Couple Anti-Swing Obstacle Avoidance Control Strategy for Underactuated Overhead Cranes

Abstract

Overhead cranes are widely used for transportation in factories. They move slowly by manual operation to prevent the payload from swinging sharply or colliding with sudden obstacles. To address these issues and enhance work efficiency, this paper proposes a couple anti-swing obstacle avoidance control method for 5-DOF overhead cranes. Time polynomial fitting is employed for trajectory planning to achieve obstacle avoidance. To achieve anti-swing of the payloads, a coupled variable incorporating both actuated and underactuated states is defined, alongside a boundary for dynamic performance. Finally, MATLAB simulation and hardware experiments are carried out to verify the reliability and compared with some existing control methods.

1. Introduction

Overhead cranes play an important role in the transportation of heavy loads. The operation of overhead cranes depends on skilled operators. However, there is always poor lighting and even high temperature in the work environment of overhead cranes, such that it is difficult to guarantee positioning accuracy anti-swing and obstacle avoidance. Therefore, it is necessary to achieve automation for overhead cranes, and there have been many studies on this topic so far [1,2,3,4].

In recent decades, many scholars have designed a large number of effective controllers for overhead cranes based on different methods, such as input shapers, proportional–integral–derivative (PID) control, optimal control, sliding mode control, intelligent control and so forth. Ref. [5] integrates input shaping methods with optimization control to achieve enhanced control of overhead cranes. Ref. [6] combines input shaping with neuro-adaptive sliding mode control. Refs. [7,8,9] adopt a trajectory planning method to achieve control of overhead cranes. Input shaping and trajectory planning are classic open-loop control methods. Although their structure is simple, they are helpless for small tracking errors and lack robustness once external disturbances suddenly appear. PID control is one of the most widely used control techniques in actual applications. In [10], a neural network adaptive PID control method is proposed. Ref. [11] introduces a unique algorithm (Stray Lion Swarm Optimization) into a fuzzy PID method, which addresses the challenging issue of parameter adjustment for the control of overhead cranes. Ref. [12] develops a PID-like coupling control for overhead cranes with input constraints. In [13], by analyzing the coupling relationships between state variables of overhead cranes, a new control strategy based on payload energy coupling is designed. PID and energy-based methods are both model-free control approaches. For nonlinear systems like overhead cranes, they cannot achieve excellent control performance. Optimal control is a renowned method for solving time optimization problems. For overhead cranes, various optimal control methods are combined with trajectory planning [8], visual feedback control [14] and integral sliding mode control [15]. Sliding mode control is also widely used for the control of overhead cranes [16,17,18]. A discrete time sliding mode predictive control for overhead cranes based on a disturbance observer is proposed in [19], which includes receding rolling optimization and feedback correction for the predictive control. Ref. [20] presents a sliding mode control strategy based on a finite-time disturbance observer for anti-swing control of overhead cranes, which can easily handle the presence of unpredictable disturbances. In recent years, intelligent control methods such as neural network control [21,22] and fuzzy control [23] have been applied to overhead crane systems. In [24], an energy-coupling output feedback control strategy is proposed for the 4-DOF overhead crane system, achieving precise trolley positioning and effective payload swing suppression while taking practical input constraints into account. Based on the energy shaping methodology and passivity based control, Ref. [25] propose an enhanced coupling control method for 4-DOF overhead cranes. The research objects of [24,25] are 4-DOF overhead crane systems, but the control of rope lifting can not be realized. Ref. [26] gives the Euler–Lagrange equations of 5-DOF overhead cranes and designs a second-order sliding mode controller for this system. However, these methods often require substantial computational resources, such as in the complexity of controller design, which is unavoidable.

Cranes often work in complex environments and there may be obstacles on transport paths. For a sudden obstacle, an emergency stop is always chosen [27]. However, swing angles of loads will increase sharply. The methods of avoiding obstacles can be divided into simple trajectory planning methods [28,29,30,31,32] and a combination of trajectory planning with other methods. Simple trajectory planning methods always define paths in initial stages and cannot respond to obstacles that suddenly appear. In [33], a control method that combines LQR (Linear Quadratic Regulator) with trajectory planning is proposed, which enables the system to jump over or bypass obstacles. In [34], based on neural network technology, an adaptive mechanism is introduced, which can update trajectory parameters in real time such that dynamic trajectory planning for payloads is achieved. By modeling the working environment of cranes, Ref. [35] shows an improved ant colony algorithm suitable for solving crane path planning, which can find routes with fewer turning points and the shortest path. Combining complex methods with trajectory planning can result in enormous computational demands. These control methods avoid collisions between obstacles and the payload through various trajectory planning techniques. However, they do not take into account collisions that might occur due to payload swinging during transportation. Most of the aforementioned obstacle avoidance strategies focus on overhead cranes with lower degrees of freedom and do not simultaneously account for the movements of the trolley, bridge and rope. Although [36] designed a spatial obstacle avoidance strategy for 5-DOF overhead cranes, there are few control strategies with an obstacle avoidance function for 5-DOF bridge cranes.

Considering the above issues, this paper presents a control method with obstacle avoidance and state constraint for overhead cranes, which can transport payloads to desired positions with or without obstacles and ensure that swing angles of payloads remain within predefined certain ranges.

The main contributions of this paper are presented as follows:

(1)

Based on a novel block-backstepping method, an anti-swing controller with obstacle avoidance is designed for overhead cranes. Different from the traditional backstepping methods, this method depends on actuated and underactuated states, such that accurate positioning and anti-swing are ensured simultaneously.

(2)

A dynamic performance boundary is defined, which can always keep swing angles of payloads within predetermined certain ranges. Besides, the differentiability of the dynamic boundary is guaranteed by a set of low-pass filters, whose parameters can be obtained systematically.

(3)

The proposed controller plans the trajectory of payloads and selects an obstacle avoidance strategy. By integrating advanced polynomial fitting technology, it ensures that the modified trajectory remains as close as possible to the original, with the primary condition being safe navigation around obstacles.

The remainder of this paper is organized as follows: In Section 2, the dynamical model of overhead cranes is given, and trajectory modification and state constraints are described. Section 3 displays the constraint function, controller design and stability analysis. The experimental results of the proposed controller are described in Section 4. Finally, a summary is presented in Section 5.

