# A Survey on Active Simultaneous Localization and Mapping: State of the Art
and New Frontiers

Julio A. Placed*, Jared Strader, Henry Carrillo,
Nikolay Atanasov, Vadim Indelman, Luca Carlone, José A. Castellanos This work
was partially supported by MINECO project PID2019‐108398GB‐I00, DGA_FSE
T45_20R, ARL DCIST W911NF-17-2-0181 and Israel Science Foundation (ISF), grant
No. 371/20.*Corresponding author.Julio A. Placed and José A. Castellanos are
with the Instituto de Investigación en Ingeniería de Aragón (I3A), Universidad
de Zaragoza, C/María de Luna 1, 50018, Zaragoza, Spain (e-mail: {jplaced,
jacaste}@unizar.es).Jared Strader and Luca Carlone are with the Laboratory for
Information & Decision Systems, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA (e-mail: {jstrader, lcarlone}@mit.edu).Henry Carrillo
is with Genius Sports, One Plaza, Avenida El Poblado #5A – 113, North Tower,
2nd & 10th Flr, Medellín, AN, COL (e-mail:
henry.carrillo@geniussports.com).Nikolay Atanasov is with the Department of
Electrical and Computer Engineering, UC San Diego, 9500 Gilman Drive, La
Jolla, CA 92093-0411, USA (e-mail: natanasov@ucsd.edu).Vadim Indelman is with
the Department of Aerospace Engineering, Technion – Israel Institute of
Technology, Haifa 32000, Israel. (e-mail: vadim.indelman@technion.ac.il).

###### Abstract

Active Simultaneous Localization and Mapping (SLAM) is the problem of planning
and controlling the motion of a robot to build the most accurate and complete
model of the surrounding environment. Since the first foundational work in
active perception appeared, more than three decades ago, this field has
received increasing attention across different scientific communities. This
has brought about many different approaches and formulations, and makes a
review of the current trends necessary and extremely valuable for both new and
experienced researchers. In this work, we survey the state-of-the-art in
active SLAM and take an in-depth look at the open challenges that still
require attention to meet the needs of modern applications. After providing a
historical perspective, we present a unified problem formulation and review
the well-established modular solution scheme, which decouples the problem into
three stages that identify, select, and execute potential navigation actions.
We then analyze alternative approaches, including belief-space planning and
deep reinforcement learning techniques, and review related work on multi-robot
coordination. The manuscript concludes with a discussion of new research
directions, addressing reproducible research, active spatial perception, and
practical applications, among other topics.

###### Index Terms:

Active SLAM, autonomous robotic exploration, active perception, optimality
criteria, belief-space planning, next best view, deep reinforcement learning.

##  I Introduction

Autonomous operation in robotics applications requires robots to have access
to a consistent model of the surrounding environment, in order to support safe
planning and decision making. Towards this goal, a robot must have the ability
to create a map of the environment, localize itself on it, and control its own
motion. Active SLAM refers to the joint resolution of these three core
problems in mobile robotics, with the ultimate goal of creating the most
accurate and complete model of an unknown environment. Active SLAM can be seen
as a decision-making process in which the robot has to choose its own future
control actions, balancing between exploring new areas and exploiting those
already seen to improve the accuracy of the resulting map model.

During the last decades, active SLAM has received increasing attention111The
number of publications on active SLAM has grown from 53 in 2010 to over 660 in
2022 (a twelve-fold increase). The number becomes almost 5500 if we extend the
search to include belief-space planning, active exploration, and simultaneous
planning, localization and mapping. Source: dimensions.ai. and has been
studied in different forms across multiple communities, with the ambition of
deploying autonomous agents in real-world applications (e.g., search and
rescue in hazardous environments, underground or planetary exploration). This
divergence has broadened the scope of the problem and provided a wider
context, yielding numerous approaches based on different concepts and theories
that have made the field flourish; but it also created a disconnect between
research lines that could mutually benefit from each other. With this survey,
we seek to fill this gap by providing a general problem statement and a
unified review of related works.

Currently, active SLAM is at a decisive point, driven by novel opportunities
in spatial perception and artificial intelligence (AI). These include, for
instance, the application of breakthroughs in neural networks to prediction
beyond line-of-sight, reasoning over novel environment representations, or
leveraging new SLAM techniques to process dynamic and deformable scenes.
Throughout this paper, we give a fresher picture of active SLAM that goes
beyond the classical —but still mainstream— entropy computation over
discretized grids. Besides, we identify the open challenges that need to be
addressed for active SLAM to have an impact on real applications, shaping
future lines of research, and describing how they can nourish from the cross-
fertilization between research fields. Among those challenges, we emphasize
the urgent need for benchmarks and reproducible research.

###  I-A Historical Perspective

Ever since the first mobile robots were built in the late 1940s, the ambition
that they could perform autonomous tasks has been one of the major focuses of
robotics research. To operate autonomously, a robot needs to form a model of
the surrounding environment —including localization and mapping— and perform
safe navigation [1]. While the former involves estimating the position of the
robot and creating a symbolic representation of the environment, the latter
refers to planning and controlling the movements of the robot to safely
achieve a goal location. Localization, mapping, and planning have been often
investigated in combination, resulting in multiple research areas such as
SLAM, active localization, active mapping, and active SLAM.

Localization and mapping were treated deterministically and solved
independently until probabilistic approaches went mainstream in the 1990s,
when researchers realized that both tasks were correlated and dependent of one
another. SLAM refers, thereby, to the problem of incrementally building the
map of an environment while at the same time locating the robot within it [2].
This problem has attracted significant attention from the robotics community
in the last decades; see [3, 4, 5] and the references therein.

SLAM, however, is a passive method and is not concerned with guiding the
navigation process. In contrast, _active_ approaches do consider the
navigation aspects of the problem. Bajcsy [6], Cowan and Kovesi [7], and
Aloimonos et al. [8] were the first to study and analyze the problem of active
perception (also referred to as active information acquisition [9]) in the
late nineties. Bajcsy [10] would later formally define it as the problem of
actively acquiring data in order to achieve a certain goal, necessarily
involving a decision-making process. For the cases in which the objective is
to improve localization, mapping, or both, the problems are respectively
referred to as active localization, active mapping, and active SLAM.

_Active mapping_ was the first problem to be addressed, dating back to the
work of Connolly [11] in 1985. Better known since then as the _next best view_
problem, active mapping tackles the search of the optimal movements to create
the best possible representation of an environment. Subsequent examples date
to the 1990s [12, 13, 14], always under the assumption of perfectly known
sensor localization. This problem has been primarily addressed in the computer
vision community to reconstruct objects and scenes from multiple viewpoints,
since the nature of the projective geometry for monocular cameras, occlusions,
and limited field of view often make impossible to do it from just one
viewpoint; see [15] and the references therein.

In a similar vein, _active localization_ aims to improve the estimation of the
robot’s pose by determining how it should move, assuming the map of the
environment is known. First relevant works can be traced back to 1998, when
Fox et al. [16] and Borgi and Caglioti [17] formulated it as the problem of
determining the robot motion so as to minimize its future expected (i.e., a
posteriori) uncertainty. In particular, it is in [16] where the foundations of
the current workflow were laid: (i) goal identification, (ii) utility
computation, and (iii) action selection (we will extensively review these
stages later in this survey). Other relevant subsequent work can be found in
[18, 19, 20, 21], but also in the related literature of perception-aware
planning [22] and planning under uncertainty [23].

Finally, _active SLAM_ unifies the previous problems, and allows a robot to
operate autonomously in an initially unknown environment. It refers to the
application of active perception to SLAM and can be defined as the problem of
controlling a robot which is performing SLAM in order to reduce the
uncertainty of its localization and the map representation [24]. Historically,
active SLAM has been referred to with different terminology, which has
significantly hindered knowledge sharing and dissemination within the robotics
community. Relevant seminal works can be found under the names of active
exploration [25], adaptive exploration [26, 27], integrated exploration [28,
29], autonomous SLAM [30], simultaneous planning, localization and mapping
[31], belief-space planning (BSP) [32], or simply robotic exploration [33,
34]. It was not until 2002 —when Davison and Murray [35] coined the term
active SLAM— that the robotics community started adopting this nomenclature.
Thrun and Möller [25] demonstrate that in order to solve robotic exploration,
agents have to switch between two opposite principles depending on the
expected costs and gains: exploring new areas and revisiting those already
seen, i.e., the so-called _exploration-exploitation dilemma_. The first
approach in which a robot chooses actions that maximize the knowledge of the
two variables of interest is attributed to Feder et al. [26], who also
separate the procedure in three major stages as in [16]. Table I contains a
subset of relevant works that have followed [26]. This table differentiates
the main aspects of each approach, including the type of sensors, the state
representation, and the theoretical foundations.

###  I-B About Previous Surveys

Only two works have previously addressed the problem of surveying active SLAM
research. The first of them, published in 2016, is a section of a more general
survey on SLAM carried out by Cadena et al. [5]. The other, by Lluvia et al.
[36], conducts a more extensive survey on “Active Mapping and Robot
Exploration”. Table II summarizes the topics they address, along with those
covered in the present survey.

Cadena et al. [5] describe both the history and the main aspects of the
problem, and identify three open challenges: the decision of when to stop
performing active SLAM, the problem of accurately predicting the effect of
future actions, and the lack of mathematical guarantees of optimality.
However, the brevity of the active SLAM section prevented delving into a
detailed discussion of the most relevant works or providing a more unified
mathematical formulation of the problem. Moreover, since [5] was published,
many relevant contributions have been proposed and new open problems have
arisen. For instance, progress has been made on the way uncertainties of the
robot location and the map are represented and quantified. Furthermore, recent
work has also opened new research endeavors, including deep learning (DL).

Lluvia et al. [36] also provide a thorough historical review and relate the
different communities that have been trying to solve this problem under
different nomenclatures. Similar to [5], they do not attempt to present a
unified mathematical formulation of active SLAM nor do they cover utility
computation, a field which has been mostly overlooked in the literature. They
delve, nevertheless, into the optimization of vantage points and the
trajectories to reach them, a new problem that has attracted significant
attention from the control community and has seen many contributions in recent
years. In [36], the authors present a comparison between representative works
in active SLAM, although with a limited scope. Contrarily to [36], we present
a more complete analysis and a broader set of open challenges, which extends
the ones identified in [5].

|  SLAM Approach |  Sensors |  Environment Representation |  Formulation |  Candidate Goal Locations |  Utility Function |  Validation Environment |  Stopping Criterion |  Publicly available
---|---|---|---|---|---|---|---|---|---
|  EKF |  Sonar |  Landmark map |  Modular |  Local vicinity |  $D\text{-}opt$ |  Sim. & real |  - |  ✗
|  EKF |  Lidar |  OG map |  Modular |  Local vicinity |  MI |  Real |  - |  ✗
|  FastSLAM [37] |  Lidar |  OG map |  Modular |  Frontiers & re-visiting |  Particle’s volume & distance |  Sim. & real |  Particle’s volume |  ✗
|  RBPF |  Lidar |  OG map |  Modular |  Frontiers & re-visiting |  MI & distance |  Sim. & real |  - |  ✗
|  EKF |  Lidar |  Landmark map |  MPC |  Unknown space & re-visiting |  $T\text{-}opt$ |  Simulation |  - |  ✗
|  Pose SLAM [41] |  Lidar |  OG map |  Modular |  Frontiers & revisiting |  Entropy |  Simulation |  - |  ✗
|  RBPF |  Lidar |  OG map |  Modular |  Frontiers & re-visiting |  KLD |  Simulation |  - |  ✗
|  GTSAM |  Camera |  Landmark map |  BSP |  - |  Robot’s $T\text{-}opt$ & distance |  Simulation |  - |  ✗
|  RGBDSLAM [45] |  RGB-D |  Octomap |  Modular |  Frontiers |  Coverage & distance |  Simulation |  - |  ✗
|  ROVIO [47] & mapping |  Stereo & IMU |  Octomap |  MPC |  RRT paths |  Coverage & distance |  Sim. & real |  Coverage |  ✓
|  ROVIO [47] & dense mapping |  Stereo & IMU |  Octomap |  MPC |  RRT paths |  $D\text{-}opt$ |  Sim. & real |  Min. utility |  ✓
|  Gmapping [50] |  Lidar |  OG map |  Modular |  Frontiers |  Map’s MI & distance |  Sim. & real |  - |  ✓
|  ICP & iSAM [52] |  Lidar |  OG map |  Modular |  Frontiers |  Shannon-Rényi entropy |  Sim. & real |  Time |  ✓
|  Pose SLAM [41] |  Lidar |  COM |  Modular |  Frontiers |  MI & distance |  Simulation |  Coverage |  ✓
|  ICP & g2o [55] |  Lidar |  Octomap |  Modular |  Random |  Coverage |  Sim. & real |  Coverage |  ✗
|  Neural Networks |  RGB |  OG map |  DRL |  Local vicinity |  Coverage |  Sim. & real |  - |  ✓
|  Gmapping [50] |  Lidar |  OG map |  DRL |  Frontiers |  Map’s MI & distance |  Sim. & real |  - |  ✗
|  GTSAM |  Range |  Virtual landmark map |  DRL |  Frontiers |  Virtual landmark’s $T\text{-}opt$ |  Simulation |  - |  ✓
|  Karto [60] & g2o [55] |  Lidar |  OG map |  DRL |  Sampled from the OG map |  Map’s MI & distance |  Sim. & real |  Coverage |  ✗
|  ICP & iSAM [52] |  Sonar |  Octomap |  MPC |  RRT paths & re-visiting |  Robot’s $D\text{-}opt$ & coverage |  Sim. & real |  - |  ✗
|  Linear SLAM [63] |  Camera |  Landmark map |  MPC |  Local vicinity |  Graph’s $D\text{-}opt$ |  Sim. & real |  Coverage |  ✗
|  Cartographer [65] |  Lidar & IMU |  Octomap |  Modular |  Frontiers |  Map’s MI & distance |  Sim. & real |  Coverage |  ✓
|  ORB-SLAM2 [67] |  Lidar & RGB-D |  Octomap |  Modular |  Frontiers |  Graph’s $D\text{-}opt$ |  Simulation |  Time |  ✓
|  RTAB-Map [69] |  Lidar, RGB-D & IMU |  Octomap |  Modular & MPC |  Frontiers |  Map’s MI, distance & visual features |  Sim. & real |  Time |  ✓
TABLE I: A comparison between representative active SLAM approaches, ordered
chronologically.

Topic |  Cadena et al. [5] |  Lluvia et al. [36] | Ours
---|---|---|---
Introduction | Historical review |  Briefly |  Yes | Yes
Problem formulation |  No |  No | Yes
Modular scheme | Env. representation |  Yes |  Yes | Yes
Goal identification |  Briefly |  Yes | Yes
Information Theory |  Briefly |  No | Yes
TOED |  Briefly |  No | Yes
Graph Theory |  No |  No | Yes
Alternative approaches | Continuous domain |  No |  Yes | Yes
Deep Learning |  No |  Briefly | Yes
Multi-robot |  No |  No | Yes
Open problems | State prediction |  Yes |  Yes | Yes
Stopping criteria |  Yes |  Briefly | Yes
Novel representations |  No |  Briefly | Yes
Data association |  No |  No | Yes
Complex environments |  No |  No | Yes
Reproducible research |  No |  No | Yes
Practical applications |  No |  No | Yes
TABLE II: Comparison between the topics and open challenges addressed in
previous surveys and the current one.

