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Technical Note GSGM movement model for cooperative robots system
\ PERGAMON
Mechatronics 7 "0887# 894Ð814
Technical Note GSGM movement model for cooperative robots system M[ Parnichkuna\ \ S[ Ozonob a
School of Advanced Technolo`ies\ Asian Institute of Technolo`y\ P[O[ Box 3 Klon`luan`\ Pathumthani 01019\ Thailand b Department of Precision Machinery En`ineerin`\ The University of Tokyo\ 6!2!0 Hon`o\ Bunkyo!ku\ Tokyo 002\ Japan Received 02 April 0886^ revised 16 February 0887^ accepted 0 June 0887
Abstract Robots cooperation means {work!accomplishment action with collaboration of multiple robots by applying shared information\ transmitted via communication network in a system|[ The de_nition implies that the e.ciency of the cooperative robots system depends directly upon two main factors] one is communication among the robots^ the other is movement of the robots[ Here the authors consider the latter e}ect on the robots system[ Robots with appropriate cooperation are expected to work e.ciently[ Cooperation action of multiple robots is viewed here as the consequence of a collection of basic group!shape generation[ Some of the basic group shapes are straight\ curve\ circle\ multi!angle\ and so on[ Thus\ the model\ which can be applied to control multiple robots to generate their group shapes generally with the least exceptions\ is necessary[ Group Shape Generating Model "GSGM# is proposed here as a general movement model for cooperative robots system[ It looks like the result of a combination of potential _eld\ rule based\ and communication based models[ It applies information\ trans! mitted through the system\ to determine accelerations of all robots[ Evaluation of the proposed model is done by applying it to many di}erent tasks to estimate its e.ciency and generality[ Þ 0887 Elsevier Science Ltd[ All rights reserved[
0[ Introduction Current research e}orts focus on creating a {smart| robot that can {see|\ {hear|\ {touch|\ and {make decisions|[ An {autonomous robot| recognizes its environment Corresponding author[ Tel[] ¦55 1 413 4118^ Fax] ¦55 1413 4586^ E!mail] manukidÝait[ac[th 9846Ð3047:87:,*see front matter Þ 0887 Elsevier Science Ltd[ All rights reserved[ PII] S 9 8 4 6 Ð 3 0 4 7 " 8 7 # 9 9 9 2 4 Ð X
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using onboard sensors and decides what to do next based on the sensor data by its own decision[ A {self!contained robot| contains everything which is needed to allow the robot to operate independently and autonomously\ such as onboard sensors\ locomotive units\ information processing units\ and power supplies\ on its body[ It is necessary that an {intelligent robot| has to behave following the two properties above[ Furthermore\ a robot can be said to be {more intelligent| by being able to cooperate with other robots\ that is\ the robot decides its action by considering not only its own objective but also the intentions of other robots or the community to which the robot belongs[ At the present time\ the trend of robot tasks seems harder and more complicated for only the ability of a single robot system to cope[ A multiple robots system is expected to be the key to solving the above problem[ In recent years\ there has been a lot of research which investigate multiple robots[ However\ the multiple robots system alone cannot improve working e.ciency so much without cooperation among the robots in the system[ This research is done in order to search for an e.cient way to control multiple robots so that they behave cooperatively[ Within this research\ the de_nition of robots cooperation is {work!accomplishment action with collaboration of multiple robots by applying shared information\ trans! mitted via communication network in a system|[ Figure 0 shows some cooperative tasks of multiple robots[ From the de_nition of robots cooperation\ there are two main factors necessary for cooperative robots system^ one is communication among multiple robots^ the other is movement of the robots[ The authors have proposed an e.cient communication method\ Code Division Carrier Sensing Multiple Accesses with Collision Detection "CDCSMA!CD# for cooperative robots system in ð0Ł[ Here\ the authors consider the e}ect of movement of multiple robots to the system[ Path planning is started from studying of movement of single robot but now many researches are concerned with movement of multiple robots[ Path planning of robots is divided into global and local path planning[ Examples of global path planning are cell decomposition ð1Ð3Ł\ retraction ð4\ 5Ł\ and priority setting ð6Ð8Ł models\ whereas examples of local path planning are potential _eld ð09Ð04Ł\ rule based ð05Ð07Ł\ and communication based ð08Ð12Ł models[ Because global path planning needs infor! mation about a working area before execution\ it is not practical in a real situation[ This research concentrated only on local path planning of multiple robots[ Although many movement models of multiple robots were studied and proposed\ most of them seem appropriate only for speci_c works or speci_c cases of cooperation[ A model\ designed for an environment\ cannot be applied to other environments[ General movement model of multiple robots\ employed for any cases of cooperation\ is proposed here as GSGM[ Many simulations on GSGM are done to evaluate its e.ciency and generality[
1[ GSGM movement model Cooperative action is the consequence of a collection of basic group!shape gener! ation[ Some of the basic group shapes are straight\ curve\ circle\ multi!angle and so on\ as shown in Fig[ 1[ Thus\ the model\ which can be applied to control multiple
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Fig[ 0[ Some cooperative tasks of multiple robots[
Fig[ 1[ Some group shapes of multiple robots[
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robots to generate their group shapes generally with the fewest exceptions\ is necessary[ GSGM "Group Shape Generating Model# is proposed here as a movement model for cooperative robots system[ GSGM looks like the result combination of potential _eld\ rule based\ and communication based models[ GSGM main equation is a potential _eld!type equation\ which includes all factors a}ecting the robots[ The important factors consist of the standing positions\ velocities\ and goal positions of all robots[ Obstacle positions detected by the robots and task speci_cation are also taken into account in the model[ Parameters in GSGM are adjusted automatically based on rule based algorithm[ The model applies information\ transmitted through the system\ to decide accelerations of the robots[ The information\ such as position\ velocity\ ques! tion\ answer\ acknowledgement\ and so on\ is transmitted by the previously proposed CDCSMA!CD ð0Ł[ Because all the robots apply only GSGM for their movement model and there is no leader in the model\ GSGM supports a decentralized system[ 1[0[ Principle of GSGM Some variables\ parameters\ and subscripts are de_ned below[ :
s Position vector of a robot s¾ Velocity vector of a robot : s Acceleration vector of a robot łI Unit vector in a direction u Relative angle between a couple of robots v Natural frequency of an element t Magnetic _eld parameter of an element g Angular parameter f Obstacle avoidance parameter z Damping ratio K Density distance between a couple of robots V Additional virtual range L Desired tracing range of a robot around an obstacle R Radius of a robot n Number of robots m Number of obstacles ` Goal element r Robot element o Obstacle element v Velocity element p Present d Destiny :
:
