Control and Optimization of a Variable-Pitch Quadrotor with Minimum Power Consumption

Abstract

Recently, there has been a rapid growth of interest in quadrotors with electric variable-pitch propellers. The control and optimization of such propellers are important factors for improving the flight performance of the vehicles. Therefore, the steady-state identification method to estimate the parameters of the mathematical model of the electric variable-pitch propeller is developed. The steady-state control and optimization scheme with minimum power consumption and the adaptive compensation scheme for the variable-pitch propeller are then proposed, based on which the response performance of the lift force produced by the variable-pitch propeller can be greatly improved by using a cascade compensation scheme. Furthermore, the direct lift-based flight control strategy is presented, which can significantly contribute to the improvement of the flight performance, precisely because the roll, pitch, yaw and vertical channels of the variable-pitch quadrotor are approximately linearized and completely decoupled from each other in this case. The experimental results demonstrate that both the endurance performance and the positioning accuracy of the variable-pitch quadrotor are improved simultaneously by using the proposed method with minimum power consumption.

1. Introduction

With the development of small and micro Unmanned Aerial Vehicles (UAVs) in recent years, small, multi-rotor UAVs equipped with electric propellers have gained considerable momentum, and are widely used as experimental and hobby platforms because of their simple mechanical structure, good operability, maneuverability and agility. Therefore, multi-rotor UAVs have recently attracted great interest. Also, considerable work exists on various modeling, design, control, and optimization schemes for multi-rotor UAVs [1,2,3,4,5,6,7,8,9].

Fixed-pitch multi-rotor designs are mechanically simple. The stability and flight control of fixed-pitch multi-rotors are well established [10,11,12,13,14,15,16,17]. However, the only way to change the lift force produced by a fixed-pitch propeller is by changing the voltage to the motor, which restricts the aggressive and aerobatic maneuvers that multi-rotor UAVs can perform, therefore limiting the applicability of fixed-pitch multi-rotor UAVs in agility-intensive missions [18]. Variable-pitch multi-rotor UAVs can largely overcome the limitations resulting from the fixed-pitch flight.

The benefits of variable-pitch propellers over fixed-pitch propellers for a quadrotor have been analyzed in [19]. The addition of variable-pitch propellers to a quadrotor platform results in an additional degree of freedom for changing the lift produced by each motor-propeller combination. With a variable-pitch propeller, the lift can be changed by either changing the propeller pitch or by changing the rotational speed. These two actuators, to a large extent, overlap, and there are many combinations of rotational speed and propeller pitch that yield an identical lift. The number of possible combinations is mainly limited by aerodynamic constraints, the maximum propeller pitch and the maximum available motor power. It can be seen from the discussion thus far that on the one hand, the combination could be adjusted to more power efficient settings as the desired lift increases or decreases, and on the other hand, only one combination can yield a desired lift with minimum power consumption. Thus, the control allocation problem of which actuator to use, propeller pitch or rotational speed or combination of the two, needs to be explored systematically under the precondition of ensuring the flight performance, which can certainly contribute to the improvement of the endurance performance of quadrotors.

The modeling and control of the variable-pitch quadrotors have been explored only recently, yet little literature or research is available until now. The design, development, and control of a variable-pitch quadrotor have been studied in [20]. The problem of characterizing the dynamics of a variable-pitch quadrotor from data has been discussed, and black-box versus grey-box models have been analyzed in [21]. The variable-pitch model and the nonlinear proportional squared control algorithm for a quadrotor have also been implemented in the quaternion space [22]. The design of a variable-pitch quadrotor with constant motor speed has been investigated, which has also proven to be most effective in increasing the maneuverability of the quadrotor while largely maintaining its mechanical simplicity [23]. Control and trajectory generation algorithms for a variable-pitch quadrotor have been presented both theoretically and experimentally. The control law is not based on near-hover assumptions, allowing for large attitude deviations from hover [24]. Nevertheless, that research work mainly focused on the aerobatic maneuvers of electric variable-pitch quadrotors. The control and optimization of quadrotors with minimum power consumption has not, to date, been addressed in detail under the precondition of ensuring the flight performance.

The purpose of this study is therefore to develop a proper control and optimization strategy with minimum power consumption for the variable-pitch quadrotor, based on which the endurance performance and the positioning accuracy can be improved. The efficiency and superiority of the proposed method is to be verified by a series of experimental tests.

The structure of the paper is as follows: first, the control and optimization problems are addressed in Section 2, and the steady-state identification method for the variable-pitch propeller is developed in Section 3. Then, the control and optimization strategy for the variable-pitch propeller is presented in Section 4, followed by the direct lift-based flight control strategy for the variable-pitch quadrotor in Section 5. Experimental results are shown in Section 6. Finally, conclusions are drawn in Section 7.

2. The Control and Optimization Problem Statement of the Variable-Pitch Quadrotor

The prototype quadrotor vehicle chosen for this study, 0.5 m in height and 1.45 m in opposite axial distance, is depicted in Figure 1. A maximum take-off mass of 15 kg is assumed including 7 kg of maximum load. The maximum climb speed is approximately 8 m/s while the maximum descent speed is about 13 m/s. The vehicle is used specifically because of the simple mechanical structure, good operability, maneuverability and agility. As the core of the vehicle, the propulsion system mainly consists of an advanced lithium battery, four brushless DC motors, four variable-pitch propellers, and torque-transmitting mechanisms. The variable-pitch propellers driven by the electric motors through the torque-transmitting mechanisms can provide sufficient lift force to lift the vehicle weighing from 8 kg for no load to 15 kg for full load according to the mission equipment.