2. Problem Formulation and Preliminaries

2.1. Dynamical Model

In this paper, a tracking controller is designed for the overhead cranes with variable rope length, whose dynamic model [37] (see Figure 1) is given as follows:

$$M_t\left(q\right)\ddot{q}+C_t\left(q,\dot{q}\right)\dot{q}+G_t\left(q\right)-F_d=U$$

(1)

where $q={\left[\eta \left(t\right),\phi \left(t\right),l\left(t\right),{\delta }_1\left(t\right),{\delta }_2\left(t\right)\right]}^T\in {\mathbb{R}}^{5\times 1}$ represents the state vector, and $M_t\left(q\right)\in {\mathbb{R}}^{5\times 5}$ and $G_t\left(q\right)\in {\mathbb{R}}^{5\times 1}$ are the inertia and gravity matrices, respectively.$C_t\left(q,\dot{q}\right)\in {\mathbb{R}}^{5\times 5}$ is the centripetal-Coriolis matrix. $U={\left[\begin{array}{ccccc}{F}_{\eta }& F_{\phi }& F_{\eta }& 0& 0\end{array}\right]}^T\in {\mathbb{R}}^{5\times 1}$ denotes the system input vector, and $F_d\in {\mathbb{R}}^{5\times 1}$ is the damping vector. The above matrices/vectors are given in Appendix A. The physical parameters of the system states are shown in

Table 1. Parameters of overhead cranes.

Parameter	Physical Meaning	Unit
$\eta$	trolley translation displacement	m
$\phi$	displacement of bridge	m
l	cable length	m
${\delta }_1$	payload swing angle in the x direction	rad
${\delta }_2$	payload swing angle in the y direction	rad
$M_x$	trolley mass	kg
$M_y$	mass of trolley and bridge	kg
m	payload mass	kg
g	gravity acceleration constant	m/s2
$F_{\eta }$	translation control force of trolley	N
$F_{\phi }$	translation control force of bridge	N
$F_l$	hoisting/lowering actuating force	N
$f_{rx}$, $f_{ry}$	coefficients of sliding friction	N
$x_l$, $y_l$, $z_l$,	position of the load center of payload	m
$d_x$, $d_y$, $d_l$	resistance coefficients	/
$d_f$	coefficients of air resistance	/

.

By multiplying (1) by $M_t^{-1}\left(q\right),$ one has

$$\ddot{q}=M_t^{-1}\left(q\right)\left(U+f_d-G_t\left(q\right)-C\left(q,\dot{q}\right)\dot{q}\right)$$

(2)

Define $M^{-1}\left(q\right)=\left[\begin{array}{cc}{M}_{11}^{*}& M_{12}^{*}\\ M_{21}^{*}& M_{22}^{*}\end{array}\right]$ and ${\left[\begin{array}{cc}{F}_1& F_2\end{array}\right]}^T=M_t^{-1}\left(q\right)\left(f_d-G_t\left(q\right)-C\left(q,\dot{q}\right)\dot{q}\right)$ with $M_{11}^{*}\in {\mathbb{R}}^{3\times 3},M_{12}^{*}\in {\mathbb{R}}^{2\times 1},$$M_{21}^{*}\in {\mathbb{R}}^{2\times 3},E_{22}\in {\mathbb{R}}^{2\times 2},F_1\in {\mathbb{R}}^{3\times 1}$ and $F_2\in {\mathbb{R}}^{2\times 1}.$ For convenience, (2) is reformulated as

$$\begin{array}{ccc}\hfill {\dot{x}}_1& =& z_1\hfill \\ \hfill {\dot{x}}_2& =& z_2\hfill \\ \hfill {\dot{z}}_1& =& f_1\left(x,z\right)+g_1\left(x\right)u\hfill \\ \hfill {\dot{z}}_2& =& f_2\left(x,z\right)+g_2\left(x\right)u\hfill \\ \hfill y& =& x_1\hfill \end{array}$$

(3)

where

$$x_1={\left[\begin{array}{ccc}\eta & \phi & l\end{array}\right]}^T,z_1={\left[\begin{array}{ccc}\dot{\eta }& \dot{\phi }& \dot{l}\end{array}\right]}^T,x_2={\left[\begin{array}{cc}{\delta }_1& {\delta }_2\end{array}\right]}^T,z_2={\left[\begin{array}{cc}{\dot{\delta }}_1& {\dot{\delta }}_2\end{array}\right]}^T,$$

and

$$f_1=F_1,f_2=F_2,g_1=M_{11}^{*},g_2=M_{21}^{*},u=F_1.$$

Assumption 1. 

The desired output $y_d={\left[\begin{array}{ccc}{\eta }_d& {\phi }_d& l_d\end{array}\right]}^T$ is a second-order continuously differentiable vector $\left(y_d\in C^3\right),$ and $y_d^{\left(i\right)}$ is bounded, where $i=0,1,2.$

Assumption 2. 

There exist constants ${\delta }_{min}\in \left(-{\displaystyle \frac{\pi }{2}},0\right)$ and ${\delta }_{max}\in \left(0,{\displaystyle \frac{\pi }{2}}\right)$ such that the swing angle of payload ${\delta }_i$ satisfies

$${\delta }_{min}\le {\delta }_i\le {\delta }_{max},i=1,2.$$

Assumption 3. 

The rope is considered to be a rigid body that is not bendable.

The control objectives of the controller studied in this paper are expressed:

(1) 

As is known, this is a tracking controller designed for obstacle avoidance, enabling the payload to avoid obstacles of various shapes and reach the desired position.

(2) 

The swing angles ${\delta }_1$ and ${\delta }_2$ are always constrained within the predefined boundaries ${\xi }_1$ and ${\xi }_2,$ and they are asymptotically stable, which means

$$\begin{array}{ccc}\hfill -{\xi }_1& <& {\delta }_1<{\xi }_1,\underset{t\to \infty }{lim}{\delta }_1=0,\hfill \\ \hfill -{\xi }_2& <& {\delta }_2<{\xi }_2,\underset{t\to \infty }{lim}{\delta }_2=0.\hfill \end{array}$$

(3) 

The displacement of the trolley, movement of the bridge and length of the rope are asymptotically stable, which satisfy

$$\underset{t\to \infty }{lim}\eta ={\eta }_d,\underset{t\to \infty }{lim}\phi ={\phi }_d,\underset{t\to \infty }{lim}l=l_d.$$

2.2. The Trajectory Modification

A straight line connecting the initial position $\left(x_0,y_0,z_0\right)$ and desired position $\left(x_d,y_d,z_d\right)$ of the payload is adopted as an ideal trajectory. However, there may be obstacles on the ideal path. To deal with the problem, a trajectory modification method that combines time polynomial fitting (TPF) with the utilization of bounding circles around obstacles is proposed, which ensures a smooth and collision-free path for the payload. For 5-DOF overhead cranes, obstacle avoidance can be realized by rope lifting and bridge movement, respectively. $\left(X,Y,Z\right)$ and $\left(\acute{X},\acute{Y},\acute{Z}\right)$ are the based and payload coordinate systems, respectively.

Step 1. Obstacle trajectory generation.

