2.1. Network Cramer-Rao Lower Bound

CRLB provides a lower boundary on the achievable variance of any unbiased location estimator for unknown parameters, which is useful for justifying how well an estimator can perform (Patwari et al., Reference Patwari, Ash, Kyperountas, Hero, Moses and Correal2005; Wymeersch et al., Reference Wymeersch, Lien and Win2009; Penna et al., Reference Penna, Caceres and Wymeersch2010). CRLB states that the variance of an unbiased estimator $\hat \theta $ should at least be as high as the inverse of a function of the expectation taken with respect to the probability density function (pdf) p(x; θ),

(1)$$CRLB = {\rm \;} \displaystyle{1 \over { - E\left[ {\displaystyle{{{\partial ^2}\ln p\left( {x;\theta} \right)} \over {\partial {\theta ^2}}}} \right]}}{\rm \;} \le {\rm v}ar\left( {\hat \theta} \right)$$

where the derivate is evaluated at the true value of θ. Assume that we are interested in ranging measurement stated as,

(2)$${r_i} = \sqrt {{{({{\hat x}_u} - {x_i})}^2} + {{({{\hat y}_u} - {y_i})}^2}} + \varepsilon $$

where $({\hat x_u},{\hat y_u})$ is the user location, (x _i, y_i) is the ith reference node, r _i is the ranging measurement centred at $h({\hat \theta})$ with a noise ε of Gaussian zero mean with covariance R. If there were m nodes in the network and $H = \displaystyle{\partial \over {\partial \theta}} h\left( \theta \right)$. CRLB at location (x, y) is given by

(3)$$CRLB(x,y) = \sqrt {tr({{({H^T}{R^{ - 1}}H)}^{ - 1}})} $$

where $R = diag(\sigma _1^2, \sigma _2^2, \ldots, \sigma _m^2 )$, $\sigma _i^2 $ is the variance of ith measurement. The resulting CRLB indicates the positioning accuracy level at each location. The performance of different networks is discussed below in detail.

2.2. Ranging accuracy

Generally, ranging measurement noise ε consists of white noise w and bias b. Thus Equation (2) can be rewritten as,

(4)$${r_i} = \sqrt {{{({{\hat x}_u} - {x_i})}^2} + {{({{\hat y}_u} - {y_i})}^2}} + \; {w_i} + {b_i}$$

The white noise is a zero mean variable with a variance of σ ², which can be modelled based on prior observation data. To testify the error bound for range-based positioning of different measurement levels, four different anchor locations are set on each corner of a 100 m × 100 m square area. The CRLB of the entire simulated area is calculated for different noise levels with variances of σ ² = 1, σ ² = 3 and σ ² = 5 while bias b = 1 m and b = 5 m respectively (Figure 1). Blue indicates low CRLB and red indicates high values.

Figure 1.

CRLB with different noise variance and bias.

While CRLB increases when the variance or bias increases, meaning a lower positioning accuracy, the effect of the variance is larger than that of the bias.

2.3. Network dilution of precision

The accuracy of GNSS positioning at a point on Earth is related to the geometry of observable satellite constellation, which is reflected by the Dilution Of Precision (DOP) (Langley, Reference Langley1999). Thus good signal geometry plays a significant role in positioning which restricts the measurement uncertainty into a smaller boundary.

The geometry of the collaborative network can be assumed as the satellite constellation projected onto a 2-D scenario. If ranging measurements between a user located at $({\hat x_u},{\hat y_u})$ and surrounding units located at (x _i, y_i) is expressed as Equation (4), the coordinate differences form the geometry matrix,

