Distributed Multi-robot Work Load Partition In Manufacturing Automation

Transcription

1 th IEEE Conference on Autoation Science and Engineering Key Bridge Marriott, Washington DC, USA August 3-, Distributed Multi-robot Work Load Partition In Manufacturing Autoation Gira S Tewolde, Changhua Wu, Yu Wang, Weihua Sheng Abstract This paper presents strategies for systeatic ways to deploy ultiple obile robots for sericing large nubers of points of interest in a distributed fashion. The obile robots in such systes are responsible for proiding seeral essential serices. For exaple, in anufacturing autoation, the tasks could include parts inspection, parts changing and data collection, etc. The efficiency, responsieness, and serice life of the obile robots depend on the balanced allocation of the entire work load to the indiidual robots. Starting fro an initial rando deployent of the robots, the proposed distributed load balancing algorith coputes the load share of each robot using irtual potential force ethod. The load ibalance is used as the irtual force to dynaically oe the robots until a ore balanced distribution is achieed. The total traelling cost to isit all the points of interest in a partition and the cost of sericing indiidual points are cobined to copute the load in each partition. To erify the effectieness of the proposed algoriths, we conducted siulations by applying the in inspection applications and obtained satisfactory results. A. Motiation I. INTRODUCTION In arious anufacturing and industrial scenarios, obile robots are used to serice a large nuber of points of interest [], []. Mobile robots can be used to inspect and aintain large nuber of equipents whose locations coer a wide area. Each robot traels back and forth between the assigned equipents periodically to ensure the proper functioning of the equipents. In the aboe entioned scenarios, the cost of perforing the assigned task has great effect on the outcoe. Miniizing the cost will result in ore efficient utilization of often liited resources and quicker response to urgent eents. If the cost to serice each point of interest is constant, then the total cost is dependent on the total traelling distance of the obile robot, which is essentially a traelling sales person proble [1]. The obile robot can trael along the optial path or approxiately optial path. Howeer, when there are ultiple robots deployed at the sae tie, achieing optial work load partition for each robot becoes a challenge. In our preious work [], we presented a distributed partition ethod for balancing the work load of ultiple Gira S Tewolde is with the Electrical & Coputer Eng at Kettering Uniersity, Flint, MI, USA gtewolde@kettering.edu Changhua Wu is with the Departent of Coputer Science at Kettering Uniersity, Flint, MI, USA cwu@kettering.edu Yu Wang is with Departent of Coputer Science, Uniersity of North Carolina at Charlotte, USA. Eail: yu.wang@uncc.edu Weihua Sheng is with the School of Electrical and Coputer Engineering at Oklahoa State Uniersity, Stillwater, OK, 77, USA weihua.sheng@okstate.edu R1 R (a) Fig. 1. The final partition of the work load depends largely on the initial position of the robots. Partition in (a) is obtained when the initial positions of the two robots are at the botto and the top respectiely. The partition in (b) is obtained when the two robots are initially positioned close to each cluster. obile robots. In that work, the initial positions of robots are randoly deterined and the work load odel only considers the cost at each point of interest. Therefore, the partition resulting fro the proposed ethod depends uch on the initial positions of the robots. The rando initialization of the robot positions works well when the distribution of the points of interest are rando. When the distribution of points of interest confors to soe ixture of ultiple clusters, then it would be better to put the robots at the center or close to the clusters as deonstrated by Figure 1. The preiously proposed ethod works well for workload partitions in applications where the points of interest are closely distributed in the working enironent and in cases where the cost of sericing the work requests is by far the doinant cost coponent copared to the traelling cost of the robots. Howeer, in widely distributed enironents, such as in large anufacturing plants, the traelling cost of the robots becoes a significant coponent of the total cost hence requiring special attention in the work load odel. If the traelling cost is not taken into account, then the partitions ay get to a state where robots hae to trael long distances in order to serice a few points of interest, as shown in Figure. In this paper, we propose a ore realistic workload odel which considers both the traelling cost and the cost of sering each point of interest. We also study the effects of the initial positions of the robots on the final load partition. B. Proble Stateent We will consider the case where obile robots are used for inspecting points of interest in a large enironent. Let X = {x 1, x,..., x k } be a set of points of interest and C = {c 1, c, c 3,..., c k } be the on-spot cost that a robot has to spend for inspecting each point of interest. We assue that the robots do not hae to go to the exact location of the points R1 (b) R //$. IEEE.

