Theseus Gradient Guide: an Indoor Transmitter Searching

Approach Using Received Signal Strength

Xiaochen Zhang, Yi Sun, Jizhong Xiao and Flavio Cabrera-Mora

Abstract—The searching for a location-unknown radio trans-mitter is a challenging task for autonomous robot. We proposean adaptive searching algorithm named theseus gradient guide(TGG) which is designed for solving the searching problemin indoor environments using received signal strength (RSS).While the RSS gradient serves as the main guide, the robotprefers to move to the places which have never been traveled.Thus the robot will not get stuck in the local maxima. Moreover,unlike the commonly used random kick strategy the TGGdrives the robot escaping the local maxima with low cost interms of travel distance. Meanwhile, TGG is not sensitive tomotion errors. Simulation results show that the searches usingTGG cost much less compared with those using other gradientbased methods in our testing indoor environment. Guided byTGG, the robot can successfully reach the location-unknownradio transmitter with a ratio over 97% when the standarddeviation of motion error is up to 20% of the step length.

Index Terms—mobile robot, radio transmitter searching,received signal strength, theseus gradient guide.

I. INTRODUCTION

Sensor networks have been broadly applied in variousfields in industry and in military. They have also stimulatedresearch in information acquisition, storage, transmissionand reachback at physical layer as well as network or-ganization and management at upper layers. A particularinterest in current research is sensor localization with threeapplication scenarios. First, sensors localize others in a pre-viously deployed sensor network [1]–[5]. Second, a numberof sensors localize a mobile agent [6]–[9]. In both cases,a number of location-aware sensors are usually employedto beacon signals to localize their neighbors. The thirdis surveillance or information reachback where a mobilerobot localizes a number of activated sensors in a sensornetwork [10]–[15]. This paper focus on the third scenario.Several gradient based searching method has been proposedin last decade: the bacterium inspired chemotaxis search [16]takes advantage of random walk in addition to its gradientsearch; the olfaction-based pheromone search [13] restrictsthe motion rules compared with chemotaxis; our previouswork [17] considers gradient over a short period appears as a

Xiaochen Zhang, Yi Sun and Jizhong Xiao are with the Departmen-t of Electrical Engineering at the City College of City University ofNew York, New York, NY 10031. Email: xzhang15@ccny.cuny.edu, y-sun@ccny.cuny.edu, jxiao@ccny.cuny.edu.

Flavio Cabrera-Mora is with the Dept. of Electrical Engineering, TheGraduate Center of the City University of New York, New York, NY 10016.Email: fcabrera-mora@gc.cuny.edu.

This work was supported in part by the U.S. Army Research Office undergrant No. W911NF-08-1-0531, W911NF-09-1-0565, and U.S. NationalScience Foundation under grants No.CNS-0619577 and No. IIS-0644127

promising solution while searching in a open space; a variantof chemotaxis [15], [18], [19] which takes advantage ofLevy walks [20] appears recently, by actively switching thebehaviors between Levy walks and Brownian random walks,the searching mission can always be accomplished. However,most of the gradient based methods suffer from the high costin terms of travel distance. A series of localization works[10], [21], [22] using received signal strength (RSS) has beenproposed recently, [10], [21] assumes the employment ofCSMA in target radio transmitters, they used the monte carlomethod to simultaneously localize the radio transmitters.[22] relaxes the CSMA assumption, while a grid based local-ization method named spatiotemporal probability occupancygrid (SPOG) is proposed. The simultaneous localizationmethods requires intensive training and calibration whichis difficult in indoor environments.