Assuming that there is an obstacle in the payload transportation trajectory, it is necessary to avoid it by lifting cables (as shown in Figure 2). In an $\left(\acute{X},O,\acute{Z}\right)$ coordinate system, the ideal trajectory of the payload in the vertical direction is constructed as $\acute{z}$

$$\acute{z}=\left\{\begin{array}{cc}0& \acute{x}\le {\acute{x}}_1\mathrm{and}\acute{x}\ge {\acute{x}}_4\\ l_{s1}\left(\acute{x}\right)& {\acute{x}}_1<\acute{x}<{\acute{x}}_2\\ S_b\left(\acute{x}\right)& {\acute{x}}_2\le \acute{x}\le {\acute{x}}_3\\ l_{s2}\left(\acute{x}\right)& {\acute{x}}_3<\acute{x}<{\acute{x}}_4\end{array}\right.$$

(4)

where ${\acute{x}}_1<{\acute{x}}_2<{\acute{x}}_3<{\acute{x}}_4$ denote the switching points along the trajectory, which can be dynamically chosen based on the payload’s actual position relative to obstacles. $l_{s1}\left(\acute{x}\right),$$S_b\left(\acute{x}\right)$ and $l_{s2}\left(\acute{x}\right)$ are different parts of ideal trajectories, and the detailed generation processes for them are provided below.

First, the Arc segment $S_b$ is designed as

$$S_b\left(\acute{x}\right)=\sqrt{r_1^2-{\left(\acute{x}-{\acute{x}}_s\right)}^2}+{\acute{z}}_s,$$

(5)

where the point $\left({\acute{x}}_s,{\acute{y}}_s,{\acute{z}}_s\right)$ is the coordinate of the geometric center of the obstacle. $r_1=r_s+d$, with $r_s$ being the maximum radius of the obstacle. d is a safety distance predefined to prevent collisions.

Figure 2. Obstacle-avoiding strategy.

Figure 2. Obstacle-avoiding strategy.

Remark 1. 

d should be greater than the maximum distance:

$$d_{max}=l\sqrt{{sin}^2{\overline{\delta }}_1{cos}^2{\overline{\delta }}_2+{sin}^2{\overline{\delta }}_2{cos}^2{\overline{\delta }}_1}$$

with ${\overline{\delta }}_i$ being the maximum angle of ${\delta }_i,$ which can be determined based on the performance boundaries later.

The TPF sections are divided into two parts: the stage of approaching the obstacle $\left(l_{s1}\left(\acute{x}\right)\right)$ and moving away from the obstacle $\left(l_{s2}\left(\acute{x}\right)\right)$. The specific calculation processes for the TPF phases are outlined as follows:

$$\begin{array}{ccc}\hfill l_{si}\left(\acute{x}\right)& =& a_{i0}+a_{i1}\acute{x}+a_{i2}{\acute{x}}^2+a_{i3}{\acute{x}}^3+a_{i4}{\acute{x}}^4+a_{i5}{\acute{x}}^5\hfill \\ & =& \left[a_{i0},\dots ,a_{i5}\right]{\left[\begin{array}{cccccc}1& \acute{x}& {\acute{x}}^2& {\acute{x}}^3& {\acute{x}}^4& {\acute{x}}^5\end{array}\right]}^T\hfill \end{array}$$

(6)

where $A_i=\left[a_{i0},\dots ,a_{i5}\right]$ are the parameter vectors, $i=1,2$.

Fitting the Arc segment and TPF sections at two points $\left({\acute{x}}_2,{\acute{z}}_2\right)$ and $\left({\acute{x}}_3,{\acute{z}}_3\right)$ yields

$$\begin{array}{ccc}\hfill l_{si}\left({\acute{x}}_{3i-2}\right)& =& {\dot{l}}_{si}\left({\acute{x}}_{3i-2}\right)={\ddot{l}}_{si}\left({\acute{x}}_{3i-2}\right)=0,\hfill \\ \hfill l_{si}\left({\acute{x}}_{i+1}\right)& =& S_b\left({\acute{x}}_{i+1}\right),\hfill \\ \hfill {\dot{l}}_{si}\left({\acute{x}}_{i+1}\right)& =& {\dot{S}}_b\left({\acute{x}}_{i+1}\right),\hfill \\ \hfill {\ddot{l}}_{si}\left({\acute{x}}_{i+1}\right)& =& {\ddot{S}}_b\left({\acute{x}}_{i+1}\right).\hfill \end{array}$$

(7)

According to (6), (7) can be written as

$$T_i{A}_0^T={\left[0,0,0,S_b\left({\acute{x}}_{i+1}\right),{\dot{S}}_b\left({\acute{x}}_{i+1}\right),{\ddot{S}}_b\left({\acute{x}}_{i+1}\right)\right]}^T$$

(8)

with

$$T_i=\left[\begin{array}{cccccc}1& {\acute{x}}_{3i-2}& {\acute{x}}_{3i-2}^2& {\acute{x}}_{3i-2}^3& {\acute{x}}_{3i-2}^4& {\acute{x}}_{3i-2}^5\\ 0& 1& 2{\acute{x}}_{3i-2}& 3{\acute{x}}_{3i-2}^2& 4x_{3i-2}^3& 5{\acute{x}}_{3i-2}^4\\ 0& 0& 2& 6x_{3i-2}& 12{\acute{x}}_{3i-2}^2& 20{\acute{x}}_{3i-2}^3\\ 1& {\acute{x}}_{i+1}& {\acute{x}}_{i+1}^2& {\acute{x}}_{i+1}^3& {\acute{x}}_{i+1}^4& {\acute{x}}_{i+1}^5\\ 0& 1& 2{\acute{x}}_{i+1}& 3{\acute{x}}_{i+1}^2& 4{\acute{x}}_{i+1}^3& 5{\acute{x}}_{i+1}^4\\ 0& 0& 2& 6{\acute{x}}_{i+1}& 12{\acute{x}}_{i+1}^2& 20{\acute{x}}_{i+1}^3\end{array}\right].$$

Multiplying both sides of (8) by $T_i^{-1},$ one has

$$A_i^T=T_i^{-1}{\left[0,0,0,S_b\left({\acute{x}}_{i+1}\right),{\dot{S}}_b\left({\acute{x}}_{i+1}\right),{\ddot{S}}_b\left({\acute{x}}_{i+1}\right)\right]}^T$$

(9)

Substituting $A_i^T$ into (6), $l_{si}\left(\acute{x}\right)$ can be obtained.

Remark 2. 

If obstacle avoidance is performed by moving the bridge, then according to Step 1, the trajectory of the payload in the Y direction within the $\left(\acute{X},O,\acute{Y}\right)$ coordinate system should be designed.

Step 2. Transformation of coordinates (Figure 3).

