Smooth Obstacle-Avoidance Trajectory Planning for Cable Cranes During Concrete Hoisting in Arch Dam Construction

Abstract

The cable crane is the core hoisting equipment for high arch dam construction, and its hoisting trajectory is critical for both operational efficiency and safety. However, current trajectory planning does not adequately consider the underactuated characteristics of the cable crane. For instance, sudden stops or abrupt changes in direction can easily induce large swings of the bucket, causing safety risks and equipment wear. To address this issue, this paper developed a trajectory planning model for obstacle avoidance with smooth transitions in cable crane hoisting for arch dams and solved the high-dimensional optimization problem using a path–velocity decoupling strategy. First, a shortest path with geometrical conciseness and free collision was generated based on an improved A* algorithm to reduce the frequency of directional changes. Next, for different hoisting scenarios, segmented S-curve and polynomial velocity functions were proposed to ensure smooth velocity transitions. Then, an orthogonal experimental design was employed to generate a cluster of candidate trajectories that meet kinematic constraints, from which the optimal trajectory was selected using a multi-objective evaluation function. The results demonstrate that the motion trajectory planned using the proposed method is notably smoother. Compared with the traditional trapezoidal velocity method, it reduces the maximum swing amplitude of the bucket by 40.78% at a modest time cost. In real-time obstacle avoidance scenarios, the approach outperforms emergency-stop strategies, reducing the bucket’s maximum swing amplitude by 30.48%. This work will provide a reference for engineers to optimize the trajectory of large lifting equipment in construction fields such as high arch dams and bridges.

1. Introduction

High arch dams are often constructed in narrow canyons, where steep terrain and limited accessibility pose significant logistical challenges [1]. Traditional lifting equipment cannot meet the wide-area demands of concrete pouring under such conditions. Owing to their large-span and high-lifting capabilities, cable cranes serve as the core transportation system, enabling full cross-sectional and high-intensity concrete delivery across the entire dam body [2,3]. This large-scale apparatus drives a loaded trolley via traction mechanisms, enabling horizontal movement along carrier cables that span several hundred to over a thousand meters. Combined with hoisting mechanisms, the bucket achieves a vertical coverage range of more than 300 m, transporting up to 9 m³ of concrete in a single lift and handling millions of cubic meters [4]. However, the construction site exhibits typical three-dimensional intersecting operations [5,6] (see Figure 1), where completed dam blocks, concrete forms, and tower cranes form a complex environment. This demands that the hoisting trajectory must precisely avoid obstacles to prevent collisions, while also maintaining transport efficiency [7,8]. Therefore, ensuring the operational safety and efficiency of the cable cranes remains a key challenge and a prominent research focus in high arch dam construction [9].

It is challenging for operators to obtain obstacle information intuitively because the cable crane operation platform is located far from the pouring block. In this case, guidance and coordination are needed from ground signal personnel. In recent years, advances in GNSS, RFID, and computer vision technologies [10,11,12] have driven the research and application of collision avoidance warning systems and bucket tracking [13,14,15]. On the one hand, these technologies remain limited in providing real-time risk alerts and data analysis afterwards, failing to optimize the hoisting process proactively during the initial stage. On the other hand, safety and efficiency are often presented only as passive outcome indicators. Therefore, it is imperative to enhance the overall operational performance of cable cranes through the proactive planning of operation trajectories.

Scholars have conducted valuable investigations into the optimization of cable crane hoisting paths and task allocation. For instance, Wu et al. [16] utilized GPS to acquire the position of the cable crane hook and treated the dam body as an obstacle. They employed the artificial immune algorithm to plan the bucket’s path offline before entering the pouring site. Wang et al. [3] developed a task allocation model for a group of cable cranes, enabling the planning of their operational time and spatial scope, improving operational efficiency and utilization. These studies primarily focused on the on-site safety constraints between lifting equipment and fixed structures (e.g., dam bodies, slopes). However, trajectory planning in three dimensions must consider not only safety but also dynamic feasibility constraints, including limitations on speed and acceleration [17,18].

In contrast, trajectory planning research in fields such as robotics [19,20,21], autonomous vehicles [22,23,24], and drones [25,26,27] has demonstrated that actively planned trajectories incorporating dynamic constraints can significantly enhance system performance [28]. However, cable cranes—characterized by large spans, flexible connections, and heavy inertial loads—are fundamentally underactuated systems with far greater sensitivity to abrupt motion commands than rigid systems, resulting in considerably more severe engineering risks. Field measurements reveal that at a hoisting height of 200 m, an emergency stop of the trolley can induce bucket swings exceeding 30 m. Similarly, rapid directional changes under inertial forces can accelerate fatigue damage in critical components such as gear reducers and couplings, thereby drastically reducing equipment lifespan. These characteristics render conventional trajectory planning methods, developed for rigid systems, unsuitable for direct application to cable crane operations. Therefore, achieving smooth geometric trajectories and seamless velocity becomes the central challenge to ensuring safe operation, improving efficiency, and enhancing equipment longevity [29].

To address the above challenges, this study aimed to develop a smooth-transition obstacle-avoidance hoisting trajectory planning model tailored for cable cranes. By proposing a path–velocity decoupling strategy, the problem of high-dimensional trajectory optimization was divided into two sub-parts: geometric path planning and dynamic adaptation. (1) For path planning, an improved A* algorithm was adopted to generate a shortest collision-free path with redundant nodes removed to address efficient obstacle avoidance and path simplification in complex environments, improving geometric smoothness; (2) for velocity planning, segmented S-shaped velocity curves and polynomial interpolation functions were, respectively, designed to accommodate both long-distance composite motion and short-range alignment for unloading, ensuring smooth velocity transitions. Using orthogonal experimental design and a multi-objective cost function that incorporated the equipment’s maximum velocity and acceleration, optimal trajectories were selected to balance safety distance and hoisting efficiency. By achieving these objectives, this approach provided a trajectory-planning solution that integrated geometric smoothness with dynamic adaptability, supporting safe and efficient cable-crane operations in high arch dam construction.