###  I-C Paper Structure

The remainder of this manuscript is organized as follows. Section II provides
a unified problem formulation for active SLAM and describes the three
subproblems (or _stages_) it has traditionally been divided into. Sections III
to V cover those three stages separately. In particular, Section III deals
with the identification of vantage points, Section IV with utility
computation, and Section V with selection and execution of the optimal action.
Sections VI and VII consider, on the other hand, alternative continuous-state
optimization and DL methods. Section VIII is devoted to multi-robot active
SLAM. Section IX outlines the open research questions in active SLAM. Finally,
Section X concludes the manuscript.

##  II The Active SLAM Problem

###  II-A Problem Formulation

Active SLAM can be framed within the wider mathematical framework of partially
observable Markov decision processes (POMDPs), after some particularization.
POMDPs model decision-making problems under both action and observation
uncertainties and can be formally defined as the 7-tuple
$(\mathcal{S},\mathcal{A},\mathcal{Z},\xi_{s},\xi_{z},r,\gamma)$. In
particular, a POMDP consists of the agent’s state space $\mathcal{S}$, a set
of actions $\mathcal{A}$, a transition function between states
$\xi_{s}:\mathcal{S}\times\mathcal{A}\mapsto\Pi(\mathcal{S})$ where
$\Pi(\mathcal{S})$ is the space of probability density functions (pdfs) over
$\mathcal{S}$, an observation space $\mathcal{Z}$, the conditional likelihood
of making any of those observations
$\xi_{z}:\mathcal{S}\mapsto\Pi(\mathcal{Z})$, where $\Pi(\mathcal{Z})$ is the
space of pdfs over $\mathcal{Z}$, a reward scalar mapping
$r:\mathcal{S}\times\mathcal{A}\to\mathbb{R}$, and the discount factor
$\gamma\in(0,1)\in\mathbb{R}$ which allows to work with finite rewards even
when planning over infinite time horizons.

Contrary to the fully observable case, agents in a POMDP cannot reliably
determine their own true state, $\boldsymbol{s}$. Instead, they maintain an
internal belief or information state, $b_{t}(\boldsymbol{s}_{t})$, which
represents the posterior probability over states at time $t$, given the
available data collected up to that time [70, 2, 71]:

$b_{t}(\boldsymbol{s}_{t})\triangleq
p(\boldsymbol{s}_{t}|\underbrace{\boldsymbol{z}_{1:t},\boldsymbol{a}_{1:t-1}}_{\text{history,
}\boldsymbol{h}})\,,$ (1)

where $\boldsymbol{z}_{1:t}$ is the set of all available observations and
$\boldsymbol{a}_{1:t-1}$ the set of past control actions (both collectively
referred as the _history_ $\boldsymbol{h}$). The _belief space_ ,
$\mathcal{B}(\mathcal{S})\equiv\Pi(\mathcal{S})$, of probability density
functions over the set $\mathcal{S}$ is defined as:

$\mathcal{B}(\mathcal{S})\triangleq\\{b:\mathcal{S}\mapsto\mathbb{R}\mid\int
b(s)ds=1,\ b(s)\geq 0\\}\,.$ (2)

In order to evaluate the effect of future actions, agents must be capable of
predicting posterior belief distributions, that is, the pdf over $\mathcal{S}$
after performing a certain action, $\boldsymbol{a}_{t}$, and taking a future
observation $\boldsymbol{z}_{t+1}$:

$\displaystyle b_{t+1}(\boldsymbol{s}_{t+1})$ $\displaystyle\triangleq
p(\boldsymbol{s}_{t+1}|\boldsymbol{z}_{t+1},\boldsymbol{a}_{t},b_{t}(\boldsymbol{s}_{t}))\,.$
(3)

Since the future measurements are unknown for the agent, their expected value
has to be studied instead. Consider that an agent in the state defined by
$b_{t}(\boldsymbol{s}_{t})$ executes a certain action $\boldsymbol{a}_{t}$,
and transitions to another state with pdf $p(\boldsymbol{s}_{t+1})$. Then, the
likelihood of making an observation will be given by [71]:

$\displaystyle
p(\boldsymbol{z}_{t+1}|b_{t}(\boldsymbol{s}_{t}),\boldsymbol{a}_{t})=\int\int$
$\displaystyle\xi_{z}(\boldsymbol{s}_{t+1})\
\xi_{s}(\boldsymbol{s}_{t},\boldsymbol{a}_{t})$ $\displaystyle\
b_{t}(\boldsymbol{s}_{t})\ d\boldsymbol{s}_{t}\ d\boldsymbol{s}_{t+1}\,,$ (4)

where
$\xi_{z}(\boldsymbol{s}_{t+1})=p(\boldsymbol{z}_{t+1}|\boldsymbol{s}_{t+1})$
is the observation model and
$\xi_{s}(\boldsymbol{s}_{t},\boldsymbol{a}_{t})=p(\boldsymbol{s}_{t+1}|\boldsymbol{s}_{t},\boldsymbol{a}_{t})$
the motion model.

Since the belief is a sufficient statistic, optimal policies for the original
POMDP may be found by solving an equivalent continuous-space MDP over
$\mathcal{B}(\mathcal{S})$ [72, 70]. Such MDP is defined by the 5-tuple
$(\mathcal{B},\mathcal{A},\xi_{b},\rho,\gamma)$, where the transition and
reward functions are
$\xi_{b}:\mathcal{B}\times\mathcal{A}\mapsto\Pi(\mathcal{B})$ and
$\rho:\mathcal{B}\times\mathcal{A}\mapsto\mathbb{R}$. To preserve consistency,
this belief-dependent reward function builds on the expected rewards of the
original POMDP:

$\rho(b_{t},\boldsymbol{a}_{t})=\int_{\mathcal{S}}b_{t}(\boldsymbol{s}_{t})\
r(\boldsymbol{s}_{t},\boldsymbol{a}_{t})\ d\boldsymbol{s}_{t}\,.$ (5)

Then, the decision at time $t$ will be provided by the (control/action) policy
$\pi_{t}$, which maps elements from the space of pdfs over $\mathcal{S}$ to
the action space:

$\pi_{t}:\mathcal{B}(\mathcal{S})\mapsto\mathcal{A}\,.$ (6)

The optimal policy, $\pi^{\star}$, that yields the highest expected rewards
for every belief state can be found via:

$\pi^{\star}(b)=\operatorname*{arg\,max}_{\pi}\sum_{t=0}^{\infty}\mathbb{E}\left[\gamma^{t}\rho(b_{t},\pi(b_{t}))\right]\,,$
(7)

where expectation is taken w.r.t.
$p(\boldsymbol{z}_{t+1}|b_{t}(\boldsymbol{s}_{t}),\boldsymbol{a}_{t})$. In
general, computing the optimal policy for MDPs with continuous state spaces is
hard and most works resort to approximate solutions or problem simplifications
[70, 73].

The active SLAM problem requires, however, some variation and
particularization of the above general POMDP formulation. Let us consider a
robot capable of moving in an unknown environment while performing SLAM. That
is, at every time step, the robot can change its own linear and angular
velocities; moreover, the robot is able to process the sensor data into a map
representation, $\boldsymbol{m}_{t}\in\mathcal{M}$, and an estimate of its own
state (e.g., pose), $\boldsymbol{x}_{t}\in\mathcal{X}$. Thus, the state space
can be defined as the joint space
$\mathcal{S}\triangleq\mathcal{X}\times\mathcal{M}$.

The evolution of both the state and the measurements in SLAM is governed by
probabilistic laws [2], as (1) and (4) express. However, two assumptions are
worth mentioning in the context of active SLAM regarding each of the
equations, that further simplify its resolution. First, the robot state is
commonly assumed Gaussian with a pdf $b(\boldsymbol{x})$ having mean
$\hat{\boldsymbol{x}}$ and covariance $\boldsymbol{\Sigma}_{r}$ (see, e.g.,
[40, 43]). Thus, the map and the robot state are usually treated
independently, although some representations allow for a joint distribution
(e.g., in sparse landmark maps or using Gaussian Processes to model dense maps
[53]). Secondly, despite less prevalent than the former, some works (e.g.,
[32]) also assume maximum likelihood (ML) observations, i.e., that executing
an action in a given belief state will always produce the same, most probable
observation. This allows to rewrite the expected measurements as:

$\boldsymbol{z}_{t+1}^{ML}=\operatorname*{arg\,max}_{\boldsymbol{z}\in\mathcal{Z}}\
p(\boldsymbol{z}_{t+1}|b_{t}(\boldsymbol{s}_{t}),\boldsymbol{a}_{t})\,.$ (8)

In addition, in active SLAM the reward typically reflects the agent’s
knowledge of the system (i.e., it involves the uncertainty in the belief
rather than focusing on reaching specific states). These reward functions are
known as _utility functions_ and may be defined mathematically as the scalar
mapping $\rho:\mathcal{B}(\mathcal{S})\times\mathcal{A}\mapsto\mathbb{R}$.
This reward mapping, however, is inconsistent with both POMDPs (where the
reward is dependent on $\boldsymbol{s}$ and $\boldsymbol{a}$) and belief MDPs
(where the reward is restricted to the form in (5)). To circumvent this
limitation, $\rho$-POMDP [73] extends the POMDP formulation to allow the
inclusion of beliefs’ uncertainty in the objective. This enables the use of
information-oriented criteria rather than control-oriented, without losing
basic properties such as Markovianity.

Finally, considering a finite-horizon and ML observations, the discount factor
and expectation over future measurements in (7) can be dropped, and active
SLAM can be reduced to the following optimization for open-loop planning
settings:

$\boldsymbol{a}_{t:t+k}^{\star}=\operatorname*{arg\,max}_{\boldsymbol{a}_{t:t+k}\in{{\mathcal{A}^{k}}}}\
{{\sum_{\tau=t}^{t+k}\rho\left(b(\boldsymbol{s}_{\tau}),\boldsymbol{a}_{\tau}\right)}}\,,$
(9)

where $\boldsymbol{a}_{t:t+k}^{\star}$ is the optimal sequence of actions to
execute over the future planning horizon ($k$ lookahead steps) and
$\mathcal{A}^{k}\triangleq\mathcal{A}\times\mathcal{A}\times...\times\mathcal{A}$
the space of sequences of actions over $k$.

###  II-B Decoupling Active SLAM into Three Subproblems

While the previous section provided a unified formulation for active SLAM, for
computational convenience active SLAM has been traditionally decoupled into
three subproblems (or _stages_) [16, 26, 28], which will be briefly described
hereafter and covered in detail in Sections III to V:

  1. 1)

_Identification of the potential actions:_ solely to reduce the computational
burden, the first stage aims to determine a reduced subset of possible actions
to execute.

  2. 2)

_Utility computation:_ the expected cost and gain of performing each candidate
action has to be estimated.

  3. 3)

_Action selection and execution:_ finally, the last stage involves finding and
executing the optimal action(s).

The entire process should be iteratively repeated until the whole environment
is accurately modeled, although in practice it is done until some stopping
conditions are met.

For clarity of presentation and because many existing works do decompose
active SLAM into these stages, we review each stage separately in Sections III
to V. However, this decoupling can produce suboptimal results and lead to
undesired behaviors. Performing the three stages simultaneously is certainly
advantageous, e.g. when optimizing over a continuous action space, or when a
control policy is optimized or learned under the umbrella of POMDPs. We review
these approaches, alternative to the modular scheme, in Sections VI and VII.

##  III Stage 1: Identification of Potential Actions

The first stage in modular active SLAM approaches consists in generating the
set of available actions the robot could execute (i.e., goals the robot can
reach); this can be understood as a way to reduce (and discretize) the search
space of potential actions. Early works simply used random goals or required
human interaction, until the concept of _frontiers_ was introduced by Yamauchi
[74]. This resulted in improved exploration strategies, and has consolidated
as the most common approach. Nevertheless, the advent of neural networks has
led to new ways of evaluating the space of potential goals. In this section,
we present the most important methods to identify goal locations. Since they
strongly depend on the representation of the environment estimated by the SLAM
pipeline, we start by providing a brief description of the different existing
representations for active SLAM.

###  III-A Representation of the Environment

We review four different types of map representations: topological, metric,
metric-semantic, and hybrid maps.

####  III-A1 Topological maps

use lightweight graphs to describe information about the topology of the
environment. Historically, vertices in this graph represent convex regions in
the free space, while edges model connections between them. The construction
of these graphs is a segmentation problem, usually done over an occupancy
grid; see [75] for a survey on these methods. Despite these maps allow
leveraging graph theory, which provides powerful tools for planning and
exploration, they are not frequently used in active SLAM [76, 77].

####  III-A2 Metric maps

are the most used representations to encode information about the environment
in active SLAM. They can be further divided into two categories: sparse and
dense maps. The former rely on a sparse set of interest points (or
_landmarks_) to represent a scene, and have been especially used in optimal
control [39, 78, 62] and belief-space planning [43] approaches. Dense maps can
be based on point clouds, meshes or, more typically, a discretization of the
environment into cells that encode a certain metric (e.g., occupancy, distance
to obstacles). Occupancy grid (OG) maps, first proposed in the late eighties
for perception and navigation by Elfes [79] and Moravec [80], assign to each
cell its probability of being occupied. They have been used in numerous active
SLAM frameworks, e.g., [51, 56, 57, 81]. Their extension to 3D include
OctoMaps [82], Supereight [83] and voxel maps [84], all of which have been
also used in active SLAM [85, 86, 87, 88]. Jadidi et al. [53] use continuous
occupancy maps (COM) to leverage continuous optimization methods. There exist
many other dense maps that encode more sophisticated metrics, such as those
based on signed distance fields (SDF), like Voxblox [89]. Still, they are
seldom used in active SLAM [90].

####  III-A3 Metric-semantic maps

go beyond geometric modeling and associate semantic information to classical
metric maps. Instead of geometric features, a sparse map can capture objects,
described by a semantic category, pose, and shape [91, 92]. Active object-
level SLAM has been considered in [93, 94]. Examples of dense metric-semantic
maps include Voxblox++ [95] and Kimera [96] (which build upon an SDF), and
Fusion++ [97] and [98] (based on voxel maps). Despite being used in some SLAM
formulations (see [5, 96] and the references therein), they have not yet been
used in active SLAM. An exception is the work of Asgharivaskasi and Atanasov
[99, 100] which develops a multi-class (semantic) OctoMap and uses a closed-
form lower bound on the Shannon mutual information between the map and range-
category observations to select informative robot trajectories.

####  III-A4 Hybrid and hierarchical maps

combine some of the previous representations to enhance the decision-making
process. Hybrid metric-topological maps have been applied to tackle either
navigation [101] or SLAM [102]. Rosinol et al. [103] combine metric, semantic,
and topological representations into a single model, a _3D scene graph_. These
hierarchical representations break down metric-semantic maps into
interconnected high-level entities, paving the way for high-level reasoning.
The use of hybrid maps in active SLAM is mostly unexplored, with [104] among
the few works that have integrated them.

