A Neural Network Model that Calculates Dynamic Distance Transform for Path Planning and Exploration in a Changing Environment

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1 Proceedngs of the 3 IEEE Internatonal Conference on Robotcs & Automaton Tape Tawan September A Neural Network Model that Calculates Dynamc Dstance Transform for Path Plannng and Exploraton n a Changng Envronment Dmtry V. Lebedev Jochen J. Stel Helge Rtter AG Neuronformatk Faculty of Technology Unversty of Belefeld P.O.-Box Belefeld Germany Abstract In ths paper we present a neural network model that realzes a dynamc verson of the dstance transform algorthm (used for path plannng n a statonary doman). The novel verson s capable of performng path generaton for hghly dynamc envronments. The neural network has dscrete-tme dynamcs s locally connected and hence computatonally effcent. No prelmnary nformaton about the world status s requred for the plannng process. Path generaton s performed va the neural-actvty landscape whch forms a dynamcallyupdatng potental feld over a dstrbuted representaton of the confguraton space of a robot. The network dynamcs guarantees local adaptatons and ncludes a set of strct rules for determnng the next step n the path for a robot. Accordng to these rules planned paths tend to be optmal n a metrc. Smulaton results n a seres of experments for varous dynamcal stuatons prove the effectveness of the proposed model. I. INTRODUCTION One of the most mportant attrbutes of a robotc system s ts ablty to plan the paths and to navgate autonomously. At the same tme an ntellgent navgaton s characterzed by the capablty of adaptng a route dynamcally n the case of sudden appearance of other objects or obstacles. There exsts a lot of research on path plannng (see e.g. [1]-[5]). A number of neural network approaches has also been proposed to solve ths problem ([6]-[17]). The capablty of a multlayer perceptron to learn successfully the navgaton task n a maze-lke envronment has been demonstrated n [6]. In [7] a self-organzng Kohonen net wth nodes of two types has been used. In [8] a descrpton of a network wth oscllatng behavor that solves the problem of path plannng for an object wth two degrees of freedom (DOFs) formulated as a dynamc programmng task s gven. The algorthm proposed n [9] uses a set of ntermedate ponts connected by elastc strngs. Gradent forces of the potental feld generated by a multlayer neural network mnmze the length of the strngs forcng them at the same tme to round the obstacles. In [1] a multlayer feed-forward network for performng real-tme path plannng was appled. The neural network for path fndng descrbed n [11] has three layers of neurons wth recurrent connectons n the local neghborhoods. The dynamcs of the network emulates the dffuson process. Most of the known neural network approaches requre however full knowledge of envronment and can be appled only for a statonary doman. Besdes that optmalty of the path s often left out of consderaton. For a non-statonary doman a topologcally-organzed Hopfeld-lke neural network for dynamc trajectory generaton has been proposed n [15] and mproved recently n [16] but some addtonal efforts are requred for tunng the network parameters. In ths paper we present a novel neural network dynamcs for fndng a path n a dynamc world. The model s based n nterweavng manner on three paradgms ( the notaton of confguraton space as a framework for a flexble object representaton [18]; ( potental feld buldng whch s a generc and elegant method for formaton/reconstructon of a path; and ( a wave expanson mechansm that guarantees an effcent constructon of the potental feld. In comparson wth our prevous results n [19] the network dynamcs has been reconsdered smplfed and yelds now shorter and smoother paths. The model has been smulated and tested n the context of autonomous path plannng and exploraton for varous types of dynamcal changes n the envronment and has demonstrated effcent and effectve path generaton capabltes. The paper s organzed as follows. In secton 2 we descrbe the general dea of the proposed algorthm and