Figure 2 illustrates the research framework of this study. The remainder of the paper is organized as follows: Section 2 provides a detailed introduction to the smooth-transition obstacle-avoidance trajectory planning method, including the operational characteristics of the cable crane and the trajectory planning model; Section 3 validates the model performance through multi-scenario simulations; Section 4 discusses the results and outlines future research directions; and Section 5 concludes the study.

2. Methods

2.1. Description of Cable Cranes’ Operation Characteristics

In trajectory control, the terminal load is governed by the motion of the cable crane bucket, which is jointly determined by the horizontal movement of the trolley and the retraction/extension of the hoisting wire rope. The actual motion trajectory has a direct influence on operational safety and efficiency. It is essential to clarify the dynamic characteristics of the bucket to understand the swing behavior throughout the hoisting process.

In high arch dam construction, cable cranes commonly employ a fixed-line unloading mode, in which the main tower and tail tower remain stationary to ensure efficient transport and the bucket is moved solely by the trolley and hoisting mechanisms. Accordingly, the system can be simplified as a model consisting of a cable crane trolley, hoisting ropes, and a bucket. The system’s dynamic behavior was modeled using Lagrangian equations, which account for the hoisting rope’s length change at different times.

In this system, the horizontal swinging of the cable crane was primarily caused by variations in trolley velocity, while vertical oscillations were mainly induced by changes in the speed of the hoisting mechanism; the two are driven independently. As the amplitude and potential hazards of horizontal swinging were significantly greater than those of vertical vibration, this study focused on the dynamics of horizontal swinging. Additionally, since the mass of the cable crane hook and ropes was negligible compared to the total payload, and the elastic elongation of the rope was minimal, the masses of the hook and rope, as well as the elastic damping coefficient, were neglected in the dynamic analysis.

A right-handed coordinate system was established with the positive $x$-axis directed from the main tower to the tail tower of the cable crane. Let the mass of the bucket be $m$, the mass of the trolley be $M$, the length of the hoisting rope be $l$. The angle between the hoisting rope and the $x$-axis during swinging is ${\theta }_x$, and the angle between rope and the $y$-axis is ${\theta }_y$. The traction force on the trolley is denoted by $F_M$ and the tension in the hoisting rope is denoted by $F_l$, as shown in Figure 3. The coordinates of the trolley and the bucket are defined as follows:

$$\left\{\begin{array}{c}{x}_M=x\\ y_M=0\\ z_M=z\end{array}\right.$$

(1)

$$\left\{\begin{array}{l}{x}_m=x-l sin {\theta }_x cos {\theta }_y\\ y_m=l sin {\theta }_y\\ z_m=z-l cos {\theta }_x cos {\theta }_y\end{array}\right.$$

(2)

The velocity of the trolley and the bucket are defined as follows:

$$\left\{\begin{array}{l}{\dot{x}}_M=\dot{x}\\ {\dot{y}}_M=0\\ {\dot{z}}_M=\dot{z}\end{array}\right.$$

(3)

$$\left\{\begin{array}{l}{\dot{x}}_m=\dot{x}+\dot{l}sin{\theta }_{x}cos{\theta }_y+l{\dot{\theta }}_{x}cos{\theta }_{x}cos{\theta }_y-l{\dot{\theta }}_{y}sin{\theta }_{x}sin{\theta }_y\\ {\dot{y}}_m=\dot{l}sin{\theta }_y+l{\dot{\theta }}_{y}cos{\theta }_y\\ {\dot{z}}_m=\dot{z}-\dot{l}cos{\theta }_{x}cos{\theta }_y+l{\dot{\theta }}_{x}sin{\theta }_{x}cos{\theta }_y+l{\dot{\theta }}_{y}cos{\theta }_{x}sin{\theta }_y\end{array}\right.$$

(4)

In the system comprising a cable crane trolley, hoisting ropes, and a bucket, the inputs are $F_M$, $F_l$, and $F_w$, while the outputs include $x$, $y$, $z$, $l$, ${\theta }_x$, and ${\theta }_y$. This formed a multivariable and highly coupled system, making it complex to construct a dynamic model using Newtonian mechanics. In contrast, the Lagrangian approach requires only the computation of the kinetic and potential energy of the trolley and bucket, offering a more concise formulation while still capturing the essential dynamic characteristics of the system [30]. Therefore, this study adopts the Lagrangian mechanics for system modeling. The general form of the Lagrange equation is given as:

$$L\left(q,\dot{q}\right)=T\left(q,\dot{q}\right)-V\left(q,\dot{q}\right)$$

(5)

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial L}{\partial {\dot{q}}_i}}-{\displaystyle \frac{\partial L}{\partial {\dot{q}}_i}}=f_i$$

(6)

where $L$ is the Lagrangian operator, $T$ is the system’s kinetic energy, $V$ is the system’s potential energy, $q$ is the variable, $g$ is the acceleration of gravity, $i$ is the variable index, and $f_i$ is the generalized external force.

$$\begin{array}{c}T={\displaystyle \frac{1}{2}}M\left({\dot{x}}^2+{\dot{z}}^2\right)+{\displaystyle \frac{1}{2}}m({\dot{x}}^2+{\dot{l}}^2+l^2{\dot{\theta }}_x^2{cos}^2{\theta }_y+l^2{\dot{\theta }}_y^2+2\dot{x}\dot{l}sin{\theta }_{x}cos{\theta }_y+2\dot{x}l{\dot{\theta }}_{x}cos{\theta }_{x}cos{\theta }_y\\ \hspace{1em}\hspace{1em}\hspace{1em} -2\dot{x}l{\dot{\theta }}_{y}sin{\theta }_{x}sin{\theta }_y+2\dot{l}l{\dot{\theta }}_{y}sin{\theta }_{x}cos{\theta }_y+{\dot{z}}^2-2\dot{z}\dot{l}cos{\theta }_{x}cos{\theta }_y+2\dot{z}l{\dot{\theta }}_{x}sin{\theta }_{x}cos{\theta }_y+2\dot{z}l{\dot{\theta }}_{y}cos{\theta }_{x}sin{\theta }_y)\end{array}$$

