ELECTROTECHNICS, ELECTRONICS, AUTOMATIC CONTROL, INFORMATICS ON ESTIMATION OF THE ORIENTATI

更新时间：2023-07-21 05:42:01 阅读量：实用文档文档下载

说明：文章内容仅供预览，部分内容可能不全。下载后的文档，内容与下面显示的完全一致。下载之前请确认下面内容是否您想要的，是否完整无缺。

electrotechnical推荐度：
相关推荐

FASCICLE III, 2003 ISSN 1221-454X

ELECTROTECHNICS, ELECTRONICS, AUTOMATIC CONTROL, INFORMATICS

ON ESTIMATION OF THE ORIENTATION OF MOBILE ROBOTS USING

TURNING FUNCTIONS AND SONAR INFORMATION

Dorel AIORDACHIOAIE *

University of Antwerp, TEW-MTT Department, Active Perception Lab

Prinsstraat – 13, Antwerp 2000, Belgium. Email: Dorel.Aiordachioaie@ua.ac.be

Abstract: SONAR systems are widely used by some artificial objects, e.g. robots, andby animals, e.g. bats, for navigation and pattern recognition. The objective of this paperis to present a solution on the estimation of the orientation in the environment of mobilerobots, in the context of navigation, using the turning function approach. The results areshown to be accurate and can be used further in the design of navigation strategies ofmobile robots.

Keywords: SONAR, Navigation, Pattern Recognition, Signal Processing, and TurningFunctions.

1. INTRODUCTION

The objective of the work is to estimate theorientation of mobile robots in navigationenvironments, based on information provided by aSONAR system. The robot is an ATRV-Jr mobilerobot, (iROBOT 2002). The work is part of EUproject CIRCE - Chiroptera Inspired RoboticCephaloid: A Novel Tool for Experiments inSynthetic Biology, (CIRCE 2002).

In navigation of a mobile robot the state of the robothas two parameters: its position (which can be codedby ordinary number or by a pair of scalars)coordinates and the orientation of the robot.Estimation of the orientation can be solved in a widevariety of methods with roots in the inverseproblems, state estimation or pattern recognitiondomain. In this work methods from patternrecognition are considered.

The problem of estimation of the orientation will besolved in two steps: first, the estimation of theposition, and, second, the estimation of theorientation with reference to the estimated position.

In section 2 the attributes of the experiments arepresented and briefly explained. In section 3computational details of turning functions and thealgorithm for classification are presented. Next, insection 4 some simulation results and, theconclusions are presented.

2. DESCRIPTION OF THE EXPERIMENTThe exploratory and working field in this experimentis a corner from our laboratory, where 35 positionswere considered. Positions from 1 to 21 areconsidered reference positions, and the rest, positionsfrom 22 till 35, are test positions. Two sets of dataare obtained: reference data and test data. Thedistribution of the positions is presented in Fig.1.The ATRV-Jr mobile robot has a SONAR systemwith 17 transducers. In every reference position, therobot is making a complete rotation around hiscenter, stopping 12 times for measurements. In everyposition measurement data from each SONAR sensorare collected and stored for later usage. The structureof reference data, in each position, is a 17 x 12 matrixwith real values. All the measurements in thereference positions are referenced to the horizontal

Contact address: “Dunarea de Jos” Galati UniversityAutomatic Control and Electronics DepartmentDomneasca-47, Galati 6200, Romania.This paper was recommended for publication by Emil CEANGA

axis (by adding the initial orientation angles to thevalues of angles in each reference position). The

data structure of test positions is the same.

The turning function is widely used in the pattern

recognition applications, like recognition of polygons

Fig. 2. SONAR data of position #2: all orientations

and the orientation 0 only(Arkin et al 1991; Alt et al 1995; Alt et al 1996) orindexing of shape images (Chiueh 2000).

Fig. 1. The environment and the distribution of the

reference positions (1:21) and test positions(22:35)Every test array is a pattern matrix of size 17 x 12,but the initial orientation of the robot is unknown,and should be estimated.

