Online Trajectory Planning Method for Double-Pendulum Quadrotor Transportation Systems

Abstract

This study investigates the trajectory planning problem for double-pendulum quadrotor transportation systems. The goal is to restrain the hook swing and payload swing while achieving precise positioning. An online trajectory planning method with two capabilities—precise positioning and swing suppression—is proposed. The stability and convergence of the system are proved using the Lyapunov principle and the LaSalle’s invariance theory. Simulation results show that the proposed method has excellent control performance.

1. Introduction

Quadrotors with vertical takeoff, landing, and hovering capabilities have become ideal flight platforms for many applications [1,2,3,4,5]. Transportation of a suspended payload is one of the important applications of quadrotors in logistics. The quadrotor can sling medicines, food, or emergency supplies for package delivery with cable, regardless of their shape or size [6]. It can also be used as a traditional manned helicopter to slung water tanks to participate in urban high-rise buildings or forest fire fighting, load and transport a large number of materials to support disaster areas, or collect samples, and even in the military [7]. America’s K-MAX helicopter, for example, is already used to transport with external slung load [8]. Therefore, it has always been a topic of great interest to international researchers [9].

In such a context, there are many representative research teams dedicated to the study of the quadrotor transport systems. The GRASP Laboratory at the University of Pennsylvania has carried out research on modeling, control, and planning of the quadrotor transport systems to enable autonomous operation for many years [10]. Professor Klausen of the Norwegian University of Science has been working on the application of UAV formations in the rescue field for many years. This year, he completed the verification of the control method based on three small quadcopters cooperative hanging load and completed the flight demonstration of the straight segment [11]. Guo Huaimin of Nanjing University of Science and Technology in China studied the flight control of double-aircraft cooperative suspension in his doctoral project and verified the effectiveness of the method through simulation [12]. Professor Han Jianda’s team [13] and Professor Fang Yongchun’s team [14] also carried out relevant research and verified the correctness of relevant theoretical methods through numerical simulation and indoor dynamic capture system.

Because the swing of the payload will reduce the transport efficiency, it will also affect the stability of the aircraft, making it easy to cause safety accidents. Therefore, how to restrain the swing of the payload while achieving precise positioning is important for improving transportation efficiency. However, the swing of the payload of an underactuated quadrotor cannot be controlled directly during transportation, which undoubtedly increases the control difficulty [15].

For this reason, many researchers have attempted to achieve such control and proposed various control methods. Most of the control methods proposed, thus far, have been for single-pendulum quadrotor transportation systems. These methods can be roughly divided into the following categories: linear control, optimal control, and regulation control. Typical methods based on the linear control strategy include proportional-integral-derivative/proportional-derivative (PID/PD) control and linear quadratic regulator control [16,17]. These methods have the advantages of a simple structure and a small computational load, but they can deal only with simplified models because of their limited application scope. Examples of methods based on the optimal control strategy include input shaping and optimal control [18,19]. Such methods can improve safety and reduce energy consumption by limiting the state or controlling the amplitude of the input. However, they are open-loop control methods and highly sensitive to external interferences. The most representative method based on the regulation control strategy is nonlinear control, which includes the energy-based method [20], back-step method [21], hierarchical control [22], sliding mode control [23,24], and adaptive control [25]. These nonlinear control methods can make a system’s state asymptotically or exponentially track the reference instruction and also achieve accurate positioning. However, such methods tend to produce larger attitudes at the initial moment, which leads to an increase in energy consumption and also causes the initial swing of the payload to be larger.

All of the abovementioned methods have been proposed for single-pendulum quadrotor transportation systems. However, the quadrotor transportation system will exhibit more complex, double-pendulum characteristics when the hook mass cannot be ignored or the payload is not assumed to be a particle [26]. Few studies have, thus far, been conducted on the double-pendulum quadrotor transportation system. This is a very challenging problem. The system was preliminarily studied, and a dynamic model of this system in the two-dimensional plane was established [27]. A nonlinear controller was designed, and the pendulum damping performance was verified via simulation. Then, the model was further extended to the three-dimensional space and a new energy function was constructed [28]. A coupling term was also introduced to improve the transient performance of the system. Similar double-pendulum characteristics are often observed in cranes, and some research results can be used as reference. The well-known input shaping control method can well suppress the hook swing and payload swing [29,30], but it is sensitive to external disturbances. Some methods based on a nonlinear control strategy have also been developed, e.g., the energy coupling method [31], sliding mode control [32], and adaptive control [33]. In addition, the trajectory planning method is simply adopted [34], but it is also an open-loop method and has poor robustness to disturbances.

From the above-presented research review, it is clear that many methods are still used for single-pendulum quadrotor transportation systems. These methods have some limitations, such as poor performance in terms of pendulum control, a complex structure, and difficulty of practical application.

Therefore, in this paper, an online trajectory planning method inspired by S-curve trajectory planning is proposed for double-pendulum quadrotor transportation systems [35]. The restraining hook and payload swing part is added to the quadrotor reference trajectory, which does not affect positioning performance. In particular, this method can theoretically ensure that the core indexes, such as the maximum acceleration and velocity of the quadrotor, are constrained. The stability and convergence of the system are proved using the Lyapunov principle and the LaSalle’s invariance theory. Simulation results show that the proposed method has excellent control performance.

The following are the important contributions of this study.

