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monthyeardate[] ,

#  Data Fusion for Radio Frequency SLAM
with Robust Sampling

Erik Leitinger13, Bryan Teague2, Wenyu Zhang4, Mingchao Liang4, and Florian
Meyer4
1TU Graz, Austria and 3Christian Doppler Laboratory for Location-aware
Electronic Systems<EMAIL_ADDRESS> 2MIT Lincoln Laboratory,
Lexington, MA, USA<EMAIL_ADDRESS> 4Scripps Institution of
Oceanography and Department of Electrical and Computer Engineering  University
of California San Diego, La Jolla, CA, USA<EMAIL_ADDRESS>
###### Abstract

Precise indoor localization remains a challenging problem for a variety of
essential applications. A promising approach to address this problem is to
exchange radio signals between mobile agents and static physical anchors that
bounce off flat surfaces in the indoor environment. Radio frequency
simultaneous localization and mapping (RF-SLAM) methods can be used to jointly
estimates the time-varying location of agents as well as the static locations
of the flat surfaces. Recent work on RF-SLAM methods has shown that each
surface can be efficiently represented by a single MVA. The measurement model
related to this MVA-based RF-SLAM method is highly nonlinear. Thus, Bayesian
estimation relies on sampling-based techniques. The original MVA-based RF-SLAM
method employs conventional “bootstrap” sampling. In challenging scenarios it
was observed that the original method might converge to incorrect MVA
positions corresponding to local maxima. In this paper, we introduce MVA-based
RF-SLAM with an improved sampling technique that succeeds in the
aforementioned challenging scenarios. Our simulation results demonstrate
significant performance advantages.

##  I Introduction

Maintaining accurate and timely situational awareness in indoor environments
is challenging in a variety of applications. This is particularly true when no
prior geometric information about the environment is available. In such
scenarios, it is desirable to generate a map and localize mobile “agents”
within that map. RF-SLAM is a promising methodology for precise localization
in the aforementioned scenarios. Here, radio signals that bounce off flat
surfaces are used to estimate the locations of mobile agents and the surfaces
themselves.

RF-SLAM methods typically represent a flat surface in the environment using
the notion of virtual anchors[1, 2, 3, 4]. A VA represents the location of the
mirror image of a PA on a reflecting surface. The reflected propagation path
agent – surface – PA can equivalently be described by a direct path agent –
VA. The goal of RF-SLAM methods is to detect and localize VAs along with the
time-varying position of the mobile agent [1, 2, 3, 4]. By detecting and
localizing the VAs, multiple radio propagation paths can be leveraged for
agent localization which increases accuracy and robustness.

###  I-A Background and Contributions

RF-SLAM follows a traditional feature-based SLAM approach [5, 6, 7], i.e., the
map is represented by static _features_ , whose unknown positions are
estimated using sequential inference. In particular, most RF-SLAM methods
consider the VAs as the features to be mapped [3, 8, 4, 9]. A complicating
factor in RF-SLAM is measurement origin uncertainty, i.e., the unknown
association of measurements with features [3, 8, 4, 10]. State-of-the-art
techniques for RF-SLAM formulate and solve a high-dimensional sequential
Bayesian estimation problem using factor graphs[3, 8, 4]. Due to the nonlinear
measurement model the resulting sum-product algorithm (SPA) relies on sampling
techniques [11, 1, 2, 3, 4]. In traditional RF-SLAM, a reflective surface can
take part in multiple propagation paths, each path is represented by a VA, and
each VA is estimated independently [1, 2, 3, 4]. This limits timeliness and
accuracy of RF-SLAM.

The notion of a unique MVA has been recently introduced to enable data fusion
across propagation paths, i.e., to allow multiple paths to be used for joint
estimation of a single reflective surface [12]. This approach has the
potential to strongly improve the accuracy and convergence time of RF-SLAM
[12]. The original MVA-based RF-SLAM method [12] employs conventional
“bootstrap” sampling. In challenging scenarios with one or two PAs where only
range measurements are available, bootstrap sampling has severe limitations.
In particular, due to geometric symmetries, probability density functions of
MVAs can be multi-modal during some initial time steps, and the sample
representations provided by bootstrap sampling may collapse in a wrong mode.
In turn, it was observed that the original method might converge to incorrect
MVA positions that are local maxima.

In this paper, we address this limitation of MVA-based RF-SLAM by introducing
an improved robust sampling technique. Our method periodically uses a proposal
distribution constructed from informative range measurements to avoid local
maxima. Sampling from this type of proposal distribution leads to new
dissemination of samples and avoids wrong modes. The key contributions of this
paper are as follows.