In Fig. 2 some measurements from a referenceposition are presented. On the upper side thecombined measurements from all orientations and, onthe bottom, the measurements from one orientationonly, corresponding to an orientation value of zerodegrees of the robot.

The considered algorithm compares the SONARpattern from the test orientation with all theorientation from the closest reference position, todecide which one is resulting in the best match. Theusage of an classifier based on Euclidian distancefailed in generation of valid results for all the testpositions and orientations.

3. THE TURNING FUNCTION

The turning function, or cumulative angle function,of a polygon represents the angles of the polygon’sedges with respect to a reference axis, of arbitraryorientation as shown in Fig. 3, for a simple polygon.As the perimeter of a polygon is traversed, the angleat each point is computed, thus effectivelytransforming a 2D shape into a 1D turning functionwhile preserving all relevant information. Thestarting point on the polygon’s contour for thetraversal corresponds to the position on the turningfunction diagram and is called the origin. The origincan be arbitrarily chosen. The above definition isgeneral enough to apply to any type of shape. Forclosed shapes, the function repeats itself shifted upby 2 * pi in each successive period, which is equal tothe length of the boundary.

To support scale invariance, the length of one cycleof the traversal around a polygon is normalized to 1in the turning function. Note that rotating thereference axis by θ degrees shifts the entire turningfunction up or down by θ, depending on the directionof rotation. The same effect occurs if the polygon is

rotated around a point on the plane. Sliding thepolygon’s origin along the perimeter effectively

shifts the turning function left or right by the amount.

Fig. 3. An example of the correspondence polygon - turning functionThese two parameters, namely the choice ofreference axis and the traversal starting point, can betuned to accomplish a “maximal” match between twopolygons’ turning functions. Their distances are thencomputed at the maximally matched configuration.For polygons the turning function is a piecewiseconstant function, increasing or decreasing at thevertices, and constant between two consecutivevertices.

The turning function is invariant to sometransformations like translation, rotation and scaling.So it seems to be the ideal candidate in defining agood similarity measure between two polygons.To compare two polygons whose turning functionsare TA(s) and TB(s), s in [0,1], a function of the areabetween the functions should be computed. Then, itis minimized over all choices of reference axis (angleθ) and origin for the second polygon (t), whilekeeping the first one’s reference axis and originfixed. First, a function h(.) is computed

(radians) and the range to vertices (meters) of thepolygon, a polygon with n vertices is represented by

(3)R=[R1(4) θ=[θ1

θ2

...Rn]

...θn]

Computation of the turning function can be done intwo ways: geometric and algebraic.A geometric solution

In Fig.4 a simple geometric scheme is presented for apolygon with two vertices, A1 and A2. The origin axisis Ox. The angle θ is the difference of angles θ1 andθ2, corresponding of the ranges R1 and R2.

The problem is to evaluate the α angle, whichcorrespond to a value of the turning function, and thelength of the first edge of the polygon, edge definedby the points A1 and A2. The following relations arevalid . For the edge A1A2

(5) θ=θ1 θ2

(6) A1A2=QA1+QA21

(1) h(θ,t)=(T(s+t) T(s)+θ)2ds

B∫A

The distance metric D(TA, TB) between two turning

functions TA(.) and TB(.) is (Veltkamp 2001):

(2) D(T,T)=min(θ,t)

θ,t

Modifying the orientation of the reference axiscorresponds to rotating the polygon or, equivalently,shifting the turning function up or down.

If the information of the polygon is available underpolar representation, which means the angles

'(7) QA=OA'1 OA2=R1cosθ1 R2cosθ21(8) QA=A2A'2 QA'2=R2sinθ Rsinθ

2211

(9) AA==R+R 2RRcosθ121212and for the angle α

(10) =180 α

(11) tg=QA

2=R2sinθ RsinθQA1R1cosθ1 R2cosθ2

Fig. 4. The computation of turning function in polar coordinates

(12) α=180 arctg R2sinθ Rsinθ

Rcosθ Rcosθ 1122 The information necessary in the turning function is

related to the angle α and to the length of the edgeA1A2.

