IEEE Int. Cnf. Evlutinary Cmputatin, Nagya, Japan, May 1996, pp. 366{371. 1 Evlutinary Planner/Navigatr: Operatr Perfrmance and Self-Tuning Jing Xia, Zbigniew Michalewicz, and Lixin Zhang Cmputer Science Department University f Nrth Carlina - Charltte Charltte, NC 28223, USA xia@uncc.edu, zbyszek@uncc.edu, lizhang@uncc.edu Abstract Based n evlutinary cmputatin cncepts, the Evlutinary Planner/Navigatr (EP/N) [4, 5] represents a new apprach t path planning and navigatin. Since its rst versin, the develpment f the EP/N system has been an ever living \evlutin" prcess itself: much new develpment and further research has been made [8, 7] t fulll the EP/N prmise f being able t (1) accmmdate dierent ptimizatin criteria, (2) achieve bth near-ptimality f paths and high planning eciency, (3) be exible t changes, and (4) be rbust t uncertainties. A mre imprtant prmise f the EP/N is its ability fr perfrmance self-tuning t adapt t dierent task envirnments, mstly thrugh the adaptiveness f its genetic peratins. This paper intrduces a methdlgy t measure the verall perfrmance f the EP/N peratrs and demnstrates hw such a measure, called `perfrmance index' fr each peratr, can be used t make the EP/N adaptive. 1 Backgrund The mtin planning prblem fr mbile rbts is typically frmulated as fllws [9]: given a rbt and a descriptin f an envirnment, plan a path f the rbt between tw specied lcatins, which is cllisin-free and satises certain ptimizatin criteria. Althugh a great deal f research has been dne in mtin planning and navigatin (see [9, 2] fr surveys), cnventinal appraches tend t be inexible in respnding t (1) dierent ptimizatin gals and changes f gals, (2) dierent envirnments r changes and uncertainties in an envirnment, and (3) dierent cnstraints n cmputatinal resurces (such as time and space). Traditinal planners either try t search fr the ptimal path based n sme xed criteria (mst cmmnly the shrtest path) by any csts r simply d nt try t ptimize a path. There is a need t have a general, exible, and even adaptive planner capable f meeting any changes in requirements and envirnments. The EP/N system [4, 5, 7, 8] has been develped t meet such a need, inspired by the fllwing ideas/bservatins: (1) randmized search can be the mst eective in dealing with NP-hard prblems and in escaping lcal minima; (2) parallel search actins nt nly increase speed but als prvide grund fr interactins amng search actins t achieve even greater eciency in ptimizatin; (3) creative applicatin f the evlutinary cmputatin cncept t incrprate heuristic knwledge is mre eective in slving practical prblems than dgmatic impsitin f a standard algrithm; (4) intelligent behavir is the result f a cllectin f simple reactins t a cmplex wrld; (5) a planner can be greatly simplied, much mre ecient and exible, and increase the quality f search, if search is nt cnned t be within sme xed abstract map structure; (6) a planner is mre useful t be able t accmmdate dierent and changing ptimizatin gals. The EP/N embdies the abve ideas by cmbining the cncept f evlutinary cmputatin with prblem-specic chrmsme structures and genetic peratrs. Unlike many ther planners which requires a discretized map fr search, the EP/N simply \searches" the riginal and cntinuus envirnment t generate paths. In additin, there is little dierence between -line planning and n-line navigatin fr the EP/N. In fact, the EP/N unies -line planning and n-line navigatin with the same evlutinary algrithm and chrmsme structure. In the evlutinary algrithm f the EP/N, a chrmsme represents a path, cnsisting f straight-line segments, as the sequence f knt pints (i.e., intersectins between tw segments) r ndes n the path (Figure 1). Each nde, apart frm the pinter t the next nde, cnsists f x and y crdinates f the knt