• The proposed method can be used for online trajectory planning without the need for offline design of the quadrotor speed and acceleration.
• The method can restrain the hook swing and payload swing without any adverse effects on the positioning performance.
• The method can ensure that the core indexes, e.g., the maximum acceleration and velocity of the quadrotor, are constrained.

The rest of this paper is organized as follows. Section 2 presents modeling of the double-pendulum quadrotor transportation. Section 3 describes the design process of the online trajectory planning method. Then, Section 4 proves the stability and convergence of the double-pendulum quadrotor transportation system. Section 5 analyzes the simulation results in detail. Finally, Section 6 summarizes the study and presents future prospects.

2. Dynamical Model

As shown in Figure 1, a dynamic model of the double-pendulum quadrotor transportation system was previously established on the basis of the Euler–Lagrange method [22].

$$\begin{array}{cc}\hfill & \left(M+m_h+m_p\right)\ddot{x}+\left(m_h+m_p\right)l_h\left(\ddot{\alpha }{C}_{\alpha }-{\dot{\alpha }}^2{S}_{\alpha }\right)+m_p{l}_p\left(\ddot{\beta }{C}_{\beta }-{\dot{\beta }}^2{S}_{\beta }\right)=Fsin\theta \hfill \\ \hfill & \left(M+m_h+m_p\right)\ddot{z}+\left(m_h+m_p\right)l_h\left(\ddot{\alpha }{S}_{\alpha }+{\dot{\alpha }}^2{S}_{\alpha }\right)+m_p{l}_p\left(\ddot{\beta }{S}_{\beta }-{\dot{\beta }}^2{S}_{\beta }\right)=Fcos\theta -g\left(M+m_h+m_p\right)\hfill \\ \hfill & \left(m_h+m_p\right)l_h\ddot{x}{C}_{\alpha }+\left(m_h+m_p\right)l_h\ddot{z}{S}_{\alpha }+\left(m_h+m_p\right)l_h^2\ddot{\alpha }+m_p{l}_h{l}_p{C}_{\alpha -\beta }\ddot{\beta }+m_p{l}_h{l}_p{S}_{\alpha -\beta }\dot{\alpha }\dot{\beta }+g{l}_h{S}_{\alpha }\left(m_h+m_p\right)=0\hfill \\ \hfill & m_p{l}_p\ddot{x}{C}_{\beta }+m_p{l}_p\ddot{z}{S}_{\beta }+m_p{l}_h{l}_p{C}_{\alpha -\beta }\ddot{\alpha }+m_p{l}_p^2\ddot{\beta }-m_p{l}_h{l}_p{S}_{\alpha -\beta }{\dot{\alpha }}^2+m_{p}g{l}_p{S}_{\beta }=0\hfill \\ \hfill & J\ddot{\theta }=\tau \hfill \end{array}$$

(1)

where $P\left(t\right)={\left[x\left(t\right),z\left(t\right)\right]}^T$ is the position of the quadrotor; M, $m_h$, and $m_p$, respectively, are the masses of the quadrotor, hook, and payload; $l_h,l_p\in {\Re }^{+}$ denote the length of the quadrotor’s center of mass to the hook and the payload to the hook, respectively; $\theta \left(t\right)$ is the attitude angle of the quadrotor; $F\left(t\right),\tau \left(t\right)$ are the control inputs; J is the moment of inertia; and $\alpha \left(t\right)$ and $\beta \left(t\right)$ are the swing angles of the hook and payload, respectively, which satisfy $-\pi /2\le \alpha \left(t\right)\le \pi /2,-\pi /2\le \beta \left(t\right)\le \pi /2$. $S_{\alpha },S_{\beta },C_{\alpha },C_{\beta },S_{\alpha -\beta }$ and $C_{\alpha -\beta }$ denote $sin\alpha ,sin\beta ,cos\alpha ,cos\beta ,sin(\alpha -\beta )$ and $cos(\alpha -\beta )$, respectively.

To avoid loss of generality, the following assumptions are made.

Assumption 1.

The hook and payload are always under the quadrotor body, and the cable is always tensioned, without any bending.

During the transportation process, the quadrotor’s flight performance is affected by the hook swing and payload swing, and the double-pendulum quadrotor transportation system has strong coupling and underactuation. As mentioned above, the research objective of this study is to make a quadrotor move stably to the desired position while inhibiting the hook swing and payload swing through design of an effective trajectory. To design the online trajectory generation system, we obtain the coupling dynamic relationship among the quadrotor, hook, and payload by rearranging Equation (1) as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}& \ddot{x}\left(l_h{C}_{\alpha }\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_p{C}_{\beta }\dot{\beta }\right)+\ddot{z}\left(l_h{S}_{\alpha }\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_p{S}_{\beta }\dot{\beta }\right)+\hfill \\ & \ddot{\alpha }\left[l_h^2\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_h{l}_p{C}_{\alpha -\beta }\dot{\beta }\right]+\ddot{\beta }\left[\frac{m_p}{m_h+m_p}{l}_h{l}_p{C}_{\alpha -\beta }\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_p^2\dot{\beta }\right]+\hfill \\ & l_{h}g\dot{\alpha }{S}_{\alpha }+\frac{m_p}{m_h+m_p}{l}_{p}g\dot{\beta }{S}_{\beta }=0.\hfill \end{array}\end{array}$$

(2)