2 R1 Fig.. If only the cost for sericing each point of interest is considered, then it ay lead to a situation in which a robot has to trael long distances in order to serice soe points of interest, which deonstrates the need to take the trael distance into consideration during the load partition. of interest for inspection. Range H = {h 1, h, h 3,..., h k } defines the axiu distance a robot can be fro the points of interest in order to perfor its job. We can safely assue that the traelling cost trael of the robots is linear to the trael distance. Therefore, the work load (cost) of a robot is the suation of its traelling cost trael and the su of all on-spot costs serice at points of interest it isits. The robots hae to periodically inspect the points of interest one by one. After finishing one round, the robots go back to their original positions. We want to find a work load partition such that each robot is only responsible for the points of interest in its partition and the work loads (cost) of all robots are approxiately balanced or equal. Intuitiely, such load balance strategy tends to extend the life tie of the distributed syste by iniizing the oer-utilization of soe of the robots and under-utilization of others. This is also expected to iproe the response tie of the robots to serice requests. In this paper, we propose a load partition algorith which cobines two load etrics. One is the traelling distance of the robot, and the other is the cost at sericing each point of interest. This paper is organized as follows. Section II presents the algorith to calculate the optiized traelling cost for a obile robot to serice a partition. Section III presents a distributed approach for partitioning the work load aong ultiple obile robots. Section IV presents our siulation experients. Conclusion and future work are discussed in Section V. II. CALCULATION OF THE OPTIMIZED TRAVELLING COST In this section, we will briefly describe the algorith for coputing the traelling cost of sericing a set of points of interest using a obile robot inside each partition. Here we only consider the set of points of interest in a partition, X = {x 1, x,, x l }. As we hae discussed in the proble stateent, if we draw a circle around each point of interest x i using its range h i, then a robot has to get into the circle of a point of interest in order to serice it. If the disks of ultiple points of interest oerlap, then the robot can siply go to the oerlapping area and serice all points of interest inoled. We denote the areas fored by the circles and their intersections by A = {a 1, a,, a l }. If an area a i intersecting the circle defined by point of interest x j, we R call x j can be coered by a i. There are two optiization probles in coputing the traelling cost. The first optiization proble is how to select the iniu nuber of areas fro A for the robot to isit to guarantee the coerage of all points of interest within a partition. This proble is actually the iniu set coer proble, which is a NP-hard proble []. We adopted a greedy heuristic for this proble. The greedy algorith is described in Algorith 1. Algorith 1 Greedy algorith to select the iniu nuber of areas for coering the points of interest within a partition Input: A set of areas A = {a 1, a,, a l } and a set of points of interest X = {x 1, x,, x } within the partition. Output: A subset of areas A S = {a s1, a s,, a sn } for a robot to isit. 1: Initially, set all points of interest uncoered and the uncoered counter k =. Let the potential coerage pc i of each area a i equal to the nuber of circles intersecting with this area. : while k! = do 3: Select area a j with the largest potential coerage pc j (using IDs to break a tie) and add it into the selected subset A S ; : Mark all points of interest coered by a j coered; : k = k pc j ; : Update the pc k for all adjacent areas a k, by setting it to the nuber of intersected circles fro uncoered points of interest. 7: end while After selecting the area for a robot to isit, we need to decide their exact positions where a robot shall isit. Since the positions can affect the total length of the path that the robot needs to isit, we hae to consider the position proble jointly with the path schedule proble. In our approach, we try to find a position in the selected area so that the total length of the path traelled by the robot is iniu. Our algorith is an iteratie algorith in which in each step we add a new turn point inside one of the unisited areas such that the distance added to the robot path is iniu. Reeber that we hae s n areas to isit (output fro Algorith 1) and initially all areas are unisited, our algorith will terinate after s n rounds, since in each round it adds a new turn point in the path and coers an unisited area. Algorith shows the detailed algorith. Algorith is actually an approxiation solution to traelling salesperson with neighborhoods (TSPN) proble [3], which is NP-hard. Notice that in Step 3 of Algorith we need to draw an ellipse which is tangent to a j. This can be done by two ways. We can start with a sall ellipse and increase its size until it reaches a j. Howeer, how to decide the initial size of the ellipse and what size to increase at each step are difficult. The second way is to use binary search. We first randoly select a point b inside a j. We

