Artificial Intelligence-Based Multi-Task Multi-Robot Cooperative Motion Planning System

Abstract

The present invention discloses a multi-level task multi-robot collaborative motion planning system based on artificial intelligence, which belongs to the field of robot motion planning technology, and includes: a work area module; a working robot module, which is used to obtain the position information and path planning information of the robot currently performing the task; an idle robot module; a task allocation module, which is used to allocate the task to be allocated to the idle robot; a constraint condition module, which is used to set the constraint condition according to the task priority; a feasible path module, which is used to obtain a feasible path set according to the robot movement speed, path length and constraint condition; a collision information module, which is used to obtain collision information according to the path in the feasible path set and the robot movement speed; a path planning module, which is used to obtain an initial population according to the feasible path set, and obtain the path planning information of the multi-robot of the multi-task level based on the genetic algorithm and the fitness value calculation. The present invention can realize the path planning of the multi-robot of the multi-task level.

Description

Technical Field

[0001] The present invention relates to the technical field of robot motion planning, and in particular to a multi-task multi-robot collaborative motion planning system based on artificial intelligence. Background Art

[0002] Motion planning is a fundamental problem in robot control and robotics. Generally, in order to complete a task without colliding with any obstacles in the operating environment or reducing the possibility of colliding with any obstacles in the operating environment, motion planning specifies the path that the robot can follow from the starting state to the target state. When two or more robots move in a work area to perform tasks, the robots may interfere with each other during the execution of the task. In addition, the tasks assigned to different robots may have different priorities, so it is necessary to ensure that time-limited tasks can be completed first according to the urgency. Moreover, in the actual working environment, different paths may have different widths. For a path with a larger width, two or more robots can pass through without collision, while a path with a smaller width may only allow one robot to pass. Therefore, when assigning tasks to multiple robots in the same work area and performing path planning, the above influencing factors need to be fully considered. Summary of the invention

[0003] The present invention aims to solve at least one of the technical problems existing in the prior art. To this end, the present invention proposes a multi-task multi-robot collaborative motion planning system based on artificial intelligence, which can realize multi-task level multi-robot collaborative path planning.

[0004] On the one hand, an embodiment of the present invention provides a multi-level task multi-robot collaborative motion planning system based on artificial intelligence, including: a work area module, used to obtain map information of the work area where multiple robots work together, including a topological map composed of several nodes and edges and the width of the edges; encoding the nodes and edges to obtain node numbers and section numbers; a working robot module, used to obtain the position information and path planning information of the robot currently performing the task; an idle robot module, used to obtain the position information of the currently idle robot; a task allocation module, used to obtain several tasks to be allocated, and allocate the tasks to be allocated to the idle robots nearby according to the starting and ending point information of the tasks to be allocated and the position information of the idle robots; a constraint condition module, used to set constraint conditions according to task priorities; the constraint conditions include task execution time thresholds; a feasible path module, used to A path set is generated according to the map information and the starting and ending point information of the task, and a feasible path set is obtained according to the robot movement speed, path length and task execution time threshold; the feasible path set includes several feasible paths, and the feasible path is a sequence of several road section numbers; a collision information module is used to obtain the position information and path planning information of the robot currently executing the task, and obtain collision information according to the path in the feasible path set and the robot movement speed; the collision information includes a robot identifier, a feasible path identifier and a road section number; a path planning module is used to randomly generate several feasible solutions for the robot assigned the task according to the feasible path set, and use the several feasible solutions as individuals to obtain the initial population of the genetic algorithm; the path planning information of multiple robots at the multi-task level is obtained based on the genetic algorithm and the fitness value calculation; wherein the fitness value T is: Where N is the total number of robots that have been assigned tasks, s i is the length of the path currently selected by the ith robot, v i is the movement speed of the ith robot, c is the coefficient, s t,i is the number of turns of the path currently selected by the i-th robot, θ i is the average turning angle of the ith robot, s L,i is the width of the road section where the ith robot may collide, t a (s L,i ) is the avoidance time required by the robot when a collision is likely to occur.

[0005] According to some embodiments of the present invention, the work area module includes: a map information acquisition unit, which is used to move the robot in the work area for one circle, and scan the surrounding environment based on the laser radar of the mobile robot to obtain laser radar scanning data, and generate map information based on the laser radar scanning data; the map information includes a topological map and the width of each edge in the topological map.