The theseus gradient guide (TGG) approach presented inthis paper is the subsequent work of [17] which focus on thesearching in open spaces. By using TGG, the searching fora radio transmitter in indoor environments can be accom-plished by a mobile robot with basic collision avoidanceand signal strength measurement capabilities. Searching ina grid map which is composed of quadrate cells, the robotonly needs to instantly measure signal strength and estimatethe direction of RSS gradient. Although the gradient is noisy,our method takes advantage of dead reckoning and alwaysprefers to move to the cells which have never been traveled.The accumulative motion error is unavoidable however nofatal since the RSS gradients are produced by short periodRSS measurements in most cases. The simulation resultsshow that the robot with theseus gradient guide alwaystravels less to reach the target radio transmitter comparedwith the chmotaxis method [16], yuragi-based searching [15]and the exhaustive search. When the standard deviation ofmotion error is up to 20% of the step length, the successrate of our approach is still greater than 97%.

The rest of the paper is organized as follows. The gradientguide and theseus traverse rules are presented in Section II.The simulation and results are stated in Section III. SectionIV concludes the paper.

II. THESEUS GRADIENT GUIDE

A. Application scenario

Consider a common application in sensor network main-tenance: a mobile robot is sent to an indoor sensor networkfield to find the target sensor transmitter. Without knowing

2011 IEEE International Conference on Robotics and AutomationShanghai International Conference CenterMay 9-13, 2011, Shanghai, China

978-1-61284-385-8/11/$26.00 ©2011 IEEE 2560

the map, the robot tries to reach the transmitter by utilizingthe RSS measurements.

We assume a mobile robot with knowledge of its orienta-tion from an onboard compass searching in an unknown 2Dworld. To be specific, the robot believes the searching is ina grid world composed of a series of cells (Figure 1). Werestrict the robot’s motion within four directions (i.e. north,south, east and west). Since the collision avoidance is not ourmain concern, we assume the robot has the ability to detectwhether the adjacent four cells are blocked by obstacles.While searching, the robot estimates RSS gradient and findsthe next cell to reach according to its travel records.

Fig. 1. The grid map of terrain.

B. The received signal strength

Consider the signal power attenuates in accordance withthe path loss prediction model [23]:

P (d′) =P0

(d′/d0)β(1)

where P0 and d0 are the power and distance scaling factors,respectively, β stands for the decay exponential, d′ = d ×10

∑f+

∑t−Ω

10·β is the pseudo distance concerning the relativeposition of receiver and transmitter, d is the correspondingeuclidean distance, f is a function related to reflection, tis a function of energy loss due to wall penetration, Ω isthe wave guiding factor [23]. The psedo distance d′ has thesame convergence trend as the euclidian distance d.

Suppose the transmitter to be searched is located atx = (x, y), the robot at discrete time i is located atxi, i = 1, 2, . . . , k. At each time, the robot measures thesignal strength from the transmitter. Since only the signalstrength shall be used, the fading effect can be reduced to theminimum if the signal strength is measured by an integratorover a time period longer than the fading coherent time. Witha static indoor environment and a specified transmitter, themeasurement ui relies on the geometric location of the robotxi:

ui = ai

√P (d′(xi +

∑i

j=1wj ,x)) + ni, i = 1, ..., k (2)

The Gaussian noise ni ∼ N(0, σ2) and the robot motionerrors wj ∼ N(0, σ2

mI) are multually independent. Dueto the accumulative motion error, the true robot location

is xi +∑i

j=1wj where xi is the nominal location the

robot believes. The fading coefficients ai can be treated asa constant.

Since the measurement needs to be reliable, we suggest totake L > 1 measurements of signal strength at each singlelocation xi. The corresponding average ui is used in thegradient estimation.

