Int. J. Mechatronics and Automation, Vol. 3, No. 1, 2013 25

Copyright © 2013 Inderscience Enterprises Ltd.

A novel radio source search algorithm using force field vectors and received signal strengths

Xiaochen Zhang, Yi Sun and Jizhong Xiao* Department of Electrical Engineering, The City College of The City University of New York, New York, NY 10031, USA E-mail: xzhang15@ccny.cuny.edu E-mail: ysun@ccny.cuny.edu E-mail: jxiao@ccny.cuny.edu *Corresponding author

Abstract: The indoor radio source search using received signal strength (RSS) is difficult due to the multipath effects. Robots driven by the RSS gradient-based searching methods are always stuck by local basins which are the areas around local maxima. In this paper, we propose a force field searching (FFS) algorithm which is cost efficient in environments with RSS local basins. The FFS algorithm fuses the geometry layout information with RSS. Virtual attraction forces weighted by RSS gradients are modelled to guide the robot to the radio source. Furthermore, the travel costs of FFS and the gradient ascent with correlated random walks (GACRW) algorithm are derived as functions of the searching space size, the local basin size and the local basin number in gradient fields. The analytical expressions are confirmed by simulations. It is shown that FFS is much faster than other algorithms in simulations driven by real data.

Keywords: force field search; FFS; gradient method; mobile robot; radio source search; received signal strength; RSS.

Reference to this paper should be made as follows: Zhang, X., Sun, Y. and Xiao, J. (2013) ‘A novel radio source search algorithm using force field vectors and received signal strengths’, Int. J. Mechatronics and Automation, Vol. 3, No. 1, pp.25–35.

Biographical notes: Xiaochen Zhang is a PhD candidate in the Department of Electrical Engineering, the City College of the City University of New York. His research interests include robotics navigation, optimisation in search, communication systems and cloud computation.

Yi Sun received his BS and MS in Electrical Engineering from the Shanghai Jiao Tong University, Shanghai, China, in 1982 and 1985, respectively, and PhD in Electrical Engineering from the University of Minnesota, Minneapolis, MN, in 1997. Since September 1998, he has been an Assistant Professor now with tenure in the Department of Electrical Engineering, City College, City University of New York. His research interests are in the areas of wireless communications and networking with focus on CDMA multi-user detection, cross-layer design for wireless networks, and information theory and coding.

Jizhong Xiao is an Associate Professor of Electrical Engineering at the City College of the City University of New York (CUNY City College). He received his PhD degree from Michigan State University in 2002; Master of Engineering degree from Nanyang Technological University, Singapore, and MS, BS degrees from East China Institute of Technology in 1999, 1993, and 1990, respectively. He started the robotics research programme at the City College in 2002 as the Founding Director of CCNY Robotics Lab. He is a recipient of USA National Science Foundation (NSF) CAREER award in 2007. His research interests include robotics and control, mobile sensor networks, robotic navigation, and multi-agent systems.

1 Introduction

Search of radio sources using received signal strength (RSS) remains an important focus of research in both robotics and wireless sensor networks communities due to vast potential applications across sensor network management (Dong et al., 2010), maintenance (Fowler, 2009) and sweeping. Moreover, the research is closely connected with the developing of hazardous media monitor in urban search and

rescue. Inspired by a wide variety tracking instinctive behaviours in biosphere, various methods based on gradient ascent (Dhariwal et al., 2004; Marques et al., 2002; Nurzaman et al., 2010; Sun et al., 2008; Zhang et al., 2011) arises. In gradient-based search, how to overcome the local maxima is crucial. Figure 1 shows a contour map of actual RSS distribution in a hallway, a robot is likely to be trapped in local basins which are areas around local maxima.

26 X. Zhang et al.

Figure 1 The contour map of the RSS distribution in a hallway (see online version for colours)

Note: The upper-right figure shows the radio source, the

hallway boundary and RSS local maxima.