The quadrotor has six degrees of freedom. Let ϕ,θ,ψ represent the roll, pitch and yaw angles of the quadrotor in the inertial frame. Let v_x, v_y, v_z represent the flight velocities of the quadrotor in the body frame. Considering that the flight velocities v_x, v_y are dependent on the attitude θ, ϕ in flight control system design, for simplicity, only the control schemes for ϕ,θ,ψ and v_z are investigated in this study.

The four propellers driven by each motor constitute the propulsion system of the quadrotor. Therefore, the quadrotor has only four control inputs to produce the desired roll, pitch and yaw moments, and the desired vertical movement, as shown in Figure 2, where α_j, ω_j and L_j denote the propeller pitch, the rotational speed and the lift force. Specially, the numerical subscript j indicates different variable-pitch propellers in this paper.

As for the jth control input, the combination of the rotational speed ω_j and the propeller pitch α_j produces the lift force L_j in the direction of the propeller axis as follows [25]:

$$L_j=b_L{\mathsf{\alpha }}_j{\mathsf{\omega }}_j^2$$

(1)

where b_L is an aerodynamic constant, whose estimate, ${\widehat{b}}_L$, can be obtained by using least-squares regression. The total lift force required to produce the movement along the body z axis is then given by:

$$T={\displaystyle \sum _{j=1}^4{L}_j}$$

(2)

Assuming the variable-pitch quadrotor is near hover, the drag produced by the motor-propeller combination can then be approximately modeled by [25,26]:

$$Q_j=b_{D_1}{\mathsf{\alpha }}_j^2{\mathsf{\omega }}_j^2+b_{D_2}{\mathsf{\omega }}_j^2+b_{D_3}{\mathsf{\alpha }}_j{\mathsf{\omega }}_j$$

(3)

where b_D1, b_D2 and b_D3 can be considered as aerodynamic constants. The torques used to control the roll, pitch and yaw moments can then be described by:

$$T_{\varphi }=l\left(L_4-L_2\right)$$

(4a)

$$T_{\mathsf{\theta }}=l\left(L_3-L_1\right)$$

(4b)

$$T_{\mathsf{\psi }}=Q_1+Q_3-Q_4-Q_2$$

(4c)

where l is the horizontal distance from the propeller center to the center of gravity of the quadrotor.

The use of control and optimization of variable-pitch propellers is desirable for a high-performance quadrotor, mainly because on the one hand, the characteristics of the variable-pitch propeller strongly affect the flight dynamics, stability and endurance performance of the vehicle; while on the other hand, any great changes in the gross flight weight with the loads and the flight speed also require the control system to be more complex and more exact. Obviously each motor and propeller has the same characteristics and can be controlled independently. To avoid repetition, the present study is focused on any pre-specified motor and propeller to demonstrate the proposed control algorithm, which, without loss of generality, can apply to others. For simplicity, the subscript j, denoting the jth motor and propeller, will be omitted in the following two sections, except when there is ambiguity.

3. The Steady-state Identification Method of the Electric Variable-pitch Propeller

The analysis of the basic principle model is crucial for the development of a proper control and optimization scheme to improve the performance of the variable-pitch quadrotor.

The motor is normally modeled by a circuit containing a resistor, inductor, and voltage generator. With regard to the aim of this study, the inductance of a small, brushless hobby motor is negligible when compared to the physical response of the system and so can be ignored. The motor terminal voltage is then approximately given by [27]:

$$u=Ri+\frac{\mathsf{\omega }}{K_V}$$

(5)

where u is the armature voltage, i is the armature current, R is the armature coil resistance, K_V is the voltage constant, ω is the rotational speed of the motor.

Generally, the motor torque, Q_m, is assumed proportional to the difference between the armature current, i, and the no-load current, i₀, through the torque constant, K_Q:

$$Q_{\mathrm{m}}=K_Q\left(i-i_0\right)$$

(6)

The dynamics of the variable-pitch propeller can then be modeled as a simple first order differential equation:

$$I\dot{\mathsf{\omega }}=Q_{\mathrm{m}}-Q$$

(7)

where I is the inertia including the motor and propeller.

Substituting Equations (3), (5) and (6) into Equation (7) yields the following nonlinear differential equation for the electric variable-pitch propeller:

$$I\dot{\mathsf{\omega }}+\left(b_{D_1}{\mathsf{\alpha }}^2+b_{D_2}\right){\mathsf{\omega }}^2+\left(b_{D_3}\mathsf{\alpha }+b_{D_4}\right)\mathsf{\omega }=bu+b_0$$

(8)

where $b_{D_4}=\frac{K_Q}{R{K}_V},b=\frac{K_Q}{R}$ and $b_0=-K_Q{i}_0$. Both the armature voltage, u, and the propeller pitch, α, have a substantial effect on the rotational speed of the propeller, thus having a large impact on the lift force of the propeller, which will be discussed in sufficient details in the following section.

However, the control and optimization of the electric variable-pitch propeller is a difficult and often not very precise task, mainly because the coefficients of the differential equation are unknown. As for the steady-state operation, we have:

$$\mathsf{\gamma }\left(\mathsf{\alpha },u\right)\mathsf{\Theta }=bu$$

(9)

where $\mathsf{\gamma }\left(\mathsf{\alpha },u\right)=\left[{\mathsf{\alpha }}^2{\varpi }^2\left(\mathsf{\alpha },u\right),{\varpi }^2\left(\mathsf{\alpha },u\right),\mathsf{\alpha }\varpi \left(\mathsf{\alpha },u\right),\varpi \left(\mathsf{\alpha },u\right),-1\right]$, $\mathsf{\Theta }={\left[b_{D_1},b_{D_2},b_{D_3},b_{D_4},b_0\right]}^{\mathrm{T}}$, and $\varpi \left(\mathsf{\alpha },u\right)=$$\underset{t\to \infty }{\text{lim}}{\mathsf{\omega }\left(t\right)|}_{\begin{array}{l}\mathsf{\alpha }=\text{constant},\\ u=\text{constant}\end{array}}$ is the steady-state response to constant inputs.

