1

A Flexible Distributed Maximum LogLikelihood Scheme for UWB Indoor Positioning Benoît Denis, Liyun He, and Laurent Ouvry

Abstract—This paper presents a distributed positioning algorithm for Ultra Wideband (UWB) indoor ad hoc networks. The proposed Distributed Maximum Log-Likelihood (DMLL) solution claims to benefit from redundancy and spatial diversity as network completeness increases, tends to mitigate the harmful effects of Non Line Of Sight (NLOS) ranging biases by incorporating refined Time Of Arrival (TOA) models, and finally exhibits fine flexibility by offering distinct implementation options regarding the integration of range measurements, or the distribution of required calculi. One important point is the use of synergetic cooperative protocol transactions that can handle simultaneously ranging, local contributions to the iterative optimization of a global objective, as well as the exchange of positional information. Depending on the retained underlying models and the amount of prior statistical information, either a “blind” approach or more advanced options (e.g. aided by a preliminary channel identification step) could be adopted within a unique generic framework. In this paper, we make a short description of the DMLL solution. Then, we provide some simulation results obtained under realistic indoor scenarios and discuss the impact of a few critical parameters on positioning precision and/or reliability. Index Terms— Ad Hoc, Biased Time Of Arrival, Distributed Positioning, Indoor Environments, Maximum Log-Likelihood, Ranging, Ultra Wideband

I. INTRODUCTION

T

adjunction of a reliable radiolocation functionality to classical communication means is clearly one of the major recent concerns in the field of wireless networks. As an example, Low Data Rate - Location and Tracking (LDR-LT) applications in Wireless Personal Networks (WPAN) or Wireless Sensor Networks (WSN) based on Impulse Radio Ultra Wideband (IR-UWB) ([1]) have been intensively investigated for the last past years in the frame of prospective HE

Manuscript received November 26th, 2006. This work was supported in part by the European integrated project PULSERS–II / WP3a (www.pulsers.eu). B. Denis is with CEA-Leti / MINATEC (www.minatec.com), 17 rue des Martyrs, 38054 Grenoble Cedex 9, France (tel: (+33) (0)4 38 78 18 21; fax: (+33) (0)4 38 78 65 86; e-mail: [email protected]). L. He was with CEA-Leti / MINATEC till September 2006. She is now with ENSIMAG / ID (www-id.imag.fr), ZIRST 51, avenue Jean Kuntzmann 38330 Monbonnot Saint Martin, France (e-mail: [email protected]). L. Ouvry is with CEA-Leti / MINATEC (e-mail: [email protected]).

normalization (e.g. [2]-[3]) and addressed by abundant academic works (e.g. [4]-[10]) or major research projects (e.g. the European Integrated project “PULSERS-II” [11]). In this context, a growing attention has been more specifically paid to positioning algorithms based on temporal radiolocation metrics, namely Time Of Arrival (TOA) or Time Difference Of Arrival (TDOA), in harmful indoor environments. Among possible algorithms, unlike more classical centralized approaches, distributed solutions ([12]-[18]) imply that blind nodes compute their own locations. One main reason motivating this choice is the necessity imposed by some applications to operate with an opportunist and perennial network in the lack of fixed infrastructure or extra-resources. The first idea is to share the computation load onto the whole network, while preserving reasonable computational complexity in each node. In addition, distributed approaches usually claim to enjoy better scalability, flexibility, and potentially reactivity to timely changes. Finally, they avoid the relay of information (e.g. measured relative distances) to a collecting point before the computation of blind locations, most probably in multi-hop scenarios. Hence, distributed solutions theoretically tend to alleviate radiolocation overhead, routing/traffic constraints, as well as network latency. Considering these last crucial points, it is worth reminding that a large majority of classical approaches would adversely require that successive pair-wise noised range measurements are averaged and relayed before applying positioning algorithms. Such an option is obviously demanding and consumptive from the medium access point of view. As a response, we describe hereafter a distributed solution that enables to perform jointly ranging and positioning, and should save protocol resources accordingly. At the price of slight degradations on the final expected accuracy, this algorithm can start converging locally whenever at least one rough measurement is available per pair-wise link and further successive cooperative transactions are performed for the exchange of positional information. Actually, these exchanges can also be advantageously coupled with appropriate ranging transactions (e.g. cooperative Two Way Ranging schemes) so that further measurements can be incorporated while communicating and converging to true locations.