A bacterium inspired method named chemotaxis (Dhariwal et al., 2004) search is one candidate solution. Motions in chemotaxis are classified as the smooth move and tumble (Adler, 1966). Chemotaxis approach is fundamentally a biased random walk method, a mathematical model can be found in Muller et al. (2002) and another simplified model is in Nurzaman et al. (2009). A study on olfaction-based search (Marques et al., 2002) describes the pheromone search which is able to achieve better performance than the ordinary chemotaxis search in noisy environments. Inspired by the study of optimisation on random walks (Viswanathan et al., 1999, 2008), a method (Nurzaman et al., 2008, 2009, 2010) using levy walks achieves better performance compared with chemotaxis especially when the gradients are not always perceivable. Compared with chemotaxis (Doyle and Snell, 1984), levy walk is preferable to a search with sparse and fixed small targets. By actively switching the behaviour between levy walk (Viswanathan et al., 2008) and biased random walk, the levy walk guided random walk is more efficient than chemotaxis in source finding. These works use varieties of biased random walks in escaping local maximum traps. However, they are suffering from the high cost in terms of travel distance or searching time. The reason of high cost is obvious: random walks are inefficient.

We previously proposed a period gradient method which fits the need of outdoor source search (Sun et al., 2008). For indoor cases, we proposed another method named Theseus gradient guide (Zhang et al., 2011). Borrowing the idea of self-avoid-walk (Dubins et al., 1988), the robot tries to avoid the visited sites in search. Since random walk is no longer involved, the searching cost is less.

In this paper, we further improve the efficiency of the radio source search by generating force field vectors weighted by RSS gradients. Specifically, the locations where the robot cannot be are estimated at each step. At the same time, each possible radio source location generates an attraction force weighted by RSS gradient on the robot.

Thus, the robot will be pulled to the target radio source. By gradually building up the map using laser range finder, the search can be planned in a global view other than ordinary local view. Moreover, since the robot learns its own location from laser reading instead of dead reckoning, the influence of motion error is minimised.

This approach has some advantages in indoor search compared with other RSS gradient-based approaches: first, the search is robust in indoor environments since attraction forces are generated by possible locations of the radio source instead of pure RSS gradients. Second, the travel distance is shorter. When RSS gradients are heading to the radio source, robot guided by force field vectors is most likely to follow RSS gradients; when RSS gradients are not heading to the radio source, the robot will not be trapped and still be able to move towards the radio source since RSS gradients do not dominate the motions.

This paper is organised as follows. The force field searching (FSS) algorithm is presented in Section 2. In Section 3, the gradient field is defined and the analytical analyses are performed. Simulation results are stated in Section 4. Section 5 concludes this paper.

2 Source search algorithms

2.1 Force field search

2.1.1 System outline

As stated in Dellaert et al. (1999) and Song et al. (2009), it is promising to localise the radio source if the real propagation is well fitted by the open space decay model. However, the indoor model is more difficult to obtain compared with the open space one (Bai and Atiquzzaman, 2003; Sarkar et al., 2003; Neskovic et al., 2000). Dozens of works localise the target directly while fewer attentions have been paid to the searches. Actually, it is better to narrow the range of the possible locations first when the true location cannot be estimated immediately. In indoor radio source searches, it is always possible to estimate part of ‘impossible locations’. The ‘impossible locations’ are termed discarded area and denoted by D, and the ‘possible locations’ are termed candidate area and denoted by C. D and C are complementary sets, the union of D and C equals the detected area S which is initially null.

Borrowing ideas of potential fields, locations in C generate attraction forces weighted by RSS gradient to the robot. After summing up all forces together a guiding vector can be obtained.

The system architecture is shown in Figure 2. The laser-based line mapping takes advantage of existing simultaneous localisation and mapping technology so as to enable the robot to locate itself. Furthermore, it provides field information in supporting the decision of discarding already searched area. The RSS gradient estimation component together with the discarded area estimation component provides the needed knowledge of the force field estimation so as to make decisions on the applied virtual force on the mobile robot. Finally, based on the current

A novel radio source search algorithm using force field vectors and received signal strengths 27

unvisited area and the virtual force, the motion decision can be made.

Figure 2 The system architecture

Line mappingLaser reading

Discarded area estimation

RSS reading

RSS Gradientestimation

Force Field estimation

Motion planning

Figure 3 A typical procedure of the line mapping (see online version for colours)

2.1.2 Line mapping

The line mapping (Diosi and Kleeman, 2005; Borges and Aldon, 2004; Nguyen et al., 2007) is indispensable for this algorithm since it gives the location of robot and geometry layout which are used in the discarded area estimation and motion planning. Typical procedure of line mapping in step k is shown in Figure 3: by fusing the line map Lk–1 (blue lines) built in step k – 1 with latest point map (dark dots) the line map Lk is obtained; a compensation line set Bk (red lines) is then generated to enclose the detected area Sk together with Lk.