[0006] According to some embodiments of the present invention, the working area module includes: a map information updating unit, used to update the map information by acquiring real-time lidar scanning data of the working robot and the real-time position information of the robot.

[0007] According to some embodiments of the present invention, the collision information module is used to obtain the position information and path planning information of the robot currently performing the task, and calculate the section number and the corresponding time period that the robot will pass through in the future; and obtain the feasible paths in the feasible path set in turn, and calculate the section number and the corresponding time period that the robot plans to pass through for each feasible path according to the robot's movement speed; obtain multiple robots in the same section within the same time period according to the section number and the corresponding time period, and obtain the collision information according to the movement directions of the multiple robots.

[0008] According to some embodiments of the present invention, the path planning module is used to obtain the path planning information of a multi-task level multi-robot based on a genetic algorithm and fitness value calculation, including: randomly generating a number of feasible solutions according to the feasible path set, taking the several feasible solutions as individuals to obtain an initial population; calculating the fitness value of the individual, and performing a selection operation to select individuals with lower fitness values ​​to enter the next generation population, and discarding individuals with higher fitness values; performing a crossover operation and a mutation operation to generate new individuals to obtain the next generation population; performing a selection operation on the next generation population according to the fitness value; determining whether to terminate according to a preset termination condition; and outputting the final multi-task level multi-robot path planning information.

[0009] According to some embodiments of the present invention, the path planning module is used to obtain the segment number sequence of the currently selected feasible path when calculating the fitness value, and obtain the number of turns based on the length of the sequence; and obtain the angle between each adjacent edge in the map information to calculate the turning angle corresponding to each segment in the feasible path.

[0010] According to some embodiments of the present invention, when the path planning module is used to calculate the fitness value, calculating the avoidance time required by the robot when a collision may occur includes: obtaining the collision information of the feasible path currently selected by the robot, obtaining the robot identifier, feasible path identifier and section number where a collision may occur; obtaining the corresponding width according to the section number, and obtaining the size information of the robot according to the robot identifier, and determining the probability of a collision according to the width and size information; when the probability of a collision is greater than a threshold, a robot with a lower task priority is selected to perform a path planning adjustment operation according to the task priority, and the waiting time of the robot on the previous section of the collision section is increased, so that the robot with a higher task priority leaves the collision section earlier than the robot with a lower task priority enters the collision section; the waiting time is used as the avoidance time required by the robot; when the probability of a collision is less than or equal to the threshold, a preset value is taken as the avoidance time required by the robot.

[0011] According to some embodiments of the present invention, the calculation of the avoidance time required by the robot when a collision may occur also includes: after executing the path planning adjustment operation, updating the section number and the corresponding time period, determining multiple robots in the same section in the same time period based on the updated section number and the corresponding time period, and obtaining updated collision information based on the movement directions of the multiple robots.

[0012] According to some embodiments of the present invention, the system also includes: a speed setting module, used to preset multiple movement speeds for the robot; a second path planning module, used to determine a feasible path and speed set for each robot based on the constraints and the movement speed; the feasible path and speed set includes a number of points, each point represents a feasible path sequence and a movement speed; a second fitness value is determined based on task priority, path length, movement speed, number of turns, turning angle and avoidance time, an initial population is obtained based on the feasible path and speed set, and path planning information for multiple robots at multiple task levels is obtained based on the second fitness value and a genetic algorithm.

[0013] According to some embodiments of the present invention, the second adaptation value T * for:

[0014] Where N is the total number of robots that have been assigned tasks, The weight corresponding to the current task priority assigned to the i-th robot, s i is the length of the path currently selected by the ith robot, v i is the movement speed of the ith robot, c is the coefficient, s t,i is the number of turns of the path currently selected by the i-th robot, θ i is the average turning angle of the ith robot, sL,i is the width of the road section where the ith robot may collide, t a (s L,i ) is the avoidance time required by the robot when a collision is likely to occur.