An algebraic solution

With reference to the bottom of Fig. 4 let be the(xi,yi) the Cartesian coordinates of the vertices of theconsidered polygon, with i going from 1 to n. and asmall drawing like in the Figure 17.b was presented.The following relations hold

(13) AA==

x2 x12+y2 y12

(14) α=arctg y2 y1

x x 21

As a final remark, the co-domain of the turningfunction in on [0:2π), so it is possible to make someoffset corrections after directly applying theexpressions for the computation of the turningangles.

For the purpose of the present work, the SONARdata from the test position are not ordered, so, foreach orientation a turning function is computed.Every position, reference and test, has a set with 12

turning functions corresponding to the 12orientations of the robot in that position. By takingthe index of the minimum distance between thereference and the test turning function the value of

the initial orientation is estimated. To increase therobustness of the estimator, the final orientation ofthe robot is computed as the mean of all the

estimations from the 12 considered orientations.

Fig. 5. Results of the estimation of the initial orientation

4. SIMULATION RESULTS

In Fig. 5 a global result for the estimation of theinitial angle is presented. On the left side of Fig. 5 forevery position a pair of two rectangles arerepresented. On the left is the real value and on theright the estimated value. On the right side of Fig. 5the absolute errors are represented.

Taking into account the fact that the rotatingincrement is about 30 degrees (360 / 12 positions) theresults are acceptable because the absolute error isless than the 30 degree for each considered position.There is only one exception, position 29, which canbe explained by observing that a “circle” covers thelocation and the rotation information cannot be codedin the sonar data and, hence, cannot be extracted bythe turning function.

5. CONCLUSIONS

The objective of the work was to present a solution tothe estimation of the orientation of mobile robots,using SONAR information and techniques frompattern recognition. The turning function is used as itproves to be sensitive in the discrimination of theorientation. Depending on the application, theestimation can be obtained immediately from theSONAR information gotten from a single orientationor can be obtained by averaging all estimates fromdifferent orientations of the robot, requiring the robotto rotate on the spot. The results are accurate foralmost cases (13 from 14 test cases) and less than thestep of moving from one orientation to the next one.

ACKNOWLEDGEMENT

The work was supported by the project CIRCE -Chiroptera-Inspired Robotic Cephaloid: a Novel Toolfor Experiments in Synthetic Biology, IST-2001-35144, as a collaborative EU-project within theProactive Initiative 2001 in Bionics entitled LIFE-LIKE PERCEPTION SYSTEMS (LPS). Financialassistance and support of professor Herbert Peremansis gratefully acknowledged.

6. REFERENCES

Alt, H. , U. Fuchs, G. Rote, and G. Weber, (1996).

Matching convex shapes with respect to thesymmetric difference. In Algorithms ESA ’96,Proceedings of the 4th Annual EuropeanSymposium on Algorithms, Barcelona, Spain,September ’96, pages 320–333. LNCS 1136,Springer.

Alt, H., B. Behrends, and J. Blomer, (1995).

Approximate matching of polygonal shapes.Annals of Mathematics and ArtificialIntelligence, p. 251–265.

Arkin, E. P. Chew, D. Huttenlocher, K. Kedem, and

J. Mitchel, (1991). An efficiently computablemetric for comparing polygonal shapes. IEEETransactions on Pattern Analysis and MachineIntelligen ce, 13(3):209–215.

CIRCE, (2002). Chiroptera - Inspired Robotic

Cephaloid: a Novel Tool for Experiments inIROBOT, (2002). iROBOT Corporation, http://www.

irobot. com/rwi/p03.asp.

Tzi-cker Chiueh, Tzi-cker, Allen Ballman, Kevin

Kreeger, (2000). Multi-Resolution Indexing forShape Images, Research Report 223, ComputerScience Departement, State University of NewYori, USA.

Veltkamp, R.C., (2001). Shape Matching: Similarity

Measures and Algorithms, Dept. Computing

Science, Utrecht University, The Netherlands,

Research Report.

本文来源：https://www.bwwdw.com/article/06u1.html

相关文章：