  * •

We introduce a proposal distribution for MVA-based RF-SLAM that can provide
robustness in challenging scenarios with range measurements.

  * •

We demonstrate significant performance advantages of the MVA-based RF-SLAM
with robust sampling compared to conventional MVA-based RF-SLAM.

##  II System model of MVA-based RF-SLAM

We consider a mobile agent and $J$ PAs with known positions
$\bm{p}_{\mathrm{pa}}^{(j)}\\!\in\mathbb{R}^{2}$, $j=1,\ldots,J$, where $J$ is
assumed to be known. At each discrete time slot $n$, the position of the
mobile agent $\bm{p}_{n}\\!\in\mathbb{R}^{2}$ is unknown. The mobile agent
transmits a radio signal and the PAs act as receivers. However, the proposed
algorithm can be easily reformulated for the case where the PAs act as
transmitters and the mobile agent acts as a receiver. Associated with the
$j$th PA, there are $K$ VAs [3] at unknown positions
$\bm{p}_{k,\mathrm{va}}^{(j)}\\!\in\mathbb{R}^{2}$, $k=1,\ldots,K\\!$. VA
position $\bm{p}_{k,\mathrm{va}}^{(j)}$ is the mirror image of
$\bm{p}_{\mathrm{pa}}^{(j)}$ at reflective surface $k$; it represents the
single-bounce propagation path agent – surface $k$ – PA $j$. In a
preprocessing stage, distance measurements $d_{m,n}^{(j)}$,
$m=1,\dots,M^{(j)}_{n}$ are extracted from the radio signal received at PA $j$
and time $n$ [3, 8, 13]. The distance measurement related to the propagation
path represented by $\bm{p}_{k,\mathrm{va}}^{(j)}$ is modeled as
$z_{m,n}^{(j)}=\big{\|}\bm{p}_{n}-\bm{p}^{(j)}_{k,\mathrm{va}}\big{\|}+\nu_{m,n}^{(j)}$
where $\|\cdot\|$ represents the Euclidean norm and $\nu_{m,n}^{(j)}$ is zero-
mean Gaussian measurement noise with standard deviation $\sigma^{(j)}_{m,n}$.
Note that $\nu_{m,n}^{(j)}$ is assumed statistically independent of
$\bm{p}_{n}$ and $\bm{p}^{(j)}_{k,\mathrm{va}}$.

A reflective surface is involved in multiple propagation paths and thus gives
rise to multiple VAs. To enable the consistent combination, i.e., “fusion” of
map information provided by distance measurement of different PAs, following
[12], we represent reflective surfaces by $K$ unique MVAs at unknown positions
$\bm{p}_{k,\mathrm{mva}}\\!\in\mathbb{R}^{2}$, $k=1,\ldots,K\\!$. The unique
MVA position $\bm{p}_{k,\mathrm{mva}}\\!\in\mathbb{R}^{2}$ is defined as the
mirror image of the global origin $[0\hskip 1.70717pt0]^{\text{T}}$ on the
reflective surface $K$.

Figure 1: A graphical depiction of a multipath-based localization scenario
involving $K=1$ reflective surfaces, $J=2$ PAs, and positions of the agent
$\bm{p}_{n}$ at two time steps $n\in\\{n_{1},n_{2}\\}$.

By using some algebra, the transformation from MVA $\bm{p}_{k,\mathrm{mva}}$
to VA position $\bm{p}^{(j)}_{\mathrm{pa}}$ can be obtained as

$\displaystyle\bm{p}^{(j)}_{k,\mathrm{va}}$
$\displaystyle=-\bigg{(}\frac{2\big{\langle}\bm{p}_{k,\mathrm{mva}},\bm{p}^{(j)}_{\mathrm{pa}}\big{\rangle}}{\big{\|}\bm{p}_{k,\mathrm{mva}}\big{\|}^{2}}-1\bigg{)}\hskip
0.85358pt\bm{p}_{k,\mathrm{mva}}+\hskip 0.85358pt\bm{p}^{(j)}_{\mathrm{pa}}$
$\displaystyle=h\big{(}\bm{p}_{k,\mathrm{mva}},\bm{p}^{(j)}_{\mathrm{pa}}\big{)}.$
(1)

where $\langle\cdot,\cdot\rangle$ represents the inner-product between two
vectors.

