We present a method to allocate multiple tasks with uncertainty to heterogeneous robots using the theory of comparative advantage: an economic theory that maximizes the benefit of specialization. In real applications, robots often must execute various tasks with uncertainty and future multirobot system will have to work effectively with people as a team. As an example, it may be necessary to explore an unknown environment while executing a main task with people, such as carrying, rescue, military, or construction. The proposed task allocation method is expected to reduce the total makespan (total length of task-execution time) compared with conventional methods in robotic exploration missions. We expect that our method is also effective in terms of calculation time compared with the time-extended allocation method (based on the solution of job-shop scheduling problems). We simulated carrying tasks and exploratory tasks, which include uncertainty conditions such as unknown work environments (2 tasks and 2 robots, multiple tasks and 2 robots, 2 robots and multiple tasks, and multiple tasks and multiple robots). In addition, we compared our method with full searching and methods that maximize the sum of efficiency in these simulations by several conditions: first, 2 tasks (carrying and exploring) in the four uncertain conditions (later time, new objects appearing, disobedient robots, and shorter carrying time) and second, many types of tasks to many types of robots in the three uncertain conditions (unknown carrying time, new objects appearing, and some reasonable agents). The proposed method is also effective in three terms: the task-execution time with an increasing number of objects, uncertain increase in the number of tasks during task execution, and uncertainty agents who are disobedient to allocation orders compared to full searching and methods that maximize the sum of efficiency. Additionally, we performed two real-world experiments with uncertainty.
Research on a multirobot task allocation system and coordinating heterogeneous robots have become a hot topic of research in the field robotics. We propose a method of allocating multiple types of tasks to heterogeneous robots, based on the theory of comparative advantage [
A key driving force of multirobot systems is their force of numbers. Multirobot systems are expected to improve the effectiveness of robotic systems in terms of the temporal efficiency [
Conventional task allocation methods are categorized as static or dynamic allocation [
Dispatching methods to reduce the calculation time have been studied. In multirobot task allocation, the methods are instances of optimal assignment problems [
In this study, the allocation based on our method reduces the makespan compared with methods that maximize the sum of efficiency. We perform simulation experiments in environments with uncertainty and find that allocation based on the proposed method reduces the makespan under any uncertainty, compared with methods that maximize the sum of efficiency. In uncertain environments, the makespan with the proposed method was almost the same compared with frequent reallocation of static allocation. We also conduct a heterogeneous robots task-execution experiment and a human-robot collaborative experiment in the real environment. We propose task allocation methods for various tasks execution by heterogeneous multirobots. The proposed method is expected to be effective in real world experiments, because uncertainty can actually appear.
Currently, the area of the human-robot symbiosis is growing rapidly. As stated in the call of this special issue, assembly, warehouses, and home services need human-robot collaboration as soon as possible. Uncertainness of human is not only each individual difference but also effort input in human-robot collaboration. The definition of the theory of comparative advantage had to be extended to adapt to increases in the number of robots and tasks in the real world including humans.
Because of the difficulties of constructing a single robot that has the required capability in real world, in recent literature, there are works on heterogeneous multirobot systems (e.g., in assembly of large-scale structures [
Task allocation to heterogeneous robots.
Under other circumstances, the problems of heterogeneous multirobot task allocation problems have been understood as instances of operations research, economics, scheduling, network flows, and combinatorial optimization [ Uncertainty in task-execution time Uncertainty increasing the number of tasks during task execution Uncertainty agent who is disobedient to allocation orders
Dynamic allocation is a method that repeatedly allocates tasks dynamically during execution. There are many studies on rescheduling in job-shop scheduling problems in factories. In these works, jobs are added during execution [
To increase the efficiency of tasks executed by heterogeneous robots through task allocation, we focus on “comparative advantage.” Comparative advantage is an economic theory referring to the ability of any given economic actor to decrease the total economic cost. According to this theory, an appropriate allocation of tasks among robots is more efficient than each robot executing all tasks. We adapt this theory to the ability of robots and reduce the cost of multirobot decision.
Let
Pareto front.
Example of an optimal allocation on the Pareto front.
For example, when two robots are allocated to two tasks by the Pareto front (Figure
Comparative advantage with multiple tasks and two agents.
The theory of comparative advantage can be extended to allocation among multiple robots or multiple tasks [
Comparative advantage with two tasks and multiple agents.
In cases of multiple robots and multiple tasks, it will be complex to solve the problem. No general method for describing all the Pareto fronts has been established yet. In economics, several methods have been used to apply the theory of comparative advantage to cases of multiple agents and multiple commodities. There are methods of finding the solutions on the Pareto front or methods in which the number of agents or commodities is restricted [
Allocation flowchart.