3 Algorith Path schedule for a robot within a partition: to select the turn points of the robot Input: A set of areas A S = {a s1, a s,, a sn }. Output: A path Π D = 1 n which the robot shall isit. 1: Initially, set all selected areas a si unisited and the unisited counter k = s n. Let the path Π D =. : while k! = do 3: For each edge on i i+1 in path Π D and eery unisited area a j, we draw an ellipse which uses i and i+1 as its foci and is tangent to a j. Let j be the tangent point. See Figure 3(a) for illustration. If select a j to isit between i and i+1, the distance added to the path Π D will be i j + j i+1 i i+1. : It is obious that we want to select the unisited area which adds the least distance to path Π D. For exaple, in Figure 3(b), a p, hence p, is a better choice than a j. Assue we select a p which is the best for all edges in Π D and all unisited areas, we ark a p isited, and insert p between i and i+1 in Π D. Thus the nuber of edges in the path increases by one. k = k 1. : end while (a) Fig. 3. (a) For each edge on i i+1 in path Π D and eery unisited area a j, we draw the ellipse which uses i and i+1 as its foci and is tangent to a j. The distance added to the path Π D by isiting a j is i j + j i+1 i i+1. (b) We select the unisited area which adds the least distance to path Π D. In this exaple, a p is a better choice than a j. use i b + i+1 b as the ajor axis to draw the ellipse which guarantees to intersect with a j. Then we reduce the ajor axis by half, if the ellipse does not intersect with a j, we increase the ajor axis, otherwise further reduce it. By recursiely doing this, we can find the ellipse which is tangent to a j efficiently. In practice, if the range r i is sall copared with the distance between all areas, we can just use the ellipse ia the center of a i to estiate the optial ellipse. Figure shows the path Π D generated by Algorith. Path Π D represented by the line in the figure is the path that the robot will follow to serice the points of interest, while the sall squares are the turn points of the robot. In this exaple, ranges of all points of interest are set to be the (b) sae. 1 3 Point of Interest Fig.. Path Π D generated by Algorith. Here, the sall squares are the turn points of the robot. III. DISTRIBUTED MULTI-ROBOT LOAD PARTITION ALGORITHM In our preious work [] a coordinated serice set partition algorith has been deeloped to diide a large working area into regions that each obile robot is responsible for. The robots are in charge of proiding the routine serices to the points of interest in their partition. The goal of the partitioning strategy was to balance the workload in each partition. A siplified workload odel was deised by considering the nuber of points of interest in the partition as the deterining factor for the load. So the ore points of interest there are the higher the load and ice ersa. This odel was easy to ipleent but it does not necessarily capture the actual work load in each partition. A ore realistic workload odel is proposed here to capture the actual aount of energy and tie the obile robots would expend in their partition to carry out their ission. This proposed odel considers not only the nuber of points of interest per partition but also their spatial distribution, as this affects the total distance the robot has to coer in order to proide the serices to each of the points of interest in its partition. Hence the actual workload L s in a partition s is coputed as the total energy or tie trael s the robot spends in traelling to all points of interest in its partition and the energy or tie serice s in sericing the. serice s = (c i x i x i s), where c i is the cost to serice x i in s as defined in the proble stateent. Thus, L s = trael s + serice s Algoriths 1 and in the preious section present strategies for the coerage of distributed points of interest and for path planning to optially coer the all. We eploy these heuristic algoriths for coputing the traelling cost coponent of the workload, trael s. In the workload odel considered in this paper we assue the cost of sericing the point of interest to be constant and the sae for all points, i.e. c i = c, for all i. The goal of the algorith described below is to partition the total area of distributed points of interest into a gien nuber of regions for each robot to take care so as to achiee a ore balanced load distribution, according to the new workload odel. As in our preious serice set partition (SSP) [] algorith, starting with initial partitions we eploy the irtual Sensor Robot