GSGM is used to search for acceleration\ s\ of the considered robot[ The model takes information into account about conditions of the considered robot and the external environment[ The information includes current position\ velocity\ and destiny position of the considered robot and also information of current positions\ velocities\ and destiny positions of other robots[ Furthermore information of obstacles positions
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within detectable range and the transmitted information by communication is con! sidered in the model also[ GSGM is expressed by eqn "0#\ whereas force elements\ being equivalent with acceleration elements\ of GSGM are shown in Fig[ 2[ :
:
:
:
:
:
"0#
s sgoal¦srobot¦sobstacle¦svelocity¦sapplication
Each element in the model can be calculated from the equations expressed below[ :
: goal
s
:
:
:
"=s−s`=¦V#"s−s`# −v : : =s−s`= 1 `
"1#
The goal element depends upon the current distance between the considered robot : : and its goal\ =s−s`=[ Additional virtual range\ V\ is added to the distance in order to : : : : decrease the time spent to the goal[ Direction of the goal element\ −"s−s`#:=s−s`=\ points from position of the robot to position of the goal[ Di}erent from natural frequency parameter of the robot element\ vr\ which is adjusted automatically using rule based model\ natural frequency parameter of the goal element\ v`\ is constant[ Figure 3 shows variable expressions of the goal element[ : robot
s
n
$
:
:
:
:
:
:
:
:
"=s−sri=−Ki#"s−sri# "=s−sri=−Ki#"s−sri# s −v −tr : : : : : : =s −s = "=s −s ri ri=−1R#"Ki−1R#=s−sri= i 0 1 r
−v1rug"upi−udi#Iłi
Fig[ 2[ Force elements of GSGM[
%
"2#
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Fig[ 3[ Variable expressions of goal element[
The robot element depends upon the di}erences between the current and the goal : : relative distances between the robot and other robots\ =s−sri=−Ki\ and also the di}er! ences between the current and the goal relative angles between the robot and other robots\ upi−udi\ in the system[ The direction of the former di}erences points between : : : : the position of the robot and positions of other robots\ "s−sri#:=s−sri=[ The latter di}erence points perpendicular direction\ łIi\ to the previous direction[ The natural frequency parameter of the robot element\ vr\ and its angular natural frequency parameter\ vru\ are adjusted automatically using the rule based model[ The parameters are set higher if an approximately _xed group shape is necessary[ If the robot is not safe from crashing with external obstacles\ the parameters are set to lower values[ To guarantee freedom from crashing with other robots\ the second term in eqn "2# exists[ This term makes in_nity acceleration in the direction that separates two robots from each other when the two robots are touching[ Magnetic _eld parameter of the robot element\ tr\ is constant[ Figure 4 shows variable expressions of the robot element[ : obstacle
s
m
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s −to j 0
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:
:
łIj "=s−soj=−L#"s−soj# : : : −f : : "=s−soj=−R#"L−R#=s−soj= "=s−soj=−R# :
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"3#
The obstacle element depends upon the distance between the robot and the detect! : : able obstacles\ =s−soj=\ and shapes of the obstacles[ Desired tracing range of the robot around the obstacles\ L\ is the range that the robot stands far away from the obstacles while tracing around the obstacles[ The obstacle element makes in_nity acceleration in the direction that separates the robot from the obstacle if the robot is touching the
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Fig[ 4[ Variable expressions of the robot element[
obstacle[ The direction of the _rst term in eqn "3# points between positions of the robot and the obstacles[ The second term in the equation makes the robot walk around the obstacles[ Its direction\ clockwise or counter!clockwise\ around the obstacles\ is considered at the positions where the robot _rst decides to avoid the obstacles[ The angle between the line of the obstacle surface and the line which connects the robot and its destiny\ is used to consider the avoidance direction[ If an approximately _xed group shape is necessary\ the direction is decided by the _rst robot which meets the obstacle[ Magnetic _eld parameter of the obstacle element\ to\ and obstacle avoidance parameter\ f\ are set constantly[ Figure 5 shows variable expressions of the obstacle element[ : velocity
s
:
"4#
−1zvvs¾
:
The velocity element depends upon the current velocity\ s¾\ of the robot[ The direction of the velocity element points in the opposite direction from the velocity direction of the robot[ Damping ratio\ z\ and natural frequency parameter\ vv are constant[ Figure 6 shows variable expressions of the velocity element[ The application element depends upon the application of the robots[ The appli! cation element is applied to improve the e.ciency of the system[ This element exists to overcome speci_c problems which need extra elements[ By substituting all the elements into eqn "0#\ the model is rewritten in eqn "5#[
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Fig[ 5[ Variable expressions of the obstacle element[
Fig[ 6[ Variable expressions of the velocity element[
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:
s −v1`
:
:
−tr
:
:
:
$
:
:
: ri
:
: ri :
"=s−s =−Ki#"s−s # 1 łi : : : −vrug"upi−udi#I "=s−sri=−1R#"Ki−1R#=s−sri=
m
$
:
¦ s −to j 0
:
"=s−sg=¦V#"s−sg# n "=s−sri=−Ki#"s−sri# ¦ s −v1r : : : : =s−sg= =s−sri= i 0
: oj
:
: oj : : oj
%
łIj "=s−s =−L#"s−s # −f : : : "=s−soj=−R#"L−R#=s−s = "=s−soj=−R#
:
:
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:
¦ð−1zvvs¾Ł¦sapplication
"5#
By GSGM\ the trajectories of all robots in the system depend upon parameters in the model and they are controllable[ Firstly the parameters are randomly selected\ then they can be adjusted dynamically by any method to obtain the desired trajectories[ A characteristics comparison between GSGM and three other pure theories of potential _eld\ rule based\ and communication based models is shown in Table 0[ Because the potential _eld model uses an equation to control the motion of multiple robots\ the trajectory continuity is excellent[ On the contrary\ the trajectory of rule based and communication based models depends upon the condition of the system at the time of execution\ so the trajectory continuity is not good*whereas GSGM\ which is the compromised model of these three models\ makes fair trajectory continuity[ The pure potential _eld model cannot solve the deadlock problem\ while the other models work well[ Rule based\ communication based\ and GSGM models detect the current condition information before deciding what to do next\ so the deadlock problem can be recognized and solved[ Because the potential _eld model uses an equation to control the motion of the robot\ it is simple compared with rule based and com! munication based models which needs memory to store a lot of rules[ Because only the parameters in GSGM need rules in order to select their appropriate values\ the number of applied rules is reduced[ Finally\ for generality characteristics\ GSGM can generate any robot group shape easily with fewer exemptions[ The other models can be applied only to some speci_c tasks[
2[ Simulation and evaluation In order to evaluate the e.ciency and the generality of GSGM\ the authors arranged _ve di}erent tasks^ load carrying^ objects rearranging^ mobile object catching^ parts
Table 0 Characteristics comparison of four movement models Model
Continuity
Deadlock solving ability
Complexity
Generality
Potential _eld model Rule based model Communication based model GSGM
! X X
X ! ! !