The ideal trajectory in Step 1  should be transformed into the basic coordinate system, which is given as

$$\begin{array}{ccc}\hfill x& =& \acute{x}cos{\theta }_{1}cos{\theta }_2-\acute{y}sin{\theta }_1+\acute{z}cos{\theta }_{1}sin{\theta }_2+x_0\hfill \\ \hfill y& =& \acute{y}cos{\theta }_1+\acute{x}cos{\theta }_{2}sin{\theta }_1+\acute{z}sin{\theta }_{1}sin{\theta }_2+y_0\hfill \\ \hfill z& =& \acute{z}cos{\theta }_2-\acute{x}sin{\theta }_2+z_0\hfill \end{array}$$

(10)

where ${\theta }_1=arctan{\displaystyle \frac{z_d-z_0}{x_d-x_0}}$ and ${\theta }_2=-arctan{\displaystyle \frac{y_d-y_0}{x_d-x_0}}$ are the angles of transformation of the payload coordinate system from the basic coordinate system.

Figure 3. Transformation of coordinates.

Figure 3. Transformation of coordinates.

Step 3. Inverse kinematics.

Based on the kinematic model of the overhead cranes, the ideal trajectory of the payload is decomposed into the trajectories of the trolley, bridge and rope. This can be demonstrated as

$${\eta }_d=x,y_d=y,l_d=z$$

(11)

with ${\eta }_d$, $y_d$ and $l_d$ being the desired trajectories of the trolley, bridge and rope, respectively.

2.3. Swing Angle Constraint

In order to ensure obstacle avoidance, the swings of the payload must be within a specific range. Based on a three-order low-pass filter, the new performance boundaries ${\xi }_{11}$ and ${\xi }_{21}$ are defined as follows:

$$\begin{array}{ccc}\hfill {\dot{\xi }}_{i1}& =& {\xi }_{i2},\hfill \\ \hfill {\dot{\xi }}_{i2}& =& {\xi }_{i3},\hfill \\ \hfill {\dot{\xi }}_{i3}& =& -a_{i1}{\xi }_{i1}-a_{i2}{\xi }_{i2}-a_{i3}{\xi }_{i3}+d_i,\hfill \end{array}$$

(12)

where $a_{i1},$$a_{i2}$ and $a_{i3}$ are positive design constants, ${\xi }_{i1}\left(0\right)>0$ and d is referred to as

$$d_i=\left\{\begin{array}{cc}{\tau }_i\left|{\delta }_i\right|+{\chi }_i{\kappa }_{iend},\hfill & \left|{\delta }_i\right|\ge {\varpi }_i{\xi }_i\hfill \\ {\chi }_i{\kappa }_{iend}\hfill & \left|{\delta }_i\right|<{\varpi }_i{\xi }_i\hfill \end{array}\right.$$

(13)

where $\tau >0$ and $0<\varpi <1$ are design constants; ${\kappa }_{iend}>0$ is the desired steady value of ${\xi }_{i1}$; and ${\chi }_i=|p_{i1}\left|\right|p_{i2}\left|\right|p_{i3}|$ with $p_{i1},p_{i2}$ and $p_{i3}$ being closed-loop poles of the system (12).

The rate of convergence of ${\xi }_{i1}$ depends on the values of $a_{i1},$$a_{i2}$ and $a_{i3}$. To avoid the complexity of trial-and-error, this paper selects $a_{i1},$$a_{i2}$ and $a_{i3}$ according to the following approach:

Step 1: The ideal settling time $t_s$ of ${\xi }_{i1}$ is determined according to the actual demand. Based on the properties of the first-order filter, the time parameter can be defined as ${\tau }_1={\tau }_2={\displaystyle \frac{t_s}{4}}.$

Step 2: The desired poles are defined as

$$p_{i1}=-{\displaystyle \frac{1}{{\tau }_i}},p_{i2}=-c_{i1}{p}_{i1},p_{i3}=-c_{i2}{p}_{i1},$$

where $c_{i1}$ and $c_{i2}$ are constants not less than 5. The desired characteristic polynomial is expressed as

$$\begin{array}{ccc}\hfill F& =& \left(s-p_{i1}\right)\left(s-p_{i2}\right)\left(s-p_{i3}\right)\hfill \\ & =& s^3-\left(p_{i1}+p_{i2}+p_{i3}\right)s^2+\left(p_{i1}{p}_{i2}+p_{i2}{p}_{i3}+p_{i2}{p}_{i3}\right)s-p_{i1}{p}_{i2}{p}_{i3}\hfill \end{array}$$

(14)

Step 3: (12) can be rewritten as

$${\left[\begin{array}{ccc}{\dot{\xi }}_{i1}& {\dot{\xi }}_{i2}& {\dot{\xi }}_{i3}\end{array}\right]}^T=A{\left[\begin{array}{ccc}{\xi }_{i1}& {\xi }_{i2}& {\xi }_{i3}\end{array}\right]}^T$$

(15)

with $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -a_{i1}& -a_{i2}& -a_{i3}\end{array}\right].$

Then, the actual characteristic polynomial is constructed as follows:

$$\left|sI-A\right|=s^3+a_{i3}{s}^2+a_{i2}s+a_{i1}$$

(16)

Step 4: Based on the combination of (14) and (16), one has

$$\begin{array}{ccc}\hfill a_{i1}& =& -p_{i1}{p}_{i2}{p}_{i3}\hfill \\ \hfill a_{i2}& =& p_{i1}{p}_{i2}+p_{i2}{p}_{i3}+p_{i2}{p}_{i3}\hfill \\ \hfill a_{i3}& =& -\left(p_{i1}+p_{i2}+p_{i3}\right)\hfill \end{array}$$

(17)

3. Controller Design and Stability Analysis

Define the new constraint function matrix as

$$\Xi \left(\delta ,{\xi }_1\right)=diag\{{\Xi }_1,{\Xi }_2,\vartheta \}$$

(18)

with

$${\Xi }_i={\displaystyle \frac{4{\Gamma }_i{\xi }_{i1}^2}{({\delta }_i+{\xi }_{i1})({\xi }_{i1}-{\delta }_i)}}$$

(19)

where $\vartheta >0$ and ${\Gamma }_i>0$ are design constants.

Based on (3), the tracking error e is designed as

$$e=y-y_d,$$

(20)

with $y_d={\left[{\eta }_d,{\phi }_d,l_d\right]}^T.$

To couple e and $x_2,$ the following new variable is constructed as

$${\varrho }_1={ϵ}_{1}e-{ϵ}_2{x}_2,$$

(21)

where ${ϵ}_1=diag\{{ϵ}_{11},{ϵ}_{12},{ϵ}_{13}\}$ and ${ϵ}_2={\left[\begin{array}{ccc}{ϵ}_{21}& 0& 0\\ 0& {ϵ}_{22}& 0\end{array}\right]}^T$, with ${ϵ}_{11},{ϵ}_{12},{ϵ}_{13},{ϵ}_{21}$ and ${ϵ}_{22}$ being positive constants.

The new coordinate transformation is given as

$${\varrho }_2=z_1-{\alpha }_1,$$

(22)

where ${\alpha }_1$ is the virtual control law and will be specified later.

