Path Finding Problem and Information Support of Mobile Robots in Uncertainty

Copyright ~ IFAC Intelligent Autonomous Vehicles, Espoo. Finland. 1995

PATH FlNDlNG PROBLEM Al~D INFORMATION SUPPORT OF MOBILE ROBOTS IN UNCERTAlNTY

S.A.

Bez~ov,

A.A. Kirikbenko. A.K. Platonov. V.E. Pranicbniko,·, V.S. Yaroshcysky

Russian , Icademy o/Sciences, Kddysh Institute o{.lpplied .\/athemalics, Afiusskaya sq. 4, Moscow, 125047 Russia

Abstract: This paper shows the reasons of an incomparability of path finding algorithms in uncertainty. An example of non-monotonous dependency of algorithm efficiency upon radius of ranger action and an example of the algorithm .idaptation in unknown lCrr.nn during the motion are given. The investigation of unstable behaviour of path flnding algorithm is presented. Some resulls of the robot information processing arc briefly discussed. Wc consider the basic algorithms of data processing for laser di!.tancc measuring systems (LDMS) and their modification for ullrasonic sensors (lJDMS). Keywords: algorithms. information analysis. mobile robots. path planning. unstable algorithms.

I . How arises the incomparability of algorithms in uncertainty?

I. INTRODUCfION

The path flnding problem for a mobile robot (MR) in environment uncertainty can be divided into two stages. The first stage is to choose of a global path finding algorithm. Such choice depends upon a priori information about the terrain. the information possibilities of MR. etc.

2. How can be realised the adaptation to unknown terrain during the MR motion?

3. Why and in which situations the algorithm efficiency can be connected in non-monotonous way with the radius of ranger action?

The second stage is the realisation of the path rmding algorithm for the MR with necessary information support. This paper deals with some important elemenlS of path finding problem investigation for both stages. It incorporales and extends resuJts describes in (Kargashin et aI. , 19<) I: Kirilchenko 1991. 1993: Kononov et al.. 1993: Platonov et al.. 1994: Pranichnikov 1993). In comparison with well known investigations in this area. such as (Lurnelsky and Skewis. 1988: Pctrov 1987. Rao 1989), we suggest new directions of investigation of the path finding algorithms in unccnain environment and find an answers to the following queslions :

... What can we say about unstable algorithms in some situations when a small variation of goal position or obstacles parameters etc. bring us to incomparable (fOT example. nonhomotopic) paths?

Discussing the problems of the second stage (information support of MR activity) we briefly consider the imponance of well-matched peculiarity of the information and motion activities of MR. The basic set of data processing algorithms for range measurements is also discussed.

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2. PATH FI:"JOING ALGORln~MS. 1l-{EIR I;-.JCOMPARABlLlTY AND ADAPTATIO;-.J IN UNCERTAINTY .

Eacb algorithm (suppose) is characterised by some structural lormuJa. which defme algorithm class. In simplest cases we have two formulas: V and C. Valgorithm can be characterised by the estimation function f for the sub-goals. The function f can be constructed on the base of ~timation the distances according to the g and h functions for minimum path cost on the path-graphs. Suppose. that h is a distance from a sub-goal to the goal point and g - distance from current (not start!) position of MR to a ~ goal. We can consider such functions as g., It. g+b. (g+h)h , etc. In the structural fonnula of algorithm the estimation function is the second clement. Than the "natur.u" V-algoritlun can be presented by formula V(g+h) . In the case of C-algorithm we can fix the direction of the motion w on the obstacle boundary after first hit point. Let it be w=+(counterclockwise) and w---{c1ockwisc): C"', C-. Cw (in last case w is fixed. but its sign is not important).

Let us consider the problem of achiC\'ing the goal point G from point S on a plane environment with finite number of obsbcles. This obstacles arc insuperable for MR and "opaque" for MR-s infomtation system. Suppose for simplicity, that MR looks like a moving point and has . no a priory intormation about obstacles. The classification of path findjng algorithms in this case is the following. We can distinguish the algorithms looking at the MR information system parameters. In this case exist two extremal classes: V-algorithms and C-algorithms. In the case Valgorithm for MR the information about each point in visual vicinity of current location of MR is accessible. We introduce the relation of "visibility" : a point x is visible from the point y (and conversely) if and only if the interval (x. y) don't cross the obstacle boundaries. The visibility is a tolerance relation (it can be suggested as reflexive. symmetrical. but not transitive). In the case of C-a1gorithm we suppose. that the information system radius of action equals zero and MR can make only two operations: move directly to the goal point (if it is possible). and follow the obstacle boundar)·.