Considering that the motor rotates in a single direction, which provides sufficient thrust force to lift the quadrotor, the commutation interval angle can be expressed as follows:

$$\vartheta \left(t_k\right)-\vartheta \left(t_{k-1}\right)=60/N_{\mathrm{p}}$$

(10)

where $\vartheta$ denotes the rotational angle; $t_{k-1},t_k$ denote the $\left(k-1\right)\text{th}$ and $k\text{th}$commutation instants, respectively, both of which are available from the electric speed controller (ESC) of the motor; $N_{\mathrm{p}}$ is the number of pole pairs. The steady-state output, $\varpi$, can therefore be accurately estimated by:

$$\varpi \approx \frac{60\ell }{N_{\mathrm{p}}\left(t_k-t_{k-\ell }\right)}$$

(11)

where $\ell$ is an optional integral number. On the basis of the discussion made thus far, we can present the steady-state identification algorithm to estimate the coefficients in Equation (8) as follows:

The Steady-state Identification Algorithm:

Input:

The steady-state sample set $S={\left\{\left({\mathsf{\alpha }}_k,u_k,\varpi \left({\mathsf{\alpha }}_k,u_k\right)\right)\right\}}_{k=1}^n$, where the subscript, k , denotes the sample number, and $n\ge 5$. Moreover, The chosen $S$ satisfies the constraint condition $\text{cond}\left({\mathit{R}}^{\mathrm{T}}\mathit{R}\right)<c_0$, where $\mathit{R}=\left[\begin{array}{c}\mathsf{\gamma }\left({\mathsf{\alpha }}_1,u_1\right)\\ ⋮\\ \mathsf{\gamma }\left({\mathsf{\alpha }}_n,u_n\right)\end{array}\right]$, and $c_0$ is an optional positive value.

Output:

$\widehat{\mathsf{\Theta }}={\left[{\widehat{b}}_{D_1},{\widehat{b}}_{D_2},{\widehat{b}}_{D_3},{\widehat{b}}_{D_4},{\widehat{b}}_0\right]}^{\mathrm{T}}$ and $\widehat{b}$, which are the estimates of $\mathsf{\Theta }$ and $b$, respectively.

Steps:

(1) Calculate ${\widehat{\mathsf{\Theta }}}_b={\left({\mathit{R}}^{\mathrm{T}}\mathit{R}\right)}^{-1}{\mathit{R}}^{\mathrm{T}}\mathit{U}$, where $\mathit{U}=\left[\begin{array}{c}{u}_1\\ ⋮\\ u_n\end{array}\right]$;

(2) Calculate $\widehat{b}=\frac{1}{n}{\displaystyle \sum _{k=1}^n\frac{\mathsf{\gamma }\left({\mathsf{\alpha }}_k,u_k\right){\widehat{\mathsf{\Theta }}}_b}{u_k}}$;

(3) Calculate $\widehat{\mathsf{\Theta }}={\widehat{\mathsf{\Theta }}}_b\widehat{b}$

As for the DUΛLSKY^® XM5060EA-9 (Shanghai Dualsky Models Co., Ltd., Shanghai, China) brushless motor and the 16 inch diameter propeller used in this study, the estimates of the coefficients of the nonlinear differential equation are shown in

Table 1. The estimates of the coefficients of the electric variable-pitch propeller.

Configuration	Variable	Value
XM5060EA-9 motor and 16 inch diameter propeller	${\widehat{b}}_{D_1}$	2.73 × 10−8
	${\widehat{b}}_{D_2}$	1.13 × 10−6
	${\widehat{b}}_{D_3}$	2.93 × 10−5
	${\widehat{b}}_{D_4}$	2.37 × 10−2
	${\widehat{b}}_0$	−2.97 × 10–2
	$\widehat{b}$	0.78

for reference.

4. The Control and Optimization Strategy for the Electric Variable-Pitch Propeller

4.1. The Steady-State Control and Optimization Scheme with Minimum Power Consumption

As shown in Equation (1), the lift force can be changed by either changing the rotational speed or by changing the propeller pitch. There are many rotational speed and propeller pitch combinations that yield the same lift force values. The optimization of the motor power consumption among the combinations is therefore essential to improve the endurance of the variable-pitch quadrotor. The motor power, P, can be given by:

$$P\approx i^{2}R+Q_p\mathsf{\omega }$$

(12)

The online optimization of the motor power is often an undecidable problem, which, however, can be approximately converted into the minimization of the input voltage to the motor and propeller under the steady-state condition [18]. The solution to the optimization problem can therefore be formulated as follows:

$$\left({\mathsf{\alpha }}_c,{\mathsf{\omega }}_c\right)\approx \text{arg}\underset{L=L_c}{\text{min}}u$$

(13)

where ${\mathsf{\alpha }}_c,{\mathsf{\omega }}_c$ are the sub-optimized propeller pitch command and the sub-optimized rotational speed command of the electric variable-pitch propeller, respectively, and $L_c$ is the lift force command from the flight controller discussed in Section 5.