$$A = \left( {\matrix{ {\matrix{ {\displaystyle{{{{\hat x}_u} - {x_1}} \over {{r_1}}}} & {\displaystyle{{{{\hat y}_u} - {y_1}} \over {{r_1}}}} & 1 \cr}} \cr {\matrix{ {\displaystyle{{{{\hat x}_u} - {x_2}} \over {{r_2}}}} & {\displaystyle{{{{\hat y}_u} - {y_2}} \over {{r_2}}}} & 1 \cr}} \cr {\matrix{ {\; \; \vdots} & {\; \; \; \; \; \; \; \vdots} && {\; \; \; \; \; \; \; \vdots} \cr}} \cr {\matrix{ {\displaystyle{{{{\hat x}_u} - {x_m}} \over {{r_m}}}} & {\displaystyle{{{{\hat y}_u} - {y_m}} \over {{r_m}}}} & 1 \cr}} \cr}} \right)$$

where ^̂ denotes estimated results. In 2-D positioning, the Horizontal DOP (HDOP) is defined as

(5)$$HDOP = \sqrt {{\rm trace}({{({A^T}A)}^{ - 1}})} $$

DOP can be applied to analyse the collaborative network from two aspects. First of all, lower DOP indicates better positioning geometry. Secondly, increasing units could also potentially reduce the DOP. Figure 2 reflects that the positioning uncertainty boundary is markedly affected by the relative position of the units. When the two anchors are close together, DOP increases as well as the positioning uncertainty.

Figure 2.

Network geometry and error boundary.

2.3.1. Network Geometry Quality

In recent works, DOP has been applied to the analysis of geometric and signal strength for GPS/Wi-Fi and cellular communications positioning system (Zirari et al., Reference Zirari, Canalda and Spies2009; Chen et al., Reference Chen, Chiu, Lee and Lin2013). In this paper, DOP is used as an indicator of the anticipated network performance. The corresponding CRLB and DOP is compared for the designated area described in Section 2.2 where two anchors placed at different locations along the side of the area, marked as red diamonds in Figures 3 and 4. Dark blue indicates low values and red indicates high values. Results indicate that low DOP areas correspond to the low CRLB areas.

Figure 3.

CRLB for different geometry settings.

Figure 4.

DOP for different geometry settings.

2.3.2. Network Capacity

Increasing the network size is also a solution to improve accuracy. As Yang (Reference Yang2014) suggests, increasing the number of units will give better positioning performance in collaborative positioning scenarios. As the network size increases, the relative location of the anchors becomes a less dominating factor. However, the number of anchors should be controlled so that computation cost is kept as low as possible without affecting the positioning performance. Therefore the number of anchors should be carefully selected to maintain the balance. In this case, a threshold should be identified where increasing the unit number begins to have less of an obvious impact on improving positioning performance.

The CRLB of the designated area corresponding to network sizes increasing from two to eight are computed, of which four scenarios are shown below. The anchors are located along the side of the designated area marked as red diamonds in Figure 5, while noise level remains σ ² = 1. Results indicate an obvious decrease in CRLB when the network size increases, however the deduction rate is reduced when the number of anchors reaches four.

Figure 5.

CRLB for different network sizes.

3.1. Gathering measurements

The proposed collaborative positioning aims to constrain inertial measurement errors by integrating external information to a Pedestrian Dead Reckoning (PDR) model. A popular method of constraining heading bias in indoor positioning is through map matching. P2P ranging is also integrated to provide further constraint. The principles of PDR are given below, as well as the characteristics of the other measurements.

3.1.1. Pedestrian Dead Reckoning

PDR obtains the current position from a relative measurement between the current state and the previous state, e.g. the distance and direction travelled. These measurements may be obtained from any inertial device that provides a step count and heading, e.g. low-cost IMUs or smartphone. The step length is usually estimated by a step recognition model or set to a constant value, e.g. 0·7 m, and later corrected through filtering. The basic PDR model is as below,

(6)$$\left[ {\matrix{ {{{\hat x}_k}} \cr {{{\hat y}_k}} \cr}} \right] = \left[ {\matrix{ {{{\hat x}_{k - 1}} + {{\hat s}_{(k \vert k - 1)}}\cos {{\hat \theta} _{(k \vert k - 1)}}} \cr {{{\hat y}_{k - 1}} + {{\hat s}_{(k \vert k - 1)}}\sin {{\hat \theta} _{(k \vert k - 1)}}} \cr}} \right]$$

where $\left[ {{{\hat x}_k},{{\hat y}_k}} \right]$ is the estimated position at time k, ${\hat s_{(k\vert k - 1)}}$ is the estimated step length from time k-1 to time k, ${\hat \theta _{(k\vert k - 1)}}$ is the measured heading. Due to gyro drifts, the heading measurement $\hat \theta $ tends to be biased which increases continuously if no correction is implemented.