###  III-B Detecting Goal Locations

The identification of all possible destinations the robot could travel to
easily proves to be intractable because of the dimensions of the map and the
action set [105]. In practice, a finite subset of them is identified, allowing
for computational tractability despite not guaranteeing global optimality
[43]. The simplest approach consists of randomly selecting the goal
destinations [106, 107]. Random exploration requires low computational
resources and works under the assumption that every spot in the environment
has the same information associated. In 1997, Yamauchi [74] revolutionized the
field by introducing the concept of _frontiers_ , i.e., the areas that lie
between known and unknown regions. Since its proposal, frontier-based
exploration has been the most used by far and has been tailored to different
map representations. Frontiers have been effectively identified for
topological maps as nodes with no neighbors in certain directions [76]. For 2D
OG maps, a plethora of geometric frontier-detection methods have been
developed to circumvent the computational cost of searching the entire space
[108]. Keidar and Kaminka [109] propose the wavefront frontier detector (WFD)
and fast frontier detector (FFD). WFD starts the search from the robot’s
location and restricts it to the free space; FFD performs the search after
each scan is collected, following the intuition that frontiers are bound to
appear in recently scanned regions. Following this idea, the same authors
present the incremental WFD [110], that restricts the search to recently
scanned areas. Quin et al. [108] improve the performance of the previous
algorithms by only evaluating a subset of the observed free space. Refer to
[111, 108] for further discussion. Umari and Mukhopadhyay [49] first present a
frontier search method over a 2D OG based on rapidly-exploring random trees
(RRTs) that grow both globally and locally to sample recently scanned regions.
This strategy, often combined with computer vision algorithms has been widely
used [112, 81]. The sample-based frontier detector algorithm [113] reduces the
computational load of the previous methods by only storing the nodes of the
search tree. Frontier identification in 3D maps is less frequent, since 3D
maps are more expensive to store and analyze, and are often incomplete due to
the sensed volume. Apart from simple search techniques [114, 88], most methods
evaluate map portions incrementally [44, 64] or along surfaces [115].
Alternatively, in [116], authors propose a method that disperse random
particles over the 3D known space. No matter the method used, after detecting
frontiers, a clustering step is frequently required to prevent the frontier
set from being high-dimensional (e.g., using K-means [117] or mean-shift
[49]).

Shortly after the concept of frontiers was proposed, Newman et al. [30] and
Stachniss et al. [29] realized that, for a robot with high uncertainty,
potential loop closure areas encode more information than frontiers; the
ultimate goal of active SLAM goes beyond simply covering the workspace: to
improve the accuracy of localization and mapping. Similarly, Grabowski et al.
[118] observe that regions of interest where sensor readings overlap may be
more informative than new frontiers. In other words, these works explicitly
account for the _exploration-exploitation dilemma_ in the frontier detection
step. It is a common practice in active SLAM to include potential loop closure
regions —along with frontiers— in the set of goal candidates [40, 54], or to
switch between exploring new frontiers and revisiting known places [29, 119,
61].

In contrast to frontier-based approaches, some active SLAM formulations allow
the identification of goal locations locally in the robot’s vicinity. However,
note that decisions will be optimal only locally and a short decision-making
horizon may induce wrong behaviors [26, 120]. This strategy is typical in deep
reinforcement learning (DRL) approaches [121, 122, 123], for which local
optimality is alleviated by network memorization. Following the idea that
evaluating larger neighborhoods would lead to more robust decisions, in [40]
authors use RRT-based paths to several configurations over the free space as
the action set; and in [43] the entire environment is considered under the
umbrella of continuous-domain optimization.

##  IV Stage 2: Utility Computation

The second and main stage in modular active SLAM approaches focuses on the
evaluation of each possible destination, in order to estimate the effect that
executing the set of actions to reach each destination would have. Naive
utility formulations using just geometric or time-dependent functions often
result in non-desirable behaviors [124, 40, 81], since they do not properly
capture the uncertainty in the belief. The _exploration-exploitation dilemma_
can be more effectively solved by quantifying the expected uncertainty of the
two target random variables: the robot location and the map. Typically, the
different objectives (e.g., travelling cost, mapping and localization
uncertainty) are aggregated into a single utility function, although there are
multi-objective approaches in which they are kept separate and Pareto optimal
solutions are sought [125, 126, 127]. There is a plethora of metrics and the
choice of which one to use mainly depends on the selected way to represent the
variables of interest. Metrics based on information theory usually aim at OG
maps, while those based on the theory of optimal design are more suitable for
Gaussian distributions. We review each choice below.

###  IV-A Naive Cost Functions

The simplest (and first-broadly-used) metrics are naive geometric functions,
such as the Euclidean distance to the goal location [74], the time required to
reach it [88], or the expected size of the are to visit [106, 107, 49]. In
fact, the latter approximates the map’s entropy, which is strongly related to
the number of known cells in an OG map [28]. Since these metrics are computed
over Euclidean or temporal spaces, they can be used regardless of the map
representation chosen [104, 54, 64]. Stachniss et al. [38] show that combining
distance and information-based functions results in better exploration
strategies, and this has since been a common approach [128]. However, manual
tuning to overcome discrepancies between the multiple terms involved is needed
[129, 49].

###  IV-B Information Theory (IT)

The most common approach to assess utility in active SLAM uses information
theory (IT) to quantify the uncertainty in the joint belief state. Within it,
there exist different metrics that allow for such quantification, although all
of them build on the same concept: entropy. The notion of entropy was
introduced by Shannon [130] and can be defined as a measure of a variable’s
uncertainty, randomness, or surprise; this is in fact strongly related to its
associated information [131].

Early exploration strategies use only the map representation as the variable
of interest [74, 132, 88], thereby assuming no error in the robot
localization. However, soon after the first of these works emerged, it was
observed that high uncertainty in the robot state estimation leads to wrong
expected map uncertainties [27]. The entropy of the SLAM posterior after
executing a candidate action can be computed as [38]:

$\displaystyle\mathcal{H}\left[p(\boldsymbol{x},\boldsymbol{m}|\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})\right]\triangleq$
$\displaystyle\underbrace{\vphantom{\int_{\boldsymbol{x}}p(\boldsymbol{x}|\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})\mathcal{H}\left[p(\boldsymbol{x}|\boldsymbol{x},\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})\right]d\boldsymbol{x}}\mathcal{H}\left[p(\boldsymbol{x}|\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})\right]}_{\text{robot's
$\mathcal{H}$}}+\underbrace{\int_{\boldsymbol{x}}p(\boldsymbol{x}|\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})\mathcal{H}\left[p(\boldsymbol{m}|\boldsymbol{x},\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})\right]d\boldsymbol{x}}_{\text{expected
conditional map's $\mathcal{H}$}}\,,$ (10)

where $\hat{\boldsymbol{z}}$ are the expected (ML) future measurements, which
may be estimated using, e.g., ray-casting techniques [24].

The computation of the previous joint entropy is intractable in general [38].
To overcome this, most approaches resort to entropy approximations that first
compute utility of the two variables independently, and then combine them
heuristically [27, 38, 133, 61]. Let us first consider the case of graph-based
SLAM, in which the problem is described using a graph representation where
nodes represent the robot poses and edges encode the the constraints between
them; see [2, 3, 4, 5]. The joint entropy in (10) can be approximated by [40]:

$\mathcal{H}\left[p(\boldsymbol{x},\boldsymbol{m}|\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})\right]\approx\mathcal{H}\left[p(\boldsymbol{x}|\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})\right]+\mathcal{H}\left[p(\boldsymbol{m}|\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})\right]\,.$
(11)

The mismatch between the magnitudes of the addends above is the main drawback
of such approximation, calling for the addition of weighting parameters to
balance the contributions of the two terms [134, 42]. Carrillo et al. [51]
circumvent this by embedding a metric of the robot’s uncertainty in a combined
Shannon-Rényi utility function; an approach that also appears in [135]. On the
other hand, the expectation-maximization (EM) algorithm [136] embeds the
impact of robot’s uncertainty directly in a virtual map. A similar
approximation can be done for particle-filter SLAM, which represents the
belief over robot trajectories as a set of particles [2, 3, 37]. The integral
in (10) will be now approximated by the weighted mean of all possible
solutions (i.e., particles) [38].

The first term in (10) refers to the robot state entropy, which can be
computed as a function of the posterior covariance log-determinant, assuming
that it is an $\ell$-dimensional Gaussian distribution with covariance
$\boldsymbol{\Sigma}_{r}\in\mathbb{R}^{\ell\times\ell}$,

$\displaystyle\mathcal{H}\left[p(\boldsymbol{x}|\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})\right]=\frac{1}{2}\ln\left((2\pi
e)^{\ell}\det\left(\boldsymbol{\Sigma}_{r}\right)\right)\,.$ (12)

On the other hand, the second term is the expected map’s entropy, and its
computation depends on the representation chosen. For instance, in landmark-
based maps it can be computed in the same way as the robot’s entropy, under
the same assumption [137]. For discrete metric maps, and assuming cells
independent from each other, it can be defined as [31]:

$\displaystyle\mathcal{H}\left[p(\boldsymbol{m}|\textbf{x},\boldsymbol{h},\hat{\boldsymbol{z}},\textbf{a})\right]=-\textstyle\sum_{c\in\boldsymbol{m}}\theta_{c}\log\theta_{c}\,,$
(13)

with $\theta_{c}=p(c)$ being the occupancy probability of cell $c$. This
entropy measure has been used in both 2D [28, 133] and 3D OG maps [85, 88,
117]. More efficient approaches that only evaluate of cells in the robot’s
vicinity have been proposed in the context of particle-filter SLAM [134, 138,
42].

The most common metric to assess utility in active SLAM is not Shannon’s
entropy of the SLAM posterior, but its expected reduction. This utility
function is known as mutual information (MI) [27, 33] and is defined as the
difference between the entropy of the actual state and the expected entropy
after executing an action, i.e., the information gain:

$\mathcal{I}(\boldsymbol{a})\triangleq\underbrace{\mathcal{H}\left[p\left(\boldsymbol{x},\boldsymbol{m}|\boldsymbol{h}\right)\right]}_{\text{current
}\mathcal{H}}-\underbrace{\mathbb{E}\left[\mathcal{H}[p(\boldsymbol{x},\boldsymbol{m}|\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})]\right]}_{\text{expected
}\mathcal{H}\text{ for candidate }\boldsymbol{a}}\,,$ (14)

where expectation is taken w.r.t. $\hat{\boldsymbol{z}}$.

Kullback-Leibler divergence (KLD) or relative entropy [139] has also been used
as utility function. KLD measures the change in the form of a pdf (as MI), but
also how much its mean has translated [140]. It is defined as follows:

$\displaystyle\mathcal{D}_{KL}\left(p_{1}|p_{2}\right)$
$\displaystyle\triangleq\mathbb{E}\left[\log\frac{p_{1}(\boldsymbol{x})}{p_{2}(\boldsymbol{x})}\right]=\sum_{\boldsymbol{x}}p_{1}(\boldsymbol{x})\log\frac{p_{1}(\boldsymbol{x})}{p_{2}(\boldsymbol{x})},$
(15)

with $p_{1}(\boldsymbol{x})$ and $p_{2}(\boldsymbol{x})$ the prior and
posterior distributions (as in MI) [141], or the estimated and true posteriors
assuming the latter can be somehow approximated [142, 143, 42].

For OG maps, the three metrics above (entropy, MI, and KLD) ultimately rely on
counting the number of cells in a map, being thus discrete and ill-suited for
optimization techniques. To mitigate this issue, Deng et al. [144, 87] propose
a differentiable _cost-utility_ function for both 2D OG and voxel maps that
can be used with continuous optimization methods (albeit the approach still
assumes perfect robot localization).

In the context of information-theoretic planning, there exists a problem
variation in which the uncertainty of only a subset of variables is reduced.
The motivation comes from the fact that maximizing information of all
variables does not always imply maximizing that of the subset of interest.
This problem variation has been referred to as focused active inference [145].
In general, focused active inference is more computationally intensive than
the standard case, since it requires marginalization of the (posterior) Fisher
information matrix via, e.g., Schur complement. Kopitkov and Indelman [146,
147] present a method based on the matrix determinant lemma that does not
require the posterior covariance to calculate entropy considering both the
unfocused (entropy over all variables) and focused (entropy over a subset of
variables) cases.

###  IV-C Theory of Optimal Experimental Design (TOED)

There exists a second group of utility functions built upon optimal design
theory (TOED) that tries to quantify uncertainty directly in the task space
(i.e., from the variance of the variables of interest). Unlike information-
theoretic metrics that target binary probabilities in the grid map, task-
driven metrics apply to Gaussian variables. Following TOED, a set of actions
to execute in active SLAM will be preferred over another if the covariance of
the joint posterior is smaller, i.e., the posterior covariance matrix,
$\boldsymbol{\Sigma}$, has to be minimized. In order to compare matrices
associated to different candidates, several functions —known as _optimality
criteria_ — have been proposed, such as the trace (originally known as
$A$-optimality) [148], its maximum/minimum eigenvalue ($E$-optimality) [149],
or its determinant ($D$-optimality) [150]. The latter was often disregarded in
active SLAM because its traditional formulation did not allow for checking
task completion and generated precision errors ($\det(\boldsymbol{\Sigma})\to
0$ rapidly when there are low-variance terms) [140, 34]. However, Carrillo et
al. [24] show these problems can be solved using Kiefer’s formulation of
$D$-optimality [151], thus re-establishing the latter as an effective measure
of uncertainty for active SLAM.

On the basis of TOED, Kiefer [151] proposes a family of mappings
$\|\boldsymbol{\Sigma}\|_{p}:\mathbb{R}^{n\times n}\to\mathbb{R}$,
parametrized by a scalar $p$:

$\|\boldsymbol{\Sigma}\|_{p}\triangleq\left(\frac{1}{n}\text{trace}(\boldsymbol{\Sigma}^{p})\right)^{\frac{1}{p}}\,,$
(16)

which can be particularized for the different values of $p$ and expressed in
terms of the eigenvalues of $\boldsymbol{\Sigma}$,
$(\lambda_{1},\dots,\lambda_{\ell})$, by leveraging the properties of the
matrix power:

$\|\boldsymbol{\Sigma}\|_{p}=\begin{cases}\left(\frac{1}{n}\sum\limits_{k=1}^{n}\lambda_{k}^{p}\right)^{\frac{1}{p}},&\text{if
}0<|p|<\infty\\\
\exp\left(\frac{1}{n}\sum\limits_{k=1}^{n}\log(\lambda_{k})\right),&\text{if
}p=0\end{cases}\,.$ (17)

In essence, utility functions are functionals of the eigenvalues of
$\boldsymbol{\Sigma}$. The boundary cases $p=\\{0,\pm\infty\\}$ and $p=\pm 1$
result in the four modern optimality criteria:

  * •

T-optimality criterion ($p=1$) captures the average variance:

$T{\text{-}}opt\triangleq\frac{1}{n}\textstyle\sum_{k=1}^{n}\lambda_{k}\,.$
(18)

  * •

D-optimality criterion ($p=0$) captures the volume of the covariance (hyper)
ellipsoid:

$D{\text{-}}opt\triangleq\exp\left(\frac{1}{n}\textstyle\sum_{k=1}^{n}\log(\lambda_{k})\right)\,.$
(19)

  * •

A-optimality criterion ($p=-1$) captures the harmonic mean variance, sensitive
to a lower-than average value:

$A{\text{-}}opt\triangleq\left(\frac{1}{n}\textstyle\sum_{k=1}^{n}\lambda_{k}^{-1}\right)^{-1}\,.$
(20)

  * •

$E$-optimality criterion ($p\to\pm\infty$) captures the radii of the
covariance (hyper) ellipsoid:

$\displaystyle E{\text{-}}opt$
$\displaystyle\triangleq\min(\lambda_{k}:k=1,...,n)\,,$ (21)
$\displaystyle\tilde{E}{\text{-}}opt$
$\displaystyle\triangleq\max(\lambda_{k}:k=1,...,n)\,.$ (22)

Optimality criteria were first used in active SLAM by Feder et al. [26], where
utility was computed as the area of the covariance ellipses describing the
uncertainty in the joint posterior. Since then, many active SLAM methods based
on TOED have been proposed, mostly based on $T\text{-}opt$ [34, 39] and,
recently, $D\text{-}opt$ [123, 61]. Even so, IT-based methods remain the most
popular. Note that both the map and robot uncertainties must be described by a
covariance matrix $\boldsymbol{\Sigma}\in\mathbb{R}^{n\times n}$, either by
using a full covariance matrix in landmark-based representations (i.e.,
$n\gg\ell$) or by including the effect of the map’s uncertainty in
$\boldsymbol{\Sigma}_{r}$ (and thus $n=\ell$) [152].