2.1.3 Discarded area estimation

In free spaces, the signal propagation follows the inverse square law stating that the power density is proportional to the quadratic inverse of the distance from the source

20( ) /P d P d= (1)

where P0 and d0 are the power and distance scaling factors, respectively, d is the distance between the radio source and the receiver. In indoor cases, the signal power can be regarded as the vectorial sum of powers reaching the receiver through multi-propagation channels. In other

words, the received power is influenced by wave reflections, scattering and diffractions. Either multipath add-up or cancellation may occur with distinct geometry layouts. Thus, the received power in a particular location can be written as

* (1 )P P ε= × + (2)

where P is the received power in line of sight propagation, as P(d) in (1), ε stands for the multipath effect factor, P and ε are determined by the locations of radio source and receiver as well as the geometry layout.

If the ε at each step can be obtained, the radio source localisation can be done under Bayes framework. Unfortunately, ε is highly non-Gaussian and differs everywhere thus be very difficult to be estimated in real time. However, it is reasonable to conservatively assume a lower bound of ε to estimate an lower bound of d. If ˆ ,ε ε< an lower bound of d can be written as

( ) 0ˆ / *ˆ1d P P dε= × <+ (3)

where ε̂ and d̂ stands for the lower bound of ε and d, respectively. Specifically, ε̂ is empirically decided according to particular applications, e.g., ε̂ is −65% in this paper. In general, d̂ will not exceed d if ε̂ ε< in the latest several steps. Particularly, we use the mean power value *zx in a small area centred at current location x with radius η1 to produce an averaged power measurement in (3), as

( )1

* 1

η

z zk +

+

<−

= ∑x

x x

x (4)

where z(x+) is the RSS measurement in location x+, k ≥ 1 is the number of z(x+) involved, x+ is chosen from visited locations. Through substituting P* by *zx in (3), the robot is able to calculate ˆ.d Recall d̂ stands for a lower bound of the distance between the robot and the radio source. In other words, the radio source is away from the robot for at least a distance ˆ.d The robot continuously estimates the impossible locations and records them as discarded area D. Specifically, in step n,

n n new= ∪D D D (5)

{ }ˆ| ,new new new k knew k d= < ∩ =−D x x x Lx x φ (6)

where xk is the location of the robot in step k, Lk is the line sets of line map, xnew is the locations to be included in the discarded area.

We would like to mention two important properties of discarded area estimation: first, the growing speed of the discarded area is fast when the robot is far away from the radio source, and becomes slower when the robot is closer to the radio source. Second, the construction of the discarded area is safe since geometry layouts are taken into consideration. For example, in Figure 4, due to the line of sight transmission is blocked, the RSS measured by the robot is weak. Although the Euclidean distance between the

28 X. Zhang et al.

robot and the radio source is smaller than the estimated discarded area radius the radio source will not be included in the discarded area according to (6).

Figure 4 The estimation on the discarded area considering geometry layout (see online version for colours)

2.1.4 RSS gradient and force field estimation

At time n, the robot is located at xn. The RSS gradient considering RSS in adjacent area within distance η2 is

( )2

2 .n i

n in n i

n iη

z z<−

−= −

−∑

x x

x xgx x

(7)

where zn is the RSS measurement at time n, zi is the RSS measurement of visited locations xi. The gradient gn will be used as ˆ ng which is the weighting vector of the force field vectors.

According to the assumption that the radio source exists in areas other than D, a force field is generated as follows. The term cell is used to indicate the sub areas in candidate area C. For the ith cell Ci, its location is -theiCx centre of Ci. Similarly, the same term is used to indicate the small line segments in the compensation line set B. For the ith cell Bi, its location is iBx which is the centre of cell Bi. These cells have the following properties: Ci ∩ Cj = φ,

,Ci =∑ C Bi ∩ Bj = φ, .Bi =∑ B

The reason we consider cells in B and C is intuitively understandable, cells in C are possible locations of the radio source and cells in B potentially hide more cells of C. Thus, both types of cells have attractions to the robot.

