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Choosing Good Distance Metrics and Local Planners forProbabilistic Roadmap Methods
Nancy M. Amato, O. Burchan Bayazit, Lucia K. Dale,Christopher Jones, and Daniel Vallejo
AbstractThis paper presents a comparative evaluation of differentdistance metrics and local planners within the context of probabilisticroadmap methods for planning the motion of rigid objects in three-dimen-sional workspaces. The study concentrates on cluttered three-dimensionalworkspaces typical of, for example, virtual prototyping applications suchas maintainability studies in mechanical CAD designs. Our results includerecommendations for selecting appropriate combinations of distancemetrics and local planners for such applications. Our study of distancemetrics shows that the importance of the translational distance increasesrelative to the rotational distance as the environment becomes morecrowded. We find that each local planner makes some connections thannone of the others doindicating that better connected roadmaps willbe constructed using multiple local planners. We propose a new localplanning method we call rotate-at- that often outperforms the commonstraight-line in C-space method in crowded environments.
Index TermsDistance metrics, local planners, motion planning, proba-bilistic roadmaps.
Automatic motion planning has application in many areas such asrobotics, virtual reality systems, and computer-aided design. Althoughmany different motion planning methods have been proposed, most arenot used in practice since they are computationally infeasible exceptfor some restricted cases, e.g., when the robot has very few degrees offreedom , . For this reason, attention has focused on random-ized methods, such as randomized potential field methods, e.g., .
Recently, a class of randomized motion planning methods, calledprobabilistic roadmap methods(PRMs), has gained much attention(see, e.g., , , , , ). These methods use randomiza-tion during preprocessing to construct a graph in C-space (aroadmap).Queries are processed by connecting the initial and goal configurationsto the roadmap, and then finding a path in the roadmap between thesetwo connection points. PRMs have been shown to perform well inpractice, answering difficult queries in fractions of seconds , .
Manuscript received May 19, 1998; revised June 28, 1999 and March 30,2000. This paper was approved for publication by Associate Editor J. Laumondand Editor V. Lumelsky upon evaluation of the reviewers comments. The workof O. B. Bayazit was supported in part by the Turkish Ministry of Education.The work of L. K. Dale was supported in part by a GE Foundation Graduate Fel-lowship and a Department of Education GAANN Fellowship. The work of D.Vallejo was supported in part by a Fulbright-CONACYT Scholarship. This workwas supported in part by a National Science Foundation (NSF) CAREER AwardCCR-9624315, by the NSF under Grant IRI-9619850, and by the Texas HigherEducation Coordinating Board under Grant ARP-036327-017. This paper waspresented in part at the IEEE International Conference on Robotics and Au-tomation, 1998.
N. M. Amato, O. B. Bayazit, and L. K. Dale are with the Department of Com-puter Science, Texas A&M University, College Station, TX 77843-3112 USA(email: email@example.com; firstname.lastname@example.org; email@example.com).
C. Jones was with the Department of Computer Science, Texas A&M Uni-versity, College Station, TX 77843-3112 USA. He is now with Sandia NationalLaboratories, Albuquerque, NM USA (email: firstname.lastname@example.org).
D. Vallejo is with the Department of Computer Science, Texas A&M Univer-sity, College Station, TX 77843-3112 USA, on leave from Universidad de lasAmericas, Puebla, Mexico (email: email@example.com).
Publisher Item Identifier S 1042-296X(00)06955-X.
1042296X/00$10.00 2000 IEEE
IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 4, AUGUST 2000 443
PRM roadmap construction is generally performed in two stages:node generation and connection. In the node generation stage, colli-sion-free configurations of the robot are generated according to somesampling strategy in C-space (e.g., uniformly , near constraint sur-faces , , or on the medial axis ). In the connection stage, con-nections (edges) are made between node pairs if a path connecting themcan be found by a local planning method.
Although PRMs may vary in terms of high-level node generationand connection strategies, they all make heavy use of primitive op-erations such as collision detection,local planners (for connectingroadmap nodes), anddistance computations(to select which connec-tions to attempt, since it is infeasible to try them all). Thus, the choiceof appropriate techniques for these operations can crucially impact theefficiency and success of the PRM.
This paper presents a comparative evaluation of distance metricsand local planners in the context of PRMs. Our study concentrateson motion planning for rigid objects in cluttered three-dimensionalworkspaces typical, for example, of virtual prototyping applicationssuch as maintainability studies in mechanical CAD designs .Such applications present difficult motion planning problems whichtypically require navigating narrow passages in the configurationspace . We believe such problems are amenable to solution byPRMsbut this will require a fine-tuning of every PRM component,including primitive operations such as distance computations andlocal planners.