Monotonicity. One of the most important assumptions in active SLAM is that
uncertainty increases as exploration takes place. However, the seminal work in
[153] notes how monotonicity is lost for some utility functions under certain
conditions, concluding that only $D\text{-}opt$ guarantees this property and
is thereby the only appropriate utility function for this task. Kim and Kim
[154] and Rodríguez-Arévalo et al. [155] demonstrate, however, that rather
than on the utility function chosen, monotonicity depends on how the error and
uncertainty are represented. In [155], the authors prove that only
differential representations guarantee monotonicity for all utility functions.
In summary, representation of uncertainty is a key issue in active SLAM, since
certain representations do not guarantee its monotonicity property during
exploration, and thus may lead to incorrect decisions.

###  IV-D The Graphical Structure of the Problem

Quantification of uncertainty via scalar mappings of the covariance matrix may
be a computationally intensive task, mostly due to the fact that the
covariance is a large and dense matrix. Therefore, most works resort to
reasoning over the Fisher information matrix (FIM), i.e., the inverse of the
covariance, which is generally sparser. Still, their evaluation is expensive,
especially for large state spaces. To circumvent this issue, some works have
proved that analyzing the connectivity (i.e., Laplacian) of the underlying
pose-graph in active graph-SLAM is equivalent to computing optimality
criteria. The link between graph and optimum design theories can be traced
back to Cheng [156], who related the number of spanning trees of concurrence
graphs with $D$-optimal incomplete block designs. Khosoussi et al. [157] show
that classical $D\text{-}$ and $E\text{-}opt$ are related to the number of
spanning trees of the SLAM pose-graph and its algebraic connectivity,
respectively, for the case of 2D graph-SLAM with constant uncertainty along
the trajectory. In [158] and [159], these results are extended to the
$\mathbb{R}^{n}\times SO(n)$ synchronization problem, and also relate
$T\text{-}opt$ to the average node degree of the graph. Placed and Castellanos
[81, 160] study the general active graph-SLAM problem formulated over the Lie
group $SE(n)$; showing the existing relationships between modern optimality
criteria of the FIM and connectivity indices when the edges of the pose-graph
are weighted appropriately, and reporting substantial reductions in
computation time. These results have been used in coverage problems [62],
multi-robot exploration [128], active visual SLAM [66], or to develop a
_stopping criterion_ [161].

The graph structure of the problem has also been recently exploited in
conjunction with IT utility functions. Kitanov and Indelman [162] relate the
number of spanning trees of the graph to entropy (which ultimately depends on
the covariance determinant) and its node degree to Von Neumann entropy. The
latter has been also applied to the focused case, thus relating the graph
topology to the marginalized FIM [163].

##  V Stage 3: Action Selection and Execution

Once every possible destination has an associated utility value, the last
stage of active SLAM involves the selection of the optimal destination. This
can be formulated as an optimization problem w.r.t. the set of actions to
reach every possible goal location, cf. (9). When the set of candidate
destinations is discrete (and typically consists in a handful of options), the
solution of the optimization can be obtained via enumeration [74, 114, 49].
For the case of TOED-based utility functions, it will be a minimization or
maximization problem depending on whether the covariance
($\boldsymbol{\Sigma}$) or the FIM ($\boldsymbol{\Phi}$) is analyzed. Since
$\boldsymbol{\Sigma}=\boldsymbol{\Phi}^{-1}$ and
$\|\boldsymbol{\Sigma}\|_{p}=\left(\|\boldsymbol{\Phi}\|_{q}\right)^{-1}\forall
p$ with $q=-p$, the optimization problem is

$\displaystyle\boldsymbol{a}^{\star}=\operatorname*{arg\,min}_{\boldsymbol{a}\in\mathcal{A}}\|\boldsymbol{\Sigma}\|_{p}={{\operatorname*{arg\,max}_{\boldsymbol{a}\in\mathcal{A}}\
\|\boldsymbol{\Phi}\|_{q}}}\,,$ (23)

where $\|\cdot\|_{p}$ refers to Kiefer’s optimality criteria, see (16).

Information-based utility functions will seek to minimize entropy (or,
equivalently, to maximize MI). Following [40], the optimal set of discrete
actions can be found as:

$\displaystyle\boldsymbol{a}^{\star}=\operatorname*{arg\,max}_{\boldsymbol{a}\in\mathcal{A}}\
\mathcal{I}_{G}=\operatorname*{arg\,min}_{\boldsymbol{a}\in\mathcal{A}}\
\mathcal{H}\left[p(\boldsymbol{x},\boldsymbol{m}|\boldsymbol{h},\hat{\boldsymbol{z}},\boldsymbol{a})\right]\,.$
(24)

In any case, after selecting the most informative destination, it all comes
down to navigating to it using, e.g., sampling-based planning methods as RRT
[164], probabilistic road maps (PRM) [165], or their asymptotically optimal
variants [166]. Note that despite selecting the optimal destination among a
discrete set of candidates, the executed path to reach it rarely represents an
optimal solution for the original problem (9); this suboptimality is caused by
decoupling the problem into first computing and evaluating a set of goal
locations, and then computing a path to one of these goals.

##  VI Belief-space Planning and
Continuous-space Optimization

As a potential solution to the suboptimality induced by classical decoupled
approaches, there exists a second family of methods in which the future
trajectory of the robot is directly optimized. These methods represent an
alternative solution to the modular scheme and may be divided into two
categories, depending on whether they discretize the action space or not. The
first category relies on path planning algorithms to generate a discrete set
of candidate paths towards the unknown space, in order to later evaluate their
utility. Works from Oriolo et al. [167] and Freda et al. [168] are among the
first to apply these algorithms for exploration, evaluating robot
configurations inside the not-previously-sensed free space. In contrast to
discrete frontier optimization, that compares utility only at candidate
locations, these methods evaluate it over the paths to reach them,
guaranteeing that the path to execute is optimal among the considered subset.
Bonetto et al. [169, 68] go one step further and optimize exploration in all
three steps of modular approaches, considering not only the destination and
the path to reach it, but also its execution.

On the other hand, globally optimal solutions have been considered under the
umbrella of continuous-state POMDPs. Despite their resolution would ideally
require to compute a policy over the infinite-dimensional space of posteriors
of the joint state space [170] and computing an exact solution is known to be
intractable in general [171], active SLAM as a continuous-state POMDP can be
approximately solved under the frameworks of belief-space planning (BSP) or
optimal control. Such optimization techniques require a continuous utility
function, which can be obtained directly from complex continuous
representations of the environment [53] or inferred from discretized
representations. For example, Vallvé and Andrade-Cetto [133] compute a dense
entropy field from the posteriors’ evaluation over the discretized
configuration space.

###  VI-A Belief-Space Planning (BSP)

Continuous-domain BSP optimizes the future trajectory of the robot without
discretizing the action space, but rather performing a continuous
optimization. Bai et al. [172] and Kontitsis et al. [143] use sampling-based
methods to maximize an objective function that rewards uncertainty reduction
and goal achievement. Platt et al. [32] apply linear quadratic regulation
(LQR) to compute locally optimal policies. Van Den Berg et al. [120] relax the
assumption that future observations are consistent with the current robot pose
belief (ML observations). Indelman et al. [43] extend [120] to the case where
the belief describes both robot poses and unknown landmarks in the
environment, while also modeling missed observations. Porta et al. [173]
generalize value iteration to continuous-state POMDPs while assuming state-
dependent reward functions. Van den Berg et al. [170] present a highly
efficient method for solving continuous POMDPs in which beliefs can be modeled
using Gaussian distributions over $\mathcal{S}$. Prentice and Roy [174]
develop a belief-space variant of the PRM algorithm called the belief road map
(BRM), incorporating predicted uncertainty of future position estimates into
the planning process. Valencia et al. [175] contribute a pose-SLAM path-
planning approach that leverages the BRM to find a path to the goal with the
lowest accumulated pose uncertainty.

###  VI-B Active SLAM as Optimal Control

Converting a POMDP formulation of active SLAM into an equivalent continuous-
space MDP, as discussed in Section II, leads to a stochastic optimal control
problem in general. Depending on the transition and observation models, noise
distribution, and the reward function, the problem may be simplified further.
Le Ny et al. [176] and Atanasov et al. [9] show that when the transition and
observation models are linear in the state $\boldsymbol{s}$ and the noise is
Gaussian, then the time evolution of the belief state $b_{t}$ may be obtained
by the Kalman filter and the covariance is _independent_ of the measurement
realizations. If the reward function $\rho$ depends only on the covariance, as
for the MI, active SLAM reduces from a stochastic to a deterministic optimal
control problem. Deterministic optimal control problems are easier to solve,
and techniques such as linear quadratic Gaussian (LQG) regulation [177] or
search-based [9, 178], and sampling-based [179, 180, 181] motion planning are
applicable. If the assumptions necessary for the deterministic reduction
cannot be satisfied, the stochastic active SLAM problem may be solved by
obtaining an open-loop control sequence under deterministic dynamics first,
followed by a closed-loop feedback policy, under stochastic dynamics
linearized around open-loop trajectory [182].

In the presence of state or action constraints, the optimal control
formulation of active SLAM can be approached using differential dynamic
programming (DDP) or model predictive control (MPC). Rahman and Waslander
[183] introduce an augmented Lagrangian formulation of iterative LQG, which
captures belief-state constraints via a penalty function. The approach
iterates between iLQG trajectory optimization in an unconstrained stochastic
optimal control problem and Lagrange multiplier updates for the penalty
function. This and several other works [184, 183, 177] develop differentiable
formulations of sensor field-of-view constraints amenable to gradient-based
optimization. Carlone and Lyons [185] split the environment into convex
regions and formulate the problem using mixed-integer programming. Chen et al.
[62] employ a spectrahedral description of the convex hull of the space of
orientations and relax non-convex obstacle constraints using a convex half-
space representation.

Striking a suitable balance between exploration and exploitation in active
SLAM is challenging because the effects of potential future loop closures are
not easy to capture in the predicted evolution of the belief
$b_{t}(\boldsymbol{s}_{t})$. Leung et al. [39, 186] introduce attractor states
to guide the robot based on three modes (explore, improve map, and improve
localization), determined using uncertainty thresholds. Attractor states were
combined with a right-invariant extended Kalman filter in [187] to achieve
active range-bearing landmark-based SLAM.

##  VII Deep-Learning-based Approaches

Advances in DL have created new opportunities in using neural networks to
solve active SLAM; these techniques follow a completely different scheme,
circumventing the split into three stages that characterizes modular
approaches. Usually, goal identification is not required due to the chosen
action set, and utility computation and selection of the best action are both
embedded in the network. In this section, we particularly focus on DRL methods
for autonomous robotic exploration and discuss the design of the state,
action, and reward spaces, as well as the problems of partial observability,
generalization, and the necessity for training environments.

###  VII-A Deep Reinforcement Learning (DRL)

A question that arose in the early work on learning-based active SLAM was
which type of learning was suitable for this decision-making problem, in which
(i) agents must directly learn from interaction with the environment, (ii)
states may not be fully observable, and (iii) policies have to generalize to
other scenarios in which _a priori_ knowledge is nonexistent. This premise
soon led the community to explore DRL, building on existing methods that
approached active SLAM with RL [188] and using neural networks to represent
the policies or value functions. Within DRL, model-free techniques have been
the center of attention, although isolated approaches that combine them with
model-based learning do exist [189]. Methods based on supervised learning can
also be found in the literature [190, 191], although they are a minority.
Contrary to model-based active SLAM, the computational effort in DRL
approaches is mostly confined to the training phase, while the testing phase
reduces to a forward pass on the network. However, the behavior depends
entirely on the model learned from training data, thus limiting its
generalization to novel operational conditions.

The great success of the work from Mnih et al. [192] boosted the research in
model-free DRL and several value- and policy-based methods emerged shortly
after. The behavior of deep Q-networks [192] improves using the double [193]
and double-dueling [194] architectures. Actor-critic techniques combine both
value-iteration and policy gradient methods, e.g., deep deterministic policy
gradient [195], asynchronous advantage actor-critic [196]. See [197] and [198]
for a survey on the methods. Although these strategies were initially proposed
for different decision-making problems (e.g., video-games), they have been
applied to robotic exploration.

###  VII-B On the Reward Function Design and the Action Set

Tai and Liu [199] are among the first to employ DRL for robotic exploration in
simulation environments, extracting the next best actions to execute from raw
observations using a 2-layer Q-network. Convergence to policies valid in more
complex and previously unseen scenarios is achieved in [200, 121] with
parallel architectures. In any case, the above works use purely extrinsic
reward functions (i.e., by instrumenting the environment), which ultimately
addresses the obstacle avoidance problem rather than active SLAM [123]. As a
response, the notions of motivation and curiosity [201] were exploited to
design intrinsic rewards, giving origin to curiosity-driven methods that
motivate agents to visit unknown configurations [202]. Chen et al. [203] and
Chaplot et al. [56] propose holistic, open-source approaches that employ a
coverage reward to explore complex 3D simulation environments. The detailed
study in [203] shows the benefits of pre-training and combining inputs from
different sources. Similarly, the idea of uncertainty minimization led to
uncertainty-aware approaches. This is the case of [204] that encourage the
visit of high-covariance states, and [122, 20] where the reward encodes the
belief accuracy. All these methods are publicly available except active target
localization. Many of the DRL-based methods, including all of the above, aim
to directly generate optimal control commands, either discrete [203] or not
[205]. They represent end-to-end solutions in which the safe navigation task
is embedded into the network and therefore do not require planning and the
SLAM estimates.