! X X
X X X !
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assembling^ and unknown environment searching and map making[ The robots were assigned to do these tasks by applying GSGM[ Before the results of the simulation are shown\ the modeling of the tasks of multiple robots is discussed[ 2[0[ Model of task of multiple robots As before\ łs is de_ned as the position vector of a robot and n is de_ned as the number of robots in the system^ furthermore k is de_ned here as the number of steps or group shapes of a task[ The group shape of multiple robots is expressed by the function of the collective position vectors of the robots as fg"sł0\ łs1\ [ [ [ \ łsn−0\ łsn#\ whereas the task of multiple robots " fT# is de_ned by the vector summation of changes of group shapes of the robots\ and is expressed by the equation below[ k
:
:
:
:
:
:
:
:
fT s ð fg"i#"s0\ s1\ [ [ [ \ sn−0\ sn#−fg"i−0#"s0\ s1\ [ [ [ \ sn−0\ sn#Ł
"6#
i 1
A feature of the model of a task of multiple robots is shown in Fig[ 7[ Within the _gure\ a simple task of three robots and two group shapes is modeled[ The number of steps or group shapes and shape of each step of a task depends directly upon the characteristics of the task and they are always designed at the task planning subprocess[ In order to control multiple robots to move accurately\ more steps are necessary and the duration between the connected steps is shorter[ General transformation of group shape of the robots for any tasks is shown below[
Fig[ 7[ Feature of model of task of multiple robots[
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Initial shape "Random shape# : [ [ [ : Intermediate shape "Random shape# : [ [ [ :Goal shape "Random shape# 2[1[ Load carryin` Details of the cooperation among multiple robots in load!carrying are listed as follows] "0# Multiple robots stand in any initial position in the working area[ "1# The robots try to move\ without crashing into each other or external objects\ towards the position where the load is located[ "2# The load is transferred to the robots[ "3# The robots try to carry the load\ with approximately _xed group shape and without crashing into external obstacles\ to the assigned position[ "4# The robots transfer the load to the assigned position[ "5# Processes 1Ð4 are repeated until all the loads are carried to the assigned positions[ Figure 8 shows a simulation result of the cooperation among multiple robots in carrying load[ At the beginning\ three robots\ Nos 0Ð2\ are standing at the bottom left corner[ The robots then move to the predetermined position of the load[ Because there is an obstacle located between the starting point and the load point\ the robots try to avoid crashing into it[ To avoid the obstacle\ the natural frequency parameter\ vr\ and the angular natural frequency\ vru\ of the robot element\ are adjusted to lower values[ The parameters are reset to the normal values again when the robots pass the obstacle and try to form the shape of the load[ When the load is transferred to the robot\ an approximately _xed group shape is necessary[ The parameters are adjusted to the higher values so the robots can still keep the group shape even if they avoid crashing into the rectangular obstacle[ When the parameters are adjusted to higher values\ the robots in the group behave as if they are a unit[ Information about the external obstacles detected by some of the robots in the group will be transmitted to all the robots[ Thus\ all the robots will behave as if they detect the obstacles by themselves[ In other words\ the obstacles have the same e}ect on all the robots in the group even if they are only detected by some of the robots[ This makes the group robots able to avoid crashing into the obstacles while still keeping their approximately _xed group shape[ When the robots reach the next destination\ they transfer the load to the predetermined point[ Then the robots repeat the process until No[ 1 load\ located at the top right corner is carried and transferred to the desired position[ From the simulation result\ we found that the robots completed the task successfully[ 2[2[ Objects rearran`in` Details of cooperation among multiple robots in objects rearranging are listed as follows] "0# Multiple robots stand in any initial position in the working area[
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Fig[ 8[ Simulation result of cooperation among multiple robots in carrying load[
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"1# The robots decide the nearest objects which will be rearranged by applying the simple priority rule[ "2# The robots try to move\ without crashing into each other or external obstacles\ to the positions where objects are located[ "3# The objects are transferred to the robots[ "4# The robots decide the positions to place the objects[ "5# The robots try to rearrange the objects without crashing into each other or external obstacles[ "6# The robots transfer the objects to the positions[ "7# Processes 1Ð6 are repeated until all the objects are rearranged[ Figure 09 shows a simulation result of the cooperation among multiple robots in rearranging objects[ Like the previous task\ at the beginning\ three robots stand in the bottom left corner[ The objects waiting for rearrangement are located at the predetermined positions[ The robots _nd the objects located at the nearest positions by applying the priority rule[ The priority rule states that the robot with the lowest number has a higher priority to _nd the nearest object _rst[ Number 0 robot decides to rearrange the object most to the left^ No[ 1 robot decides the bottom most object\ while No[ 2 robot decides on object located on the right of the bottom most object[ After all robots have decided their objects\ they try to move to the objects| positions[ Because there is an obstacle obstructing the robots to their objects positions\ the robots are not safe from crashing into the obstacle[ The natural frequency parameter\ vr\ and the angular natural frequency\ vru\ of the robot element\ are adjusted to lower values[ When the robots have passed the obstacle\ the parameters are reset to the normal values[ When the robots move to the objects positions\ the objects are trans! ferred to the robots[ The robots then decide the positions to place the objects\ by determining the robot which is located near the left most placing position _rst[ Like at the beginning\ there is another obstacle located on the way to the placing positions[ The parameters are adjusted to lower values during the unsafe situation\ then reset to the normal values when the robots are safe[ The robots repeat the process until all objects are rearranged to the new placing positions[ From the simulation result\ we found that the robots completed the task successfully[ 2[3[ Mobile object catchin` Details of cooperation among multiple robots in mobile object catching are listed as follows] "0# Multiple robots stand in any initial position in the working area while a mobile object moves with a constant speed and a constant direction[ The mobile object changes its direction while the speed is still kept constant only when it moves close to external objects[ The re~ection angle of the mobile object is the same as the incident angle[ "1# The robots search for temporary catching positions surrounding the mobile object[ "2# The robots try to surround the mobile object without crashing into each other or external obstacles[
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Fig[ 09[ Simulation result of cooperation among multiple robots in rearranging objects[