II. DISTANCE METRICS
Distance metrics are used in PRMs to determine which nodes oneshould attempt to connect using a local planner. A good metric willlimit the number of calls to the local planner by classifying enoughconnectable nodes as close. It must also be fast to compute, since dis-tance calculations are one of the most numerous operations in a PRM.
A. C-Space Representation
We represent configurations of rigid objects in three-space by six-tuplesci = (xi1; x
3), where the first three coordinatesdefine the position and the last three the orientation in Euler angles.The orientation coordinates are represented as values in [0, 1). Orien-tational differences are measured in the shortest direction. To obtaingeneralizable results, we normalize the orientation coordinates [range[0, 1)] with respect to the position coordinates (no fixed range); see for details.
B. Distance Metrics Evaluated
Our study considered five C-space and two workspace distance met-rics. See Table I; times shown are averages of 10 000 computations.
Five C-space metrics were selected for evaluation based on effective-ness and efficiency concerns (see, e.g.,  and ). TheEuclideandistance in C-space considers C-space as a Cartesian space and givesboth position and orientation the same importance. Thescaled Eu-clideandistance changes the relative importance of the position andorientation components through the scale parameters. We note thatthe value of the metric parameters will depend on the representationchosen for C-space. TheMinkowskidistance is the generalized form ofthe Euclidean distance which uses a parameterr in place of the 2; aswith Euclidean, both position and orientation are given the same impor-tance. Our so-calledmodified Minkowskimetric distinguishes betweenthe position and orientation coordinates using the parametersr1 (posi-tion) andr2 (orientation). TheManhattanmetric is the usual Manhattandistance inIR6. Note the Minkowski metric approaches the Manhattandistance asr tends to infinity.
TABLE IDISTANCE METRICSSTUDIED
The workspace metrics we chose to study are both simple metricsbased on Euclidean distances in the workspace. The first is the Eu-clidean distance between thecenter of massof the robot in the twoconfigurations (the center of mass is approximated by averaging allobject vertices). Although this is a very rough estimation, it is fast andit gives physical distance in workspace. The second workspace metricmakes use of thebounding boxof the robot, and finds the maximumEuclidean distance between any vertex of the bounding box in one con-figuration and its corresponding vertex in the other configuration. Notethat while the center of mass metric is invariant with respect to orienta-tion, the bounding box metric is not. Hence, while the scaled Euclideanand modified Minkowski metrics enable us to examine the importanceof translation and rotation in C-space, these metrics will help us studythese issues in the workspace.
Although there are many other types of metrics, such as Riemannianmetrics , Hausdorff distance , rotation distance , growthdistance , or the minimum distance between any point on the robotin the two configurations (e.g.,  and ), they were not selectedbecause they were either too expensive for use in PRMs or were onlyapplicable to convex objects.
III. L OCAL PLANNERS
Local planners are used in PRMs to make connections betweennodes when building the roadmap. They must be fast (so many con-nections can be attempted) and deterministic (so paths dont have to besaved).
A. Local Planner Resolution Issues
Start and goal configurations of the robot are denoted byc1 andc2,respectively. The paths tested by our local planners consist of a se-quence of configurations that differ from their predecessors and suc-cessors by at most some fixedresolution(in at least one coordinate).The resolution differed for position and orientation coordinates, andalso varied according to the environment. Given(c1; c2), and a reso-lution for each coordinate, our planners calculate anincrement vectorI which is used to determine the neighboring configurations tested bythe planners.
B. Local Planners Evaluated
Our first planner is the commonstraight-line in C-space method(see, e.g., ), which interpolates without bias along a six-dimen-sional straight line from configurationc1 to c2, and checks all points atsome fixed resolution on that line for collision.
We call our second (parameterized) family of local plannersrotate-at-s, 0 s 1. This planner first translates fromc1 to anintermediate configurationc0, rotates to a second intermediate config-urationc00, and finally translates toc2. The parameters represents thefractional part of the translational distance betweenc1 andc2 that therobot travels fromc1 to c0. The straight-line planner is used to plan
444 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 4, AUGUST 2000
Fig. 1. Environments: (a) six-cube (hard) and (b) alpha-puzzle (hard).
between (c1; c0), between(c0; c00), and between(c00; c2). Note thatthe paths often will have smaller swept-volumes than the straight-linepaths. Our study considereds = 0, (1=2), 1.