[0015] The artificial intelligence-based multi-level task multi-robot collaborative motion planning system of the embodiment of the present invention includes at least the following beneficial effects: the embodiment of the present invention determines the length and width information of each road section by obtaining map information of the working area, and determines the turning angle required for the robot to pass through adjacent road sections, and evaluates the feasible path through the above actual information of the road sections; by assigning tasks to idle robots, determining the execution time threshold of the task level, and obtaining the constraint conditions when the robot selects the path when performing the task, ensuring that the selected path allows the robot to complete the task within the specified time; screening feasible paths through constraint conditions, and selecting the optimal path combination based on the genetic algorithm, which can enable multiple robots to complete the task within the specified time, and minimize the time loss caused by path turning, and minimize the robot's avoidance time, so that multiple robots can avoid interfering with each other when performing tasks at the same time.

[0016] Additional aspects and advantages of the present invention will be given in part in the following description and in part will be obvious from the following description, or will be learned through practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] The above and / or additional aspects and advantages of the present invention will become apparent and easily understood from the description of the embodiments in conjunction with the following drawings, in which:

[0018] Figure 1 The module schematic block diagram of the system according to the embodiment of the present invention.

[0019] Reference numerals:

[0020] Working area module 100 , working robot module 200 , idle robot module 300 , task allocation module 400 , constraint condition module 500 , feasible path module 600 , collision information module 700 , and path planning module 800 . DETAILED DESCRIPTION

[0021] Embodiments of the present invention are described in detail below, examples of which are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and cannot be understood as limiting the present invention.

[0022] In the description of the present invention, "several" means one or more, "more" means two or more, "greater than", "less than", "exceed" etc. are understood as not including the number itself, and "above", "below", "within" etc. are understood as including the number itself. If there is a description of "first" or "second", it is only used for the purpose of distinguishing the technical features, and cannot be understood as indicating or implying the relative importance or implicitly indicating the number of the indicated technical features or implicitly indicating the order of the indicated technical features.

[0023] The embodiments of the present invention are applicable to the motion planning of multiple robots that collaboratively perform different tasks in the same work area. When two or more robots move in a work area to perform tasks, the robots may interfere with each other during the execution of the tasks. In addition, the tasks assigned to different robots may have different priorities, so it is necessary to ensure that time-limited tasks can be completed first according to the urgency. Moreover, in an actual working environment, different paths may have different widths, so the number of robots that can be allowed to pass at the same time is also different. Therefore, when assigning tasks to multiple robots in the same work area and performing path planning, the above influencing factors need to be fully considered.

[0024] Reference Figure 1 The embodiment of the present invention provides a multi-task multi-robot collaborative motion planning system based on artificial intelligence, comprising:

[0025] The working area module 100 is used to obtain map information of the working area where multiple robots work together, including a topological map composed of several nodes and edges and the width of the edges; encode the nodes and edges to obtain node numbers and section numbers.

[0026] The working robot module 200 is used to obtain the position information and path planning information of the robot currently performing the task.

[0027] The idle robot module 300 is used to obtain the position information of the currently idle robot.

[0028] The task allocation module 400 is used to obtain a number of tasks to be allocated, and allocate the tasks to be allocated to the nearest idle robots according to the starting point and end point information of the tasks to be allocated and the position information of the idle robots.

[0029] The constraint condition module 500 is used to set the constraint condition according to the task priority; the constraint condition includes the task execution time threshold.

[0030] The feasible path module 600 is used to generate a path set based on map information and the starting and ending point information of the task, and obtain a feasible path set based on the robot movement speed, path length and task execution time threshold; the feasible path set includes several feasible paths, and the feasible path is a sequence of several road section numbers.

[0031] The collision information module 700 is used to obtain the position information and path planning information of the robot currently performing the task, and obtain the collision information according to the path in the feasible path set and the robot movement speed; the collision information includes the robot identification, the feasible path identification and the section number.

[0032] The path planning module 800 is used to randomly generate several feasible solutions for the robots assigned tasks according to the feasible path set, and use the several feasible solutions as individuals to obtain the initial population of the genetic algorithm; based on the genetic algorithm and fitness value calculation, the path planning information of the multi-robots at the multi-task level is obtained; wherein the fitness value T is: Where N is the total number of robots that have been assigned tasks, s i is the length of the path currently selected by the ith robot, v i is the movement speed of the ith robot, c is the coefficient, s t,i is the number of turns of the path currently selected by the i-th robot, θ i is the average turning angle of the ith robot, s L,i is the width of the road section where the ith robot may collide, t a (s L,i ) is the avoidance time required by the robot when a collision is likely to occur.