(7)

$$V=Mgz+mg\left(z-lcos{\theta }_{x}cos{\theta }_y\right)$$

(8)

In the absence of wind, the Lagrangian equations of the cable crane trolley, hoisting ropes, and bucket system can be established according to Equations (7) and (8):

$$\left\{\begin{array}{l}{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial {\dot{\theta }}_x}}\right)-{\displaystyle \frac{\partial L}{\partial {\dot{\theta }}_x}}=0\\ {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial {\dot{\theta }}_y}}\right)-{\displaystyle \frac{\partial L}{\partial {\dot{\theta }}_y}}=0\end{array}\right.$$

(9)

The dynamic model of this cable crane trolley, hoisting ropes, and bucket system is:

$$\ddot{x}cos{\dot{\theta }}_x+2\dot{l}{\theta }_x+l{\ddot{\theta }}_x+\ddot{z}sin{\theta }_x+gsin{\theta }_x=0$$

(10)

2.2. Cable Crane Trajectory Planning Model with Smooth Hoisting

2.2.1. Model Framework and Strategy

A complete concrete hoisting cycle using the cable crane includes four steps: bucket hoisting, compound motion, unloading, and return (Figure 4). Trajectory planning in this process faces dual challenges of the changing environment and dynamic constraints. The construction site contains not only static obstacles, such as placed blocks and fixed trestles, but also dynamic equipment like tower cranes. Offline planning alone cannot address dynamic obstacle risks, while fully online global planning requires excessive computations, making it difficult to ensure both efficiency and real-time responsiveness.

To address the above issues, this study proposed a strategy of offline global baseline planning and online real-time local adjustment. Before hoisting, a three-dimensional static environment was constructed based on the BIM model and a collision-free trajectory was planned to establish a foundation for transport efficiency. During hoisting, real-time data from existing monitoring equipment, such as GPS and vision systems, was integrated to enable local adjustments when the planned path conflicts with dynamic obstacles [31]. This strategy balanced trajectory optimality and environmental adaptability under limited computational resources by decoupling offline global optimization from online lightweight response.

In implementing the strategy mentioned above, if a spatiotemporally coupled planning approach was adopted, it would require the simultaneous resolution of obstacle avoidance and dynamic constraints in both spatial and temporal dimensions, resulting in complex computations. To address this, a path–velocity decoupling framework (Figure 5) was adopted, decomposing the high-dimensional spatiotemporal planning into two low-dimensional steps. Firstly, a path in three dimensions was planned to emphasize obstacle avoidance and path length optimization. Secondly, a one-dimensional velocity profile was designed along the fixed path to constrain parameters such as velocity and acceleration, thereby satisfying the dynamic constraints. Compared to the coupled approach, this framework significantly reduced computations, enhanced planning efficiency, and was better suited for complex hoisting scenarios in high arch dam construction.

2.2.2. Path Planning

• Environment map building

During the concrete hoisting for arch dam construction, the cable crane unloads in a fixed line, allowing the bucket to be modeled as a mass point moving in a two-dimensional plane (refer to the xz plane in Figure 3). In this study, a grid map model was employed to represent the environment efficiently [32]. The operational space of the cable crane was divided into two regions: obstacle areas and accessible areas (Figure 6a). To further account for the influence of factors like the precise shape of the dam structure, bucket swing, and positioning errors, virtual obstacles were defined around the periphery of obstacle areas (Figure 6b). By integrating the static BIM model with real-time data, the grid states were dynamically labeled as occupied (obstacle areas) or free (accessible areas). This grid-based method effectively abstracted the complex environment into a computer-processable mathematical model, providing a reliable data foundation for subsequent path planning.

2.

Optimized obstacle-avoidance path based on improved A* algorithm

The A* algorithm [33] is a commonly used global path planning search algorithm. When combined with a two-dimensional grid map, it is well-suited for arch dam cable crane construction, which features a complex static environment and a semi-dynamic environment with non-persistent changes in obstacles. It can generate safe and efficient paths. Although the A* algorithm offers high computational efficiency, the resulting path often includes redundant nodes and excessive turning points [34]. Such overly detailed paths increase the complexity of subsequent velocity planning, whereas in practice, only the simplest obstacle-avoidance path is required. Therefore, the A* algorithm requires improvement. This study introduced improvements from two aspects: the construction of the directed graph and the criteria for adding and removing path nodes.

(1) Construction of directed graphs

The traditional A* algorithm typically adopts a 4- or 8-directional search mode, where the number of directional angles influences the number of turning points and the overall path performance. In this study, four construction schemes, namely 4-, 8-, 16-, and 24-directional schemes, were selected, as shown in Figure 7, and the Euclidean distance was used to calculate the path cost [35]. The test results for global and local path planning are presented in Figure 8 and

Table 1. Statistical index comparison in the algorithm test.

	Planning Type	Global Planning		Local Planning
Number of Directions		Path Length (m)	Number of Corners	Path Length (m)	Number of Corners
4		437.18	1	137.47	1
8		349.24	10	118.45	7
16		324.32	6	113.35	5
24		324.32	6	112.12	5

. It can be observed that the path length generally decreased as the number of directions increased, with minimal difference between the 16- and 24-direction schemes. Moreover, while the overall path contours obtained through 16- and 24-direction searches were relatively simple, redundant details still existed in the finer segments. Considering the hoisting requirements for path length and smoothness, the 16-direction scheme was selected as the basis for further contour simplification.