Our last assumption about C-algorithms - MR contact the first hit-point with obstacle. follow obstacle boundary and find leave-point from obsbcle to the goal direction. This leave-point must be only the obstacle vertex. Let us accept. that there is a basic set of known path finding algorithms A and a set possible "elements of problem" M. Each clement of M is cbaract.crised by a map of obstacles. start and goal points. Now we accept for simplicity, that the obstacles in the map (the terrain) consist of lines (intervals. polygons), parallel to one of the axis in coordinate system and the length of each interval is integer-valued (according to the axis step of this coordinate system). The set M now we can characterised with integral value N. which is equal to the maximal length of boundar)' rectangular frame for elements of M (each element can be placcd in the frame with side length N). The number N we shall call "the level of variety of terrain" . We suppo5C. that each algorithm from A is admissible. i.e., for each element of problem for any fixed N it prm.idcs a path from initial to a goal point.

Suppose the following : - the obstacles can be only polygons or polygonal lines. - the next sub-goal for V-algorithm after scanning the MR environment is onJy a vertex of obstacle. - the "leave" point for C-a1gorithm (i.e., the poinL where MR changes the motion mode from the obstacle boundary following to the motion to the goal) is the obstacle vertex too. The possible "leave" and "hit" points for a concrete C-algorithm we can consider as possible sub-goals. as well as all obstacle vertexes for V-algorithms. Thus the path. produced by V- or C-algorithm. presents a polygonal line. Most of known and unknown V- and C-algorithms (according with these assumptions) can be presented in the following logical scheme with -+ steps: 1. Make scanning and select the possible ~goals (for C-algorithm - switching from one motion mode to another passing hit and leave points); 2. Choicc the vertex on previous path. which had been marked "open" for the next attempt to move inside an unknown region of terrain through some sub-goal; 3. If this vertex is not currenL than return (move) into this vertex: -t. Move to unknown region of terrain: if this attempt is unsuccessfuL than mark this vertex of the path as

By these assumptions we can show the effect of algorithms' incomparability in uncertainty. It occurs. when the basic set of algorithms A and terrain have enough variety. In this conditions there is no optimal algorithm for any set (A. M). i.e. we can't find an algorithm e. which provides the shorter (or equal) path than any algorithm from A for any clement from M. That bring as necessarily to keep and use all incomparable (on the set M) algorithms from A. Now we can formulate

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Fig. I. Theorem 1. If thc basic set of algorithm A contains two algorithms ( structural formulas arc V(g+h) and Cw ) and M consists of all possible elements with the level of vari~' of terrain greater than 5, than there is no optimal algorithm for the pair (A.M).

3. THE ALGORITHM EFFICIENCY AND THE RADIUS OF RANGER ACfION. The algorithm efficiency (length of path) can be connected with the radius of rangcr action in nonmonotonous way. To explain this fact. we consider R-algorithm for path finding. In this case infonnation system of MR collects measurements in R-vicinity around the current position U of MR. Rvicinity is the subset of visual environment that consists of all points, visible from U and being on the distance less or equal R from U. Suppose, that the logical scheme of algorithm is fixed, R is the parameter that can be ehanged. The increase of R usually' reduces the length of produced path. but not always. Such increase can "open new ways", whieh can be more suitable for estimation function (whieh is based on local features of terrain). Sometimes the "new ways" can tx: worse (for example, path can lead into dead ends). Therefore we can obtain here a nonmonotonous dependence for same problem clement. as is shown on Fig. Ill: Fig. Ul.a shows clement · of problem and clusters of paths. as we increase R from I to 10: Fig. IIl.b shows the number of mcasuring seances and Fig. IIl.c - the length of paths. where I is equal to the distance between start and goal points. The start point is on the left (below). and the goal point - on the right The estimation function has two criteria (as described in previous topic) The fIrst part of the function is (g+h)/s. where s is thc distance from a current position to the goal point. the second function is cid. where c is the \'aluc of observed jump of thc distance mcasured in the \-;cinit~ of sub-goal. d - some threshold. The weights for functions were respectively (I.g and 0.1.

The proof is obvious from Fig. I. which represents the situation "balance". M consists of two elements, each clement eonsists of onc obstacle with two horizontal and one vertical intervals (lincs). The horizontal intervals have a constant length and the vertical linc - "the pointer of balance" can be removed and "close" the path to the goal point from one side (left or right). Now for onc clement of M C+ -algorithm provides shortest path and for another - V(g+h}-algorithm. But the visual vieinity in initial position is equal for both cases and therefore there is no algorithm E. which is optimal for (A. M). Wc can cx1end such results for other terrains. i.e., for the cases. when the orientation of obstacles is arbitrary. Let us consider two types of obstacles "walls" and "pits". MR is presented as a polygon. Sub-goals are not obstacle vertexes only in that case. The adaptation to a prio~ unknown (but structured terrain) can be realised by man,· ways. One of them characterised Fig. n. We use two criteria for Valgorithm: one criterion is tied with g and h.. the second - estimates opportunity to scan the environment in the sub-goal position and the width of passes. Those two cnteria are used with the weight-coefficients to construct the estimation function. The weights are mocW:ving in accordance the motion results. That procedure is the base for environment adaptation - the transfonnation of "the a priori optimism" 10 "the careful pessimism" - gIve preference to wider passes after dead cnd situations