pint and a state variable b, prviding infrmatin such as (1) if the knt pint is feasible (i.e., utside bstacles) r nt, and (2) if the path segment cnnecting the knt pint t the next knt pint is feasible (i.e., withut intersecting bstacles) r nt. Thus, a path (r chrmsme) can be either feasible r infeasible. A feasible path is cllisin-free, i.e., has nly feasible ndes and path segments. A path (r chrmsme) x y b x y b 1 1 1... n n n Figure 1: A chrmsme representing a path can have a varied number f ndes. The initial ppulatin f chrmsmes is randmly generated s that each chrmsme has a randm number f ndes and randmly-generated crdinates fr each nde. Chrmsmes are then evaluated and selected (based n tness) t be applied by ne f eight genetic peratrs fr pssible imprvement. This sequence f evaluatin and applicatin f genetic peratrs crrespnds t a single generatin in the evlutin prcess. The prcess terminates after sme number f generatins, which can be either xed by the user r determined dynamically by the prgram itself, and the best chrmsme represents the near-ptimum path fund. Tw key factrs aect the perfrmance f the EP/N: its evaluatin functin and its peratrs. Our design f the evaluatin functin fr path tness has gne thrugh a lng prcess f develpment as detailed in [7, 8]. Since a path can be either feasible (i.e., cllisin-free) r infeasible, we adpt tw separate evaluatin functins, eval f and eval u, t handle the feasible and infeasible cases, respectively. Fr feasible paths, ur current eval f is designed t accmmdate three dierent ptimizatin gals: shrtness, smthness, and clearness (i.e., away frm bstacles) f a path. Specically, eval f is a linear cmbinatin f these three factrs. Fr infeasible paths, ur design f eval u takes int accunt several factrs: the number f intersectins f a path with bstacles, the depth f intersectin (i.e., hw deep a path cuts thrugh bstacles), the rati between the numbers f feasible and infeasible segments, the ttal lengths f feasible and infeasible segments, and s n, as detailed in [8]. In ranking all paths, we assume that the wrst feasible path is better (r tter) than the best infeasible path. Our design f the EP/N's genetic peratrs has als gne thrugh numerus changes and new develpments t take the maximum advantage f heuristic knwledge. This resulted in the eight genetic peratrs used in the current system. Hwever, an imprtant issue that has nt been addressed s far is hw t apply these peratrs eectively in dierent task envirnments, in ther wrds, hw t determine the prbabilities gverning their peratins, and furthermre, hw t make the EP/N capable f self-tuning these prbabilities s that the system is adaptive t dierent task envirnments r changes in task envirnments. In this paper, we fcus n this issue by intrducing an autmatic methd t measure the perfrmance f each peratr, taking int accunt the peratr's eectiveness in imprving paths and its cntributin t system eciency. We then demnstrate hw such a measure, called perfrmance index, can be used t tune the prbabilities f the peratrs twards better system perfrmance. It is wrth nting that there have already been attempts t explre certain self-adaptive features in evlutinary systems, as surveyed in [1]. Hwever, ur wrk is unique in that it cvers many dierent peratrs and thrughly cnsiders the inuence f peratin time and the eect f ne peratin upn anther (i.e., the interactin f results), which n existing wrk ever cnsidered 1. 2 Genetic Operatrs Nw we intrduce each f the eight peratrs, which are als illustrated in Figure 2: Crssver: it recmbines tw (parent) paths int tw new paths. The parent paths are divided randmly int tw parts respectively and recmbined: the rst part f the rst path with the secnd part f the secnd path, and the rst part f the secnd path with the secnd part f the rst path. Nte that there can be dierent number f ndes in the tw parent paths. Mutate 1: it is used fr ne tuning nde crdinates in a feasible path fr shape adjustment. The peratr randmly adjusts nde crdinates within sme lcal clearance f the path s that the path remains feasible afterwards. Mutate 2: it is used fr large randm change f nde crdinates in a path, which can be either feasible r infeasible. Insert-Delete: it perates n an infeasible path by inserting randmly generated new ndes int infeasible 1 In additin, existing wrk fcused usually n a single peratr and ccasinally n tw peratrs at mst. 2