The optimal assignment problem [
Example in which maximizing sum of efficiency does not minimize makespan.
task 0 | task 1 | |
---|---|---|
Robot 0 | 4 | 10 |
Robot 1 | 1 | 4 |
We can regard this multirobot task allocation problem as an extension of the traveling salesman problem (TSP) [
In the uncertain real environment, our proposed method should be able to solve large-scale problems with a large number of robots, tasks, people, animals, and so on. Our method should also be able to compare with different algorithms and the trajectories of robots. There are many things to clarify if our proposed method is feasible or not. But first and foremost, we think the full searching method is plausible to compare with the proposed method.
In this paper, we compare our method with existing methods and show the effectiveness of our method on situations: 2 task and 2 robots (Section
To verify the effectiveness of the proposed method, we first simulate allocating two types of tasks to two types of robots (Figure Carrying task: carrying objects to a target point (Figure Exploring task: exploration of the unknown areas for mapping and discovering objects. Robots do not know where the objects are initially. Unknown areas, indicated by white, turn into known areas, indicated by black, when robots pass through the areas. This task is finished once the robots have passed through all the areas.
Simulation: two types of tasks by two types of robots. The field consists of
The two types of robots explore unknown areas and carry discovered objects. Table
Moving speeds of each robot.
Not carrying ( | Carrying ( | |
---|---|---|
Agent 0 | 0.33 | 0.10 |
Agent 1 | 1.00 | 0.033 |
In this section, we compare the proposed method with conventional methods, i.e., maximizing the sum of efficiency [ Proposed method: the theory of comparative advantage (Pareto front). Maximizing the sum of efficiency: each robot executes each different task; if Exploring first method: both robots execute the exploring task at first and then execute the carrying task after the exploring task is finished.
Moving velocity.
Not carrying | Carrying | |
---|---|---|
Robot 0 | 0.4 | 0.1 |
Robot 1 | 1.0 | 0.4 |
We simulate three conditions of the number of objects
Results of proposed and conventional methods.
In unknown environments, robots have to execute many tasks which are manifold and uncertain. We propose a task allocation method based on the theory of comparative advantage in the case where there are multiple tasks in unknown environments. This allocation method is expected to minimize the makespan by executing in order of the comparative advantage between robots. There are many studies about situations consisting of multiple tasks (e.g., restaurants [
As noted above, we simulated the single-type carrying task allocated to heterogeneous robots by the Pareto front. We evaluate the Pareto front for allocation tasks of carrying multiple types of objects to heterogeneous robots. We define the efficiency of the carrying task
We compare the Pareto front with an allocation method based on the full searching of carrying orders. The full searching method calculates the task-execution times of all the carrying orders in the case of
We did a simulation to evaluate the proposed method by comparing it with the full searching method. Table
Moving velocity.
Not carry | Carry (0) | Carry (1) | ⋯ | |
---|---|---|---|---|
Robot 0 | 1.0 | 0.10 | 0.11 | ⋯ |
Robot 1 | 1.0 | 0.20 | 0.21 | ⋯ |
The numbers in brackets represent the object IDs.
Simulation: heterogeneous robots and multiple carrying tasks.
We set simulation conditions defining unexpected events that may occur, including the following conditions: Later condition: the time required for a robot to carry is later than predicted. Adding condition: new objects appear. Unreasonable condition: some robots are disobedient to allocation orders. Speed-up condition: the time to reach object positions is shorter than the carrying time.
We hypothesize that the proposed method based on the theory of comparative advantage performs better under these conditions than the full searching methods. In the proposed method, robots execute tasks that are given high priority according to the execution time difference among robots. Thus, the remaining tasks have shorter execution time difference among robots. If unexpected events occur, our method can avoid the situation in which an inappropriate robot executes a task and increases the total task-execution time.
Beta distribution (
We simulated two types of full searching methods: (1) full searching (static), in which the robots carry objects in the order that is decided at first and (2) full searching (dynamic), in which the robots redetermine the order every time a robot finishes carrying an object. We ran simulations to compare the proposed method and the two types of full searching methods for certain patterns of
Carrying velocity of added objects.
Appearing time [s] | 1st | 2nd | 3rd | 4th |
---|---|---|---|---|
100 | 200 | 300 | 400 | |
Robot 0 | 0.11 | 0.19 | 0.13 | 0.17 |
Robot 1 | 0.21 | 0.29 | 0.23 | 0.27 |
Initial objects in Table
Moving velocity (object ID in brackets).