Step 1. The first Lyapunov function candidate is chosen as $V_1={\displaystyle \frac{1}{2}}{\varrho }_1^T{\varrho }_1.$

Differentiating $V_1$ yields

$$\begin{array}{ccc}\hfill {\dot{V}}_1& =& {\varrho }_1^T{\dot{\varrho }}_1\hfill \\ & =& {\varrho }_1^T{ϵ}_1{\varrho }_2+{\varrho }_1^T\left({ϵ}_1{\alpha }_1-{ϵ}_1{\dot{y}}_d-{ϵ}_2{z}_2\right).\hfill \end{array}$$

(23)

The virtual controller ${\alpha }_1$ is constructed as

$${\alpha }_1=-{ϵ}_1^{-1}\left(\Xi \left(\delta ,{\xi }_1\right){\varrho }_1-{ϵ}_1{\dot{y}}_d-{ϵ}_2{z}_2\right).$$

(24)

Then, ${\dot{\alpha }}_1$ can be computed as follows:

$${\dot{\alpha }}_1=\sum _{i=1}^2{\displaystyle \frac{\partial {\alpha }_1}{\partial {\dot{x}}_i}}{\dot{x}}_i+{\displaystyle \frac{\partial {\alpha }_1}{\partial z_2}}{\dot{z}}_2+\sum _{i=0}^1{\displaystyle \frac{\partial {\alpha }_1}{\partial y_d^{\left(i\right)}}}{y}_d^{\left(i+1\right)}+{\displaystyle \frac{\partial {\alpha }_1}{\partial {\xi }_1}}{\dot{\xi }}_1.$$

(25)

Substituting (24) into (23) results in

$${\dot{V}}_1={\varrho }_1^T{ϵ}_1{\varrho }_2-{\varrho }_1^T\Xi \left(\delta ,\xi \right){\varrho }_1.$$

(26)

Step 2. By introducing Equation (25), the time derivative of ${\varrho }_2$ is given by

$$\begin{array}{ccc}\hfill {\dot{\varrho }}_2& =& {\dot{z}}_1-{\dot{\alpha }}_1\hfill \\ & =& g_{1}u-{\displaystyle \frac{\partial {\alpha }_1}{\partial z_2}}{g}_{2}u+f_1-\sum _{i=1}^2{\displaystyle \frac{\partial {\alpha }_1}{\partial x_i}}{z}_i\hfill \\ & & -{\displaystyle \frac{\partial {\alpha }_1}{\partial z_2}}{f}_2-\sum _{i=0}^1{\displaystyle \frac{\partial {\alpha }_1}{\partial y_d^{\left(i\right)}}}{y}_d^{\left(i+1\right)}-{\displaystyle \frac{\partial {\alpha }_1}{\partial {\xi }_1}}{\dot{\xi }}_1\hfill \\ & =& \Lambda u+\Delta \hfill \end{array}$$

(27)

with

$$\begin{array}{ccc}\hfill \Lambda & =& g_1-{\displaystyle \frac{\partial {\alpha }_1}{\partial z_2}}{g}_2,\hfill \\ \hfill \Delta & =& f_1-\sum _{i=1}^2{\displaystyle \frac{\partial {\alpha }_1}{\partial x_i}}{z}_i-{\displaystyle \frac{\partial {\alpha }_1}{\partial z_2}}{f}_2-\sum _{i=0}^1{\displaystyle \frac{\partial {\alpha }_1}{\partial y_d^{\left(i\right)}}}{y}_d^{\left(i+1\right)}-{\displaystyle \frac{\partial {\alpha }_1}{\partial {\xi }_1}}{\dot{\xi }}_1.\hfill \end{array}$$

(28)

The next Lyapunov function candidate is taken as

$$V_2=V_1+{\displaystyle \frac{1}{2}}{\varrho }_2^T{\varrho }_2$$

(29)

According to (26) and (27), we have

$$\begin{array}{ccc}\hfill {\dot{V}}_2& =& {\dot{V}}_1+{\varrho }_2^T{\dot{\varrho }}_2\hfill \\ & =& -{\varrho }_1^T\Xi \left(\delta ,{\xi }_1\right){\varrho }_1+{\varrho }_2^T\left(\Lambda u+\Delta +{ϵ}_1{\varrho }_1\right)\hfill \end{array}$$

(30)

The controller is designed as

$$u=-{\Lambda }^{-1}\left({ϵ}_1{\varrho }_1+\Delta +\sigma {\varrho }_2\right)$$

(31)

where $\sigma >0$ is a design parameter.

Substituting the control signal u into (30) leads to

$${\dot{V}}_2=-{\varrho }_1^T\Xi \left(\delta ,{\xi }_1\right){\varrho }_1-{\varrho }_2^T\sigma {\varrho }_2$$

(32)

$\Lambda$ can be expressed in detail as

$$\begin{array}{ccc}\hfill \Lambda & =& g_1-{\displaystyle \frac{\partial {\alpha }_1}{\partial z_2}}{g}_2\hfill \\ & =& g_1-{ϵ}_1^{-1}{ϵ}_2{g}_2\hfill \end{array}$$

(33)

Remark 3. 

$M_t\left(q\right)$ is a positive definite matrix and $M_t{\left(q\right)}^{-1}$ is also a positive definite matrix. $M_{11}^{*}$ is the principal subordinate of B, so $M_{11}^{*}$ is positive definite and nonsingular. In addition, the elements in $M_t\left(q\right)$ only include $sin{\delta }_i,$$cos{\delta }_1,cos{\delta }_i$ and $l,i=1,2.$ This is so that the change of $M_t\left(q\right)$ in practical application is restrained. It is feasible to select parameters ${ϵ}_1$ and ${ϵ}_2$ in such a way that $\mid M_{11}^{*}-{ϵ}_1^{-1}{ϵ}_2{M}_{21}^{*}\mid \ne 0$ is satisfied.

Theorem 1. 

For the overhead crane in (1), ifAssumption 1and ${\ddot{y}}_d=0$ are satisfied, the controllers (24) and (31) guarantee locally asymptotic stability at the equilibrium point, which means that

$$\underset{t\to \infty }{lim}{\left[\begin{array}{cccccccccc}\eta & \phi & l& {\delta }_1& {\delta }_2& \dot{\eta }& \dot{\phi }& \dot{l}& {\delta }_1& {\dot{\delta }}_2\end{array}\right]}^T={\left[\begin{array}{cccccccccc}{\eta }_d& {\phi }_d& l_d& 0& 0& 0& 0& 0& 0& 0\end{array}\right]}^T.$$

(34)

Proof. 