Under consideration of the fact that the swing angles of the hook and payload are small in the actual transportation process, a reasonable approximation is made as follows:

$$C_{\alpha -\beta }\approx 1.$$

Then, Equation (2) can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{cc}& \ddot{x}\left(l_h{C}_{\alpha }\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_p{C}_{\beta }\dot{\beta }\right)+\ddot{z}\left(l_h{S}_{\alpha }\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_p{S}_{\beta }\dot{\beta }\right)+\hfill \\ & \ddot{\alpha }\left[l_h^2\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_h{l}_p\dot{\beta }\right]+\ddot{\beta }\left[\frac{m_p}{m_h+m_p}{l}_h{l}_p\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_p^2\dot{\beta }\right]+\hfill \\ & l_{h}g\dot{\alpha }{S}_{\alpha }+\frac{m_p}{m_h+m_p}{l}_{p}g\dot{\beta }{S}_{\beta }=0.\hfill \end{array}\end{array}$$

(3)

3. Online Trajectory Generation

In this section, an online trajectory generation method is provided in detail. The goal is to restrain the hook swing and payload swing while achieving precise positioning. Specifically, the proposed method with two capabilities—precise positioning and swing suppression. A swing-damping term is added into the real-time smooth trajectory of the quadrotor, which eliminates swings of the hook and payload without additional tracking controller, and does not affect the quadrotor positioning performance.

To achieve the above control objective, the following acceleration trajectory is designed:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\ddot{x}}_f\left(t\right)={\ddot{x}}_{\rho }\left(t\right)+{\ddot{\delta }}_x\left(t\right)\\ {\ddot{z}}_f\left(t\right)={\ddot{z}}_{\rho }\left(t\right)+{\ddot{\delta }}_z\left(t\right)\end{array}\right.\end{array}$$

(4)

where ${\ddot{x}}_{\rho }\left(t\right)$ and ${\ddot{z}}_{\rho }\left(t\right)$ are the acceleration trajectories of the quadrotor’s forward positioning and vertical positioning, respectively, which are chosen beforehand. ${\ddot{\delta }}_x\left(t\right)$ and ${\ddot{\delta }}_z\left(t\right)$ are the terms for elimination of the forward swing and vertical swing, respectively. ${\ddot{x}}_f\left(t\right)$ and ${\ddot{z}}_f\left(t\right)$ are the final acceleration trajectories of the system.

Therefore, the design of ${\ddot{\delta }}_x\left(t\right)$ and ${\ddot{\delta }}_z\left(t\right)$ is the focus of this paper. First, we make ${\ddot{x}}_{\rho }\left(t\right)={\ddot{z}}_{\rho }\left(t\right)=0$. By substituting Equation (4) into Equation (3), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& -{\ddot{\delta }}_x\left(l_h{C}_{\alpha }\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_p{C}_{\beta }\dot{\beta }\right)-{\ddot{\delta }}_z\left(l_h{S}_{\alpha }\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_p{S}_{\beta }\dot{\beta }\right)=\hfill \\ & \ddot{\alpha }\left[l_h^2\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_h{l}_p\dot{\beta }\right]+\ddot{\beta }\left[\frac{m_p}{m_h+m_p}{l}_h{l}_p\dot{\alpha }+\frac{m_p}{m_h+m_p}{l}_p^2\dot{\beta }\right]+\hfill \\ & l_{h}g\dot{\alpha }{S}_{\alpha }+\frac{m_p}{m_h+m_p}{l}_{p}g\dot{\beta }{S}_{\beta }.\hfill \end{array}\end{array}$$

(5)

Next, we can define a Lyapunov function as

$$\begin{array}{c}\hfill \begin{array}{cc}& V\left(t\right)=\frac{1}{2}{l}_h^2\dot{\alpha }{\left(t\right)}^2+\frac{1}{2}\frac{m_h}{m_h+m_p}{l}_p^2\dot{\beta }{\left(t\right)}^2+\frac{m_h}{m_h+m_p}{l}_h{l}_p\dot{\alpha }\left(t\right)\dot{\beta }\left(t\right)\hfill \\ & +l_{h}g(1-C_{\alpha })+\frac{m_h}{m_h+m_p}{l}_{p}g(1-C_{\beta })\ge 0.\hfill \end{array}\end{array}$$

(6)

By taking the time derivative of Equation (6) and substituting it into Equation (3), we obtain

$$\begin{array}{c}\hfill \dot{V}\left(t\right)=-\ddot{x}\left(t\right)\left(l_h{C}_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}{l}_p{C}_{\beta }\dot{\beta }\left(t\right)\right)-\ddot{z}\left(t\right)\left(l_h{S}_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}{l}_p{S}_{\beta }\dot{\beta }\left(t\right)\right).\end{array}$$

(7)

According to Equation (7), the system can converge stably when $\dot{V}\left(t\right)\le 0$ is guaranteed. Therefore, ${\ddot{\delta }}_x\left(t\right)$ and ${\ddot{\delta }}_z\left(t\right)$ can be designed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\ddot{\delta }}_x\left(t\right)=\kappa (C_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{C}_{\beta }\dot{\beta }\left(t\right))\\ {\ddot{\delta }}_z\left(t\right)=\kappa (S_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{S}_{\beta }\dot{\beta }\left(t\right))\end{array}\right.\end{array}$$

(8)

where $\kappa \in {\Re }^{+}$ is the control gain, which needs to satisfy $\kappa >l_h/2$. Substitution of Equation (8) into Equation (7) gives

$$\begin{array}{c}\hfill \dot{V}\left(t\right)=-\kappa l_h{\left(C_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{C}_{\beta }\dot{\beta }\left(t\right)\right)}^2-\kappa l_h{\left(S_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{S}_{\beta }\dot{\beta }\left(t\right)\right)}^2\le 0.\end{array}$$

(9)

Furthermore, on the basis of the LaSalle invariance theory, it is proved that the swing elimination terms can guarantee asymptotic convergence of $\alpha \left(t\right)$, $\beta \left(t\right)$, $\dot{\alpha }\left(t\right)$, $\dot{\beta }\left(t\right)$, $\ddot{\alpha }\left(t\right)$, and $\ddot{\beta }\left(t\right)$ to zero.

