Implementation of two-roller scheduling path planning under road construction scenarios | Scientific Reports

Scientific Reports volume 15, Article number: 6767 (2025) Cite this article

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Previous studies have verified the feasibility of single unmanned roller tracking paths and have effectively evaluated the performance of pavement compaction. Nevertheless, the issue of scheduling faulty rollers (e.g., insufficient oil pressure) during collaborative pavement construction with multiple rollers has not been fully investigated, despite its prevalence in engineering practice. Although there are some patents that propose solutions for several cases, there is a lack of comprehensive, detailed and process-oriented scheduling strategies for specific scenarios. This paper introduces a framework for pavement construction comprising one paver and six rollers, and proposes a scheduling strategy for idle and faulty robots with the objective of addressing the problem of scheduling faulty rollers. Specifically, the traditional path planning task requires known beginning and end positions, while the scheduling position in this paper can be designed in advance. Consequently, this paper presents a methodology that leverages the ratio of discrete Bezier curve path points to discrete idle region points at distinct scheduling positions to ascertain the start and end positions of faulty and idle robots. In the scheduling process implementation phase, the paper considers the scene constraints and the model constraints of the rollers, and proposes a novel cost function that balances path length and safety distance. Furthermore, to address the issue of driving in opposite directions in the narrow passage, this paper proposes the interleaved scheduling scheme with the objective of enhancing the performance upper bound of the algorithm by significantly increasing the probability of finding a feasible solution. Moreover, the implementation of a discrete sampling of curve and position points ensures that the algorithm runs in an acceptable time. The results of comparative simulations demonstrate that the path planning algorithm proposed in this paper is more effective than alternative algorithms in addressing the specific application scenarios of this project. Furthermore, the integration of the interleaved scheduling framework has a considerable impact on the enhancement of the quality of the generated paths. The algorithm is capable of successfully completing the scheduling task of faulty and idle rollers with a road length of more than 100 m in a road width of 10.5 m, while ensuring that the other rollers are not affected in the completion of their respective tasks. The path search time is 24 s, the generated planning path length is 178.1 m and the safety distance is approximately 1.6 m.

Path planning is a fundamental technology for autonomous robots, as it is a prerequisite for robots to perform a range of tasks. In the early stages of research, scientists investigated the generation of the shortest or fastest paths in static obstacle scenarios by treating robots as plasmas. To illustrate, the A* algorithm combines heuristic and greedy strategies to guarantee the generation of the shortest path1. The quality of path generation of this algorithm is contingent upon the heuristic function, which is challenging to design with precision, and the hyper-parameters involved are contingent upon the expertise of the experts. Furthermore, the A* algorithm itself is characterized by a high spatial complexity, which has contributed to a gradual decline in the utilization of A*-improvement strategies for path-planning tasks in recent years. The RRT algorithm combines the search and random sampling characteristics, although the generated paths are not optimal. However, it greatly accelerates the efficiency of path generation, which is an excellent path planning algorithm that can balance the quality and efficiency of path generation. As a consequence of the random sampling and node selection properties of the algorithm, a considerable number of improved RRT algorithms have emerged. Notable examples of these include RRT*2, RRT*-Smart3,4, Quick-RRT∗5, and Informed-RRT*6. These algorithms have demonstrated that path planning algorithms are capable of handling certain large-scale scenarios, uncertain environments, and good real-time performance. However, these improvement strategies are not specific to a specific class of robots with incomplete kinematic constraints, so it is uncertain whether the designed paths can be tracked by certain types of robots. The Bezier curve is also a classical class of path planning algorithms. In contrast to A* and RRT, Bezier curve can directly generate curves that conform to C2 continuity, thereby enabling the generated paths to be directly tracked as long as the curvature constraints of the robot are satisfied. The development of artificial intelligence has led to the emergence of model-driven and data-driven path planning algorithms. Representative examples of these algorithms include recurrent neural network–based policy7, adversarial sampling-based planning8, deep Q learning9,10, actor-critic11, and deep deterministic policy gradient12. These algorithms are characterized by the necessity for extensive a priori learning experience, which is not available for some tasks.