Based on the nonlinear transformation in (1), the distance measurement related
to the propagation path agent – surface $k$ – PA $j$ can now alternatively be
expressed by [12]

$\displaystyle z_{m,n}^{(j)}$
$\displaystyle=\big{\|}\bm{p}_{n}-h\big{(}\bm{p}_{k,\mathrm{mva}},\bm{p}^{(j)}_{\mathrm{pa}}\big{)}\big{\|}+\nu_{m,n}^{(j)}.$
(2)

Using (2) as well as the known PA position $\bm{p}^{(j)}_{\mathrm{pa}}$, we
can directly obtain the MVA-based likelihood function as
$f\big{(}z_{m,n}^{(j)}\big{|}\bm{p}_{k,\mathrm{mva}},\bm{p}_{n}\big{)}\vspace{.8mm}$.
Note that with the proposed MVA-based measurement model, at time $n$, the
measurements collected by all PAs $j=1,\ldots,J$ can provide information on
the same MVA and agent positions $\bm{p}_{k,\mathrm{mva}},k\in\\{1,\dots,K\\}$
and $\bm{p}_{n}$, respectively. The number of MVAs $K$ is unknown. As an
example, Fig. 1 shows a scenario with two PAs $j\in\\{1,2\\}$ and two time
steps $n_{1}$ and $n_{2}$. Note that due to measurement origin uncertainty
there is an unknown association of measurements with MVAs. Furthermore there
can be missed detections, i.e., at some time steps $n$ actual MVAs may not
produce a measurement at some PA $j$; and false alarms, i.e., there may be
clutter measurements not generated by any MVA (see [3, 8, 4, 10] for details).

At each time $n$, the state of the agent $\bm{x}_{n}$ consists of it’s
position $\bm{p}_{n}$ and possibly further motion related parameters. As in
[14, 15, 3], we account for the unknown number of MVAs by introducing
potential s $k\in\\{1,\dots,K_{n}\\}$. The number $K_{n}$ of PMVAs is the
maximum possible number of actual MVA that produced a measurement so far [15]
(where $K_{n}$ increases with time). PMVA states are denoted as
$\bm{y}_{k,n}=\big{[}\bm{p}^{\text{T}}_{k,\mathrm{mva}}\;r_{k,n}\big{]}^{\text{T}}$.
The existence/nonexistence of PMVA $k$ is modeled by the existence variable
$r_{k,n}\in\\{0,1\\}$ in the sense that PMVA $k$ exists if and only if
$r_{k,n}\\!=\\!1$. It is considered formally also if PMVA $k$ is nonexistent,
i.e., if $r_{k,n}\\!=0$. The states $\bm{p}^{\text{T}}_{k,\mathrm{mva}}$ of
nonexistent PMVAs are obviously irrelevant. Therefore, all PDFs defined for
PMVA states, $f(\bm{y}_{k,n})=f(\bm{p}_{k,\mathrm{mva}},r_{k,n})$, are of the
form $f(\bm{p}_{k,\mathrm{mva}},0)$
$=f_{k,n}f_{\text{d}}(\bm{p}_{k,\mathrm{mva}})$, where
$f_{\text{d}}(\bm{p}_{k,\mathrm{mva}})$ is an arbitrary “dummy PDF” and
$f_{k,n}\\!\in[0,1]$ is a constant. Further details for the system model of
MVA-based RF-SLAM are provided in [12].

##  III Problem Formulation and Proposed Method

We aim to estimate the agent state $\bm{x}_{n}$ using all available
measurements $\bm{z}_{1:n}$ from all PAs up to time $n$. In particular, we
calculate an estimate $\hat{\bm{x}}_{n}$ by using the minimum mean-square
error (MMSE) estimator [16, Ch. 4]

$\hat{\bm{x}}_{n}\hskip 0.85358pt\triangleq\,\int\bm{x}_{n}\hskip
0.85358ptf(\bm{x}_{n}|\bm{z}_{1:n})\hskip
0.85358pt\mathrm{d}\bm{x}_{n}.\vspace{.5mm}$ (3)

For the mapping of reflection surfaces, detection of PMVAs
$k\\!\in\\!\\{1,\dots,K_{n}\\}$ and estimation their positions
$\bm{p}_{k,\mathrm{mva}}$ is considered. This relies on the marginal posterior
existence probabilities $p(r_{k,n}\\!=\\!1|\bm{z}_{1:n})$ and the marginal
posterior PDFs $f(\bm{x}_{k,n}|r_{k,n}\\!=\\!1,\bm{z}_{1:n})$. A PMVA $k$ is
declared to exist if $p(r_{k,n}\\!=\\!1|\bm{z}_{1:n})>p_{\text{de}}$, where
$p_{\text{de}}$ is a detection threshold [16, Ch. 2]. The number $\hat{K}_{n}$
of PMVA states that are considered to exist is the estimate of total number
$K$ of MVAs. For existing PMVAs, an estimate of it’s position
$\bm{p}_{k,\mathrm{mva}}$ can again be calculated by the MMSE [16, Ch. 4]

