end{array}$$

(23)

where $\rho$ is the air density, and S is the area of aircraft. The coefficients $C_L$ and $C_D$ denote the aerodynamic lift and drag coefficients, respectively, and are modeled as follows:

$$\begin{array}{cc}\hfill C_L\left(\alpha \right)& =C_{L\alpha }(\alpha -{\alpha }_0),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill C_D\left(\alpha \right)& =C_{D0}+\kappa C_L^2.\hfill \end{array}$$

(25)

This aerodynamic model is applicable only in the non-stall regime. Flight experiments in the fixed-wing mode indicate that stable flight without stall is achievable for angles of attack up to $10\mathrm{deg}$. Considering CFD (Computational Fluid Dynamics) results as well, the non-stall region is assumed to be $-15\le \alpha \le 10\mathrm{deg}$, and the analysis is therefore limited to this range.

The second term on the right-hand side of (23) is introduced to account for the rotor drag experienced by MC rotors during forward flight. This drag is assumed to be linearly proportional to the airspeed and dependent on the generated thrust [13,14]. If the rotor is surrounded by the airframe, the drag called momentum drag would be applied [15,16], which can also be represented in the form of the second term. The coefficient $C_{DT}$ is chosen as a constant in this study. The values of these coefficients are summarized in

Table 1. Parameters of an aerodynamic model.

${\mathit{\alpha }}_{\mathbf{0}}\mathbf{\left[}\mathbf{deg}\mathbf{\right]}$	${\mathit{C}}_{\mathit{L}\mathit{\alpha }}\mathbf{[}\mathbf{1}\mathbf{/}\mathbf{rad}\mathbf{]}$	${\mathit{C}}_{\mathit{D}\mathbf{0}}\mathbf{[}\mathbf{-}\mathbf{]}$	$\mathit{\kappa }\mathbf{[}\mathbf{-}\mathbf{]}$	${\mathit{C}}_{\mathit{DT}}\mathbf{[}\sqrt{\mathbf{N}}\mathbf{s}/\mathbf{m}\mathbf{]}$
$-8.6$	$2.2$	$0.020$	$0.14$	$0.09$

.

3. Maneuverability Analysis for Steady Climb

This section analyzes the maneuverability of UAV in steady climb using the dynamic model derived in the previous section, under the following assupmtions.

Assumption 3.

Thrust forces $T_i$ for MC rotors are approximated to be along the $z_b$ axis, that is, ${\theta }_{T_i}=\pi /2\mathrm{rad}(i=1,\dots ,4)$.

Assumption 4.

All the MC rotors generate the same thrust force, that is, $T_1=T_2=T_3=T_4$.

Assumption 5.

The battery voltage $E_{\mathrm{B}}$ is constant.

From Assumption 3, the total force caused by MC rotors can be regarded as a single upward force $T^{\mathrm{mc}}$ represented as follows:

$$\begin{array}{cc}\hfill T^{\mathrm{mc}}& =\sum _{i=1}^4{T}_i.\hfill \end{array}$$

(26)

Assumptions 3 and 4 are introduced, because we analyze the steady climb state while ignoring the pitch moment balance under Assumption 2. This allows us to clearly demonstrate the impact of the total upward force $T^{\mathrm{mc}}$ in (26) on the climb rate. Although the maximum thrust $T_{max}$ of each rotor depends on $E_{\mathrm{B}}$ from (20), it is determined only by $V_{\mathrm{p}}$ under Assumption 5. In the numerical analysis, $E_{\mathrm{B}}$ is set to $23.0\mathrm{V}$.

Denoting the FW rotor thrust $T_5$ as $T^{\mathrm{fw}}$ and using the total MC rotor thrust $T^{\mathrm{mc}}$, (9) and (10) can be replaced as follows:

$$\begin{array}{cc}\hfill {\mathit{F}}_t& ={\mathit{H}}^{\prime }{\mathit{T}}^{\prime },\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\mathit{H}}^{\prime }\left(\alpha \right)& =\left[\begin{array}{cc}cos(\alpha +\pi /2)& cos(\alpha +{\theta }_{T_5})\\ -sin(\alpha +\pi /2)& -sin(\alpha +{\theta }_{T_5})\end{array}\right],{\mathit{T}}^{\prime }=\left[\begin{array}{c}{T}^{\mathrm{mc}}\\ T^{\mathrm{fw}}\end{array}\right],\hfill \end{array}$$

(28)

where ${\theta }_{T_5}$ is $\pi /30\mathrm{rad}$ for the UAV in Figure 1.