Not considering the estimation error of the coefficients of the nonlinear differential in Equation (8), the steady-state component of the input voltage can be derived as follows:

$$u_0={\widehat{b}}^{-1}\left\{L_c{\widehat{b}}_L^{-1}\left({\widehat{b}}_{D_1}{\mathsf{\alpha }}_c+{\widehat{b}}_{D_2}{\mathsf{\alpha }}_c^{-1}\right)+L_c^{0.5}{\widehat{b}}_L^{-0.5}\left({\widehat{b}}_{D_3}{\mathsf{\alpha }}_c^{0.5}+{\widehat{b}}_{D_4}{\mathsf{\alpha }}_c^{-0.5}\right)-{\widehat{b}}_0\right\}$$

(14)

According to Equations (1) and (14), the steady-state optimization problem can then be formulated as:

$${\mathsf{\alpha }}_c\approx \text{arg}\underset{L=L_c}{\text{min}}\left\{L_c^{0.5}\left({\widehat{b}}_{D_1}{\mathsf{\alpha }}_c+{\widehat{b}}_{D_2}{\mathsf{\alpha }}_c^{-1}\right)+{\widehat{b}}_L^{0.5}\left({\widehat{b}}_{D_3}{\mathsf{\alpha }}_c^{0.5}+{\widehat{b}}_{D_4}{\mathsf{\alpha }}_c^{-0.5}\right)\right\}$$

(15a)

$${\mathsf{\omega }}_c=\sqrt{\frac{L_c}{{\widehat{b}}_L{\mathsf{\alpha }}_c}}$$

(15b)

We therefore conclude that the sub-optimized propeller pitch command, α_c, satisfies the following condition:

$$2{\ell }_c\left({\widehat{b}}_{D_1}{\mathsf{\beta }}_c^4-{\widehat{b}}_{D_2}\right)={\widehat{b}}_{\ell }\left({\widehat{b}}_{D_4}{\mathsf{\beta }}_c-{\widehat{b}}_{D_3}{\mathsf{\beta }}_c^3\right)$$

(16)

where ${\ell }_c=L_c^{0.5},{\widehat{b}}_{\ell }={\widehat{b}}_L^{0.5}$ and ${\mathsf{\beta }}_c={\mathsf{\alpha }}_c^{0.5}$. It is worth noting that α_c can be obtained online by using the iterative algorithm discussed below. Figure 3 shows the change of the sub-optimized propeller pitch command ${\mathsf{\alpha }}_c$ with the lift force command L_c.

The steady-state estimates of the lift force L and the motor power P, as functions of the rotational speed ω and the pitch angle α, can be obtained by using numerical analyses, as shown in Figure 4. The plots show how the lift can be increased by either increasing the rotational speed, increasing pitch, or by increasing both. Given a curve of constant lift force, only one curve of minimum constant motor power remains tangent to it. The black curve indicates the optimal $\mathsf{\alpha }-\mathsf{\omega }$ trajectory with minimum power consumption, and the sub-optimized propeller pitch command ${\mathsf{\alpha }}_c$ displayed with “X” is comparatively shown in Figure 4. We can conclude that the difference between the sub-optimized propeller pitch and the optimal propeller pitch is insignificant.

Therefore, the motor power consumption is approximately the lowest while changing the propeller pitch and the rotational speed according to Equation (15). Referring to Equation (16) and Figure 3, we then present the online iterative algorithm to optimize the propeller pitch command and the rotational speed command quickly:

The Online Iterative Algorithm:

Input:

$L_c\left(t_k\right),{\mathsf{\alpha }}_c\left(t_{k-1}\right)$ and $\Delta {\mathsf{\alpha }}_c$, where $t_{k-1},t_k$ denote the $\left(k-1\right)\text{th}$ and $k\text{th}$ control instants, respectively; $\Delta {\mathsf{\alpha }}_c$ is the optional minimum increment of ${\mathsf{\alpha }}_c$.

Output:

${\mathsf{\alpha }}_c\left(t_k\right),{\mathsf{\omega }}_c\left(t_k\right)$.

Steps:(1)

(1) ${\mathsf{\alpha }}_c\left(t_k\right)\leftarrow {\mathsf{\alpha }}_c\left(t_{k-1}\right)$, and $i_{+},i_{-}=0$;

(2) If $i_{+}>0$ and $i_{-}>0$, then

Go to Step 4;

Else

Go to Step 3;

End

(3) If $2{\ell }_c\left({\widehat{b}}_{D_1}{\mathsf{\beta }}_c^4-{\widehat{b}}_{D_2}\right)$ > ${\widehat{b}}_{\ell }\left({\widehat{b}}_{D_4}{\mathsf{\beta }}_c-{\widehat{b}}_{D_3}{\mathsf{\beta }}_c^3\right)$, then

${\mathsf{\alpha }}_c\left(t_k\right)\leftarrow {\mathsf{\alpha }}_c\left(t_k\right)-\Delta {\mathsf{\alpha }}_c,i_{-}\leftarrow i_{-}+1$;

Else

${\mathsf{\alpha }}_c\left(t_k\right)\leftarrow {\mathsf{\alpha }}_c\left(t_k\right)+\Delta {\mathsf{\alpha }}_c,i_{+}\leftarrow i_{+}+1$;

End

Return to Step 2;

(4) Calculate ${\mathsf{\omega }}_c\left(t_k\right)$.

4.2. The Adaptive Compensation Scheme

We now assume that:

$$u=u_0+u_a$$

(17)

where the adaptive compensation component of the input voltage, $u_a$, is given by:

$$u_a={\widehat{b}}^{-1}\left({\widehat{b}}_{D_1}{\mathsf{\alpha }}_c^2+{\widehat{b}}_{D_2}\right)\left({\mathsf{\omega }}^2-{\mathsf{\omega }}_c^2\right)+{\widehat{b}}^{-1}\left({\widehat{b}}_{D_3}{\mathsf{\alpha }}_c+{\widehat{b}}_{D_4}-k\right)\left(\mathsf{\omega }-{\mathsf{\omega }}_c\right)-{\widehat{b}}^{-1}\tilde{k}\mathsf{\omega }$$