$\hat{\bm{p}}_{k,\mathrm{mva}}\,\triangleq\int\bm{p}_{k,\mathrm{mva}}\hskip
0.85358pt\hskip 0.85358ptf(\bm{p}_{k,\mathrm{mva}}\hskip 0.85358pt|\hskip
0.85358ptr_{k,n}\\!=\\!1,\bm{z}_{1:n})\hskip 0.85358pt\hskip
0.85358pt\mathrm{d}\bm{p}_{k,\mathrm{mva}}.\vspace{.5mm}$ (4)

The calculation of $f(\bm{x}_{n}|\bm{z}_{1:n})$, $p(r_{k,n}\\!=\\!1|\bm{z})$,
and $f(\bm{p}_{k,\mathrm{mva}}|$ $r_{k,n}=1,\bm{z}_{1:n})$ from the joint
posterior PDF [12, Eq. (4)] by direct marginalization is not feasible.

By performing sequential sample-based message passing by means of the SPA
rules [17, 18, 14, 3] on the factor graph in [12, Fig. 3], approximations
(“beliefs”) of the marginal posterior PDFs $q(\bm{x}_{n})\approx
f(\bm{x}_{n}|\bm{z}_{1:n})$ and $q(\bm{p}_{k,\mathrm{mva}},r_{k,n})\approx
f(\bm{p}_{k,\mathrm{mva}},r_{k,n}|\bm{z}_{1:n})$, $k\in\\{1,\dots,K_{n}\\}$
can be obtained in an efficient ways. More specifically, representations of
beliefs that consist of $I$ weighted random samples or particles denoted as
$\big{\\{}\bm{x}^{(i)}_{n},\mathpzc{w}_{n}^{(i)}\big{\\}}^{I}_{i=1}\sim
q(\bm{x}_{n})$ and
$\big{\\{}\big{(}\bm{p}^{(i)}_{k,\mathrm{mva}},w_{k,n}^{(i)}\big{)}\big{\\}}^{I}_{i=1}\sim
q(\bm{p}_{k,\mathrm{mva}},r_{k,n})$, $k\in\\{1,\dots,K_{n}\\}$ are computed
[18, 14, 3]. Note that $\sum^{I}_{i=1}\mathpzc{w}_{n}^{(i)}=1$ while
$\sum^{I}_{i=1}w_{k,n}^{(i)}\approx p(r_{k,n}\\!=\\!1|\bm{z})$ (cf. [14, Sec.
VI]). These sample-based representations can be used for approximate MMSE
estimation by evaluating (3) and (4) based on Monte Carlo integration [19]. To
avoid the number of PMVA states growing indefinitely, PMVAs states with
$p(r_{k,n}=1\hskip 0.85358pt|\hskip 0.85358pt\bm{z}_{1:n})$ below a threshold
$p_{\text{pr}}$ are removed from the state space (“pruned”). Pruning is
performed at each time $n$, after the measurements of all PAs have been
processed.

##  IV Review of Bootstrap Sampling for RF-SLAM

Existing RF-SLAM methods employ the bootstrap sampling strategy [11] where
predicted beliefs are employed as proposal PDF. In particular, the proposal
distribution for calculating the belief $q(\bm{p}_{k,\mathrm{mva}},r_{k,n})$
of MVA, $k\in\\{1,\dots,K_{n}\\}$ at time $n$ is given by (cf. [18, 3, 12])

$f_{\mathrm{pred.}}(\bm{x}_{n},\bm{p}_{k,\mathrm{mva}})\propto\alpha(\bm{p}_{k,\mathrm{mva}},r_{k,n}=1)\hskip
0.85358pt\alpha(\bm{x}_{n})$ (5)

where $\propto$ indicates equality up to a normalization factor and the
“prediction messages” $\alpha(\bm{p}_{k,\mathrm{mva}},r_{k,n})$ and
$\alpha(\bm{x}_{n})$ can be obtained as

