Genetic algorithm based optimal placement of PIR ...

Genetic algorithm based optimal placement of PIR sensors for human motion localization Guodong Feng, Min Liu & Guoli Wang

Optimization and Engineering International Multidisciplinary Journal to Promote Optimization Theory & Applications in Engineering Sciences ISSN 1389-4420 Volume 15 Number 3 Optim Eng (2014) 15:643-656 DOI 10.1007/s11081-012-9209-z

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Author's personal copy Optim Eng (2014) 15:643–656 DOI 10.1007/s11081-012-9209-z

Genetic algorithm based optimal placement of PIR sensors for human motion localization Guodong Feng · Min Liu · Guoli Wang

Received: 29 May 2011 / Accepted: 24 December 2012 / Published online: 19 January 2013 © Springer Science+Business Media New York 2013

Abstract This paper studies the optimal placement of pyroelectric infrared (PIR) sensors in developing the infrared motion sensing system for human motion localization. In particular, we explore the use of genetic algorithm (GA) in optimizing both the deployment and the modulated field of view (FOV) of the PIR sensors for improving the localization performance. Two criteria, the average and maximum localization errors, are used to evaluate the localization performance. In addition, the numerical analysis is presented to offer a guidance on the searching spaces of the design parameters in implementing GA optimization. The proposed GA-based design approach is validated by means of both simulation and experimental studies in the context of human-following mobile robots. Keywords Optimal sensor placement · Pyroelectric infrared sensors · Human motion localization · Optimization design · Genetic algorithm

1 Introduction Optimal sensor placement has aroused considerable interests in robotics (Chen and Li 2004), wireless sensor network (Cheng et al. 2008; Chen et al. 2005; Boginski et al. 2011), computer vision (Horster and Lienhart 2006; Olague and G. Feng · M. Liu · G. Wang () School of Information Science and Technology, Sun Yat-sen University, Waihuan East Rd. 132, Guangzhou Higher Education Mega Center, Guangzhou 510006, China e-mail: [email protected] G. Feng e-mail: [email protected] M. Liu e-mail: [email protected]

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Mohr 2002; Bodor et al. 2007; Ercan et al. 2006; Erdem and Sclaroff 2004), structural health monitoring (Beal et al. 2008). The aim of the optimal sensor placement in these studies is to find appropriate sensing configurations so as to achieve one or more prespecified performance criteria in a cooperative fashion. For example, the studies in computer vision (Carr et al. 2005; Chen and Hanrahan 2002; Chen and Davis 2008) have shown that the optimal placement of multiple cameras can provide the best resolution and views for improving the localization performance. This paper concerns the optimal placement issue of multiple PIR sensors in developing our infrared motion sensing system for human motion localization. We focus on a GA-based optimal design approach for improving the system performance. The development of our infrared motion sensing system (Feng et al. 2012) is motivated by exploring a lightweight yet efficient implementation of the reference structure tomography (RST) (Brady et al. 2004) enhanced human motion localization with PIR sensors. The key to the RST enhanced human motion localization is the reference structure design, including both the deployment configurations and the visibility modulations of the PIR sensors involved. We addressed this issue partially by employing a two-layer structured sensing paradigm that facilitates the bearing-only localization. To fully address the reference structure design for our infrared motion sensing system, this study will take into account the issue of the optimal placement of PIR sensors under the developed two-layer structured sensing paradigm. It is followed from the recent studies in Bishop et al. (2010, 2009, 2008) that the reference structures associated with the bearing-sensitive visibility patterns can affect considerably the localization performance. The goal of this work is to find an optimal solution of the reference structure design for improving the localization performance. We will explore the use of the GA-based optimization approach in seeking appropriate deployment configurations as well as the modulated FOVs of PIR sensors. Two criteria, the average and maximum localization errors, are used as the GA-optimization objectives. In addition, the numerical localization error analysis is presented to provide a guidance on the searching spaces of optimization parameters. Finally, the proposed GA-based design approach is validated by means of both simulation and experimental studies. This paper is organized as follows. In Sect. 2, we will outline the sensing model of the infrared motion sensing system for human motion localization. In Sect. 3, the problem of optimal placement of bearing-sensitive PIR sensors is formulated and the GA-based design approach is presented as well. Two design examples are given in Sect. 4 and the conclusion is followed in Sect. 5.