(2) Design of node addition and deletion criteria

To address the abovementioned issues, redundant nodes in the path must be removed to eliminate unnecessary turns. Meanwhile, smooth transitional nodes should be inserted at large-angle turns to meet the cable crane’s operational smoothness requirements and avoid abrupt changes in path curvature. Therefore, the following node addition and deletion rules were applied to optimize the path:

• Node Deletion Criterion: If the current node was colinear with its adjacent nodes, or if the interior angle $\theta$ formed with them satisfied ${120}^{\circ }<\theta <{180}^{\circ }$ and the direct line between adjacent nodes was free of obstacles, the node was deemed redundant and removed. After each modification, the updated path was re-traversed to refresh the node list.
• Node Addition Criterion: If the angle between the current node and its adjacent segments exceeded a threshold angle ($\phi$), a new node was inserted by shifting the midpoint of the segment by grid cells ($\omega$) along the normal line. Based on the direction of the path, the side with fewer obstacles was automatically selected.

A sensitivity analysis determined the optimal parameters as $\phi ={30}^{\circ }$ and $\omega =1$ grid cell. The test results are shown in Figure 9, where the optimized path significantly reduced the number of turning nodes compared to the unprocessed path. The number of turns in global and local paths was decreased to 1 and 2, respectively, avoiding multiple direction changes during hoisting. The global and local path lengths were 322.26 m and 112.09 m, respectively.

2.2.3. Velocity Planning

Based on the turning points obtained from the path planning, it was necessary to incorporate time to construct a spatiotemporal trajectory that satisfied dynamic constraints. The design of the velocity curve should strike a balance between smoothness (to suppress bucket swing), efficiency (to shorten cycle time), and real-time response. Among typical curves, the trapezoidal velocity curve exhibits abrupt acceleration changes, which can easily cause an impact from inertial loads. In addition, polynomial curves, with the advantage of high-order continuity, lack constant velocity segments, resulting in insufficient utilization of maximum velocity. Traditional seven-segment S-shaped curves ensure continuity in acceleration; their multi-segment model, however, involves numerous parameters, making program implementation complex and less suitable for dynamic obstacle avoidance.

An analysis of the cable crane trajectory characteristics (Figure 4) revealed that scenarios can be categorized into two types: long distance and short distance, based on the point-to-point distances. Long-distance trajectories primarily occurred during the round-trip composite motion, requiring constant velocity to enhance transport efficiency. Short-distance trajectories included operations such as heavy bucket hoisting, unloading, and real-time obstacle avoidance, which was constrained by short travel distances and high real-time requirements. Accordingly, this paper proposed a scenario-adaptive velocity planning strategy. For long-distance trajectories, an S-curve was adopted to balance efficiency and smoothness. For short-distance trajectories, polynomial interpolation was used to ensure both smoothness and real-time performance. The following sections will elaborate on the construction of the velocity function.

• Improved S-curve function

Although the conventional S-curve can achieve continuous acceleration and is suitable for cable cranes’ composite motion with large-distance nodes, it has too many parameters and slow real-time calculation. Therefore, this paper proposed a simplified three-segment S-curve, which merged the traditional seven-segment process of increasing acceleration, constant acceleration, decreasing acceleration, constant velocity, increasing deceleration, constant deceleration, and decreasing deceleration into three phases of acceleration, constant velocity, and deceleration, as shown in Figure 10. This reduced computational complexity but retained key dynamic constraint boundaries. The curve function expression is as follows:

$$\left\{\begin{array}{lll}{S}_1\left(t\right)=a_{10}+a_{11}t+a_{12}{t}^2+a_{13}{t}^3+a_{14}{t}^4\hspace{1em}\hspace{1em}\hspace{1em}& & t\in \left[0,{\gamma }_1\right]\\ S_2\left(t\right)=a_{20}+a_{21}t\hspace{1em}\hspace{1em}\hspace{1em}& & t\in \left[{\gamma }_1,{\gamma }_2\right]\\ S_3\left(t\right)=a_{30}+a_{31}t+a_{32}{t}^2+a_{33}{t}^3+a_{34}{t}^4+a_{35}{t}^5\hspace{1em}\hspace{1em}\hspace{1em}& & t\in \left[{\gamma }_2,{\gamma }_3\right]\end{array}\right.$$

(11)

The coefficients are:

$$\left\{\begin{array}{l}{a}_{10}=s_0\\ a_{11}=v_s\\ a_{12}={\displaystyle \frac{{\alpha }_s}{2}}\\ a_{13}={\displaystyle \frac{3\left(v_m-v_s\right)-2{\alpha }_s{\gamma }_1}{3{\gamma }_1^2}}\\ a_{14}={\displaystyle \frac{-2\left(v_m-v_s\right)+2{\alpha }_s{\gamma }_1}{4{\gamma }_1^3}}\\ a_{20}=a_{10}+a_{11}{\gamma }_1+a_{12}{\gamma }_1^2+a_{13}{\gamma }_1^3+a_{14}{\gamma }_1^4\\ a_{21}=v_m\\ a_{30}=a_{20}+a_{21}{\gamma }_2\\ a_{31}=v_m\\ a_{32}=0\\ a_{33}={\displaystyle \frac{20\left(s-a_{30}\right)-8v_e{\gamma }_3-12a_{30}{\gamma }_3}{2{\gamma }_3^3}}\\ a_{34}={\displaystyle \frac{-15\left(s-a_{30}\right)+7v_e{\gamma }_3+8a_{30}{\gamma }_3}{2{\gamma }_3^3}}\\ a_{35}={\displaystyle \frac{6\left(s-a_{30}\right)-3v_e{\gamma }_3-3a_{30}{\gamma }_3}{{\gamma }_3^3}}\end{array}\right.$$

(12)

where $s_i$ is the displacement curve for the $i$-th segment, $a_{ij}$ is the $j$-th coefficient of the $i$-th segment displacement curve, and ${\gamma }_i$ is the time duration of the $i$-th segment. $s_0$ and $s$, $v_s$ and $v_e$, ${\alpha }_s$ and ${\alpha }_e$ represent the displacement, velocity, and acceleration at the start and end of the entire S-curve. $v_m$ is the maximum operational velocity.