4 R1 R Fig.. Voronoi diagra for area partition and the irtual forces caused by load ibalances between neighboring partitions. force ethod to iteratiely adjust the partitions for each robot until the process leads to a ore balanced load distribution. If a net non-zero irtual force acts on a robot, it will cause it to oe in the direction of the force so as to odify the load distribution for a better balance. After the robots oe in response to the applied forces the working area is repartitioned based on their new positions and the new load distribution is coputed. This process is repeated iteratiely until a balanced load distribution is achieed aong all the robots. Howeer, it is ery hard to achiee a perfect load balance between the partitions due to the discrete nature of the point of interest distribution. Initial attepts to run the algorith for a perfectly balanced load distribution led to unstable or oscillatory perforances. Hence to aoid instability behaior the goal of the algorith is odified to a goal of iniizing the load differences between the different partitions. The perforance etric used to easure the quality of the load partition is coputed as the standard deiation of the load distribution fro the aerage load. The detailed algorith is gien in Algorith 3. With this approach the algorith could be run until no ore iproeents are seen on the standard deiation of the load distribution for a pre-specified nuber of iterations. When the knowledge of the cluster distribution of the points of interest is aailable, at the initial step of the load partition process the points of interest could be classified into separate clusters. For exaple, if we hae q robots, then the points of interest can be classified into q clusters. So we can put the q robots at the centers of the q clusters. Howeer, as shown in Figure 1, the initial position of the robots has influence on the final partition. The initial robot deployents and its associated Voronoi diagra result in initial partitions gien by P 1, P,, P q, where q is the nuber of partitions. Figure illustrates the Voronoi diagra and the irtual forces attributed to the load ibalances. R IV. EXPERIMENTS To ealuate the perforance of the ulti-robot load partitioning algoriths with the ai of approxiately balancing the loads in the different partitions we perfored a series of siulation experients as described in this section. Matlab is used in the siulation due to its coputation and R3 Algorith 3 Load balancing algorith 1: Calculate the serice load L Pi of the subset S Pi, for all i = 1,,, q. : Exchange serice load with adjacent robots R i1, R i,, R iri, where r i is the nuber of neighboring robots of robot R i 3: Calculate the cobined irtual force f i on R i r i f i = α(l sj L si ) n ij. (1) j=1 Here α is a scale factor and n ij is the unit ector fro R i to R j. : If f i f th then change the elocity of R i according to the following equation V i = V i + (f i λv i )/β t () Here f th is a sall threshold alue, the ter λ i introduces soe iscous force so that the deployent can be quickly stabilized, β is a constant for the robot ass, and t is the tie step size per iteration. : Using i update the new location of R i as follows: x Ri = x Ri + V i t (3) : Exchange new location inforation with neighbors and calculate the new Voronoi diagra and update the point of interest subsets in partition P i 7: Check the stopping criteria, using the standard deiation of the load distributions aong the different partitions as a quality etric : If the stopping criteria is et stop, else go to step 1 graphic capabilities. We consider seeral test configurations, with different diensions of the working area and different nuber of points of interest randoly distributed in a large enironent. Table I shows the list of the test configurations considered in the siulation experients and the initial & final load distribution and the standard deiations of the corresponding load partitions. In each test scenario (except test cases and ), the points of interest are generated randoly at the start of the siulation. In all cases the locations of the points of interest are assued to be known. Based on the initial deployent positions of the robots the progra sets out by first diiding the entire working area into different partitions (using Voronoi diagra, see Figure ) and coputing the initial load distribution in each partition. Then using the inforation about the initial load distribution the load balancing algorith is executed to dynaically oe the robots in the area in such a way that the difference in the load distribution aong the different partitions is iniized. Virtual forces attributed to the load ibalances between neighboring partitions are responsible for the oeent of the robots until equilibriu conditions are approxiately achieed. Table I shows the results for the different test runs. As can be obsered fro the results in the table (and isualized fro Figure ), the initial rando deployent of 7