Note that the robot at each time does not need to know itstrue location since it is not necessary in the indoor searching;instead, the robot attempts to find the direction to reducethe pseudo distance d′ to the transmitter. Hence, even therobot moves with error or sometimes moves farther withrespect to the euclidian distance d, it can eventually reachthe transmitter since the pseudo distance convergence withthe euclidian distance

1) Gradient guide on pseudo distance: The receivedsignal power monotonically decreases as the pseudo distancefrom the transmitter to the robot increases. The powergradient at the robot always directs towards to the loca-tion with smaller virtual distance. Thus, a straight forwardmethod to guide the robot is to move along the directionwhich the power gradient points at. The power gradientcannot be determined by the signal strength at xk alone;but it can be estimated by previous h measurements ui

and xi, i = k − h + 1, k − h + 2, . . . , k. It shall be notedthat an accurate estimation of the transmitter direction isunnecessary and difficult. On the other hand, even when theestimated gradient is inaccurate, the robot can be consideredapproaching the transmitter if moving a sufficient smalldistance in a direction along which the pseudo distancedecreases. Hence, in the worst case, the robot needs onlyto estimate the sign of gradient along a direction. On thebasis of this, we have the power gradient estimator at xk by

gk =∑k−1

i=k−h(uk − ui)

xk − xi

||xk − xi||. (3)

In accordance with the meaning of RSS gradient, itsmagnitude increases as the difference between uk and ui

increases. To normalize the effect of distance ||xk − xi||on the estimate, the unit vector in the direction xk − xi isemployed in (3).

In the gradient estimate, only the latest h measurementsare used. When h is large, the estimate is more reliable byaveraging out noise; however, then the direction of gradientrelies more on the past measurements thus the gk changesslowly. On the other hand, using small h can allow gk

changing fast; but the estimate is noisier and less reliableespecially when the multi-path effect is strong. Hence, theoptimized h shall be large initially and gradually decreasewhen the robot approaches the transmitter, so as to speed upthe search.

According to gradient guide, the robot is able to approachthe radio source if the local maximum traps can be wellconquered.

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C. Theseus Traverse

Theseus is an attica hero in Greek mythology, he usesa thread to mark the traveled path so as to find the wayout of Minotaur’s labyrinth. The gradient guide itself cannotguarantee the escaping of local maxima (Figure 2). Consid-ering this, a few heuristic algorithms can be applied, suchas simulated annealing, tabu search, evolutionary algorithmetc. However, in indoor search using RSS, there are massesof small local maximum areas; such heuristic approachesare cost inefficient in terms of travel distance since escapingeach single local maximum requires a tedious judge-decisionprocedure.

Fig. 2. The contour map of RSS distribution from the single transmitterin a hallway. The upper-right shows the layout of the environment: thetransmitter, the hall way and the local maxima of RSS.

We decide to use a simple but efficient approach toguarantee escaping the local maximum. The robot builds upthe grid map believes gradually while moving and marks thetraveled cells as unavailable. While deciding which cell toreach next, the answer is always chosen from the cells thathave not been traveled. This approach prevents the robotfrom being trapped by local maxima.

Figure 3 illustrates the basic behaviors of robot withtheseus gradient guide. The left figure shows that if thediscrete estimated gradients on each cell directs to the target,the robot will just follow it and accomplish the searchingas fast as the pure gradient method. When the gradientleads erratically or even provides an opposite direction tothe target, as shown on the a1 and a2 column of the figureon the right, the robot will traverse the certain area first. Forillustration, we artificially assign the gradient on each cellin Figure 3.

We use a stack S to store the traveled records. Tobe specific, each element Si in S contains the followinginformation: xi is the location of the cell traveled by ithstep, xj

i , j = 1, 2, 3, 4. denotes the four adjacent cellslocations of cell xi, a

jk is the markers denoting whether the

corresponding adjacent cells have been traveled or blocked( sensor reading ljk = 1). While the robot is on cell xk andthe unit gradient is gk, the problem can be solved by a twophase recursive approach.

1) Update phase: The phase has two basic stages:

Fig. 3. The behaviors of theseus traverse with given gradient guides.(Thehollow arrows denotes the gradient directions, the solid arrow denotes themoving trajectory.)