True uncertainty metrics inherited from classic theories have also been
introduced in the reward function design, seeking more robust foundations. The
robot’s $D\text{-}opt$ is incorporated in [123] and $T\text{-}opt$ of virtual
landmarks in [58], whereas the map’s MI is used in [57, 59]. Agents trained
under this new perspective perform active SLAM in complex scenes, albeit only
targeting location or mapping uncertainties. Designing effective reward
functions that account for both is still an open problem. In addition, this
new family of methods has promoted the use of learning as a part of the
solution rather than a replacement to well-established planning algorithms.
Utilizing planning and learning together, may make policies easier to learn,
generalize better and transfer across platforms. In this vein, Niroui et al.
[57] and Chen et al. [58] employ DRL to extract the best candidate among
previously-detected frontiers, thereby creating a link with modular
approaches. Li et al. [59] and Lodel et al. [206] use nearby sampled locations
instead, but they also leave the motion planning task out of the scope of
learning. Chaplot et al. [56] use different policies to infer long-term
(i.e.., frontiers) and short-term (i.e., control commands) goals, linked
through a model-based trajectory planner.

###  VII-C Partial Observability and Generalization

Partial observability and generalization are two inherent and often-forgotten
concepts in active SLAM. First of all, the uncertainty about the observations
and actions taken, and the limited observations make the problem not fully
observable. Consequently, agents are unable to distinguish their own true
state based on single observations, and learned policies are bound to be
suboptimal [207]. Mirowski et al. [200] alleviate this by expanding the
network inputs with previous observations and rewards. Hausknecht and Stone
[208] demonstrate that recurrent architectures can also handle partial
observability, teaching agents to learn about previous data on their own. Long
short-term memory units are used for robotic exploration in [57, 200, 203],
and Karkus et al. [189] embed the computation structure of the belief (and
thus the history) in a recurrent neural network.

The second element intrinsic to active SLAM is the lack of prior knowledge of
the environment. Learning policies that generalize to unseen scenarios is
therefore crucial, and currently represents a key limiting factor for
learning-based methods. Overfitting can be mitigated by expanding the sample
space (e.g., using random starting locations [204, 57], considering noise in
the observations [209]) or by using sparser network inputs [207]. For example,
agents trained in [121, 210] learn policies generalizable to real environments
after reducing sensory data to a sparse range input. Similarly, Shi et al.
[211] specifically use sparse range measurements to reduce the simulation-to-
reality (_sim-2-real_) gap. Lodel et al. [206] improve generalization by
feeding the network with egocentric limited observations, following [203].
Chen et al. [58] leverage graph neural networks, in which the inputs are
already compressed representations. The task of transferring trained agents to
real scenarios is still an open research problem, and few efforts have been
made in this direction [121, 211, 59].

###  VII-D Training Environments

The use of DRL introduced a major challenge during training: the need of a
simulation environment to acquire data online. Unlike supervised methods,
training with offline data is not possible and real-world training seems
infeasible. To overcome this problem, some works use their own simplified
simulation scenarios, thus limiting the network inputs to ground-truth data or
range perfect observations. To use more realistic data that bridge the gap
from simulation to physical robots, more complex simulators need to be used in
training.

_Stage_ [212] is one of the simplest engines used in the literature [57],
although it restricts perception to two-dimensional bitmapped environments.
_Gazebo_ [213] is a much more complete simulator which allows for 3D
simulations, realistic rendering, visual sensors, etc. In addition, it is
tightly integrated into the widespread Robotic Operating System (ROS), which
makes its use commonplace [199, 121, 123]. _CoppeliaSim/V-REP_ [214] also
allows for online mesh manipulation, but it is not an open-source solution and
is less integrated into ROS, limiting its adoption. Combination of a physics
engine (i.e., robot motion and sensor models) with a DRL framework is not
always straightforward. Zamora et al. [215] present a powerful framework by
integrating the RL toolkit OpenAI Gym [216] with ROS and Gazebo.

In contrast to the above platforms, initially designed for robotics and later
adapted to DRL, there is a second family of simulators born in the age of AI.
They tend to prioritize training speed over the breadth of simulation
capabilities. _DeepMind Lab_ [217] allows agents to move discretely in low-
textured, game-like scenarios, and provides access to a visual sensor and
velocity. _Habitat-Sim_ [218] takes a leap forward by supporting physics
simulation and different robot and visual sensor models. More interestingly,
it has the powerful capability of rendering simulation environments from image
datasets, e.g., Replica [219]. _iGibson_ [220] also provides fast visual
rendering and physics simulation, and includes simulation of lidar and optical
flow sensors. The ROS ecosystem is already integrated in [220], whereas [218]
requires the use of external libraries. Despite their potential, none of these
platforms has yet been used for DRL in the context of active SLAM.

##  VIII Multi-robot Active SLAM

The active SLAM problem can be extended to a multi-agent setting, where $n$
robots optimize their sensing trajectories collaboratively to estimate a
common map $\boldsymbol{m}\in\mathcal{M}$ of the environment. Each robot has
its own state space $\mathcal{X}_{i}$ and action space $\mathcal{A}_{i}$.
Applying an active SLAM algorithm to the joint state space
$\mathcal{S}=\mathcal{X}_{1}\times\cdots\times\mathcal{X}_{n}\times\mathcal{M}$
and joint action space
$\mathcal{A}=\mathcal{A}_{1}\times\cdots\times\mathcal{A}_{n}$ can generate
desirable behavior but becomes computationally infeasible as the number of
robots increases because the complexity of centralized algorithms scales
exponentially with $n$ [78]. Such algorithms also require collecting all robot
measurements and performing joint optimization at a centralized server before
communicating the planned actions back to the individual robots. If the robot
team is small and connectivity is maintained at all times, centralized
algorithms can be used to plan all robot trajectories simultaneously. For
example, Charrow et al. [221] achieve multi-robot target tracking by
maximizing the MI between the target location and range-only observations over
a set of motion primitives. However, larger teams with intermittent
communication and limited onboard computation require decentralized
algorithms, where individual robots solve smaller instances of the active SLAM
problem, or fully distributed algorithms, where the robots exchange
information only with their neighbors. Kantaros et al. [181, 222] propose an
informative planning222 Informative path planning can be considered a
generalization of active SLAM to include objectives beyond the quality of
localization and mapping, e.g., for target tracking or environmental
monitoring. technique which constructs random trees of control sequences and
is particularly simple to distribute. The algorithm scales to very large
numbers of sensors and targets and is probabilistically complete and
asymptotically optimal.

A particularly important instance of the problem is collaborative multi-robot
exploration, where the robots aim to coordinate how to efficiently explore
different regions of the environment. Early works such as [223, 105] present
an approach for choosing appropriate frontiers, while simultaneously taking
into account their utility and the cost of reaching them. Each time a target
point is assigned to some specific robot, the utility of the unexplored area
visible from that frontier is reduced. This mechanism is used to assign
different frontiers to different robots. Colares and Chaimowicz [132] develop
a decentralized multi-robot formulation of the classical frontier-based
exploration method. The authors use an objective function that captures the
frontier entropy and distance, and a robot coordination factor that penalizes
regions that other robots are already exploring. Atanasov et al. [78] consider
a multi-agent active information acquisition problem, in which an information
measure is maximized over a discrete space of agent trajectories, and propose
a decentralized planning scheme using coordinate descent in the space of agent
trajectories. Schlotfeldt et al. [224] introduce an anytime search-based
planning formulation that progressively reduces the suboptimality of the
multi-agent plans while respecting real-time constraints. Instead of using
search-based planning, Ossenkopf et al. [225] generate candidate robot actions
using RRT*. The sampling is biased to prioritize exploration, map improvement,
or localization improvement. The map and robot state entropy is evaluated
along the planned trajectories in two stages: short-horizon exact computation
using filter updates, and long-horizon approximation using predicted loop
closures. Lauri et al. [226] introduce a decentralized $\rho$-POMDP, allowing
the specification of an information-theoretic objective. The authors show that
a multi-agent A* algorithm that searches the joint policy space can be applied
to belief-dependent rewards to achieve cooperative target tracking with
periodic communication. Hu et al. [205] design a hierarchical control approach
for cooperative exploration, combining a high-level region-assignment layer
and a low-level safe-navigation layer. The former uses dynamic Voronoi
partitions to assign different regions to individual robots; the latter
achieves collision-free navigation to successive frontier points using DRL.

Another important instance is collaborative multi-robot active estimation,
where the goal is to seek actions that actively reduce the uncertainty over
relevant random variables. For instance, Kontitsis et al. [143] develop a
multi-robot active SLAM method that uses a relative entropy optimization
technique [227] to select trajectories which minimize both localization and
map uncertainties. Indelman [228] develops a collaborative multi-robot BSP
framework, which incorporates reasoning about future observations of
environments that are unknown at planning time. That approach has been
extended in [229] to a decentralized setting. Best et al. [230] propose the
_self-organizing map_ algorithm, considering the problem of multi-robot path
planning for active perception and data collection tasks. Chen et al. [128]
leverage graph connectivity indices and their relationship to optimality
criteria to achieve multi-robot active graph-SLAM. Each robot aims to improve
the pose graphs of the other agents by sharing its observations when it moves
near areas where they have low connectivity.

##  IX Open Research Questions

Active SLAM still requires much research in order to support the deployment of
fully autonomous robots in complex environments. Many are the challenges and
research fields involved, so cooperation between them is crucial to achieve
real-world impact. In this section, we present some of what we consider the
most important research questions. Although some of them are long-known
challenges and are already under intense investigation, others have not
received such attention.

###  IX-A Prediction Beyond Line-of-sight

Resolution of active SLAM relies on fast and precise predictions of future
states for the variables of interest. The accurate prediction of the scene and
robot pose after executing a set of candidate actions can be the difference
between making the right decision or not. The expected sensed space and the
resulting map representation have traditionally been predicted using a sensor
model together with ray-casting techniques [105, 51]. Recent related work,
however, addresses the problem of scene completion and occupancy anticipation
from a DL perspective. Fehr et al. [231] use a neural network to augment the
measurements of a depth sensor and Ramakrishnan et al. [232] directly predict
augmented OG maps beyond the sensor’s field-of-view using auto-encoders (AE).
Rather than using raw sensor measurements, Katyal et al. [233] and Hayoun et
al. [234] extend an input OG map beyond the line-of-sight also using AE.
Shrestha et al. [235] predict maps of occupancy probabilities instead with
variational AE. Dai et al. [236] perform scene prediction over 3D SDF-based
maps. All these methods seem promising for fast and precise online map
prediction beyond line-of-sight, although their integration into active SLAM
is yet to be done and brings with it numerous challenges. How does scene
prediction behave in unstructured environments? How to account for
uncertainty? Is measurement prediction more reliable and informative than map
prediction? How to predict the effect of only a certain set of non-myopic
actions in the map rather than augmenting the whole scene? Regarding the
latter, [237, 238] and [239] subordinate predictions to candidate actions.

On the other hand, the robot state is straightforwardly estimated using motion
models or path planners. However, the prediction of its associated uncertainty
is not trivial and requires more attention. Work from Asraf and Indelman [239]
is among the very few efforts made to combine data-driven scene prediction
with BSP. In addition, they use the predicted observations to forecast the
posterior uncertainty over the robot trajectory. Besides the robot’s movement,
it is the appearance of loop closures (exploitation) that significantly
affects the new states’ uncertainty, thus making its forecast critical.
Despite some isolated works have partially studied this problem [29, 240], it
still remains as an open challenge.

###  IX-B From Active SLAM to Active Spatial Perception

Most active SLAM approaches reason over geometric representations of the
environment (e.g., OG maps). However, when we explore new environments as
humans, we are mostly interested in semantic elements of the environment
(e.g., presence of objects, rooms) rather than the shape of the environment
_per se_. Modern SLAM systems are now able to build 3D metric-semantic maps in
real-time from semantically labeled images, see [96] and the references
therein. These maps include both occupancy information and semantic labels of
entities (e.g., chairs, tables, humans, etc.) in the environment. Very recent
work goes even further and develops spatial perception systems that infer
hierarchical map representations, in the form of 3D scene graphs [103, 241,
242]. They symbolize high-level representations of an environment, that
capture from low-level geometry (e.g., a 3D mesh of the environment) to high-
level semantics (e.g., objects, people, rooms, buildings, etc.). While there
is a growing amount of work in estimating these high-level representations
from sensor data, their use in active SLAM is still uncharted territory. Very
early effort in this direction includes the work from Ravichandran _et al._
[243], which focuses on object search using 3D scene graphs.

Active metric-semantic information acquisition, or _Active Spatial Perception_
, has the potential to impact many aspects of robot autonomy: (i) by
leveraging semantic knowledge, a robot can more effectively predict unseen
space (e.g., predict the presence of rooms or objects in each room), (ii) the
use of semantics can further enhance SLAM performance (e.g., allowing for
novel loop closure detection methods [242]), and (iii) hierarchical
representations may enable novel and more computationally efficient planning
methods. However, each opportunity comes with many open research questions,
for instance: How to quantify uncertainty over metric-semantic or even
hierarchical scene representations? How to leverage hierarchical structures to
improve computation? How to perform spatial prediction in hierarchical
representations?

###  IX-C Robust Online Belief Space Planning and Active SLAM

Another key aspect is data association, i.e., association between measurements
and the corresponding landmarks (or other entities in the map representation).
In perceptually aliased and ambiguous environments, determining the correct
data association is challenging, and incorrect associations may lead to
catastrophic failures. The research community has been investigating
approaches for robust perception to allow reliable and efficient operation in
ambiguous environments (see, e.g. [244, 245, 246, 247, 248, 249]). Yet, these
approaches focus on inference (rather than planning), i.e., actions are given.
Only recently, ambiguous data association was considered also within BSP and,
in particular, active SLAM. Pathak et al. [250] incorporate, for the first
time, reasoning about future data association hypotheses within a BSP
framework, enabling autonomous hypotheses disambiguation. Another related work
in this context is [251], that also reasons about ambiguous data association
in future beliefs while utilizing the graphical model presented in [248]. A
first-moment approximation to Bayesian inference with random sets of targets,
known as the probability hypothesis density (PHD) filter, has been
successfully applied to active target tracking problems [252]. However,
explicitly considering all possible data associations leads to an exponential
growth of the number of hypotheses, and determining the optimal action
sequence quickly becomes intractable. Shienman and Indelman [253] recently
presented an approach that utilizes only a distilled subset of hypotheses to
solve BSP problems while reasoning about data association and providing
performance guarantees considering a myopic setting. Nevertheless, BSP and
active SLAM in these challenging settings remain open problems. More
generally, finding an appropriate simplification of the original BSP or active
SLAM problem, which is easier to solve, with no, or bounded, loss in
performance, is an exciting and novel direction [254, 255, 253, 256].

###  IX-D Reasoning in Dynamic and Deformable Scenes

One of the most common assumptions in active SLAM is to consider the
environment static —or slightly dynamic, at best. Real scenes, however,
contain moving agents most of the times, and even deformable elements (e.g.,
clothes, water). Handling these elements would greatly impact the robot’s
autonomy, its reasoning ability and awareness, and would facilitate its
deployment in real-world scenarios.

The study of dynamic environments has long been a topic of interest for the
path planning [257] and the SLAM [258] communities; but its investigation in
the context of active SLAM has been typically restricted to the action
execution step (i.e., re-planning) [259, 260]. However, many other aspects
emerge when reasoning with dynamic elements: How to foresee their effects in
planning? How to integrate them in the utility function? How to maintain a
lightweight representation?