3.1. Force Set and Maneuverability

We denote a throttle for FW rotor by $u^{\mathrm{fw}}\in [0,1]$ and a throttle for MC rotor by $u^{\mathrm{mc}}\in [0,1]$. From the force model (8), we can define the set of force vector $\mathit{F}$ that can be generated at a given airspeed $V_a$ and a given AoA $\alpha$ as follows:

$$\begin{array}{c}\hfill \mathcal{F}(V_a,\alpha )=\left\{(F_{x_s},F_{z_s})\right|(u^{\mathrm{fw}},u^{\mathrm{mc}})\in [0,1]\times [0,1]\}.\end{array}$$

(29)

Using (6) and (7), the set of possible acceleration for the UAV, which is called maneuverability, is also defined based on $\mathcal{F}$ as follows:

$$\begin{array}{c}\hfill \mathcal{A}(V_a,\alpha ;{\gamma }_a)=\left\{(a_{x_s},a_{z_s})\right|\mathit{F}\in \mathcal{F}\}.\end{array}$$

(30)

Note that both of $\mathcal{F}$ and $\mathcal{A}$ are represented in the stability frame. The set $\mathcal{A}$ depends on the climb angle ${\gamma }_a$ in addition to $(V_a,\alpha )$, because it includes the acceleration due to the gravity, ${[-gsin{\gamma }_a,gcos{\gamma }_a]}^{\top }$.

Numerical examples of the force set $\mathcal{F}$ are presented in Figure 5. In Figure 5a, $\mathcal{F}$ is shown for $V_a=10\mathrm{m}/\mathrm{s}$, with the AoA $\alpha$ varying from $-15\mathrm{deg}$ to $10\mathrm{deg}$ in increments of $5\mathrm{deg}$. The lower-left point of each set that corresponds to the smallest ${\mathit{F}}_t$ shifts with $\alpha$ because the aerodynamic force ${\mathit{F}}_a$ strongly depends on $\alpha$. The overall orientation of the set also changes with $\alpha$, as the direction of ${\mathit{F}}_t$ rotates according to Equation (27). Figure 5b illustrates $\mathcal{F}$ for $V_a=20\mathrm{m}/\mathrm{s}$ under the same conditions. Since ${\mathit{F}}_a$ increases with $V_a$ as given by Equations (22) and (23), the sets are more widely separated at a higher airspeed.

In a steady climb, the force components $F_{x_s}$ and $F_{z_s}$ must satisfy the following equations, derived from Equations (6) and (7).

$$\begin{array}{cc}\hfill F_{x_s}& =mgsin{\gamma }_a,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill F_{z_s}& =-mgcos{\gamma }_a.\hfill \end{array}$$

(32)

In Figure 5, a dotted circle centered at the origin with a radius of $mg$ is depicted. If a point $(F_{x_s},F_{z_s})$ in the set $\mathcal{F}(V_a,\alpha )$ lies on the circle, then in the flight state $(V_a,\alpha )$, steady climb is possible at a climb angle given by ${\gamma }_a=-{tan}^{-1}(F_{x_s}/F_{z_s})$. The thick arc on the circle represents the portion of the circle that overlaps with $\mathcal{F}$ within the AoA range $-15\le \alpha \le 10\mathrm{deg}$. This arc illustrates the range of climb angles achievable at the given airspeed $V_a$ when all rotors are simultaneously used. From Figure 5a,b, the results indicate that steady flight is possible with ${\gamma }_a\in [-13.0,38.0]\mathrm{deg}$ at $V_a=10\mathrm{m}/\mathrm{s}$, as well as with ${\gamma }_a\in [-10.3,26.9]\mathrm{deg}$ at $V_a=20\mathrm{m}/\mathrm{s}$.