The majority of the aforementioned path planning algorithms are theoretical in nature or necessitate a significant degree of prior experience to guide the generation of paths, which presents a challenge in many real-world tasks. In recent years, researchers have accordingly proposed targeted solutions for specific application scenarios. Bio-inspired neural networks have been developed for use in search and rescue tasks13. Reinforcement learning has been employed to generate paths that adhere to social behavioral norms14. AM-RRT* has been applied to unmanned vehicles overtaking tasks15. A modified RRT* has been used in the context of surgical robotics16. Bezier curves have been developed for the planning of footstep trajectories of exoskeleton robots in complex terrain17. Multi-objective cross-entropy optimization has been employed in the context of dual-robot cooperative arc welding paths18. A coronavirus optimization algorithm (COVIDOA) has been developed for the planning of automotive casting burr removal trajectories19. The results of this stage and future directions are summarized below: Firstly, the effective design of metrics and constraints has driven the development of robotics in various fields20,21,22, including obstacle avoidance and robot models, as well as evaluation metrics such as energy consumption, path smoothness, path length, and path generation efficiency. These evaluation metrics have been applied in a vast amount of literature and provide guidance for generating high-quality paths. Nevertheless, the metrics in question present certain challenges in terms of their design, which hinders the evaluation of the quality of the generated paths. For example, there is a debate as to how self-driving cars should weigh their own interests against the rules of public transport when overtaking23,24,25. Furthermore, there are sociological, ethical, legal and moral issues involved when confronting the tram problem, which makes the design of metrics even more challenging. Secondly, the collaboration of multiple robots or multiple types of robots will remain the dominant trend in path planning26,27. This is due to the fact that there are numerous scenarios that necessitate the collaboration of multiple robots on specific tasks from a requirements perspective28,29,30. However, multi-robot collaboration also encounters significant challenges from an algorithmic implementation standpoint, including the selection of centralized or distributed architectures31,32, task allocation strategies, and communication constraints33. Thirdly, it is anticipated that human-assisted robots will continue to flourish. For instance, it has been demonstrated that human-assisted robotic arms are more efficient, robust and better able to cope with unexpected situations34,35. Surgical robots provide an illustrative example. Although surgical robots are capable of autonomously planning satisfactory paths in complex environments, including those filled with blood vessels, tissues and organs, at this stage, surgical robots are still assisted by humans due to concerns about their long-term stability, accuracy and failure rate16,36,37. Such safety-critical application scenarios have the potential to result in catastrophic outcomes in the event of an accident. Consequently, human-robot collaboration will likely remain a prevalent strategy for the foreseeable future. Fourth, the results of the simulation demonstrate that path planning algorithms can be applied to large-scale scenarios and uncertain environments with satisfactory real-time performance38. However, upon verification in the field, it became evident that the limited hardware performance constrains the full exploitation of the algorithm’s potential in terms of stability, accuracy, and rapidity39. Furthermore, the large-model-based path planning strategy is of low reliability, poorly generalized40 and difficult to be interpreted41,42. This renders it challenging to provide guidance for algorithm optimization. Consequently, the hardware and the large models continue to exert an influence on the long-term stability of path planning in real scenarios. The issue of scheduling two rollers travelling in opposite directions is also one of the specific problems in the specific scenarios, and is therefore of great importance in the field of pavement construction. In addition, this is a collaborative task that requires the coordination of multiple rollers, making it a topic worthy of further research in the field of robotics. The motivation for investigating this problem is a continuation of previous work. The authors’ team previously constructed a four-layer test system, consisting of an evaluation layer, a guidance layer, a control layer, and a sensor layer and had implemented the evaluation of path tracking quality, the ratio of compacted area, and the compaction repeatability43. The preceding work is limited in that only a single roller was considered in the context of pavement construction, and the possibility of roller failure was not addressed. This paper is to address the issue of fault scheduling in multi-roller collaboration with the intention of enhancing pavement construction efficiency and improving system stability. Furthermore, the BECKHOFF controller utilized in the previous task is capable of communicating with the SIMULINK module in MATLAB. Consequently, the algorithms implemented in MATLAB can be directly guided for the generation of the paths and deployed on hardware to fulfill the engineering needs.