It can be shown. that such eITects can tx: demonstrated on tht: structures only with cOn\'c\;it\ 73

4. lJt\ST ABLE DOMINATION

obstacles (sce fig. Ill). TIle reducing of algonthm efficIency by mcreasmg radius of the ranger actIOn can be demonstrated for R-algorithms in terrains with the level of \"arie~ greater or equal than 8.

In some cases path finding algorithms ean have unstable beha,iour. when small variations of the problem clement (i.e .• co-ordinates of initial or goal points or obstacles contours) bring us lO incomparable lfor example. nonhomotopic) paths. This situation can occur in points with equal estimation of possible sub-goals. Four types of unstable situations are presented on Fig. IV. three of them - for V-algorithms (a). (b). (d) and onc - for C-algorithm (c). Parts g and h in the estimation function are equal for two sub-goals (xl and x4) in the case (a). The points on obstacles. the start and the goal points are marked with x or y coordinates respectively on all figures. Namely. the point x..J has x-coordinate ..J. y-coordinate of y6 is equal to 6 and so on. We sec the unstable behaviour for Vg-, Vh-. V(g+b)-algorithms. In the situation (b) - "oblique cross" - the sum g+h is equal for two subgoals and here V(g+b)-algorithm is unstable. but Vg-. Vh-algorithms arc stable. The situation (c) "contact" is unstable for C+ -algorithm (variation of goal point MR must follow the boundary of obstacle with vertex xl). lbc situation (d) - "haunt" demonstrates bomotopic unstable behaviour for Valgorithms. Suppose. we compare two algorithms of one element of the problem m. Suppose. algorithm a is better. than algorithm b (a produces shorter path). but after variations of m. b became better. than a. Such situation is called "unstable domination". We can characterise this situation as follO\\'S. Each algorithm is described in "unstable domination" by the pare (wl. w2), where wl="+", if the path is stable: wl="", if the path is unstable: w2="+", if the length of the path is stable; w2="-", if the length of the path is unstable. The pair (+-) for w1. w2 is evidently impossible. Now the situation of "unstable domination" can be described in the terms of two such pairs. This description is called later "shon description" . The "full description" is given by triples « wl. w2),k>. where k~). ifw2="+". k=1. if w2=" -". and the length of the path after variation is longer. than before variation: 1c2=-1, if w2="-" , and the path length after variation is shorter. than before variation. Theorem 2. There arc 5 situations of "unstable domination" , which arc characterised by "shon description" and 7 situations. which arc characterised by "full description" . The proof can be given by simplest testing. The situations in "short description" are: 04: (-... ),(-); 06 : (-),(-+) ; OX : (-).(-l:

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5. fNFORMATION SUPPORT OF ManON.

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Realisation of the MR motion is closely tied with some important aspects of the selected path finding algorithms and coordination problem for the information and motion activities of MR. We suppose. that MR is provided with distance measuring systcm (DMS) - laser (LDMS) and/or ultrasonic (UDMS). Coordination conditions define the maximum possible speed of MR when some parameters of DMS and desired discreteness of scanning ranger are fixed. The discreteness should

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warranty MR the way to find all obstacles in proper types and sizes. In simplest case we have the inequality (I) 75

~:'Wmplc. a dependence of dist.ance signal 011 texture and object slope can be ye~' high. but in such environmenL as prismatic obstacles on honl.ontal fiat by using of high accuracy of system angles measuring. the resulting accuracy can be also Yery high. For such purpose wc use the liltration algorithm.. which can fix the dispersion of a distance signal in lines of distance matnx. By this characteristics wc can distinguish lines. which are hitting on the object. After using algorithms (I )-(7) in this case wc can determine position of the objcct on a flat with accuracy 1.5 cm Instcad of average distance error 20 em (distance range is 2~ m). Sce for detail (Kononov et a1.. 11)1)3: Pranichnikov.

where e IS desired discreteness on honzont.al nat of motion. n - mcasunng frequency. \" - speed of MR motion. W - angular tilt speed of LDMS finder line. H - height of pomt 0 in DMS. R - distance on the horizontal fiat from point 0 projcction to the measuring pomt on the nat. TIlis is the condition for "pit's" finding (",;th typical dimension e). In the case of lugs (stones) \\1th height h we can use this inequality together with h estimation:

h:::; (H*e)lR.

(2)

The corresponding maximum possible MR speed have variations in several times. depending upon scanning scheme during MR motion. (8arbashova et al .. 11)88).