$\displaystyle\alpha(\bm{x}_{n})$ $\displaystyle=\int
f(\bm{x}_{n}|\bm{x}_{n-1})\hskip 0.85358ptq(\bm{x}_{n-1})\hskip
0.85358pt\mathrm{d}\bm{x}_{n-1}$
$\displaystyle\alpha(\bm{p}_{k,\mathrm{mva}},r_{k,n})$
$\displaystyle=\sum_{r_{k,n-1}\in\\{0,1\\}}\int
f\big{(}\bm{p}_{k,\mathrm{mva}}\hskip 0.85358pt,r_{k,n}\hskip
0.85358pt\big{|}\hskip 0.85358pt\bm{p}_{k,\mathrm{mva}}\hskip
0.85358pt,r_{k,n-1}\big{)}$ $\displaystyle\hskip 42.67912pt\times
q\big{(}\bm{p}_{k,\mathrm{mva}},r_{k,n-1}\big{)}\hskip
0.85358pt\mathrm{d}\bm{p}_{k,\mathrm{mva}}\hskip 0.85358pt.\vspace{-1.2mm}$
(6)

Here, $f(\bm{x}_{n}|\bm{x}_{n-1})$ and $f\big{(}\bm{p}_{k,\mathrm{mva}}\hskip
0.85358pt,r_{k,n}\hskip 0.85358pt\big{|}\hskip
0.85358pt\bm{p}_{k,\mathrm{mva}}\hskip 0.85358pt,r_{k,n-1}\big{)}$ are the
state-transition functions for the agent state and the MVA state, respectively
[12, Sec. II]. Furthermore, the beliefs of the agent state, $q(\bm{x}_{n-1})$,
and of the MVA states $q\big{(}\bm{p}_{k,\mathrm{mva}},r_{k,n-1}\big{)}$, were
calculated at the preceding time $n-\\!1$. Note that for the proposal
$f_{\mathrm{pred.}}(\bm{x}_{n},\bm{p}_{k,\mathrm{mva}})$ in (5), we only use
the functional form of $\alpha(\bm{p}_{k,\mathrm{mva}},r_{k,n})$ for the case
$r_{k,n}=1$, since the functional form for the case $r_{k,n}=0$ is always
equal to the dummy PDF $f_{\text{d}}(\bm{p}_{k,\mathrm{mva}})$. Samples of
$f_{\mathrm{pred.}}(\bm{x}_{n},\bm{p}_{k,\mathrm{mva}})$ can be obtained as
discussed in [14, 3].

The proposal distribution in (5) has to be a function of both MVA position
$\bm{p}_{k,\mathrm{mva}}$ and agent state $\bm{x}_{n}$, since the measurement
model (2) also involves both MVA state and agent state. Furthermore, note that
the proposal distribution in (5) is also used to calculate a factor that
represents the contribution of MVA $k\in\\{1,\dots,K_{n}\\}$ to the agent
weights $\mathpzc{w}_{n}^{(i)}$, $i\in\\{1,\dots,I\\}$ (cf. [18, 3, 12]).

Bootstrap sampling is suitable for RF-SLAM methods that consider VAs as the
features to be mapped [3, 8, 4]. However, its use for the fast and more
accurate MVAs-based RF-SLAM is problematic in challenging RF-SLAM scenarios
with one or two PAs where only range measurements are available. This is
because, due to certain geometric symmetries, for some initial time steps
after an MVA has been introduced, the PDFs of an MVA can be multi-modal. In
addition, during these initial time steps, often, the dominant mode might not
be the one located at the correct MVA position. Thus, the sample
representations provided by bootstrap sampling may collapse in a wrong mode,
i.e., converge to incorrect MVA positions corresponding to local maxima.

##  V Robust Sampling for MVA-based RF-SLAM

To address these limitations, we introduce an alternative strategy for
obtaining a proposal distribution. In what follows, we will again discuss the
proposal distribution for the belief $q(\bm{p}_{k,\mathrm{mva}},r_{k,n})$ of
MVA, $k\in\\{1,\dots,K_{n}\\}$. As discussed later this proposal will just be
used at certain time steps $n$.

The main idea is to construct a weighted mixture that consists of MVA position
information that is predicted and MVA position information that is related to
the best measurement for each PA $j=1,\ldots,J$ obtained at time step $n-1$. A
measurement is the best if it has the largest probability of association with
MVA $k$ at time $n-1$. The index of this best measurement is denoted by
$m^{(j)}_{k,n-1}\in\big{\\{}0,1,\dots,M^{(j)}_{n-1}\big{\\}}$, where
$m^{(j)}_{k,n-1}=0$ indicates that no measurement is the best, i.e., the
probability of no measurement being associated to the MVA (“missed detection”)
is larger than the probability of association with any measurement. To
construct the proposal distribution at time $n$, we use the measurements from
time $n-1$ because association probabilities at time $n-1$ have already been
calculated [3]; on the other hand, calculation of association probabilities at
time $n$ is based on the proposal distribution at time $n$.