Figure 6 shows numerical examples of $\mathcal{A}$ for $V_a=20\mathrm{m}/\mathrm{s}$ at ${\gamma }_a=14\mathrm{deg}$, which correspond to $\mathcal{F}$ in Figure 5b. Each set in Figure 5b is scaled by $1/m$ and translated by ${[-gsin{\gamma }_a,gcos{\gamma }_a]}^{\top }$. In this figure, when the origin is included in the set $\mathcal{A}(V_a,\alpha ;{\gamma }_a)$, the flight state $(V_a,\alpha )$ can achieve a steady climb at the climb angle ${\gamma }_a$. For steady climbing flight in the fixed-wing mode, the lower boundary of each maneuverability set must lie above the dotted circle. The case shown in magenta, corresponding to $\alpha =2.1\mathrm{deg}$, marginally satisfies this fixed-wing mode condition with nearly the maximum FW rotor thrust, indicating that the climb performance limit in the fixed-wing mode is reached at approximately this climb angle for $V_a=20\mathrm{m}/\mathrm{s}$. In contrast, the maneuverability sets for $\alpha =0,-5,-10,-15\mathrm{deg}$ include the origin, implying that steady flight is feasible for all of these flight states while using MC rotors simultaneously. This observation suggests that rotor redundancy provides additional degrees of freedom in selecting feasible flight states.

3.2. Steady Climb State

In this subsection, the flight state $(V_a,\alpha )$ where a steady climb is possible for a given ${\gamma }_a$ is revealed based on the maneuverability. When ${\gamma }_a$ is given, the thrust force ${\mathit{T}}^{\prime }$ required for the steady climb can be calculated from (8), (27), (31), and (32) as ${\mathit{T}}^{\prime }={\mathit{T}}_d(V_a,\alpha ;{\gamma }_a)\equiv {[T_d^{\mathrm{fw}},T_d^{\mathrm{mc}}]}^{\top }$. The steady climb at the angle ${\gamma }_a$ is possible, if and only if the two components of ${\mathit{T}}_d$ fall within the thrust range of each rotor, which means that the maneuverability $\mathcal{A}(V_a,\alpha ;{\gamma }_a)$ contains the origin.

The condition for each component of ${\mathit{T}}_d$ leads to the following two sets in the $(V_a,\alpha )$ space:

$${\mathcal{B}}^{\mathrm{fw}}({\gamma }_a;E_{\mathrm{B}})=\left\{(V_a,\alpha )\right|T_{\min }^{\mathrm{fw}}(V_a,\alpha ;E_{\mathrm{B}})\le T_d^{\mathrm{fw}}(V_a,\alpha ;{\gamma }_a)\le T_{\max }^{\mathrm{fw}}(V_a,\alpha ;E_{\mathrm{B}})\},$$

(33)

$${\mathcal{B}}^{\mathrm{mc}}({\gamma }_a;E_{\mathrm{B}})=\left\{(V_a,\alpha )\right|T_{\min }^{\mathrm{mc}}(V_a,\alpha ;E_{\mathrm{B}})\le T_d^{\mathrm{mc}}(V_a,\alpha ;{\gamma }_a)\le T_{\max }^{\mathrm{mc}}(V_a,\alpha ;E_{\mathrm{B}})\}.$$

(34)

Note that $T_{\min }^{\mathrm{fw}}$, $T_{\max }^{\mathrm{fw}}$, $T_{\min }^{\mathrm{mc}}$, and $T_{\max }^{\mathrm{mc}}$ are determined by $(V_a,\alpha )$ from (12), (20) and (21). The intersection of these two sets represents a set of flight states that allow for steady climb at the angle ${\gamma }_a$ and can be denoted as

$$\begin{array}{cc}\hfill \mathcal{B}({\gamma }_a;E_{\mathrm{B}})& ={\mathcal{B}}^{\mathrm{fw}}({\gamma }_a;E_{\mathrm{B}})\cap {\mathcal{B}}^{\mathrm{mc}}({\gamma }_a;E_{\mathrm{B}}).\hfill \end{array}$$

(35)

Figure 7 presents numerical examples of $\mathcal{B}$ for ${\gamma }_a=10\mathrm{and}20\mathrm{deg}$, plotted over the range $0\le V_a\le 30\mathrm{m}/\mathrm{s}$ and $-15\le \alpha \le 10\mathrm{deg}$. The set $\mathcal{B}$ is illustrated as the green-shaded region. The solid and dotted lines within $\mathcal{B}$ indicate the corresponding flight states when the throttle values $u^{\mathrm{fw}}$ and $u^{\mathrm{mc}}$ are varied in increments of 0.1. The throttle values $u^{\mathrm{fw}}$ and $u^{\mathrm{mc}}$ are obtained from $T_d^{\mathrm{fw}}(V_a,\alpha ;{\gamma }_a)$ and $T_d^{\mathrm{mc}}(V_a,\alpha ;{\gamma }_a)$, respectively, using the thrust model described in Section 2.3.