2 Sensing model for human motion localization The task of the infrared motion sensing system is to localize human motion within its sensing area of interest. It consists of two layers: the geometric sensing layer and the cooperative sensing layer. At the geometric sensing layer, the bearing-sensitive PIR sensors are deployed on three semicircular bases for generating the bearing measurements of human motion from three different perspectives. At the cooperative sensing

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Fig. 1 (a) The bearing-sensitive PIR sensor, and (b) the bearing measurement model

layer, the human motion location is inferred through fusing the bearing measurements. In what follows, the geometric sensing layer and the cooperative sensing layer are outlined briefly, respectively. 2.1 Geometric sensing layer There are two stages in building the geometric sensing layer. The first stage is the development of the bearing-sensitive PIR sensors and the associated bearing measurement model. The second stage is the deployment of multiple bearing-sensitive PIR sensors for generating the bearing measurements. The design goal of a bearing-sensitive PIR sensor is to generate the bearing measurement of human target motion. To this end, the FOV of the PIR sensor is shaped as a fan-shaped cell so that only the human motion within the fan-shaped FOV is visible to the PIR sensor. As shown in Fig. 1(a), the associated FOV modulation can be easily achieved by applying a bearing-sensitive visibility mask to the corresponding Fresnel lens. Consider the bearing measurement model shown in Fig. 1(b), the bearing-sensitive PIR sensor is placed at s and the fan-shaped FOV is modulated to be θ . When the human motion occurred in the fan-shaped FOV, the bearing-sensitive PIR sensor will be triggered. Then the bearing measurement generated by the bearing-sensitive PIR sensor is ϕ = ∠b, where b and ∠b denote the bisector vector of the fan-shaped FOV and its angle to the positive X-axis, respectively. The deployment of the bearing-sensitive PIR sensors is to find the sensing configuration for the bearing-only localization. Consider the infrared motion sensing model in Fig. 2, we have A1 the circular sensing area of interest is centered at O0 with radius r0 ; A2 there are three semicircular bases for deploying the bearing-sensitive PIR sensors, where the kth semicircular base is centered at Ok with radius rk , k = 1, 2, 3; A3 L bearing-sensitive PIR sensors are deployed on each base circumference; A4 the lth bearing-sensitive PIR sensor on the kth semicircular base is deployed at skl (xkl , ykl ), and its modulated FOV is θkl , k = 1, 2, 3, l = 1, . . . , L.

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Fig. 2 The infrared motion sensing model

In short, there are totally 3L bearing-sensitive PIR sensors involved in the infrared motion sensing system. When the three bases are specified as a prior, the placement configurations of the 3L bearing-sensitive PIR sensors can be characterized by  T p = xT , yT , θ T (1) where x = [x11 , x21 , x31 , . . . , x1L , x2L , x3L ]T and y = [y11 , y21 , y31 , . . . , y1L , y2L , y3L ]T are the deployment parameters, θ = [θ11 , θ21 , θ31 , . . . , θ1L , θ2L , θ3L ]T modulated FOVs. Since the bearing-sensitive PIR sensors are deployed on the circumference of each semicircular base, the following geometric constraints are satisfied (xkl − xOk )2 + (ykl − yOk )2 = rk2

(2)

where (xOk , yOk ) is the coordinate of Ok . With this in mind, the design parameter p in (1) can be equivalently represented as:  T p = xT , θ T . (3) Remark 1 The bearing measurement generated by the lth bearing-sensitive PIR sensor on the kth semicircular base, ϕkl = ∠bkl , is given by yO − ykl . (4) ϕkl = arctan k xOk − xkl Remark 2 When m bearing-sensitive PIR sensors are triggered by the human motion within the sensing area, the associated bearing measurements are ϕki , i = l, . . . , l + m − 1. Then the bearing measurement ϕk generated by the kth semicircular base is given by ϕk =

l+m−1 1  ϕki . m i=l

(5)

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Fig. 3 The localization errors with respect to the modulated FOV θ

