Resource-Aware Localization in Sensor Networks
6.2 Distance Estimation
Localization requires information about the interrelation between a node and its environment. This information may comprise knowledge about neighboring nodes, connectivity metrics, positions of reference nodes, the gradient and rotation of the node, and other data. Most localization algorithms require distances between nodes and beacons in a Cartesian coordinate system.
Usually, distances cannot be measured directly. Therefore, a lot of methods have been developed to determine a distance out of other conditions. They can be divided at least into three different types of methods (Figure .). In the following, the most common methods to estimate a distance are described.
6.2.1 Time of Arrival
In vacuum, a transmitted radio signal of an isotropic source propagates circularly with the speed of light at cVacuum = , km/s. In matter, the speed decreases significantly due to the mate-rial properties permittivity e = e⋅er and permeability µ = µ⋅µr. In air, the speed of light equals
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Time of arrival (ToA)
FIGURE . Classification of distance estimation methods.
Δt2
FIGURE . Distance estimation with time of arrival (ToA) at (a) global synchronization time stamp or (b) local synchronization time stamp.
cAir=, km/s. Hence, if material properties as well as propagation speed are known, the dis-tance can be calculated out of the measured transmission time ∆t of the signal between a sender and its receivers as well as the speed of the signal d = cAir⋅∆t (Figure .a).
Thetransmission time ∆t is the difference between the time receiving the signal and the start time the transmission was initiated at the sender (ToA [PJ]). Therefore, each signal contains a time stamp at which the transmission was started. After receiving the entire signal, the transmission time (time of flight, ToF) is calculated by the current time subtracted by the start time.
The calculation of the transmission time requires highly synchronized senders and receivers [Sto]. For example, a small distance (d = cm) between two nodes leads in air to a very short transmission time (∆t = d/cAir = . m/, km/s = ns). Hence, the synchronization error (skew) between these two nodes must be smaller than ts< ns to detect distances around d = cm.
Further, small distances require high time resolutions and, therefore, high clock rates to determine the transmission time very precisely.
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Ignoring the sampling theorem proposed by Shannon [Sha], the clock rate in the given example increases to at least Clockrate = /ts = GHz. Besides the significant protocol overhead to syn-chronize all sensor nodes, the strong timing conditions are hard to meet in resource-aware sensor networks.
Lanzisera et al. presented [LLP] a prototypical system to determine a distance by a reconfig-urable hardware. The averaged distance error amounts to f ≈ m. But especially at small distances (d < m), the distance error inceases drastically (f ≫ m).
A method to avoid extensive synchronization between senders and receivers is the detection of time lags caused by different types of signals as described by Savvides et al. [SHS]. Therefore, a beacon Bi transmits a radio signal with c = cAirfollowed by a second signal, e.g., an ultrasound signal with a lower transmission speed (Figure .b) with c = m/s. After receiving the first radio signal, the sensor node starts a time measurement until the second signal is detected. Thus, the radio signal acts as local synchronization point at the sensor node.
Due to the slower transmission speed of the second signal, the time difference ∆t between the reception of both signals increases linearly with the distance (Equation .). In reality, ∆t must be decreased by the ToF teof the radio signal which is mostly approximated to zero.
d = (∆t − te) ⋅c (.)
But unfortunately, this technique requires two transmitters at every node and is, therefore, unreasonably expensive.
6.2.2 Time Difference of Arrival
Theclock and time synchronization of the whole sensor network with hundreds or thousands of sensor nodes is hard to realize. In contrast to the ToA method, the TDoA reduces the synchroniza-tion effort significantly due to synchronizing all senders only [SG,GG]. Synchronizasynchroniza-tion of the receivers is not necessary.
To determine a distance, two senders, Band B, transmit a radio signal at the same time (Figure
.a). A receiver, e.g., Sreceives depending on the distance dor done of the signals first and starts a time measurement at tStart. If the second signal is received at tEnd, the time measurement is stopped.