Figure 7a shows the flight states $(V_a,\alpha )$ that allow steady climb at a climb angle of ${\gamma }_a=10\mathrm{deg}$. The fixed-wing mode corresponds to the upper red contour, where the MC throttle $u^{\mathrm{mc}}$ is zero. Since the feasible flight states are limited to those lying on this contour, steady climb in fixed-wing mode is possible only for $15.0\le V_a\le 24.0\mathrm{m}/\mathrm{s}$. At these states, as shown in Figure 8a, the aerodynamic lift L primarily balances gravity. By contrast, the rotary-wing mode corresponds to the lower-left blue contour, along which the FW throttle $u^{\mathrm{fw}}$ is zero. For $-15\le \alpha \le 10\mathrm{deg}$, this contour indicates that the allowable $V_a$ for steady climb at ${\gamma }_a=10\mathrm{deg}$ lies between 0 and $11.3\mathrm{m}/\mathrm{s}$. At these states, as depicted in Figure 8b, the UAV advances by tilting its nose downward to directly thrust forward. In this configuration, the wings generate negative aerodynamic lift, and thus the thrust $T_d^{\mathrm{mc}}$ required for steady climb in rotary-wing mode is considerably greater than that required for hovering.

Compared to the limited flight states in the conventional two flight modes, set $\mathcal{B}$ shows that a wide range of flight states are possible for the climb angle ${\gamma }_a$ by allowing the simultaneous use of FW rotor and MC rotors. In set $\mathcal{B}$, the possible airspeed range is expanded beyond $30.0\mathrm{m}/\mathrm{s}$, and various AoA $\alpha$ can be allowed at a given airspeed in the range. One of the flight states in $\mathcal{B}$ is illustrated in Figure 8c, where both the aerodynamic force L and the MC thrust $T^{\mathrm{mc}}$ support the gravity at a smaller AoA than in fixed-wing mode.

Figure 7b presents a numerical example of $\mathcal{B}$ for ${\gamma }_a=20\mathrm{deg}$. The red solid line with $u^{\mathrm{mc}}=0$ (corresponding to the fixed-wing mode) does not appear in the figure. That is, a steady climb at ${\gamma }_a=20\mathrm{deg}$ cannot be realized in the fixed-wing mode. Large thrust $T_d^{\mathrm{mc}}$ is demanded for all flight states $(V_a,\alpha )$ in set $\mathcal{B}$ to achieve steady flight.

The results shown in Figure 7 indicate that, by employing both the FW rotor and the MC rotors simultaneously, it becomes possible to select flight states within the green-shaded region of $\mathcal{B}$. The simultaneous use of both rotors is expected to enable the UAV to continue climbing steeper slopes while maintaining the cruising airspeed of the fixed-wing mode, as well as to allow operation in stronger winds compared to the conventional rotary-wing mode.

It should be noted that, although all the flight states in $\mathcal{B}$ are possible, their flight efficiency may be especially low for a smaler AoA $\alpha$. Note that a negative aerodynamic lift would occur for $\alpha <{\alpha }_0=-8.6\mathrm{deg}$ based on the lift coefficient model. Although we do not analyze the efficiency of each flight state in this study, it would be included in our future work.

4. Experimental Demonstration

In this section, the analytical results obtained in Section 3.2 are validated through flight experiments. First, flight tests in both the rotary-wing and fixed-wing modes are conducted to verify that the model is consistent with the actual steady states obtained in each mode. Then, climbing flight tests are performed using the MC rotors and the FW rotor simultaneously to demonstrate that the use of the MC rotors during high-speed climbing flight alters the flight state in set $\mathcal{B}$ and even increases the achievable climb angle.

