A Three-Dimensional Wireless Indoor Localization System

Abstract

Indoor localization, an emerging technology in location based service (LBS), is now playing a more and more important role both in commercial and in civilian industry. Global position system (GPS) is the most popular solution in outdoor localization field, and the accuracy is around 10 meter error in positioning. However, with complex obstacles in buildings, problems rise in the “last mile” of localization field, which encourage a momentum of indoor localization. The traditional indoor localization system is either range-based or fingerprinting-based, which requires a lot of time and efforts to do the predeployment. In this paper, we present a 3-dimensional on-demand indoor localization system (3D-ODIL), which can be fingerprint-free and deployed rapidly in a multistorey building. The 3D-ODIL consists of two phases, vertical localization and horizontal localization. On vertical direction, we propose multistorey differential (MSD) algorithm and implement it to fulfill the vertical localization, which can greatly reduce the number of anchors deployed. We use enhanced field division (EFD) algorithm to conduct the horizontal localization. EFD algorithm is a range-free algorithm, the main idea of which is to dynamically divide the field within different signature area and position the target. The accuracy and performance have been validated through our extensive analysis and systematic experiments.

1. Introduction

The next evolutionary in smart environments develops fast in building, utilities, industrial, home, shipboard, and transportation system automation. One of the crucial technologies and scientific researches of smart environments is based on location-aware systems. Location-aware systems have improved the quality of life and provide people with convenience in working, outing, shopping, dining, and many other activities.

In recent years, with the open-sourced Android operating system and the continuous cost-reduction in hardware, smart phone is getting more and more popular. Android/IOS based smart phones can be seen everywhere, and we can easily get the location of our friends with the widely used location-aware systems, like global position system (GPS). GPS is the most popular solution in outdoor localization field, and the accuracy is around 10 meter error in positioning. It provides good user-experience in outdoor scenarios. However, when we try to use GPS in indoor environment, like in an office or at home, we can barely receive the signals of satellites, and the error of accuracy will be out of tolerance. The root cause is that with complex obstacles in buildings the satellite signals cannot transmit the walls and floors.

Thus, we need an indoor localization system to do the “last mile” localization. The “last mile” or “last kilometer” is a phrase used by the telecommunications, cable television, and Internet industries to refer to the final leg of the telecommunications networks delivering communications connectivity to retail customers. In localization field, we also have the “last mile” problems, and we need the indoor localization systems that can make up the weakness of GPS system.

Over the past decade, indoor-localization-based technology and algorithms are flourishing, and almost all indoor localization systems are based on signal analysis of wireless sensor networks (WSNs). Generally speaking, indoor location mechanisms can be separated into two parts: range-based method and range-free method. Range-based methods are the mainstream ways to do indoor location in the early years, such as TOA [1], TDOA [2], AOA [3], and RSSI [4]. Since the range-based method requires expensive hardware and precise statistics, the range-free algorithms develop fast in recent years, such as RSD, EZ, BT, and UnLoc. However, some limitations restrain these algorithms from generalized indoor location, for fingerprinting technology needs lots of precollection of data.

Thus, we propose the idea of 3-dimensional on-demand indoor localization system (3D-ODIL), which can be fingerprint-free and deployed effectively in a multistorey building. The contributions are as follows.

In this paper, we propose the idea of 3-dimensional on-demand indoor localization system (3D-ODIL), which can be fingerprint-free and deployed rapidly in a multistorey building. The 3D-DDIL consists of two parts, horizontal localization and vertical localization.

On horizontal direction, we use enhanced field division (EFD) algorithm to conduct the horizontal localization. EFD algorithm is a range-free algorithm, which implements the idea that the whole map is dynamically divided into fields that with signature sequences mark and locate the target device. The key idea of EFD provides a new developing direction for indoor localization. Traditional ways focus on the actual value of the received signal strength (RSS), but the value of which is fluctuant and greatly impacted due to the complex environments with obstacles. The EFD algorithm lays emphasis on the possible position on the map and utilizes the high-low relationship between the RSSs from each anchor to decide which areas the receiver should lies in. This allows for the minimization of the RF fading effect. What is more, taking advantage of field division with optimal anchor placements, we avoid the issues of intensive fingerprint sampling. Fingerprinting technology is a highly precise way to do indoor localization, but the weakness is also obvious: it need more time to find optimal anchor placements.

On vertical direction, we implement multistorey differential (MSD) algorithm to fulfill the vertical localization, which can greatly reduce the number of anchors deployed. Vertical localization, or called multistorey positioning, is an underestimated aspect to be discussed in indoor localization field. With anchors predeployed in the building in typical system, we can equip anchors with the same allocation plan on each floor, and it is simple to determine the floor: the floor of the anchor which has the strongest signal. However, in on-demand situation, we need to do rapid deployment, as few anchors as possible. We need to divide more floors with limited anchors. The MSD algorithm provides an efficient way to do vertical localization with few anchors.