2

A, refers to the neighbourhood of node i, if d m is the maximum achievable range tolerated by the physical layer (e.g. d m =20m), and dij the Euclidean distance separating nodes i and j. Anch. Nodes 4 / Mob. Nodes 18 / Average Connectivity 9.4545 35

A04

A03 Anchor Nodes Mobile Nodes

M11 M21

30

M05 M18

25 M09

M22

20 Y (m)

M19 M14 M12 M08

M10 M07

15

M13 10

M20

M15

M06 M17

5 M16 0

A01 0

A02 5

10

15

20

25

30

35

X (m)

Fig. 1: Example of 2D network topology, including Na=4 surrounding anchor nodes and Nm=18 mobile nodes, with a maximum radio range of dm=20m Adjacency Matrix (A) 1 2

0.9

4 0.8 6 0.7 8 Node index i

On the other hand, most of positioning solutions, a fortiori distributed ones, rely on suboptimal approaches in the sense they usually do not take into consideration realistic underlying ranging error models. However, solutions based on a gradient descent approaches (e.g. [18]) can be enhanced if prior knowledge about ranging error statistics is available and judiciously incorporated. For instance, the Distributed Maximum Log-Likelihood (DMLL) algorithm we describe can mitigate the impact of severe ranging errors by locally and “asynchronously” maximizing the Log-Likelihood of joint range measurements conditioned upon estimated blind locations. Depending on the amount of information that is available a priori, several choices can be made for the objective function to maximize. Finally, another interesting point is that the preferred “distributed / asynchronous” option can be easily derived into “locally centralized / partially asynchronous”, or even “centralized / synchronous” implementations, depending on the application needs. As a summary, the DMLL solution described hereafter claims to be: – Flexible, by offering distinct strategies for the distribution of calculi, and the incorporation of available prior information and/or measurements – Low-traffic/low-overhead in the preferred distributed implementation strategy, by enabling joint ranging and positioning procedures – Efficient in harmful obstructed environments, by mitigating the effects of Non-Line Of Sight (NLOS) ranging errors on location accuracy After this short introduction, the basic models we use and the DMLL positioning algorithm are described respectively in Parts II and III. Then, we present in Part IV some simulation results obtained under realistic indoor scenarios. At this occasion, the effects of a couple of critical parameters on accuracy and robustness are illustrated and briefly discussed. Finally, Part V concludes the paper.

0.6

10

0.5

12 14

0.4

16

0.3

18

0.2

20

0.1

22

II. MODEL DESCRIPTION

5

10 Node index j

15

20

0

A. Network Model One network of interest typically comprises N nodes, which are assumed to be equipped with similar radio functionalities and embedded computational capabilities, including Na anchor nodes set at known locations (with arbitrary indexes i=1..Na), and Nm blind nodes whose positions are to be estimated (with arbitrary indexes i=Na+1..N). The availability of a pair-wise communication link between nodes i and j is associated with the link capability variable aij, which is equal to 1 whenever a measurement of the communication quality (PER, QoS, estimated ranging accuracy, etc…) reaches an arbitrary threshold, and 0 elsewhere. A = aij is then defined as the

{ }

i =1.. N , j =1.. N

connectivity set N e (i ) =

{j

(or

d ij ≤ d m }

adjacency) matrix. The , which can be easily derived from

Fig. 2: Connectivity (or Adjacency) matrix A (Same example as on Fig. 1)

B. Ranging Schemes and Measured Metrics The positioning algorithm we describe here is fed by TOA metrics, or equivalently relative ranges, that are measured between communicating nodes. Considering the abovementioned network topology, a preferred ranging transaction is based on a simple TWR scheme, if necessary coupled with further clock drift estimation/compensation procedures (sometimes denoted as Three-Way Ranging schemes). This kind of procedures can be proved to deliver range measurements

~ d ij of the form:

~ d ij ≈ d ij′ + nij where

(1)

d ij′ is the apparent distance travelled between i and j

3 (“radio” distance),

nij can be defined as a residual noise term

(potentially after drift estimation/compensation) with a variance

σ n2,ij .

