In this section, we focus on the positioning problem in a WSN. Particularly, we address the range-based approach in which a node that aims at determining its position first estimates its distance to a reference node using a ranging technique (e.g., RSSI or TOA). Afterward, these ranges are used to solve a geometrical problem referred to as trilateration.
There is a plethora of techniques for ranging in WSNs, see, for instance, the studies in [8, 21]. In this article, we consider RSSI- and TOA-based techniques. While RSSI techniques use measurements from the signal power at the receiver, TOA is based on the estimation of the travel time that a signal takes from the transmitter to the receiving node. The advantage of the RSSI approach versus TOA techniques is that it requires no additional hardware. However, while RSSI is greatly affected by multipath fading, TOA with UWB is a much more robust ranging technique since the large bandwidth of a UWB signal provides high time resolution [4].
Concerning TOA-based ranging, the IEEE 802.15.4a [2] standard defines a mandatory ranging protocol called two-way time-of-arrival (TW-TOA) and an optional Symmetric Double Sided (SDS) TW-TOA protocol that reduces the effect of the finite crystal tolerances at the local oscillator [2]. In order to start the ranging protocol, an upper layer of 802.15.4a standard (the protocol stack is shown in Figure 1) calls a primitive of the MAC layer. This primitive is named MCPS-DATA.request and it is used for requesting data. Also, this primitive has to be called with a corresponding attribute to start ranging [2].
The problem under study involves the positioning of N
n
target coordinators in a WSN that contains N
r
reference coordinators emitting ranging signals to allow positioning of the latter. In this article, we consider the topologies defined in the two standards, namely, the Zigbee standard with RSSI-based ranging and the 802.15.4a standard with TOA-based technique for UWB devices. We address the following WSN topology configurations: 802.15.4a mesh WSN; Zigbee mesh and Zigbee cluster-tree WSNs; and a WSN based on 802.15.4a PHY layer with cluster-tree topology similar to Zigbee. We assumed a uniform random deployment of the N
n
+ N
r
nodes in the simulation results that are provided in Section 6. Let us define the three-dimensional coordinates of target and reference nodes as
(1)
(2)
respectively. Notice that for the trilateration procedure to be valid, we need to assume that positions of the reference nodes are known. The geometrical distance between the j th node and the i th anchor is defined as
(3)
with ||·|| being the Euclidean norm in ℝ.
We can identify different range models depending on the technology used. On the one hand, we consider the following observation equation for TOA-based ranging
(4)
where j = 1, 2,..., N
n
and , with being the set of reference nodes from which the j th node receives ranging information. is additive measurement noise, independent among measures.
On the other hand, RSSI-based ranging measures are commonly modeled using the log-normal path loss model [22], defined as
(5)
where ρ
o
is the reference distance, L
o
is the attenuation at such reference distance in dB, ρ
j,i
is the distance between nodes j and i, L
j,i
the path loss for the distance ρ
j,i
in dB, and p the path loss exponent (typ. 3 in our scenarios). Notice that L
j,i
= P
T x,j
− P
Rx,i
, where P
T x, j
and P
Rx,i
are the transmitted and received powers in dBm for the pair {j, i}, respectively. If the randomness due to received power estimation is modeled by in dBm, then we can write
(6)
resulting in
(7)
and recognizing the terms, we obtain the observation equation for RSSI-based ranging:
(8)
Once range measurements are available, either resorting to TOA or RSSI techniques, the target node computes its position based on a simple algorithm to solve the trilateration problem. A least square (LS) algorithm suffices for the purpose of this article, which is to propose and analyze network topologies formations that improve the overall positioning performance. Thus, the well-known LS method is used here as a comparison tool among topology creation algorithms. Particularly, we consider one-hop ranging, meaning that ranging is performed considering only those reference nodes which are in view. As commented earlier, a set of ranging measurements from at least four reference nodes can be seen as a geometrical problem, where each (3) defines a sphere centered at the corresponding and with radii the measured range. The optimal positioning solution is given by the point in space where all the spheres intersect. Since the accuracy of range estimates is affected by noise (among other phenomena such as multipath components), the spheres are not likely to intersect at one single point and instead an uncertainty area is obtained in which the node can be found. The LS method provides an appealing solution to the problem, where the coordinates of the j th node are those which minimize the squared-error
(9)
The optimization admits a closed-form solution [23] based on the Moore-Penrose pseudoinverse
(10)
with the following definitions:
(11)
(12)
where is a constant, and indicates the cardinality of the set , i.e., the number ranging measurements at the j th node.
The next section describes algorithms for topology formation. Indeed, this comes before the position solution described above, from a practical point of view. Current topology formation criteria are focused on purposes other than localization; therefore, an effort is made in the sequel to the study existing methods and investigate clustering techniques aiming toward providing localization quality of service to the WSN.