4.1. Steady Flight Performance in Conventional Flight Mode

The results of steady climb analysis in Section 3.2 depend on the mathematical model of the UAV presented in Section 2. In this subsection, level flight in the conventional rotary-wing and fixed-wing modes, together with a steep climb flight in the fixed-wing mode, are performed to verify that the model captures steady flight conditions.

In the flight experiments, the flight log containing sensor measurements, estimated vehicle states, and control signals was recorded from the onboard flight controller at a sampling rate of approximately $40\mathrm{Hz}$. The position, velocity, and attitude data are provided as EKF estimates based on GNSS, IMU, and magnetometer measurements. Although the flight log included measured airspeed values, these were found to contain both bias and scale errors. Therefore, under Assumption 1, the wind vector ${\mathit{V}}_w$ was estimated based on the ground velocity vector, vehicle attitude, and measured airspeed during each flight trial, and the relationship in Equation (1) was then used to estimate the airspeed vector. Since the airspeed sensor is not incorporated into the EKF, this adjustment does not influence the estimated states. Consequently, we obtained the estimated or logged values of $({\widehat{V}}_a,{\widehat{\gamma }}_a,\widehat{\alpha })$ during each flight from the flight log.

Figure 9 presents the experimental results of level flights at ${\gamma }_a=0\mathrm{deg}$. The logged values of $({\widehat{V}}_a,\widehat{\alpha })$ for several flight trials are shown as red, cyan, and green “+” markers. The center of each “+” marker indicates the mean value for each trial, while the horizontal and vertical extents represent the standard deviations of ${\widehat{V}}_a$ and $\widehat{\alpha }$, respectively. The background illustrates the feasible flight region $\mathcal{B}$ evaluated at $E_{\mathrm{B}}=23.0\mathrm{V}$. The red and cyan “+” markers correspond to experiments conducted in the rotary-wing mode, with two trials performed at a low airspeed (approximately $7\mathrm{m}/\mathrm{s}$) and a high airspeed (approximately $15\mathrm{m}/\mathrm{s}$), respectively. The green “+” markers represent experiments conducted in the fixed-wing mode at an airspeed of approximately $20\mathrm{m}/\mathrm{s}$.

Figure 10 shows the results of a steep climb flight conducted in the fixed-wing mode. Empirically, the UAV can maintain steady flight in the fixed-wing mode at speeds around $20\mathrm{m}/\mathrm{s}$ for climb angles up to approximately 14 degrees, which corresponds to the climb performance limit indicated in Figure 6. In actual flight, since the vehicle is subject to disturbances and operates under feedback control, the performance limit is not always achieved. In this experiment, a climb angle of ${\gamma }_a=14\mathrm{deg}$ was attained at speeds of approximately $V_a=16.9\mathrm{and}18.4\mathrm{m}/\mathrm{s}$ at a battery voltage of about $E_{\mathrm{B}}=23.0\mathrm{V}$. The flight states for two flight trials are illustrated by the green markers with the background of the corresponding feasible flight region $\mathcal{B}$.

These results in Figure 9 and Figure 10 are consistent with the analytical predictions. Possible sources of error include wind disturbances and unmodeled effects such as rotor–airflow interactions. Nevertheless, the dynamic model presented in Section 2 demonstrates sufficient capability to represent steady flight conditions.

4.2. Experimental Results for Steady Climb with MC Rotors

This subsection demonstrates, through flight experiments, that the assistance of the MC rotors during high-speed climb flight alters the steady-flight state as predicted by the analysis in Section 3.2, and even enhances the climb capability of the UAV. In the flight experiments, both the MC rotors and the FW rotor had to be activated during high-speed cruise, and the flight needed to be stabilized to maintain a steady climb against disturbances. To accomplish this, we employed the fixed-wing mode controller of the onboard autopilot for feedback stabilization, while simultaneously applying constant throttle commands to the MC rotors. It should be noted that the objective of these experiments is not to realize the target state exactly, but rather to obtain steady climb states in which the analytical results can be validated. To this end, we present the steady states that were obtained in the experiments by employing the fixed-wing mode controller without modification. Although fluctuations in the flight states and the FW throttle $u^{\mathrm{fw}}$ were inevitable due to disturbances, the logged values of $({\widehat{V}}_a,{\widehat{\gamma }}_a,\widehat{\alpha })$ are presented in a manner similar to that in Section 4.1.