Thus, the time difference results in ∆t = tEnd−tStart.
x y
S1 S1
d1 d1
d2 d2
B1 B1 B2
B3 d3
B2 h΄(B1; B2)
h(B1; B3)
M(B1; B3)
M(B1; B2) M(B1; B2)
h(B1; B2) y h(B1; B2)
x
(a) (b)
FIGURE . (a) Implicit distance estimation with time difference of arrival (TDoA) and (b) localization based on TDoA at intersection of h(B; B) and h(B; B).
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Due to the constant signal speed in homogenous media, the difference of both distances ∆d = d−dis proportional to the time difference ∆t of the incoming signals.
∆d = ∆t ⋅ c = d−d (.)
Dividing d−d(Equation .) by ∆d results in the hyperbolic function of the TDoA method.
=
√(x − x)+ (y − y)
∆d −
√(x − x)+ (y − y)
∆d (.)
According to Equation ., a sensor node is located on one of the arms of the hyperbola with the foci Band B. The correct arm depends on the time stamp of the incoming signal. As shown in Figure .a, the first incoming signal of sender Bimplies that the distance BSis shorter than BS
(d<d). Therefore, Scan only be located on h(B; B).
To detect the correct position among the hyperbola, another pair of beacons must be evaluated to definea second hyperbola h(B; B). The interception of both hyperbolas, h(B; B)and h(B; B) defines the position of S(Figure .b). In some cases, both hyperbolas cut twice and deliver two possible positions. Then, a third hyperbola must be validated to prevent the ambiguities.
6.2.3 Round Trip Time
Thedistance estimation based on the introduced ToF methods requires highly synchronized senders or receivers and very precise clocks. Therefore, the hardware effort increases compared to other tech-niques. Thus, ToF methods are mostly contradictory to the design objective of developing small and resource-aware sensor networks.
A method to prevent synchronization is used by measuring the round trip time (RTT). At the beginning, a sensor node Stransmits a message to a target node S(Figure .a). This message con-tains the time stamp at the beginning of the transmission. The target replies to the received message immediately with an acknowledgment (ACK) after a reaction time tR.
Afterreception of the ACK at S, the time difference of the round trip is equal to ∆t = ∆tF+
∆tF+tR, where ∆tF+∆tF represents the ToF between sender and receiver. Additionally, ∆t contains the reaction time of Sand the time ∆tCSi representing the clock skew as well as the clock drift. Disregarding the usual unknown ∆tCSi, a distance between Sand S can be estimated with Equation ..
FIGURE . (a) Round trip time RTT and (b) symmetric double-sided two-way ranging SDS-TWR.
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TheRTT method requires a deterministic and constant reaction time to guarantee the accuracy.
Therefore in most systems, the ACK is automatically generated by a specialized Media Access Control (MAC)-layer without passing higher communication layers [Sta].
6.2.4 Symmetric Double-Sided Two-Way Ranging
The unavoidable clock drift between sender and receiver limits the accuracy of the distance es-timation noticeable. To increase the accuracy, Nanotron proposed an improved method called symmetric double-sided two-way ranging (SDS-TWR) [Tec]. Likewise in RTT, a message is sent from Sto Sand is acknowledged immediately. Based on this round trip, Sdetermines a distance d=.c ⋅ (∆t − ∆tR). Then, node Sinitiates a round trip by transmitting a second message to S. Afterreception of S’s ACK, Scomputes dand sends it to S. Finally, node Scalculates the arith-metic average of both distances dand dand, therefore, reduces clock skew effects compared to the RTT method.
6.2.5 Hop Count
In some sensor networks, the required accuracy of the determined position is very low. In these cases, also imprecise distance estimations are sufficient to approximate a position. In a multihop network with many uniformly distributed nodes, a distance can be estimated by the number of hops (hop count) a message is traveling through the network as shown in Figure .. The beacons Band B
are in line of sight of each other. Therefore, a message between both nodes must be forwarded by S. Thus, the estimated hop distance between Band Bis h = in contrast to the absolute distance de,= m. Hence, this distance estimation has a scaling error of eS,=/ = and has another unit than the coordinates. If there exist several alternative routes between two nodes, the minimal hop distance is chosen.