We also study two A*-like planners:A*-clearanceandA*-distance.The basic A* strategy is to compute a set of neighbors ofc1, selectthe most promising neighborc0, and iterate withc0. The process ter-minates whenc2 is reached, no further movement is possible, or aftersome set number of iterations (see, e.g.,  and ). To make ourA*-like methods faster, we limit the number of steps to some small mul-tiple (e.g., six) of the steps taken by the straight-line planner, and weconsider just three neighborsconfigurations where: i) both the posi-tion and orientation; ii) only the position; and iii) only the orientation,coordinates are incremented toward the goal. Both methods move toneighbor i) if it is collision-free. Otherwise, A*-clearance selects theneighbor with maximum clearance from the workspace obstacles, andA*-distance selects the neighbor that has minimum distance to the goal.
IV. EXPERIMENTAL DESIGN
Our experiments were coded in C++ and conducted on SGI O2 work-stations. Although there exist several collision detection algorithms (, , ), we used the C-space toolkit  since it was avail-able to us and satisfied our computational requirements. In this paper,we show only an abstract of the results. Complete experimental resultswith more environments can be found in . Intuitively, we expect theA*-like planners to make more connections, but to require more time,than the other local planners. We also expect the relative importanceof the rotational and translational distances to depend on the environ-ments.
Our study considers two basic environments representative of clut-tered three-dimensional workspaces, and three difficulty levels of eachbasic environment.
The first environment consists of seven unit cubes (12 triangleseach); six obstacle cubes and one movable cube [see Fig. 1(a)]. Thecenters of the obstacle cubes were placed on three parallel planes,one each on the front and back planes, and four in the middle plane,arranged so that they surround a cubical region centered on the middleplane. The hardness of the problem was controlled by varying the
TABLE IILOCAL PLANNER CONNECTIONS BYENVIRONMENTS
TABLE IIICOMPUTATIONAL REQUIREMENTS OFLOCAL PLANNING
distance between the front, middle, and back planes: 1.1 (hard), 1.6(moderate), and 2 (easy).
The second environment contains two twisted ( shaped) tubes(1008 triangles each); one obstacle tube and one movable tube [seeFig. 1(b)]. The reader might be familiar with the puzzle where theobjective is to separate the intertwined tubes. The hardness of thisproblem was controlled by scaling the obstacle tube in one dimension,which widened the gap between the two prongs of the. The scalefactors were 1 (hard), 1.5 (moderate), and 2.5 (easy).
To calibrate the relative difficulty of our environments, we comparedthe total number of connections made in each version. Table III showsthese statistics for three local planners; the other planners showed sim-ilar trends.
Our experiments were designed to: i) select parameters for distancemetrics; ii) select metrics for local planners; iii) select local plannersfor environments; and iv) study the benefits of using multiple localplanners.
We generated 600 free configurations (RdmpCfgs) near C-obstaclesurfaces (simulating roadmap nodes). We also generated 100 free con-figurations (TestCfgs ) as test nodes; 50 of theTestCfgs werenear C-obstacle surfaces and 50 were generated at random (generallynot near C-obstacles, referred to as free configurations). We used themethod described in  to generate configurations near the surfacesof C-space obstacles, and the method described in  for free spacenodes.
For each local planner, we tried to connect each configuration inTestCfgs to every configuration inRdmpCfgs. To rate a metric (fora given local planner), we computed the distances using that metric be-tween each node inTestCfgs and every node inRdmpCfgs, sortedthese distances, andanalyzed those connections made to the closestknodes. In , we analyze results fork = 25; 50; 100; in this paper, weshow only the results fork = 25 since they are the most relevant toPRMs.
V. EXPERIMENTAL RESULTS
A. Computational Requirements
Since PRMs perform a huge number of distance computations andlocal planning queries, their computation times are important factors to
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TABLE IVBEST SCALED EUCLIDEAN AND MODIFIED MINKOWSKI VALUES
consider when choosing among them. Distance metric evaluation timesare shown in Table I; values are averages of 10 000 computations.