Figure 1 depicts the asphalt paving and pavement compaction process. The paver is positioned at the front of the queue, evenly apportioning the asphalt mixture, followed by two rollers (Roller 1 and Roller 2) responsible for the initial compaction. Roller 3 and Roller 4 are responsible for the secondary pavement compaction and Roller 5 is responsible for the final pavement compaction. In order to address the situation where the asphalt pavement compaction tasks are put on hold due to the low oil pressure in one of the rollers, it is necessary to ensure that the idle roller follows the entire queue. This allows for the replacement of the faulty roller at the appropriate time, thus enabling the task that has been put on hold to be completed. In comparison to the aforementioned tasks, the scenario presented in this paper is considerably more constrained. Primarily, the asphalt compaction process necessitates that the rollers operate at a uniform speed in a single direction. Consequently, the trajectories of the rollers during the construction process are severely constrained, and the fault scheduling positions can only be designed at the boundaries of the sub-areas. With regard to Roller 2 and its task partition, for instance, the scheduling position can only be designed at the orange circle marked in Fig. 1. Secondly, the restricted road width may necessitate that scheduling rollers and idling rollers traverse the aisle in sequence when travelling in opposite directions. This may result in the two rollers departing at different times to complete the scheduling task. Fortunately, however, since the speed of the rollers can be set in advance and the lack of oil for the malfunctioning roller can be predicted in advance. Consequently, the starting position of the scheduling of the malfunctioning roller and the end position of the scheduling of the idle roller can also be designed in advance. Furthermore, it is necessary that the designed paths are safe and can be tracked by the rollers, and that the idle roller is not too far away from the construction area for timely scheduling. Due to the specificity of the scenarios mentioned above, it is evident that traditional, optimization-based and model-driven path planning algorithms are not applicable to this study for the following reasons: Traditional path planning algorithms do not take into account various types of constraints, including model constraints and curvature constraints1,2,6. Optimization-based path planning algorithms are also unsuitable because they are unable to quickly obtain a feasible solution in the presence of numerous equation constraints, including fixed construction sequences, fixed construction paths, and fixed travelling speeds for non-scheduled rollers37,38. Model-driven path planning algorithms lack the versatility to adapt to dynamic scenarios9,12,40. The objective of this paper is to investigate the design of scheduling paths for faulty and idle rollers without affecting the performance of other well-working rollers. In light of the aforementioned objective and the specific scenario delineated in the preceding section, this paper initially proposes a novel strategy for determining the start position of the idle roller and the end position of the dispatched roller. This strategy entails calculating the ratio of discrete Bezier curve path points to discrete idle region points at different scheduling positions. Subsequently, multiple alternative paths that align with the constraints of the scenario are designed by employing the Bezier curves. The following contributions are made: (1) A novel start/end scheduling position selection strategy is proposed. (2) A novel cost function for the trade-off between path security and path length is proposed. (3) An interleaved scheduling scheme is proposed.

The asphalt paving and pavement compaction process.

In the event that the oil pressure of the roller in operational condition is insufficient, a dual roller scheduling strategy is proposed. This entails the transfer of the faulty roller from the construction site, while the idle roller assumes the role of the faulty roller to complete the pavement compaction task. From a process perspective, the implementation of the strategy entails the determination of the starting and ending points of the two rollers, followed by the generation and selection of the most appropriate paths. From the perspective of constraints, the scheduling strategy is constrained by a number of factors, including the model, scenario, and cost functions. The dual roller scheduling framework, as designed, is depicted in Fig. 2.

The framework for two-roller scheduling strategy.

The autonomous articulated vehicle model can be simplified with two wheels and an articulated frame, as illustrated in Fig. 3. The kinematic equation is expressed as follows44:

Where \({\dot {\xi }_f}=({\dot {x}_f},{\dot {y}_f},{\dot {\Theta }_f},\dot {\gamma })\) is the change over time in the horizontal and vertical coordinates, heading angle and articulated steering angle of the roller as it moves forward. \({l_{{\text{AR}}}}\) is the distance from the front or rear axle to the hinge point, \({v_f}\) is the front wheel speed, and \({l_{{\text{AR}}}}\) and \({v_f}\) are both constants, and \({\omega _\gamma }=\dot {\gamma }\).

The steering principle of the articulated steering roller used is illustrated in Fig. 3. Given the potential safety hazards associated with the articulated position of the roller and the distinctive characteristics of convex polygons, the roller is expanded into a rectangular area of M*N.

Definition of the roller size.

As illustrated in Fig. 3, at the point where the articulated steering angle reaches its maximum value \({\gamma _{\hbox{max} }}\), the curvature is at its greatest. Consequently, each sampling point on the designed curve must be less than the maximum curvature:

.

Where \({\kappa _p}\) is the curvature on the curve.

The primary focus of the scene constraint is on obstacle avoidance in confined spaces. Given that the roller has been treated as a rectangular region, the constraint can be interpreted as a judgement on whether two rectangular regions are in contact in the two-dimensional plane or not. This type of problem can be classified as a convex hull problem.