Finally, we build a localization system platform, as shown in Figure 7, from off-the-shelf commercialized products. The system includes three major components: the mobile anchor platform, a mobile localization unit, and a back-end server. The mobile anchor platform can be a portable LTE/GSM mini-radio station or a traditional Wi-Fi router, which act as a radio signal generator. The mobile localization unit is an Android-based smart phone, which is a commercial device and we can use it to receive signals and communicate with the server. The back-end server plays the role of data storage and computing unit.

The rest of the paper is organized as follows. Multistorey differential algorithm is presented in Section 2. Enhanced field division (EFD) algorithm is discussed in Section 3. Analysis data of MSD algorithm is illustrated in Section 4. The performance of the prototype system in practical environments is evaluated in Section 5. Section 6 briefly discusses related work, and Section 7 concludes the paper.

2. Multistorey Differential Algorithm

Three-dimensional localization system should provide both horizontal and vertical localization functions as a comprehensive system. In contrast to horizontal location, vertical localization has quite different model. The insertion loss of the reinforced wall (about 20–40 dB) is larger comparing to the brick wall (about 10 dB to 20 dB) [5]. Since the separation of reinforced concrete walls between all floors, it is inaccurate to use typical trilateral RSSI localization system to do vertical localization. Wireless indoor localization without site survey (WILL) [6] provides a fingerprinting system to do 3-dimensional localization. Unfortunately, there is no mature and general vertical localization which can be rapidly deployed on-demand. Thus, we designed multistorey differential (MSD) algorithm.

2.1. Fundamental Theory of Vertical Localization

Figure 1 shows the basic building model we use. Assume that the height of the building is 𝐻(𝑚), and the height of each floor is ℎ(𝑚). We set up two anchors on the first floor (𝑆1) and top floor (𝑆2), which are at the same horizontal position. Since we focus on the vertical localization, we can simplify the 3-dimensional system of coordinate to 2-dimensional system of coordinate, of which axis 𝑍 is the vector from 𝑆1 to 𝑆2 and axis 𝑋 is the distance between our test point and axis 𝑍. The position of the test point (𝑃1) is (𝑥,𝑧).

Figure 1 

Basic model of MSD.

Definition 1 (FSPL). FSPL is the abbreviation of free-space path loss, which is the difference between transmitter power and received signal power, and also takes the gain of two antennas into consideration: FSPL=𝑃_𝑇+𝐺_𝑇+𝐺_𝑅−𝑃_𝑅, (1) where 𝑃_𝑅 is the received signal power, 𝑃_𝑇 is the base station (BS) transmitter power, 𝐺_𝑇 is the transmitting antenna gain, and 𝐺_𝑅 is the receiving antenna gain.

Definition 2 (MSPL). MSPL is the abbreviation of multistorey path loss, which is the extension of FSPL with the insertion loss of reinforced concrete wall: MSPL=FSPL+𝑁∗𝐿_𝐼, (2) where 𝑁 is the number of walls inserted in the path and 𝐿_𝐼 is the insertion loss of reinforced concrete wall (about 20–40 dm) [5].

Theorem 3. Consider 𝑀𝑆𝑃𝐿=20log𝑑+20log𝑓+20log4𝜋𝑐+𝑁∗𝐿_𝐼. (3)

Proof of Theorem 3. Corresponding values of path losses were computed from the measured received signal power using the expression [7] 𝐿_𝑝=EIRP−𝐺_𝑅−𝑃_𝑅, (4) where EIRP is the effective isotropic power of the BS transmitting antenna. For the free-space model, the expression for the received signal power, with directional base station antenna, is [8, 9] 𝑃_𝑅=𝑃_𝑇∗𝐺_𝑇∗𝐺_𝑅∗𝜆4𝜋𝑑², (5) where 𝜆 is the wavelength and 𝑑 is the distance between the BS antenna and mobile station (MS) receiver antenna. If 𝑃_𝑅 and 𝑃_𝑇 are expressed in dBm and the gains are in dB, (2) becomes [10] 𝑃_𝑅=𝑃_𝑇+𝐺_𝑇+𝐺_𝑅𝜆+20log.4𝜋𝑑 (6)
Consider the FSPL, which is defined in (1). When substituting 𝜆=𝑐/𝑓, (3) and (4) become FSPL=20log4𝜋𝑑𝜆=20log𝑑+20log𝑓+20log4𝜋𝑐. (7)
Also, we can get the MSPL equation MSPL=20log𝑑+20log𝑓+20log4𝜋𝑐+𝑁∗𝐿_𝐼. (8)

For typical radio applications, it is common to find 𝑓 measured in units of MHz and 𝑑 in meter. Assuming that we use Wi-Fi routers as the anchors (𝑓=2400 MHz), the MSPL equation becomes MSPL=20log4𝜋𝑑𝜆+𝑁∗𝐿_𝐼=20log𝑑+𝑁∗𝐿_𝐼+𝑘_MSPL, (9) where 𝑘_MSPL is a constant value, and we can calculate the exact value as 40.05 in our assumption.