3.1.3. Interior Map Information

Map matching is commonly applied to constrain measurement errors in indoor navigation by forcing the user to stay within the reasonable path, i.e. pedestrians can only walk along corridors and travel through doors (Pinchin et al., Reference Pinchin, Hide and Moore2012). As shown in Figure 6, the user could only enter Room 2 by going out of the door into the corridor (c1) and then go through the door linking c1 and Room 2. The trouble with interior maps is that they must be available prior to use.

Figure 6.

Implementation of room polygons.

3.2. Particle filtering-based collaborative positioning

Particle filtering (PF) is a recursive Bayesian filtering method that handles non-linear and non-Gaussian systems. It has been widely applied to positioning and navigation problems due to its ability to integrate different measurements (Gustafsson et al., Reference Gustafsson, Gunnarsson, Bergman, Forssell, Jansson, Karlsson and Nordlund2002). PF predicts the system states through sequential Monte Carlo estimation from a large set of particles with associated weights that represent the state probability density function (pdf). The system state vector x _k is a discrete time stochastic model:

(7)$${x_k} = {\,f_k}({x_{k - 1}},{v_{k - 1}})$$

where k is the time index, f _k is the non-linear function of the state x _k−1 and process noise v _k−1. The state vector x _k is recursively updated from observation z _k:

(8)$${z_k} = {h_k}({x_k},{w_k})$$

where h _k is usually a non-linear function with measurement noise w _k. PF looks into estimating the state x _k at time k, given observations z _1:k up to time k. At each epoch, the predicted pdf is updated through measurements to represent the posterior pdf of the current state. However it is usually impossible to obtain the true posterior pdf. Therefore N particles are generated to represent a discrete approximation p(x),

(9)$$p(x) \approx \mathop \sum \limits_{i = 1}^N {w^i}\delta (x - {x^i})$$

where w ⁱ is the weight of the ith particle. As N → ∞, the approximation should approach the true posterior pdf (Arulampalam et al., Reference Arulampalam, Maskell, Gordon and Clapp2002).

A basic PF-based Collaborative Positioning (CP) is outlined below:

1. i. Initialisation: generate N _p particles around the initial position of each Rx [x ₀,y ₀], all particles are assigned an equal weight $w_k^i = 1 / N_p$.

2. ii. Prediction: particles propagate forward based on the PDR prediction model Equation (6). The step length is a constant value sl with a uniformly distributed random noise ${\rm {\cal U}}\sim ( - {n_s},{n_s})$, the heading ${\hat \theta _{(t\vert t - 1)}}$ consists of a constant heading bias b _h and a uniformly distributed random noise ${\rm {\cal U}}\sim( - {n_h},{n_h})$.

3. iii. Update and weighting: particles that cross walls are “killed”, i.e. $w_k^i = 0$. The collaborative constraint is then implemented by obtaining the ranging measurements ${\hat r_m}$ between the current rover to each other unit, as well as the distance $\hat d_m^i $ calculated from each particle of the rover to the other N _m units. For a particular particle i, if the difference between the two is over a collaborative constraint threshold thres _r

   (10)$$dif\,f_i = {\left\vert{\hat d}_m^i - {\hat r}_m\right\vert}_{m = 1,2, \ldots, {N_m}} \ge thres_{r}$$
   the particle is “killed”. thres _r is given based on the anticipated accuracy level of ranging measurements.