Based on (21), (22) and (24), the derivative of ${\varrho }_1$ can be written as

$$\begin{array}{ccc}\hfill {\dot{\varrho }}_1& =& {ϵ}_1\dot{e}-{ϵ}_2{\dot{x}}_2\hfill \\ & =& {ϵ}_1{\dot{x}}_1-{ϵ}_1{\dot{y}}_d-{ϵ}_2{\dot{x}}_2\hfill \\ & =& {ϵ}_1{\varrho }_2-\Xi \left(\delta ,{\xi }_1\right){\varrho }_1.\hfill \end{array}$$

(35)

By virtue of (31), Equation (3) can be represented as

$$\begin{array}{ccc}\hfill {\dot{x}}_2& =& z_2\hfill \\ \hfill {\dot{z}}_2& =& f_2-g_2{\left(g_1-{ϵ}_1^{-1}{ϵ}_2{g}_2\right)}^{-1}\left(f_1-{ϵ}_1^{-1}{ϵ}_2{f}_2-{\ddot{y}}_d\right)\hfill \\ & & +g_2\left(g_1-{ϵ}_1^{-1}{ϵ}_2{g}_2\right)\Psi \left({\varrho }_1,{\varrho }_2,{\dot{\varrho }}_1\right)\hfill \\ \hfill {\dot{\varrho }}_1& =& {ϵ}_1{\varrho }_2-\Xi \left(\delta ,{\xi }_1\right){\varrho }_1\hfill \\ \hfill {\dot{\varrho }}_2& =& \Lambda u+\Delta \hfill \end{array}$$

(36)

with

$$\Psi =\sigma {\varrho }_2+{ϵ}_1{\varrho }_1+{ϵ}_1^{-1}{\Phi }_{\delta }{\varrho }_1{z}_2+{ϵ}_1^{-1}{\Phi }_{\xi }{\varrho }_1\dot{\xi }+{ϵ}_1^{-1}\Xi {\dot{\varrho }}_1.$$

(37)

By setting ${\varrho }_1={\varrho }_2=0,$ it also follows that ${\dot{\varrho }}_1={\dot{\varrho }}_2=0,$ and (36) can be expressed as

$$\begin{array}{ccc}\hfill {\dot{x}}_2& =& z_2,\hfill \\ \hfill {\dot{z}}_2& =& f_2-g_2{\left({ϵ}_{1}g-{ϵ}_2{g}_2\right)}^{-1}\left({ϵ}_1{f}_1-{ϵ}_2{f}_2-{\ddot{y}}_d\right).\hfill \end{array}$$

(38)

Then, after linearizing around the initial values $\left(0,0,l_0,0,0,0,0,0,0,0\right)$, (38) can be formulated as

$$\begin{array}{ccc}\hfill {\dot{x}}_2& =& z_2\hfill \\ \hfill {\dot{z}}_2& =& A_{x2}{x}_2+A_{z2}{z}_2+A_{yd}{\ddot{y}}_d,\hfill \end{array}$$

(39)

with $A_{x2}=\left[\begin{array}{cc}-{\displaystyle \frac{g{ϵ}_{11}}{{ϵ}_{21}+{ϵ}_{11}{l}_0}}& 0\\ 0& -{\displaystyle \frac{g{ϵ}_{12}}{l_0\left({ϵ}_{12}+{ϵ}_{22}\right)}}\end{array}\right],$ $A_{z2}=\left[\begin{array}{cc}-{\displaystyle \frac{{ϵ}_{11}{d}_{{\delta }_1}}{m{l}_0\left({ϵ}_{21}+{ϵ}_{11}{l}_0\right)}}& 0\\ 0& -{\displaystyle \frac{{ϵ}_{12}{d}_{{\delta }_2}}{m{l}_0^2\left({ϵ}_{12}+{ϵ}_{22}\right)}}\end{array}\right],A_{yd}=\left[\begin{array}{ccc}{\displaystyle \frac{{ϵ}_{11}}{{ϵ}_{21}+{ϵ}_{11}{l}_0}}& 0& 0\\ 0& {\displaystyle \frac{{ϵ}_{12}}{{ϵ}_{12}+{ϵ}_{22}}}& 0\end{array}\right].$

Furthermore, it can be obtained that

$$\left[\begin{array}{c}{\dot{x}}_2\\ {\dot{z}}_2\end{array}\right]=\left[\begin{array}{cc}{0}_{2\times 2}& I_{2\times 2}\\ A_{x2}& A_{z2}\end{array}\right]\left[\begin{array}{c}{x}_2\\ z_2\end{array}\right]-\left[\begin{array}{ccc}-{\displaystyle \frac{{ϵ}_{11}}{{ϵ}_{21}+{ϵ}_{11}{l}_0}}& 0& 0\\ 0& -{\displaystyle \frac{{ϵ}_{12}}{{ϵ}_{12}+{ϵ}_{22}}}& 0\end{array}\right]{\ddot{y}}_d,$$

(40)

By calculating the following determinant

$$det\left(\lambda I_{4\times 4}-\left[\begin{array}{cc}{0}_{2\times 2}& I_{2\times 2}\\ A_{x2}& A_{z2}\end{array}\right]\right)$$

(41)

We use Matlab to calculate (41), and it is easy to determine that its eigenvalue has a complex real part. Then, since ${ϵ}_{11}$ and ${ϵ}_{21}$ are positive constants, when ${\ddot{y}}_d=0$, the system (3) is locally asymptotically stable.

According to (29) and (32), ${\dot{V}}_2$ can be transformed into

$$\begin{array}{ccc}\hfill {\dot{V}}_2& =& -{\varrho }_1^T\Xi \left(\delta ,\xi \right){\varrho }_1-{\varrho }_2^T\sigma {\varrho }_2\hfill \\ & \le & -2{\lambda }^{*}{V}_2,\hfill \end{array}$$

(42)

with ${\lambda }^{*}=min\{\left|\left|\Xi \left(\delta ,\xi \right)\right|\right|,\sigma \}.$ It means that when $t\to \infty$, ${\varrho }_1$ and ${\varrho }_2$ converge to 0. Based on [38], (36) is asymptotically stable at the origin. Based on (36), (38) and (40), it can be seen that

$$\underset{t\to 0}{lim}{\alpha }_1={\dot{y}}_d,\underset{t\to 0}{lim}e=0.$$

(43)

That means

$$\underset{t\to 0}{lim}y=y_d,\underset{t\to 0}{lim}\dot{y}={\dot{y}}_d.$$

(44)

To sum up, the following conclusion is drawn as

$$\underset{t\to \infty }{lim}{\left[\begin{array}{cccccccccc}\eta & \phi & l& {\delta }_1& {\delta }_2& \dot{\eta }& \dot{\phi }& \dot{l}& {\delta }_1& {\dot{\delta }}_2\end{array}\right]}^T={\left[\begin{array}{cccccccccc}{\eta }_d& {\phi }_d& l_d& 0& 0& {\dot{\eta }}_d& {\dot{\phi }}_d& {\dot{l}}_d& 0& 0\end{array}\right]}^T.$$

(45)

Now, the proof of Theorem 1 is completed. □

Remark 4. 