Non-rigid environments present an even greater challenge. Planning for mobile
robots in deformable environments started receiving some attention a couple of
decades ago [261, 262]. Medical applications have also stimulated progress on
SLAM in deformable environments [263, 264]. However, to date, no efforts have
been made towards developing a deformable active SLAM framework. We believe
this is partly due to the unavailability and complexity of simulators for
mobile robots in deformable environments, and partly due to the difficulty in
extending the current map representations to deformable scenes. Given the
importance of obtaining accurate robot trajectory estimates towards mapping
deformable environments, active SLAM can play a major role in this area.

###  IX-E Towards Meaningful and Autonomous Stopping Criteria

Unlike with coverage and exploration in known environments, determining the
moment in which the task of active SLAM is complete is non-trivial. Performing
active SLAM is known to be a computationally expensive process: a vast amount
of resources is required to estimate and optimize utility online, thereby
conditioning the execution of other tasks; therefore, it is crucial to
understand when such process can be considered complete and other tasks can be
prioritized. Cadena et al. [5] already identified this issue as an open
research question, but little research has been done on the topic. Even recent
active SLAM works still rely on traditional temporal [51, 81] or spatial [62,
265] constraints to decide when exploration has terminated. These metrics,
however, cannot be used in truly unknown environments nor do they assert task
completion (see [161]). The use of TOED-based metrics has been identified as a
promising tool [5, 36, 161] to determine when a given exploration strategy is
no longer adding information. Nevertheless, many questions remain to be
answered: How to guarantee task completion? How to transition between
exploration strategies? Also, the advent of DRL approaches raises a new
question: when to stop training?

###  IX-F Reproducible Research in Active SLAM

The increasing attention towards active SLAM creates the need for new
benchmarks to objectively evaluate new findings against existing research.
This has long been a challenge in the robotics community [266], where real-
life robotics experiments are often difficult to replicate across research
groups. In related problems, such as SLAM, static datasets are commonly used
for benchmarking. However, in active SLAM, the agent must interact with the
environment, making the use of datasets impractical. In recent years, a
significant effort has been made in robotics to address challenges in
benchmarking [267] and reproducibility [268]. Despite these efforts, such
benchmarks are still lacking in active perception.

Typically, in active SLAM, researchers select a set of scenarios (e.g.,
platform, task, and environment) representative of the desired application,
and experiments are conducted in simulation via customized simulators or in
the real world via specialized hardware. While such an evaluation is adequate
for validation, the specified scenario may not be general enough or
sufficiently specified to be reproduced. Consequently, one-to-one comparisons
are rarely made between approaches. While targeting more general embodied
agents, several open-source datasets [269] and simulators [218, 270, 265] show
promise for active SLAM research. Also, open-source frameworks (see Table I)
make the comparison and testing of new algorithms straightforward, only by
modifying the decision-making portion. While some works take advantage of
these simulators and datasets [56], establishing a proper methodology for
evaluating active SLAM when it comes to generalization from simulation to the
real world remains an open question. Besides, there is a dire need to
establish adequate performance metrics for active SLAM that go beyond
commonly-used exploration time and coverage. Improving the quality of
estimates is the main objective of active SLAM, and should therefore be
measured.

###  IX-G Practical Applications

Although active SLAM methods have practical relevance in many real-world
problems such as search and rescue, where constructing a sound representation
of the environment is time critical, very few practical implementations and
deployments of active SLAM have been described in the literature. Walter et
al. [271] propose a partially autonomous system for underwater ship hull
inspection. Kim and Eustice [119] deploy a complete active SLAM system.
Palomeras et al. [85, 54] report the autonomous exploration of complex
underwater environments for environmental preservation purposes. Fairfield and
Wettergreen [240] investigate terrestrial applications and autonomous mapping
of abandoned underground mines. A roughly similar application but in the
archaeological context of catacomb exploration is presented in [272]. Strader
et al. [22] report experiments of active perception in a Mars-analogue
environment. Finally, assistive mapping examples for office-like environments
can be found in [30, 273, 251]. Aerial applications of active SLAM are
significantly less common. Chen et al. [62] propose an MPC framework to
address coverage tasks while maintaining low uncertainty estimates.

Overall, there are very few reports of field experiments involving active SLAM
systems. Besides, by 2022, there is a large imbalance between the patents
using the terms SLAM and active SLAM333We used “simultaneous localization and
mapping” after:priority:19920101, and ”active slam” OR ”active simultaneous
localization and mapping” after:priority:19920101 as queries search in the
Google patents search platform., about 39,000 for the former and 31 for the
latter. This indicates that the technology readiness level of active SLAM is
not in a deployment phase but in early development. Furthermore, it raises the
question of whether active SLAM is important for all applications or whether
human supervision is still preferred. Among the roadblocks preventing the
transition from theory to applications (including the challenges mentioned in
the previous sections), we also remark that the high computational complexity
of active SLAM often clashes with application constraints, e.g., the low
computational budget available on aerial robots.

##  X Conclusions

The active SLAM problem, which consists in actively controlling a robot such
that it can estimate the most accurate and complete model of the environment,
has been a topic of interest in the robotics community for more than three
decades, and is now receiving renewed attention —also thanks to the novel
opportunities offered by learning-based methods. Despite the role of active
SLAM in many applications, the disparity and lack of unification in the
literature has prevented the research community from providing a cohesive
framework, bringing algorithms to maturity, and transitioning them to real
applications. In this paper, we take a step toward this goal by taking a fresh
look at the problem and creating a complete survey to serve as a guide for
researchers and practitioners.

In particular, we present a unified active SLAM formulation under the umbrella
of POMDPs, highlighting the most common assumptions in the literature. Then,
we discuss the modular resolution scheme, which decouples the problem into
goal identification, utility computation, and action selection. We delve into
each stage, reviewing the most important theories and presenting state-of-the-
art techniques. We then review alternative approaches that have drawn great
interest and have undergone major advances in recent years, including
(continuous) BSP and learning-based approaches. Finally, we discuss relevant
work in multi-robot active SLAM.

Besides discussing the historical evolution and current trends in active SLAM,
we also identify the most relevant open challenges in this field. These
include prediction beyond line-of-sight and active spatial perception, among
others. We also emphasize the need for a unified formulation and evaluation
metrics that allow for direct comparison between works. Reproducibility and
benchmarking need to be addressed for this field to mature and achieve real-
world impact.