An attraction force from cell Ci ∈ C is modelled as

( ) 2 ,i

i

C Ci

n C

w ζf C×

=−x x

(8)

( )( )ˆ /21 Tn Cii

UC Cw τ − ⋅= g x (9)

where ζC is the attraction constant for cells in C, iCw is produced by the transfer function which maps ( )i

Tn C n× −g x x

to the range ~1, ( )iC Cτ U x is the unit vector of ( ).iC n−x x

The attraction force caused by C is

( ) ( )( ) .iCiF f UC= ×∑C x (10)

Similarly, an attraction force from cell Bi ∈ B is modelled as

( ) 2 ,i

i

B Bi

n B

w ζf B×

=−x x

(11)

( )( )ˆ /21 Tn Bii

UB Bw τ − ⋅= g x (12)

where ζB is the attraction constant for cells in B, iBw is produced by the transfer function which maps

( )iTn B n× −g x x to the range ~1, ( )iB Bτ U x stands for the

unit vector of ( ).iB n−x x The attraction force caused by B is

( ) ( )( ) .iBiF f UB= ×∑B x (13)

The overall attraction force vector is

( ) ( ) .( ) ( )

F B F CGF B F C

+=

+ (14)

Upon this, the input of motion planning xm chosen from set C can be obtained as

( )( )/21

maxTn Ci

i

UGm

C n

τ − ⋅⎧ ⎫⎪ ⎪= ⎨ ⎬−⎪ ⎪⎩ ⎭

G x

xx x

(15)

where τG is the transition parameter similar to τB and τC.

2.1.5 Motion planning

The motion planning starts right after the xm is obtained. Unlike the free space cases, the indoor search has to guarantee collision avoidance. The bug algorithm (Lumelsky and Stepanov, 1986; Ng and Braunl, 2007) is chosen as the basis of the motion planner due to its two merits:

1 the bug algorithm provides a robust solution as long as a path exists

2 it is simple however efficient in collision avoidance.

Note the robot is not going to reach xm. Instead, the robot just follows the direction towards xm and moves for a given step length.

2.2 Gradient ascent with correlated random walks

Biased and correlated random walks (BCRW) contain consistent biases in the preferred directions with certain persistence (Codling et al., 2008). For comparison purpose, the gradient ascent with correlated random walks (GACRW) algorithm, which is essentially a BCRW in source search, is applied under the same context as FFS. The robot does not use the information of geometry layout while driven by GACRW. In brief, a robot follows signal

A novel radio source search algorithm using force field vectors and received signal strengths 29

strength gradients when the projection of the previous movements on the latest gradient is positive.

Algorithm 1 GACRW

1: while undone do 2: if the inner product of current RSS gradient and

previous motion is positive then 3: move forward one step along current RSS gradient. 4: else 5: make a rotation with an angle subject to N(0, σ2)

and move forward one step. 6: end if 7: end condition test. 8: end while

N(0, σ2) denotes the Gaussian distribution with mean 0 and standard deviation σ. Here, σ is arbitrarily taken as 2 .π

The algorithm is composed of two phases: one is the gradient ascent; another is the correlated random walks (CRW). Apparently, in ideal case, a robot using GACRW can reach the source with a cost equals the Euclidean distance from the starting point to the radio source. However, if the robot is drawn to a local basin, it will arrive at the local maximum first.

3 Gradient fields and analysis

3.1 Local basins and gradient fields

In conventional RSS gradients-based source searches, two challenges namely noisy gradient and local basins are inevitable. Noisy gradients indicate situations where gradients of magnitudes are more likely pointing to a position around the true source instead of exactly to it; while local basins indicate situations where gradients are always towards local maxima. Fortunately, noisy gradients can be easily dealt with smoothing gradients using multiple readings (Sun et al., 2008; Nurzaman et al., 2008; Russell, 2004; Russell et al., 2003). And local basins can be overcome by either heuristic methods (Edwin and Chong, 2001), random walks, or an arbitrary exhaustive method (Zhang et al., 2011). Lots of efforts have been made to alleviate the influence of noisy gradients, while much less have been done against local basins. Actually, the local basins bring the most overhead in radio source search. Thus, we propose an analytical tool named gradient field where the local basins are highlighted.

• Gradient field: A gradient field is a square vector field where the gradient at any point always points to the unique maximum.

• Ideal searching space: The ideal searching space is a gradient field with side length N at the centre of which the global maximum is located.

• Local basin: A local basin is a gradient field with side length m.

• Searching space: The searching space is an N × N ideal searching space with k of m × m local basins, as shown in Figure 5.

In practice, the local basin side length m is related to the type of source, geometry layout of the environment and diffusion time. Without loss of generality, the starting points of the robot and locations of local basins are regulated as follows.

Assumption 1: The searching robot with step length 1 starts from a random point, in accordance with a uniform distribution.