2.

Construction of velocity curve function based on multi-segment polynomials

For short-distance scenarios such as lifting, positioning, and real-time obstacle avoidance, trajectory planning is constrained by short motion distances and high real-time requirements. Using a three-stage S-shaped curve in these cases leads to redundant calculations and reduced real-time planning efficiency. Therefore, polynomial functions were used to balance smoothness and real-time performance. A 1–2 segment polynomial was normally sufficient to meet the requirements and more complex cases are decomposed into combinations of multiple polynomials with one to two segments. Based on the initial, final, and process motion states, it can be determined that the maximum degree of a polynomial displacement curve is 5, and the maximum degree of two polynomial displacement curves is 4.

The equation for a single-segment polynomial displacement curve is as follows:

$$S=a_0+a_{1}t+a_2{t}^2+a_3{t}^3+a_4{t}^4+a_5{t}^5$$

(13)

The coefficients are:

$$\left\{\begin{array}{l}{a}_0=s_0\\ a_1=v_s\\ a_2={\displaystyle \frac{{\alpha }_s}{2}}\\ a_3={\displaystyle \frac{20\left(s-s_0\right)+\left(8v_e-12v_s\right)\gamma -\left(3{\alpha }_s-{\alpha }_e\right){\gamma }^2}{2{\gamma }^3}}\\ a_4={\displaystyle \frac{-30\left(s-s_0\right)+\left(14v_e+16v_s\right)\gamma +\left(3{\alpha }_s-2{\alpha }_e\right){\gamma }^2}{2{\gamma }^4}}\\ a_5={\displaystyle \frac{12\left(s-s_0\right)-\left(6v_e+6v_s\right)\gamma +\left({\alpha }_e-{\alpha }_s\right){\gamma }^2}{2{\gamma }^5}}\end{array}\right.$$

(14)

The two-segment polynomial displacement curve equation is as follows:

$$\left\{\begin{array}{l}{S}_1=a_{10}+a_{11}t+a_{12}{t}^2+a_{13}{t}^3+a_{14}{t}^4\\ S_2=a_{20}+a_{21}t+a_{22}{t}^2+a_{23}{t}^3+a_{24}{t}^4\end{array}\right.$$

(15)

The coefficients are:

$$\left\{\begin{array}{l}{a}_{10}=s_0\\ a_{11}=v_s\\ a_{12}={\displaystyle \frac{{\alpha }_s}{2}}\\ a_{13}={\displaystyle \frac{2\left(s_1-s_2\right)}{{\gamma }_1{\gamma }_2\left({\gamma }_1+{\gamma }_2\right)}}+{\displaystyle \frac{\left(24{\gamma }_1+12{\gamma }_2\right)\left(s_1-s_0\right)-v_s\left(18{\gamma }_1^2-12{\gamma }_1{\gamma }_2\right)-{\alpha }_s\left(5{\gamma }_1^2{\gamma }_2+6{\gamma }_1^3\right)+{\gamma }_1^2\left(6v_e-{\alpha }_e{\gamma }_2\right)}{6h_1^3\left({\gamma }_1+v_2\right)}}\\ a_{14}={\displaystyle \frac{s_1-s_0-v_s{\gamma }_1-\frac{{\alpha }_s{\gamma }_2^2}{2}-a_{13}{\gamma }_1^3}{{\gamma }_1^4}}\\ a_{20}=s_1\\ a_{21}={\displaystyle \frac{4\left(s_1-s_0\right)-3v_s{\gamma }_1-{\alpha }_s{\gamma }_1^2-a_{13}{\gamma }_1^3}{{\gamma }_1}}\\ a_{22}={\displaystyle \frac{12\left(s_1-s_0\right)-12v_s{\gamma }_1-5{\alpha }_s{\gamma }_1^2-6a_{13}{\gamma }_1^3}{2{\gamma }_1^2}}\\ a_{23}={\displaystyle \frac{4\left(s_1-s_0\right)-v_e{\gamma }_2-3a_{21}{\gamma }_2-2a_{22}{\gamma }_2^2}{{\gamma }_2^3}}\\ a_{24}={\displaystyle \frac{-13\left(s_1-s_0\right)+v_e{\gamma }_2+2a_{21}{\gamma }_2+2a_{22}{\gamma }_2^2}{{\gamma }_2^4}}\end{array}\right.$$

(16)

2.2.4. Trajectory Optimization Based on Orthogonal Experiments

After parameterizing the path–velocity smooth transition curve of the cable crane hoisting trajectory, these parameters can generate a trajectory cluster. At this point, the parameters corresponding to an optimized trajectory must be solved based on certain constraint conditions and trajectory performance evaluation metrics. Due to the difficulty in directly optimizing the higher-order non-convexity of polynomial interpolation and the insufficient real-time performance of global evolutionary algorithms, this paper introduced an orthogonal experimental design to achieve fast discrete optimization. This method set the desired lifting time range and discretizes the process, generating time parameter combinations to substitute into the trajectory equation to form a trajectory cluster. Then, feasible solutions were filtered according to a multi-objective cost function.

The cost function was constructed by comprehensively considering safety, efficiency, and smoothness. The geometric path planning ensured basic obstacle avoidance. The focus was on quantifying the safety distance between the trajectory and obstacles, as well as the running time. Thus, when the distance between a trajectory point and an obstacle was smaller than the minimum safety threshold, a penalty term was applied to balance safety risks and lifting efficiency. Meanwhile, the constraints of maximum velocity and acceleration were satisfied by discretizing the time domain and solving the constraint metrics at discrete points. The trajectory cost function is as follows:

$$J={\omega }_1{\displaystyle \sum _{i=1}^m{\gamma }_i}+{\omega }_2\left({\displaystyle \frac{d_i-D_{min}}{d_i}}\right)$$

(17)

where $m$ is the index of the trajectory segment, ${\gamma }_i$ is the corresponding duration, $d$ is the minimum safety distance between the bucket and obstacles, and $D_{min}$ is the minimum distance between the trajectory point and the obstacle. The weight coefficient satisfies: when $d_i-D_{min}\le 0$, ${\omega }_1+{\omega }_2=1$, and ${\omega }_1>0$, ${\omega }_2>0$; when $d_i-D_{min}>0$, ${\omega }_1={\omega }_2$, and a large penalty value $\mathrm{J}$ is applied.