As we can see, 𝑁 equals the floor integer of 𝑧/ℎ, which is [𝑧/ℎ]. At test point, we can receive the signals from both 𝑆1 and 𝑆2. The MSPL for 𝑆1 is MSPL_𝑆1=20log√𝑥²+𝑧²+𝑘_MSPL+𝑧ℎ∗𝐿_𝐼. (10)

And the MSPL for 𝑆2 is MSPL_𝑆2=20log𝑥²+(𝐻−𝑧)²+𝑘_MSPL+(𝐻−𝑧)ℎ∗𝐿_𝐼. (11)

Considering the variant of MSPL_𝑆1 and MSPL_𝑆1, ΔMSPL=MSPL_𝑆2−MSPL_𝑆1=20log𝑥²+(𝐻−𝑧)²𝑥²+𝑧²+(𝐻−2𝑧)ℎ∗𝐿_𝐼. (12)

In our scenario, our target is to conduct localization in multistorey building, and we should combine our math equation with the given conditions. Thus, we focus on the exact floor rather than the precise height.

Theorem 4. When we are on different floors of a multistorey building, we can distinguish the Δ𝑀𝑆𝑃𝐿 value regardless of the horizontal point we stay at.

Proof of Theorem 4. Since we are aiming to distinguish the floor, we assume that the vertical coordinates are evenly spaced, and the absolute height is (𝑁_floor−1)∗ℎ, where 𝑁_floor is the floor number. To prove our theorem, we receive the ΔMSPL at two different points, which are noted as 𝑃1 (𝑥₁,𝑧₁) and 𝑃2 (𝑥₂,𝑧₂): ΔMSPL_𝑃1−ΔMSPL_𝑃2=20log𝑥²₁+𝐻−𝑧₁²𝑥²₁+𝑧²₁+𝐻−2𝑧₁ℎ∗𝐿_𝐼−20log𝑥²₂+𝐻−𝑧₂²𝑥²₂+𝑧²₂−𝐻−2𝑧₂ℎ∗𝐿_𝐼. (13)
Consider the relationship between 𝑧₁ and 𝑧₂. Obviously, if we can distinguish different points of the floor 𝑛 and floor 𝑛+1, we can easily distinguish each floor of the building. Thus, we need to make research on the extreme condition that 𝑧₂=𝑧₁+ℎ, and we generalize 𝑧₁ as 𝑦. Thus, (13) becomes ΔMSPL_𝑃1−ΔMSPL_𝑃2=20log𝑥²₁+𝐻−𝑧₁²𝑥²₁+𝑧²₁+𝐻−2𝑧₁ℎ∗𝐿_𝐼−20log𝑥²₂+𝐻−𝑧₂²𝑥²₂+𝑧²₂−𝐻−2𝑧₂ℎ∗𝐿_𝐼=20log𝑥²₁+𝐻−𝑧₁²𝑥²₁+𝑧²₁∗𝑥²₂+𝑧²₂𝑥²₂+𝐻−𝑧₂²−2∗𝐿_𝐼𝑥=10log²¹+𝐻−𝑧₁²𝑥²₁+𝑧²₁∗𝑥²₂+𝑧²₂𝑥²₂+𝐻−𝑧₂²−2∗𝐿_𝐼. (14) If we want to use this theorem to localize the floor, we need to decrease the impact of 𝑥. We separate it into two parts:
(1)  𝑥₁=𝑥₂.
We name Δ_DMSPL from ΔMSPL_𝑃1−ΔMSPL_𝑃2. Equation (14) becomes Δ_DMSPL=ΔMSPL_𝑃1−ΔMSPL_𝑃2𝑥=10log²+𝐻−𝑧₁²𝑥²+𝑧²₁∗𝑥²+𝑧²₂𝑥²+𝐻−𝑧₂²−2∗𝐿_𝐼. (15)
Take 𝑥 as variable; we can get the derivative of the function: 𝑑Δ_DMSPL=𝑑𝑥20𝑥∗1ln10𝑥²+𝐻−𝑧₁²−1𝑥²+𝐻−𝑧₂²+1𝑥²+𝑧²₂−1𝑥²+𝑧²₁. (16)
When 𝑥>0, the derivative 𝑑Δ_DMSPL/𝑑𝑥<0, which means that Δ_DMSPL is a monotonic decreasing function with variable 𝑥 when 𝑥>0. When 𝑥→0, (15) becomes Δ_DMSPL=20log𝐻∗ℎ(𝐻−𝑧−ℎ)∗𝑧+1−2∗𝐿_𝐼. (17)
When 𝑥→∞, (15) becomes Δ_DMSPL=−2∗𝐿_𝐼. (18)
(2)  𝑥₁≠𝑥₂.
We assume that 𝑥₁<𝑥₂. If we need to evaluate the impact of variables 𝑥₁,  𝑥₂, we can analyze the extreme condition: 𝑥₁→0, 𝑥₂→∞. Equation (14) becomes Δ_DMSPL=10log𝐻−𝑧₁²𝑧²₁∗1−2∗𝐿_𝐼𝐻=20log𝑧−1−2∗𝐿_𝐼. (19)
Equations (17), (18), and (19) demonstrate the different condition of Δ_DMSPL, the first item of them has minor impact (about 0–10 dB), and the second item has a major impact (about 40–50 dB).
With the deduction above, we can use Δ_DMSPL as the reference value to distinguish different floors.