Next, we need to select the appropriate target trajectory to drive the quadrotor and combine it with the swing elimination parts to form the final trajectory. The desired trajectory of an S-curve with a smooth and continuous speed is set as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_{\rho }\left(t\right)=\frac{x_d}{2}+\frac{s_{vx}^2}{4s_{ax}}ln\left(\frac{cosh(2s_{ax}t/s_{vx}-\sigma )}{cosh(2s_{ax}t/s_{vx}-\sigma -2P_{dx}{s}_{ax}/s_{vx}^2)}\right)\\ z_{\rho }\left(t\right)=\frac{z_d}{2}+\frac{s_{vz}^2}{4s_{az}}ln\left(\frac{cosh(2s_{az}t/s_{vz}-\sigma )}{cosh(2s_{az}t/s_{vz}-\sigma -2P_{dz}{s}_{az}/s_{vz}^2)}\right)\end{array}\right.\end{array}$$

(10)

where $x_d$ and $z_d$ are the desired positions and $s_v$ and $s_a$ are the maximum speed and acceleration, respectively, permitted by the quadrotor’s hanging transport flight. $\sigma \in R^{+}$ is the adjustable initial acceleration. It should be noted that neither the velocity nor the acceleration of the quadrotor—which is a physical system—can exceed a usually known limit. On the basis of these facts, the selected desired trajectory should have the following properties:

Property 1. The selected trajectory converges asymptotically to $(x_d,z_d)$, whereas the velocity and acceleration converge asymptotically to zero:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \underset{t\to \infty }{lim}}{x}_{\rho }\left(t\right)=x_d\begin{array}{cc},& {\displaystyle \underset{t\to \infty }{lim}}{s}_{vx}=0\begin{array}{cc},& {\displaystyle \underset{t\to \infty }{lim}}{s}_{ax}=0\end{array}\end{array}\\ {\displaystyle \underset{t\to \infty }{lim}}{z}_{\rho }\left(t\right)=z_d\begin{array}{cc},& {\displaystyle \underset{t\to \infty }{lim}}{s}_{vz}=0\begin{array}{cc},& {\displaystyle \underset{t\to \infty }{lim}}{s}_{az}=0.\end{array}\end{array}\end{array}\right.\end{array}$$

(11)

Property 2. The velocity, acceleration, and added acceleration of the quadrotor are bounded:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}0\le {\dot{x}}_{\rho }\left(t\right)\le s_{vx}\begin{array}{cc},& \left|\ddot{x}\left(t\right)\right|\le s_{ax}\begin{array}{cc},& \left|\stackrel{⃛}{x}\left(t\right)\right|\le s_{jx}\end{array}\end{array}\\ 0\le {\dot{z}}_{\rho }\left(t\right)\le s_{vz}\begin{array}{cc},& \left|\ddot{z}\left(t\right)\right|\le s_{az}\begin{array}{cc},& \left|\stackrel{⃛}{z}\left(t\right)\right|\le s_{jz}\end{array}\end{array}\end{array}\right.\end{array}$$

(12)

where $s_{jx}=4s_{ax}^2/s_{vx},s_{jz}=4s_{az}^2/s_{vz}$ are added accelerations in the x-direction and z-direction, respectively.

Property 3. The quadrotor does not have an initial displacement and velocity:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_{\rho }\left(0\right)={\dot{x}}_{\rho }\left(0\right)=0\\ z_{\rho }\left(0\right)={\dot{z}}_{\rho }\left(0\right)=0.\end{array}\right.\end{array}$$

(13)

By combining Equations (4) and (8), we can express the final expected acceleration, velocity, and position trajectory of the system as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\ddot{x}}_f\left(t\right)={\ddot{x}}_{\rho }\left(t\right)+\kappa (C_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{C}_{\beta }\dot{\beta }\left(t\right))\\ {\ddot{z}}_f\left(t\right)={\ddot{z}}_{\rho }\left(t\right)+\kappa (S_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{S}_{\beta }\dot{\beta }\left(t\right))\end{array}\right.\end{array}$$

(14)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_f\left(t\right)={\dot{x}}_{\rho }\left(t\right)+\kappa (S_{\alpha }+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{S}_{\beta })\\ {\dot{z}}_f\left(t\right)={\dot{z}}_{\rho }\left(t\right)-\kappa (C_{\alpha }-\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{C}_{\beta })\end{array}\right.\end{array}$$

(15)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_f\left(t\right)=x_{\rho }\left(t\right)+\kappa {\int }_0^t\left(S_{\alpha }+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{S}_{\beta }\right)dt\\ z_f\left(t\right)=z_{\rho }\left(t\right)-\kappa {\int }_0^t\left(C_{\alpha }-\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{C}_{\beta }\right)dt.\end{array}\right.\end{array}$$

(16)

4. Convergence Analysis

Theorem 1.

