Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised By: Patricio La Rosa ad Paul Mi Departmet of Electrical ad Systems Egieerig Washigto Uiversity i St. Louis
Abstract: Locatio Based Services (LBS) deliver the user positioig data for the purpose of avigatio i ukow eviromets. Withi the comig years, the idustry is expected to experiece sigificat growth i reveue ad market presece. The more promiet techology withi the LBS idustry is the Global Positioig System (GPS). GPS is highly effective for avigatig outdoor areas but lacks the precisio ecessary for effectively mappig ad egotiatig idoor eviromets. Idoor mappig techology geerally utilizes wireless idoor devices that emit referece sigals to the referet devices (sometimes statioary ad sometimes mobile) which, i tur, map the surroudig area. Over the past few years, there has bee growig demad for precise ad relatively cheap idoor mappig techology. This is largely due to the variety of its commercial applicatios such as helpig hadicapped persos avigate withi their household, givig household robotic devices the ability to map idoor eviromets, trackig products i a warehouse, ad maagig retail ivetory. I desigig a idoor mappig system, the mai goals are precisio ad accuracy. A acillary goal would be to desig a simplistic system which would prove less costly. This project takes aim at both the aforemetioed primary ad acillary goals. We seek to develop a precise method for idoor positioig ad avigatio by first selectig a appropriate commuicatio protocol ad the selectig a efficiet trackig algorithm. After surveyig available methods, we determied that the Wireless Local Area Network (WLAN) is the appropriate protocol due to its simplicity ad low costs. Before choosig a trackig
algorithm for our movig source, we eeded to be able to accurately estimate the positio of a static source both with ad without measuremet error. Without measuremet error, we could coveietly calculate a Liear ad Noliear solutio for our source positio usig the three WLAN APs ad the trilateratio method. With measuremet error, we could still use trilateratio ad the three APs but did ot have a perfect three-way itersectio that could be ascertaied mathematically. Thus, we had to devise our ow method of estimatig a source positio which simply ivolved takig the average of three closest itersectio poits. Oce we established methods for estimatig the positio of a static source, we decided to use a Kalma Filter as our trackig algorithm for a mobile source due to its precisio ad relatively simple implemetatio. 1.0 Descriptio of Projects ad Accomplishmets: 1.1 Commuicatio Protocol The first phase of the project ivolved selectig a efficiet commuicatio protocol betwee our sigal emitters (APs) ad sigal receiver (source termial). At the oset of this project the two cadidates were the WLAN ad Bluetooth protocols. After surveyig both methods, we ultimately determied that WLAN protocol was more viable because it is easier to extract pertiet data such as sigal stregth ad roud trip time (RTT). RTT i particular would be of great importace to this project. Uder the IEEE 802.11 WLAN Stadard, RTT is measured as the time elapsed betwee the Request-to-Sed (RTS) frame ad the Clear-to-Sed (CTS) frame. RTS ad CTS represet a frame set by the trasmitter (source) ad a respose frame set by the receiver (AP). Therefore, RTT ca be
estimated as the time betwee whe the source asks the AP for a sigal ad whe the AP respods. Usig the simple formula distace = rate x time where time is represeted by RTT ad rate is represeted by the speed of light, we ca calculate distace which represets the sigal radius of a AP. Whe we kow the sigal radius of each poit, we ca mathematically calculate the itersectio of all three sigals ad this itersectio provides us with a reasoable estimatio of our source termial. 1.2 Trilateratio without Measuremet Error or Noise The experimet ivolves three WLAN access poits (APs) ad a mobile termial that receives the sigals from each of three APs. Each AP emits a uique sigal which forms a uique circle with a uique radius. Usig the method of trilateratio, we fid the itersectio of these three circles mathematically ad the result is our positio estimatio. We ca use a fourth AP to measure the height of the mobile termial but, for this project we assume a flat surface ad thus oly three APs are eeded to measure the termial s locatio i two dimesios. This assumptio is made for the purpose of simplifyig calculatios later o. The trilateratio process is visualized i the figure below: Access Poit sigals Source Estimatio via Trilateratio Method Access Poit 1 Access Poit 2 Access Poit 3 Figure 1.2.1: Trilateratio Visualized