Assumption 2: Local basins are located randomly in a searching space under the following conditions:

1 Local basins do not cover global maximum.

2 Local basins do not cover areas outside the searching space.

3 Coordinates of local basin centre (a, b) satisfy mod(a, m) = mod(b, m) = 0 where mod stands for modulo operation.

Figure 5 An N × N searching space with six m × m local basins (see online version for colours)

N

m

Obviously, the local basins are highlighted ⊂ in gradient fields while the noisy gradients are omitted. Henceforth, the term size is used to represent the side length for convenience. A searching space is fully defined by searching space size N, local basin size m, and local basin number k, as shown in Figure 5. For simplicity of calculation, N, m and k are considered as integers. The average travel cost of each algorithm is investigated in the following section.

3.2 Costs using FFS

One of the most significant merit in FFS is the travel cost has limited relationship with the RSS local basins. According to the rule of discarding impossible locations, the local basin will lose its attraction ability to the robot when the robot travels nearby. In most cases, the robot will just travel through the local basin with limited reiterations. In other words, an upper bound of the overhead in travel for each local basin met by the robot is m. When k local basins

30 X. Zhang et al.

are presented, a robot starting from a random point will meet α local basins on average, where 0 < α < k. The searching cost FFS

kS can be obtained by adding all the overhead of escaping local basins on the travel cost without a local basin 0 ,FFSS where the subscript k in FFS

kS indicates that k local basins are presented. If a robot starts from a random point, the probability that the robot access a local basin can be written as

( )

( )

2cos 2cos22

2 20

2cos

2 2

2 2

2 2

2

8 (cos sin )( 2 )

( 2 )8( 2 ) 16

2 ( 2 )

N Nπθ θ

mθ

mP r θ θ drdθN N m

N mm N m mNN N mm N m mN

N N m

−

= × +−

− + −= ×−

+ −=−

∫ ∫

(16)

where m denotes the size of a local basin, N denotes the size of the searching space.

Thus, the number of local basins encountered by a searching robot α ≈ k × P.

If there is no local basin in the searching space, the searching cost equals to the distance between a starting point and the target. Regarding the searching space centre as the origin in polar coordinates, the expected travel cost

0FFSS can be written as

2cos4

0 20 0

42

0

8

sec3

0.383 .

Nπθ

FFS

π

S r rdrdθN

N θdθ

N

= ×

=

≈

∫ ∫

∫ (17)

Therefore, the upper bound of the expected overall cost is

0 .FFS FFSkS S kP= + (18)

3.3 Costs using GACRW

Similarly, the travel cost using GACRW in gradient fields is derived in this section. Since local basins have notable influence to the total travel cost, it is necessary to perform the analysis step by step.

3.3.1 Costs of escaping a local basin

Turning angle after n continuous CRW

When a robot is trapped in a local basin, the CRW phase is triggered. Before the robot escapes the local basin, it may go back to the centre of local basin a few times. For example, after a few steps of CRW, the robot has a high chance to face back to the local basin centre. In this case, the phase of gradient ascent draws the robot to the local basin centre for one more time. Obviously, the escaping consists of CRW

which drives the robot outward, and the gradient ascent which pulls the robot back to the local basin centre.

The distribution of turning angle ω at each step can be approximated by a truncated Gaussian distribution

1

( )

2 2

ωσ σf ωπ πσ σ

⎛ ⎞⎜ ⎟⎝ ⎠=

⎛ ⎞ ⎛ ⎞Φ −Φ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

φ (19)

where φ denotes the probability density function of the standard normal distribution and Φ is its cumulative distribution function, σ denotes the standard deviation of the turning angle at each step.

Then, for continuous n steps of CRW, the distribution of turning angle can be approximated by a Gaussian variable with mean zero and variance 3nπ2 / 40. This approximation is good enough in finding expected displacement as shown in the follows.

Expected displacement after n continuous CRW

A common way to find the expected displacement after n continuous CRW is to find the root of mean square displacement (MSD) (Kareiva and Shigesada, 1983) instead. The approximation is accurate when n is not a large number (Byers, 2001). Let θi be defined as the turning angle at the ith step compared with the initial orientation, ωi as the turning angle at the ith step compared with that of the previous step,

as in Figure 6. Obviously, 1 1.

ii i i jjθ θ ω ω−

== + =∑

Then, the MSD after n continuous CRW can be written as

( ) ( )

( )

2 cos cos sin sin

cos .

i j i jn

i j

i ji j

θ θ θ θnE ER

θ θn E

≠

≠

⎛ ⎞++= ⎜ ⎟⎝ ⎠

⎛ ⎞−= + ⎜ ⎟⎝ ⎠

∑

∑ (20)

Figure 6 Sequential movements start from the origin

1 1

2 1 2+

3 2 3+

4 3 4+

Since max( , ) 1

min( , ),

i ji j ll i jθ θ ω

−

=− =∑ an approximated value of

2( )nE R can be obtained.