The Separating Axis Theorem (SAT) is an overlap detection method that is applicable solely to convex hulls. In the event that two convex packets in a two-dimensional plane do not overlap, a separating axis will always exist. This axis separates the two packets and is defined as the separating axis. The following three-stage detection process is employed to perform SAT: (1) the calculation of rectangular vertex coordinates, (2) the calculation of the projection coordinates, and (3) the recognition of the overlapping area.

Since the roller’s heading changes in real time as it moves, the coordinates of the rotated rectangular vertices \(({x_d}^{\prime},{y_d}^{\prime})\) can be calculated using the known default vertex coordinates \(({x_d},{y_d})\) when the conditions of the roller’s size (M*N), center position \(({x_c},{y_c})\) and orientation \(\theta\) are known:

Given that the dimensions of the roller are defined as M*N, the default vertex coordinates \(({x_d},{y_d})\) are \(({x_c} - M/2,{y_d}+N/2)\),\(({x_c}+M/2,{y_d}+N/2)\), \(({x_c}+M/2,{y_d} - N/2)\) and \(({x_c} - M/2,{y_d} - N/2)\) respectively, are established. In particular, when the rotation angle is\(\theta =0\), \(({x_d}^{\prime},{y_d}^{\prime})=({x_d},{y_d})\).

To calculate the projected coordinates, the projection is plotted in accordance with the illustration presented in Fig. 4, where \(\overrightarrow u {\text{ }}\) is the unit vector of the vector \({\omega _1}\) and the mathematical expression is \(\overrightarrow u {\text{ = }}{\omega _1}/\left| {{\omega _1}} \right|\). Based on the aforementioned illustration, some expressions can be derived as follows:

The simultaneous application of Eqs. (4) and (5) yields the result \(\overrightarrow {{\text{OM}}} {\text{ }}={\text{ }}\left| {\overrightarrow {{\text{OM}}} } \right|{\text{ }}\overrightarrow u {\text{ }}={\text{ }}\left| {\overrightarrow {{\text{OP}}} } \right|{\text{ }}\overrightarrow u {\text{ }}\overrightarrow u\).

A schematic diagram of the vector projection method.

As illustrated in Fig. 5, when the larger starting position of the two intervals is larger than the smaller ending position of the two intervals, these two intervals have no intersection, and the expression is \({\text{max(}}{a_1},{b_1})>\hbox{min} ({a_2},{b_2})\).

Five cases of the relative positions of interval \([{a_1},{a_2}]\) and \([{b_{1i}},{b_{2i}}]\)

The two-roller scheduling strategy encompasses the determination of start and end positions, the generation of scheduling paths, and the selection of optimal scheduling paths.

A novel strategy for calculating the ratio of the discrete Bezier curve path points to the discrete spare area points for different scheduling positions is proposed to determine the optimal scheduling position. The expression is as follows:

The indicator function, represented by the symbol \(\phi (\Delta )\), takes on the values 1 and 0 when the condition \(\Delta\) is true and false, respectively. The symbol \({P_0}\) represents all points in the spare area, while the symbol \(p({\rm T})\) represents all feasible solution path points in the Bezier curve. The Bezier curve will be described in the sub-section titled “Scheduling path generation.”

The generated scheduling paths must exhibit continuous curvature and be straightforward to compute. The polynomial curve can be derived to any order, and interpolation of the curve allows the curvature between any two points to be calculated, thus satisfying the aforementioned properties. The expression for the polynomial curve is as follows:

The analysis of the results obtained in43 indicates that the optimal path is defined by a curve with four control points, with a parametric equation as follows:

Equation (8) is also known as the cubic Bezier curve, where \({b_0}({x_0},{y_0})\)、\({b_1}({x_1},{y_1})\)、\({b_2}({x_2},{y_2})\) and \({b_3}({x_3},{y_3})\) represent the four control points of the cubic Bezier curve. The following equations can be derived from Eq. (8):

Assuming that the initial heading \(P^{\prime}(0)\) and the final heading \(P^{\prime}(1)\) of the scheduling rollers are known, the variable parameters \({b_1}\) and \({b_2}\) are controlled by a single variable each. This is because the tangent vectors at the end position of the cubic Bezier curve, expressed as \(\overrightarrow {{b_1} - {b_0}}\) and \(\overrightarrow {{b_3} - {b_2}}\), are directed along the first and last span of the polygon, as also shown in Fig. 6. In order to more accurately represent curves in the two-dimensional space, the four control points of the cubic Bezier curve are then defined as follows:

Where \({b_0}=({b_{0x}},{b_{0y}})\), \(b_{1}^{i}=(b_{{1x}}^{i},b_{{1y}}^{i})\), \(b_{2}^{i}=(b_{{2x}}^{i},b_{{2y}}^{i})\) and \({b_3}=({b_{3x}},{b_{3y}})\) are the coordinates of the four control points of the cubic Bezier curve on the two-dimensional plane. The superscripts i for b1 and b2 indicate the fact that the diversity of the curves is due to the control points b1 and b2. Figure 6 illustrates two different sets of control points and the curves they generate from these control points, marked in red and blue respectively.

Cubic Bezier curve with two different sets of control points.

In the scheduling path generation section, a multitude of cubic Bezier curves will be generated that conform to both roller model constraints and scenario constraints. The objective of this section is to identify the optimal curve. This paper proposes a novel cost function that incorporates a safety function and a path length function.

Cost function The cost function for two scheduling rollers is defined as follows:

Where \(X=\{ {x_1},{x_2},\ldots{x_n}\}\) represents a series of discrete points of the cubic Bezier curve. \(\alpha\) and \(\beta\) are constant, \(S(X)\) and \(L(X)\) are the safety function and the path length function.

Safety function The safety function can be calculated using the following formulae:

Where \(D{\text{(}}{x_i}{\text{) = min(}}D({q_{m\mathop m\limits^{\sim } }}))\) is the distance between the m-th roller being scheduled and the other n-1 rollers, \({D_0}\) is the safety threshold constant.

Path length function The path length function can be calculated using the following formulae:

Where \(l({x_i},{x_{i+1}})\) represents the distance between adjacent discrete points.

Pseudocode The pseudo-code of the novel two-roller scheduling framework based on multi-roller cooperative construction proposed in this paper is presented in Algorithm 1. The inputs to the algorithm include the pose of the well-conditioned rollers (u = 1,2,…) at the construction site, the end position of the idle roller \(I{R_{{\text{end}}}} \in IR\) and the end position of the malfunctioning roller. The algorithmic flow is as follows: firstly, the proposed position selection strategy is employed to determine the end position of the malfunctioning roller \(M{R_{{\text{end}}}}\) and the start position of the idle roller \(I{R_{{\text{init}}}}\); secondly, the scheduling of the malfunctioning and idle rollers is realized sequentially. When planning the path of the malfunctioning roller, the Bezier curves are first used to generate multiple compliant paths, designated as Multi_Path1_MR; Subsequently, curvature constraints and collision constraints are introduced to screen out the compliant curves, designated as Multi_Path3_MR. Lastly, the proposed novel cost function that incorporates the safety function and the path length is used to select the optimal path, designated as Optimal_Path_MR. In the event that it is impossible to generate a single path that satisfies all constraints, a five-second wait is initiated until at least one path is generated that meets all constraints. The same strategy is employed to plan the path of the idle roller, with the exception that collision constraints between the idle and malfunctioning rollers are also introduced in the collision detection phase.

The proposed two-roller scheduling framework based on multi-roller collaborative construction.

The pavement width of a secondary highway in China is 10.5 m. A schematic of a multi-roller pavement construction is presented in Fig. 7, which employs a 30-meter grinded strip with flat joints construction strategy. The pavement construction area is divided into six sub-areas, comprising two initial pavement compaction areas, two secondary pavement compaction areas, one final pavement compaction area and one spare area. Each of these areas is equipped with a roller, which is denoted by the letters A to F. The entire construction queue moves forward along the positive half-axis of the y-axis, while the roller queue advances along the positive half-axis of the y-axis.

Multi-roller pavement construction schematic.

In order to calculate the ratio of discrete Bezier curve path points to discrete idle region points, the following steps are taken. Firstly, the curve and the spare area are sampled. Secondly, the number of discrete Bezier curve path points falling on discrete spare area points and the total number of discrete spare area points are calculated. Lastly, the ratio of discrete Bezier curve path points to discrete spare area points is calculated using the following Eq. (6). In Fig. 8, the blue and green asterisks represent the sampling points for the Bezier curve and the area, respectively. The sampling accuracy is 0.1 m. The red rectangles represent the spare areas. The scheduling start and end points are (7.69,120) and (2, − 30) in Fig. 8a, and (7.69, 120) and (6, − 100) in Fig. 8b. A comparison of Fig. 8a and b reveals that the greater the number of blue asterisks, the wider the distribution of the generated paths and the higher the probability of bypassing the obstacle. This provides the basis for determining the start and end positions.