gve a formalzaton of the problem. Secton 3 contans the descrpton of the neural network model. We llustrate smulaton results n secton 4 and conclude wth a dscusson n secton 5. A. The General Idea II. THE PROPOSED ALGORITHM The orgnal verson of the dstance transform algorthm was presented frst n [] and then exploted extensvely for path plannng and navgaton tasks n a statonary doman (see n [21]-[23]). In the essence of ths algorthm les the dea of dstance propagaton n the workspace around the goal poston such that the value of a cell after applcaton of the algorthm corresponds to the cost of the path to the goal (for more detals see [] [21]). The path tself s found by followng the steepest descent wth respect to the calculated dstance values. One could notce that the dea of dstance transform s very smlar to the generaton of dscretzed numercal potental felds (see e.g. [17] and [24]). So n [17] the model of a wave expanson neural network has been proposed that computes a dstance transform (named grd potentals ) over a dscretzed /3/$17. 3 IEEE 49

2 A A E representaton of the confguraton space of a robot. The methods [17] [21] and [24] can be characterzed as one-tme-gothrough. Ths means that the procedure of potental feld generaton s performed only one tme and s fnshed usually as soon as each poston n the whole workspace (or confguraton space) a numercal value has been assgned. Thus these methods can not be appled effectvely for a dynamc doman. In [23] e.g. a replannng of the whole path s ntated each tme when the robot encounters an obstacle. However n an envronment populated densely by obstacles or other agents the replannng of the whole path can become a restrctng factor for real-tme navgaton capabltes of a robot. For plannng a path n a tme-varyng world we propose a novel neural network model that calculates dynamc dstance transform or dynamc grd potentals. To form the desred grd potentals a wave-expanson mechansm s used. Durng the process of potental generaton the actvty s spread around the source of exctaton and the mnmum value of the generated potental feld stays always at the exctory pont whch n turn attracts the robot. The most mportant challenge dstngushng our model from the approach descrbed n [17] s the proposed neural network dynamcs whch makes an effectve combnaton of (1) repettve wave expansons and (2) rules to cope wth dynamcally-generated waves of neural actvty. These rules are based on a set of threshold-lke functons and are ncluded nto the network dynamcs to ensure the proper formaton of a dynamc dstance transform and to guarantee that a robot wll move only along a safe route. To provde the repettve wave expansons a regular exctaton source s fed at the target neuron. Ths results n orgnaton at each tme step of a new wave of neural actvty n the network feld. Neural actvty therefore changes locally whle propagatng through the network feld and adapts to the dynamcal status of envronment. Snce n the case of a statonary envronment wave fronts yeld paths whch are optmal n a metrc (see [17]) dynamcal paths whch are generated by the proposed model also tend to keep such optmalty. Consequently longer paths to the target are cut off automatcally by the algorthm. B. Formalzaton of the Problem Wthout loss of generalty we can defne the confguraton space to be a regularly dscretzed hypercube where s the number of DOFs of a robot. For a robot n the startng and the fnal confguratons denoted accordngly and are defned. Suppose at the tme there s a number of obstacles (.e. of forbdden confguratons) n. At that moment of tme postons of all obstacles n form the obstacle regon! where the obstacle coordnates n are denoted by vectors! "$#&%'#($. Let )*! + -!!! defne the confguraton of a robot n at the tme. The task s to fnd a safe (.e. a collson-free) path ) that satsfes the condtons: )*.! / A@B = < <> = < <> =@?>? =@?>? :$; < A D FG; < Fg. 1. The model of the neuron; Network neghborhood nh I. )- J! )*!-K $(L.