2.2. Modeling with Multiple Anchors on Each Floor

We can separately do the vertical localization with the value of ΔMSPL theoretically, but, in real conditions, with the complex indoor environment and the fluctuation feature of wireless signal, we need to design a stable and multianchor model to enhance the accuracy of vertical localization.

The fundamental theory of vertical localization analyzes the condition of 1-pair-anchor deployment, and we want to generalize it to multianchor model.

Consider the actual deployment scenario. We need to deploy wireless anchors in the whole building. For floor 𝑖, we put 𝑚_𝑖 anchors on it.

As we name our system on-demand, we should minimize the number of anchors deployed with less impact on the accuracy of the whole system. With the fundamental theory of vertical localization, we can distribute the anchors on some specific floors, which have an interval of 𝑛 floors: 𝑚_𝑖=𝑚_𝑖,if𝑖%𝑛=1,0,otherwise. (20) The 𝑚_𝑖 of each floor is decided by the anchors needed for horizontal localization system.

If a building has more storeys than 𝑛+1, we can degrade this building to several subbuildings with 𝑛+1 storeys and filter the receiving wireless signal only from floor 𝑡 (the first floor of subbuilding) and floor (𝑡+𝑛).

Now we have a (𝑛+1) storey subbuilding model, and we need to decide the interval 𝑛 and figure out the 𝑚-anchor algorithm.

The interval 𝑛 is decided by the value that how many anchors on (𝑛+1)th floor can be detected on the 𝑖th floor. We note the value as 𝑟_𝑖. Obviously, 𝑟_𝑖 must be a nonnegative number, and 𝑟₁ is the minimum of 𝑟_𝑖. We name 𝑅 the threshold of the subbuilding model, which equals 𝑟₁.

To do the vertical localization, we need to guarantee that 𝑅≥1. Referring to (3) and (8) and substituting 𝑁=𝑛−1: 𝑃_𝑅=𝑃_𝑇+𝐺_𝑇+𝐺_𝑅−20log𝑑−𝑘_MSPL−(𝑛−1)∗𝐿_𝐼. (21) Consider the normal usage conditions: 𝑃_𝑇=20 dBm, 𝐺_𝑇=17 dBm, and 𝐺_𝑅=2 dBm. The smart phone can detect the wireless signals that 𝑃_𝑅 is more than −99 dBm. Thus, (𝑛−1)∗𝐿_𝐼<97.95−20log𝑑. (22) If 𝑑_max=30 m and 𝐿_𝐼=20 dB, we can get the 𝑛_max=4.

When we receive 𝑟_𝑖 wireless signals on the 𝑖th floor, we need to take advantage of all received signals to enhance the accuracy and the success rate of vertical localization.

We assume that the value of ΔMSPL on each floor is in a narrow interval, for we can separate the value from different floors. Let us define the weight assigned to different anchors as 𝑊={𝑤₁,𝑤₂,…𝑤_𝑖,…,𝑤_𝑘}; is the weight for anchor {𝑖}. As such, the ΔMSPL value of multianchor mode will be determined as ΔMSPL_WAVG=_𝑖𝑤_𝑖∗ΔMSPL_𝑖. (23)

Now we should consider the determination of 𝑊: 𝑈𝜌,𝜎²_𝜌=𝜌−𝛾𝜎²_𝜌, (24) where 𝜌 is defined as the average of the signal, 𝜎²_𝜌 is the standard deviation of 𝜌, and 𝛾 defines a relative interference coefficient that describes the loss in signal strength due to environmental interference. The 𝑟 will be higher for a signal unfriendly environment such as a concrete wall building than for an outdoor environment known for less interface. Therefore, to improve the accuracy, a signature must be selected. Let us define the weight assigned to different anchors as 𝑊={𝑤₁,𝑤₂,…,𝑤_𝑛}; 𝑤_𝑖 is the weight for anchor_𝑖. As such, the optimization problem will be to determine 𝑊 so that the total utility is maximized; that is, max_𝑊𝑖𝑈𝜌_𝑖,𝜎²_𝜌𝑖,s.t.𝑊^′𝑖=1,𝑤_𝑖>0. (25)

Lemma 5. The optimal solution of  𝑊 will be 1𝑤=𝛾𝑄⁻¹𝜌+𝜆₁. (26)