After trajectory optimization, the path must be converted into mechanism motion parameters (Figure 11). Based on the two-dimensional reference coordinate system (with the $x$-axis representing horizontal traction and the $y$-axis representing vertical hoisting), the reference position and velocity of the bucket were planned. The trolley elevation angle $\beta$ was then calculated using the cable crane structural parameters and the carrying rope state equations.

$$tan\beta =(P+G_X){\displaystyle \frac{2xl}{2l{\displaystyle \sum H_X}}}$$

(18)

$P$ represents the concentrated load when the rated lifting capacity is applied, $G_X$ is the total gravitational force of the rope in kilonewtons ($KN$); $l$ is the span in meters ($m$); $H_X$ is the horizontal tension of the carrying rope at each state, with $x$ representing the position of the trolley.

Let $v_c$, $v_l$, $v_b$, and $a_c$, $a_l$, $a_b$ represent the velocities and accelerations of the trolley, hoisting ropes, and bucket, respectively. The relationship between the motion parameters of the cable crane trolleybus, hoisting rope, and bucket can be expressed as:

$$v_c={\displaystyle \frac{v_{bx}}{cos\beta }}$$

(19)

$$v_l=v_{bx}\left(1-tan\beta \right)$$

(20)

$\beta$ denotes the trolley’s lift angle [36], $\beta \in \left[-\alpha ,\alpha \right],\alpha \ll {90}^{\circ }$. The inequality constraints discretize the time domain into $N$ equal intervals and the constraint indices are solved at $N+1$ discrete points. The constraint conditions are as follows:

$$\begin{array}{l}\left|S^{\prime }\right|\le v_m\\ \left|S^{″}\right|\le a_m\end{array}$$

(21)

where $S^{\prime }$ represents velocity, $S^{″}$ represents acceleration, $a_m$ is the maximum acceleration limit, and $v_m$ is the maximum velocity limit.

3. Model Validation

To validate the engineering applicability of the cable crane’s trajectory planning model for smooth transition and obstacle avoidance, a high arch dam project in Southwest China was selected as a case study. In this project, seven 30-ton parallelly traveling cable cranes were deployed to undertake concrete hoisting tasks, utilizing a dual-layer layout with high and low lines (three on the high line and four on the low line), with spans of 1110 m and 1180 m, respectively, and design lifting heights of 330 m and 372 m. The spatial arrangement and load capacity were sufficiently simulated to represent the complex scenario of three-dimensional cross-operation during construction. The core dynamic parameters of the cable cranes in this project were configured as follows: the hoisting and traction mechanisms supported five speeds (see

Table 2. Speed gears for the cable crane in a certain project (m/s).

Power Mechanisms	First Gear	Second Gear	Third Gear	Fourth Gear	Fifth Gear
Traction mechanism	0.60	1.50	3.00	5.00	7.50
Hoisting mechanism	0.20	0.60	1.50	2.50	3.50

) and the actual operating speed was dynamically adjusted based on lifting distance, wind speed, and other meteorological conditions. The maximum acceleration of the traction mechanism was 0.75 m/s² and that of the hoisting mechanism was 0.50 m/s². These parameters served as the boundary conditions for dynamic constraints in the model’s velocity planning. For safety control, the minimum safe distances were set to 3 m between the bucket and the dam body and 15 m between the bucket and the tower crane to validate the effectiveness of obstacle avoidance in trajectory planning.

3.1. Offline Global Baseline Trajectory Planning

3.1.1. Results of Offline Global Baseline Trajectory Planning

The typical concrete pouring block in the riverbed section of the dam was selected for validation. As the core zone for concrete pouring, the complexity of trajectory planning in this region adequately represents conditions encountered in other construction areas.

The planning process was as follows. First, an obstacle-avoidance path was generated based on the start and end points of the hoisting task, constructing an environment map under an offline setting without considering mobile obstacles. Next, the maximum horizontal and vertical lifting velocities were set to 7.5 m/s and 3.5 m/s, respectively. Based on the transport distance, the expected duration range for each operation phase was estimated to be [8,30] s. An orthogonal experimental combination was generated with a time step of 1 s. Finally, trajectory optimization was completed in 0.08 s (based on an i7-11700K CPU and 32 GB RAM hardware) and the resulting motion parameters were converted into execution parameters for the traction and hoisting mechanisms. The outbound and return trajectories are shown in Figure 12 and the corresponding motion parameter curves are illustrated in Figure 13.

Figure 12 shows that the planned trajectory is smooth and continuous. Although there is a slight offset from the geometric path, it accurately reaches the start and end points as well as the intermediate control points, while satisfying all safety constraints. Figure 13 illustrates that the velocity profiles of the traction and hoisting mechanisms are smooth and the dynamic constraints are strictly met, ensuring stability throughout hoisting. It should be noted that the continuity of acceleration leads to more time spent in the acceleration and deceleration phases. Its impact on the total operation time must be quantified in subsequent analysis.

3.1.2. Comparison with Trapezoidal Velocity Planning

A comparative analysis was conducted using the full-load compound motion during outbound as the subject, with the motion parameter curves of trapezoidal velocity planning shown in Figure 14. The results indicated that the compound motion durations of the proposed method and the trapezoidal velocity planning were 62 s and 51 s, respectively. The trapezoidal velocity achieved the optimal operating state quickly through a large initial acceleration, demonstrating clear efficiency. However, for hoisting tasks with a single cycle of 5–10 min, the time difference had a limited impact.