@# #& J where. and J are the tme of start and the tme of reachng the goal respectvely. III. THE NEURAL NETWORK MODEL A. The Network Archtecture The neural network has a parallel locally connected structure of cellular type. Dependng on the dmensonalty of the network feld may consst ether of a sngle layer (for a 2D confguraton space) or a set of layers wth locally connected neurons. The arrangement of the neurons concdes wth the dscretzed representaton of.e. each dscrete poston n s assocated wth a neuron n the network feld. Fg. 1a shows the neuron model. The% -th neuron s connected wth MNO mmedate neghbors where s the dmensonalty of. We wll denote the set of neghbors of the % -th neuron as P.e. P / % % Q and an enumeraton of neurons n the local neghborhoods s fxed. A neuron neghborhood for an example 2D confguraton space s depcted n Fg. 1b. The network can be vewed as a dscrete-tme dynamcal system whch can be fully descrbed by a set of neuron state vectors SRT UV. W XY Z [ Q \ [. The frst element T of vector s the actvty level or the output of the % -th neuron whch s a real scalar quantty. The second element vector UV. W]SR^ ^ _ ` ^ ` ^ ` ^ _ X conssts of two sets of connecton weghts defnng the synaptc strengths of the connectons between neuron% and ts mmedate neghbors. Notce that ^'a ^'a ` ^ a and^ ` a are four connecton weghts between neurons % and b and that weghts ^'a and ^ a ` follow dfferent update rules. Actvty levels of the neurons n the neghborhood of the % comprse the vector Y. W]cRT T d T _ X. The neuron% s actve f T feg and nactve otherwse. Addtonally the actvty of the % -th neuron can be nfluenced by the exctory and nhbtory nputs from the neghborng neurons whch form accordngly vectors h. W Rh h _ h X andj. Wf/Rj j _ j X whose elements can be of value zero or one only. B. Path Plannng Process In ths part we present the underlyng dea of the dynamc generaton of dstance potentals and ts formal descrpton. A C 421

3 State equatons (1)-(3) contan the rules provdng the actvty evoluton of the neuron k and of the connecton weghts assocated wth the latter. l m n o*pq r'st mvu n lvm n o r pq r*p(n q wnt m r x y n z({ } ~ n o r u n l { n o r*p& r r (1) n o-p(q rfsƒ &n * n } ~ n o r r r- ˆŠ m (2) Œ n o*pq r'sl { n o r ˆŠ ] m Ž] k (3) The functons y n r and n r are defned as y n vr's Y n vr's q ] & ]š& where denotes the ntegral part of. Intally actvty levels of all neurons and all connecton weghts are zero. Avalable nformaton about the current status of the envronment s appled to the external neuron nputs. The neurons correspondng to statonary or dynamcal obstacles n are nhbted by nputs œ m*s/q. The neuron assocated wth the target poston n s excted by ts nput t mfs q and ntates the propagaton of actvty n the neural feld. As can be seen from (1) the actvty level of the target neuron s ncremented by one at each tme step and remans the global mnmum of the potental feld durng the network evoluton. The actvty of other neurons depends on the current state of the neurons n the local neghborhoods. Equatons (2) guarantee that among the ncomng connectons from the local neghbors only one wll be postve. Ths neghborng neuron s chosen by the functon * n l m } ~ ž } ~ œ } ~ r : n r's Ÿ@ Ysƒq Ÿ p& n z Ÿ r cqg š (4) Ths functon queres the neghborng neurons accordng to the preassgned enumeraton and selects the frst neuron whch corresponds to a postve value of the expresson: Ÿ $s( n œ r*u n lv n o r r u Šn l n o r Œ m n o r r x n ª-n l m n o r m Œ m n o r r r x (5) n n l m n o r r u «n lvm n o r l n o r r p Šn l m n o r r r Here ª-n l m n o r m Œ m n o r rs n lvm n o r r uv n m Œ m n o r r and functons n r Šn r «n r and n r are gven by vn r's w@ š ˆ q $ (q Šn r's q s& s& «n rfs q wn Ḡ± wn Ḡ and n vr's/q w n vr. From the formulas (5) (4) and (2) follows that the connecton weght s assgned the value one f and only f the followng condtons are fulflled: the neuron has changed ts actvty level at the prevous tme step; ths means that the neuron can be consdered as a canddate from whch the connecton weght can receve a postve value; t has zero on ts nhbtory nput;.e. the neuron does not correspond to an obstacle at the current tme step; the k -th neuron had already a non-zero actvaton at the prevous tme step