The considered proposal distribution at time $n$, is given by

$\displaystyle f_{\mathrm{mixture}}(\bm{x}_{n},\bm{p}_{k,\mathrm{mva}})$
$\displaystyle\hskip 17.07164pt\triangleq
f_{\mathrm{pred.}}(\bm{x}_{n},\bm{p}_{k,\mathrm{mva}})+\sum_{j\in{\cal{J}}}f^{(j)}_{\mathrm{meas.}}(\bm{x}_{n},\bm{p}_{k,\mathrm{mva}})$
(7)

where ${\cal{J}}$ consists of all indexes $j$ with $m^{(j)}_{k,n-1}\neq 0$.
Furthermore, the component related to the best measurements reads

$\displaystyle
f^{(j)}_{\mathrm{meas.}}(\bm{x}_{n},\bm{p}_{k,\mathrm{mva}})\propto\alpha(\bm{x}_{n})f\big{(}z_{m^{(j)}_{k,n-1},n-1}^{(j)}\big{|}\bm{p}_{k,\mathrm{mva}},\bm{p}_{n}\big{)}.$

Samples of $f^{(j)}_{\mathrm{meas.}}(\bm{x}_{n},\bm{p}_{k,\mathrm{mva}})$ can
be obtained by following the importance sampling principle [19]. In
particular, samples
$\big{\\{}(\tilde{\bm{x}}^{(i)}_{n},\tilde{\bm{p}}^{(i)}_{k,\mathrm{mva}})\big{\\}}^{I^{\prime}}_{i=1}$
are drawn from $\alpha(\bm{x}_{n})\hskip
0.85358ptf_{\mathrm{uni.}}(\bm{p}_{k,\mathrm{mva}})$ first, where
$f_{\mathrm{uni.}}(\bm{p}_{k,\mathrm{mva}})$ is a PDF that is uniform on the
area of interest. Then corresponding unnormalized weights are calculated as

$\tilde{w}^{(i)}_{k,n}=\frac{f\Big{(}z_{m^{(j)}_{k,n-1},n-1}^{(j)}\hskip
0.85358pt\big{|}\hskip
0.85358pt\tilde{\bm{p}}^{(i)}_{k,\mathrm{mva}},\tilde{\bm{p}}^{(i)}_{n}\Big{)}}{f_{\mathrm{uni.}}\big{(}\tilde{\bm{p}}^{(i)}_{k,\mathrm{mva}}\big{)}}.$
(8)

The samples
$\big{\\{}(\bm{x}^{(i)}_{n},\bm{p}^{(i)}_{k,\mathrm{mva}})\big{\\}}^{I^{\prime}}_{i=1}$
representing $f^{(j)}_{\mathrm{meas.}}(\bm{x}_{n},$ $\bm{p}_{k,\mathrm{mva}})$
are computed from
$\big{\\{}(\tilde{\bm{x}}^{(i)}_{n},\tilde{\bm{p}}^{(i)}_{k,\mathrm{mva}},\tilde{w}^{(i)}_{k,n})\big{\\}}^{I^{\prime}}_{i=1}$
by first normalizing the weights $\tilde{w}^{(i)}_{k,n}$,
$i\in\\{1,\dots,I^{\prime}\\}$ and then performing a resampling step [19].
Finally, $I$ samples of the considered proposal distribution
$f_{\mathrm{mixture}}(\bm{x}_{n},\bm{p}_{k,\mathrm{mva}},r_{k,n})$ in (7) are
obtained as follows: (i) $I^{\prime}$ is selected as small as possible such
that $I^{\prime}(|{\cal{J}}|+1)\geqslant I$; (ii) $I^{\prime}$ samples from
$f_{\mathrm{pred.}}(\bm{x}_{n},\bm{p}_{k,\mathrm{mva}})$ and from each
$f^{(j)}_{\mathrm{meas.}}(\bm{x}_{n},\bm{p}_{k,\mathrm{mva}})$,
$j\in{\cal{J}}$, are obtained; and (iii) $I$ of the resulting
$I^{\prime}(|{\cal{J}}|+1)$ samples are selected randomly.