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"3# Processes 1Ð2 are repeated until the mobile object is caught[ Figure 00 shows a simulation result of the cooperation among multiple robots in catching a mobile object[ At the beginning\ three robots stand at the bottom right corner[ The robots try to catch the mobile object by trying to surround it[ Because the destination points change when the mobile object moves to a new position\ destination points around the mobile object are searched dynamically all the time[ At the position where obstacle locates\ the natural frequency parameter\ vr\ and the angular natural frequency\ vru\ of the robot element are adjusted to lower values[ At the other position\ the parameters are set to normal values[ The robots repeat the process until the mobile object is surrounded by the three robots[ The mobile object is _nally caught at the mid!right position[ 2[4[ Parts assemblin` Details of cooperation among multiple robots in parts assembling are listed as follows] "0# Multiple robots stand in any initial position in the working area[ "1# The robots try to move\ without crashing into each other or external obstacles\ to the positions where parts are located[ "2# The parts are transferred to the robots[ "3# The robots try to assemble the parts together without crashing into each other or external obstacles[ Figure 01 shows a simulation result of the cooperation among multiple robots in assembling parts[ There are four robots standing at the bottom left corner[ The robots separate into groups[ The number of the groups depends upon the number of parts needing to be assembled[ There are two parts at the top left corner and at the bottom right corner[ After the robots have moved to the parts and the parts have been transferred\ the robots of each group have to keep their approximately _xed group shape[ Thus\ the natural frequency parameter\ vr\ and the angular natural frequency\ vru\ of the robot element are adjusted to higher values[ As in the load carrying task\ when the parameters are adjusted to higher values\ the robots in the group behave as if they are a unit[ Information of the external obstacles detected by some of the robots in the group will be transmitted to all the robots[ All the robots will behave as if they detect the obstacles by themselves[ Thus\ the group robots can avoid crashing into the obstacles while still keeping their approximately _xed group shape[ The destination points of the robots carrying parts located at the parts assembling point\ are deter! mined from the current positions of all the robots[ The assembling point is the position at the mid of all the parts[ Thus\ the destination points change dynamically[ From the simulation result\ the two parts are _nally assembled together[ 2[5[ Unknown environment searchin` and map makin` Details of cooperation among multiple robots in unknown environment searching and map making are listed as follows]
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Fig[ 00[ Simulation result of cooperation among multiple robots in catching a mobile object[
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Fig[ 01[ Simulation result of cooperation among multiple robots in assembling parts[
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"0# Multiple robots stand in any initial position in the working area[ "1# The robots try to move\ without crashing into each other\ to the positions on the route equation[ The route equation is designed for the robots in order that the robots will search and move completely within the unknown environment[ The authors select a variable radius circle model as the route equation here[ "2# If the robots detect new objects\ they try to move\ without crashing into each other or external obstacles\ to the objects[ Then the robots search around the objects and make the search map[ "3# Processes 1Ð2 are repeated until the working area is completely searched[ Figure 02 shows a simulation result of the cooperation among multiple robots in searching an unknown environment and making a map[ At the beginning\ the three robots\ Nos 0Ð2\ stand at the bottom left corner with empty maps[ The robots move to the positions on the variable radius route equation[ The destination positions of the robots change dynamically all the time[ When the robots detect new obstacles\ they search around the obstacles and make the search map[ When the robots complete searching for the obstacles\ they send the information about the searched obstacle to the other robots so that the other robots can _ll the obstacle in their maps also[ The map is made\ little by little\ until all the environment is searched and the completed map is obtained[ For unknown environment searching and map making task\ the application element in the GSGM model is needed to force the robots to move around the searched obstacles[ Natural frequency parameter\ vr\ and angular natural frequency\ vru\ of the robot element are set to lower values when the robots are searching around the obstacles or when they are avoiding crashing into the obstacles[ 2[6[ Discussion of simulation results From the simulation results\ all tasks were completed successfully[ These show the e.ciency and the generality of the proposed GSGM[ Robots see all tasks as only a vector summation of changes of their group shapes*there is no di}erence among the tasks to the robots[ The key of GSGM is the determination of destination points[ For some tasks\ e[g[\ load carrying and objects rearranging\ the destination points are predetermined and _xed[ But for some other tasks\ e[g[\ mobile object catching\ parts assembling\ and unknown environment searching and map making\ the destination points vary and depend upon the situation at the time of execution[ The destination points are programmed directly in the former case\ while in the latter case\ the method of determination of destination points which depends strictly upon the task\ is instructed by operators[ Since any tasks of multiple robots can be modeled as the vector summation of changes of group shapes of the robots\ GSGM can handle the modeled tasks[ The conclusion is that the proposed GSGM is an e}ective general movement model for multiple robots[ 3[ Conclusion The main objective of this paper is to realize e}ective robot cooperation[ The movement model which is used to control robots in cooperating with each other is a
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Fig[ 02[ Simulation result of cooperation among multiple robots in searching an unknown environment and making a map[
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key point of this realization[ GSGM was proposed in this paper[ It is a combination model of potential _eld\ rule based\ and communication based models[ GSGM con! siders the situation in the system and information\ transmitted through the system\ to decide the acceleration of each robot[ The model comprises _ve elements] goal\ robot\ obstacle\ velocity\ and application elements[ Furthermore\ in this paper\ the cooperative task was modeled as the vector summation of changes of group shapes of the robots[ By the evaluation of many simulation cases\ GSGM found that it could handle the tasks successfully[ Because any cooperative tasks can be modeled and GSGM can handle the tasks being able to be modeled\ the conclusion from this paper is that GSGM is a general model for a cooperative robots system[
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ð06Ł Noborio H\ Hashime J[ A Feasible Collision!Free and Deadlock!Free Planning Algorithm in a Certain Workspace where Multiple Robots Move Flexibly[ Proceedings of 0880 IEEE:RSJ International Workshop on Intelligent Robots and Systems\ 0963Ð68\ 0880[ ð07Ł Kato S\ Nishiyama S\ Takeno J[ Coordinating Mobile Robots by Applying Tra.c Rules[ Proceedings of 0881 IEEE:RSJ International Conference on Intelligent Robots and Systems\ 0424Ð30\ 0881[ ð08Ł Noreils FR[ Coordinated Execution of Trajectories by Multiple Mobile Robots[ Proceedings of 0880 IEEE:RSJ International Workshop on Intelligent Robots and Systems\ 0950Ð57\ 0880[ ð19Ł Asama H\ Ozaki K\ Itakura H\ Matsumoto A\ Ishida Y\ Endo I[ Collision Avoidance among Multiple Mobile Robots Based on Rules and Communication[ Proceedings of 0880 IEEE:RSJ International Workshop on Intelligent Robots and Systems\ 0104Ð19\ 0880[ ð10Ł Fukuda T\ Sekiyama K[ Network Type Robot System Toward Intelligent Communication[ Journal of the Robotics Society of Japan 0883^01"5#]679Ð3[ ð11Ł Premvuti S\ Yuta S[ Consideration on the Cooperation of Multiple Autonomous Mobile Robots[ Proceedings of 0889 IEEE International Workshop on Intelligent Robots and Systems\ 48Ð52\ 0889[ ð12Ł Fukuda T\ Sekiyama K\ Ueyama T\ Arai F[ E.cient Communication Method in the Cellular Robotic System[ Proceedings of 0882 IEEE:RSJ International Conference on Intelligent Robots and Systems\ 0980Ð85\ 0882[