(18)

where $k$ and $\tilde{k}$ are adjustable parameters. Considering the estimates of the coefficients in Table 1 and the small change of ${\mathsf{\alpha }}_c$ in Figure 3, $u_a$ should be insignificant, as long as $k\approx {\widehat{b}}_{D_4},\tilde{k}\approx 0$, and $\mathsf{\omega }$ is close to ${\mathsf{\omega }}_c$ within the normal operating rotational speed range, thus having little effect on the optimization of the electric variable-pitch propeller. Substituting Equations (14), (17) and (18) into Equation (8) yields:

$$\dot{\mathsf{\omega }}+\left(k_a+{\tilde{k}}_a+{\mathsf{\epsilon }}_b\right)\mathsf{\omega }=k_a{\mathsf{\omega }}_{\mathrm{c}}$$

(19)

where $k_a={\widehat{I}}_b^{-1}k$, ${\tilde{k}}_a={\widehat{I}}_b^{-1}\tilde{k}$, ${\mathsf{\epsilon }}_b=\left\{\left({\widehat{b}}_{D_1}{\mathsf{\alpha }}_c^2+{\widehat{b}}_{D_2}\right)\mathsf{\omega }+\left({\widehat{b}}_{D_3}{\mathsf{\alpha }}_c+{\widehat{b}}_{D_4}\right)-b{\mathsf{\omega }}^{-1}\right\}\left(I^{-1}-{\widehat{I}}_b^{-1}\right)$, and ${\widehat{I}}_b=I\widehat{b}{b}^{-1}$. Note that ${\mathsf{\epsilon }}_b$ is also insignificant.

Considering that $u_a$ should be insignificant within the normal operating rotational speed range, the ideal model is therefore defined as:

$${\dot{\mathsf{\omega }}}_m+{\widehat{k}}_m{\mathsf{\omega }}_m={\widehat{k}}_m{\mathsf{\omega }}_c$$

(20)

where ${\widehat{k}}_m={\widehat{I}}^{-1}{\widehat{b}}_{D_4}$, and $\widehat{I}$ is the estimate of I.

The tracking error between the outputs of the plant and the model is then defined as:

$$e={\mathsf{\omega }}_m-\mathsf{\omega }$$

(21)

Now, we consider the Lyapunov function candidate:

$$V=e^2+{\mathsf{\lambda }}_a{\mathsf{\delta }}_a^2+{\tilde{\mathsf{\lambda }}}_a{\tilde{\mathsf{\delta }}}_a^2$$

(22)

where ${\mathsf{\delta }}_a={\widehat{k}}_m-k_a,{\tilde{\mathsf{\delta }}}_a={\tilde{k}}_a+{\mathsf{\epsilon }}_b$, and ${\mathsf{\lambda }}_{\mathrm{a}},{\tilde{\mathsf{\lambda }}}_a$ are optional positive values. The derivative of the Lyapunov function candidate is then negative definite as the adaptive laws are presented as follows:

$$\{\begin{array}{l}\dot{k}={\mathsf{\gamma }}_a\left(e+\mathsf{\mu }\right)e_0\\ \dot{\tilde{k}}=-{\tilde{\mathsf{\gamma }}}_a\left(e+\mathsf{\mu }\right)\mathsf{\omega }\end{array}$$

(23)

where ${\mathsf{\gamma }}_a={\mathsf{\lambda }}_a^{-1}{\widehat{\mathrm{I}}}_b$, ${\tilde{\mathsf{\gamma }}}_a={\tilde{\mathsf{\lambda }}}_a^{-1}{\widehat{\mathrm{I}}}_b$, $e_0={\mathsf{\omega }}_c-\mathsf{\omega }$, and $\mathsf{\mu }$ satisfies the constraint condition:

$$\text{sgn}\left(\mathsf{\mu }\right)=\text{sgn}\left(\dot{e}+{\widehat{k}}_{m}e\right)$$

(24)

Note that the term μ can greatly contribute to the convergence rates of adjustable parameters, as can be seen from:

$$\dot{V}=0.5\left\{-{\widehat{k}}_m{e}^2-\left(\dot{e}+{\widehat{k}}_{m}e\right)\mathsf{\mu }\right\}$$

(25)

The adaptive compensation given by Equation (23) can reduce the tracking error such that ω is, as far as possible, close to ${\mathsf{\omega }}_m$, thus improving the response performance of the rotational speed.

4.3. The Cascade Compensation Scheme

As described thus far, referring to Equations (1) and (20), we approximately have:

$$\frac{L\left(s\right)}{L_c\left(s\right)}\approx \frac{a_m}{s+a_m}$$

(26)

where $a_m=2{\widehat{k}}_m$. We can further improve the response performance of the variable-pitch propeller through a single cascade compensation element:

$$\frac{L_c\left(s\right)}{L_p\left(s\right)}\approx \frac{a_m^{-1}s+1}{a_p^{-1}s+1}$$

(27)

where $a_p$ is an optional positive value, and $L_p$ is the control signal from the flight controller discussed in Section 5. Therefore:

$$\frac{L\left(s\right)}{L_p\left(s\right)}\approx \frac{a_p}{s+a_p}$$

(28)

It is finally noted that both the adaptive compensation and the cascade compensation can greatly contribute to the improvement of the control quality of the variable-pitch propeller. The control scheme of the variable-pitch propeller, including the steady-state optimization, the adaptive compensation and the cascade compensation, is shown in Figure 5.

5. The Direct Lift-based Flight Control Strategy for the Variable-Pitch Quadrotor

5.1. The Direct Lift-based Height Control Scheme

Considering that each motor and propeller has the same input-output properties, according to Equations (2) and (28), the total lift produced by the combined effect of the forces produced by the four propellers can also be expressed by:

$$\frac{T\left(s\right)}{T_p\left(s\right)}\approx \frac{a_p}{s+a_p}$$

(29)

where:

$$T_p={\displaystyle \sum _{j=1}^4{L}_{pj}}$$

(30)

and $L_{pj}$ denotes the lift command signal corresponding to the $j\text{th}$ propeller, as shown in Figure 5.