1993),

5.2.Parti('1tiar jeatures o/ultrasonic rangers.

Some eITects and coordination conditions were investigated analytically and numerically. For example we find "measurement interference" in repeating scanning measuring points measurements are placed between measuring points of previous scanning so that the discrete step requirements can be rcdu.ccd (Kargashin et al.. 11)91). Another effect - the all possible MR speeds can be connected with scanning process frequency in nonmonotonous way with jumps. It occurs for the problem "reference point tracking" . MR navigation can use such points to calculate the coordinates of MR in current position.

The set of suggested basic algorithms is completely applicable to UDMS. but it should operate with the considerable beam angle. For example search and calculation of minimum and maximum obstacle distance in the measurements sector for LDMS is reduced to onc direct measurement of this parameters with the help of UDMS. Logical comparison of ranges and jumps of ranges in neighbour beams allow to identifY relief of objects in 64 classes. Tests of implemented ultrasonic sensors were prm-idcd on 3 and 4-wheels kJudges. 2 level control algorithm was suggested to switch the control unit of MR to follow the linear referenceobjects. Sce for detail (Pranichnikov. 1993).

5. 1 /,aser rangers Several examples of LDMS were used for information support algorithms realisation (Kononov et al.. 11)93. Platonov et al.. 1994). They were implemented in BaJtic State Technical university (Sl.-Petersburg). These LDMS have different kinematic schemes and parameters: maximum range distance 5- \(I ID. average error 5-20 cm. scanning frequencies 1.8-4 kHz etc. LDMS and algorithms were t.cstcd in laboratory equipment with 2 wheeled and 2 legged MR-kJudges. moving with path planning on the average speed 2() cm1scc. As a resulL 7 basic algorithms were constructed: () ) statistical processing; (2) threshold filtration: (3) priority processing (in this case each range measurement is assigned with some priority indication): (4) logical filtration: (5) data generalisation (conjunction of low level elements into I high level clement): (6) sharing out singular object elements (vertical edges. silboucne e(j;.): (7) improvement of distance measuring by using information about laser-Signal attributcs. such as signal fluctuation depending on texture and object slope. In the last case the characteristics of LDMS can be improved by special signal flltration . For

6 . CONCLUSIONS.

The new directions of investigation of path planning algorithms in uncertainty arc: - incomparability of algorithms; - unstable behaviour of algorithms: - dependence of the algorithm efficiency upon ranger type and complexity of terrain: - adaptation of algorithms. There arc 2 very important topics in the field of infonnation support and realisation of the robots' motion in uncertainty: - to find coordination conditions of information and motion activities of the robot: - to select the basic set of data processing algorithms.

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Function. I'roc. "~'FE 1nl. ('onl Roh. and . lulom .. Philadelphia, Pa. Vol. 2. pp. n-l-731J . Petrov AA (1987). Path Finding Algorithms for Mobile Robots. Rcviews of scicnce and technolo&,.v. In: Ser. Technical cyhernetics. VoL 21. pp. 92-lJU. VINITI, Moscow. Platonov AK.. Bczbogo\" SA. Kirilchenko AA.. Yaroshcvsky V.S. (1994). The Invcstigation of Laser Distancc Measuring Systcm. Pre-print: KcIdysh Inst. of Applied. Math.. N 9. 28 pp. Moscow. Pranichnikov V.E. (1993). Information Support and Navigation of Robototechnical Systems with Ultrasonic and Optical Sensors. Kcldysh Inst. of Applied. Math.. Moscow. Rao N.S.V. (1989). Algorithmic Framework for Learned Navigation in Unknown Terrains. Computer, VoL 22. N 6. pp. 37-43 .

Kargashin A.Yu .. t-jrilchenko ·AA. Yaroshcvsky V.s. (1991). The Estimation of Discreteness in Distance Measuring Systems for Terrain Scanning. Pre-prinl: Kcldysh Inst. of Applied Math .. N 97. 26 pp.. Moscow. Kirilchenko AA (199 I). The Basic Path Finding Algorithms in Uncertainty. Prc-print: Keldysh Inst. of Applied Math.. N 108. 24 pp.. Moscow. Kirilchenko AA (1993). The Investigation of Path Finding Algorithms in Uncertainty: Property of Incomparability and Comparison of Classes. Prc-prinl: Keldysh Inst. of Applied. Math.. N 61 , 25 pp., Moscow. Kononov O.A. Kirilchcnko A.A , Yaroshevsky v.s. (1993). Laser Distance Measuring Systems for Mobile Robots. Technology, Series FMS and Robototechnical Systems. N 1-2. pp.3944. Lwnclsky V.. Skewis T. (1988). A Paradigm for Incorporating Vision in the Robotic Navigation

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