A novel radio source search algorithm using force field vectors and received signal strengths 31

Absorbing Markov chain in CRW

Before getting the overall cost, it is necessary to obtain the cost in escaping a single local basin. A difficulty is that the location of each step in CRW is not analytically tractable since it is not Markovian (Codling et al., 2008; Johnson et al., 2008). However, the moving average process (Lloyd, 1974) in the system can be viewed as a Markov chain if we define states as follows:

0 {Initial state: at the centre of the local basin},{Transient state : after consecutive CRW},

{Absorbing state: escapes the local basin}.i

esc

xx i ix

≡≡≡

Specifically, it is an absorbing Markov chain since the transition probability of the self loop in xesc is 1, as shown in Figure 7. To enter the state xesc, the cost χ can be resolved into two parts: the forward cost χf and backward cost χb. Forward cost χf denotes the transition cost from state i – 1 to i, *i∈ where * denotes the non-zero natural number; backward cost χb denotes the cost from state i to state 0.

.f bχ χ χ= + (21)

Figure 7 The absorbing Markov chain of the CRW

x0 x1 x2 xn

1 1p np

1q2q

nq

xesc

1s

2s

ns

In an absorbing Markov chain, all states except the absorbing state are termed transient states. At transient state xi, the robot has a probability pi to go to transient state xi+1, qi to go back to the initial state x0, si to enter the absorbing state xesc, and pi + qi + si = 1. For computational convenience, we regard a robot leaves a local basin if the displacement of robot from the centre of local basin is no less than / .m π

If we use e and f to denote the events of escaping a local basin and move to next state respectively, the corresponding transition probabilities of each state can be written as follows:

• In state x0, the transition probabilities are s0 = 0, p0 = 0.

• In state x1, s1 = 1 if m ≥ π2, otherwise s1 = 0.

• In state x2, s2 = Pr(e2 | ē1f1) 2 1 1

1 1

Pr( ) ,Pr( )

e e fe f

p2 = Pr(ē2 | ē1f1)

× Pr(f2 | ē1ē2f1) = Pr(ē2f2 | ē1f1) = 1 1 2 2

1 1

Pr( ) .Pr( )e f e f

e f

• In state x3, s3 = Pr(e3 | ē1f1ē2f2), p3 = Pr(ē3 | ē1f1ē2f2) ×

Pr(f3 | ē1f1ē2f2ē3) = Pr(ē3f3 | ē1f1ē2f2) = 1 1 2 2 3 3

1 1 2 2

Pr( ) .Pr( )e f e f e f

e f e f

All these probabilities can be written in multi-fold integral forms, for example

( )

( ) ( ) ( )1 1 2 2 3 3

1 1 2 2 3 3

1 2 331 2

Pr

e f e f e f

e f e f e f

p p p dω dω dωωω ω

=

∫ ∫ ∫ (22)

where p denotes the distribution function, ω1, ω2, ω3 denotes the turning angle at step 1, 2, 3, respectively, ē1 indicates m ≥ π2, ē2 indicates 2 + cos(ω2) ≤ 2 ,π

m ē3 indicates 3 + 2[cos(ω2) + cos(ω3) + cos(ω2 + ω3)] ≤

2 ,πm f1 is always true, f2 indicates 2kπ – 2

π ≤ tg–1

( )1 1 2

1 1 2

sin( ) sin( )cos( ) cos( )

ω ω ωω ω ω

+ ++ + ≤ 2kπ + 2 ,π f3 indicates 2kπ – 2

π ≤

tg–1 ( )1 1 2 1 2 3

1 1 2 1 2 3

sin( ) sin( ) sin( )cos( ) cos( ) cos( )

ω ω ω ω ω ωω ω ω ω ω ω

+ + + + ++ + + + + ≤ 2kπ + 2 ,π k ∈

where denotes the integer set.

To find the forward cost χf and backward cost χb, it is necessary to know the expected number of times the chain is in each state. For this purpose, we need to investigate on the transition matrix.