Usually, distributed nodes are not in line with each other. Therefore, hop distances and real dis-tances do not correlate linearly. Thus, the scaling error differs as exemplarily visualized for the connections B −B (h = , de, = m, eS, = ) and B−B (h = , de, = m, eS,=.). To determine a distance with correct units, Langendoen et al. proposed the Sum_Dist method [LR,BWHF]. In this algorithm, all single distances between beacons are simply accu-mulated. But besides the unhandled scaling errors, even all measurement errors are accumulated too, and finally reduce the accuracy.
FIGURE . Hop count between several nodes.
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With the presented distance vector (DV)-hop method, Niculescu and Nath decreased the impact of the scaling error and the linearization error noticeable [NN]. They calculate an average hop size (AHS) at each beacon Bjusing hop distances and real distances to all k known remote beacons Bi(i ≠ j).
cj= ∑ki=
√(xi−xj)+ (yi−yj)
∑ki=hi (.)
During distance estimation, the calculated AHS cj, the actual hop count to beacon Bj, and the bea-con’s position are stored within the DV. A sensor node i selects the AHS cjof the nearest beacon Bj
to compute all distances dilto all l remote nodes. Thus, ci=cj.
dil=hil⋅ci (.)
The unmotivated selection of the nearest beacon and its AHS leads to a loss of potential useful information. Huang et al. therefore improved the DV-hop method by weighting all m known AHS at a sensor node i (weighted DV, WDV-hop [HS]). According to Equation ., the AHS of node Sis wc = ( ⋅ . + ⋅ . + ⋅ . m)/( + + ) = . m compared to c =. m of the original DV-hop method (Figure . ).
wci =∑mj=hj⋅cj
∑mj=hj (.)
Thedistance information based on hop counts is predestinated for start values of iterative algo-rithms, e.g., hop terrain [SRL] or collaborative multilateration [SPS]. Further, DV-hop can be used to approximate positions. But in environments with many obstacles, the accuracy decreases drastically due to huge hop count produced by bypassing the holes. Another disadvantage is the decreasing connectivity to remote sensor nodes [TRV+].
6.2.6 Lighthouse Localization System
Thelighthouse localization system was introduced by Römer [Röm]. In this system, light beams rotate around a source point with a fixed width b and constant angular speed. The time, a turn takes is defined as tTurn. Due to the rotating light, a sensor node Sidetects depending on its distance to the source of light for a finite time tBeam,per round (Figure .a).
Parallel beams of light b
(b) (a)
Rotating source of light
t = 0 t = tBeam
, 1
Rotation
α1 α1
S1 S2
α2
d1 d2
S1
d1
FIGURE . Lighthouse localization system: (a) Angle detection and (b) distance estimation depending on angles.
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Thetime period to receive the rotating light depends on the distance between a node and the source point. Hence, the angle aidecreases also with the distance (a <a) as visualized in Figure
.b. This angle is defined as ratio of tBeam,i to tTurnand results after the conversion in radian in Equation . [SHS].
α = π tBeam,i
tTurn (.)
As in the right triangle, the sinus of the half angle ai/ is defined as ratio of the beam width b divided by two and the distance between the source of light and the sensor node.
sin ( αi) = b
di (.)
Inserting Equation . in Equation . and rearranging to di results in the final equation to determine the distance (Equation .).
di= b
sin (πtBeam,itTurn ) (.)
In reality, an uninterruptable beam of light as visualized in Figure .a is hard to realize. Römer therefore proposed a realization with two single rays of light in parallel. To distinguish the first from the second ray, he marked the rays.
Thelighthouse localization system is characterized by very low complexity of computation and communication, has a very high accuracy and is only slightly affected by diffraction effects. Thus in a two-dimensional area, each sensor node can determine its own position based on a trilateration provided that three distances to three different rotating beams of light are measured.
But in practice, a system of three rotating rays is hard to realize because of the high effort to install and calibrate light transceivers on all nodes. In particular, all nodes require direct line of sight to the rotating source which restricts the case of operations significantly.