5 TABLE I SIMULATION RESULTS OF THE LOAD BALANCING ALGORITHM Test no. Serice area n ns Initial load distribution Final load distribution Load in each partition st. d. Load in each partition st. d. 1 x x x x (c) x (c) x x x x x standard deiation of load distribution iterations Fig.. Progress of the load-balancing algorith. the robots results in load distributions with large differences in the serice load aong the different partitions. This load ibalance is expressed quantitatiely by the standard deiations of the loads carried by each robot. As the load balancing algorith iteratiely proceeds the load ibalance gradually decreases. The algorith continues running until soe terination condition is satisfied, which could be specified by either a iniu standard deiation or a axiu nuber of iterations without appreciable iproeents. The standard deiation of the final load distribution indicates how closely the loads between the different partitions hae been balanced. The graph in Figure shows the progressie iproeent of the standard deiation of the load distribution with increasing nuber of iterations. To further understand how the distribution of the points of interest ipacts the load balancing algorith, we prepared two experiental configurations that contain points of interest distributed in clusters (experiental setups and ). The working areas for both setups are x. In experiental setup we generated clusters with 1 points of interest randoly placed in each cluster, resulting in a total of points of interest. In experiental setup, we also hae clusters but with 1 points of interest randoly distributed in each cluster, resulting in a total of points of interest in the entire working area. The clusters are well separated spatially fro each other. Figure 7 shows the (a) before load balance (b) after load balance Fig. 7. The effect of the initial position of robots on the load balancing, as applied to points of interest distributed in cluster forations (test ). test configuration in the experiental setup. The initial rando placeent of the obile robots in the working areas results in non-optial load distribution with large alues of standard deiations. Howeer, after the application of the load balancing algorith, the resulting load distributions along the clearly defined clusters was produced with the expected sallest standard deiations. V. CONCLUSIONS This paper presented strategies for work load balancing of obile robots in autoated anufacturing enironents, where a large nuber of points of interest need to be sericed. Starting with initial rando deployent positions of the robots the algoriths were applied iteratiely to redeploy the robots to positions that resulted in ore balanced load distributions. A nuber of test runs hae been perfored

6 (test 1-a) (test -a) (test 1-b) (test -b) (test 9-a) (test 9-b) Fig.. Three test cases (test nubers 1,, & 9, in Table I). The left colun shows the original partition. The right colun shows the partition after load balance. to erify the effectieness of the algoriths. Future work will further inestigate effects of clustering of the points of interest on the load balancing strategy and also closely study the relationship between the goals of iniizing load ibalances and that of iniizing total serice load in the entire network. Since a iniu load ibalance does not necessarily lead to a iniu total serice load this optiization goal requires techniques of cobining the two iniization criteria. REFERENCES [1] D. Applegate, R. Bixby, V. Chaatal, and W. Cook. On the solution of traeling salesan probles. Docueenta Matheatica Extra Colue, (3):, 199. [] T. H. Coren, C. E. Leiserson, R. L. Riest,, and C. Stein. Introduction to Algoriths. MIT Press, Cabridge, Massachusetts, 1. [3] A. Duitrescu and J. S. B. Mitchell. Approxiation algoriths for TSP with neighborhoods in the plane. In Syposiu on Discrete Algoriths, pages 3, 1. [] Lars Kock Jensen, Bent Bruun Kristensen, and Yes Deazeau. Flip: Prototyping ulti-robot systes. Robotics and Autonoous Systes, 3(3-):3 3,. Intelligent anufacturing;agent architecture;multi-robot systes;autonoous transportation syste;. [] W. Sheng and G.S. Tewolde. Robot workload distribution in actie sensor networks. In Proc. of 7th IEEE International Syposiu on Coputational Intelligence in Robotic and Autoation, Jacksonille, Florida, 7. [] C. Son. Intelligent control planning strategies for obile base robotic part acro- and icro-assebly tasks. International Journal of Robotics and Autoation, 1(3):7 17,. Macro and icroassebly;intelligent control planning;fuzzy coordinator;optiality criterion (fuzzy entropy);machine intelligence;vision sensors;. 9