4. iv. Resampling: if the number of “live” particles falls below a threshold, i.e. $N_p / 2$, new particles are generated to maintain a total number of N particles.

5. v. Return to ii or end iteration.

This algorithm is applied to the simulated network as shown in Figure 7. Eight potential locations are indicated along the sides of a square area of 100 m × 100 m where anchors could be placed. North direction points upwards along the y-axis, East points rightwards along the x-axis. A single trajectory is defined and plotted in the green line while the magenta line indicates the PDR result. Five different location pairs for two anchors are simulated at five different locations, producing different geometries. Figure 8(a) indicates the positioning error of the networks with different DOPs, Figure 8(b) shows the DOP for each network while the rover is moving.

Figure 7.

Simulated positioning network.

Figure 8.

(a) Positioning errors for different networks. (b) DOP of each network.

To examine the positioning accuracy for different network sizes, the performance of networks consisting of between 2-10 anchors (shown in Figure 7) is examined. The mean positioning errors for different ranging accuracy levels, i.e. ranging error standard deviation σ _err of 3, 5 and 15 respectively, are plotted in Figure 9. We see a distinct improvement in positioning when the number of anchors increases from three to four for networks when σ _err = 3 and σ _err = 5 m, and four to five when σ _err = 15. The improvement in positioning becomes less evident after this size is reached. An increase in positioning error can actually be spotted when the number of anchors increases from 7 to 8 when σ _err = 3 m and σ _err = 15 m, and from 8 to 9 when σ _err = 5 m. This is due to the inaccuracy in the ranging measurement; when the number of ranging constraints increases, so does the inaccuracy in the constraint. Hence such increases are more likely when the ranging measurement itself is uncertain, e.g. the increase happens twice when σ _err = 15 m. Therefore, it is not a good idea to use more than the necessary number of units in a collaborative network, especially when the measurements themselves contain error or bias. Yet the optimal accuracy cannot be achieved if not enough units are used. Thus keeping a balance of the network size is important.

Figure 9.

Positioning errors for different network sizes.

3.3. Modified DOP

Although DOP demonstrates the relationship between the geometry and positioning performance, it cannot reflect all details inside a collaborative network. The first factor that is not reflected in DOP is the ranging accuracy, which directly influences the effectiveness of the constraint in collaborative positioning. The constraint threshold thres _r is defined as the anticipated accuracy level of the ranging measurements plus an “error boundary”. However, if this threshold is smaller than the measurement error itself, i.e. the constraint is too “tight”, the positioning estimation would be pushed towards a wrong location. On the other hand, if the bound is much larger than the error, i.e. constraint too “weak”, then the observation noise and error may not be sufficiently eliminated.

Therefore, a Modified DOP (MDOP) that integrates the ranging quality is proposed here and the geometry matrix A _mod is computed as below,

(11)$${A_{{\rm mod}}} = \left( {\matrix{ {\matrix{ {\displaystyle{{{{\hat x}_u} - {X_1}} \over {a \cdot {r_1}}}} & {\displaystyle{{{{\hat y}_u} - {Y_1}} \over {a \cdot {r_1}}}} & 1 \cr}} \cr {\matrix{ {\displaystyle{{{{\hat x}_u} - {X_2}} \over {a \cdot {r_2}}}} & {\displaystyle{{{{\hat y}_u} - {Y_2}} \over {a \cdot {r_2}}}} & 1 \cr}} \cr {\matrix{ {\; \; \vdots} & {\; \; \; \; \; \; \; \vdots} & {\; \; \; \; \; \; \; \vdots} \cr}} \cr {\matrix{ {\displaystyle{{{{\hat x}_u} - {X_n}} \over {a \cdot {r_n}}}} & {\displaystyle{{{{\hat y}_u} - {Y_n}} \over {a \cdot {r_n}}}} & 1 \cr}} \cr}} \right)$$

where a is a measurement accuracy coefficient derived from accuracy detection. The detection method provides the user with how likely the measurement is to be reflecting the true distance, which is given by a, a value between 0 and 1. Hence reliable measurements produce a closer to 1 and A _mod would be close to A. MDOP is computed from A _mod as in Equation (12), thus the produced MDOP is usually larger than the original DOP.