Proof of (26). Under our definition, the total utility can be written as ^𝑛_𝑖=1𝑤_𝑖𝑈𝜌₁,𝜎²_𝜌𝑖=^𝑛_𝑖=1𝑤_𝑖𝜌_𝑖−𝑟^𝑛_𝑖=1𝑤²_𝑖𝜎²_𝜌𝑖+2^𝑛𝑛_{𝑖=1𝑗≠𝑖}𝑤_𝑖𝑤_𝑗Cov𝜎_𝜌𝑖,𝜎_𝜌𝑗=𝑊𝜌−𝑟𝑊^′𝑄𝑊, (27) in which, 𝜎𝑄=^2𝜌1𝜎_𝜌1,𝜌2⋯𝜎_{𝜌1,𝜌𝑛}𝜎_𝜌2,𝜌1𝜎²_𝜌2⋯𝜎_{𝜌1,𝜌𝑛}𝜎⋮⋮⋱⋮_{𝜌𝑛,𝜌1}𝜎_{𝜌𝑛,𝜌2}⋯𝜎²_𝜌𝑛, (28)𝜎_{𝜌𝑖,𝜌𝑗} is the degradation correlation between anchors 𝑖 and 𝑗, and 𝑛 is the number of anchors. The numerical solution of 𝑊 can be solved by Lagrangian derivation. Let us form 𝐶=𝑊^′𝜌−𝛾𝑊^′𝑄𝑊−𝜆₁𝑊^′1−1, (29) where 𝜆₁ is a Lagrange multiplier. Then, taking derivatives of the Lagrangian with respect to vector 𝑊^′, we get 𝛿𝐿𝛿𝑊^′∶𝜌−𝛾𝑄𝑊−𝜆₁=0. (30) As a result, this proves (26).

The optimal 𝑊={𝑤₁,𝑤₂,…,𝑤_𝑛} can then be applied to our measurement of RSS_𝑡 as WRSS_𝑡 to derive the optimal field signature 𝑠^∗(𝑡).

3. The Algorithm of Enhanced Field Division

3.1. Field Division

During our previous research, we find that the relative signal strength rather than the absolute one can be employed to mark different unique regions in an environment. We then introduce the concept of field division with each divided subarea owning a unique signature (which will be described later). Without loss of generality, we start the discussion within a two-dimensional paradigm although our algorithm can be applied into a three-dimensional paradigm. We focus on how to use relative signal strength to determine subarea. For convenience, we first take a look at one region without interference. (Note that the practical factors will be considered in detail later on.) Ignoring the environmental noise, the received signal strength (RSS) will demonstrate a monotonic effect on the geographic distance. Though using the relative signal strength, we consider that a RSS model can facilitate our discussion. A well-studied RSS distribution model in db (decibel) as in (31) [11] meets the purpose: RSS𝑛_𝑖=𝑝^𝑛𝑖₀𝑟+10𝑛log^𝑛𝑖0𝑟_𝑖, (31) in which, 𝑛, 𝑟^𝑛𝑖₀, and 𝑝^𝑛𝑖₀ are the constants parameters for anchor 𝑛_𝑖 and 𝑟_𝑖 is the distance between the current location and anchor 𝑛_𝑖. Letting 𝑘 be the number of anchor nodes in a map, we define a high-dimensional location signature function ℱ as follows: 𝑠=ℱrss𝑛₁,rss𝑛₂,…,rss𝑛_𝑘, (32) where 𝑠 is signature of the target location and rss(𝑛_𝑖), 1≤𝑖≤𝑘, is the target’s RSSs relative to the anchor node 𝑛_𝑖. The function ℱ is modeled by a descending sorting function. Therefore, we have rss(𝑛_𝑖)>rss(𝑛_𝑗) for all 𝑖<𝑗 for every signature 𝑠. Given the modified RSS transmission model, we denote the subarea in Definition 6 for field division.

Definition 6 (subarea). Subarea is the point set whose members own the same signature in the field segmentation.

And the boundary is defined as a curve that is separating two adjacent subareas. Given any paired anchors, 𝐴_𝑛𝑖, 𝐴_𝑛𝑗, the dedicated part of the boundary attributed to them consists of positions, where rss(𝑛_𝑖)=rss(𝑛_𝑗). Taking a region with two anchors, 𝐴 and 𝐵, as an example, the positions on the boundary can be derived as 𝑝^𝐴₀𝑟+10𝑛log^𝐴0𝑟₁=𝑝^𝐵₀𝑟+10𝑛log^𝐵0𝑟₂, (33) where 𝑟₁ and 𝑟₂ represent the relative distance to anchors 𝐴 and 𝐵. Solving this equation, we can get a relationship of 𝑟₁ and 𝑟₂ as (𝑟₂/𝑟₁)=𝑘, where 𝑘=2^{(𝑝𝑏0−𝑝𝐴0)/10𝑛}.