Renumber the states of a transition matrix so that the transient states come first. If there are r absorbing states and t transient states, the transition matrix can be written in the canonical form (Grinstead and Snell, 2006)

⎛ ⎞= ⎜ ⎟⎝ ⎠

Q RP

0 I (23)

Here, I is a r by r identity matrix, 0 is a r by t zero matrix, Q is a t by t matrix and R is a non-zero t by r matrix. In the absorbing Markov chain, r = 1 and t is infinity. Q and R are shown as follows.

1 1

2 2

3

1

0 1 0 0 00 0 00 0 00 0 0 0

0 0 0 0t

q pq pq

q −

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟

= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

Q (24)

( )1 2 10 Tts s s −=R (25)

Recall pi denotes the transition probability from xi to xi+1, qi denotes the transition probability from xi to x0, and si is the transition probability from xi to xesc. Due to the fact that si ≈ 1 when i is large, t can be truncated as a large number instead of infinity. A fundamental matrix N Grinstead and Snell (2006) can then be obtained as

1( )−= −N I Q (26)

The entry nij of N is the expected number of times that the chain is in xj, if it starts from xi. A vector L = (n12 n13 n14 ···

32 X. Zhang et al.

n1t) is picked up from the first row of N for the consequential calculation of travel cost.

Then, the forward cost χf is

fχ c= ×L (27)

where c is a column vector of all 1’s. The backward cost is

bχ v= ×L (28)

where v is a column vector whose entry vi = qi+1 × 2( ).iE R The expected cost of escaping a local basin can be found by (21).

However, as soon as a robot leaves a local basin, it has a probability to move back to the same local basin again. The probability depends on the position where the robot escapes. For example, if a robot escapes the leftmost local basin in the searching space in Figure 5 from the left edge, it will immediately enter the same local basin again. When m is much larger than 1, possible escaping positions can be regarded as uniformly distributed on the corresponding local basin perimeter. Under the Assumption 1 that local basins are square gradient fields, the probability of re-entering the same local basin is either 50% or 75%.

3.3.2 Costs with k local basins

If there is no local basin in the searching space, the searching cost 0

GACRWS will be the same as 0FFSS in (17).

Similarly, the probability that the robot access a local basin is the same as the probability in (16), and the number of local basins encountered by a searching robot is α ≈ k × P. However, as soon as a robot leaves a local basin, it has a probability of either 50% or 75% to re-enter the same local basin right away. Furthermore, for the probability of 50%,

the expected visiting time is 1

1 (1/ 2) 2;ii

∞

=+ =∑ for the

probability of 75%, the expected visiting time is

11 (3 / 4) 4.i

i

∞

=+ =∑ The probabilities are decided by the

locations of local basins. By taking the repetitive counts into consideration, the times of escaping local basins can be obtained as

2 6 .N mH kPN m+

= ×+

(29)

Therefore, the searching cost GACRWkS is

0 .GACRW GACRWχkS S H= + (30)

4 Simulation and results

4.1 Travel costs in gradient fields

The derived upper bound of FFS FFSkS and expected

searching cost GACRWkS are compared with the travel costs

obtained in numerical simulations. For each N, m, k set, 500

trials for each algorithm are simulated. A robot starts from a random point, will be regarded as arriving at the target when the global maximum is within one step. In this subsection, the costs are in terms of the unit step walk.

Figure 8 Travel cost versus local basin size m (N = 100, k = 15) (see online version for colours)

2 3 4 5 6 7 8 9 1020

40

60

80

100

120

140

160

180

mC

ost

S15FFS

Simulation cost of FFS

S15GACRW

Simulation cost of GACRW

Figure 9 Travel cost versus local basin number k (N = 100, m = 10) (see online version for colours)

2 3 4 5 6 7 8 9 1030

40

50

60

70

80

90

100

110

120

130

k

Cos

t

SkFFS

Simulation cost of FFS

SkGACRW

Simulation cost of GACRW

In Figure 8, it is shown that FFSkS and GACRW

kS fits the costs from simulation in gradient fields defined by N, m, and k. As the local basin size m grows, GACRW

kS increases in a quasi-quadratic ratio while FFS

kS does not change much. Similarly in Figure 9, when local basin number grows,

GACRWkS increase much faster that .FFS

kS The results are not surprising, since the relationships between the travel costs and the environment defined by N, m, and k are tractable through (16) (30). Note, the algorithm of FFS has a chance to fail if the robot starts within a local basin near the global maximum. In these simulations, the ratio of failure is 1.4%. Fortunately, in practical scenarios the search will not fail since the gradients can be calculated using several sequential RSS measurements. Furthermore, the random kick mechanism with limited overhead can be used to guarantee the success in gradient fields.