(12)$$MDOP = \sqrt {{\rm trace}({{(A_{{\rm mod}}^T {A_{{\rm mod}}})}^{ - 1}})} $$

While the rover is always moving during navigation, it is hard for the DOP to reflect the dynamic directional information, e.g. the direction of the bias of the current rover relative to the anchors. Figure 10 shows the error in both the East and North directions when CP is applied to the rover simulated in Figure 7. As Tx5 and Tx6 are located on either side of the rover, the network constrains the error in the North direction better than the East direction. Tx7 and Tx8 are both located to the north of the rover, thus they constrain the error in the East direction better than the North direction. Measurements coming from different directions will constrain error along different directions. The selected units should consider the dynamic situation of the rover as directions change.

Figure 10.

CP Positioning error of different networks.

MDOP is not just a value that reflects the geometry with the ranging accuracy, but rather a concept of considering all the relevant information of a dynamic collaborative network, i.e. the ranging accuracy, the network geometry and the relative positions of the units.

3.4. ARCP

When collaborative units are available, the appropriate units should be selected to form a network with the optimal MDOP to produce the best positioning results. The Adaptive Ranging Constraint collaborative Positioning (ARCP) method is developed here and its procedure outlined in Figure 11. Compared to CP, the adaptivity of the ARCP is defined from three aspects: adaptability to varying measurement accuracies, the flexibility to select different network size and unit locations. More specifically, the adaptivity is reflected in the selection of the appropriate units.

Figure 11.

Flowchart of ARCP.

Once the rover takes a step based on PDR, it will look for local units to form the collaborative network. The optimal size of the network is considered to be four according to the simulation results presented in Section 3.2. Integrating too many or not enough units will both result in reduced positioning performance. If more than four units (including rovers and anchors) are available, the estimated accuracy level of the ranging measurement from each unit is obtained. Those units with an associated a larger than 0·5 are considered as potential units. They are then combined with the current rover to form a network of four units and the MDOP of each possible network is computed. The relative positions of the units are also considered by sharing the position of the anchors and the estimated position of the other rovers. The network with the smallest MDOP value is selected as the optimal network. The constraint thres _r for each ranging measurement is set according to MDOP, which reflects both a and DOP. If less than four units are available, the units would simply be included in the collaborative network and thres _r set according to MDOP.

a can be converted to the estimated standard deviation of the measurement σ _r, which is then applied as the constraint threshold. a is mapped onto three categories of thres _r,

(13)$$thre{s_r}(a) = \left\{ {\matrix{ {1,} & {a \ge 0.8,} \cr {2,} & {0.5 \le a \lt 0.8,} \cr {3,} & {{\rm \; \; \;} a \lt 0.5.} \cr}} \right.$$

The values are selected based on real indoor measurement error levels and indicate the expected error in metres. Most measurements should fall within category 1 or 2. A threshold of 3 indicates a very loose constraint, where the rover mostly depends on PDR propagation. The thres _r is further derived from DOP based on Equation (14) which multiplies a coefficient to thres _r(a). Simulations have shown that if the threshold were set to the same value as the real measurement standard deviation, the constraint would be too tight. Hence the final threshold is always larger than the expected error standard deviation. The threshold categories can be adjusted but the values applied here are selected from the combination that gives the best constraint performance for the simulations in this paper.

(14)$$thre{s_r}({\rm DOP}) = \left\{ {\matrix{ {thre{s_r}\left( a \right){\ast}1.5,} & {{\rm DOP} \lt 5,} \cr {thre{s_r}\left( a \right){\ast}2,} & {5 \le {\rm DOP} \lt 10,} \cr {thre{s_r}\left( a \right){\ast}3,} & {{\rm \; \; DOP} \ge 10.} \cr}} \right.$$

By applying the ARCP, the possibility of selecting units with a low ranging accuracy is reduced. The constraint threshold is then also set according to the estimated measurement accuracy. Hence a collaborative network consisting of the optimal units will more likely output positions with higher accuracy and reliability. Less optimal network positioning mostly depends on inertial measurements.