Not carry | Carry (0) | Carry (1) | ⋯ | |
---|---|---|---|---|
| | | ⋯ | |
Robot 0 | 0.6 | 0.010 | 0.011 | ⋯ |
Robot 1 | 1.0 | 0.020 | 0.021 | ⋯ |
Moving velocity of added objects.
Appearing time [s] | 1st | 2nd | 3rd | 4th |
---|---|---|---|---|
1000 | 1500 | 2000 | 2500 | |
Robot 0 | 0.011 | 0.019 | 0.013 | 0.017 |
Robot 1 | 0.021 | 0.029 | 0.023 | 0.027 |
Makespan and calculation time in 100 simulations. There are 7, 8, 9, and 10 objects.
Makespan in 100 simulations.
Makespan in the later condition.
Makespan in the adding condition.
Makespan in the unreasonable condition.
Each cell in Tables
Comparison between proposed method and full searching (with no unreasonable agents).
| | | | |
---|---|---|---|---|
| full searching | - | full searching | - |
| full searching | - | - | - |
| full searching | - | - | - |
| full searching | - | - | - |
| - | - | - | - |
Comparison between proposed method and full searching (with an unreasonable agent).
| | | | |
---|---|---|---|---|
| proposed | - | - | - |
| proposed | proposed | proposed | - |
| proposed | proposed | - | proposed |
| proposed | proposed | proposed | - |
| proposed | proposed | - | - |
(
Makespan in the speed-up condition. (1/10 carrying speed condition).
We think the Pareto front can shorten the makespan of many robots executing two tasks (exploring and carrying). In this theory, we must nominate a benchmark robot to determine how many robots should be allocated to each task (exploring or carrying) and the benchmark robot is the only one that executes both tasks. For the adjustment to many robots, the following steps are added to the allocation method based on the theory of comparative advantage. To decide a benchmark robot, we performed a simulation. Nominate a benchmark robot in some way. Allocate one task to robots that have a comparatively greater degree of advantage than the benchmark robot, according to ( Allocate the other task to the other robots.
The whole-period Pareto front has a limitation for allocating tasks to many types of robots. When one task is finished first, every agent will be executing the other task. This means that there are robots that execute both tasks and the whole-period Pareto front is unadaptable to this situation. The whole-period Pareto front can be adapted to the situation in which each robot executes a single task. Hence, to allocate two tasks to many types of robots, it is unadaptable to shorten the overall makespan. However, we think that the particular makespan (e.g., until one task is finished) will be shortened by the whole-period Pareto front. We propose a new method to shorten the overall makespan through the simulation results.
We performed simulation experiments to allocate a carrying and an exploring task to four types of robots by the theory of comparative advantage (Figure Determine a benchmark robot in number order of robots (decide how many robots to allocate the exploring task). Allocate exploring task to robots that have a comparatively greater degree of advantage than the benchmark robot. Allocate the carrying task to other robots in the order of the degree of comparative advantage. Allocate the exploring task to the remaining robots.
Moving velocity (object ID in brackets).
Not carrying | Carrying | |
---|---|---|
Robot 0 | 0.2 | 0.1 |
Robot 1 | 0.3 | 0.08 |
Robot 2 | 0.4 | 0.06 |
Robot 3 | 0.5 | 0.04 |
Simulation example: allocate the exploring and carrying tasks to four robots.
We simulated five robot conditions (benchmark robots are 1–4; exploring robots are 0–4) versus the object condition (number of objects are 4 or 16) on the instantaneous Pareto front. Figure
Results based on our method. The legend shows the benchmark agent id: minimal numbers of exploring agents.
We propose a method to allocate many types of tasks to many types of robots, based on the theory of comparative advantage proposed by Tian [
Example of allocation based on the proposed method.
We verify the proposed method to allocate carrying tasks to many robots by a simulation, as shown in Figure Robot condition: the robots have different comparative advantages. We set the moving velocities of four robots as the comparative advantage (Table Environment condition: the system knows or does not know about the environment. Unreasonable condition: this is the same as in Section
Moving velocity.
Not carry | Carry (0) | Carry (1) | ⋯ | |
---|---|---|---|---|
| | | ⋯ | |
Robot 0 | 1.0 | 0.10 | 0.11 | ⋯ |
Robot 1 | 1.0 | 0.20 | 0.21 | ⋯ |
Robot 2 | 1.0 | 0.30 | 0.31 | ⋯ |
Robot 3 | 1.0 | 0.40 | 0.41 | ⋯ |
Simulation of carrying objects with four robots.