In most cases,

σ n2,ij

C={LOS, NLOS, NLOS²} (e.g. Fig. 3) is given by: ~ π (BG2C −4 AGC CGC ) (4 AGC ) pC d d , C = WGC DGC e + AGC

[

is analytically tractable as a function of

the protocol durations involved in ranging transactions (e.g. the prescribed response delays) and the power of the initial detection noise (i.e. unitary uncertainties on detection events). In addition, we deliberately make the distinction between

WEC DEC

WEC

1 2π σ C

1{d ′>d } d

λC e

e

(

(

(3)

⎞ ⎟ ⎟ ⎠

)

(

)

NLOS2 at d=14m under n∼G(0,0.3m) 0.8 0.7

True pdf Dispersion of Erroneous pdfs (ε = +/-40% of Nominal Parameters)

0.6 0.5

2

pdf

1 d

AEC

e

⎛ BE − 4 AEC C EC ) (4 AEC ) C erfc⎜ ⎜2 A EC ⎝

⎧ 1⎛ 1 1 ⎞ 1 1 ⎪ AGC = ⎜⎜ 2 + 2 2 ⎟⎟ = AEC + 2 ⎝σn d σC ⎠ 2 d 2σ C2 ⎪ ~ ⎪ λ d −d B = = BEC − C ⎪ GC 2 d σn ⎪ ⎨ ~2 d −d ⎪ CGC = = C EC ⎪ 2σ n2 ⎪ 2 DEC 1 ⎪ DGC = = ⎪⎩ 2πdσ C σ n 2π λC σ C

~

⎛ d′ ⎞ ⎜ −1 ⎟ d ⎠ −⎝ 2σ C2

π

BE2 C

where

the measured distance d , the Euclidean distance d , and finally the radio distance d ′ . The main idea is that d ′ is actually a random variable at the initialization of the network, conditioned upon the actual Euclidean distance d (for the purpose of simplification, subscripts corresponding to node indexes have been removed in this sub-section). Moreover, d ′ is most probably a biased version of d , since ranging algorithms often select secondary paths due to severe blockage situations in NLOS environments and/or, based on the first observable path, may overestimate d due to successive transmissions through dense materials, even under satisfactory SNR conditions. Conditioned upon d and a particular channel configuration C={LOS, NLOS, NLOS²}, one commonly used pdf for d ′ is based on a mixture description ([19]-[22]):

pC [d ′ d , C ] = WGC

]

+

0.3

(2)

⎛ d′ ⎞ − λC ⎜ −1 ⎟ ⎝d ⎠

where 1{x> y} equals 1 whenever x>y, and 0 otherwise. In this example, the underlying model assumes an exponential component in the mixture that accounts for the largest excess delays. Other judiciously tailed densities may be adopted as well. Now, concerning the noise term in (1), several options can be taken. One possible solution consists in assuming a Gamma distribution ([23]) centred around the biased distance d ′ . However, another widely used model is the centred Gaussian one. In the simulations presented hereafter, we use both Gamma and Gaussian distributions, with constant or random variances for generating measured distances. Generally speaking, this choice clearly impacts the tractability and the practicability of optimal Bayesian estimators. However, given a certain noise variance, a centred Gaussian assumption can always be made, leading to suboptimal estimators in the presence of actual Gamma noise occurrences. As an example of tractable calculation, using a normal centred assumption for n in (1), the resulting pdf of

~ d conditioned upon d and a particular channel configuration

0.4

0.2 0.1 0 13

14

15

16 17 18 19 Measured Distance (m)

20

21

22

Fig. 3: Example of (true and erroneous) prior ranging model for a NLOS² channel configuration at d=14m under σn=0.3m. Erroneous pdfs are obtained with a relative error up to ε=±40% of nominal parameters ([19], [21]) WGc,

σC

and λC in (3)

Finally, independently on the chosen conditional pdf for

~ d , we propose another refinement by taking into account the channel configuration at d . Consequently, the pdf associated ~ with the range measurement d conditioned upon d should ~ be described as a mixture between pC d d , C distributions with distinct weights WC (d ) , C={LOS, NLOS, NLOS²}. These weights obviously depend on the actual distance d ,

[

]

and represent the probability to suffer from a particular channel configuration at d . As an example, it appeared from an indoor measurement campaign ([19]) that these weights could be described as Gaussian-like functions. As a

4 consequence, we will assume in the following for C={LOS, NLOS, NLOS²} that: 2 C

(4)

σC

ensure

that

WLOS (d ) + W NLOS (d ) + W NLOS 2 (d ) = 1, ∀d ∈ [0 : 20 m ]

[

~ ~ p d d = ∑WC (d ) pC d d , C

]

(5)

C

In the following simulations, when generating measured distances under the centered Gaussian noise assumption, “MIX” will refer to (5), and “LOS”, “NLOS” or “NLOS²” to (3), with the same model parameters as in [19] and [21]. III. POSITIONING ALGORITHM