(a) The discrete Bezier curve path points (blue asterisk) and the discrete spare area points (green asterisk) when scheduling start and end position are (7.69, 120) and (2, − 30). (b) The discrete Bezier curve path points (blue asterisk) and the discrete spare area points (green asterisk) when scheduling start and end position are (7.69, 120) and (6, − 100).

The enumeration process is employed to ascertain the ratio of discrete Bezier curve path points to discrete idle region points. The results of the simulations indicate that if the malfunctioning scheduling position is [7.69 m, 118 m], the x-axis search ranges from 1.06 m to 9.44 m in steps of 0.2 m. Similarly, the y-axis search ranges from 30 m to 0 m in steps of − 5 m for the malfunctioning roller. When the end position of the malfunctioning roller and the start position of the idle roller are [1.06 m, − 60 m] and [1.26 m, 0 m], respectively, the ratio of the discrete Bezier curve path points to the discrete spare area points reaches its maximum value, which are 36.36% and 27.86%, respectively, as illustrated in Fig. 9. The planned paths corresponding to the aforementioned ratio of 36.36% are illustrated in Fig. 10. It can be observed that the results presented in Fig. 10 are not as readily apparent as might be expected. As the Euclidean distance between the start and end points increases, the ratio also increases, thereby further illustrating the importance of identifying the optimal ratio.

The ratio of the discrete Bezier curve path points to the discrete spare area points at different initial positions.

The discrete Bezier curve path points (blue asterisk) and the discrete spare area points (green asterisk).

As previously stated in the theory section, it is anticipated that multiple sets of b1 and b2 will be selected on the roller’s initial and final heading in order to generate multiple Bezier curves. Figure 11 illustrates the selection of 20 control points b1 (purple asterisks) and 20 control points b2 (red asterisks) to obtain a total of 400 smooth curves (blue curves). Subsequently, the aforementioned curves are then selected using the roller’s curvature constraint in order to identify those that can be tracked by the roller. To calculate the curvature, 1000 points are uniformly interpolated from the beginning and end positions of the curve. The curvature of the three neighboring points is then sequentially solved using the three-point method. Figure 11a displays 361 curves that satisfy the curvature constraints. Figure 11b illustrates the curvature values of these curves at each point. Figure 11b also demonstrates that the generated paths exhibit greater curvature at the start and end points, and less curvature in the middle of the curve. It is therefore recommended that the speed of the roller be reduced at the start and end points when reaching the end point during path tracking. It is fortuitous that this curvature property is also well suited to the kinematics of autonomous rollers. Even when the roller is given to move at a certain constant speed, it must accelerate during the start-up phase and decelerate during the termination phase.

Bezier curve generation and curvature analysis. (a) The selected 361 curves. (b) The curvature of these 361 curves.

In order to ascertain the viability and efficacy of the strategy proposed in this paper, the scenario described in Sect. 3.1 is employed, with a focus on one of the most challenging scenarios: the scheduling task between the malfunctioning roller at the front of the convoy and the idle roller at the rear of the convoy. The six graphs in Fig. 12 represent the scheduling times from 1 to 15 s (Fig. 12a), 16 to 30 s (Fig. 12b), 31 to 45 s (Fig. 12c), 46 to 60 s (Fig. 12d), 61 to 75 s (Fig. 12e), and 76 to 90 s (Fig. 12f). The figures illustrate the positions of the scheduling rollers at different times, with the blue rectangular box and the red rectangular box representing these positions. The blue asterisks and the red asterisks represent the midpoints of the scheduling rollers (the center of the rectangular boxes), while the black rectangular boxes indicate the position of other rollers under construction. When the end position of the malfunctioning roller B and the start position of the idle roller F are [1.06 m, − 60 m] and [1.26 m, 0 m] respectively, it can be observed that the idle roller F commences movement after the roller B has been in motion for 20 s. The planned path satisfies the roller curvature and collision-free constraints, and ultimately completes the scheduling task of the malfunctioning roller in the initial pavement compaction area and the idle roller in the spare area within 90 s. The results of the simulation demonstrate that the proposed framework has a running time of 24 s and that the planned paths of the idle roller are 178.1 m in length. In this example, due to the limited width of the road, the two rollers that need to be dispatched must avoid each other when moving in opposite directions. This issue is effectively addressed by the interleaved path planning strategy designed in this paper.