6.2.7 Three/Two Neighbor Algorithm
Most of the localization algorithms require three distances di( < i = ) to reference nodes to cal-culate a position deterministically, e.g., using a trilateration. Geometrically, the position Sis defined as interception of circles around these three references with the given distances.
In sensor networks, a node may know less than three distances to reference nodes. Therefore, a trilateration is plurivalent as visualized in Figure .. The visualized sensor node Sdetermines its own positions at Sand Sif it knows two distances dto beacon Band dto beacon B, only. Hence, Sand Sare in the neighborhood of Band B(S∈N(B) ∩N(B), S∈N(B) ∩N(B)). Further, beacons Band Bare in the neighborhood of Sand S(B∈N(S) ∩N(S), B∈N(S) ∩N(S)).
Both beacons divide the plane into two half-planes E∣E = S, B, B∣E ∉ Sand E∣E = S, B, B∣E ∉ S
at g.
Barbeau et al. proposed the three/two neighbor algorithm to determine a position in such con-figurations by determining implicit distances [BKKM]. They assume there is a third beacon Bin a network (Figure .b) in range of S(S ∈N(B) ∩N(B) ∩N(B)). Thus, node Sknows three distances and can compute its position correctly. If Bis as well in range of S, trilateration of Sis unique (S∈N(B) ∩N(B) ∩N(B)).
If it is not in range (S∈N(B) ∩N(B) ∩ /N(B)) and Sdoes not know anything about B, the localization is plurivalent anymore. But if Sis able to communicate with Band is therefore able to detect the correct half-plane E or E where Band Sare placed, the localization will be unique. In
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d1
d1 d1
d1
S1
S2 S2
S1 d2 d2 d3
B2 E
E d2
d2B2
g12
B1 B1
B3
Beacon Biwith known position
Maximal transmission range
Circles with radius di around Bi with intersections at S1 and S2
Sensor node Si with unknown position
(a) (b)
FIGURE . (a) Localization of Band Bresulting in two positions each. (b) full localization of Band B. the visualized example, Bis located in plane E and therefore Smust be E (B∈N(B) ∩N(S) → B, S ∈ E → S ∈ E). In the end, this algorithm calculates a third distance implicitly, if enough neighboring information is provided.
6.2.8 Neighborhood Intersection
Buschmann et al. proposed another algorithm [BPF] using neighboring information called neigh-borhood intersection distance estimation scheme (NIDES). It requires uniformly distributed sensor nodes only. A synchronization of nodes and explicit distance measurements are not necessary.
Figure . demonstrates the estimation of distance d between two sensor nodes S and S. If both nodes are in range to each other, the transmission ranges form an overlapping region A. The
S1 S2
l
r
x y
d2
Overlapping region A Transmission
range ATR
FIGURE . Neighborhood intersection distance estimation schema.
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maximum width of the resulting overlapping region A is l. According to the figure, the transmission range r is equal to l + (d/).
d = (r − l) (.)
In an ideal environment, both sensor nodes Sand Stransmit a message. This message can be received within the radius r around the nodes. In the overlapping region A (dashed region), all nA
enclosed sensor nodes receive both messages. These sensor nodes Si( = i = nA) are neighbors of both sending nodes. Due to the uniform distribution of nodes, the number of nodes ˜n in range of one sensor node (ATR=π⋅r) is constant. Since the node density µ is constant in the whole network, the following equation is formed:
Thecalculation of width l can be approximated by
l = r + √
Aftercalculation of l, the distance d can be determined by Equation .. The achieved accuracy of NIDES oscillates between .r and .r.
6.2.9 Received Signal Strength
In vacuum, a radio signal of a circular source propagates circularly. Generally, the transmission of radio signals is adherent with the transport of energy into the surrounding of the sender. This en-ergy level is flatten with the distance d to the sender, but it can be detected by a receiver. In several publications, this method to detect the energy level is called received signal strength (RSS).
Friis presented an equation to compute the distance based on transmission power PT X, received power PRX, antenna gains (GT X, GRX), system losses L, and the wave length λ provided ideal conditions (no reflection, no diffraction, no obstacles, etc.) are met [Fri].