As for the near-hover flight, we suppose that:

$$T_{ph}=T_p-m\mathrm{g}$$

(31a)

$$T_h=T-m\mathrm{g}$$

(31b)

where $T_{ph}$ denotes the control signal from the height controller discussed below, m denotes the mass of the quadrotor, and g denotes the gravitational acceleration. After offsetting the gravity acting on the variable-pitch quadrotor, Equation (29) can then be rewritten by:

$$\frac{T_h\left(s\right)}{T_{ph}\left(s\right)}\approx \frac{a_p}{s+a_p}$$

(32)

Let $z,z_c$ be the actual height and the height command signal, respectively. We propose the direct lift-based height control law as follows:

$$T_{ph}=k_{zp}{\mathsf{\epsilon }}_z+k_{zi}{\displaystyle \int {\mathsf{\epsilon }}_z\mathrm{d}t}-m{a}_p{v}_z-T_h$$

(33)

where $k_{zp},k_{zi}$ are optional parameters, and:

$${\mathsf{\epsilon }}_z=m{a}_p\left(z_i-z\right)+m\left(v_{zi}-v_z\right)$$

(34)

where $z_i$ is the output of a cascade compensation element in response to $z_c$; $v_{zi}$ is correspondingly the derivative of $z_i$.

It is worthy to note that the direct-lift feedback term in Equation (33) can contribute to the stability of the height. The transfer function, from the input $z_i$ to the output $z$, can then be derived by using the knowledge of rigid-body kinematics and by substituting Equations (33) and (34) into Equation (32) as follows:

$$\frac{z\left(s\right)}{z_i\left(s\right)}\approx \frac{a_p{k}_{zp}s+a_p{k}_{zi}}{s^3+a_p{s}^2+a_p{k}_{zp}s+a_p{k}_{zi}}$$

(35)

The cascade compensation element is further chosen as:

$$\frac{z_i\left(s\right)}{z_c\left(s\right)}\approx \frac{{\mathsf{\omega }}_z^2{s}^3+{\mathsf{\omega }}_z^2{a}_p{s}^2+{\mathsf{\omega }}_z^2{a}_p{k}_{zp}s+{\mathsf{\omega }}_z^2{a}_p{k}_{zi}}{a_p{k}_{zp}{s}^3+\left(2{\mathsf{\xi }}_z{\mathsf{\omega }}_z{a}_p{k}_{zp}+a_p{k}_{zi}\right)s^2+\left({\mathsf{\omega }}_z^2{a}_p{k}_{zp}+2{\mathsf{\xi }}_z{\mathsf{\omega }}_z{a}_p{k}_{zi}\right)s+{\mathsf{\omega }}_z^2{a}_p{k}_{zi}}$$

(36)

where ${\mathsf{\xi }}_z$ and ${\mathsf{\omega }}_z$ are optional parameters. Thus:

$$\frac{z\left(s\right)}{z_c\left(s\right)}\approx \frac{{\mathsf{\omega }}_z^2}{s^2+2{\mathsf{\xi }}_z{\mathsf{\omega }}_{z}s+{\mathsf{\omega }}_z^2}$$

(37)

where ${\mathsf{\xi }}_z$ and ${\mathsf{\omega }}_z$ can be considered as the damping and natural frequency. The dynamics of the height is therefore determined by the choices of $k_{zp},k_{zi},{\mathsf{\xi }}_z,{\mathsf{\omega }}_z$ and $a_p$.

5.2. The Direct Lift-based Attitude Control Scheme

As described in Section 2, the moments are produced by generating a differential lift across the two propellers on the same arm of the quadrotor, thus providing the input necessary for the quadrotor to move longitudinally or laterally; for making the quadrotor yaw to a particular orientation, the lifts of a set of opposite propellers are changed simultaneously and by the same amount.

It can be seen from Table 1 that the second term on the right-hand side of Equation (3) remains dominant in the drag produced by the motor-propeller combination. We therefore suppose that:

$$T_{\varphi c}=L_{p4}-L_{p2}$$

(38a)

$$T_{\mathsf{\theta }c}=L_{p3}-L_{p1}$$

(38b)

$$T_{\mathsf{\psi }c}=L_{p1}+L_{p3}-L_{p4}-L_{p2}$$

(38c)

The roll, pitch and yaw channels can then be considered to be approximately linear and decoupled as $T_{\varphi c},T_{\mathsf{\theta }c}$ and $T_{\mathsf{\psi }c}$ are chosen as the control input signals without considering other aerodynamic effects.

We then propose the direct lift-based attitude control laws as follows:

$$T_{\varphi c}=k_{\varphi p}{e}_{\varphi }+k_{\varphi i}{\displaystyle \int e_{\varphi }\mathrm{d}t}+k_{\varphi d}{\dot{e}}_{\varphi }-k_{\dot{\varphi }}\dot{\varphi }$$

(39a)

$$T_{\mathsf{\theta }c}=k_{\mathsf{\theta }p}{e}_{\mathsf{\theta }}+k_{\mathsf{\theta }i}{\displaystyle \int e_{\mathsf{\theta }}\mathrm{d}t}+k_{\mathsf{\theta }d}{\dot{e}}_{\mathsf{\theta }}-k_{\dot{\mathsf{\theta }}}\dot{\mathsf{\theta }}$$