2.2 Cooperative sensing layer Firstly, we describe the fusion method of the bearing measurements for inferring the human motion location. Then we present the numerical localization error analysis as a guidance on the searching design parameters. When the human motion triggers one or more bearing-sensitive PIR sensors in each base, the bearing measurements, {ϕ1 , ϕ2 , ϕ3 }, can be generated accordingly, the least square estimation for the human motion location p(x, y) is given by  −1 pe = A T A A T b (6) where pe = [xe ye ]T , A and b are given by ⎡ sin ϕ1 A = ⎣ sin ϕ2 sin ϕ3 ⎡ xO1 sinϕ1 b = ⎣ xO2 sinϕ2 xO3 sinϕ3

⎤ − cos ϕ1 − cos ϕ2 ⎦ − cos ϕ3

⎤ −yO1 cos ϕ1 −yO2 cos ϕ2 ⎦ −yO3 cos ϕ3

respectively. Consider the localization errors due to the deployment configurations and modulated FOVs, defined by  e(p) = (xe − x)2 + (ye − y)2 . (7) Figure 3 plots the relation between the localization error e(p) and θ , where the result is the average localization errors of all 22400 samples within the sensing area of interest. Here the modulated FOVs of all bearing-sensitive PIR sensors are the same, that is, θkl = θ for all k and l; {∠sk1 Ok skL /L, k = 1, 2, 3} are the same, denoted as γ .

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It can be seen from Fig. 3 that the localization error is the smallest when θ = γ /L. Motivated by the above numerical analysis, the searching space of θ is in the form of θ ∈ [γ /L − θ , γ /L + θ ].

(8)

where θ is half size of the searching space of θ . It should be noted that (8) can provide a guidance on determining the searching space of the design parameter θ .

3 GA-based design approach Firstly, the problem of optimal placement of bearing-sensitive PIR sensors is formulated. Then, the GA-based design approach is developed. 3.1 Statement of the optimal placement problem Recalling (1), the placement configurations of 3L bearing-sensitive PIR sensors are characterized by p = [xT , θ T ]T . It is followed from (2) and (8), the searching space of xkl in x and θkl in θ are xkl ∈ [xOk − rk , xOk + rk ]

(9)

θkl ∈ [γk /L − θ , γk /L + θ ]

(10)

and

respectively. In this paper, the criteria to evaluate the localization performance are chosen as average and maximum localization errors, which are widely used to evaluate the localization performance, for example in references (Bulusu et al. 2000; Shen et al. 2005; Cuffin et al. 2001; Feng et al. 2011). We denote the average and maximum localization errors as eavg and emax , then the optimization criteria for the design of infrared motion sensing model are denoted with a vector g: g = [eavg , emax ]T .

(11)

The average and maximum localization errors eavg and emax can be approximated by eavg ≈

N 1  e(pn ) N

(12)

n=1

and emax ≈ max e(pn ) n

(13)

where e(pn ) denotes the localization error of the human motion occurred at the sample pn . In this way, the problem of the optimal sensor placement is formulated as minimize p

g.

In what follows, GA is applied to find a solution of (14).

(14)

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3.2 GA-based optimal design approach The formats of the chromosome C in a population for p are as follows: C = [x11 , x21 , x31 , . . . , x1L , x2L , x3L | θ11 , θ21 , θ31 , . . . , θ1L , θ2L , θ3L ].

(15)

C is coded as vectors of floating numbers for the solution vector. It is followed from (9) and (10), the searching space of xkl is [xOk − rk , xOk + rk ] and the searching space of θkl is [γk /L − θ , γk /L + θ ], k = 1, 2, 3, l = 1, . . . , L. The fitness function is used to evaluate individual chromosome in a population, and the chromosome with larger fitness value has a higher probability to be selected into the next generation. Reproductive success varies with fitness function. Having considered the fact that we aim to minimize eavg and emax , the fitness function F for evaluating each chromosome C can be denoted as: F = K1 /eavg + K2 /emax ,