Scheduling between the malfunctioning roller in the initial pavement compaction area and the idle roller in the spare area. (a) Scheduling time from 1 to 15 s. (b) Scheduling time from 16 to 30 s. (c) Scheduling time from 31 to 45 s. (d) Scheduling time from 46 to 60 s. (e) Scheduling time from 61 to 75 s. (f) Scheduling time from 76 to 90 s.

In order to demonstrate the superiority of the proposed framework, a variety of path planning algorithms are applied as comparisons. These include traditional path planning methods, such as A*, RRT, B-spline, ACO, Voronoi, Inform RRT*, as well as machine learning-based path planning algorithms, such as support vector machines and other supervised learning-based path planning algorithms. Additionally, one deep learning-based path planning algorithm, such as deep Q-learning and some supervised learning-based path planning algorithms are included. Table 1 illustrates whether the various types of algorithms satisfy the constraints of the scenario, as well as two quantitative results, the running time of the algorithms and the length of the scheduling paths of the idle robots.

Table 1 illustrates that some traditional path planning methods as well as deep Q-learning algorithms, are unable to satisfy the conditions of curve smoothing and curvature constraints. For instance, A* and deep Q-learning algorithms divide the map into a grid of small rectangles, while Voronoi diagrams divide the map into multiple polygons by using some specific points. The characteristics of rectangles or polygons, however, present challenges in guaranteeing the smoothness of curves. The fundamental concept of the RRT algorithm is to branch from nodes in a manner analogous to the branching of a tree. Consequently, it is also challenging to guarantee the smoothness of the curve. The aforementioned algorithms are applicable to autonomous robots that are able to adjust their direction in situ. These algorithms are distinguished by their capacity to achieve the theoretically optimal solutions, as evidenced by the outcomes of the A* algorithm. Some path planning algorithms, such as Bezier and Inform RRT*, are capable of generating smooth curves. However, these two algorithms do not consider certain properties of non-holonomic kinematic models of certain types of robots, such as radius of curvature constraints. Consequently, they are not applicable to the articulated robots discussed in this study. The framework proposed in43 designs several types of improved Bezier curves for soccer robots. The algorithm generates smooth curves and screens inappropriate curve curvature, and also realizes dynamic obstacle avoidance, which is well applied to the scenario of soccer robots. However, the objective of the scenario presented in this paper is to complete the scheduling task within a limited space. The obstacle avoidance strategy employed in these scenarios involves a strategy of bypassing obstacles, which, in our scenarios, will result in the design of paths that deviate from the boundaries of the map. It is possible that running out of the viaduct may cause safety accidents during real construction processes, such as highway compacted pavement sections. Therefore, although the strategy in43 is applicable to the robot studied in this paper, it is not applicable to the scenario. Literature45 also discusses the use of Bezier curves to plan the paths of soccer robots, but this paper lacks an interleaved scheduling strategy, as it aims at optimizing the paths of a single robot. In the last two rows of Table 1, the paper also compares the results with and without the interleaved scheduling strategy, demonstrating the advantages of the interleaved scheduling strategy. The proposed framework in46 presents a fast path planning algorithm applicable to a variety of car park scenarios. However, the algorithm relies on static obstacle modelling, which is not applicable to the scenarios of this paper. In the scenarios, when the malfunctioning robot is dispatched, the other well-functioning robots continue to work. Therefore, the algorithm is also not applicable to the scenarios required in this paper.