and the actvty level of the neuron s not larger than that of the k -th neuron;.e. at the next tme step the actvty of neuron k wll be larger than the actvty of neuron what provdes a proper gradent formaton of the grd potentals; d) the current actvty of the k -th neuron s not zero and the Œ connecton m s not postve smultaneously;.e. f ths condton s false all connecton weghts wll receve zero value that results n zero actvty value for the k -th neuron (see (1)). Ths s done to prohbt actvty oscllatons n the local neghborhoods and to guarantee therefore a correct generaton of grd potentals. e) the neuron s actve;.e. the condtons d) and e) are fulflled also for ths neuron. If one of these condtons s false the correspondng weght Œ becomes zero. Connecton weghts (3) are responsble for storng the actvty values of the neghborng neurons calculated at the prevous tme step. Permanent exctaton of the target neuron va ts external nput t m leads at each tme step to generaton of a new wave of neural actvty n the network feld. These waves carry updated nformaton about the envronment status. Therefore the dynamcal actvty landscape accounts for changes n the envronment and adapts to them. A robot starts movng as soon as the frst wave of neural actvty has reached ts ntal poston. Due to the strct rules whch are ncluded nto the network state equatons and provde a proper potental feld buldng the next step rule for a robot becomes rather smple: t should move n the drecton of the postve weght from the neghborng neuron or formally -n o } p( *rys+ ² {N³ Y ( / m where ² { s the next poston of a robot n the workspace (or the next confguraton n ) assocated wth the neuron o } s the startng tme and / denotes the -th dscrete tme step. Ths rule ensures that a robot always moves along a safe and shortest path. IV. SIMULATION EXAMPLES In ths secton we llustrate smulaton results for varous types of dynamcal changes n the envronment. To demonstrate the dynamc nature of the proposed algorthm more clearly all the experments have been done for a pont robot n a 2D workspace that however does not restrct general applcablty of the model. Statonary obstacles n the workspace are shown on µ - plots n a lght-gray color and dynamcal obstacles are colored black. Paths of the robot are represented by contnuous curves. Black squares settlng spontaneously the workspace denote randomly appearng obstacles. and stand for Start and Target Poston. 4211

4 Fg. 2. The open gate stuaton: - two ntermedate stages; The whole path; d) Actvty landscape at the moment of reachng the target. We used for our experments a network feld consstng of ¹ (º ¹G» º ¹ ) neurons over the dscretzed workspace representaton. The borders of the workspace were treated as obstacles. For test smulatons we chose the neghborng neurons labeled as shown n Fg. 1b. Therefore whenever possble the robot prefers frst to move horzontally. A. The Open Gate Stuaton Ths model stuaton s shown n Fg. 2. The workspace s cluttered wth statc obstacles. After 5 path steps of the robot the dynamc obstacle starts to move n the drecton shown by the arrow see Fg. 2a. It stops at the poston shown n Fg. 2c leavng a small gate open. The robot traverses through the gate whle avodng random obstacles whch appear n the workspace at each tme step. The resultng path of the robot and the potental feld correspondng to the tme of reachng the goal are llustrated n Fg. 2c and 2d accordngly. B. The Closed Gate Stuaton The ntal setup for ths test example s very smlar to the open gate stuaton. The robot travels through the same ntermedate stages as shown n Fg. 2a and 2b. But n the new stuaton the movng obstacle closes the gate before the robot can pass through t see Fg. 3a Then the robot reacts dynamcally to the change n the envronment. The fnal path and the actvty landscape at the moment of reachng the target are depcted n Fg. 3a and 3b respectvely. 