• Push the current cell element Sk in stack S, the Sk

contains xk, xjk and ajk, j = 1, 2, 3, 4. To be specific, ajk = 0

denotes the corresponding adjacent cell has been traveled oris blocked by obstacle (ljk = 1).• Use the latest cell xk to update the rest elements in stack

S. For instance, for element Si, if xji = xk, then aji = 0.

2) Decision phase: After the updating, the robot willmake decisions on next cell to reach. The decision phasehas two alternative options.• If ∃ajk = 1, j = 1, 2, 3, 4, the next cell xk+1 will be

chosen from the available adjacent cells depending on gk:

xk+1 = argmaxxjk

(gTk ·(x

jk−xk)+1)×ajk, j = 1, 2, 3, 4. (4)

• Otherwise, the stack will pop the elements from Suntil the latest stack top Sτ satisfies the condition ∃ajτ =1, j = 1, 2, 3, 4. The next cell xk+1 will be chosen from theadjacent cells of xτ :

xk+1 = argmaxxjk

(gTk · (xj

τ − xk)

||xjτ − xk||

+ 1)× ajτ , j = 1, 2, 3, 4.

(5)On one hand the robot with theseus gradient escapes the

local maxima locally since the motion errors in a short periodis non-fatal. On the other hand, without being entangledin the local maxima the robot approaches the transmitterglobally. The long term inconsistency of robot’s belief andground truth does not prevent the robot approaching the radiosource since the true location of robot itself is unessentialto the search.

D. Provisional behavior switch

While the theseus gradient method takes advantage ofdead-reckoning in escaping local maxima, some drawbacksoccur. One of the special case is the virtual wall dilemma: theprior path may isolate the robot from the radio transmitter.Figure 4 illustrates one instance of virtual wall: a robotstarts from a3b3 and reaches a2b1 through a3b2, a3b1, itcannot move across the column a3 based on theseus gradientuntil all the cells on the left of column a3 are traveled.Although this phenomena may not deter the final discoveryof transmitter, the overhead is not favorable.

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Fig. 4. The virtual wall.

Concerning this, a provisional behavior switch is adopted.The robot continuously monitors the inner product betweenits motion vector xk−xk−1 and the estimated RSS gradientvector gT

k−1, if the inner product is lower than a threshold γ,the robot believes the move is not following the gradient, orwe say roaming motion. Note the vectors are unitized beforecalculation. In Tcon most recent moves, if the number ofroaming motion ratio λk exceeds a threshold R, the robotwill temporarily switch to another behavior mode. To bespecific, the robot tends to move to a new cell xk+1 withoutcomplying the rule of theseus gradient.

At step k the roaming motion ratio λk can be computedas:

λk =1

Tcon

∑k

i=k+1−Tcon

di (6)

where

di =

1 if gT

i · (xi − xi−1) < γ

0 else. (7)

If λk exceeds the threshold R, a xk+1 is obtained using

xk+1 = arg maxxp∈S

[u(xp) +

∑u(xj

p)]× Ip, if λk > R

(8)where u(xp) is the RSS reading on cell xp, u(xj

p) denotesthe RSS reading of adjacent cells of cell xp and Ip is anbinary indicator equals to

∪ajp, j = 1, 2, 3, 4. After reaching

xk+1, the robot will switch back to the original mode.By using (4)(5)(8), we can always get a next cell xk+1 to

reach when the robot is on xk, as Algorithm 1:

III. SIMULATION AND RESULTS

In this section, we focus on the searching cost of theseusgradient guide in terms of the travel distance. Control groupsof four different methods is used for comparison. A studyon tolerance property to motion error is presented as well.Simulations are all driven by real RSS data collected inadvance.

Algorithm 1: Theseus gradient guideInput: xk , uk , lkOutput: xk+1

Estimate gk using (3)Push Sk to SUpdate elements in S

if ∃ajk = 1 thenFind xk+1 using (4)

elsewhile @ajτ = 1 do

Pop stack top xτ from S

Find xk+1 using (5)

Compute dk using (7)Compute λk using (6)if λk > R then

Replace xk+1 with new xk+1 using (8)

return xk+1

Fig. 5. Scenario of Hallway in Steinmen hall building.