(39b)

$$T_{\mathsf{\psi }c}=k_{\mathsf{\psi }p}{e}_{\mathsf{\psi }}+k_{\mathsf{\psi }i}{\displaystyle \int e_{\mathsf{\psi }}\mathrm{d}t}+k_{\mathsf{\psi }d}{\dot{e}}_{\mathsf{\psi }}-k_{\dot{\mathsf{\psi }}}\dot{\mathsf{\psi }}$$

(39c)

where $e_{\varphi }={\varphi }_i-\varphi ,e_{\mathsf{\theta }}={\theta }_i-\theta ,e_{\mathsf{\psi }}={\psi }_i-\psi$; ${\varphi }_i,{\mathsf{\theta }}_i,{\mathsf{\psi }}_i$ are the roll, pitch and yaw command signals, respectively; $k_{\varphi p},k_{\varphi i},k_{\varphi d},k_{\dot{\varphi }},k_{\mathsf{\theta }p},k_{\mathsf{\theta }i},k_{\mathsf{\theta }d},k_{\dot{\mathsf{\theta }}},k_{\mathsf{\psi }p},k_{\mathsf{\psi }i},k_{\mathsf{\psi }d},k_{\dot{\mathsf{\psi }}}$ are optional parameters.

Compared with conventional flight control methods for variable-pitch quadrotors, the direct lift-based flight control method can improve the flight performance, precisely because all channels are approximately linearized and completely decoupled from each other in this case. It is finally noted that the lift command signal to each motor and propeller can be obtained from Equations (30), (31), (33), (38) and (39).

The longitudinal movement is constituted by the forward velocity caused by the pitch angle θ produced by the moment $T_{\mathsf{\theta }}$, and so does the lateral movement. The longitudinal and lateral trajectory controllers are commonly referred to as the outer-loop controllers and can also be easily designed by using conventional linear control techniques, which have been studied in many literatures, such as [4,5,6], so won't be covered here.

6. Experimental Tests

In this section, we first describe the details of the experimental setup used for verification purposes. Finally, extensive experimental tests are carried out along with necessary discussions and evaluations.

6.1. The Experimental Setup and Description

Apart from the variable-pitch propeller described in Section 2, the component parts of the control system of the quadrotor include a DSP-based hardware platform and a high integrity navigation unit. The software of the proposed control scheme runs at 200 Hz on the DSP-based hardware platform that can provide reliable support for high precision timer and synchronization operations. The flight controller, performing the closed-loop attitude and altitude control module, is employed to generate the command as inputs to the control system of the variable-pitch propeller. The high integrity navigation system, based on the combined use of the Global Positioning System (GPS) and an inertial measurement unit (IMU), is used to provide the flight controller with the accurate attitude, position and velocity information. The discrete estimate of the rotational speed can be obtained by the use of the commutation instant from the ESC since the commutation interval angle is constant, shown in Equation (10), and the rotational speed is high within the normal operating rotational speed range.

To test and verify the efficiency of the proposed control method, the prototype quadrotor is first conducted to repeatedly perform climb and descent flights in attitude hold mode. Figure 6 shows the experimental test scenes for the present study. For simplicity, only the experimental results related to the pre-specified variable-pitch propeller are given to demonstrate the proposed method under the aerodynamic constraints and the power constraints.

6.2. Vertical Flight and Endurance Tests

6.2.1. Experimental Tests of the Proposed Method

The vertical flight tests are conducted to comparatively assess the endurance performance of the variable-pitch quadrotor by using the proposed method and the conventional method. All predefined parameters of the flight control system of the variable-pitch quadrotor are first described as follows:

(1)

The estimates of aerodynamic parameters: ${\widehat{b}}_L=1.75\times {10}^{-5},\widehat{I}=1.37\times {10}^{-3}$;

(2)

Adaptive compensation in the rotational speed loop: ${\mathsf{\gamma }}_a=1.0\times {10}^{-3},{\tilde{\mathsf{\gamma }}}_a=1.0\times {10}^{-5},\mathsf{\mu }=\dot{e}+$${\widehat{k}}_{m}e$;

(3)

Cascade compensation in the lift force loop: $a_p=5$;

(4)

Direct lift-based height control scheme: $k_{zp}=2.0,k_{zi}=1.6,{\mathsf{\xi }}_z=2.0,{\mathsf{\omega }}_z=0.75$;

(5)

Direct lift-based attitude control scheme: $k_{\varphi p},k_{\theta p}=2.1,k_{\varphi i},k_{\theta i}=0,k_{\varphi d},k_{\theta d}=0.5,k_{\dot{\varphi }},k_{\dot{\theta }}=0.1$, $k_{\psi p}=9.0,k_{\psi i}=0.5,k_{\psi d}=3.0,k_{\dot{\psi }}=0.5$;

(6)

Longitudinal position control scheme: ${\mathsf{\theta }}_i=k_p{e}_x+k_i{\displaystyle \int e_x\mathrm{d}t}+k_d{\dot{e}}_x-k_v{v}_x$, where $e_x$ is the position error along the body $x$ axis, $k_p=1.0,k_i=0.1,k_d=0.2,k_v=1.0$, and so does the lateral position control scheme. A series of real-time experimental tests are then conducted based on the above assumptions.

Figure 7 shows the experimental results of the proposed method in the case of the vertical flight with constant velocity, where $z_c\left(t\right)={\displaystyle {\int }_0^t{v}_{zc}\left(\mathsf{\tau }\right)\mathrm{d}\mathsf{\tau }}$; $v_{zc}$ is the vertical velocity command signal; $e_{vz}$ is the error between $v_{zc}$ and $v_z$; $e_{L1}$ is the error between $L_{p1}$ and $L_1$.

