# Towards Energy Efficient Autonomous Exploration of Mars Lava Tube with a
Martian Coaxial Quadrotor

Akash Patel<EMAIL_ADDRESS>Samuel Karlsson Björn Lindqvist Christoforos
Kanellakis Ali-Akbar Agha-Mohammadi4 George Nikolakopoulos Robotics & AI Team,
Department of Computer, Electrical and Space Engineering,
Luleå University of Technology, Luleå SE-97187, Sweden

###### Abstract

Mapping and exploration of a Martian terrain with an aerial vehicle has become
an emerging research direction, since the successful flight demonstration of
the Mars helicopter Ingenuity. Although the autonomy and navigation capability
of the state of the art Mars helicopter has proven to be efficient in an open
environment, the next area of interest for exploration on Mars are caves or
ancient lava tube like environments, especially towards the never-ending
search of life on other planets. This article presents an autonomous
exploration mission based on a modified frontier approach along with a risk
aware planning and integrated collision avoidance scheme with a special focus
on energy aspects of a custom designed Mars Coaxial Quadrotor (MCQ) in a
Martian simulated lava tube. One of the biggest novelties of the article stems
from addressing the exploration capability, while rapidly exploring in local
areas and intelligently global re-positioning of the MCQ when reaching dead
ends in order to to efficiently use the battery based consumed energy, while
increasing the volume of the exploration. The proposed novel algorithm for the
Martian exploration is able to select the next way point of interest, such
that the MCQ keeps its heading towards the local exploration direction where
it will find maximum information about the surroundings. The proposed three
layer cost based global re-position point selection assists in rapidly
redirecting the MCQ to previously partially seen areas that could lead to more
unexplored part of the lava tube. The Martian fully simulated mission
presented in this article takes into consideration the fidelity of physics of
Mars condition in terms of thin atmosphere, low surface pressure and low
gravity of the planet, while proves the efficiency of the proposed scheme in
exploring an area that is particularly challenging due to the subterranean-
like environment. The proposed exploration-planning framework is also
validated in simulation by comparing it against the graph based exploration
planner. Intensive simulations with true Mars conditions are carried out in
order to validate and benchmark our approach in a utmost realistic Mars lava
tube exploration scenario using a Mars Coaxial Quadrotor.

###### keywords:

Frontier, Mars Exploration, Mars lava tube, Global re positioning

††journal: Journal of Advances in Space Researchfn1fn1footnotetext: This work
has been partially funded by the European Unions Horizon 2020 Research and
Innovation Programme under the Grant Agreement No. 869379
illuMINEation.fn2fn2footnotetext: The video link of the presented work:
https://youtu.be/9mc0UzAJC3Ufn3fn3footnotetext: This work was carried out at
Luleå University of Technology, Lulea, Sweden and not in the author’s capacity
as an employee of JPL, California Institute of Technology.fn4fn4footnotetext:
The author is an employee of Jet Propulsion Laboratory, California Institute
of Technology Pasadena, CA, 91109, USA

##  1 Introduction

For many years, Mars has been a prime candidate for terrestrial exploration in
the solar system. The findings of Mars exploration lenders and rovers have
paved a way for the future exploration of the planet with much ambitious
missions that also include aerial surveying and mapping of the terrain with a
helicopter and with increased levels of autonomy. The Mars exploration rovers
have presented limitations in terms of their pace, traversability, as well as
exploration capability from the ground. To address some of these limitations
and mainly for the technology demonstration purposes, a Mars helicopter named
Ingenuity (Balaram, Canham, Duncan, Grip, Johnson, Maki, Quon, Stern & Zhu,
2018) was sent to Mars with the Perseverance rover (Farley, Williford, Stack,
Bhartia, Chen, de la Torre, Hand, Goreva, Herd, Hueso et al., 2020). Recent
observations (Cushing, Titus, Wynne & Christensen, ; Cushing, 2012) from
operational satellites in Mars orbit have shown evidence of possible cave
skylights, subsurface voids and potential lava tubes like structures on Mars.
Exploration of such environment is particularly interesting because upon
surveying the environment with a robot, the structural integrity can be
inspected for possible habitation inside a cave or void. Traversing through a
cave like environment poses multiple challenges to wheel based rovers due to
the uneven ground, rocks and boulders. MAVs are particularly interesting
platforms because of the increased degrees of freedom, while providing a
faster exploration independent of the terrain but limited in time. Due to the
reason that very little or no information about the surrounding is available
at the start of the exploration, it is necessary that the MCQ is backed by a
sophisticated on board autonomy for guidance, navigation and control (GNC).

Fig. 1: Mars Lava Tube exploration mission configuration and overall MCQ
autonomy architecture

Among the essential components of such GNC architecture is the exploration
algorithm that feeds in the information about where the robot should navigate
next. Exploration task also involves incrementally building a map of the
environment as the robot navigates further. Occupancy grids composed of Voxels
or cells are used to represent the environment around the robot. Each cell has
a value of occupancy probability which represents if the cell is free,
occupied or unknown. Based on the occupancy probability of a particular cell
is defined to be frontier if it falls within the bounds set by the algorithm.
The exploration algorithm also iterates over frontiers which selects an
optimal frontier cell out of the list of frontiers. The optimal frontier is
selected such that if a robot navigates to that point, it will acquire maximum
information about the surrounding environment to extend the map. The method by
which an occupancy probability is assigned mainly depends on the sensor used
to map the area. Mainly these sensor feed point cloud data to the frontier
based exploration algorithm and an occupancy probability values is assigned to
each cell. This article presents a fully realistic simulated mission at a
Martian atmosphere and environment, in which a coaxial Quadrotor, designed
specifically to operate in Mars condition, explores a cave or lava tube like
environment consisting of multiple junctions, wide as well as confined
passages and unstructured interior that closely resembles an actual Martian
cave. The overall mission architecture with proposed autonomy workflow is
presented in Figure 1.