Some times for representative local planning queries are shown inTable II; values shown are averages of approximately 100 computa-tions. For these tests, we considered the hard alpha-alpha environment.We generated tenStartCfgs of the robot on C-obstacle surfaces.For eachcs StartCfgs we randomly generated 20GoalCfgs atapproximately the same distance fromcs (so mostGoalCfgs are re-moved from C-obstacle surfaces). We then tried to connectcs to eachcg GoalCfgs using each local planner. We report the successful andunsuccessful queries separately in Table II. Running times are averagesover all relevant connection attempts. The average number of collisiondetection calls for the set of 200 local planning queries (successful andunsuccessful) in the hard alpha puzzle environment is shown in TableIII. As can be seen, the A*-like local planners are about three timesslower than the other planners, while the straight-line planner is slightlyfaster than therotate-at-s planners. Table II also shows the averagefailing time for the local planners. Since there are many unsuccessfulconnection attempts in a typical PRM application, it is desirable for alocal planner to fail fast.
B. Selecting Parameters for Distance Metrics
Three of our metrics require parameters: scaled Euclidean (s),Minkowski (r), and modified Minkowski (r1, r2, andr3). To deter-mine good values for these parameters, we tested several and selectedthe best for further evaluation.
For each local planner, we evaluated each distance metric param-eter. We defined a scoring mechanism intended to give higher scoresto parameter values which identified more close nodes that could beconnected by that local planner. For eachTestCfg , theRdmpCfgswere partitioned into five sets according to relative distance as deter-mined by the metric/parameter(s) under consideration: 1 (closest 10),2 (next 15), 3 (next 25), 4 (next 50), and 5 (remaining 500). The scoreof a metric/parameter(s) for a specific local planner was
whereni is the number of connections that planner made in seti. Whiledifferent scoring mechanisms may lead to different recommendations,this one was selected based on our experience with PRMs.
A selected summary of our results for the scaled Euclidean andMinkowski metrics is contained in Table IV. A general observationwas that the relative importance of the translational (rotational)
distance between the two configurations increased (decreased) as theenvironments became harder. Also, we noted that the straight-line androtate-at-(1=2) were similar, as were the two A*-like planners.
For the scaled Euclidean metric, the optimals value increases as theenvironment gets harder for all local planners and environments. Forthe surface-to-surface connections, the optimals is usually higher inthe alpha puzzle than in the six-cube environments. Overall,s = 0:75ands = 0:90 performed well. Our results for the modified Minkowskivalues were generally supportive of these results as well.
For the Minkowski metric in the six-cube environments, most localplanners reached their peak atr = 1:5 for surface-to-surface connec-tions, andr = 4 for free-to-surface connections. In the alpha puzzleenvironments, the best values werer = 1:5 for surface-to-surface con-nections andr = 2:5 or r = 3 for free-to-surface connections. Wenote the optimal parameter for a particular metric will depend on therepresentation chosen for C-space. However, the general trend shouldbe invariant to this choice.
C. Selecting Metrics for Local Planners
After selecting interesting parameter values for the various metrics,we were left with a total of 12 different distance metrics for each en-vironment. All four nonparameterized metrics (Euclidean, Manhattan,center of mass, and bounding box) were analyzed in both environments.In the six-cube environment, we selected the scaled Euclidean withs = f0:25; 0:5; 0:75; 0:9g, the Minkowski withr = f1:5; 4g, andthe modified Minkowski with(r1; r2; r3) = f(2;0:5; 2); (2; 2:5; 2)g.In the alpha puzzle environment, we selected scaled Euclidean withs= f0.25, 0.75, 0.9g, Minkowski with r = f1.5, 2.5g, and modifiedMinkowski with (r1; r2; r3) = f(2, 1.5, 2), (2, 2.5, 2), (2.5, 2, 2)g. Ineach of the six environments, each local planner was evaluated with the12 selected metrics.
A selected summary of distance metric recommendations for threelocal planners in the easy and hard environments is shown in Table V.Generally, the best metrics placed more importance on the translationaldistance than on the rotational distance. Our recommendations takeboth efficiency and effectiveness into account (we defined efficiencyas the speed of a distance metric and effectiveness as its score, i.e.,the characterization of more connectable nodes as close). When resultsdiffered for free-to-surface and surface-to-surface connections, prefer-ence is given to surface-to-surface connections as those are consideredmost difficult. When the best metric is computationally expensive, wesuggest a more efficient alternative (with comparable effectiveness).For example, it can be seen in Table V that in many cases when the
446 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 4, AUGUST 2000
TABLE VDISTANCE METRIC RECOMMENDATIONS BY PLANNER/ENV
Fig. 2. Comparison of local planners (LPs) in six-cube environments.
best performer was the Minkowski or modified Minkowski metric, weinstead recommend using the scaled Euclidean. Our recommendationswere based on the fact that the scaled Euclideans performance was al-most as good and its computational requirements are about 20% and40% those of the Minkowski and modified Minkowski, respectively(see Table I). The individual scores for each metric/planner/environ-ment combination can be found in .