(16)

where K1 and K2 denote the weight of eavg and emax , respectively. The role of K1 and K2 is controlling the importance of criteria eavg and emax . It can be seen from (16) that with the decrease of eavg and emax , the fitness value F will increase rapidly. In other words, the optimal solution is obtained when F is maximized by GA. The pseudo-code of GA-based design procedure is as follows 1. Begin 2. Initialization: Set the population size Npop , the maximum number of generations Mgen , the probability of crossover Pc and mutation Pm , and generation counter gen = 0. Randomly generate a population {Ci (0), i = 1, . . . , Npop } with Npop chromosomes. Calculate the fitness value F (Ci (0)) of each chromosome Ci (0), i = 1, . . . , Npop . Find the best fitness value Fbest (0), the corresponding chromosome Cbest (0), the worst fitness value Fworst (0), and the corresponding chromosome Cworst (0). 3. Selection: According to the roulette wheel rule, select Npop chromosomes from {Ci (gen − 1), i = 1, . . . , Npop } to form {Ci (gen), i = 1, . . . , Npop }. 4. Crossover: Select every two chromosomes from {Ci (gen), i = 1, . . . , Npop } to exchange genes according to crossover strategy until all chromosomes are considered. 5. Mutation: Select a chromosome from {Ci (gen), i = 1, . . . , Npop } to mutate one gene according to the mutation strategy until all chromosomes are considered. 6. Calculation and Preservation: Calculate the fitness value F (Ci (gen)) of each chromosome Ci (gen), i = 1, . . . , Npop . Find the best fitness value Fbest (gen), the corresponding chromosome Cbest (gen), the worst fitness value Fworst (gen), and the corresponding chromosome Cworst (gen). If Fbest (gen − 1) < Fbest (gen), replace Cworst (gen) with Cbest (gen − 1), else replace Cbest (gen) with Cbest (gen − 1). gen = gen + 1

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Fig. 4 PIR sensing model in the design example

7. If gen ≤ Mgen , go to Step: Selection; 8. End 9. Output: the best chromosome Cbest (Mgen ). The details about the steps of GA can be found in (Zhang et al. 2001; Houck et al. 1995; Whitley 1994).

4 Design examples The above design approach is illustrated in the context of the infrared motion sensing system with human-following mobile robots. The schematic of the infrared motion sensing system is shown in Fig. 4, of which the known system parameters are listed in Table 1. The bearing-sensitive PIR sensors are deployed on the three semicircular bases. The GA-based design approach is used to optimize the placement of the bearing-sensitive PIR sensors. Specifically, two design examples are presented, that is, • Type one: 3L = 15 • Type two: 3L = 9 The chromosome C of p for the two design examples are Type one : C = [x11 , x21 , x31 , . . . , x15 , x25 , x35 | θ11 , θ21 , θ31 , . . . , θ15 , θ25 , θ35 ]. (17) Type two : C = [x11 , x21 , x31 , . . . , x13 , x23 , x33 | θ11 , θ21 , θ31 , . . . , θ13 , θ23 , θ33 ]. (18)

Author's personal copy Genetic algorithm based optimal placement of PIR sensors Table 1 The values of known system parameters

651

Known system parameters

Value

r0

40 cm

r1 = r2 = r3

20 cm

O0

(0, 150)

O1

(−40, 20)

O2

(0, 0)

O3

(40, 20)

A1

[π/3, π/2]

A2

[5π/12, 7π/12]

A3

[π/2, 2π/3]

Table 2 The searching space of each parameter to be optimized by GA Type one

Type two

Variables

Range (cm)

Variables

Range (cm)

x11 , . . . , x15

[−40.5, −29.5]

x11 , . . . , x13

[−40.5, −29.5]

x21 , . . . , x25

[−5.5, 5.5]

x21 , . . . , x23

[−5.5, 5.5]

x31 , . . . , x35

[29.5, 40.5]

x31 , . . . , x33

[29.5, 40.5]

θkl , k = 1, 2, 3, i = 1, . . . , 5

[π/36, π/9]

θkl , k = 1, 2, 3, i = 1, 2, 3

[π/36, π/9]

The searching spaces of the parameters xkl and θkl are listed in Table 2. The optimization criteria for both design examples are