PRX
PT X = GRXGT X
L ( λ
πd) (.)
If a sensor node Sis able to detect the received power PRXof a message, the distance between the transmitting node Sand Scan be calculated by rearranging Equation ..
According to Equation ., wave length λand distance d affect PRX quadratically [Rap]. The attenuation of a signal is defined as path loss (PL) which is the logarithm of transmitting power to received power in decibel.
PL (dB) = log PPT XRX = − log (GRXGT X
L ( λ
πd)) (.)
To detect a signal of a transmitting node at a receiving node correctly, the receiver sensitivity PLmax
must be higher than the PL depending on the specific distance d between both nodes. For instance,
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433 MHz CC1010 868 MHz CC1010, ESB 915 MHz CC1010 2,4 GHz Bluetooth, WLAN 3,5 GHz WiMax
1 10 100 1000
Distance (m)
Path loss (dB)
110 100 90 80 70 60 50 40 30 20
dmax= 297 m
PLmax= 80.7
FIGURE . PL of radio signals over distance between sender and receiver at different frequencies, e.g., used by WiMax, WLAN, Chipcon CC, and ESB.
the receiver sensitivity of the embedded sensor boards (ESB) is PLmax = . dB [Sch,Dei].
Therefore, signals can be detected in a range < d < dmax= m (Figure .).
dmax= λ
π−PLmax (.)
The equation of Friis (Equation .) is an approximation of the received power, but it is valid only outside the Fraunhofer region df=D/λof the sender. The Fraunhofer distance dfdepends on the maximum expansion of the antenna D (df>>D) and the wave length λ(df>>λ).
6.2.10 Minimal Transmission Range
In systems without any analog to digital converters (A/D converter) or without circuits to con-vert signals, the RSS cannot be determined. In those devices, the signal strength can be measured indirectly.
In ideal environments, the antenna gains (GT X = , GRX = ) and system losses L can be dis-regarded. Thus, Equation . simplifies to
PL (dB) = − log λ
πd (.)
Usually, a sender transmits with the maximum transmission power PT X,max. This signal can be detected within range dmaxby measuring the received power if it is higher or equal than PRX,min. TheESB for example transmits a signal at PT X,max=. mW. Thus, the minimal received power is PRX,min=. pW at dmax= m.
PRX,min= PT X,max
PLmax (.)
RSS-based methods determine the distance by solving Equation .. Figure .a visualizes the ideal relation between PRX and d. Within range < d < dmax, the signal can be detected and the distance can be calculated along the graph.
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d PTX
PTX,2 PTX,1 PTX,max
d PTX
PTX,max
Measuring received signal
strength PRXalong the graph Measuring minimal transmission power PTXby scaling the graph
d1 d2 dmax d1 d2 dmax
PRX,1 PRX,2
(a) (b)
PRX,min PRX,min
FIGURE . Distance estimation between sender and receiver with (a) RSS and (b) MTP.
A new approach to determine a distance was introduced by Blumenthal et al. in [BTB+] called minimal transmission power (MTP). This method detects the MTP PT X,minrequired at the sender to transmit a signal to the receiver. PT X,mincan be calculated by rearranging Equation . using PRX,min
(Equation .).
In MTP, a sender continuously transmits a signal with an increasing transmission power (PT X ∶ PT X,min<PT X(RT X) <PT X,max). The transmitted signal contains the sender’s adjusted transmission power PT Xin terms of a register value RT X,ito setup the transceiver. At a receiver, the signal can be detected first at PRX,mindepending on the distance d between both nodes. In most cases, the signal is received more than once with increasing RT X,i. Then, only the smallest RT X,iis valid to determine d by solving Equation ..
d =
PT X(RT X)GT XGRX
PRX,minL (λ
π) (.)
In contrast to RSS where received power and distance are related along a graph with a maximum at PT X,max(Figure .a), MTP scales and moves the whole graph with its maximum at PT X(RT X) (Figure .b).
Thedistance estimation using MTP is very precise, especially in very short ranges due to preventing multipath effects. Figure . shows typical results using ESB.