the whle evlutin prcess. Clearly dierent values f these prbabilities aect the verall perfrmance f the EP/N. In the next sectin, we intrduce a systematic methd which enables self-tuning f these prbabilities t achieve the best results. 3 Operatr Perfrmance Index Crssver Mutatin_1 Mutatin_2 Insert_Delete We evaluate the perfrmance f an peratr taking int accunt three essential aspects: 1. its eectiveness in imprving the tness f a path, 2. its peratin time (r time cst), and Delete Swap Smth Repair Figure 2: The rles f the genetic peratrs path segments and deleting infeasible ndes (i.e., path ndes that are inside bstacles). Delete: it deletes ndes frm a path, which can be either feasible r infeasible. If the path is infeasible, the deletin is dne randmly. Otherwise, the peratr decides whether a nde shuld denitely be deleted based n sme heuristic knwledge, and if a nde is nt denitely deletable, its deletin will be randm. Swap: it swaps the crdinates f randmly selected adjacent ndes in a path, which can be either feasible r infeasible. Smth: it smths turns f a feasible path by \cutting crners", i.e., fr a selected nde, the peratr inserts tw new ndes n the tw path segments cnnected t that nde respectively and deletes that selected nde. The ndes with sharper turns are mre likely t be selected. Repair: it repairs a randmly selected infeasible segment in a path by \pulling" the segment arund its intersecting bstacles. It can be seen that except fr the purely randm peratrs Crssver and Mutate 2, all the ther peratrs (which are varied frms f mutatins) are designed with sme heuristic knwledge t make them mre eective. Nte that since mst knwledge needed is available frm the evaluatin f path tness (Sectin 1), the peratrs mstly use the knwledge with little extra cmputatin. The ring prbability p i (i = 1; :::; 8) f each peratr gverns the cntributin r rle f the peratr t 3. its peratin side eect t future generatins. While the rst tw aspects are self-explanatry, the third aspect refers t the fact that ve peratrs, i.e., crssver, insert-delete, delete, smth, and repair, tend t change the number f ndes (r the length) f a chrmsme after their applicatin. Since the length f a chrmsme aects bth the prcessing time and the strage space needed by the chrmsme the mre ndes are in the chrmsme, the mre space and time (i.e., evaluatin time and ften peratin time) are needed, if an peratr alters the number f ndes in a chrmsme, the eect will be felt in future prcessing. Such eect can be either psitive r negative n the prcessing cst f future generatins 2, depending n if the peratr reduces r increases the number f ndes in the chrmsme. Nte that including the last tw aspects in evaluating peratrs is particularly useful when cnstraints n peratin resurces (i.e., time and space) fr the EP/N are stringent. Nw we describe the perfrmance measures in detail. The three aspects are rst measured individually and then cmbined t frm a cmpund perfrmance index fr an peratr. Since the rle f an peratr ften varies in dierent stages f an evlutin prcess (e.g., sme peratrs apply nly t infeasible paths while sme nly apply t feasible nes), each aspect is measured as a functin f generatin interval [T1; T2], where T1 and T2 are the starting and ending generatins f the interval. Fr an peratr i, i = 1; :::; 8, its eectiveness in imprving the tness f a path is measured by the rati e i (T1; T2) between the number f times it imprves a path and the ttal number f times it is applied; 2 It is imprtant t dierentiate the prcessing cst (in terms f time and space) and the tness f a chrmsme; the latter is ften imprved as the chrmsme has mre ndes (such as after the peratr repair r smth is applied). 3
its peratin time t i (T1; T2) is measured as the average time per its peratin ; its peratin side eect s i (T1; T2) is measured as the average time cst f all peratrs n the average change f ndes by the peratr i: s i (T1; T2) = n i t(n) n where n i is the average change in the number f ndes f a chrmsme by peratr i per its peratin during the generatins in [T1; T2], such that n i is negative if the number f ndes is decreased n average and is psitive therwise, n is the average number f ndes in a chrmsme ver the generatins in [T1; T2], and t(n) is the weighted average peratin time (n an average chrmsme) f all peratrs during [T1; T2]: 8X m i t(n) = i=1 T2? T1 t i(t1; T2) where m i is the number f times (i.e., generatins) the peratr i is applied during [T1; T2]. Nte that the frmulatin f s i (T1; T2) takes int accunt the fact that the nde number change in a chrmsme by peratr i has an eect n any future peratin, nt necessarily by the same peratr i. The verall perfrmance f an peratr i is measured by the fllwing perfrmance index I i (T1; T2), which cmbines the three aspects f perfrmance: I i (T1; T2) = e i (T1; T2) + c t i (T1; T2) + s i (T1; T2) where c 0 is a small cnstant. Nte that greater value f I i means better perfrmance. In additin, when s i (T1; T2) is negative (i.e., when n i is negative), it cntributes psitively t I i, which can be shwn t be nn-negative. The peratr perfrmance index I i has great significance because f the fllwing: It can be autmatically cmputed by the EP/N since it is based n statistics which the EP/N can accumulate during its run. Thus, it can be used by the EP/N fr autmatic determinatin f the peratr prbability p i, dened as: I i p i = P 8 (1) i=1 I i Since I i is a functin f generatin interval [T1; T2], fr dierent generatin intervals, the EP/N can cmpute dierent I i 's and accrdingly dierent p i 's. That is, a lk-up table that maps dierent generatin intervals t dierent peratr prbabilities can be built by the EP/N autmatically. Next, the peratr prbabilities can be changed during dierent stages f evlutin t achieve greater eectiveness and eciency. Mre imprtantly, the EP/N can be self-adaptive as fllws. Let T be a suciently small number f generatins. Assign all initial peratr prbabilities randmly (fr example, unifrmly). After the rst T generatins, use the cmputed I i (0; T ), i = 1; :::; 8, t cmpute new prbabilities p i (I i ) and use the new prbabilities in the next T generatins. Afterwards, cmpute the next I i (T; 2T ) and again reset the prbabilities accrdingly. Repeat the prcedure until the whle evlutin prcess terminates. 4 System Perfrmance Measures We use the fllwing measures t evaluate the perfrmance f the whle EP/N system ver [0; T ] generatins: eectiveness index in terms f the average path cst avg T r the best path cst best T in the ppulatin f the nal generatin T. eciency index in terms f the prduct f the average path cst avg T r the best path cst best T in the nal generatin T and the ttal time t T spent ver [0; T ] generatins: avg T t T r best T t T. Clearly smaller values f bth indices mean better effectiveness and better eciency respectively. T imprve system perfrmance is t reduce the values f thse indices. 5 Experiments and Results We implemented the prcedure f cmputing the peratr perfrmance indices I i 's in the EP/N. We then did tw kinds f experiments n the EP/N fr dierent tasks in dierent envirnments (Figure 3): A. Optimizing peratr prbabilities based n I i 's. First, we assigned equal prbabilities t all peratrs and run the EP/N fr sme reasnable number f generatins T. We divided the number f generatins int several intervals, and during the run, we let the 4
EP/N cmpute the I i 's and, by equatin (1), the crrespnding peratr prbabilities p i 's fr each interval f generatins. Then, we used the cmputed peratr prbabilities (fr dierent generatin intervals) t rerun the EP/N fr the same number f generatins as in the rst run and cmpared the system eectiveness index and eciency index with thse achieved in the rst run. B. Adapting peratr prbabilities based n I i 's. We used the same T number f generatins t run the EP/N as in the experiment A and als divided the number f generatins int the same intervals as in A. T begin the run, we let the EP/N assign equal prbabilities t all peratrs initially. Then, after the rst interval f generatins, the EP/N cmputed the crrespnding I i 's and prbabilities p i 's (by equatin (1)) and reset the peratr prbabilities t the newly cmputed p i 's t run the next interval f generatins. At the end f interval 2, the EP/N again reset the peratr prbabilities based n the I i 's crrespnding t that interval and used the new prbabilities t run interval 3, and s n. Thus, adaptiveness is achieved by applying the prbabilities cmputed based n the peratr perfrmance in generatin interval n t the next interval n + 1. We nally cmpared the system eectiveness index and eciency index achieved frm such adaptive run f the system with thse achieved by the run with equal peratr prbabilities. Tables 1 and 2 shw sme f ur results, where, fr each task envirnment in Figure 3, T = 400 and was divided int 4 intervals f 100 generatins each. Als, we chse the ppulatin size t be 30. We did every experiment fr 30 times and averaged the utputs. As expected, in mst f the cases, values f bth system eectiveness and eciency indices are reduced, which means that we can achieve better eectiveness and eciency by ptimizing the peratr prbabilities r making the system t