[0033] In some embodiments, the work area module includes: a map information acquisition unit, which is used to move the robot in the work area for one circle, and scan the surrounding environment based on the laser radar of the mobile robot to obtain laser radar scanning data, and generate map information based on the laser radar scanning data; the map information includes a topological map and the width of each edge in the topological map.

[0034] In some embodiments, the work area module includes: a map information updating unit, which is used to update the map information by acquiring the real-time lidar scanning data of the working robot and the real-time position information of the robot.

[0035] In some embodiments, the collision information module is used to obtain the position information and path planning information of the robot currently performing the task, calculate the section number and the corresponding time period that the robot will pass through in the future; and obtain the feasible paths in the feasible path set in turn, and calculate the section number and the corresponding time period that the robot plans to pass through for each feasible path according to the robot's movement speed; obtain multiple robots in the same section within the same time period according to the section number and the corresponding time period, and obtain collision information according to the movement directions of the multiple robots.

[0036] In some embodiments, a path planning module is used to obtain path planning information for a multi-task level multi-robot based on a genetic algorithm and fitness value calculation, including: randomly generating a number of feasible solutions according to a set of feasible paths, taking the several feasible solutions as individuals to obtain an initial population; calculating the fitness value of the individuals, and performing a selection operation to select individuals with lower fitness values ​​to enter the next generation population, and discarding individuals with higher fitness values; performing crossover operations and mutation operations to generate new individuals to obtain the next generation population; performing a selection operation on the next generation population according to the fitness value; determining whether to terminate according to a preset termination condition; and outputting the final multi-task level multi-robot path planning information.

[0037] In some embodiments, the path planning module is used to obtain the segment number sequence of the currently selected feasible path when calculating the fitness value, and obtain the number of turns based on the sequence length; and obtain the angle between each adjacent edge in the map information to calculate the turning angle corresponding to each segment in the feasible path.

[0038] In some embodiments, when the path planning module is used to calculate the fitness value, calculating the avoidance time required by the robot when a collision may occur includes: obtaining the collision information of the feasible path currently selected by the robot, obtaining the robot identifier, feasible path identifier and section number where a collision may occur; obtaining the corresponding width according to the section number, and obtaining the size information of the robot according to the robot identifier, and determining the probability of a collision according to the width and size information; when the probability of a collision is greater than a threshold, a robot with a lower task priority is selected for path planning adjustment operations according to the task priority, and the waiting time of the robot on the previous section of the collision section is increased, so that the robot with a higher task priority leaves the collision section earlier than the robot with a lower task priority enters the collision section; the waiting time is used as the avoidance time required by the robot; when the probability of a collision is less than or equal to the threshold, a preset value is taken as the avoidance time required by the robot.

[0039] In some embodiments, calculating the avoidance time required by the robot when a collision may occur also includes: after executing the path planning adjustment operation, updating the section number and the corresponding time period, determining multiple robots in the same section in the same time period based on the updated section number and the corresponding time period, and obtaining updated collision information based on the movement directions of the multiple robots.

[0040] In some embodiments, the system of the embodiment of the present invention also includes: a speed setting module, used to preset multiple movement speeds for the robot; a second path planning module, used to determine the feasible path and speed set of each robot according to the constraints and movement speed; the feasible path and speed set includes a number of points, each point represents a feasible path sequence and movement speed; the second fitness value is determined according to the task priority, path length, movement speed, number of turns, turning angle and avoidance time, the initial population is obtained based on the feasible path and speed set, and the path planning information of multi-robots at multiple task levels is obtained based on the second fitness value and the genetic algorithm.

[0041] In some embodiments, the second adaptation value T * for:

[0042]

[0043] Where N is the total number of robots that have been assigned tasks, The weight corresponding to the current task priority assigned to the i-th robot, s i is the length of the path currently selected by the ith robot, v i is the movement speed of the ith robot, c is the coefficient, s t,i is the number of turns of the path currently selected by the i-th robot, θ i is the average turning angle of the ith robot, s L,i is the width of the road section where the ith robot may collide, t a (s L,i ) is the avoidance time required by the robot when a collision is likely to occur.