First, we present the experimental results of climbing flight at a climb angle of ${\gamma }_a=10\mathrm{deg}$ with $u^{\mathrm{mc}}=0\mathrm{and}0.3$. Figure 11 shows the results obtained with a commanded airspeed of $20\mathrm{m}/\mathrm{s}$, where the cases of $u^{\mathrm{mc}}=0\mathrm{and}0.3$ are indicated by green and cyan markers, respectively. By comparing these two cases, the AoA $\alpha$ with $u^{\mathrm{mc}}=0.3$ is clearly lower than that with $u^{\mathrm{mc}}=0$ by about $5\mathrm{deg}$. This indicates that the flight state was switched from (a) to (c) in Figure 8 by using MC rotors. The green markers lie nearly on the red solid curve corresponding to the fixed-wing mode flight, indicating good agreement with the commanded value of $u^{\mathrm{mc}}=0.0$. In contrast, the cyan markers are located between the contours for $u^{\mathrm{mc}}=0.3\mathrm{and}0.4$, suggesting a slight discrepancy from the commanded value of $u^{\mathrm{mc}}=0.3$. This discrepancy may be attributed to a higher actual thrust of the MC rotors or to a higher effective lift than predicted by the model in Section 2.

Next, we present the experimental results obtained with a larger thrust of MC rotors at $u^{\mathrm{mc}}=0.75$. When the vehicle was commanded to follow a trajectory with a climb angle of $20\mathrm{deg}$ at a target airspeed of $20\mathrm{m}/\mathrm{s}$, stable climbing flight was observed at a climb angle of ${\widehat{\gamma }}_a=18\mathrm{deg}$. In Figure 12, the airspeed ${\widehat{V}}_a$ and angle of attack $\widehat{\alpha }$ of the steady-flight states obtained from two trials are represented by the cyan markers. As discussed in Section 4.1, this climb angle cannot be achieved using the FW rotor alone. This is also clearly illustrated in Figure 12, where the red solid line corresponding to the fixed-wing mode does not appear. These results therefore validate that the use of the MC rotors enables an improvement in climb performance. It should be noted that, during this experiment, the battery current increased, resulting in a reduced battery voltage of approximately $20.2\mathrm{V}$ due to the internal resistance of the battery. The background of the figure illustrates the feasible flight region $\mathcal{B}$ evaluated at a climb angle of ${\gamma }_a=18\mathrm{deg}$ and a battery voltage of $20.2\mathrm{V}$. The experimental results are therefore in reasonable agreement with the conditions corresponding to $u^{\mathrm{mc}}=0.75$.

4.3. Discussion

From an energy perspective, a steady climb corresponds to an increase in the aircraft’s mechanical energy by raising its potential energy while maintaining its kinetic energy. By moving the gravity term in Equation (6) to the left-hand side and multiplying both sides by the airspeed $V_a$, the following equation is obtained:

$$m{V}_a{\dot{V}}_a+mg{V}_{a}sin{\gamma }_a=V_a{F}_{x_s}.$$

(36)

The two terms on the left-hand side of Equation (36) respectively represent the rates of change of kinetic and potential energy; thus, the entire left-hand side expresses the rate of change of mechanical energy, while the right-hand side represents the rate of energy supply. Accordingly, in order to increase the climb angle ${\gamma }_a$ while maintaining a constant airspeed $V_a$, additional energy input is required, which can be achieved by increasing the forward force $F_{x_s}$. For a conventional fixed-wing aircraft, increasing the upper limit of the forward force $F_{x_s}$ requires either selecting a propulsion system with a larger maximum thrust $T_{\max }^{\mathrm{fw}}$ or designing the airframe to reduce aerodynamic drag D. In contrast, when employing MC rotors in combination, the angle of attack $\alpha$ can be chosen more flexibly, allowing for an increase in forward force even at negative angles of attack. Furthermore, by selecting an appropriate $\alpha$, the induced drag can be reduced, thereby increasing the resultant $F_{x_s}$. It should be noted that the thrust generated by the MC rotors inherently produces rotor drag, represented by the second term on the right-hand side of Equation (23). To achieve a larger climb angle, it is therefore desirable to design the airframe in a manner that minimizes the drag associated with the MC rotor thrust. The analytical and experimental results presented in this study clearly demonstrate that, despite the presence of rotor drag, the employment of MC rotors effectively enhances steady climb performance.