The quadrotor ultimate trajectory can ensure that the swing angle, angular velocity, and angular acceleration of the hook and load converge to zero asymptotically.

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\left[\begin{array}{cccccc}\alpha \left(t\right)& \beta \left(t\right)& \dot{\alpha }\left(t\right)& \dot{\beta }\left(t\right)& \ddot{\alpha }\left(t\right)& \ddot{\beta }\left(t\right)\end{array}\right]}^T={\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\end{array}\right]}^T.\end{array}$$

(17)

Proof of Theorem 1.

Consider the Lyapunov function $V\left(t\right)$; by substituting the final acceleration trajectories $\ddot{x}\left(t\right)$, $\ddot{z}\left(t\right)$ in (14) into Equation (7), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \dot{V}\left(t\right)=-\left(l_h{C}_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}{l}_p{C}_{\beta }\dot{\beta }\left(t\right)\right){\ddot{x}}_f\left(t\right)-\left(l_h{C}_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}{l}_p{C}_{\beta }\dot{\beta }\left(t\right)\right){\ddot{z}}_f\left(t\right)\hfill \\ & =-{\ddot{x}}_{\rho }\left(t\right)\left(l_h{C}_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}{l}_p{C}_{\beta }\dot{\beta }\left(t\right)\right)-\kappa l_h{\left(C_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{C}_{\beta }\dot{\beta }\left(t\right)\right)}^2\hfill \\ & -{\ddot{z}}_{\rho }\left(t\right)\left(l_h{S}_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}{l}_p{S}_{\beta }\dot{\beta }\left(t\right)\right)-\kappa l_h{\left(S_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{S}_{\beta }\dot{\beta }\left(t\right)\right)}^2.\hfill \end{array}\end{array}$$

(18)

□

On the basis of the average algebraic-geometric inequality, Equation (18) can be written as

$$\begin{array}{c}\hfill \begin{array}{cc}& \dot{V}\left(t\right)\le -l_h(\kappa -\frac{1}{2}{l}_h){\left(C_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{C}_{\beta }\dot{\beta }\left(t\right)\right)}^2\hfill \\ & -l_h(\kappa -\frac{1}{2}{l}_h){\left(S_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{S}_{\beta }\dot{\beta }\left(t\right)\right)}^2\hfill \\ & +\frac{1}{2}{\ddot{x}}_{\rho }^2\left(t\right)+\frac{1}{2}{\ddot{z}}_{\rho }^2\left(t\right).\hfill \end{array}\end{array}$$

(19)

By taking the time integral of Equation (19), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& V\left(t\right)\le -l_h(\kappa -\frac{1}{2}{l}_h){\int }_0^t{\left(C_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{C}_{\beta }\dot{\beta }\left(t\right)\right)}^{2}dt\hfill \\ & -l_h(\kappa -\frac{1}{2}{l}_h){\int }_0^t{\left(S_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{S}_{\beta }\dot{\beta }\left(t\right)\right)}^{2}dt\hfill \\ & +\frac{1}{2}{\int }_0^t{\ddot{x}}_{\rho }^2\left(t\right)dt+\frac{1}{2}{\int }_0^t{\ddot{z}}_{\rho }^2\left(t\right)dt+V\left(0\right).\hfill \end{array}\end{array}$$

(20)

According to Property 3, the last two items in Equation (20) can be rewritten as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{1}{2}{\int }_0^t{\ddot{x}}_{\rho }^2\left(t\right)dt=\frac{1}{2}{\ddot{x}}_{\rho }\left(t\right){\dot{x}}_{\rho }\left(t\right)+\frac{1}{2}{}_{\rho }\left(t\right){\dot{x}}_{\rho }\left(t\right)\in L_{\infty }\\ \frac{1}{2}{\int }_0^t{\ddot{z}}_{\rho }^2\left(t\right)dt=\frac{1}{2}{\ddot{z}}_{\rho }\left(t\right){\dot{z}}_{\rho }\left(t\right)+\frac{1}{2}{}_{\rho }\left(t\right){\dot{z}}_{\rho }\left(t\right)\in L_{\infty }.\end{array}\right.\end{array}$$

(21)

When $\kappa >l_h/2$ is satisfied, the following inequality relation holds:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-l_h(\kappa -\frac{1}{2}{l}_h){\int }_0^t{\left(C_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{C}_{\beta }\dot{\beta }\left(t\right)\right)}^{2}dt\le 0\\ -l_h(\kappa -\frac{1}{2}{l}_h){\int }_0^t{\left(S_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{S}_{\beta }\dot{\beta }\left(t\right)\right)}^{2}dt\le 0.\end{array}\right.\end{array}$$

(22)

The following conclusion can be drawn on the basis of Equations (4), (12), and (20)–(22):

$$\begin{array}{c}\hfill V\left(t\right)\in L_{\infty }\Rightarrow \dot{\alpha }\left(t\right),\dot{\beta }\left(t\right)\in L_{\infty }\Rightarrow {\ddot{x}}_f\left(t\right),{\ddot{z}}_f\left(t\right)\in L_{\infty }.\end{array}$$

(23)