[0044] Although specific embodiments are described herein, those of ordinary skill in the art will recognize that many other modifications or alternative embodiments are also within the scope of the present disclosure. For example, any of the functions and / or processing capabilities described in conjunction with a particular device or component may be performed by any other device or component. In addition, although various exemplary implementations and architectures have been described according to embodiments of the present disclosure, those of ordinary skill in the art will recognize that many other modifications to the exemplary implementations and architectures described herein are also within the scope of the present disclosure.

[0045] It will be appreciated by those skilled in the art that all or some of the steps and systems in the disclosed method above may be implemented as software, firmware, hardware and appropriate combinations thereof. Some physical components or all physical components may be implemented as software executed by a processor, such as a central processing unit, a digital signal processor or a microprocessor, or may be implemented as hardware, or may be implemented as an integrated circuit, such as an application specific integrated circuit. Such software may be distributed on a computer-readable medium, which may include a computer storage medium (or a non-transitory medium) and a communication medium (or a temporary medium). As known to those skilled in the art, the term computer storage medium includes volatile and non-volatile, removable and non-removable media implemented in any method or technology for storing information (such as computer-readable instructions, data structures, program modules or other data). Computer storage media include, but are not limited to, RAM, ROM, EEPROM, flash memory or other memory technologies, CD-ROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tapes, disk storage or other magnetic storage devices, or any other medium that may be used to store desired information and may be accessed by a computer. Furthermore, it is well known to those skilled in the art that communication media typically embodies computer readable instructions, data structures, program modules, or other data in a modulated data signal such as a carrier wave or other transport mechanism, and may include any information delivery media.

[0046] The above is a specific description of the preferred implementation of the present application, but the present application is not limited to the above-mentioned implementation mode. Technical personnel familiar with the field can also make various equivalent deformations or substitutions without violating the spirit of the present application. These equivalent deformations or substitutions are all included in the scope defined by the claims of the present application.

Claims

1. A multi-task multi-robot collaborative motion planning system based on artificial intelligence, characterized in that: include: A working area module is used to obtain map information of the working area where multiple robots work together, including a topological map composed of several nodes and edges and the width of the edges; Encode the nodes and edges to obtain node numbers and road segment numbers; The working robot module is used to obtain the position information and path planning information of the robot currently performing the task; Idle robot module, used to obtain the location information of the currently idle robot; A task allocation module is used to obtain a number of tasks to be allocated, and allocate the tasks to be allocated to the nearest idle robot according to the starting point and end point information of the tasks to be allocated and the position information of the idle robots; A constraint condition module, used to set constraint conditions according to task priority; the constraint conditions include a task execution time threshold; A feasible path module is used to generate a path set according to the map information and the starting and ending point information of the task, and obtain a feasible path set according to the robot movement speed, path length and task execution time threshold; the feasible path set includes a plurality of feasible paths, and the feasible path is a sequence of a plurality of road section numbers; A collision information module, used to obtain the position information and path planning information of the robot currently performing the task, and obtain collision information according to the paths in the feasible path set and the robot movement speed; the collision information includes the robot identification, the feasible path identification and the section number; The path planning module is used to randomly generate several feasible solutions for the robots assigned tasks according to the feasible path set, and use the several feasible solutions as individuals to obtain the initial population of the genetic algorithm; based on the genetic algorithm and fitness value calculation, the path planning information of the multi-robots at the multi-task level is obtained; wherein the fitness value T is: Where N is the total number of robots that have been assigned tasks, s i is the length of the path currently selected by the ith robot, v i is the movement speed of the ith robot, c is the coefficient, s t,i is the number of turns of the path currently selected by the i-th robot, θ i is the average turning angle of the ith robot, s L,i is the width of the road section where the ith robot may collide, t a (s L,i ) is the avoidance time required by the robot when a collision is likely to occur.

2. The multi-task multi-robot collaborative motion planning system based on artificial intelligence according to claim 1 is characterized in that: The working area module includes: The map information acquisition unit is used to move the robot in the working area for one circle, scan the surrounding environment based on the laser radar of the mobile robot, obtain the laser radar scanning data, and generate map information based on the laser radar scanning data.

3. The multi-task multi-robot collaborative motion planning system based on artificial intelligence according to claim 1 is characterized in that: The working area module includes: The map information updating unit is used to update the map information by acquiring the real-time laser radar scanning data of the working robot and the real-time position information of the robot.