Although the aerodynamic lift L does not directly contribute to the rate of energy supply, it is associated with the force balance (32) that must be satisfied to maintain the climb angle, thereby constraining the feasible flight states $(V_a,\alpha )$. As the airspeed increases, the negative lift generated at smaller angles of attack becomes more pronounced. When the MC rotor thrust is insufficient to counteract this negative lift, additional power is required to maintain the climb angle. This condition contradicts the earlier requirement that the angle of attack must be reduced to negative values to obtain additional energy supply from the MC rotor thrusts. Furthermore, the feasible flight region could be expanded by adjusting the wing incidence angle or by introducing mechanisms that mitigate the generation of negative lift, thereby allowing more effective utilization of the MC rotor thrusts.

The analytical results of this study can be extended to wind-resistance performance evaluation, because the flight states during steady flight without wind are equivalent to those during hovering in a corresponding steady wind. Although the UAV in the conventional rotary-wing mode hovers at a nose-down pitch angle under headwind conditions, as shown in Figure 8b, this flight state degrades wind-resistance performance. Since the aerodynamic lift acts downward in the flight state, the required thrust forces tend to suffer from their saturation. The analytical results presented in this paper indicate that employing the FW rotor in conjunction can significantly improve the wind-resistance capability during hovering by enabling a change in the pitch angle [17]. Furthermore, although this paper focuses on steady climb, the use of MC rotors has the potential to enhance unsteady maneuvers such as rapid ascent and deceleration. In addition to longitudinal maneuvers, lateral motion performance could also be improved, as suggested by numerical simulations for steep turns in [10].

The assumptions adopted in this study were introduced to simplify the analysis. Although the experimental results presented in this section are consistent with the analysis even under these assumptions, deviations from them may arise in practical implementations. In Assumption 2, the changes in lift and drag caused by the elevons are neglected; however, these effects cannot be considered entirely negligible, and the effectiveness of the elevons may vary with the angle of attack. Assumption 4 does not account for the fact that the thrusts generated by the fore and aft rotors may differ due to interference with the airflow around the UAV’s body even when identical command inputs are applied. To clarify these effects and incorporate them into the analysis, the development of more detailed models based on wind tunnel experiments or CFD analyses would be necessary, which is beyond the scope of this paper.

It should be emphasized that the experimental results in Figure 12 clearly demonstrate that steady climb flight can be achieved at climb angles that are unattainable in a conventional fixed-wing mode by utilizing the MC rotors. The ability to fly at larger climb angles expands the range of feasible missions for VTOL UAVs, as they are often required to perform missions such as surveying along mountain slopes. VTOL UAVs can exhibit significantly enhanced maneuverability when their onboard actuators are fully exploited.

For future work, it is necessary to establish control strategies for utilizing MC rotors during high-speed flight. Although this study represents the feasible steady-flight states as a set of possible flight conditions, it does not determine which state should be selected as the optimal one. The use of MC rotors may introduce a trade-off between increased maneuverability and reduced flight range due to higher power consumption. Future studies should examine the energy efficiency associated with the use of MC rotors and develop control strategies that enable efficient flight while considering both mission requirements and vehicle performance.

5. Conclusions

This paper investigated the potential of a dual-propulsion VTOL UAV to achieve greater climb angles during high-speed flight by utilizing the MC rotors, based on the maneuverability analysis. The analysis examined the range of total force that the UAV can generate as the sum of thrust and aerodynamic forces. By mapping the total force to an acceleration space through the equations of motion, the feasibility of steady flight at a given climb angle was evaluated. Employing all the rotors of the dual-propulsion UAV enables various flight states and enhances the maximum attainable climb angle. The experimental results demonstrated that the flight states in steady climb with and without the MC rotors were different as predicted by the analysis, and that steady climb at large climb angles that are unattainable without employing the MC rotors can indeed be realized. As a future research direction, developing control strategies that effectively exploit the enhanced climb performance according to mission requirements remains an important challenge.