A further comparison of the time histories of the swing angle, angular velocity, and angular acceleration of the bucket within 200 s (Figure 15) revealed that under trapezoidal velocity planning, the swing angle of the bucket during both transport and stabilization was greater than that of the proposed method (

Table 3. Characteristic statistics of swing angles and amplitudes.

Motion Phase	Method	Maximum Absolute Value of the Swing Angle (rad)	Corresponding Pendulum Length (m)	Corresponding Swing Amplitude (m)
Compound motion	Traditional Method	0.1425	57.21	8.12
	Proposed Method	0.1154	54.40	6.26
Stabilization	Traditional Method	0.1084	197.00	21.31
	Proposed Method	0.0641	197.00	12.62

). In the early stage of compound motion, the maximum swing angle under the trapezoidal velocity exceeded that of the proposed method by 0.0271 rad. This short pendulum length led to a low safety risk. However, at the end of motion, the maximum swing angle difference reached 0.0443 rad, corresponding to a swing amplitude of 21.31 m (compared to 12.62 m with the proposed method). This excessive swing amplitude not only prolonged the swing suppression time but also significantly increased the risk of collision. According to Figure 15b,c, the angular velocity and angular acceleration curves of the proposed method exhibit no obvious inflection points, with only minor oscillations caused by time discretization. In contrast, the trapezoidal velocity shows abrupt changes in angular velocity and sharp transitions in angular acceleration at the start and end of the acceleration and deceleration. These discontinuities exacerbate bucket oscillations and increase fatigue wear on key components such as reducer gears.

In summary, while trapezoidal velocity planning offers higher efficiency, it compromises safety and smoothness, potentially increasing the hidden costs of swing suppression and equipment maintenance. The proposed method, albeit with a slightly longer execution time, significantly enhances transportation stability and reduces safety risks, making it more suitable for underactuated cable crane systems.

3.1.3. Analysis of the Influence of the Traction Hoisting Mechanism’s Maximum Velocity on the Cable Cranes’ Hoisting Process

According to Table 2, when the maximum horizontal and vertical hoisting velocities were 7.5 m/s and 3.5 m/s, respectively, the maximum bucket swing amplitude during positioning reached 12.62 m, unfavorable for both safe and efficient operation. To address this, multiple peak velocity combinations were designed (

Table 4. Combinations of maximum horizontal and vertical hoisting velocities.

Scheme Number	Maximum Horizontal Hoisting Velocity (m/s)	Maximum Vertical Hoisting Velocity (m/s)
1	3.0	1.5
2	5.0	3.0
3	7.5	3.5

) to compare and analyze the impact of different velocity parameters on trajectory performance.

The experimental results indicated that the transportation duration for both the outbound and return trips decreases as velocity increases (

Table 5. Comparison of hoisting parameters for different schemes.

Hoisting Stage	Operating Conditions	Average Horizontal Hoisting Velocity (m/s)	Average Vertical Hoisting Velocity (m/s)	Hoisting Duration (s)
Outbound	Scheme 1	2.13	1.09	134.00
	Scheme 2	3.14	1.58	91.00
	Scheme 3	3.86	1.89	74.00
Return	Scheme 1	1.98	1.17	143.00
	Scheme 2	2.80	1.57	102.00
	Scheme 3	3.25	1.66	88.00

). Since the motion states across different schemes were consistent except during the compound motion, the analysis primarily focused on the swing response characteristics of the bucket to velocity parameters during this phase. The swing angle curves for the different schemes are shown in Figure 16. Specifically, the maximum swing angle of the bucket decreased as the maximum horizontal and vertical hoisting velocities were reduced. When the trolley’s maximum velocity increased, the peak acceleration within the expected transport time window also increased accordingly, intensifying the swing of the bucket. Statistical analysis of the characteristic parameters (

Table 6. Characteristic statistics of swing angles and amplitudes.

Hoisting Stage	Operating Conditions	Maximum Swing Angle During Positioning/Empty Bucket Descending (rad)	Corresponding Pendulum Length (m)	Corresponding Swing Amplitudes (m)
Outbound	Scheme 1	0.0248	197.00	4.89
	Scheme 2	0.0398	197.00	7.84
	Scheme 3	0.0641	197.00	12.62
Return	Scheme 1	0.1242	45.00	5.57
	Scheme 2	0.1821	45.00	8.15
	Scheme 3	0.3069	45.00	13.59

) shows that the maximum outbound swing amplitudes for the three velocity combinations were 4.89 m, 7.84 m, and 12.62 m. Accordingly, the return trip amplitudes were 5.57 m, 8.15 m, and 13.59 m. This result suggests that selecting excessively high maximum velocities for short transport distances can shorten or even eliminate the constant-velocity period. This leads to significant acceleration fluctuations due to frequent acceleration and deceleration during transport, which notably compromises operational smoothness.

Comparison with measured data reveals that, in actual operations, the cable crane tends to operate at medium to low average velocities, with the primary consideration being operational smoothness. The smooth transition trajectory proposed in this study can expand the applicable range of velocity by suppressing swing and it is recommended that the maximum horizontal and vertical velocities be limited to those specified in Scheme 1.

3.2. Real-Time Trajectory Planning for Obstacle Avoidance

During a nighttime operation of this project, a cable crane hook bucket deviated from its intended trajectory due to operator fatigue, resulting in a collision between the hook pulley and the tower crane (Figure 17). Post-incident analysis indicated that, with a detection frequency of 0.5 s, the real-time collision warning system had already issued a warning at point A and generated an obstacle avoidance strategy (Figure 18).