Mechatronics 7 "0887# 894Ð814
Technical Note GSGM movement model for cooperative robots system M[ Parnichkuna\ \ S[ Ozonob a
School of Advanced Technolo`ies\ Asian Institute of Technolo`y\ P[O[ Box 3 Klon`luan`\ Pathumthani 01019\ Thailand b Department of Precision Machinery En`ineerin`\ The University of Tokyo\ 6!2!0 Hon`o\ Bunkyo!ku\ Tokyo 002\ Japan Received 02 April 0886^ revised 16 February 0887^ accepted 0 June 0887
Abstract Robots cooperation means {work!accomplishment action with collaboration of multiple robots by applying shared information\ transmitted via communication network in a system|[ The de_nition implies that the e.ciency of the cooperative robots system depends directly upon two main factors] one is communication among the robots^ the other is movement of the robots[ Here the authors consider the latter e}ect on the robots system[ Robots with appropriate cooperation are expected to work e.ciently[ Cooperation action of multiple robots is viewed here as the consequence of a collection of basic group!shape generation[ Some of the basic group shapes are straight\ curve\ circle\ multi!angle\ and so on[ Thus\ the model\ which can be applied to control multiple robots to generate their group shapes generally with the least exceptions\ is necessary[ Group Shape Generating Model "GSGM# is proposed here as a general movement model for cooperative robots system[ It looks like the result of a combination of potential _eld\ rule based\ and communication based models[ It applies information\ trans! mitted through the system\ to determine accelerations of all robots[ Evaluation of the proposed model is done by applying it to many di}erent tasks to estimate its e.ciency and generality[ Þ 0887 Elsevier Science Ltd[ All rights reserved[
0[ Introduction Current research e}orts focus on creating a {smart| robot that can {see|\ {hear|\ {touch|\ and {make decisions|[ An {autonomous robot| recognizes its environment Corresponding author[ Tel[] ¦55 1 413 4118^ Fax] ¦55 1413 4586^ E!mail] manukidÝait[ac[th 9846Ð3047:87:,*see front matter Þ 0887 Elsevier Science Ltd[ All rights reserved[ PII] S 9 8 4 6 Ð 3 0 4 7 " 8 7 # 9 9 9 2 4 Ð X
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using onboard sensors and decides what to do next based on the sensor data by its own decision[ A {self!contained robot| contains everything which is needed to allow the robot to operate independently and autonomously\ such as onboard sensors\ locomotive units\ information processing units\ and power supplies\ on its body[ It is necessary that an {intelligent robot| has to behave following the two properties above[ Furthermore\ a robot can be said to be {more intelligent| by being able to cooperate with other robots\ that is\ the robot decides its action by considering not only its own objective but also the intentions of other robots or the community to which the robot belongs[ At the present time\ the trend of robot tasks seems harder and more complicated for only the ability of a single robot system to cope[ A multiple robots system is expected to be the key to solving the above problem[ In recent years\ there has been a lot of research which investigate multiple robots[ However\ the multiple robots system alone cannot improve working e.ciency so much without cooperation among the robots in the system[ This research is done in order to search for an e.cient way to control multiple robots so that they behave cooperatively[ Within this research\ the de_nition of robots cooperation is {work!accomplishment action with collaboration of multiple robots by applying shared information\ trans! mitted via communication network in a system|[ Figure 0 shows some cooperative tasks of multiple robots[ From the de_nition of robots cooperation\ there are two main factors necessary for cooperative robots system^ one is communication among multiple robots^ the other is movement of the robots[ The authors have proposed an e.cient communication method\ Code Division Carrier Sensing Multiple Accesses with Collision Detection "CDCSMA!CD# for cooperative robots system in ð0Ł[ Here\ the authors consider the e}ect of movement of multiple robots to the system[ Path planning is started from studying of movement of single robot but now many researches are concerned with movement of multiple robots[ Path planning of robots is divided into global and local path planning[ Examples of global path planning are cell decomposition ð1Ð3Ł\ retraction ð4\ 5Ł\ and priority setting ð6Ð8Ł models\ whereas examples of local path planning are potential _eld ð09Ð04Ł\ rule based ð05Ð07Ł\ and communication based ð08Ð12Ł models[ Because global path planning needs infor! mation about a working area before execution\ it is not practical in a real situation[ This research concentrated only on local path planning of multiple robots[ Although many movement models of multiple robots were studied and proposed\ most of them seem appropriate only for speci_c works or speci_c cases of cooperation[ A model\ designed for an environment\ cannot be applied to other environments[ General movement model of multiple robots\ employed for any cases of cooperation\ is proposed here as GSGM[ Many simulations on GSGM are done to evaluate its e.ciency and generality[
1[ GSGM movement model Cooperative action is the consequence of a collection of basic group!shape gener! ation[ Some of the basic group shapes are straight\ curve\ circle\ multi!angle and so on\ as shown in Fig[ 1[ Thus\ the model\ which can be applied to control multiple
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Fig[ 0[ Some cooperative tasks of multiple robots[
Fig[ 1[ Some group shapes of multiple robots[
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robots to generate their group shapes generally with the fewest exceptions\ is necessary[ GSGM "Group Shape Generating Model# is proposed here as a movement model for cooperative robots system[ GSGM looks like the result combination of potential _eld\ rule based\ and communication based models[ GSGM main equation is a potential _eld!type equation\ which includes all factors a}ecting the robots[ The important factors consist of the standing positions\ velocities\ and goal positions of all robots[ Obstacle positions detected by the robots and task speci_cation are also taken into account in the model[ Parameters in GSGM are adjusted automatically based on rule based algorithm[ The model applies information\ transmitted through the system\ to decide accelerations of the robots[ The information\ such as position\ velocity\ ques! tion\ answer\ acknowledgement\ and so on\ is transmitted by the previously proposed CDCSMA!CD ð0Ł[ Because all the robots apply only GSGM for their movement model and there is no leader in the model\ GSGM supports a decentralized system[ 1[0[ Principle of GSGM Some variables\ parameters\ and subscripts are de_ned below[ :
s Position vector of a robot s¾ Velocity vector of a robot : s Acceleration vector of a robot łI Unit vector in a direction u Relative angle between a couple of robots v Natural frequency of an element t Magnetic _eld parameter of an element g Angular parameter f Obstacle avoidance parameter z Damping ratio K Density distance between a couple of robots V Additional virtual range L Desired tracing range of a robot around an obstacle R Radius of a robot n Number of robots m Number of obstacles ` Goal element r Robot element o Obstacle element v Velocity element p Present d Destiny :