A. Setup

In this section, we briefly introduce the settings in simu-lation.

Scenario:Four transmitters are placed in the hallway of 6th floor of

Steinmen Hall at CCNY, as shown in Fig. 5.RSS Data:To validate the results in high fidelity, the simulation

is driven by real data. The Hallway is divided into 343uniformly distributed grids. On each center of the grid, 10readings for each radio source are collected.

Motion error:An accumulative motion error w˜N(0, σ2

mI) is involvedin every single move, where σm stands for the standarddeviation of motion error.

Control groups:Four different methods are compared.(1) The ideal case (IDEAL): The robot knows the exact

location of target transmitter, and just goes to the transmitterwith the least number of steps. This is an ideal searchingcase.

(2) The exhaustive search: The robot tries to travel thewhole space exhaustively until the target transmitter is found.

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(3) The chemotaxis search: This approach is a gradientapproach [16] inspired by bacteria behaviors [24]. Two basicbehaviors form the search: the run which is based on thegradient and the tumble which is a variance of random walk.

(4) The Yuragi-based adaptive searching (YBAS) [15]:This approach is an optimized combination of levy walk[20] and chemotaxis [24].

Finding criterion:The radio transmitter is found if the robot is within the

unit step length.

B. Efficiency comparison

We simulated 50 trails for each approach. The step lengthis 0.9144 meter and the standard deviation of motion erroris 6.7% of the step length.

Fig. 6. The searching success ratio versus travel distance

Figure 6 shows the efficiency comparison using differentsearching methods. One trail will be regarded as succeedif all four radio sources are found. The term success ratioin y-axis denotes that the success trails over the grouptrail volume (50 in this simulation) with a travel distanceshown in x-axis. The unit of travel distance is the steplength. From the numerical experiments, we can find that theaverage accomplishing speed of TGG is 65%, 370%, 450%faster than that of the Exhaustive, YBAS and Chemotaxismethods, respectively. Compared with the ideal case, TGGtakes approximate 150% more steps in average. The result isnot surprising since the ideal case indicates the lower bound.Note the results are based on limited numerical simulationin the given indoor scenario.

C. Motion error influence on success ratio

We also concern about the influence of motion error onsearching success ratio. In this scenario, one search trail issucceed if the robot can find all four radio transmitters within1000 steps.

For this purpose, we measured the success ratio of findingthe radio source in Fig. 5 with different standard deviationof motion errors σm. A numerical experiment of 200 trailsis launched for each approach (50 for each σm), an averagedsuccess rate is listed in the Table. I.

TABLE ISUCCESS RATIO (%) WITH DIFFERENT σm

σm(meter) IDEAL TGG Exhaustive YBAS Chemotaxis0 100 100 100 96.4 87.80.06096 100 99.8 99.8 94.2 86.20.12192 100 99 98.6 95.8 83.60.18288 100 97.2 96.8 95.2 81.2

From the table, we can find that the success rate of TGG is100% when there is no motion error. Few trails failed afterslight motion error is presented. In a case that when theσm is up to 20% of the step length (0.9144m), the successratio is still over 97%. Notice that the high success ratio ofExhaustive, chemotaxis and YBAS approaches come withrelative high costs in travel distance.

IV. CONCLUSION

We have proposed a simple yet efficient searching algo-rithm taking advantage of both gradient and dead-reckoning.Since the motion is well restricted and prior trajectory is wellrecorded, the robot will not be trapped by local maxima.Thus the indoor search is safe and fast. Simulations drivenby real data show that TGG is 65%, 370%, 450% fasterthan the Exhaustive, YBAS and Chemotaxis approaches ingiven indoor scenario. Moreover, the robot can reach thetransmitter with a rate of 97.2% when the standard deviationof step motion error is 20% of the step length. We arecurrently testing our algorithm using physical experiment.

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