## References

  * [1] R. Siegwart, I. R. Nourbakhsh, and D. Scaramuzza, _Introduction to Autonomous Mobile Robots_. MIT press, 2011\.
  * [2] S. Thrun, W. Burgard, and D. Fox, _Probabilistic Robotics (Intelligent Robotics and Autonomous Agents)_. MIT Press, 2005.
  * [3] H. Durrant-Whyte and T. Bailey, “Simultaneous localization and mapping: Part I,” _IEEE Robotics & Automation Mag._, vol. 13, no. 2, pp. 99–110, 2006\.
  * [4] G. Grisetti, R. Kummerle, C. Stachniss, and W. Burgard, “A tutorial on graph-based SLAM,” _IEEE Intelligent Transportation Systems Mag._ , vol. 2, no. 4, pp. 31–43, 2010.
  * [5] C. Cadena, L. Carlone, H. Carrillo, Y. Latif, D. Scaramuzza, J. Neira, I. Reid, and J. J. Leonard, “Past, present, and future of simultaneous localization and mapping: Toward the robust-perception age,” _IEEE Trans. on Robotics_ , vol. 32, no. 6, pp. 1309–1332, 2016.
  * [6] R. Bajcsy, “Active perception vs. passive perception,” in _IEEE Workshop on Computer Vision_ , 1985.
  * [7] C. K. Cowan and P. D. Kovesi, “Automatic sensor placement from vision task requirements,” _IEEE Trans. on Pattern Analysis and machine intelligence_ , vol. 10, no. 3, pp. 407–416, 1988.
  * [8] J. Aloimonos, I. Weiss, and A. Bandyopadhyay, “Active vision,” _Int. J. of Computer Vision_ , vol. 1, no. 4, pp. 333–356, 1988.
  * [9] N. Atanasov, J. Le Ny, K. Daniilidis, and G. Pappas, “Information acquisition with sensing robots: Algorithms and error bounds,” in _IEEE Int. Conf. on Robotics and Automation_ , 2014, pp. 6447–6454.
  * [10] R. Bajcsy, “Active perception,” _Proceedings of the IEEE_ , vol. 76, no. 8, pp. 966–1005, 1988.
  * [11] C. Connolly, “The determination of next best views,” in _IEEE Int. Conf. on Robotics and Automation_ , vol. 2, 1985, pp. 432–435.
  * [12] J. Maver and R. Bajcsy, “Occlusions as a guide for planning the next view,” _IEEE Trans. on Pattern Analysis and Machine Intelligence_ , vol. 15, no. 5, pp. 417–433, 1993.
  * [13] P. Whaite and F. P. Ferrie, “Autonomous exploration: Driven by uncertainty,” _IEEE Trans. on Pattern Analysis and Machine Intelligence_ , vol. 19, no. 3, pp. 193–205, 1997.
  * [14] R. Pito, “A solution to the next best view problem for automated surface acquisition,” _IEEE Trans. on Pattern Analysis and Machine Intelligence_ , vol. 21, no. 10, pp. 1016–1030, 1999.
  * [15] R. Zeng, Y. Wen, W. Zhao, and Y.-J. Liu, “View planning in robot active vision: A survey of systems, algorithms, and applications,” _Computational Visual Media_ , pp. 1–21, 2020.
  * [16] D. Fox, W. Burgard, and S. Thrun, “Active Markov localization for mobile robots,” _Robotics and Autonomous Systems_ , vol. 25, no. 3-4, pp. 195–207, 1998.
  * [17] G. Borghi and V. Caglioti, “Minimum uncertainty explorations in the self-localization of mobile robots,” _IEEE Trans. on Robotics and Automation_ , vol. 14, no. 6, pp. 902–911, 1998.
  * [18] P. Jensfelt and S. Kristensen, “Active global localization for a mobile robot using multiple hypothesis tracking,” _IEEE Trans. on Robotics and Automation_ , vol. 17, no. 5, pp. 748–760, 2001.
  * [19] C. Mostegel, A. Wendel, and H. Bischof, “Active monocular localization: Towards autonomous monocular exploration for multirotor MAVs,” in _IEEE Int. Conf. on Robotics and Automation_ , 2014, pp. 3848–3855.
  * [20] S. K. Gottipati, K. Seo, D. Bhatt, V. Mai, K. Murthy, and L. Paull, “Deep active localization,” _IEEE Robotics and Automation L._ , vol. 4, no. 4, pp. 4394–4401, 2019.
  * [21] Q. Xie and Y. Wang, “A survey of filtering based active localization methods,” in _4th Int. Conf. on Big Data and Internet of Things_ , 2020, pp. 69–73.
  * [22] J. Strader, K. Otsu, and A.-a. Agha-mohammadi, “Perception-aware autonomous mast motion planning for planetary exploration rovers,” _J. of Field Robotics_ , vol. 37, no. 5, pp. 812–829, 2020.
  * [23] N. Roy, W. Burgard, D. Fox, and S. Thrun, “Coastal navigation-mobile robot navigation with uncertainty in dynamic environments,” in _IEEE Int. Conf. on Robotics and Automation_ , vol. 1, 1999, pp. 35–40.
  * [24] H. Carrillo, I. Reid, and J. A. Castellanos, “On the comparison of uncertainty criteria for active SLAM,” in _IEEE Int. Conf. on Robotics and Automation_ , 2012, pp. 2080–2087.
  * [25] S. B. Thrun and K. Möller, “Active exploration in dynamic environments,” _Advances in Neural Information Processing Systems_ , vol. 4, pp. 531–538, 1991.
  * [26] H. J. S. Feder, J. J. Leonard, and C. M. Smith, “Adaptive mobile robot navigation and mapping,” _The Int. J. of Robotics Research_ , vol. 18, no. 7, pp. 650–668, 1999.
  * [27] F. Bourgault, A. A. Makarenko, S. B. Williams, B. Grocholsky, and H. F. Durrant-Whyte, “Information based adaptive robotic exploration,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , vol. 1, 2002, pp. 540–545.
  * [28] A. A. Makarenko, S. B. Williams, F. Bourgault, and H. F. Durrant-Whyte, “An experiment in integrated exploration,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , vol. 1, 2002, pp. 534–539.
  * [29] C. Stachniss, D. Hahnel, and W. Burgard, “Exploration with active loop-closing for FastSLAM,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , vol. 2, 2004, pp. 1505–1510.
  * [30] P. Newman, M. Bosse, and J. Leonard, “Autonomous feature-based exploration,” in _IEEE Int. Conf. on Robotics and Automation_ , vol. 1, 2003, pp. 1234–1240.
  * [31] C. Stachniss, _Robotic Mapping and Exploration_. Springer, 2009, vol. 55.
  * [32] R. Platt Jr, R. Tedrake, L. Kaelbling, and T. Lozano-Pérez, “Belief space planning assuming maximum likelihood observations,” in _Robotics: Science and Systems_ , 2010.
  * [33] C. Stachniss and W. Burgard, “Mapping and exploration with mobile robots using coverage maps,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , vol. 1, 2003, pp. 467–472.
  * [34] R. Sim and N. Roy, “Global A-optimal robot exploration in SLAM,” in _IEEE Int. Conf. on Robotics and Automation_ , 2005, pp. 661–666.
  * [35] A. J. Davison and D. W. Murray, “Simultaneous localization and map-building using active vision,” _IEEE Trans. on Pattern Analysis and Machine Intelligence_ , vol. 24, no. 7, pp. 865–880, 2002.
  * [36] I. Lluvia, E. Lazkano, and A. Ansuategi, “Active mapping and robot exploration: A survey,” _Sensors_ , vol. 21, no. 7, p. 2445, 2021.
  * [37] M. Montemerlo, S. Thrun, D. Koller, B. Wegbreit _et al._ , “FastSLAM: A factored solution to the simultaneous localization and mapping problem,” in _18th AAAI Conf. on Artificial Intelligence_ , 2002, pp. 593–598.
  * [38] C. Stachniss, G. Grisetti, and W. Burgard, “Information gain-based exploration using Rao-Blackwellized particle filters,” in _Robotics: Science and Systems_ , vol. 2, 2005, pp. 65–72.
  * [39] C. Leung, S. Huang, and G. Dissanayake, “Active SLAM using model predictive control and attractor based exploration,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2006, pp. 5026–5031.
  * [40] R. Valencia, J. Valls Miro, G. Dissanayake, and J. Andrade-Cetto, “Active pose SLAM,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2012, pp. 1885–1891.
  * [41] V. Ila, J. M. Porta, and J. Andrade-Cetto, “Information-based compact pose SLAM,” _IEEE Trans. on Robotics_ , vol. 26, no. 1, pp. 78–93, 2009.
  * [42] L. Carlone, J. Du, M. K. Ng, B. Bona, and M. Indri, “Active SLAM and exploration with particle filters using Kullback-Leibler divergence,” _J. of Intelligent & Robotic Systems_, vol. 75, no. 2, pp. 291–311, 2014\.
  * [43] V. Indelman, L. Carlone, and F. Dellaert, “Planning in the continuous domain: A generalized belief space approach for autonomous navigation in unknown environments,” _The Int. J. of Robotics Research_ , vol. 34, no. 7, pp. 849–882, 2015.
  * [44] C. Zhu, R. Ding, M. Lin, and Y. Wu, “A 3D frontier-based exploration tool for MAVs,” in _IEEE 27th Int. Conf. on Tools with Artificial Intelligence_ , 2015, pp. 348–352.
  * [45] F. Endres, J. Hess, J. Sturm, D. Cremers, and W. Burgard, “3-D mapping with an RGB-D camera,” _IEEE Trans. on Robotics_ , vol. 30, no. 1, pp. 177–187, 2013.
  * [46] A. Bircher, M. Kamel, K. Alexis, H. Oleynikova, and R. Siegwart, “Receding horizon next-best-view planner for 3D exploration,” in _IEEE Int. Conf. on Robotics and Automation_ , 2016, pp. 1462–1468.
  * [47] M. Bloesch, S. Omari, M. Hutter, and R. Siegwart, “Robust visual inertial odometry using a direct EKF-based approach,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2015, pp. 298–304.
  * [48] C. Papachristos, S. Khattak, and K. Alexis, “Uncertainty-aware receding horizon exploration and mapping using aerial robots,” in _IEEE Int. Conf. on Robotics and Automation_ , 2017, pp. 4568–4575.
  * [49] H. Umari and S. Mukhopadhyay, “Autonomous robotic exploration based on multiple rapidly-exploring randomized trees,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2017, pp. 1396–1402.
  * [50] G. Grisetti, C. Stachniss, and W. Burgard, “Improved techniques for grid mapping with Rao-Blackwellized particle filters,” _IEEE Trans. on Robotics_ , vol. 23, no. 1, pp. 34–46, 2007.
  * [51] H. Carrillo, P. Dames, V. Kumar, and J. A. Castellanos, “Autonomous robotic exploration using a utility function based on Rényi’s general theory of entropy,” _Autonomous Robots_ , vol. 42, no. 2, pp. 235–256, 2018.
  * [52] M. Kaess, A. Ranganathan, and F. Dellaert, “iSAM: Incremental smoothing and mapping,” _IEEE Trans. on Robotics_ , vol. 24, no. 6, pp. 1365–1378, 2008\.
  * [53] M. G. Jadidi, J. V. Miro, and G. Dissanayake, “Gaussian processes autonomous mapping and exploration for range-sensing mobile robots,” _Autonomous Robots_ , vol. 42, no. 2, pp. 273–290, 2018.
  * [54] N. Palomeras, N. Hurtós, E. Vidal, and M. Carreras, “Autonomous exploration of complex underwater environments using a probabilistic next-best-view planner,” _IEEE Robotics and Automation L._ , vol. 4, no. 2, pp. 1619–1625, 2019.
  * [55] G. Grisetti, R. Kümmerle, H. Strasdat, and K. Konolige, “g2o: A general framework for (hyper) graph optimization,” in _IEEE Int. Conf. on Robotics and Automation_ , 2011, pp. 9–13.
  * [56] D. S. Chaplot, D. Gandhi, S. Gupta, A. Gupta, and R. Salakhutdinov, “Learning to explore using active neural SLAM,” in _Int. Conf. on Learning Representations_ , 2020.
  * [57] F. Niroui, K. Zhang, Z. Kashino, and G. Nejat, “Deep reinforcement learning robot for search and rescue applications: Exploration in unknown cluttered environments,” _IEEE Robotics and Automation L._ , vol. 4, no. 2, pp. 610–617, 2019.
  * [58] F. Chen, J. D. Martin, Y. Huang, J. Wang, and B. Englot, “Autonomous exploration under uncertainty via deep reinforcement learning on graphs,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2020, pp. 6140–6147.
  * [59] H. Li, Q. Zhang, and D. Zhao, “Deep reinforcement learning-based automatic exploration for navigation in unknown environment,” _IEEE Trans. on Neural Networks and Learning Systems_ , vol. 31, no. 6, pp. 2064–2076, 2020.
  * [60] K. Konolige, G. Grisetti, R. Kümmerle, W. Burgard, B. Limketkai, and R. Vincent, “Efficient sparse pose adjustment for 2D mapping,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2010, pp. 22–29.
  * [61] S. Suresh, P. Sodhi, J. G. Mangelson, D. Wettergreen, and M. Kaess, “Active SLAM using 3D submap saliency for underwater volumetric exploration,” in _IEEE Int. Conf. on Robotics and Automation_ , 2020, pp. 3132–3138.
  * [62] Y. Chen, S. Huang, and R. Fitch, “Active SLAM for mobile robots with area coverage and obstacle avoidance,” _IEEE/ASME Trans. on Mechatronics_ , vol. 25, no. 3, pp. 1182–1192, 2020.
  * [63] L. Zhao, S. Huang, and G. Dissanayake, “Linear SLAM: Linearising the SLAM problems using submap joining,” _Automatica_ , vol. 100, pp. 231–246, 2019\.
  * [64] A. Batinovic, T. Petrovic, A. Ivanovic, F. Petric, and S. Bogdan, “A multi-resolution frontier-based planner for autonomous 3D exploration,” _IEEE Robotics and Automation L._ , vol. 6, no. 3, pp. 4528–4535, 2021.
  * [65] W. Hess, D. Kohler, H. Rapp, and D. Andor, “Real-time loop closure in 2D lidar SLAM,” in _IEEE Int. Conf. on Robotics and Automation_ , 2016, pp. 1271–1278.
  * [66] J. A. Placed, J. J. G. Rodríguez, J. D. Tardós, and J. A. Castellanos, “ExplORB-SLAM: Active visual SLAM exploiting the pose-graph topology,” in _Iberian Robotics Conf._ Springer, 2023, pp. 199–210.
  * [67] R. Mur-Artal and J. D. Tardós, “ORB-SLAM2: An open-source slam system for monocular, stereo, and RGB-D cameras,” _IEEE Trans. on Robotics_ , vol. 33, no. 5, pp. 1255–1262, 2017.
  * [68] E. Bonetto, P. Goldschmid, M. Pabst, M. J. Black, and A. Ahmad, “iRotate: Active visual SLAM for omnidirectional robots,” _Robotics and Autonomous Systems_ , vol. 154, p. 104102, 2022.
  * [69] M. Labbé and F. Michaud, “RTAB-Map as an open-source lidar and visual simultaneous localization and mapping library for large-scale and long-term online operation,” _J. of Field Robotics_ , vol. 36, no. 2, pp. 416–446, 2019.
  * [70] L. P. Kaelbling, M. L. Littman, and A. R. Cassandra, “Planning and acting in partially observable stochastic domains,” _Artificial Intelligence_ , vol. 101, no. 1-2, pp. 99–134, 1998.
  * [71] O. Sigaud and O. Buffet, _Markov Decision Processes in Artificial Intelligence_. John Wiley & Sons, 2013\.
  * [72] K. J. Åström, “Optimal control of Markov processes with incomplete state information,” _J. of Mathematical Analysis and Applications_ , vol. 10, no. 1, pp. 174–205, 1965.
  * [73] M. Araya, O. Buffet, V. Thomas, and F. Charpillet, “A POMDP extension with belief-dependent rewards,” _Advances in Neural Information Processing Systems_ , vol. 23, 2010.
  * [74] B. Yamauchi, “A frontier-based approach for autonomous exploration,” in _IEEE Int. Symp. on Computational Intelligence in Robotics and Automation_ , 1997, pp. 146–151.
  * [75] R. Bormann, F. Jordan, W. Li, J. Hampp, and M. Hägele, “Room segmentation: Survey, implementation, and analysis,” in _IEEE Int. Conf. on Robotics and Automation_ , 2016, pp. 1019–1026.
  * [76] B. Mu, M. Giamou, L. Paull, A.-a. Agha-mohammadi, J. Leonard, and J. How, “Information-based active SLAM via topological feature graphs,” in _IEEE Conf. on Decision and Control_ , 2016, pp. 5583–5590.
  * [77] L. Fermin-Leon, J. Neira, and J. A. Castellanos, “TIGRE: Topological graph based robotic exploration,” in _European Conf. on Mobile Robots_ , 2017, pp. 1–6.
  * [78] N. Atanasov, J. Le Ny, K. Daniilidis, and G. J. Pappas, “Decentralized active information acquisition: Theory and application to multi-robot SLAM,” in _IEEE Int. Conf. on Robotics and Automation_ , 2015, pp. 4775–4782.
  * [79] A. Elfes, “Using occupancy grids for mobile robot perception and navigation,” _Computer_ , vol. 22, no. 6, pp. 46–57, 1989.
  * [80] H. P. Moravec, “Sensor fusion in certainty grids for mobile robots,” in _Sensor Devices and Systems for Robotics_. Springer, 1989, pp. 253–276.
  * [81] J. A. Placed and J. A. Castellanos, “Fast autonomous robotic exploration using the underlying graph structure,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2021, pp. 6649–6656.
  * [82] A. Hornung, K. M. Wurm, M. Bennewitz, C. Stachniss, and W. Burgard, “OctoMap: An efficient probabilistic 3D mapping framework based on octrees,” _Autonomous Robots_ , vol. 34, no. 3, pp. 189–206, 2013.
  * [83] E. Vespa, N. Nikolov, M. Grimm, L. Nardi, P. H. Kelly, and S. Leutenegger, “Efficient octree-based volumetric SLAM supporting signed-distance and occupancy mapping,” _IEEE Robotics and Automation L._ , vol. 3, no. 2, pp. 1144–1151, 2018.
  * [84] M. Muglikar, Z. Zhang, and D. Scaramuzza, “Voxel map for visual SLAM,” in _IEEE Int. Conf. on Robotics and Automation_ , 2020, pp. 4181–4187.
  * [85] N. Palomeras, M. Carreras, and J. Andrade-Cetto, “Active SLAM for autonomous underwater exploration,” _Remote Sensing_ , vol. 11, no. 23, p. 2827, 2019\.
  * [86] M. Selin, M. Tiger, D. Duberg, F. Heintz, and P. Jensfelt, “Efficient autonomous exploration planning of large-scale 3D environments,” _IEEE Robotics and Automation L._ , vol. 4, no. 2, pp. 1699–1706, 2019.
  * [87] D. Deng, Z. Xu, W. Zhao, and K. Shimada, “Frontier-based automatic-differentiable information gain measure for robotic exploration of unknown 3D environments,” _arXiv e-prints:2011.05288_ , 2020.
  * [88] A. Dai, S. Papatheodorou, N. Funk, D. Tzoumanikas, and S. Leutenegger, “Fast frontier-based information-driven autonomous exploration with an MAV,” in _IEEE Int. Conf. on Robotics and Automation_ , 2020, pp. 9570–9576.
  * [89] H. Oleynikova, Z. Taylor, M. Fehr, R. Siegwart, and J. Nieto, “Voxblox: Incremental 3D Euclidean signed distance fields for on-board MAV planning,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2017, pp. 1366–1373.
  * [90] K. Saulnier, N. Atanasov, G. J. Pappas, and V. Kumar, “Information theoretic active exploration in signed distance fields,” in _IEEE Int. Conf. on Robotics and Automation_ , 2020, pp. 4080–4085.