We compare the proposed method with the full searching method and the method that maximizes the sum of efficiency. The full searching method in this section refers to the “dynamic full searching” method: robots are reallocated to each task every time a robot finishes carrying an object.
Figure
Makespan of some number of robots.
We show the following results of the conditions in an unknown environment.
Makespan in situations where carrying objects can be late (three robots).
Makespan in situations where carrying objects can be late (four robots).
Makespan in situations where carrying objects are added later (three robots).
Makespan in situations where carrying objects are added later (four robots).
Makespan in situations where there is an unreasonable agent (three robots).
Makespan in situations where there is an unreasonable agent (four robots).
The task-execution time for the proposed method is slightly longer than that of the full searching method from Figures
These results are similar to those in Figure
Additionally, beyond the simulations, we conducted an allocation experiment in a real environment. Figure Box: both robots can easily carry this object by pushing Roomba. Ball on the wall: it is difficult for Roomba to push this object. Roomba cannot move the object to the target area directly, unless Roomba pushes it off the wall. Roomba needs to go around it and move it from behind. NAO can scrape objects from the wall using its hand, which costs approximately 5 s. Table: it has four legs. Roomba is not wide enough to span the legs of the long side. Therefore, Roomba must alternately push the left and right legs in order to move the object. NAO has a comparative advantage to carry the table.
Moving ability of each robot.
Proceeding | Rotating | |
---|---|---|
Roomba | 100 mm/s | 100°/s |
NAO | 60 mm/s | 60°/s |
Experiment of real robots based on the proposed method. (1) Initial state. The robot decided the carrying object based on the list of carrying time. (2) Trying to move each object. (3) Roomba fails to carry the box and try to do again. (4) NAO finishes carrying the stand and begins to carry the ball. (5) Roomba finishes carrying the box. The method then determines which robot carries the ball, and NAO continues carrying the ball. (6) Roomba finished tidying, and NAO continues carrying the ball.
We calculated the carrying time using the moving path and moving speeds (Table
Experiment of the robot with a human.
In this research, to allocate tasks to robots, we used the economic theory of comparative advantage as a method to reduce the total makespan. We compared the proposed method with conventional methods used in robotic exploration missions. These results have been enabled by the recent developments in computing power and the ability of robots. In the future, high computing power will reduce the total makespan, particularly the calculating time, and standardized robots that are able to execute multitasks will be developed. However, our method contributes to the efficiency of robots’ labor output. Monofunctional robots can practically execute tasks other than their main task (e.g., cleaning task for the Roomba). Real robots can execute tasks that they should not or that their developers did not consider, but only at a low level (e.g., carrying task for the Roomba). In real environments, we cannot perceive everything that occurs. In such cases, our method can allocate the tasks to robots that include these low-level performances on a real-time basis according to the comparative advantage.
We conducted simulation in a field that is free (without obstacles) and small (
There have been sparse improvements to the methods of allocating many tasks to many robots. We have a forward-looking approach in which the allocation of next tasks is preliminarily based on the theory of trade with the comparative advantage [
This study investigated the allocation methods using simulations. We must comparatively verify these results in a real environment with other conventional methods such as market-based techniques inclusive of TraderBots [
We investigated a method of using the theory of comparative advantage to allocate tasks to robots with uncertainty including humans. The proposed method is a dynamic sharing algorithm to allocate uncertainty tasks in an unknown environment, assuming timely reallocation. First, we confirmed that the proposed method reduces the total makespan (the total task-execution time) compared with conventional methods used in robotic exploration missions. We expect that our method is also effective in terms of calculation time when compared with the time-extended allocation method. We simulated carrying tasks and exploring tasks, which include uncertainty conditions of the work in an unknown environment. The proposed method is also more effective in dealing with uncertainty in task-execution time, uncertainty in the increasing number of tasks during task-execution, and uncertainty agents who are disobedient to allocation orders, compared to existing methods (the sum of efficiency and full searching methods). Finally, through experiments in a real environment, we confirmed that the proposed method can reduce the makespan.
This paper makes several contributions to human-robot interaction. First, the effectiveness of a new economic theory was shown for heterogeneous robots. In robot-robot collaboration, it is important to execute tasks even if the robot has inferior ability to accomplish a task. Second, the theory was effective in the uncertain environments including a human. Human-robot collaboration is receiving a lot of attention. The reallocation corresponding to uncertainness of people is a critical issue.
The full text is not published; it was accepted and presented in ROBOMECH2015 conference [
The authors declare that there are no conflicts of interest regarding the publication of this article.
The authors would like to thank Editage (