For simplicity, only the results related to the first propeller are shown in the figure. It is seen from Figure 7a–f that the resulting controlled system can achieve good tracking performances, even though the voltage and the propeller pitch exhibit significant variations in accordance with the vertical velocity command signal shown in Figure 7g,h. Meanwhile, the adjustable parameters of the adaptive compensation element also have good convergence properties as shown in Figure 7i,j during the maneuver flights. It is, however, to note that the rotational speed tracking error demonstrates a remarkable difference during the climb flight mode and the descent flight mode due to the nonlinear characteristics of the propeller.

To further demonstrate the positive effect of the propeller pitch on the endurance performance and comparatively assess the efficiency and superiority of the proposed method, the propeller pitch $\mathsf{\alpha }$ involved in the proposed method is set equal to ${\mathsf{\alpha }}_c$ and different constant values, respectively, and simultaneously, the variable-pitch quadrotor is particularly equipped with the same type of Lithium battery with a capacity of 5000 mAh for each experimental test.

Table 2. The endurance performance versus propeller pitch angles.

Battery Capacity	$\mathsf{\alpha }$(deg)	Endurance (s)
5000 mAh	Sub-optimized value	267.30
	3	155.36
	4	197.50
	5	226.61
	6	246.72
	7	251.65
	8	249.22
	9	246.77
	10	243.27
	11	239.70
	12	234.91
	13	229.19
	14	220.52
	15	212.58

and Figure 8 show the endurance performance versus propeller pitch angles using the proposed method.

The proposed method with minimum power consumption has the most significant improving effect on the endurance performance when compared to the fixed propeller pitches. According to the test results, the improvement of the endurance performance of the quadrotor ranges from 6.2% to 72.1%.

6.2.2. Experimental Tests of the Conventional Method with Constant Rotational Speed

In order to make a fair comparison between the proposed method and the conventional method with constant rotational speed implemented through the widely used Pixhawk^® autopilot module, the experimental tests of the conventional method are then conducted for this purpose, where the autopilot employs one optimal proportional-integral controller to obtain the armature voltage according to the rotational speed tracking error, and another optimal proportional-integral- differential controller to determine the propeller pitch according to the vertical velocity tracking error.

The predefined parameters are described as follows:

(1)

Rotational speed control scheme (optimal proportional-integral control): proportional and integral coefficients equal to 0.85 and 1.95;

(2)

Vertical velocity control scheme (optimal proportional-integral-differential control): proportional, integral and differential coefficients equal to 2.5, 1.0 and 0.2.

Figure 9 shows the experimental results with the rotational speed command signal ω_cj = 400 (j = 1, 2, 3, 4). The propeller pitch exhibits a significant variation in accordance with the vertical velocity command signal during the maneuver flights, as shown in Figure 9f, although the vertical velocity tracking performance deteriorates slightly and is basically acceptable when compared to the proposed method.

Similarly, to demonstrate the effect of the rotational speed on the endurance performance in this test case, the rotational speed command signal is set equal to different constant values. The experimental tests are conducted with the same equipment as mentioned above, and the resulting endurance performance versus rotational speeds is shown in

Table 3. The endurance performance versus rotational speeds.

Battery Capacity	${\mathsf{\omega }}_c$(rad/s)	Endurance (s)
5000 mAh	350	230.95
	400	240.09
	450	239.89
	500	232.52
	550	220.43
	600	205.31
	650	189.06

and Figure 10.

In contrast to what has been shown in Table 2 and Figure 8, the endurance with fixed rotational speed decreases distinctly, although the control quality is basically satisfactory in practice, which also comparatively demonstrates that the proposed method can contribute to the significant improvement of the endurance performance.

6.3. Positioning Accuracy Tests

All predefined parameters of the proposed algorithm are defined in Section 6.2.1. Figure 11 shows the experimental results of the proposed algorithm during near-hover operations, where $v_n,v_e,v_h$ are the northward, eastward and vertical velocities in the Earth frame. The proposed method can achieve a good trajectory tracking performance when the variable-pitch quadrotor is conducted to perform a low-velocity trajectory tracking task, as shown in Figure 11a–c. Furthermore, according to the experimental data, the deviation distance is less than 0.08 m during the straight line flight, and the height tracking error is less than 0.12 m in the presence of disturbances and uncertainties.

Figure 12 shows the experimental results of the conventional method during near-hover operations. According to the experimental data, the deviation distance is more than 0.23 m during the straight line flight, and the height tracking error is more than 1.3 m, as shown in Figure 12a–c. The experimental results of the conventional method also demonstrate that the horizontal position error reaches a maximum value of 0.20 m, and height error also reaches a maximum value of 0.55 m in the presence of disturbances and uncertainties during the hovering flight.

Compared with the results of the proposed method, the resulting performance of the conventional method deteriorates significantly precisely because of the nonlinearity and the coupling interactions mentioned above, as demonstrated in [3,21]. From the comparison of the experimental results of two flight control methods, we can therefore conclude that the proposed method contributes to the significant improvement of the flight performance of the variable-pitch quadrotor, such as the endurance performance and the positioning accuracy.

7. Conclusions

To address the control and optimization problems of the electric variable-pitch quadrotor, a steady-state identification method to estimate the parameters of the mathematical model of the electric variable-pitch propeller has been first developed. A control and optimization strategy for the variable-pitch propeller, mainly including the steady-state control and optimization scheme with minimum power consumption, the adaptive compensation scheme and the cascade compensation scheme, is then proposed, which greatly improves the response performance of the lift force produced by the variable-pitch propeller. Furthermore, the direct lift-based flight control strategy is presented, which contributes to the improvement of the flight performance of the variable-pitch quadrotor, precisely because the roll, pitch, yaw and vertical channels are approximately linearized and completely decoupled from each other in this case. The experimental test results have comparatively demonstrated the efficiency and superiority of the proposed method. These achievements can also apply to other micro air vehicles with electric variable-pitch propellers.