N 1  g= e(pn ), max e(pn ) . (19) n N n=1

The fitness function is F = K1 /

N 1  e(pn ) + K2 /max e(pn ), n N

(20)

n=1

where K1 and K2 are selected to be 100 and 20. In the two design examples, we use the same parameters of GA, where Npop = 100, Mgen = 1000, Px = 0.85, and Pm = 0.1. The computer program stops when the generation counter gen is larger than the maximum generation Mgen . The proposed method requires about 3.06 hours for Type one and 2.53 hours for Type two, where the computer is a Pentium(R) Dual-Core CPU E5300 2.60 GHz RAM:2G machine. The fitness values of the two design examples are shown in Fig. 5. It can be observed from Fig. 5 that the fitness values have come to a better level after the GA-based optimization. The eavg and emax of the two design examples in each iteration are shown in Figs. 6 and 7. It can be seen from Figs. 6 and 7 that for Type one, the eavg and emax are 9.8 cm and 28.6 cm after GA-based optimization; for Type two, the eavg and emax are 12.8 cm and 35.5 cm after GA-based optimization. However, with uniform sensor

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Fig. 5 The fitness values in each generation

Fig. 6 The best average localization error in each generation

placement (as mentioned in Sect. 2), for Type one, the eavg is about 13.5 cm; for Type two, the eavg is about 20 cm. In order to test the result of GA-based design approach, a system prototype with 3L = 9 bearing-sensitive PIR sensors is designed in the context of human-following mobile robot, which is shown in Fig. 8. The sensor placement parameters use the optimal solution from Type two, which are listed in Table 3. The results of the human motion localization experiments are listed in Table 4. It can be seen from Table 4 that the average and maximum localization errors are 13.8 cm and 24 cm, respectively. These results are in close agreement with the results (eavg = 12.8 cm, emax = 35.5 cm) of GA-based design approach. The localization errors within the sensing area of interest are shown in Fig. 9. It can be seen from Fig. 9 that the average and maximum localization errors are 13 cm and 35 cm, respectively. These results are agree with the GA-based optimization results (eavg = 12.8 cm, emax = 35.5 cm).

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Fig. 7 The best maximum localization error in each generation

Fig. 8 The system prototype with nine bearing-sensitive PIR sensors

5 Conclusion The optimal PIR sensors placement is studied for improving the localization performance, and the schematic implementation of GA-based design approach is presented in this paper. The process entails the selection of the deployment configurations and the modulated FOV of PIR sensors. The experimental results validate the proposed design approach. Currently, most PIR sensing systems (Hao et al. 2006; Shankar et al. 2006; Fang et al. 2006; Lee et al. 2006; Liu et al. 2012) are designed by manually placing the PIR sensors according to the designers’ experience. Obviously, when the sensor network becomes larger, the system design will be more difficult and less flexible. However, the proposed GA-based design approach enables flexible and near-optimal design with little prior knowledge and experience.

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Fig. 9 The localization errors within the sensing area

Table 3 The optimal placement parameters for the system prototype

Table 4 The experimental results, the p and pe denote the real and estimated human motion location

Parameters Value (cm) Parameters Value (cm) Parameters Value x11

−37.781

y11

39.8765

θ11

0.2007

x12

−36.075

y12

39.6111

θ12

0.2909

x13

−33.0222 y13

38.7433

θ13

0.3141

x21

−2.6001

y21

19.8303

θ21

0.1891

x22

−1.5903

y22

19.9367

θ22

0.1811

x23

3.4001

y23

19.7089

θ23

0.2247

x31

30.2926

y31

37.4862

θ31

0.2651

x32

34.8583

y32

39.3278

θ32

0.2306

x33

37.5671

y33

39.8515

θ33

0.2707

Number

p (cm)

pe (cm)

Localization error (cm)

1

(−25, 120)

(−17.5, 132.5)

14.6

2

(0, 120)

(−6.1, 108.9)

12.6

3

(25, 120)

(24.8, 144.4)

24

4

(−25, 150)

(−23.2, 163.4)

13.5

5

(0, 150)

(6.7, 142.3)

10.2

6

(25, 150)

(23.7, 159.2)

9.2

7

(−25, 175)

(−23.2, 163.4)

11.8

8

(0, 175)

(−12.6, 176.3)

12.6

9

(25, 175)

(23.7, 159.2)

15.9

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Acknowledgements This work was partly supported by the National Nature Science Foundation under Grant 60775055 and 61074167.

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