For this reason, Equation (20) can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{cc}& l_h(\kappa -\frac{1}{2}{l}_h){\int }_0^t{\left(C_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{C}_{\beta }\dot{\beta }\left(t\right)\right)}^{2}dt\hfill \\ & +l_h(\kappa -\frac{1}{2}{l}_h){\int }_0^t{\left(S_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{S}_{\beta }\dot{\beta }\left(t\right)\right)}^{2}dt\hfill \\ & \le \frac{1}{2}{\int }_0^t{\ddot{x}}_{\rho }^2\left(t\right)dt+\frac{1}{2}{\int }_0^t{\ddot{z}}_{\rho }^2\left(t\right)dt+V\left(0\right)-V\left(t\right)\in L_{\infty }.\hfill \end{array}\end{array}$$

(24)

From this, the following conclusion can be drawn:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{C}_{\beta }\dot{\beta }\left(t\right)\in L_{\infty }\bigcap L_2\\ S_{\alpha }\dot{\alpha }\left(t\right)+\frac{m_p}{m_h+m_p}\frac{l_p}{l_h}{S}_{\beta }\dot{\beta }\left(t\right)\in L_{\infty }\bigcap L_2.\end{array}\right.\end{array}$$

(25)

Next, we substitute Equation (4) into the last two terms of Equation (1) and take into account the conditions in Equation (3), which can be sorted as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& \ddot{\alpha }\left(t\right)=-\frac{1}{m_h{l}_h}\underset{f_1\left(t\right)}{\underbrace{[{\ddot{x}}_f\left(t\right)\left(m_h{C}_{\alpha }+m_p{C}_{\alpha }-m_p{C}_{\beta }\right)+{\ddot{z}}_f\left(t\right)\left(m_h{S}_{\alpha }+m_p{S}_{\alpha }-m_p{S}_{\beta }\right)}}\hfill \\ & +\underset{f_2\left(t\right)}{\underbrace{m_p{l}_p{S}_{\alpha -\beta }\dot{\alpha }\left(t\right)\dot{\beta }\left(t\right)+m_p{l}_h{S}_{\alpha -\beta }\dot{\alpha }{\left(t\right)}^2+g{S}_{\alpha }(m_h+m_p)-m_{p}g{S}_{\beta }]}}\hfill \\ & \ddot{\beta }\left(t\right)=-\frac{1}{m_h{l}_p}\underset{f_3\left(t\right)}{\underbrace{[{\ddot{x}}_f\left(t\right)\left(m_h+m_p\right)\left(C_{\beta }-C_{\alpha }\right)+{\ddot{z}}_f\left(t\right)\left(m_h+m_p\right)\left(S_{\beta }-S_{\alpha }\right)}}\hfill \\ & -\underset{f_4\left(t\right)}{\underbrace{\left(m_h+m_p\right)l_h{S}_{\alpha -\beta }\dot{\alpha }{\left(t\right)}^2-m_p{l}_p{S}_{\alpha -\beta }\dot{\alpha }\left(t\right)\dot{\beta }\left(t\right)+\left(m_h+m_p\right)g\left(S_{\beta }-S_{\alpha }\right)}}].\hfill \end{array}\right.\end{array}$$

(26)

From Equation (23), we obtain

$$\begin{array}{c}\hfill \ddot{\alpha }\left(t\right),\ddot{\beta }\left(t\right)\in L_{\infty }.\end{array}$$

(27)

On the basis of the Barbara theorem, the following conclusion can be drawn for Equations (25) and (27):

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\dot{\alpha }\left(t\right)=0\begin{array}{cc}& {\displaystyle \underset{t\to \infty }{lim}}\dot{\beta }\left(t\right)=0.\end{array}\end{array}$$

(28)

It can be obtained from Equations (14) and (28):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \underset{t\to \infty }{lim}}{\ddot{x}}_{\rho }\left(t\right)={\displaystyle \underset{t\to \infty }{lim}}{\ddot{\delta }}_x\left(t\right)=0\Rightarrow {\displaystyle \underset{t\to \infty }{lim}}{\ddot{x}}_f\left(t\right)=0\\ {\displaystyle \underset{t\to \infty }{lim}}{\ddot{z}}_{\rho }\left(t\right)={\displaystyle \underset{t\to \infty }{lim}}{\ddot{\delta }}_z\left(t\right)=0\Rightarrow {\displaystyle \underset{t\to \infty }{lim}}{\ddot{z}}_f\left(t\right)=0.\end{array}\right.\end{array}$$

(29)

Further, Equations (23), (26), and (29) give

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \underset{t\to \infty }{lim}}{f}_1\left(t\right)={\displaystyle \underset{t\to \infty }{lim}}{f}_3\left(t\right)=0\\ {\dot{f}}_2\left(t\right)\in L_{\infty },{\dot{f}}_4\left(t\right)\in L_{\infty }.\end{array}\right.\end{array}$$

(30)

Therefore, by introducing the extended Barbara theorem, we obtain

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\ddot{\alpha }\left(t\right)=0\begin{array}{cc},& {\displaystyle \underset{t\to \infty }{lim}}\ddot{\beta }\left(t\right)=0.\end{array}\end{array}$$

(31)

Next, the following conclusions can be drawn from the constraint conditions of the quadrotor in the transportation process:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \underset{t\to \infty }{lim}}sin\alpha \left(t\right)=0\Rightarrow {\displaystyle \underset{t\to \infty }{lim}}\alpha \left(t\right)=0\\ {\displaystyle \underset{t\to \infty }{lim}}sin\beta \left(t\right)=0\Rightarrow {\displaystyle \underset{t\to \infty }{lim}}\beta \left(t\right)=0.\end{array}\right.\end{array}$$

(32)

Finally, Theorem 1 is proved by combining the conclusions in Equations (28), (31), and (32).