ARCP is applied to the same networks as those in Figure 8 and the positioning error of ARCP and CP is compared in Figure 12. An obvious improvement can be seen when the adaptive selection is applied.

Figure 12.

Positioning error comparisons for ARCP and non-adaptive CP.

4.1. Simulations

The proposed ARCP algorithm is applied to two sets of trials, denoted as Trial A and Trial B, to validate its application within real environments. Data are collected and post-processed in real-time mode on Matlab 2013 and different algorithms are implemented for comparison. For both trials, the inertial data are collected using MicroStrain 3DM-GX3®-25 IMU, which is connected to a Raspberry Pi for data logging. The step and heading information are then extracted and applied to the PDR model. The interior building map was surveyed by Leica TS30 total station and loaded into Matlab as polygons (rooms and corridors) and points (doors).

In Trial A1, all real inertial data was collected in the Nottingham Geospatial Building (NGB), University of Nottingham. Three anchors (Tx1, Tx2 and Tx3) were simulated at different locations inside the building to provide extra ranging constraints. The ranging data between the rovers and anchors were simulated based on indoor ranging performance of wireless signals, where the error variance is larger when the two rovers are in NLOS and smaller when there is no obstruction. The basic CP algorithm is applied in Trial A1 by integrating one of the anchors into the network. The measurement error of the rover is constrained by integrating the ranging measurement from the other rover and one anchor at every epoch by applying a constant threshold. The non-adaptive result of the network consisting Rx1, Rx2 and Tx1 is shown in Figure 13. The green line indicates the ground truth for both rovers, the cross dot line indicates the position estimation of Rover 1 and the circle dash line indicates the position estimation of Rover 2.

Figure 13.

CP Positioning result with wall constraint (Trial A1-Tx1).

The ARCP algorithm is applied in Trial A2 where each rover selects one anchor to form a collaborative network with the other rover that produces the optimal MDOP at every epoch. thres _r is adjusted according to MDOP. Results are shown in Figure 14. The plot indications are the same as Figure 13.

Figure 14.

ARCP Positioning result (Trial A2).

In Trial A3, the ARCP is applied while eliminating the building map information with results as shown in Figure 15. Here, the particles are no longer restricted from crossing walls and the measurement error is bounded only by the ranging constraint.

Figure 15.

ARCP Positioning result without map matching (Trial A3).

4.2. Real data

Trial B was carried out in the Business School Building, University of Nottingham. PDR data was collected using the same equipment worn on two pedestrians, Rover 1 and Rover 2. A UWB network was set up in the building as indicated with a red star in Figure 16 to act as anchors and provide ranging measurement. Each rover also carried a mobile UWB unit to receive ranging measurements from other units.

Figure 16.

Trial B Ground truth for Rover 1 and Rover 2.

Data was collected for ten minutes. In every epoch, each rover selected a number of the anchors to form a collaborative network with the other rover. The network size and the ranging constraint threshold were adjusted according to the actual network quality.

The ground truth is plotted in Figure 16. Due to lack of equipment, the ground truth of Rover 2 was provided by the UWB system, whose outdoor performance was disrupted (light blue part of the trajectory in Figure 16(b)), as all units are set up indoors. The positioning results for Rover 1 and Rover 2 are shown in Figure 17 (a) and (b) respectively. The green solid line indicates the ground truth, the cyan dashed line shows the PDR output from raw inertial data. The blue line represents the ARCP output with wall constraint, and the magenta line represents the ARCP result without wall constraint.

Figure 17.

ARCP Positioning result for Rover 1 and Rover 2 (Trial B).