The main advantage of using (7) as the proposal distribution is that particles
are again spread out over a wider area of potential MVA locations and thus
collapsing to a wrong mode is avoided. On the other hand, using (7) at each
time step would significantly reduce the speed of convergence to the correct
mode. In our numerical evaluation of MVA-based RF-SLAM, we found it useful to
only use the proposal distribution (7) at certain time steps $n^{\prime}$ that
are determined randomly. Let $n_{1}$ be the last time steps where (7) has been
used. The next time step $n_{2}$ where (7) is used can now be obtained by
sampling from the uniform probability mass function (PMF)
$p_{\mathrm{uni.}}\big{(}n_{2};n_{1}+N_{\mathrm{1}},n_{1}+N_{\mathrm{2}}\big{)}$.
Finally, the proposal (7) is not used anymore after the PMVA has existed for a
certain number of time steps $N_{\mathrm{max}}$. A possible choice for the
hyperparameters $N_{\mathrm{1}}$, $N_{\mathrm{2}}$ are $N_{\mathrm{max}}$ is
discussed in Section VI. When we calculate the weights of the samples drawn
from the proposal (7), we perform an approximation and avoid the costly
evaluation of (7). As demonstrated in Section VI, despite this approximation
the proposed robust sampling can yield convincing MVA-based RF-SLAM
performance.

##  VI Simulation Results

Figure 2: Considered scenario for performance evaluation with three PAs, four
reflective surfaces and corresponding MVAs and VAs, as well as agent
trajectory.

The proposed robust sampling for MVA-based RF-SLAM is validated in the indoor
scenario shown in Figure 2. The scenario consists of four reflective surfaces,
i.e., $K=4$ MVAs, as well as two PAs at positions
$\bm{p}_{\mathrm{pa}}^{(1)}=[-0.5\hskip 2.84526pt\hskip
2.84526pt6]^{\text{T}}$ and $\bm{p}_{\mathrm{pa}}^{(3)}=[-0.5\hskip
2.84526pt\hskip 2.84526pt1.3]^{\text{T}}$. We compare the proposed method with
MVA-based RF-SLAM that relies on bootstrap sampling [12].

The agent’s state-transition PDF $f(\bm{x}_{n}|\bm{x}_{n-1})$, with
$\bm{x}_{n}=[\bm{p}_{n}^{\text{T}}\;\bm{v}_{n}^{\text{T}}]^{\text{T}}$, is
defined by a linear, near constant-velocity motion model [20, Sec. 6.3.2],
i.e., $\bm{x}_{n}=\bm{A}\bm{x}_{n-1}+\bm{B}\bm{\omega}_{n}$. Here,
$\bm{A}\in\mathbb{R}^{4\times 4}$ and $\bm{B}\in\mathbb{R}^{4\times 2}$ are as
defined in [20, Sec. 6.3.2] (with sampling period $\Delta T=1\,\text{s}$), and
the driving process $\bm{\omega}_{n}$ is iid across $n$, zero-mean, and
Gaussian with covariance matrix $\sigma_{\omega}^{2}\mathbb{I}_{2}$, where
$\mathbb{I}_{2}$ denotes the $2\times 2$ identity matrix and
$\sigma_{\omega}=0.0032\,\text{m}/\text{s}^{2}$. For the sake of numerical
stability, we introduced a small regularization noise to the PMVA state
$\bm{p}_{k,\mathrm{mva}}$ at each time $n$, i.e.,
$\underline{\bm{p}}_{k,\mathrm{mva}}=\bm{p}_{k,\mathrm{mva}}+\bm{a}_{k}$,
where $\bm{a}_{k}$ is iid across $k$, zero-mean, and Gaussian with covariance
matrix $\sigma_{\bm{a}}^{2}\mathbb{I}_{2}$ and
$\sigma_{\bm{a}}=10^{-5}\,\text{m}$.

We performed 300 simulation runs using 30,000 samples, each using the floor
plan and agent trajectory shown in Fig. 2. In each simulation run, we
generated noisy distance measurements $z_{m,n}^{(j)}$ according to (2) with
noise standard deviation $\sigma_{m,n}^{(j)}=0.1\,$m and detection probability
$p_{\mathrm{d}}=0.95$. In addition, a mean number $\mu_{\mathrm{fa}}=1$ of
false alarm measurements $z_{m,n}^{(j)}$ were generated according to a false
alarm PDF $f_{\mathrm{fa}}(z_{m,n}^{(j)})$ that is uniform on
$[0\,\text{m}\hskip 0.85358pt\hskip 0.85358pt\hskip 0.85358pt30\,\text{m}]$.
The samples for the initial agent state are drawn from a 4-D uniform
distribution with center
$\bm{x}_{0}=[\bm{p}_{0}^{\text{T}}\;0\;\,0]^{\text{T}}$, where $\bm{p}_{0}$ is
the starting position of the actual agent trajectory. The support of each
position component about the respective center is given by
$[-0.1\,\text{m},0.1\,\text{m}]$ and of each velocity component is given by
$[-0.01\,\text{m/s},0.01\,\text{m/s}]$. At time $n=0$, the number of MVAs is
$K_{0}=0$, i.e., no prior map information is available. The prior distribution
for new PMVA states $f_{\mathrm{n}}(\overline{\bm{y}}_{m,n})$ is uniform on
the square region given by $[-15\,\text{m},\hskip 0.85358pt15\,\text{m}]\hskip
0.85358pt\times\hskip 0.85358pt[-15\,\text{m},\hskip 0.85358pt15\,\text{m}]$
around the center of the floor plan shown in Fig. 2 and the mean number of new
PMVA at time $n$ is $\mu_{n}=0.01$. The probability of survival is
$p_{\mathrm{s}}=0.999$, the detection threshold is $p_{\mathrm{de}}=0.5$, and
the pruning threshold is $p_{\mathrm{pr}}=10^{-3}$. The parameters for robust
sampling are $N_{1}=5$, $N_{2}=10$, and $N_{\text{max}}=120$, respectively.