Generally, the metrics performed comparably in all environments.An exception was the bounding box, which performed worse in thealpha puzzle than in the six-cube environments, likely because thebounding box is not a good approximation of the shape.
D. Selecting Local Planners for Environments
After determining which distance metrics were best suited for eachlocal planner in each type of environment, we then compared the localplanner and distance metric combinations. The straight-line and ro-tate-at-(1=2) planners behaved similarly, while the A*-like plannerswere generally the most effective (and expensive).
The six-cube environment results are shown in Fig. 2. In the graphs,there is one bar for each local planner/distance metric combination, andthe black (white) portion of each bar represents the surface-to-surface(free-to-surface) connections made. We see that the best local plan-ners in all six-cube environments are the A*-like planners, followedby the rotate-at-(1=2) planner. In general, rotate-at-(1=2) outperformsthe straight-line planner, probably because its swept volume is oftensmaller. The dramatic difference between the rotate-at-0 and rotate-at-1for the free-to-surface connections (white portion) is because the goalconfiguration is always near a C-obstacle surface (i.e., rotation at thegoal,s = 1, will likely cause collision).
Fig. 3. Comparison of LPs in alpha environments.
TABLE VICONNECTIONSMADE BY ONLY ONE LOCAL PLANNER
The alpha puzzle environment results, shown in Fig. 3, are similarto the six-cube, with the advantage of rotate-at-(1=2) over thestraight-line being more pronounced.
E. Using Multiple Local Planners
The results shown in Table VI indicate that different local plannersdo indeed make different connections. In each row (environment), thepercentage of the connections madeonlyby that local planner is shown.The straight-line is not shown because all its connections were madeby the A*-like methods. Similarly, the rotate-at-(1=2) values are lowin easier environments because most of its connections were also madeby straight-line or one of the A*-like methods. However, as the en-vironments gets higher, the performance of rotate-at-(1=2) increases.Finally, the values for both A*-like methods are low since they mademany of the same connections (they are both biased to neighbor i),which increments the position and orientation coordinates toward thegoal). When we included only one of the A*-like methods in the anal-ysis, in every environment at least 22% of the total connections weremadeonly by the A*-like planner; this percentage increased with thedifficulty.
Table VII shows the comparison of two local planners. In the top halfof the table, column one shows connections made by the straight-linebut not the rotate-at-(1=2) planner, and vice versa in column two.Column four shows the percentage of the total connections made bythe combination of the two planners, which was generally more than50%. The results of the rotate-at-(1=2) and A*-clearance combination(bottom half of the table) show that more than 80% of all connectionsare made by this pair. This implies that using both the rotate-at-(1=2)and A*-like planners would result in good connectivity. A goodalgorithm should try the faster local planner first, only if the first onefails it should try to use second local planner. In this schema, thefailing time also plays an important role (see Table II) since we would
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TABLE VIICOMBINATIONS OF TWO LOCAL PLANNERS
prefer the first local planners should fail fast so that we could try thenext local planners in reasonable time.
The main goal of our study was to determine good combinations ofdistance metrics and local planners for use by PRMs for planning themotion of rigid objects in cluttered environments. Our results (TableV), include recommendations for selecting distance metrics for variouslocal planners in different types of environments. Generally, a goodchoice is the scaled Euclidean metric, where more weight is placed onthe position coordinates as the environment becomes more cluttered.Although it was not always the absolute best metric, its performancewas comparable and it is quite efficient to compute.
The most powerful local planners were the A*-like planners. Amongthe less expensive planners, the rotate-at-(1=2) planner was the best,outperforming the common straight-line local planner. However, wealso found that each of the tested local planners made some connectionsthat the others did not. Thus, roadmap connectivity will be enhancedif the PRM uses multiple local planners. Based on our experience, wewould recommend the following order: first, the rotate-at-(1=2) andstraight-line planners, next, the rotate-at-0 and rotate-at-1 planners, andfinally, theA planners.
The alpha puzzle was designed by B. Yamrom of the ComputerGraphics and Systems Group at GE CRD.
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