A novel radio source search algorithm using force field vectors and received signal strengths 33

4.2 Travel costs in a hallway

In this section, simulation results driven by real RSS data are illustrated. The scenario is the hallway of 6th floor of Steinmen Hall in CCNY, three radio sources are placed in different locations as shown in Figure 1. For collecting the RSS data, the hallway is partitioned into 343 uniform distributed grids. Ten readings from each radio source are collected in each grid. The simulated laser range finder has a detection range of 10 metres with angular resolution 1°. In simulation, we use step length 0.488 m, ε̂ = –65%, η1 = 1 m, η2 = 1 m, ζC = 10, ζB = 100, τC = 0.5, τB = 0.5, τG = 0.1. A search for a radio source will be regarded as succeed if the distance between robot and source is smaller than 0.6096 metres.

4.2.1 A sample of searching trajectory in a hallway

Here, a typical searching trajectory (Figure 10) is shown to illustrate two properties of FFS. First, the robot is neither loop nor trapped in small areas although the robot movements are not always towards the radio source. The reason is that the force field is generated by candidate area C. The local maxima of RSS cannot dominate the motions. Secondly, the search is flexible that is the robot is able to turn back fast when it is moving towards a wrong direction. This property is very important for a searching algorithm.

Figure 10 A typical searching trajectory (see online version for colours)

4.2.2 Comparison of travel costs

Simulations driven by real RSS data are carried out to compare FFS with three other algorithms: the GACRW, the Theseus gradient search (TGS) (Zhang et al., 2011) and the simplified chemotaxis (Nurzaman et al., 2009). While it is proposed that the starting positions are random, the direct comparison between travel distances holds limited meaning. Instead, the cost indicators, which are the ratios between travel distances and initial Euclidean distances, are collected in simulations for illustration purpose. For a robot initially 3 metres away from the radio source, if it travels 5 metres until reaching the radio source, the corresponding cost indicator is 5/3. In our simulation, 50 trials for each radio

source are launched. The starting positions are random selected from all vacant positions at least 2 metres away from the each radio source.

Figure 11 The histogram of cost indicator (see online version for colours)

0

5

10

15

Fre

quen

cy

Force Field Search

0

5

10

15

Fre

quen

cy

Gradient ascent with correlated random walks

0

5

10

Fre

quen

cy

Thesues gradient search

0 5 10 15 20 25 30 35 400

2

4

6

8

Fre

quen

cy

Cost indicator

Chemotaxis

Figure 11 shows the histogram of cost indicator in simulation. The vertical axis indicates the frequency of occurrence of cost indicators. For example, the frequency in the interval 5˜5.4 equals to 1 in FFS indicates that, in all 50 × 3 trials using FFS the cost indicator fall into interval 5˜5.4 once. 55% of FFS trials fall into intervals 1˜5, compared with 24% GACRW trials, 42% TGS trials and 19% chemotaxis trials fall into the same interval. This result is not surprising since GACRW uses CRW, TGS uses self-avoiding walk and chemotaxis uses biased random walk in escaping the RSS local basins while FFS is relatively invulnerable to the local basins. However, the FFS has a probability to fail: in our test, three failure trials are presented. Although robots use FFS are unlikely to be trapped in local basins, they have a chance to be trapped in force field locals. Specifically, by very particular chance the robot may loop its movements between two points forever. The FFS will also fail if the true radio source location is discarded by mistake. Nine GACRW, two TGS and 16 chemotaxis trials are out of the range in Figure 11 since cost indicators larger than 40 are not shown.

5 Conclusions

The FFS algorithm uses RSS and geometry layout information was proposed in this paper. Virtual attraction forces caused by possible radio source locations were modelled so as to guide the robot. Because the impossible radio source locations are gradually marked as discarded areas which have no virtual attraction force, the influences of local basins are alleviated. Compared with the algorithms using only RSS, the FFS costs much less travel distance in search. The gradient field which highlights the local basins was designed as a tool to model the standard searching

34 X. Zhang et al.

scenario. The average travel costs using the FFS and the GACRW in gradient fields were derived as functions of the searching space size, the local basin size and the local basin number. The derived expression revealed that FFS was much more efficient than GACRW and were further confirmed by simulations in gradient fields. Moreover, simulations driven by real RSS data are performed for the purpose of comparing FFS with other three source searching algorithms. It was shown that FFS was the most efficient.

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