Theorem 2.

The designed online trajectory can drive the quadrotor to the desired position and ensure convergence of its flight speed and acceleration to zero.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \underset{t\to \infty }{lim}}\left[\begin{array}{ccc}{x}_f\left(t\right)& {\dot{x}}_f\left(t\right)& {\ddot{x}}_f\left(t\right)\end{array}\right]=\left[\begin{array}{ccc}{x}_d& 0& 0\end{array}\right]\\ {\displaystyle \underset{t\to \infty }{lim}}\left[\begin{array}{ccc}{z}_f\left(t\right)& {\dot{z}}_f\left(t\right)& {\ddot{z}}_f\left(t\right)\end{array}\right]=\left[\begin{array}{ccc}{z}_d& 0& 0\end{array}\right].\end{array}\right.\end{array}$$

(33)

Proof of Theorem 2.

First, from Equations (10), (15), and (32), we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \underset{t\to \infty }{lim}}{\dot{x}}_f\left(t\right)=0\\ {\displaystyle \underset{t\to \infty }{lim}}{\dot{z}}_f\left(t\right)=0.\end{array}\right.\end{array}$$

(34)

□

Next, consider again the swing angles of the hook and payload are small enough in the transportation process, $S_{\alpha }\approx 0$, $S_{\beta }\approx 0$, $C_{\alpha }\approx 1$, $C_{\beta }\approx 1$, the last two terms of Equation (1) can be summarized as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& \alpha \left(t\right)=-\frac{1}{{\ddot{x}}_{\rho }\left(t\right)+g}\left[{\ddot{x}}_{\rho }\left(t\right)+l_h\ddot{\alpha }\left(t\right)+\frac{m_p{l}_p}{\left(m_h+m_p\right)}\ddot{\beta }\left(t\right)\right]\hfill \\ & \beta \left(t\right)=-\frac{1}{{\ddot{z}}_{\rho }\left(t\right)+g}\left[{\ddot{x}}_{\rho }\left(t\right)+l_h\ddot{\alpha }\left(t\right)+l_p\ddot{\beta }\left(t\right)\right].\hfill \end{array}\right.\end{array}$$

(35)

We take the time integral of the above expression and also take the limit value. We simultaneously consider the initial conditions, i.e., $\alpha \left(0\right)=\beta \left(0\right)=\dot{\alpha }\left(0\right)=\dot{\beta }\left(0\right)=0$, and Property 3. The following conclusions can be drawn on the basis of Equations (28) and (34):

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\int }_0^t\alpha \left(t\right)dt=0\begin{array}{cc},& {\displaystyle \underset{t\to \infty }{lim}}{\int }_0^t\beta \left(t\right)dt=0.\end{array}\end{array}$$

(36)

Then, the following conclusions can be drawn on the basis of Equations (11), (16), and (36):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \underset{t\to \infty }{lim}}{x}_f\left(t\right)={\displaystyle \underset{t\to \infty }{lim}}{x}_{\rho }\left(t\right)=x_d\\ {\displaystyle \underset{t\to \infty }{lim}}{z}_f\left(t\right)={\displaystyle \underset{t\to \infty }{lim}}{z}_{\rho }\left(t\right)=z_d.\end{array}\right.\end{array}$$

(37)

Theorem 2 is therefore proved.

5. Simulation Analysis

In this section, we verify the control performance of the proposed online trajectory planning method through numerical simulation. First, we demonstrate the excellent performance of the proposed method by comparing it with nonlinear control and PD control. Then, we verify the robustness of the proposed method by using different parameters and applying an external disturbance.

The following nonlinear controller is designed to make the quadrotor move according to the planned trajectory [27]:

$$\left\{\begin{array}{c}{u}_x=-k_{nx}{e}_x-k_{mx}{\dot{e}}_x\\ u_z=-k_{nz}{e}_z-k_{mz}{\dot{e}}_z\end{array}\right.$$

where the control inputs are defined as $u_x=Fsin\theta$, $u_z=Fcos\theta$; the error signals are defined as $e_x=x-k_1{S}_{\alpha }-k_2{S}_{\beta }-x_d$, $e_z=z-k_1(1-C_{\alpha })-k_2(1-C_{\beta })-z_d$; $k_{nx}=0.8,k_{mx}=4.5,k_{nz}=0.8,k_{mz}=4.5,k_1=3,k_2=0.16$ are positive control gains; and x and z denote the actual positions.

The following PD controller is designed to make the quadrotor move according to the planned trajectory:

$$\left\{\begin{array}{c}A=k_p{e}_x+k_d\dot{x}\hfill \\ B=k_p{e}_z+k_d\dot{z}-\left(M+m_h+m_p\right)g\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{c}{u}_x=-Acos\gamma -Bsin\gamma \hfill \\ u_z=Asin\gamma -Bcos\gamma \hfill \end{array}\right.$$

where $k_p=3,k_d=10$ are positive control gains, which are obtained by repeated testing.