3 1 d) Fg. 3. The closed gate stuaton: The robot follows dynamcally another route; Actvty landscape at the moment of arrvng at the goal. C. Freezng up Dynamc Obstacles In ths model stuaton the dynamc obstacles start to move n the drecton of the arrows when the robot makes ts frst step (see Fg. 4. After path steps of the robot the obstacles are frozen as shown n Fg. 4c. The actvty landscape then adapts quckly to the status of the envronment. The resultng actvty landscape reflectng the structure of the statonary workspace at the moment of arrval at the goal s shown n Fg. 4d. The path of the robot s depcted n Fg. 4c. D. Warmng up Dynamc Obstacles In ths example stuaton the start and the target postons for the robot are as n the prevous example. The obstacles appear n the workspace n the postons shown n Fg. 5a. The robot starts movng and after 5 path steps the obstacles drft n the drectons of the arrows (Fg. 5. The robot adapts dynamcally ts path n ths complex stuaton and approaches the goal successfully (see Fg. 5c and 5d). E. Occupaton by Random Obstacles We performed wth our model a seres of experments wth randomly appearng obstacles. Some examples for 15 and 25 obstacles are llustrated n Fg. 6. These model examples demonstrate clearly the tendency of the path to be optmal n a ¼ ½ metrc. The path length ncreases gradually wth the number of obstacles. F. Autonomous Workspace Exploraton In ths example the task s to reach a goal n the envronment wth statonary obstacles. It s assumed that the robot can detect an obstacle only f the latter les mmedately n front of t. The gven workspace wth obstacles s shown n Fg. 7a. Durng the path plannng process the postons of obstacles are treated as free. Unlke the algorthm reported n [23] where the path plannng s done from scratch each tme as a new obstacle s detected by the robot n the proposed approach the dscovered obstacles are ntegrated dynamcally nto the path generaton process. Ths allows the robot to move more contnuously n real tme and not to wat untl the whole path s 4212

5 d) d) Fg. 4. Freezng up dynamc obstacles: Movng obstacles and the actvty landscape at an ntal stage of navgaton; The planned path; d) Actvty landscape at the moment of reachng the target. Fg. 5. Warmng up dynamc obstacles: Arrangement of the obstacles and actvty landscape at the start of the traverse; Reachng the goal; d) Actvty landscape at the moment of arrval at the target. replanned. The resultng path and the obstacles detected by the robot durng the traverse are llustrated n Fg. 7c. V. CONCLUSIONS We have presented a novel model of a neural network whch s capable of calculatng a dynamc dstance transform (or dynamc grd potentals) useful for route plannng n a changng envronment. The proposed neural network dynamcs combnes n an effcent manner a wave expanson mechansm wth a set of rules for detecton of the next canddate path step for a robot. Due to local nteractons between neurons and a regular exctaton at the target neuron the neural actvty landscape adapts and accounts for envronmental changes. Ths provdes a proper formaton of the grd potentals. The target pont stays always at the mnmum value of the potental feld and attracts the robot to the goal. Snce the network has a hghly parallel and locally connected structure (all the neurons n the network feld update ther states smultaneously) the generaton of grd potentals s an extremely fast process. The man propertes of the proposed neural network model are: no a pror knowledge of the envronment s needed; no learnng process s requred; the network s locally connected and hghly parallel; computatonal complexty grows lnearly n the number of neurons n the feld; tendency to gan optmal paths n a À metrc; fast actvty propagaton makes possble real-tme plannng. The presented neural network dynamcs has been tested n the context of autonomous navgaton and exploraton on varous types of complex dynamcal changes n the envronment ncludng appearance dsappearance and drft of obstacles avodance of random obstacles occupyng the workspace. It has shown both the capabltes of fast adaptaton to dynamcal changes and a fast actvty stablzaton n the absence of the latter. The planned paths are safe and have the tendency to be optmal n a À metrc. Due to the fast dynamcal updatng of potental feld robot navgates actvely wthout watng untl the envronment presents good-traversal opportuntes. Hence one can consder the proposed approach as a compromse between tendency to path optmalty and actve and moble reacton to envronmental changes. The descrbed approach could be ap Fg. 6. Plannng paths avodng randomly appearng obstacles: Wth obstacles (the path length s 18 steps); Wth 15 obstacles (the path length s 13 steps); Wth 25 obstacles (the path length s 15 steps). 4213