Some studies are competent for our path planning task. To illustrate, the goal-directed RRT with articulated model47 proposes the ‘head correction with fixed wheel position’ strategy as a means of achieving the kinematic model of articulated robots. However, this results in a series of repeated movements, which in turn increases the planned path. The proposed framework in48 successfully accomplishes the path planning task with the hyperplane of support vector machine, achieving favorable results under the application requirements of articulated robots approaching the site. Fortunately, this algorithm can satisfy the articulated robot model constraints and scenario constraints. However, the performance of the algorithm is not satisfactory due to several factors. Firstly, the support vector machine treats every 2D coordinate on the map as labelled training data, which results in a lengthy learning process for the zero-potential decision curve lines (planning path). Secondly, it is possible that the algorithm generates the zero-potential decision curve lines that do not satisfy the kinematic model constraints of the robots and therefore require further optimization. Additionally, this type of supervised learning strategy learns data from the entire map and is therefore only suitable for path planning tasks under static obstacles or when waiting for moving obstacles to drive away from the planned path. Consequently, this inevitably results in a time delay for the two-robot scheduling task. Similarly, reinforcement learning is also susceptible to this issue and is challenging to identify an optimal direction for optimization, rendering it unsuitable for this task10,22,49. In comparison to the aforementioned algorithms, the path planning algorithm proposed in this paper demonstrates superior performance in both performance metrics, namely, algorithm running time and path lengths of idle robots. Additionally, it obviates the necessity to model obstacles or analyze robots’ trajectory beforehand, thus circumventing the issues associated with insufficient data and time-consuming problems caused by data-driven path planning.

It is not possible to simultaneously satisfy the requirements of curve smoothing, curvature constraints and direct applicability to articulated robots using traditional path planning methods and deep learning algorithms. Some augmented algorithms address these challenges, but they are unable to directly provide a reasonable set of scheduling solutions for narrow environments that also satisfy the requirements of pavement construction. Some augmented algorithms are capable of satisfying all of the aforementioned constraints. Nevertheless, the planned paths designed for pavement construction characteristics in this study demonstrate superior performance in terms of both algorithm runtime and path length for idle robots. This is due to the fact that there is no necessity to model obstacles or analyze the trajectories of the robots, thus circumventing the issues associated with inadequate data and time-consuming data-driven path planning. Moreover, the algorithm does not utilize a randomization strategy, thereby ensuring its runtime stability.

It is also crucial to emphasize that a more pivotal evaluation metric than path length and algorithm execution time should be the assurance of the construction scenario’s safety. It is essential to maintain a specific distance between the two rollers to enable the potential for avoiding a collision between them by remotely controlling the cessation of the roller’s movement, even in the event of an erroneous algorithmic execution by the roller. However, as this factor is not considered in the existing comparative literature, it has not been included in the evaluation metrics in the ‘3.4 Algorithm performance comparison. Nevertheless, this factor is reflected in the cost function and in Algorithm 1. Furthermore, the conclusions regarding the quantified safe distance are also presented in the abstract.

The paper presents a construction site planning path that accounts for potential roller failures. Simulation results demonstrate that the proposed algorithm outperforms other path planning algorithms in terms of efficiency and fault tolerance. In previous work, our team employed a BECKHOFF controller for roller autonomy, which was also capable of communicating with the SIMULINK module in MATLAB44. This module has the capacity to encapsulate customized MATLAB sub-functions, thereby enabling the potential for guiding path generation, which is in alignment with the project’s practical requirements. However, due to a lack of funding, experimental validation of multi-roller pavement construction scheduling is not currently feasible. Consequently, the efficacy of the algorithms cannot be adequately validated. Furthermore, the following special cases have not been considered, and idealized assumptions have been made in this paper. Firstly, it is assumed that the road width remains constant over time, which is a highly idealized assumption. Consequently, the generation of scheduling paths in response to changes in road width represents another promising avenue for future research. Secondly, in certain road conditions, it is unavoidable to require the hand-push micro-roller for final compaction. Therefore, the use of a combination of pavers, rollers and hand-push micro-rollers to facilitate the cooperative pavement construction process on specific road sections also represents an interesting avenue for future research.

The data are available from the corresponding author upon reasonable request.

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The authors are very grateful to the experienced reviewers for their valuable suggestions for improving the quality of the paper.

This study is funded by a research start-up grant from Tong Xu.

School of Information Technology, Jiangsu Open University, Nanjing, 210000, China

Tong Xu, Junhao Liu & Zhixuan Zhou

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Conceptualization, Tong Xu; Formal analysis, Tong Xu; Methodology, Tong Xu, Junhao Liu; Software, Tong Xu, Junhao Liu, Zhixuan Zhou; Validation, Tong Xu, Junhao Liu, Zhixuan Zhou; Writing-original draft, Tong Xu; Writing – review & editing, Tong Xu.

Correspondence to Tong Xu.

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Xu, T., Liu, J. & Zhou, Z. Implementation of two-roller scheduling path planning under road construction scenarios. Sci Rep 15, 6767 (2025). https://doi.org/10.1038/s41598-025-91107-8

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DOI: https://doi.org/10.1038/s41598-025-91107-8

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