It can be seen from the structure of the nonlinear controller and the PD controller that trajectory planning is not included. Therefore, it is easy to compare the advantages of the proposed method.

To ensure the integrity of the control system, the attitude controller is designed based on the feedback linearization method [20].

$$\tau =-J\left(k_{p\theta }{e}_{\theta }+k_{d\theta }{\dot{e}}_{\theta }\right)$$

where, $e_{\theta }=\theta -{\theta }_d$ is attitude angle tracking error, $k_{p\theta }=10$, $k_{p\theta }=20$ are the positive control gains.

For ease of verification of the simulation results, the following are considered as the system parameters for all the simulation tests: M = 7 kg, $m_h$ = 3 kg, $m_p$ = 1.5 kg, $l_h$ = 1 m, $l_p$ = 0.6 m, g = 0.98 m/s ${}^2$.

5.1. Comparison Test

In this test, nonlinear control and PD control are selected for comparison with the proposed method. The simulation results are shown in Figure 2, Figure 3 and Figure 4, and the corresponding quantization results are listed in

Table 1. Compare the quantitative results of the tests.

Control Methods	$(\mathit{x},\mathit{z})\left(\mathit{m}\right)$	$\mathit{t}\left(\mathit{s}\right)$	${\mathit{\alpha }}_{max}{(}^{\circ })$	${\mathit{\beta }}_{max}{(}^{\circ })$	$\mathit{F}\left(\mathit{N}\right)$
PD control	(5.0, 5.0)	12.0	9.0	16.1	128.6
Nonlinear control	(5.0, 5.0)	13.5	2.2	3.9	116.8
Online trajectory planning	(5.0, 5.0)	17.4	0.7	0.7	114.0

. The following performance metrics are compared.

• Final position of the quadrotor $(x,z)\left(m\right)$.
• Transportation time $t\left(s\right)$ (the time when the quadrotor reaches the target position).
• The maximum swing angles of the hook and payload during transportation ${\alpha }_{max}{(}^{\circ })$, ${\beta }_{max}{(}^{\circ })$.
• Consumption of driving energy in the entire transportation process $F\left(N\right)$.

The results show that all the three control methods can drive the quadrotor to the desired position. Since online trajectory planning constrains the maximum acceleration and velocity of the quadrotor, the transportation time in this method is slightly longer than those in the other two methods. However, online trajectory planning can effectively prevent a large change in the attitude at the start time of the quadrotor to achieve smooth start and stop and consequently prevent larger initial swings of the hook and payload. The ranges of the swing angles of the hook and payload in the case of the proposed method are smaller than those in the cases of the nonlinear control and PD control methods, and the residual swing can be rapidly eliminated when the proposed method is applied. In addition, from the perspective of the generated thrust and flight attitude, the maximum driving force and the energy consumption are smaller and lower, respectively, when the proposed method is applied. Therefore, the proposed online trajectory planning method has superior control performance to the other two methods.

5.2. Robustness Test

Case 1: Different target positions

To further verify the performance of the proposed online trajectory planning method, the following three situations are considered for the control performance of different target positions without changing controller parameters.

Situation 1: $(x_d,z_d)$ = (5, 5)

Situation 2: $(x_d,z_d)$ = (15, 12)

Situation 3: $(x_d,z_d)$ = (20, 20)

In this test, both the hook and the payload have an initial swing angle of 0${}^{\circ }$. Simulation results of Figure 5 show that the proposed method can complete online trajectory planning for different target positions. It should be pointed out that, regardless of the target position, precise positioning is achieved and violent swings of the hook and payload are well inhibited; this result proves that the proposed method has good online planning ability.

Case 2: Different initial swing angles

To further verify the robustness of the proposed method, we consider the following three scenarios with different initial swing angles of the hook and payload, in which the controller parameters remain unchanged.

Situation 1: $\alpha$ = 5${}^{\circ }$$\beta$ = 0${}^{\circ }$

Situation 2: $\alpha$ = 0${}^{\circ }$$\beta$ = 5${}^{\circ }$

Situation 3: $\alpha$ = 5${}^{\circ }$$\beta$ = 5${}^{\circ }$

It can be seen from Figure 6 that the quadrotor can be adjusted rapidly, and the swing is well restrained even when the hook and load have different initial swing angles. Furthermore, accurate positioning is achieved, and the residual swings of the hook and payload are rapidly eliminated.

Case 3: External disturbance

To verify the anti-disturbance performance of the proposed method, we apply a 5${}^{\circ }$ impulse disturbance to the payload at 8 s and 60 s. It can be seen from Figure 7 that the proposed method rapidly suppresses and eliminates the external disturbance, which demonstrates its high robustness.

6. Conclusions

This paper focuses on the elimination swing angles and the positioning of the double-pendulum quadrotor transportation systems. A swing-damping term is added into the real-time smooth trajectory of the quadrotor, which eliminates swings of the hook and payload without additional tracking controller and does not affect the quadrotor positioning performance. Subsequently, the asymptotic stability and convergence of the system are proved using the Lyapunov principle and the LaSalle invariance theory. Simulation tests demonstrate that the proposed method has excellent control performance and high robustness to uncertainties and external disturbances. In addition, most of the existing research on the quadrotor transport systems are focused on the single-pendulum. The proposed method in this paper solves the problem of swing and positioning in the presence of double pendulum and is expected to be extended to more complex suspension systems. In future work, cooperative transportation using multiple quadrotors will be considered.