A. Log-Likelihood Maximization Under the double assumption of independence and symmetry, the Log-Likelihood of joint pair-wise range measurements can be written as: N

Λ=

∑

([

~ ∑ ln p d ij d ij

i = N a +1 j∈N e ( i ) j >i

])

j ∈ N e (i ) and receives xˆ j ,k −1 and

yˆ j ,k −1 from this neighbor: xˆ i ,k = xˆ i ,k −1 + δ ijd,k

(7)

where

In (7),

∂d ij ∂xi

=

(x

d ij = d ij , k −1

i

− xj ) d ij

⎛ ∂d ij ⎜⎜ ⎝ ∂xi

may represent the confidence in neighbor j (typically

equal to 1 if it is an anchor and to 0.1 if it is a mobile, like in [18] and [21]) or the importance of the link i-j with respect to the consistency of the overall Euclidean graph. In addition, γ ij ,k can be linearly determined so as to satisfy a particular local test (e.g. Armijo’s test). We only represent the x coordinate in (7) for the purpose of simplification, but note that similar expressions are straightforwardly obtained for y. In the locally centralized realization, one node i waits for collecting a complete (or even partial) set of xˆ j ,k −1 and

yˆ j ,k −1 from its available neighbors j ∈ N e (i ) , before

xˆi ,k −1 into xˆi ,k :

⎞ ⎟⎟ xi = xˆi , k −1 ⎠ x j = xˆ j , k −1 d ij = dˆij , k −1

~* , d ij ,k is a function of all the

available range measurements between i and j up to step k,

(8)

where

∑( ) β

δ ijlc,k =

To solve the positioning problem, (6) is maximized with respect to blind locations as optimization variables. Like in [18], we also decline here a classical iterative gradient ascent approach into distinct strategies: distributed (or asynchronous), locally centralized (or partially synchronous), and finally centralized (or synchronous). The distributed asynchronous scheme, which is the th preferred realization mode here, consists in updating at the k step the position of one particular node i. The update can be realized each time node i communicates with one of its

⎛ ∂Λ ij ⎞ ⎟ ⎟ d~ij = d~ij*,k ∂ d ij ⎝ ⎠ ˆ

2

xˆ i ,k = xˆ i ,k −1 + δ ijlc,k (6)

δ ijd,k = β ij ,k ⎜⎜

2

confidence coefficient used to mitigate or inflect the ascent direction and γ ij ,k a dynamic step.

updating

1 N 1 N = ∑ ∑ Λ ij = ∑ Λ i 2 i = N a +1 j∈N e ( i ) 2 i = N a +1

available neighbors

− xˆ j,k ) + ( yˆi,k − yˆ j,k ) is the estimated distance

yˆi,k , xˆ j ,k and yˆ j,k , and β ij ,k = α ij γ ij ,k , where α ij is a

α ij

Finally, the complete model can be written as:

[ ]

i, k

between nodes i and j, based on estimated coordinates xˆi,k ,

( d − d C )2

− ζ WC (d ) ≈ e 2σ 2π σ C where and ζ

(xˆ

dˆij,k =

j∈N e i

⎛ ∂Λ ij ⎜ ij , k ⎜ ⎝ ∂d ij

⎞ ⎟ ⎟ d~ij = d~ij*, k ⎠ ˆ

d ij = d ij , k −1

⎛ ∂d ij ⎜⎜ ⎝ ∂xi

⎞ ⎟⎟ xi = xˆi , k −1 ⎠ x j = xˆ j ,k −1 d ij = dˆij , k −1

Now, extending the concept to a purely centralized synchronous scheme, all the coordinates can be computed simultaneously, according to

Xˆ k = Xˆ k −1 + Δ k where

[

Xˆ k = xˆ N a +1,k

[ = [δ

(9)

xˆ N a + 2,k

]

... xˆ N ,k ,

] ]

Xˆ k −1 = xˆ N a +1,k −1

xˆ N a + 2,k −1 ... xˆ N ,k −1 , and

Δ k −1

δ Nlc + 2,k −1 ... δ Nlc,k −1

lc N a +1, k −1

a

Hereafter, we principally evaluate the performance of the distributed scheme, defining one “iteration” as the update of all the blind nodes relatively to all their available neighbours. As an example, one iteration represents approximately 40 effective updates if Nm=10 and the average number of neighbours per node is 4.