4. The multi-task multi-robot collaborative motion planning system based on artificial intelligence according to claim 1 is characterized in that: The collision information module is used to obtain the position information and path planning information of the robot currently performing the task, calculate the section number and the corresponding time period that the robot will pass in the future; and sequentially obtain the feasible paths in the feasible path set, and calculate the section number and the corresponding time period that the robot plans to pass for each feasible path according to the robot's movement speed; obtain multiple robots in the same section within the same time period according to the section number and the corresponding time period, and obtain the collision information according to the movement directions of the multiple robots.

5. The multi-task multi-robot collaborative motion planning system based on artificial intelligence according to claim 1 is characterized in that: The path planning module is used to obtain path planning information of multiple robots at multiple task levels based on genetic algorithm and fitness value calculation, including: According to the feasible path set, several feasible solutions are randomly generated, and the several feasible solutions are used as individuals to obtain an initial population; the fitness values ​​of the individuals are calculated, and a selection operation is performed to select individuals with lower fitness values ​​to enter the next generation population, and individuals with higher fitness values ​​are discarded; crossover operations and mutation operations are performed to generate new individuals to obtain the next generation population; a selection operation is performed on the next generation population according to the fitness value; whether to terminate is determined according to a preset termination condition; and the final multi-task level multi-robot path planning information is output.

6. The multi-task multi-robot collaborative motion planning system based on artificial intelligence according to claim 1 is characterized in that: The path planning module is used to obtain the segment number sequence of the currently selected feasible path when calculating the fitness value, and obtain the number of turns according to the sequence length; And obtain the angle between each adjacent edge in the map information to calculate the turning angle corresponding to each section in the feasible path.

7. The multi-task multi-robot collaborative motion planning system based on artificial intelligence according to claim 4 is characterized in that: When the path planning module is used to calculate the fitness value, the calculation of the avoidance time required by the robot when a collision may occur includes: Obtain the collision information of the feasible path currently selected by the robot, and obtain the robot ID, feasible path ID and section number where a collision may occur; Acquire the corresponding width according to the road section number, acquire the size information of the robot according to the robot identifier, and determine the probability of collision according to the width and size information; When the probability of collision is greater than a threshold, a robot with a lower task priority is selected to perform path planning adjustment operations according to the task priority, and the waiting time of the robot on the previous section of the collision section is increased, so that the robot with a higher task priority leaves the collision section earlier than the robot with a lower task priority enters the collision section; the waiting time is used as the avoidance time required by the robot; When the collision probability is less than or equal to the threshold, the preset value is taken as the avoidance time required by the robot.

8. The multi-task multi-robot collaborative motion planning system based on artificial intelligence according to claim 7 is characterized in that: The calculation of the avoidance time required by the robot when a collision may occur also includes: After executing the path planning adjustment operation, the section number and the corresponding time period are updated, multiple robots in the same section within the same time period are determined according to the updated section number and the corresponding time period, and updated collision information is obtained according to the movement directions of the multiple robots.

9. The multi-task multi-robot collaborative motion planning system based on artificial intelligence according to claim 1 is characterized in that: The system further comprises: Speed ​​setting module, used to preset multiple movement speeds for the robot; The second path planning module is used to determine the feasible path and speed set of each robot according to the constraint conditions and the movement speed; the feasible path and speed set includes a number of points, each point represents a feasible path sequence and a movement speed; the second fitness value is determined according to the task priority, path length, movement speed, number of turns, turning angle and avoidance time, an initial population is obtained based on the feasible path and speed set, and path planning information of multi-robots at a multi-task level is obtained based on the second fitness value and a genetic algorithm.

10. The artificial intelligence-based multi-task multi-robot collaborative motion planning system according to claim 9, characterized in that: The second fitness value T * for: Where N is the total number of robots that have been assigned tasks, The weight corresponding to the current task priority assigned to the i-th robot, s i is the length of the path currently selected by the ith robot, v i is the movement speed of the ith robot, c is the coefficient, s t,i is the number of turns of the path currently selected by the i-th robot, θ i is the average turning angle of the ith robot, s L,i is the width of the road section where the ith robot may collide, t a (s L,i ) is the avoidance time required by the robot when a collision is likely to occur.

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