:
GSGM is used to search for acceleration\ s\ of the considered robot[ The model takes information into account about conditions of the considered robot and the external environment[ The information includes current position\ velocity\ and destiny position of the considered robot and also information of current positions\ velocities\ and destiny positions of other robots[ Furthermore information of obstacles positions
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within detectable range and the transmitted information by communication is con! sidered in the model also[ GSGM is expressed by eqn "0#\ whereas force elements\ being equivalent with acceleration elements\ of GSGM are shown in Fig[ 2[ :
:
:
:
:
:
"0#
s sgoal¦srobot¦sobstacle¦svelocity¦sapplication
Each element in the model can be calculated from the equations expressed below[ :
: goal
s
:
:
:
"=s−s`=¦V#"s−s`# −v : : =s−s`= 1 `
"1#
The goal element depends upon the current distance between the considered robot : : and its goal\ =s−s`=[ Additional virtual range\ V\ is added to the distance in order to : : : : decrease the time spent to the goal[ Direction of the goal element\ −"s−s`#:=s−s`=\ points from position of the robot to position of the goal[ Di}erent from natural frequency parameter of the robot element\ vr\ which is adjusted automatically using rule based model\ natural frequency parameter of the goal element\ v`\ is constant[ Figure 3 shows variable expressions of the goal element[ : robot
s
n
$
:
:
:
:
:
:
:
:
"=s−sri=−Ki#"s−sri# "=s−sri=−Ki#"s−sri# s −v −tr : : : : : : =s −s = "=s −s ri ri=−1R#"Ki−1R#=s−sri= i 0 1 r
−v1rug"upi−udi#Iłi
Fig[ 2[ Force elements of GSGM[
%
"2#
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Fig[ 3[ Variable expressions of goal element[
The robot element depends upon the di}erences between the current and the goal : : relative distances between the robot and other robots\ =s−sri=−Ki\ and also the di}er! ences between the current and the goal relative angles between the robot and other robots\ upi−udi\ in the system[ The direction of the former di}erences points between : : : : the position of the robot and positions of other robots\ "s−sri#:=s−sri=[ The latter di}erence points perpendicular direction\ łIi\ to the previous direction[ The natural frequency parameter of the robot element\ vr\ and its angular natural frequency parameter\ vru\ are adjusted automatically using the rule based model[ The parameters are set higher if an approximately _xed group shape is necessary[ If the robot is not safe from crashing with external obstacles\ the parameters are set to lower values[ To guarantee freedom from crashing with other robots\ the second term in eqn "2# exists[ This term makes in_nity acceleration in the direction that separates two robots from each other when the two robots are touching[ Magnetic _eld parameter of the robot element\ tr\ is constant[ Figure 4 shows variable expressions of the robot element[ : obstacle
s
m
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s −to j 0
:
:
:
:
łIj "=s−soj=−L#"s−soj# : : : −f : : "=s−soj=−R#"L−R#=s−soj= "=s−soj=−R# :
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"3#
The obstacle element depends upon the distance between the robot and the detect! : : able obstacles\ =s−soj=\ and shapes of the obstacles[ Desired tracing range of the robot around the obstacles\ L\ is the range that the robot stands far away from the obstacles while tracing around the obstacles[ The obstacle element makes in_nity acceleration in the direction that separates the robot from the obstacle if the robot is touching the
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Fig[ 4[ Variable expressions of the robot element[
obstacle[ The direction of the _rst term in eqn "3# points between positions of the robot and the obstacles[ The second term in the equation makes the robot walk around the obstacles[ Its direction\ clockwise or counter!clockwise\ around the obstacles\ is considered at the positions where the robot _rst decides to avoid the obstacles[ The angle between the line of the obstacle surface and the line which connects the robot and its destiny\ is used to consider the avoidance direction[ If an approximately _xed group shape is necessary\ the direction is decided by the _rst robot which meets the obstacle[ Magnetic _eld parameter of the obstacle element\ to\ and obstacle avoidance parameter\ f\ are set constantly[ Figure 5 shows variable expressions of the obstacle element[ : velocity
s
:
"4#
−1zvvs¾
:
The velocity element depends upon the current velocity\ s¾\ of the robot[ The direction of the velocity element points in the opposite direction from the velocity direction of the robot[ Damping ratio\ z\ and natural frequency parameter\ vv are constant[ Figure 6 shows variable expressions of the velocity element[ The application element depends upon the application of the robots[ The appli! cation element is applied to improve the e.ciency of the system[ This element exists to overcome speci_c problems which need extra elements[ By substituting all the elements into eqn "0#\ the model is rewritten in eqn "5#[
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Fig[ 5[ Variable expressions of the obstacle element[
Fig[ 6[ Variable expressions of the velocity element[
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M[ Parnichkun\ S[ Ozono:Mechatronics 7 "0887# 894Ð814 :
:
s −v1`
:
:
−tr
:
:
:
$
:
:
: ri
:
: ri :
"=s−s =−Ki#"s−s # 1 łi : : : −vrug"upi−udi#I "=s−sri=−1R#"Ki−1R#=s−sri=
m
$
:
¦ s −to j 0
:
"=s−sg=¦V#"s−sg# n "=s−sri=−Ki#"s−sri# ¦ s −v1r : : : : =s−sg= =s−sri= i 0
: oj
:
: oj : : oj
%
łIj "=s−s =−L#"s−s # −f : : : "=s−soj=−R#"L−R#=s−s = "=s−soj=−R#
:
:
%
:
¦ð−1zvvs¾Ł¦sapplication
"5#
By GSGM\ the trajectories of all robots in the system depend upon parameters in the model and they are controllable[ Firstly the parameters are randomly selected\ then they can be adjusted dynamically by any method to obtain the desired trajectories[ A characteristics comparison between GSGM and three other pure theories of potential _eld\ rule based\ and communication based models is shown in Table 0[ Because the potential _eld model uses an equation to control the motion of multiple robots\ the trajectory continuity is excellent[ On the contrary\ the trajectory of rule based and communication based models depends upon the condition of the system at the time of execution\ so the trajectory continuity is not good*whereas GSGM\ which is the compromised model of these three models\ makes fair trajectory continuity[ The pure potential _eld model cannot solve the deadlock problem\ while the other models work well[ Rule based\ communication based\ and GSGM models detect the current condition information before deciding what to do next\ so the deadlock problem can be recognized and solved[ Because the potential _eld model uses an equation to control the motion of the robot\ it is simple compared with rule based and com! munication based models which needs memory to store a lot of rules[ Because only the parameters in GSGM need rules in order to select their appropriate values\ the number of applied rules is reduced[ Finally\ for generality characteristics\ GSGM can generate any robot group shape easily with fewer exemptions[ The other models can be applied only to some speci_c tasks[
2[ Simulation and evaluation In order to evaluate the e.ciency and the generality of GSGM\ the authors arranged _ve di}erent tasks^ load carrying^ objects rearranging^ mobile object catching^ parts
Table 0 Characteristics comparison of four movement models Model
Continuity
Deadlock solving ability
Complexity
Generality
Potential _eld model Rule based model Communication based model GSGM
! X X
X ! ! !