For this phase of our calculatios, we assume there is o measuremet error or oise ad thus all three of our AP sigals ca coveietly itersect at oe poit. This assumptio allows us to mathematically derive a estimate for the itersectio poit first through the Liear Least Squares method ad the a refied estimate through the Noliear Least Squares method. The first step i solvig for this itersectio poit is idetifyig the pertiet system variables. For the system visualized i Figure 1.2.1, we defie the followig parameters: (x,y): source positio (x i, y i ), r i : ceter ad radius of APs sigals for,2,3 Figure 1.2.2: Trilateratio System Parameters 1.2.1 Liear Least Squares Method as such: With system defied system parameters from Figure 1.2.2, our system ca be visualized AP 1 (x 1, y 1 ) Source Positio (x, y) r 1 r 2 r 3 (x 2, y 2 ) (x 3, y 3 ) AP 2 AP 3 Figure 1.2.1.1: Trilateratio Visualized with System Parameters
Kowig that the distace betwee each idividual AP ad the source is the AP s sigal radius, we ca derive the followig system of equatios usig the distace formula: (1) (x 1 -x) 2 + (y 1 -y) 2 2 = r 1 (2) (x 2 -x) 2 + (y 2 -y) 2 2 = r 2 (3) (x 3 -x) 2 + (y 3 -y) 2 2 = r 3 3 equatios, 2 ukows ad (x i, y i ), r i for,2,3 are give Figure 1.2.1.2: Trilateratio System of Equatios I order to obtai a solutio to the aforemetioed system of equatios, we apply the Liear Least Squares (LLS) method. Though the method is ot the most accurate, it provides a decet meas for estimatig the source, a estimate we ca later improve via the Noliear Least Squares method. To liearize the system, we eed to remove oe costrait ad so we arbitrarily choose AP 1 which gives us a system of two equatios ad two ukows. The source estimatio is set up as follows: 1. Calculate the distace from AP 1 to the other APs via the followig formula: d ij = (x i -x j ) 2 + (y i -y j ) 2, (i=2,3 ad j=1) 2. Calculate the system costraits (b ij ) give by: b ij = ½(r 2 2 j r i + d 2 ij ), (i=2,3 ad j=1) Figure 1.2.1.3: LLS Calculatio (Part 1)
3. Simplify the system ito matrix form: Ax = b where x A = 2 -x 1 y 2 -y 1 x-x x= 1 b b= 21 x 3 -x 1 y 3 -y 1 y-y 1 b 31 4. Sice the radii, r i, are oly approximate, the problem requires the determiatio of x such that Ax b: A T Ax = A T b 5. Assumig the APs are ot placed i a straight lie, we kow that A T A is o-sigular ad ca solve for x: x= (A T A) -1 A T b Figure 1.2.1.3: LLS Calculatio (Part 2) Usig Matlab, we implemeted the LLS method above ad effectively estimated the iitial static source positio with the radii ad locatio of the three AP sigals as our iputs. 1.2.2 Noliear Least Square Method As metioed before, the LLS method is ot the most accurate method of estimatio but, usig the result from the LLS method as a iitial estimate, we ca improve that estimate by miimizig the sum of the squared error for each distace usig the Noliear Least Squares (NLS) method. NLS is a iterative algorithm that rus util the differece betwee the curret ad previous iteratios meets some modifiable ad pre-specified threshold. The NLS is set up as follows:
(1) R k+1 = R k (J k T J k ) -1 J k T f k *R k deotes the k th approximate solutio (x, y) T, J k represets the Jacobia matrix, ad f k represets error betwee the give radius ad the measured distace at the k th iteratio for each access poit. Figure 1.2.2.1: NLS Algorithm The above algorithm ca easily be programmed i Matlab give the followig equatios: (x-x i ) 2 (x-x i )(y-y i ) (f i +r i ) 2 (f i +r i ) 2 J T J= J T f= (x-x (y-y i ) 2 i )(y-y i ) (f (f i +r i ) 2 i +r i ) 2 *measuremet error f i = ((x-x i ) 2 + (y-y i ) 2 ) 1/2 r i for,2,3 (x-x i ) 2 f i (f i +r i ) 2 (y-y i ) 2 f i (f i +r i ) 2 Figure 1.2.2.2: NLS Algorithm Simplified //Still to do: explai ad defie our NLS method, show graphical results from Matlab fuctios //Explai our cluster method for source estimatio with measuremet oise