3.2. Model Adjustment

Note that it is not the absolute accuracy of the RSSs, but rather the relative strength that EFD algorithm relies on in localization. Based on our measurements, we discovered the RSS degradation due to the impact of interference toward different anchors to be highly correlated. To accommodate RF fading and the multipath effects in environments, we can make an adjustment to the RSS model in (31) for the sake of convenience. Therefore, we model the effects by a factor 𝜆 to 𝑝₀ so that 𝜆_𝑖𝑝₀ absorbs the environmental interference pattern. The ideal RSS model will be adjusted as RSS(𝑖)=𝜆_𝑖𝑝₀𝑟+10𝑛log⁰𝑟_𝑖+𝜖, (34) where 𝜆_𝑖 is the interference factor of anchors and 𝜖 is the adjustment error. Because factors 𝜆_𝑖,𝑖=1,2,…,𝑁, are highly correlated, the huge interference does not actually make that much difference in the division signatures after adjusting the model. As a result, the problem turns into estimating 𝜆_𝑖 to minimize the estimation 𝜎(𝜖). If we denote 𝑝^′=𝜆_𝑖𝑝₀ and assume that the 𝑝^′ follows a normal distribution, 𝑝^′∼𝑁(𝜇₀,𝜎²₀), and there are prior beliefs about the 𝐸(𝑝^′)=𝜇₀ and 𝜎(𝑝^′)=𝜎₀, 𝜇₀ and 𝜎₀ represent the best guess for 𝑝^′ and the uncertainty of the guess, which comes from prior experiments or specification from service providers; we can modify the inputs to (34) based on the in situ estimation of 𝑝^′ and 𝜎(𝑝^′) [12]. A standard estimator of expectations is the sample mean 𝜎(𝜖): 1𝜇=𝑙^𝑙_𝑡=1𝑋_𝑇∼𝑁𝜇₁,𝜎²₁𝑙, (35) where 𝑙 is the number of available sampling series. However, the sample mean is a highly inefficient estimator as the sampling estimation varies widely when different sampling series are fed into the estimation process [12]. One way to cope with this issue is to use a more efficient balance estimator: 𝜇^(𝑏)≡(1−𝑏)𝜇+𝑏𝜋⁰, (36) where 𝜋⁰ is our best guess, 𝜋⁰∼𝑁(𝜇₀,𝜎²₀), and 0≤𝑏≤1 is the balance factor. The purpose is then to minimize balance 𝜎_𝑏 under any given 𝜇^(𝑏). We can formalize our model as Minimize𝜎²_𝑏=𝑏^′𝜇𝑉𝑏Subjectto𝐸₀=𝑏^′𝑈=𝜇_𝑛𝑖=1𝑏_𝑖=1, (37) where 𝑉 is covariance matrix between 𝜇 and 𝜋⁰ and 𝑈 is the vector [𝜇₀,𝜇₁].

The first solution to that formalization is 𝑉𝑏=𝜆𝑈,⟶𝑏=𝑉⁻¹𝑈. (38)

Thus, the optimal balance factors given by the answer are used to derive the 𝑢^(𝑏), as the optimal representation of 𝑝^′ in (34).

3.3. Grid Tracking Strategy

As mentioned previously, one of the biggest challenges that the EFD system faces is the status quo issue. In this section, we will discuss another key technique used in EFD, the grid tracking strategy, in which ordinal gridding based on in situ information such as the magnetic field is used to assist localization within a subarea. Denote 𝑉_𝑡 by the estimated velocity of the target at time 𝑡 and 𝐷_𝑡 as the moving direction of the target. These two properties, importantly describing the motion characteristics, can be estimated precisely by approaches such as the geomagnetic field analysis by the EFD location platform.

3.3.1. Magnetic Theory

Geomagnetic field is relatively stable and easy to use in giving a direction, so we estimate the directions via the geomagnetic field. To preprocess the magnetic field data, we adapt a recursive method to track the dynamic measurement by assuming that it is approximately stationary. Denote 𝑠(𝑡) by the measured signal at time slot 𝑡, which corresponds to 𝑥-, 𝑦-, or 𝑧-component of the magnetic field data and may suffer from the measurement noise or other uncertainties. The recursive method intends to track the noise-free data by updating 𝑠̂(𝑡)=𝛼∗𝑠̂(𝑡)+(1−𝛼)𝑠(𝑡), (39) where 𝑠(̂𝑡) denotes the estimated signal at time slot 𝑡 and 𝛼 is a weight factor that balances between earlier data prior to time 𝑡 and the current measurement 𝑠(𝑡). Clearly, the larger 𝛼 is, the more dependent 𝑠(̂𝑡) is on earlier measurements. By this way, it is possible to remove the noise effects by relying on the historical data. However, there is a clear trade-off on the value of 𝛼, since a larger 𝛼 would introduce difficulties in capturing sudden changes in 𝑠(𝑡). To this end, the smoothed magnetic fields with various choices of 𝛼 are plotted in Figure 2 to illustrate the trade-off in selecting an appropriate value for 𝛼. Lastly, by initializing 𝑠(̂1)=𝑠(1), the recursive update in (39) is extremely simple to implement since it involves only linear updates. Interestingly, this preprocessing step can effectively smooth out the perturbation present in the magnetic field data, as shown in Figure 2 with 𝛼=0.75.

(a)

(b)

(c)

(d)

Figure 2 

Magnetic data processed with different 𝛼.