adapt the peratr prbabilities based n the perfrmance index I i 's f the peratrs. Nte als that, thugh the methd A (i.e., experiment A) yielded greater eectiveness in many cases than the methd B did, which was what we expected, the methd B achieved better results in bth eectiveness and eciency in the cmplex task envirnment 5 (the last rws f Tables 1 and 2). This fact is very remarkable since the methd A is almst twice as expensive as the methd B in that it requires an initial run f the EP/N (with peratr prbabilities being equal) fr the entire T generatins, which the methd B des nt need. The pwer f adaptiveness (in methd B) is quite bvius. Hwever, it is easy t ntice the increases in the ef- ciency index fr the task envirnment 3 (the middle ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) Figure 3: Task envirnments and typical paths fund by EP/N 5
Table 1: Gain n system eectiveness against the case with equal peratr prbabilities env avg T %change best T %change A B A B 1-3.27% -2.37% -1.64% -1.04% 2-15.14% -6.06% -9.89% -2.02% 3-6.05% -1.82% -3.61% -0.57% 4-7.69% -4.34% -5.74% -2.80% 5-6.02% -6.52% -4.97% -6.51% Table 2: Gain n system eciency against the case with equal peratr prbabilities env avg T t T %change best T t T %change A B A B 1-13.97% -17.25% -12.52% -16.12% 2-11.71% -6.36% -6.24% -2.34% 3-0.20% 13.11% 2.38% 14.55% 4 3.99% -4.50% 6.19% -2.96% 5-9.39% -15.57% -8.38% -15.56% rw f Table 2), which means decreases in the system eciency. We currently d nt have a gd explanatin fr that. Cmparing envirnments 3 and 5, we see that the bjects in 3 are mre cmplex than thse in 5, and cmparing envirnments 3 and 4, we see that the diversity f feasible paths in 3 is much greater than that in 4. These may cntribute t the phenmenn. The phenmenn may als suggest that fr envirnment 3, it is better t use dierent T r dierent intervals frm what were used. There is denite a need fr further investigatin (see next sectin). 6 Cnclusins We intrduced a strategy fr autmatically determining peratr prbabilities in the EP/N by the system itself based n the perfrmances f the genetic peratrs t achieve greater system eectiveness and eciency. The strategy can be used fr ptimizing peratr prbabilities, and mst imprtantly, fr enabling the peratr prbabilities t self-adapt. Experimental results cnrmed the usefulness f the strategy and als demnstrated the impressive pwer f adaptiveness. Mre signicantly, the strategy is very general and can be used in any evlutinary system. Our further research includes studying hw dierent selectins f generatin intervals (i.e., [T1; T2]'s) impact the system perfrmance in bth ptimizing and adapting peratr prbabilities and hw such selectins are aected by dierent characteristics f task envirnments. In ther wrds, we will try t address questins such as hw frequently the peratr prbabilities shuld be adjusted r adapted and whether the generatin intervals shuld als be self-adaptive t t dierent envirnments. References [1] Angeline, P.J., \Adaptive and Self-Adaptive Evlutinary Cmputatin," Cmputatinal Intelligence, IEEE Press, 1995, pp.152{161. [2] Latmbe, J.C., Rbt Mtin Planning. Kluwer Academic Publishers, 1991. [3] Lin, H.-S., \Dynamic Path Planning fr a Mbile Rbt Using Evlutin Prgramming", Master Thesis, UNCC, 1993. [4] Lin, H.-S., Xia, J., and Michalewicz, Z., \Evlutinary Navigatr fr a Mbile Rbt," Prc. IEEE Int. Cnf. Rbtics & Autmatin, San Dieg, May 1994, pp. 2199-2204. [5] Lin, H.-S., Xia, J., and Michalewicz, Z., \Evlutinary Algrithm fr Path Planning in Mbile Rbt Rbt Envirnment," Prc. IEEE Int. Cnf. Evlutinary Cmputatin, Orland, Flrida, June 1994, pp. 211-216. [6] Michalewicz, Z., Genetic Algrithms + Data Structures = Evlutin Prgrams. Third editin, Springer, 1996. [7] Michalewicz, Z., and Xia, J., \Path Evaluatin in Evlutinary Planner/Navigatr," Prc. IEEE/SOFT Int. Wrkshp n Bi. Inspired Evlutinary Systems, Tky, Japan, May 30-31, 1995, pp. 45-52. [8] Xia, J., \Evlutinary Planner/Navigatr in a Mbile Rbt Envirnment," t appear in the Handbk f Evlutinary Cmputatin, (T.Back, D. Fgel, and Z. Michalewicz, Editrs), Oxfrd University Press and Institute f Physics Publishing. [9] Yap, C.-K., \Algrithmic Mtin Planning", Advances in Rbtics, Vl.1: Algrithmic and Gemetric Aspects f Rbtics, J.T. Schwartz and C.- K. Yap Ed., Lawrence Erlbaum Assciates, 1987, pp. 95-143. 6