When applying the proposed method for obstacle-avoidance trajectory planning, a collision-free path was first generated from point A to the alignment start point. The maximum horizontal and vertical velocities were set to 7.5 m/s and 3.5 m/s, respectively. After estimating the required time interval, the expected horizontal duration was discretized into 15 segments. The remaining planning steps followed the baseline trajectory procedure described in Section 3.1. The results showed that the optimal trajectory planning took only 0.01 s. Given this extremely short time, the motion states of the cable crane trolley and hoisting rope can be considered unchanged. Therefore, the initial state for trajectory adjustment remained identical to point A (Figure 19).

Figure 19 and Figure 20 demonstrate that the smooth obstacle-avoidance trajectory planned using the proposed method does not intrude upon the tower crane’s minimum safety boundary. The path transition is smooth and both the cable crane trolley and the hoisting rope exhibit smooth velocity, satisfying the dynamic constraints without any abrupt changes in velocity or acceleration. A comparison of the swing angle curves of the bucket under immediate braking and smooth obstacle avoidance (Figure 21) reveals that immediate braking results in a rapid increase in the swing angle. In contrast, the proposed method results in a smaller amplitude variation at the onset of obstacle avoidance and maintains a lower swing angle thereafter. This indicates that the proposed method not only effectively avoids obstacles but also ensures the stable operation of the cable crane, thereby validating its effectiveness.

4. Discussion

In cable crane trajectory planning, bucket swing induced by non-smooth obstacle-avoidance trajectories is a critical factor limiting hoisting efficiency and safety. Although GNSS-based real-time obstacle avoidance technologies can reduce the risk of collision, such methods do not account for the underactuated characteristics of cable cranes and emergency stops or abrupt direction changes tend to cause severe bucket swing. To address this issue, this study develops a smooth transition planning model for real-time obstacle-avoidance trajectories of cable cranes and its effectiveness is validated through an engineering case study.

Section 3.1 systematically analyzes the performance of offline global reference trajectories under different working conditions. The results show that, compared with the traditional trapezoidal velocity planning, the proposed method achieves smoother bucket velocity and continuous acceleration, which significantly suppresses the swing amplitude of the bucket and ensures the stable operation of the cable crane. Furthermore, Section 3.2 validates the effectiveness of the proposed real-time obstacle-avoidance planning method through a case study between a tower crane and a cable crane. The bucket swing angle under the obstacle-avoidance trajectory execution is significantly reduced compared to immediate braking. Overall, the results demonstrate that the planned trajectories offer both smoothness and dynamic adaptability, effectively meeting the requirements of cable crane dynamic obstacle avoidance in arch dam construction, and providing technical support for crane obstacle-avoidance operations in the construction of other large-scale structures such as bridges and prefabricated buildings.

This study has some limitations:

(1) The arch dam construction site is subject to variable climatic conditions and strong winds may induce bucket swing. However, the influence of wind has not been considered in this research.

(2) Although the proposed trajectory smoothing method can reduce bucket swing, it does not involve active anti-swing control.

(3) Necessary assumptions were made during the establishment of the kinematic model and discrepancies may exist between the actual motion and the planned trajectory.

(4) While large-scale dam projects typically involve collaborative operations of multiple cable cranes, this study has not yet extended to multi-crane scenarios.

(5) This study primarily compares the proposed approach with the conventional trapezoidal velocity planning method, without incorporating systematic comparisons with other state-of-the-art trajectory planning techniques.

Future research can be advanced along two key directions. First, within the dynamically coupled system of trolley, hoisting ropes, and bucket, greater attention should be paid to uncertainties such as wind loads, variations in concrete properties, sensor noise, and unmapped dynamic obstacles. By acquiring high-precision, high-frequency motion state data, it is possible to design an active anti-sway module, integrate robust control algorithms, and establish adaptive error compensation mechanisms to mitigate external disturbances. Second, the research can be extended to multi-machine cooperative scenarios, where spatiotemporal resource scheduling enables real-time collision avoidance and dynamic task allocation among multiple cable cranes. This approach holds promise for optimizing overall construction workflows and enhancing the integrated operational performance of large-scale engineering projects. Third, systematic comparisons with other state-of-the-art trajectory planning methods should be conducted, considering the large-span, flexible connection, and underactuated characteristics of cable cranes, to further validate and refine the proposed trajectory planning model.

5. Conclusions

In cable crane operations, the smoothness of trajectory planning directly affects both safety and operational efficiency. Conventional methods often suffer from excessive changes in path direction and unsmooth velocity, where emergency braking upon encountering obstacles can induce severe bucket swings and accelerate equipment wear. This study proposes a decoupled path–velocity trajectory planning model for cable cranes, validated through engineering case simulations, yielding the following conclusions:

(1)

To address the challenge of balancing efficiency and real-time responsiveness in conventional planning, a hierarchical strategy of offline global baseline planning and online real-time local adjustment is proposed. In the offline stage, a static environmental map is constructed from a BIM model to generate the global baseline trajectory. In the online stage, real-time monitoring data are used to rapidly adjust the local trajectory. The total computation time for optimal trajectory generation is only 0.08 s (offline) and 0.01 s (online), satisfying real-time obstacle avoidance requirements in complex environments and overcoming the limitation of static paths in adapting to dynamic obstacles.

(2)

The proposed path–velocity decoupling model substantially enhances trajectory performance. In geometric path planning, an improved A* algorithm is adopted to reduce redundant nodes and lower the frequency of changes in direction. In velocity planning, a simplified S-curve is designed for long-distance composite motions. At the same time, polynomial interpolation is employed for short-range scenarios, effectively suppressing bucket oscillations and ensuring smooth velocity.

(3)

Engineering trials confirm that the model is well-adapted to cable crane dynamics and produces geometrically smooth trajectories. Compared with conventional trapezoidal velocity planning, the maximum bucket swing amplitude during composite motion is reduced by 40.78%, with only a minor increase in composite motion duration, resulting in negligible impact on single-cycle tasks. In real-time obstacle avoidance scenarios, the maximum bucket swing amplitude is reduced by 30.48% relative to an emergency stop strategy, demonstrating the method’s effectiveness in safeguarding operational safety and efficiency.