3.3.2. Details of the Grid Tracking

While the target is moving, fictitious edges for four two-dimensional directions, S, N, W, and E, are dynamically created to gauge the motion range of a target at each cycle 𝑡. By default, the edges matrix 𝐿 is set to the boundary (𝔹_𝕟) of the subarea 𝐴_𝑛: 𝐿(0)=𝔹_𝕟=(𝑥)(𝑥)𝑦𝑦,minmaxminmax𝑥,𝑦∈𝐴_𝑛, (40) where (𝑥,𝑦) is the coordinates of the subarea 𝐴_𝑖 when the target enters initially. Then, the 𝐿 at time 𝑡 is updated by 𝐿(𝑡)=𝐿(𝑡−1)+𝑉⋅𝐷, (41) where 𝐿 is a 2×2 matrix; for example, 𝐿_1,1 ← min(𝑥)and 𝐿_2,2 ← limit to the north.

If the edge 𝐿 exceeds the boundary of subarea 𝐴_𝑖, this 𝐿 will be replaced with the boundary. Note that the process continues until a detection of a signature change, which indicates that the target is crossing the boundary of the subarea. To move the 𝐿, we define a velocity matrix 𝐿𝑣 as 𝐿𝑣=𝑉⋅cos𝜃𝑉⋅cos𝜃𝑉⋅sin𝜃.𝑉⋅sin𝜃 (42)

If a target moves within one subarea, 𝐿(𝑡)=𝐿(𝑡−1)+𝐿𝑣. However, once it crosses the boundary, the limit 𝐿 will make a specific change.

3.4. Correction with Boundary Passing

As the target crosses the boundary separating two subareas 𝑆_𝑖 and 𝑆_𝑗, the corresponding signatures from EFD’s mobile localization platform will change. This actually offers a great opportunity for EFD to calibrate the location estimation. Denote 𝐷_𝑣 by the direction indicator as 𝐷_𝑣=𝑤𝑒𝑠𝑛, (43) in which 𝑤𝑒=((1/2)⋅𝜋<𝜃<(3/2)⋅𝜋)?1∶2,𝑠𝑛=(𝜋<𝜃<2⋅𝜋)?1∶2, and 𝜃=∠𝐷. As the signatures change, the correction algorithm will be carried out immediately as described in Algorithm 1.

Output: 𝐿→ the limit for estimated position (1)  Input area sequence number 𝑛 (2)  Get 𝔹𝕟 as defined in (40) (3)  Get the estimated angle for current moving (𝜃) to calculate the Dir and 𝐿𝑣 as is defined in (43) and (42) (4)  temp=𝐿+𝐿𝑣 (5)  for  𝑖=1;𝑖≤2  do (6)   if  temp𝑖,𝑗, 𝑗=3−Dir1,𝑖  inside subarea  then (7)   𝐿𝑖𝑗=temp𝑖,𝑗 (8)    else (9)   𝑈𝑠𝑒𝑑𝐵𝑜𝑟𝑑𝑒𝑟←𝔹𝕟(𝑖,3−Dir1,𝑖) (10)    𝐿𝑖,𝑗=𝑈𝑠𝑒𝑑𝐵𝑜𝑟𝑑𝑒𝑟 (11)   𝐿𝑖,Dir1,𝑖=𝐿𝑖,3−Dir1,𝑖  +  2𝐿𝑣𝑖,Dir1,𝑖; (12) 𝑒𝑠𝑡𝑃𝑜𝑠←∑(𝑥𝑘,𝑦𝑘)/𝑛𝑢𝑚𝑏𝑒𝑟𝑜𝑓(𝑥𝑘,𝑦𝑘) where (𝑥𝑘,𝑦𝑘)∈𝑠𝑢𝑏𝑎𝑟𝑒𝑎(𝑛)&&(𝑥𝑘,𝑦𝑘)∈𝐿𝑖𝑚𝑖𝑡(𝑡)

Algorithm 1 

Correction at boundary crossing.

4. System Analysis

To illustrate the performance of the MSD system, we executed extensive computer analysis and conducted localization tests based on our software-hardware 3D-ODDIL system platform. In this section, the analysis results are demonstrated.

4.1. MSD with One Anchor Pair

First, we design the analytical environment: we plot 1 pair of anchors on the 1st and (𝑛+1)th floor and calculate the ΔMSPL values under this circumstance. The anchor pair is at the same point vertically, and we evaluate the ΔMSPL value when the horizontal distance 𝑥 varies from 0 to 20 meter. We repeat the computing with different 𝑁 values to see whether our algorithm is universal. And the results are shown in Figure 3. We define the floor height ℎ=3.5 meters and the insertion loss of reinforced concrete wall 𝐿_𝐼=25 dB. The max distance from test point to the anchors horizontally is 20 meter.

(a) Interval 𝑛=3

(b) Interval 𝑛=4

(c) Interval 𝑛=5

(d) Interval 𝑛=10

Figure 3 

Different ΔMSPL when interval varies.