! X X
X X X !
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assembling^ and unknown environment searching and map making[ The robots were assigned to do these tasks by applying GSGM[ Before the results of the simulation are shown\ the modeling of the tasks of multiple robots is discussed[ 2[0[ Model of task of multiple robots As before\ łs is de_ned as the position vector of a robot and n is de_ned as the number of robots in the system^ furthermore k is de_ned here as the number of steps or group shapes of a task[ The group shape of multiple robots is expressed by the function of the collective position vectors of the robots as fg"sł0\ łs1\ [ [ [ \ łsn−0\ łsn#\ whereas the task of multiple robots " fT# is de_ned by the vector summation of changes of group shapes of the robots\ and is expressed by the equation below[ k
:
:
:
:
:
:
:
:
fT s ð fg"i#"s0\ s1\ [ [ [ \ sn−0\ sn#−fg"i−0#"s0\ s1\ [ [ [ \ sn−0\ sn#Ł
"6#
i 1
A feature of the model of a task of multiple robots is shown in Fig[ 7[ Within the _gure\ a simple task of three robots and two group shapes is modeled[ The number of steps or group shapes and shape of each step of a task depends directly upon the characteristics of the task and they are always designed at the task planning subprocess[ In order to control multiple robots to move accurately\ more steps are necessary and the duration between the connected steps is shorter[ General transformation of group shape of the robots for any tasks is shown below[
Fig[ 7[ Feature of model of task of multiple robots[
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Initial shape "Random shape# : [ [ [ : Intermediate shape "Random shape# : [ [ [ :Goal shape "Random shape# 2[1[ Load carryin` Details of the cooperation among multiple robots in load!carrying are listed as follows] "0# Multiple robots stand in any initial position in the working area[ "1# The robots try to move\ without crashing into each other or external objects\ towards the position where the load is located[ "2# The load is transferred to the robots[ "3# The robots try to carry the load\ with approximately _xed group shape and without crashing into external obstacles\ to the assigned position[ "4# The robots transfer the load to the assigned position[ "5# Processes 1Ð4 are repeated until all the loads are carried to the assigned positions[ Figure 8 shows a simulation result of the cooperation among multiple robots in carrying load[ At the beginning\ three robots\ Nos 0Ð2\ are standing at the bottom left corner[ The robots then move to the predetermined position of the load[ Because there is an obstacle located between the starting point and the load point\ the robots try to avoid crashing into it[ To avoid the obstacle\ the natural frequency parameter\ vr\ and the angular natural frequency\ vru\ of the robot element\ are adjusted to lower values[ The parameters are reset to the normal values again when the robots pass the obstacle and try to form the shape of the load[ When the load is transferred to the robot\ an approximately _xed group shape is necessary[ The parameters are adjusted to the higher values so the robots can still keep the group shape even if they avoid crashing into the rectangular obstacle[ When the parameters are adjusted to higher values\ the robots in the group behave as if they are a unit[ Information about the external obstacles detected by some of the robots in the group will be transmitted to all the robots[ Thus\ all the robots will behave as if they detect the obstacles by themselves[ In other words\ the obstacles have the same e}ect on all the robots in the group even if they are only detected by some of the robots[ This makes the group robots able to avoid crashing into the obstacles while still keeping their approximately _xed group shape[ When the robots reach the next destination\ they transfer the load to the predetermined point[ Then the robots repeat the process until No[ 1 load\ located at the top right corner is carried and transferred to the desired position[ From the simulation result\ we found that the robots completed the task successfully[ 2[2[ Objects rearran`in` Details of cooperation among multiple robots in objects rearranging are listed as follows] "0# Multiple robots stand in any initial position in the working area[
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Fig[ 8[ Simulation result of cooperation among multiple robots in carrying load[
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"1# The robots decide the nearest objects which will be rearranged by applying the simple priority rule[ "2# The robots try to move\ without crashing into each other or external obstacles\ to the positions where objects are located[ "3# The objects are transferred to the robots[ "4# The robots decide the positions to place the objects[ "5# The robots try to rearrange the objects without crashing into each other or external obstacles[ "6# The robots transfer the objects to the positions[ "7# Processes 1Ð6 are repeated until all the objects are rearranged[ Figure 09 shows a simulation result of the cooperation among multiple robots in rearranging objects[ Like the previous task\ at the beginning\ three robots stand in the bottom left corner[ The objects waiting for rearrangement are located at the predetermined positions[ The robots _nd the objects located at the nearest positions by applying the priority rule[ The priority rule states that the robot with the lowest number has a higher priority to _nd the nearest object _rst[ Number 0 robot decides to rearrange the object most to the left^ No[ 1 robot decides the bottom most object\ while No[ 2 robot decides on object located on the right of the bottom most object[ After all robots have decided their objects\ they try to move to the objects| positions[ Because there is an obstacle obstructing the robots to their objects positions\ the robots are not safe from crashing into the obstacle[ The natural frequency parameter\ vr\ and the angular natural frequency\ vru\ of the robot element\ are adjusted to lower values[ When the robots have passed the obstacle\ the parameters are reset to the normal values[ When the robots move to the objects positions\ the objects are trans! ferred to the robots[ The robots then decide the positions to place the objects\ by determining the robot which is located near the left most placing position _rst[ Like at the beginning\ there is another obstacle located on the way to the placing positions[ The parameters are adjusted to lower values during the unsafe situation\ then reset to the normal values when the robots are safe[ The robots repeat the process until all objects are rearranged to the new placing positions[ From the simulation result\ we found that the robots completed the task successfully[ 2[3[ Mobile object catchin` Details of cooperation among multiple robots in mobile object catching are listed as follows] "0# Multiple robots stand in any initial position in the working area while a mobile object moves with a constant speed and a constant direction[ The mobile object changes its direction while the speed is still kept constant only when it moves close to external objects[ The re~ection angle of the mobile object is the same as the incident angle[ "1# The robots search for temporary catching positions surrounding the mobile object[ "2# The robots try to surround the mobile object without crashing into each other or external obstacles[
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Fig[ 09[ Simulation result of cooperation among multiple robots in rearranging objects[