  * [91] S. L. Bowman, N. Atanasov, K. Daniilidis, and G. J. Pappas, “Probabilistic data association for semantic SLAM,” in _IEEE Int. Conf. on Robotics and Automation_ , 2017, pp. 1722–1729.
  * [92] L. Nicholson, M. Milford, and N. Sünderhauf, “Quadricslam: Dual quadrics from object detections as landmarks in object-oriented SLAM,” _IEEE Robotics and Automation L._ , vol. 4, no. 1, pp. 1–8, 2018.
  * [93] R. Eidenberger and J. Scharinger, “Active perception and scene modeling by planning with probabilistic 6d object poses,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2010, pp. 1036–1043.
  * [94] N. Atanasov, B. Sankaran, J. L. Ny, G. Pappas, and K. Daniilidis, “Nonmyopic view planning for active object classification and pose estimation,” _IEEE Trans. on Robotics_ , vol. 30, no. 5, pp. 1078–1090, 2014.
  * [95] M. Grinvald, F. Furrer, T. Novkovic, J. J. Chung, C. Cadena, R. Siegwart, and J. Nieto, “Volumetric instance-aware semantic mapping and 3D object discovery,” _IEEE Robotics and Automation L._ , vol. 4, no. 3, pp. 3037–3044, 2019.
  * [96] A. Rosinol, M. Abate, Y. Chang, and L. Carlone, “Kimera: An open-source library for real-time metric-semantic localization and mapping,” in _IEEE Int. Conf. on Robotics and Automation_ , 2020, pp. 1689–1696.
  * [97] J. McCormac, R. Clark, M. Bloesch, A. Davison, and S. Leutenegger, “Fusion++: Volumetric object-level SLAM,” in _IEEE Int. Conf. on 3D Vision_ , 2018, pp. 32–41.
  * [98] L. Zheng, C. Zhu, J. Zhang, H. Zhao, H. Huang, M. Niessner, and K. Xu, “Active scene understanding via online semantic reconstruction,” in _Computer Graphics Forum_ , vol. 38, no. 7, 2019, pp. 103–114.
  * [99] A. Asgharivaskasi and N. Atanasov, “Active Bayesian multi-class mapping from range and semantic segmentation observations,” in _IEEE Int. Conf. on Robotics and Automation_ , 2021, pp. 1–7.
  * [100] ——, “Semantic octree mapping and shannon mutual information computation for robot exploration,” _arXiv e-print:2112.04063_ , 2022.
  * [101] S. Thrun and A. Bücken, “Integrating grid-based and topological maps for mobile robot navigation,” in _13th AAAI Conf. on Artificial Intelligence_ , 1996, pp. 944–951.
  * [102] N. Tomatis, I. Nourbakhsh, and R. Siegwart, “Hybrid simultaneous localization and map building: A natural integration of topological and metric,” _Robotics and Autonomous Systems_ , vol. 44, no. 1, pp. 3–14, 2003.
  * [103] A. Rosinol, A. Violette, M. Abate, N. Hughes, Y. Chang, J. Shi, A. Gupta, and L. Carlone, “Kimera: From SLAM to spatial perception with 3D dynamic scene graphs,” _The Int. J. of Robotics Research_ , vol. 40, no. 12–14, pp. 1510–1546, 2021.
  * [104] C. Gomez, A. C. Hernandez, and R. Barber, “Topological frontier-based exploration and map-building using semantic information,” _Sensors_ , vol. 19, no. 20, p. 4595, 2019.
  * [105] W. Burgard, M. Moors, C. Stachniss, and F. E. Schneider, “Coordinated multi-robot exploration,” _IEEE Trans. on Robotics_ , vol. 21, no. 3, pp. 376–386, 2005.
  * [106] H. H. González-Banos and J.-C. Latombe, “Navigation strategies for exploring indoor environments,” _The Int. J. of Robotics Research_ , vol. 21, no. 10-11, pp. 829–848, 2002.
  * [107] B. Tovar, L. Munoz-Gómez, R. Murrieta-Cid, M. Alencastre-Miranda, R. Monroy, and S. Hutchinson, “Planning exploration strategies for simultaneous localization and mapping,” _Robotics and Autonomous Systems_ , vol. 54, no. 4, pp. 314–331, 2006.
  * [108] P. Quin, D. D. K. Nguyen, T. L. Vu, A. Alempijevic, and G. Paul, “Approaches for efficiently detecting frontier cells in robotics exploration,” _Frontiers in Robotics and AI_ , vol. 8, p. 1, 2021.
  * [109] M. Keidar and G. A. Kaminka, “Robot exploration with fast frontier detection: Theory and experiments,” in _11th Int. Conf. on Autonomous Agents and Multiagent Systems_ , 2012, pp. 113–120.
  * [110] ——, “Efficient frontier detection for robot exploration,” _The Int. J. of Robotics Research_ , vol. 33, no. 2, pp. 215–236, 2014.
  * [111] D. Holz, N. Basilico, F. Amigoni, and S. Behnke, “Evaluating the efficiency of frontier-based exploration strategies,” in _41st Int. Symp. on Robotics_ , 2010, pp. 1–8.
  * [112] C.-Y. Wu and H.-Y. Lin, “Autonomous mobile robot exploration in unknown indoor environments based on rapidly-exploring random tree,” in _IEEE Int. Conf. on Industrial Technology_ , 2019, pp. 1345–1350.
  * [113] W. Qiao, Z. Fang, and B. Si, “Sample-based frontier detection for autonomous robot exploration,” in _IEEE Int. Conf. on Robotics and Biomimetics_ , 2018, pp. 1165–1170.
  * [114] C. Dornhege and A. Kleiner, “A frontier-void-based approach for autonomous exploration in 3D,” _Advanced Robotics_ , vol. 27, no. 6, pp. 459–468, 2013.
  * [115] P. Senarathne and D. Wang, “Towards autonomous 3D exploration using surface frontiers,” in _IEEE Int. Symp. on Safety, Security, and Rescue Robotics_ , 2016, pp. 34–41.
  * [116] S. Shen, N. Michael, and V. Kumar, “Stochastic differential equation-based exploration algorithm for autonomous indoor 3D exploration with a micro-aerial vehicle,” _The Int. J. of Robotics Research_ , vol. 31, no. 12, pp. 1431–1444, 2012.
  * [117] L. Lu, C. Redondo, and P. Campoy, “Optimal frontier-based autonomous exploration in unconstructed environment using RGB-D sensor,” _Sensors_ , vol. 20, no. 22, p. 6507, 2020.
  * [118] R. Grabowski, P. Khosla, and H. Choset, “Autonomous exploration via regions of interest,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , vol. 2, 2003, pp. 1691–1696.
  * [119] A. Kim and R. M. Eustice, “Active visual SLAM for robotic area coverage: Theory and experiment,” _The Int. J. of Robotics Research_ , vol. 34, no. 4-5, pp. 457–475, 2015.
  * [120] J. Van Den Berg, S. Patil, and R. Alterovitz, “Motion planning under uncertainty using iterative local optimization in belief space,” _The Int. J. of Robotics Research_ , vol. 31, no. 11, pp. 1263–1278, 2012.
  * [121] L. Tai, G. Paolo, and M. Liu, “Virtual-to-real deep reinforcement learning: Continuous control of mobile robots for mapless navigation,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2017, pp. 31–36.
  * [122] D. S. Chaplot, E. Parisotto, and R. Salakhutdinov, “Active neural localization,” in _Int. Conf. on Learning Representations_ , 2018.
  * [123] J. A. Placed and J. A. Castellanos, “A deep reinforcement learning approach for active SLAM,” _Applied Sciences_ , vol. 10, no. 23, p. 8386, 2020.
  * [124] B. Charrow, G. Kahn, S. Patil, S. Liu, K. Goldberg, P. Abbeel, N. Michael, and V. Kumar, “Information-theoretic planning with trajectory optimization for dense 3D mapping,” in _Robotics: Science and Systems_ , vol. 11, 2015, pp. 3–12.
  * [125] F. Amigoni and A. Gallo, “A multi-objective exploration strategy for mobile robots,” in _IEEE Int. Conf. on Robotics and Automation_ , 2005, pp. 3850–3855.
  * [126] W. Chen and L. Liu, “Pareto monte carlo tree search for multi-objective informative planning,” in _Robotics: Science and Systems_ , 2019.
  * [127] A. Soragna, M. Baldini, D. Joho, R. Kümmerle, and G. Grisetti, “Active slam using connectivity graphs as priors,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2019, pp. 340–346.
  * [128] Y. Chen, L. Zhao, K. M. B. Lee, C. Yoo, S. Huang, and R. Fitch, “Broadcast your weaknesses: Cooperative active pose-graph SLAM for multiple robots,” _IEEE Robotics and Automation L._ , vol. 5, no. 2, pp. 2200–2207, 2020.
  * [129] M. Juliá, A. Gil, and O. Reinoso, “A comparison of path planning strategies for autonomous exploration and mapping of unknown environments,” _Autonomous Robots_ , vol. 33, no. 4, pp. 427–444, 2012.
  * [130] C. E. Shannon, “A mathematical theory of communication,” _The Bell System Technical J._ , vol. 27, no. 3, pp. 379–423, 1948.
  * [131] T. Cover, _Elements of Information Theory_. John Wiley & Sons, 1999.
  * [132] R. G. Colares and L. Chaimowicz, “The next frontier: Combining information gain and distance cost for decentralized multi-robot exploration,” in _ACM Symp. on Applied Computing_ , 2016, pp. 268–274.
  * [133] J. Vallvé and J. Andrade-Cetto, “Dense entropy decrease estimation for mobile robot exploration,” in _IEEE Int. Conf. on Robotics and Automation_ , 2014, pp. 6083–6089.
  * [134] J.-L. Blanco, J.-A. Fernandez-Madrigal, and J. González, “A novel measure of uncertainty for mobile robot SLAM with Rao—Blackwellized particle filters,” _The Int. J. of Robotics Research_ , vol. 27, no. 1, pp. 73–89, 2008.
  * [135] M. Popović, T. Vidal-Calleja, J. J. Chung, J. Nieto, and R. Siegwart, “Informative path planning for active field mapping under localization uncertainty,” in _IEEE Int. Conf. on Robotics and Automation_ , 2020, pp. 10 751–10 757.
  * [136] J. Wang and B. Englot, “Autonomous exploration with expectation-maximization,” in _Robotics Research_. Springer, 2020, pp. 759–774.
  * [137] A. J. Davison, “Active search for real-time vision,” in _IEEE Int. Conf. on Computer Vision_ , vol. 1, 2005, pp. 66–73.
  * [138] J. Du, L. Carlone, M. K. Ng, B. Bona, and M. Indri, “A comparative study on active SLAM and autonomous exploration with particle filters,” in _IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics_ , 2011, pp. 916–923.
  * [139] S. Kullback and R. A. Leibler, “On information and sufficiency,” _The Annals of Mathematical Statistics_ , vol. 22, no. 1, pp. 79–86, 1951.
  * [140] L. Mihaylova, T. Lefebvre, H. Bruyninckx, K. Gadeyne, and J. De Schutter, “A comparison of decision making criteria and optimization methods for active robotic sensing,” in _Int. Conf. on Numerical Methods and Applications_ , 2002, pp. 316–324.
  * [141] R. Houthooft, X. Chen, Y. Duan, J. Schulman, F. De Turck, and P. Abbeel, “VIME: Variational information maximizing exploration,” _Advances in Neural Information Processing Systems_ , vol. 29, 2016.
  * [142] D. Fox, “Adapting the sample size in particle filters through KLD-sampling,” _The Int. J. of Robotics Research_ , vol. 22, no. 12, pp. 985–1003, 2003.
  * [143] M. Kontitsis, E. A. Theodorou, and E. Todorov, “Multi-robot active SLAM with relative entropy optimization,” in _American Control Conf._ , 2013, pp. 2757–2764.
  * [144] D. Deng, R. Duan, J. Liu, K. Sheng, and K. Shimada, “Robotic exploration of unknown 2D environment using a frontier-based automatic-differentiable information gain measure,” in _IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics_ , 2020, pp. 1497–1503.
  * [145] D. S. Levine and J. P. How, “Sensor selection in high-dimensional Gaussian trees with nuisances,” _Advances in Neural Information Processing Systems_ , vol. 26, 2013.
  * [146] D. Kopitkov and V. Indelman, “No belief propagation required: Belief space planning in high-dimensional state spaces via factor graphs, the matrix determinant lemma, and re-use of calculation,” _The Int. J. of Robotics Research_ , vol. 36, no. 10, pp. 1088–1130, 2017.
  * [147] ——, “General-purpose incremental covariance update and efficient belief space planning via a factor-graph propagation action tree,” _The Int. J. of Robotics Research_ , vol. 38, no. 14, pp. 1644–1673, 2019.
  * [148] H. Chernoff, “Locally optimal designs for estimating parameters,” _The Annals of Mathematical Statistics_ , pp. 586–602, 1953.
  * [149] S. Ehrenfeld, “On the efficiency of experimental designs,” _The Annals of Mathematical Statistics_ , vol. 26, no. 2, pp. 247–255, 1955.
  * [150] A. Wald, “On the efficient design of statistical investigations,” _The Annals of Mathematical Statistics_ , vol. 14, no. 2, pp. 134–140, 1943.
  * [151] J. Kiefer, “General equivalence theory for optimum designs (approximate theory),” _The Annals of Statistics_ , pp. 849–879, 1974.
  * [152] L. Carlone and S. Karaman, “Attention and anticipation in fast visual-inertial navigation,” in _IEEE Int. Conf. on Robotics and Automation_ , 2017, pp. 3886–3893.
  * [153] H. Carrillo, Y. Latif, M. L. Rodriguez-Arevalo, J. Neira, and J. A. Castellanos, “On the monotonicity of optimality criteria during exploration in active SLAM,” in _IEEE Int. Conf. on Robotics and Automation_ , 2015, pp. 1476–1483.
  * [154] Y. Kim and A. Kim, “On the uncertainty propagation: Why uncertainty on Lie groups preserves monotonicity?” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2017, pp. 3425–3432.
  * [155] M. L. Rodríguez-Arévalo, J. Neira, and J. A. Castellanos, “On the importance of uncertainty representation in active SLAM,” _IEEE Trans. on Robotics_ , vol. 34, no. 3, pp. 829–834, 2018.
  * [156] C.-S. Cheng, “Maximizing the total number of spanning trees in a graph: Two related problems in graph theory and optimum design theory,” _J. of Combinatorial Theory, Series B_ , vol. 31, no. 2, pp. 240–248, 1981.
  * [157] K. Khosoussi, S. Huang, and G. Dissanayake, “Novel insights into the impact of graph structure on SLAM,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2014, pp. 2707–2714.
  * [158] K. Khosoussi, M. Giamou, G. S. Sukhatme, S. Huang, G. Dissanayake, and J. P. How, “Reliable graphs for SLAM,” _The Int. J. of Robotics Research_ , vol. 38, no. 2-3, pp. 260–298, 2019.
  * [159] Y. Chen, S. Huang, L. Zhao, and G. Dissanayake, “Cramér–Rao bounds and optimal design metrics for pose-graph SLAM,” _IEEE Trans. on Robotics_ , vol. 37, no. 2, pp. 627–641, 2021.
  * [160] J. A. Placed and J. A. Castellanos, “A general relationship between optimality criteria and connectivity indices for active graph-SLAM,” _arXiv e-prints:2110.01289_ , 2022.
  * [161] ——, “Enough is enough: Towards autonomous uncertainty-driven stopping criteria,” _IFAC-PapersOnLine_ , vol. 55, no. 14, pp. 126–132, 2022, 11th IFAC Symp. on Intelligent Autonomous Vehicles.
  * [162] A. Kitanov and V. Indelman, “Topological information-theoretic belief space planning with optimality guarantees,” _arXiv e-prints:1903.00927_ , 2019\.
  * [163] M. Shienman, A. Kitanov, and V. Indelman, “FT-BSP: Focused topological belief space planning,” _IEEE Robotics and Automation L._ , vol. 6, no. 3, pp. 4744–4751, 2021.
  * [164] S. M. LaValle and J. J. Kuffner Jr, “Randomized kinodynamic planning,” _The Int. J. of Robotics Research_ , vol. 20, no. 5, pp. 378–400, 2001.
  * [165] L. E. Kavraki, P. Svestka, J.-C. Latombe, and M. H. Overmars, “Probabilistic roadmaps for path planning in high-dimensional configuration spaces,” _IEEE Trans. on Robotics and Automation_ , vol. 12, no. 4, pp. 566–580, 1996\.
  * [166] S. Karaman and E. Frazzoli, “Sampling-based algorithms for optimal motion planning,” _The Int. J. of Robotics Research_ , vol. 30, no. 7, pp. 846–894, 2011.
  * [167] G. Oriolo, M. Vendittelli, L. Freda, and G. Troso, “The SRT method: Randomized strategies for exploration,” in _IEEE Int. Conf. on Robotics and Automation_ , vol. 5, 2004, pp. 4688–4694.
  * [168] L. Freda and G. Oriolo, “Frontier-based probabilistic strategies for sensor-based exploration,” in _IEEE Int. Conf. on Robotics and Automation_ , 2005, pp. 3881–3887.
  * [169] E. Bonetto, P. Goldschmid, M. J. Black, and A. Ahmad, “Active visual SLAM with independently rotating camera,” in _European Conf. on Mobile Robots_ , 2021, pp. 1–8.
  * [170] J. Van den Berg, S. Patil, and R. Alterovitz, “Efficient approximate value iteration for continuous Gaussian POMDPs,” in _26th AAAI Conf. on Artificial Intelligence_ , vol. 26, no. 1, 2012.
  * [171] O. Madani, S. Hanks, and A. Condon, “On the undecidability of probabilistic planning and infinite-horizon partially observable Markov decision problems,” in _16th AAAI Conf. on Artificial Intelligence_ , 1999, pp. 541–548.
  * [172] H. Bai, D. Hsu, and W. S. Lee, “Integrated perception and planning in the continuous space: A POMDP approach,” _The Int. J. of Robotics Research_ , vol. 33, no. 9, pp. 1288–1302, 2014.
  * [173] J. M. Porta, M. T. Spaan, and N. Vlassis, “Robot planning in partially observable continuous domains,” in _Robotics: Science and Systems_ , 2005, pp. 217–224.
  * [174] S. Prentice and N. Roy, “The belief roadmap: Efficient planning in belief space by factoring the covariance,” _The Int. J. of Robotics Research_ , vol. 28, no. 11-12, pp. 1448–1465, 2009.
  * [175] R. Valencia, M. Morta, J. Andrade-Cetto, and J. M. Porta, “Planning reliable paths with pose SLAM,” _IEEE Trans. on Robotics_ , vol. 29, no. 4, pp. 1050–1059, 2013.
  * [176] J. Le Ny and G. Pappas, “On trajectory optimization for active sensing in Gaussian process models,” in _IEEE Conf. on Decision and Control_ , 2009, pp. 6286–6292.
  * [177] S. Koga, A. Asgharivaskasi, and N. Atanasov, “Active exploration and mapping via iterative covariance regulation over continuous $SE(3)$ trajectories,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2021, pp. 2735–2741.
  * [178] B. Schlotfeldt, N. Atanasov, and G. J. Pappas, “Maximum information bounds for planning active sensing trajectories,” in _IEEE/RSJ Int. Conf. on Intelligent Robots and Systems_ , 2019, pp. 4913–4920.
  * [179] G. Hollinger and G. Sukhatme, “Sampling-based robotic information gathering algorithms,” _The Int. J. of Robotics Research_ , vol. 33, no. 9, pp. 1271–1287, 2014.