4.2. MSD with Multipair Anchors

In this section, we analyze the condition that there are multianchors in MSD algorithm. The test environment is an expanded condition of MSD with 1 pair of anchors. Figure 4 illustrates different conditions with 3 anchor pairs. Figure 5 illustrates different conditions with 5 anchor pairs. We define 𝑑_𝑖 as the horizontal distance for anchor pair 𝑖. We set interval 𝑛=4. The figures illustrate the value curve when multipair anchors are detected.

(a) d1 = 0, d2 = 10, and d3 = 20

(b) d1 = 5, d2 = 10, and d3 = 20

(c) d1 = 10, d2 = 15, and d3 = 20

(d) d1 = 12, d2 = 15, and d3 = 18

Figure 4 

ΔMSPL with 3 anchor pairs.

(a) d1 = 0, d2 = 5, d3 = 10, d4 = 15, and d5 = 20

(b) d1 = 6, d2 = 8, d3 = 16, d4 = 18, and d5 = 18

Figure 5 

ΔMSPL with 5 anchor pairs.

4.3. Effects of Multipair Anchors

As we can see in the results, the analytical points on each floor fluctuate in narrow space, and the ΔMSPL line of each floor is almost parallel to each other. When more anchors are detected, the result curve is smoother and improves to distinguish the floors. We can conclude that, under ideal circumstance, the MSD algorithm can fulfill the task of floor localization.

5. Experiment and Analysis

To quantify the performance of the proposed MSD algorithm, we carried out extensive vertical location tests by using our developed localization system platform. In this section, we categorize the experiments we have done into two sets in different environments with increasing complexity.

5.1. Localization System Platform

As a generic localization system, the 3D-ODDL systems can be suitable to any types of anchors with different radio signal frequencies. For the convenience deployment of experiments, we build our platform with Wi-Fi routers as the anchors and Android-based smart phones/tablets as the signal receivers. We choose Tenda W3000R routers as our anchors and use Google Nexus 7 as the signal receiver. The whole platform is shown in Figure 6.

Figure 6 

3D-ODDL platform.

(a)

(b)

Figure 7 

Dormitory building overview.

5.2. Experiments in a Dormitory Building

5.2.1. Received Signals in the Building

Figure 8 is the signal received map when we jog on the first floor. We set one anchor on each floor from floor 1 to floor 3. All the anchors are on the same horizontal position. We can see that the signals of each anchor fluctuate as time flies.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8 

MSD experiment data with 2 anchor pairs.

5.2.2. MSD Experiments with Two Anchor Pairs

In this part, we set two anchor pairs on floor 1 and floor 3 of the building and receive signals on floors 1, 2, and 3. The distance between each anchor pair on the same floor is 10 meters. The sampling points on horizontal direction are 2 m, 4 m, 6 m, 8 m, and 10 m. We record 20 times for all sampling points with every 5 seconds. Figure 8 shows the test data, with the coordinates: axis 𝑥 sampling time in sec, axis 𝑦 distance on horizontal direction in meter, and axis 𝑧 received signals in dB.

Then, we use our weighting model to calculate the weighted-average of received signals on each floor: ΔMSPL_WAVG=_𝑖1/𝛿_𝑖∑_𝑖1/𝛿_𝑖∗ΔMSPL_𝑖. (44) Finally, we use the MSD algorithm to calculate the values of ΔMSPL. We can see the key feature in Table 1.

Table 1 

Average and standard deviation.

6. Related Work

Localization in WSN has become an active research topic again recently, and a continuum of algorithms has been proposed [13–25]. Sensors and beacons are utilized to detect the movements for range-based localization in [14, 16, 22, 23]. To undermine the effect from physical channel, range-free algorithms based on a relative distance using trilateration and grid-based methods based on landmarks in school and office contexts emerged in RSD [26], RND [27], UnLoc [28], and EZ [29]. Both range-free and grid-based methods make great improvement in roubustness and reliability, and each of them owns unique merits in indoor localization. However, they understate the complexity of implementing 3-dimensional localization, albeit they depict reliable methods for 2-dimensional localization. In this paper, we focus on the 3-dimensional localization, analyze the characteristics of indoor vertical distribution, and propose a distinctive vertical localization and a horizontal localization in a hybrid idea of range-free and grid-based methods.

7. Conclusion

This paper introduces a new mechanism of localization called 3-dimensional on-demand indoor localization system (3D-ODIL), which can increase the accuracy and stability of localization of multistorey buildings. On horizontal direction, we use enhanced field division (EFD) algorithm to conduct the horizontal localization. On vertical direction, we implement multistorey differential (MSD) algorithm to fulfill the vertical localization, which can greatly reduce the number of anchors deployed. To test the mechanism of the MSD, we conduct a series of experiments. And we also build a localization system platform to conduct real-environmental experiments. The result proves that the MSD is more reliable than other approaches for vertical localization. More importantly, the low-cost design and on-demand deployment allow the 3D-ODIL for large-scale applications and wide utilization.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported in part by the National Key Basic Research Program of China (2013CB329603), the National Natural Science Foundation of China (nos. 61271220 and 61170164), and NSF Grant CNS-1217791.