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"3# Processes 1Ð2 are repeated until the mobile object is caught[ Figure 00 shows a simulation result of the cooperation among multiple robots in catching a mobile object[ At the beginning\ three robots stand at the bottom right corner[ The robots try to catch the mobile object by trying to surround it[ Because the destination points change when the mobile object moves to a new position\ destination points around the mobile object are searched dynamically all the time[ At the position where obstacle locates\ the natural frequency parameter\ vr\ and the angular natural frequency\ vru\ of the robot element are adjusted to lower values[ At the other position\ the parameters are set to normal values[ The robots repeat the process until the mobile object is surrounded by the three robots[ The mobile object is _nally caught at the mid!right position[ 2[4[ Parts assemblin` Details of cooperation among multiple robots in parts assembling are listed as follows] "0# Multiple robots stand in any initial position in the working area[ "1# The robots try to move\ without crashing into each other or external obstacles\ to the positions where parts are located[ "2# The parts are transferred to the robots[ "3# The robots try to assemble the parts together without crashing into each other or external obstacles[ Figure 01 shows a simulation result of the cooperation among multiple robots in assembling parts[ There are four robots standing at the bottom left corner[ The robots separate into groups[ The number of the groups depends upon the number of parts needing to be assembled[ There are two parts at the top left corner and at the bottom right corner[ After the robots have moved to the parts and the parts have been transferred\ the robots of each group have to keep their approximately _xed group shape[ Thus\ the natural frequency parameter\ vr\ and the angular natural frequency\ vru\ of the robot element are adjusted to higher values[ As in the load carrying task\ when the parameters are adjusted to higher values\ the robots in the group behave as if they are a unit[ Information of the external obstacles detected by some of the robots in the group will be transmitted to all the robots[ All the robots will behave as if they detect the obstacles by themselves[ Thus\ the group robots can avoid crashing into the obstacles while still keeping their approximately _xed group shape[ The destination points of the robots carrying parts located at the parts assembling point\ are deter! mined from the current positions of all the robots[ The assembling point is the position at the mid of all the parts[ Thus\ the destination points change dynamically[ From the simulation result\ the two parts are _nally assembled together[ 2[5[ Unknown environment searchin` and map makin` Details of cooperation among multiple robots in unknown environment searching and map making are listed as follows]
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Fig[ 00[ Simulation result of cooperation among multiple robots in catching a mobile object[
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Fig[ 01[ Simulation result of cooperation among multiple robots in assembling parts[
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"0# Multiple robots stand in any initial position in the working area[ "1# The robots try to move\ without crashing into each other\ to the positions on the route equation[ The route equation is designed for the robots in order that the robots will search and move completely within the unknown environment[ The authors select a variable radius circle model as the route equation here[ "2# If the robots detect new objects\ they try to move\ without crashing into each other or external obstacles\ to the objects[ Then the robots search around the objects and make the search map[ "3# Processes 1Ð2 are repeated until the working area is completely searched[ Figure 02 shows a simulation result of the cooperation among multiple robots in searching an unknown environment and making a map[ At the beginning\ the three robots\ Nos 0Ð2\ stand at the bottom left corner with empty maps[ The robots move to the positions on the variable radius route equation[ The destination positions of the robots change dynamically all the time[ When the robots detect new obstacles\ they search around the obstacles and make the search map[ When the robots complete searching for the obstacles\ they send the information about the searched obstacle to the other robots so that the other robots can _ll the obstacle in their maps also[ The map is made\ little by little\ until all the environment is searched and the completed map is obtained[ For unknown environment searching and map making task\ the application element in the GSGM model is needed to force the robots to move around the searched obstacles[ Natural frequency parameter\ vr\ and angular natural frequency\ vru\ of the robot element are set to lower values when the robots are searching around the obstacles or when they are avoiding crashing into the obstacles[ 2[6[ Discussion of simulation results From the simulation results\ all tasks were completed successfully[ These show the e.ciency and the generality of the proposed GSGM[ Robots see all tasks as only a vector summation of changes of their group shapes*there is no di}erence among the tasks to the robots[ The key of GSGM is the determination of destination points[ For some tasks\ e[g[\ load carrying and objects rearranging\ the destination points are predetermined and _xed[ But for some other tasks\ e[g[\ mobile object catching\ parts assembling\ and unknown environment searching and map making\ the destination points vary and depend upon the situation at the time of execution[ The destination points are programmed directly in the former case\ while in the latter case\ the method of determination of destination points which depends strictly upon the task\ is instructed by operators[ Since any tasks of multiple robots can be modeled as the vector summation of changes of group shapes of the robots\ GSGM can handle the modeled tasks[ The conclusion is that the proposed GSGM is an e}ective general movement model for multiple robots[ 3[ Conclusion The main objective of this paper is to realize e}ective robot cooperation[ The movement model which is used to control robots in cooperating with each other is a
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Fig[ 02[ Simulation result of cooperation among multiple robots in searching an unknown environment and making a map[
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key point of this realization[ GSGM was proposed in this paper[ It is a combination model of potential _eld\ rule based\ and communication based models[ GSGM con! siders the situation in the system and information\ transmitted through the system\ to decide the acceleration of each robot[ The model comprises _ve elements] goal\ robot\ obstacle\ velocity\ and application elements[ Furthermore\ in this paper\ the cooperative task was modeled as the vector summation of changes of group shapes of the robots[ By the evaluation of many simulation cases\ GSGM found that it could handle the tasks successfully[ Because any cooperative tasks can be modeled and GSGM can handle the tasks being able to be modeled\ the conclusion from this paper is that GSGM is a general model for a cooperative robots system[
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