(a)

(b)

Figure 3: Preliminary performance results: (a) RMSE position errors of the
MVAs and (b) RMSE of the mobile agent.

As an example, Fig. 2 depicts for one simulation run the posterior PDFs
represented by samples of the MVA positions and corresponding reflective
surfaces as well as estimated agent tracks. Fig. 3a shows the mean optimal
subpattern assignment (MOSPA) errors [21] of the MVA positions, all versus
time $n$. The MOSPA errors are based on the Euclidean metric with cutoff
parameter $c=5\,$m and order $p=1$. The red line shows the MOSPA errors of the
proposed MVA-based SLAM algorithm and the black line shows the MOSPA errors of
the algorithm presented in [12]. The MOSPA error of the proposed MVA-based
SLAM algorithm converges to a much smaller mapping error than that of the
algorithm presented in [12]. This can be explained by the symmetric multi-
modality of the marginal posterior PDFs. The proposed robust sampling avoids
the behavior where all samples collapse into the wrong mode. However, since
the robust sampling “excites” the multi-modality of the marginal PDF of the
MVAs, the MOSPA error remains quite large until robust sampling is disabled at
time $n=120$. Finally, Fig. 3b shows the RMSEs of the agent positions of the
converged simulation runs versus time $n$. We define a simulation run to be
converged if $\\{\forall n:||\hat{\bm{x}}_{n}-\bm{x}_{n}||<0.5\\}$. For the
converged runs both methods have an agent RMSE below $0.1\,$m, however, the
agent RMSE provided by the proposed algorithms is still significantly smaller.
More importantly, only one of the $300$ simulation runs diverged for the
proposed algorithm, but $38\,$% of the simulation runs diverged for the
algorithm in [12].

##  VII Conclusions and Future Work

In this paper, we introduced MVA-based RF-SLAM with an improved sampling
technique that is suitable for challenging scenarios where only range
measurements are available, and only one or two PAs are deployed. Our
numerical evaluation demonstrated significant performance advantages of the
proposed method compared to the recently introduced conventional MVA-based RF-
SLAM. Promising directions for future research are an extension of MVA-based
RF-SLAM to (i) angle measurements provided by antenna arrays and (ii) higher-
order reflections from flat surfaces.

## Acknowledgement

DISTRIBUTION STATEMENT A: Approved for public release. This work was supported
in part by the Under Secretary of Defense for Research and Engineering under
Air Force Contract No. FA8702-15-D-0001. Any opinions, findings, conclusions,
or recommendations expressed in this material are those of the author(s) and
do not necessarily reflect the views of the Under Secretary of Defense for
Research and Engineering. This work was also supported in part by the
Christian Doppler Research Association, the Austrian Federal Ministry for
Digital and Economic Affairs and the National Foundation for Research,
Technology and Development.

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  *[RF-SLAM]: Radio frequency simultaneous localization and mapping
  *[MVA]: master virtual anchor
  *[VA]: virtual anchor
  *[PA]: physical anchor
  *[VAs]: virtual anchor
  *[SPA]: sum-product algorithm
  *[PAs]: physical anchor
  *[MVAs]: master virtual anchor
  *[]:
  *[PMVAs]: potential   ()
  *[PMVA]: potential   ()
  *[PDFs]: probability density function
  *[PDF]: probability density function
  *[MMSE]: minimum mean-square error
  *[PMF]: probability mass function
  *[RMSE]: root mean squared error
  *[MOSPA]: mean optimal subpattern assignment
  *[RMSEs]: root mean squared error