If ${\ddot{y}}_d\ne 0,$ we can see from (40) that the conclusion ofTheorem 1will be changed from asymptotically stable to bounded, and the boundary depends on ${ϵ}_{11}$ and ${ϵ}_{21}.$

Remark 5. 

It is noteworthy that the developed controller is a nonlinear block-backstepping-based controller. Linearization is only used for the stability analysis.

4. Experiment Result

To verify the feasibility of the proposed control method, several sets of experiments were conducted, including MATLAB R2022b simulation and hardware experiments. The system parameters of the experiments are given as

$$\begin{array}{ccc}\hfill M_x& =& 3.5\mathrm{kg},m=0.5\mathrm{kg},M_y=m+1.5\mathrm{kg}=5\mathrm{kg},\hfill \\ \hfill g& =& 9.8{\mathrm{m}/\mathrm{s}}^2,d_x=d_y=d_l=0.2,d_f=0.1,f_{rx}=f_{ry}=0.05.\hfill \end{array}$$

The state constraint parameters are defined as

$$\begin{array}{ccc}\hfill {\xi }_{1i}\left(0\right)& =& 0.2\mathrm{rad},{\xi }_{2i}\left(0\right)=-0.06\mathrm{rad}/\mathrm{s},{\xi }_{3i}\left(0\right)=0.024{\mathrm{rad}/\mathrm{s}}^2,\hfill \\ \hfill {\kappa }_{iend}& =& 0.115\mathrm{rad},t_s=10\mathrm{s},\varpi =0.51.\hfill \end{array}$$

In the experiments, the proposed controller is compared with backstepping and LQR controllers. The parameters of backstepping controller are described as

$$\begin{array}{ccc}\hfill {\alpha }_b& =& -k_{1}e+{\dot{y}}_d\hfill \\ \hfill u_b& =& -g_1^{-1}\left(k_{1}e+{\rm Y}{k}_2{z}_1+e+f_1-k_1{\alpha }_b-{\ddot{y}}_d\right)\hfill \end{array}$$

(46)

with $k_1=diag\{k_{11},k_{12},k_{13}\}$ and $k_2=diag\{k_{21},k_{22},k_{23}\}$ being the positive parameters, where ${{\rm Y}}_1\in {\mathbb{R}}^{3\times 3}$ is a custom function matrix.

The parameters of the LQR controller are given by

$$u=-K{\left[\left(x_1-{\eta }_d\right),\left(x_2-{\phi }_d\right),\left(x_3-l_d\right),x_4,x_5,\left(x_6-{\dot{\eta }}_d\right),\left(x_7-{\dot{\phi }}_d\right),\left(x_8-{\dot{l}}_d\right),x_9,x_{10}\right]}^T$$

where $K={{\rm Y}}_2{\left[\begin{array}{ccc}{K}_1& K_2& K_3\end{array}\right]}^T$ with $K_1\in {\mathbb{R}}^{1\times 10},K_2\in {\mathbb{R}}^{1\times 10}$ and $K_2\in {\mathbb{R}}^{1\times 10}$, and ${{\rm Y}}_2\in {\mathbb{R}}^{3\times 3}$ is a custom function matrix.

4.1. Experiment 1

The straight line connecting the initial and desired positions represents the shortest distance between two points, which is a common trajectory. Experiment 1 is the comparison between the proposed and traditional controllers (backstepping and LQR) without obstacle.

The initial and desired positions of the trolley, bridge and rope are chosen as follows:

$${\eta }_0=0\mathrm{m},{\phi }_0=0\mathrm{m},l_0=-0.8\mathrm{m},{\eta }_d=-0.6\mathrm{m},{\phi }_d=0.6\mathrm{m},l_d=-0.2\mathrm{m}.$$

In Experiment 1, the chosen controller parameters are $\sigma =0.15$, ${ϵ}_1=diag\left\{\begin{array}{ccc}12.8& 16& 11\end{array}\right\}$ and ${ϵ}_2={\left[\begin{array}{ccc}0.86& 0& 0\\ 0& 0.816& 0\end{array}\right]}^T,$ and the constraint function parameters are adjusted as ${\Gamma }_1=10$, ${\Gamma }_2=8$ and $\vartheta =3.1$. The control parameters of backstepping are set as

$$\begin{array}{lll}\hfill k_1& =& diag\left\{\begin{array}{ccc}1.9289& 1.9289& 0.876\end{array}\right\},k_2=diag\left\{\begin{array}{ccc}16.0711& 16.0711& 0.4\end{array}\right\},\\ \hfill {{\rm Y}}_1& =& diag\left\{\begin{array}{ccc}{\displaystyle \frac{1}{35\left(e^{^(-t)}+1\right)}}& {\displaystyle \frac{1}{35\left(e^{^(-t)}+1\right)}}& {\displaystyle \frac{1}{1650\left(e^{^(-t)}+1\right)}}\end{array}\right\}.\hfill \end{array}$$

For the LQR controller, the parameters are

$$\begin{array}{ccc}\hfill K_1& =& \left[\begin{array}{cccccccccc}-10.4& 0& 0& 0& 0.51681& -9.5391& 0& 0& 0& 1.3475\end{array}\right],\hfill \\ \hfill K_2& =& \left[\begin{array}{cccccccccc}0& -24.721& 0& -27.3686& 0& 0& -12.5226& 0& -15.923& 0\end{array}\right],\hfill \\ \hfill K_3& =& \left[\begin{array}{cccccccccc}0& 0& -555.12& 0& 0& 0& 0& -455.5& 0& 0\end{array}\right],\hfill \\ \hfill {{\rm Y}}_2& =& diag\left\{\begin{array}{ccc}{\displaystyle \frac{1}{5\left(e^{^(-t)}+1\right)}}& {\displaystyle \frac{1}{30\left(e^{^(-t)}+1\right)}}& {\displaystyle \frac{1}{10\left(e^{^(-t)}+1\right)}}\end{array}\right\}.\hfill \end{array}$$

The results of Experiment 1 are shown in Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8. Figure 4 illustrates the translational path of the trolley, movement of the bridge and lifting of the rope. From Figure 4a, we can see that during the translational motion, the proposed method approaches the desired trajectory more closely compared with backstepping and LQR. Besides, the steady-state error of the proposed method is $0.0002$ m, which is significantly smaller. In Figure 4b, the method proposed in this paper makes the bridge have the best tracking effect during the translational motion. After reaching the desired position, the running trajectory almost coincides with the expected trajectory, which is obviously better than the other two methods. Figure 4c shows the lifting error of the rope. The proposed control method has a good tracking effect throughout the experiment, which is similar to the backstepping method, but the backstepping method has certain fluctuations in the preliminary trial stage. The LQR method has a large error, and the steady-state error is close to 0.1 m. Figure 5 shows the displacement speed of the trolley, bridge and rope. Figure 6 represents the motion trajectory of the payload. The proposed control method ensures that the payload tracks the ideal trajectory accurately, with an error that is smaller compared with the other methods. Due to the relatively slow operating speed, the swing angle of the payload (in Figure 7) is minimal under all three control methods, especially using the control method proposed in this paper. Additionally, the performance function proposed in this paper generates the performance boundaries (${\kappa }_1$ and ${\kappa }_2)$ with convergence effect, which can also be observed in Figure 7, and the payload swing angles remain consistently within the boundaries. Finally, the control signals for the trolley, bridge and rope under each of the three control methods are illustrated in Figure 8.