B. Incorporation of New Measurements

~*

In the previous sub-section, we have seen that d ij ,k is incorporated at each step of the optimization procedure as a transformation of all the pair-wise measurements available before updating the coordinates of blind nodes.

5

~*

As an example, d ij can be the average of K preliminary range measurements that were performed before optimizing. This would be the preferred solution if ranging and positioning are disjoint and independent procedures, most probably in the centralized optimization scheme. However, considering the distributed case (where prior ranging would be clearly meaningless from the medium access point of view), one interesting point is that the optimization procedure requires explicitly the exchange of positional information (namely xˆ j ,k −1 and yˆ j ,k −1 ). As a consequence, these exchanges can be advantageously coupled with cooperative ranging transactions so that one new measurement is incorporated at each new optimization step. In such a case, ranging and positioning are viewed as joint procedures and

~ d ij*,k can be the average of all the range measurements that are

available up to step k. Whatever the chosen option (i.e. joint or independent ranging and positioning procedures), note that the standard deviation of the residual noise (i.e. the average of successive measurements) attached to each pair-wise link can be empirically determined from the observed samples as

σˆ n2,ij K

2 or σˆ n ,ij k , and properly injected afterwards into

(1), (3) and subsequent analytical results. Theoretically, the observation of a set of pair-wise

{~ } or a set of average measurements {d~ } * ij

measurements d ij

would justify that one single global objective function is defined and a complete optimization procedure is carried out till convergence. However, in the case one new measurement is incorporated at each step of the distributed scheme, the remarkable point is that a new objective function is defined at each optimization step. Hence, each novel update must be viewed as the “first step” of one particular optimization procedure associated to the current objective function.

C. Prior Knowledge As other Bayesian approaches, the DMLL algorithm necessitates a priori models about the observed data. An interesting point concerning this algorithm is the possibility to particularize these underlying models, depending on the availability (physical layer capability) and/or reliability of prior statistical information. In particular, the conditional pdf of range measurements can be selected accordingly. The first option consists in using the conditional model (3), where biased distances are conditioned upon d and C, after removing the dependency on C, i.e. by performing a preliminary identification of the channel configuration i-j:

[

~ ⎡ ∂pC d ij d ij , C 1 =⎢ ~ ∂d ij ⎢⎣ pC d ij d ij , C ∂d ij

∂Λ ij

[

]

]⎤⎥

⎥⎦ C =Cˆij

(10)

Another strategy relies on the complete model (5), assuming the overall mixture between LOS, NLOS and

NLOS² components. Then, measured distances are only conditioned upon d and one can write:

[

]

⎡ ∂WC (dˆij ) ⎤ ~ pC d ij dˆij + ⎥ ⎢∑ ∂dˆij ∂Λ ij ⎢C ⎥ 1 = ~ ⎢ ⎥ ~ ∂d ij p d ij dˆij ⎢ ∂pC d ij dˆij ⎥ ˆ WC (d ij ) ⎥ ⎢ ˆ ∂d ij ⎣ ⎦

[

]

[

]

(11)

Finally, if no refined model is available (e.g. assuming no bias but uniquely Gaussian centered noise affecting range measurements), the DMLL algorithm can be proved to reduce to a Distributed Weighted Least-Squares (DWLL) algorithm. For instance, under the assumption

σ n2,ij = σ n2 = cst ,

we

obtain (like in [18]):

∂Λ ij ∂d ij

(

~ =2 d −d

)

(12)

IV. BRIEF OVERVIEW OF CRITICAL PARAMETERS

A. Network Completeness Network completeness (or equivalently, network connectivity) is obviously one of the most critical parameters affecting the performance of the DMLL algorithm. We define network completeness as the ratio between the number of active pair-wise links and the total number of links that would be active under full connectivity. As an illustration, we present on Fig. 4 and Fig. 5 the 2D Circular Error Probability (defined as the cdf of positioning errors over 100 network realizations) obtained with the DMLL algorithm after 20 iterations and the average of K=20 prior measurements (assuming prior ranging procedure), biased distances d ′ following (2), and noise terms n in (1) following a Gamma distribution with a standard deviation σn,ij uniformly distributed within [0.1,3]m, respectively for 4 surrounding anchors / 10 blind nodes and 8 random anchors / 20 blind nodes. As it could be expected, results tend to show that the DMLL algorithm benefits from redundant information and spatial diversity as completeness and the number of nodes increase, whatever the considered channel configuration or the network topology. Moreover, the general trend is that ErrorLOS < ErrorMIX