6 d) Fg. 7. Autonomous task-orented exploraton: The vew of the gven maze-lke workspace; Potental feld at the begnnng of the navgaton; The found path and the obstacles dscovered durng the traverse; d) Actvty landscape at the moment of comng to the goal. pled for path plannng of both moble autonomous systems and robotc manpulators. VI. ACKNOWLEDGMENTS The work of the frst author has been done wth fnancal support from the DFG (German Research Councl) the grant GRK Dmtry Lebedev would lke to thank also Prof. V.V. Mayorov for the dscusson and supportng of the deas presented n ths paper. VII. REFERENCES [1] J.C. Latombe Robot Moton Plannng Kluwer Acad. Publ [2] Y.K. Hwang and N. Ahuja Gross Moton Plannng - A Survey ACM Computng Surveys vol. 24(3) 1992 pp [3] D. Henrch Fast Moton Plannng by Parallel Processng - A Revew J. of Intel. and Robotc Syst. vol pp [4] Jacob T. Schwartz and Mcha Sharr Algorthmc Moton Plannng n Robotcs Handbook of theoretcal computer scence (ed. J. van Leeuwen) Elsever 199 pp [5] B. Kroese and J. van Dam Neural Vehcles Neural Systems for Robotcs (eds. O. Omdvar P. van der Smagt) Academc Press [6] D.C. Dracopoulos Neural Robot Path Plannng: The Maze Problem Neural Computng & Applcatons vol pp [7] Jules M. Vleugels Joost N. Kok and Mark H. Overmars Moton Plannng Usng a Colored Kohonen Network Techncal Reports Dep. of Computer Scence Utrecht Unv Report RUU-CS [8] M. Lemmon Á -Degree-of-freedom Robot Path Plannng usng Cooperatve Neural Felds Neural Comput. vol pp [9] S. Lee and G. Kardaras Collson-Free Path Plannng wth Neural Networks n Proc Int. Conf. on Robotcs and Automaton pp [1] J. Park and S. Lee Neural Computaton For Collson-free Path Plannng n Proc. 199 IEEE Conf. on Neural Networks pp [11] Th. Kndermann H. Cruse and K. Dautenhahn A fast threelayer neural network for path fndng Network: Computaton n Neural Systems vol pp [12] D.A. Ageev and A.Yu. Istratov Neural Network Implementaton for the Optmal Path Problem J. of Computer and Systems Scences Int. vol. 37(1) 1998 pp [13] G. Bugmann J.G. Taylor and M. Denham Route Fndng by Neural Nets Neural Networks (ed. J.G. Taylor) Alfred Waller Ltd 1995 pp [14] F.A. Kolushev and A.A. Bogdanov Neural Algorthms of Path Plannng for Moble Robots n Transport Systems n Proc IEEE Int. Jont Conf. on Neural Networks. [15] R. Glasus A. Komoda and S. Gelen Neural Network Dynamcs for Path Plannng and Obstacle Avodance Neural Networks vol. 8(1) 1995 pp [16] Smon X. Yang and Max Meng An effcent neural network approach to dynamc robot moton plannng Neural Networks vol. 13(2) pp [17] Ashraf A. Kassm and B.V.K. Vjaya Kumar Path Plannng for Autonomous Robots Usng Neural Networks Journal of Intellgent Systems vol. 7(1-2) 1997 pp [18] T. Lozano-Perez Spatal plannng: a confguraton space approach IEEE Transactons on Computers 1983 pp. C-32:18-1. [19] D.V. Lebedev J.J. Stel and H. Rtter A New Wave Neural Network Dynamcs for Plannng Safe Paths of Autonomous Objects n a Dynamcally Changng World n Proc. 2 Int. Conf. on Neural Networks and Applcatons pp [] R.A. Jarvs and J.C. Byrne Robot Navgaton: Touchng Seeng and Knowng n Proc st Austral. Conf. on Artfcal Intellgence. [21] A. Zelnsky Usng path transforms to gude the search for fndpath n 2D Int. J. of Robotc Research vol. 13(4) 1994 pp [22] Y.T. Chn H. Wang L. P. Tay and Y. C. Soh Vson Guded AGV Usng Dstance Transform n Proc. 1 32nd Int. Symp. on Robotcs (ISR 1) pp [23] A. Zelnsky A Moble Robot Navgaton Exploraton Algorthm IEEE Trans. of Robotcs and Autom. vol. 8(6) 1992 pp [24] J. Barraquand and J.C. Latombe Robot moton plannng: A dstrbuted representaton approach Int. J. of Robotcs Research vol. 1(6) 1991 pp