4.2. Experiment 2

Experiment 2 considered obstacle avoidance and was a hardware experiment. The hardware experiments were realized on a laboratory platform of overhead crane (Figure 9). The trolley transfer motion is driven by a drive motor with a built-in coaxial encoder, which measures the trolley displacement. The lifting motion of the rope is driven by a lifting motor, and the length is measured by an encoder. The swing angle of the payload is measured by an encoder. The host PC with Matlab/Simulink R2022b calculates the control commands and transfers them to the motor drivers by the motion control board.

Figure 4. Transport trajectories of overhead crane.

Figure 4. Transport trajectories of overhead crane.

Remark 6. 

Experiment 2 simulates an overhead crane transporting a load to the desired position and passing over obstacles using rope lifting without requiring the movement of the bridge. Therefore, it is feasible to use the experimental platform shown in Figure 9 to simulate the obstacle avoidance experiment.

The initial and desired positions of Experiment 2 and the position and size of obstacles are set as

$$\begin{array}{ccc}\hfill {\eta }_0& =& 0m,{\phi }_0=0m,l_0=-0.8m,{\eta }_d=-0.6m,{\phi }_d=0m,l_d=-0.2m.\hfill \\ \hfill {\acute{x}}_s& =& -0.4m\left(x_s=-0.2828m\right),{\acute{z}}_s=0m\left(z_s=-0.51m\right),{\acute{y}}_s=0m(y_s=0m),d=0.02m.\hfill \end{array}$$

The positions of switching points (in $(X,O,Z)$ coordinate system, $y=0$) are defined as

$$\begin{array}{ccc}\hfill \left({\acute{x}}_1,{\acute{z}}_1\right)& =& \left(0m,0m\right),\left({\acute{x}}_2,{\acute{z}}_2\right)=\left(-0.3m,0.45m\right),\hfill \\ \hfill \left({\acute{x}}_3,{\acute{z}}_3\right)& =& \left(-0.5m,0.45m.\right),\left({\acute{x}}_4,{\acute{z}}_4\right)=\left(-0.8m,0m\right).\hfill \end{array}$$

The parameters of the proposed controllers in Experiment 2 are selected as $\sigma =0.5$, ${ϵ}_1=diag\left\{\begin{array}{ccc}5.39& 5.39& 26.3\end{array}\right\}$ and ${ϵ}_2={\left[\begin{array}{ccc}0.001& 0& 0\\ 0& 0.01& 0\end{array}\right]}^T,$ and the constraint function parameters are ${\Gamma }_1={\Gamma }_2=0.22$ and $\vartheta =20.85$. The control parameters of backstepping are set as

$$\begin{array}{ccc}\hfill k_1& =& diag\left\{\begin{array}{ccc}0.4887& 1.0826& 3.586\end{array}\right\},k_2=diag\left\{\begin{array}{ccc}9.7174& 9.7174& 68.02\end{array}\right\},\hfill \\ \hfill {{\rm Y}}_1& =& diag\left\{\begin{array}{ccc}{\displaystyle \frac{1}{35\left(e^{^(-t)}+1\right)}}& {\displaystyle \frac{1}{35\left(e^{^(-t)}+1\right)}}& {\displaystyle \frac{4}{30\left(e^{^(-t)}+1\right)}}\end{array}\right\}.\hfill \end{array}$$

Meanwhile, the parameters for LQR are

$$\begin{array}{ccc}\hfill K_1& =& \left[\begin{array}{cccccccccc}-38.5721& 0& 0& 0& 0.51681& -5.5391& 0& 0& 0& 1.3475\end{array}\right],\hfill \\ \hfill K_2& =& \left[\begin{array}{cccccccccc}0& -16.4721& 0& -7.3686& 0& 0& -6.5226& 0& -2.5923& 0\end{array}\right],\hfill \\ \hfill K_3& =& \left[\begin{array}{cccccccccc}0& 0& -5.6228& 0& 0& 0& 0& -0.2214& 0& 0\end{array}\right],\hfill \\ \hfill {{\rm Y}}_2& =& diag\left\{\begin{array}{ccc}{\displaystyle \frac{1}{500\left(e^{^(-t)}+1\right)}}& {\displaystyle \frac{1}{20\left(e^{^(-t)}+1\right)}}& {\displaystyle \frac{1}{35\left(e^{^(-t)}+1\right)}}\end{array}\right\}.\hfill \end{array}$$

Figure 5. The velocities of movement.

Figure 5. The velocities of movement.

Figure 6. Linear transportation trajectories in space.

Figure 6. Linear transportation trajectories in space.

Figure 7. The swings of payload.

Figure 7. The swings of payload.

Figure 8. The control signals.

Figure 8. The control signals.

Figure 9. The laboratory platform of overhead crane.

Figure 9. The laboratory platform of overhead crane.

The results of the obstacle avoidance experiment are illustrated in Figure 10, Figure 11 and Figure 12. First, as shown in Figure 10a, the proposed method with a $0.002$ m error demonstrates smaller errors compared to backstepping and LQR, which is about $0.001$ m and is much smaller than $0.01$ m and $0.018$ m of backstepping and LQR, respectively. Similarly, as shown in Figure 10b, it can be seen that the developed controller not only achieves smaller errors but also exhibits smoother rope movement. The errors of rope trajectory are about $0.015$ m, $0.002$ m and $0.006$ m under three controllers, respectively. Figure 10c shows the swing of the payload. The obstacle avoidance performance of the three control methods is drawn in Figure 11. The proposed control method allows the payload to avoid obstacles while maintaining a safe distance. In contrast, for the trajectory under the LQR method, collisions may occur within the maximum radius of the obstacle, which is unacceptable in real-world operations. For backstepping method, the trajectory intersects the predefined safe zone, which means collisions are possible. Moreover, poor tracking of the desired trajectory results in an increased transport path.

5. Conclusions

This article has introduced a coupled underactuated tracking control method for 5-DOF overhead cranes. First, a trajectory planning strategy for obstacle avoidance was designed based on polynomial fitting. Then, a performance boundary was defined to ensure the limits on the payload swing angles. In addition, according to the dynamic model of overhead cranes, the controller with coupled block-backstepping was designed and the stability was analyzed. Finally, the effectiveness of the controller was verified through the MATLAB simulation and hardware experiments in a laboratory platform. In the future, we will focus on integrating obstacle avoidance with visual analysis.