The proposed approach is also tested intensively in simulated exploration
missions and is compared against state of the art global exploration planner
for validating the energy efficient nature of the proposed scheme. The
exploration volume gain and ground covered by the MCQ is presented in Figure
2. The same is visible in Figure 3 where using the proposed approach the MCQ
explores larger part of the lava tube. The video of the proposed exploration
mission and multimedia material can be found at https://youtu.be/9mc0UzAJC3U.

Fig. 2: Exploration metrics comparison with state of the art global
exploration planner. Fig. 3: Explored lava tube and tracked trajectory of MCQ
using the proposed autonomy architecture. Compared against the graph based
planner. The sampling based planning approach shows instances of loops in one
place and in contrast, the proposed framework allows MCQ to efficiently
explore new areas.

##  2 Related Works

Mars exploration have been prime interest in terrestrial exploration of solar
system bodies due to its place in solar system and relatively easy access to
send autonomous robots for exploration and surveying of Martian terrain. As
mentioned earlier in section 1, the Mars helicopter’s ground breaking success
on first powered flight in thin Martian atmosphere has attracted ideas and
designs for future Unmanned Aerial Vehicles (UAVs) based surveying of Martian
terrain. In (Tzanetos, Aung, Balaram, Grip, Karras, Canham, Kubiak, Anderson,
Merewether, Starch et al., 2022a) A detailed study on the Mars helicopter’s
mission profiles, operations, anomalies of the design as well as lessons
learned are presented based on retrieved flight data. The studies presented
in(Tzanetos, Aung, Balaram, Grip, Karras, Canham, Kubiak, Anderson,
Merewether, Starch et al., 2022a) shows that the helicopter as an aerial scout
has helped the Perseverance rover in multiple instances to refine its planned
route to go around rocks and boulders that could be a potential hazard to
mission. The aerial images taken by the helicopter are laid over low
resolution images from satellite in Mars orbit to further evaluate future path
planning and steps for the rover. More interpretations of the flight data and
details on Ingenuity’s operations are presented in (Grip, Conway, Lam,
Williams, Golombek, Brockers, Mischna & Cacan, 2022), (Alibay, Koch, Verma,
Bean, Toupet, Petrizzo, Chamberlain-Simon, Lange & Hogg, 2022) and (von
Ehrenfried, 2022). More recent designs for future aerial exploration of
Martian terrain are presented in (Pipenberg, Langberg, Tyler & Keennon, 2022)
and (Lorenz, 2022) that include designs inspired from Ingenuity but with the
idea of hybrid motion of aerial exploration as well as wheeled locomotion for
sample retrieving purposes. Although the Mars helicopter has surpassed
exprectations in terms of performance and operations for a technology
demonstrator vehicle, the lessons learned from Ingenuity also show that it
lacks advanced autonomy architecture to perform exploration and surveying
independent of the commands from mother ship rover. In (Tzanetos, Bapst,
Kubiak, Tosi, Sirlin, Brockers, Delaune, Grip, Matthies, Balaram et al.,
2022b) a design for a Mars Science Helicopter vehicle is proposed with the
focus on advancing the operations carried out by Ingenuity but with a bigger
science payload and a longer range of flight. The vehicle proposed in
(Tzanetos, Bapst, Kubiak, Tosi, Sirlin, Brockers, Delaune, Grip, Matthies,
Balaram et al., 2022b) is a hex-coaxial based multi rotor vehicle designed to
operate in Martian atmosphere. However, from the best of our knowledge, the
autonomy enabling sensor suite and control architecture proposed in (Tzanetos,
Bapst, Kubiak, Tosi, Sirlin, Brockers, Delaune, Grip, Matthies, Balaram et
al., 2022b) is still using the same duo of Laser Range Finder (LRF) and down
facing gray-scale navigation camera as Ingenuity (Bayard, Conway, Brockers,
Delaune, Matthies, Grip, Merewether, Brown & San Martin, 2019) for guidance
and navigation purpose. In contrast, in our previous work (Patel, 2021),
(Patel, Banerjee, Lindqvist, Kanellakis & Nikolakopoulos, 2022a) a Mars
coaxial quadrotor vehicle named MCQ is proposed with advanced sensor suite for
navigation in subsurface void or potential lava tube on Mars. The subterranean
areas are not always well lit and therefore, vision-only based navigation
might be challenging to back a sophisticated on board GNC architecture. The
autonomy architecture on Ingenuity helicopter also relies on PID based casced
control for low level actuation and monocular vision + inertial measurements
unit based localization for performing autonomous flights. The design aspect
of rotor crafts for Mars condition is comprehensively discussed in the
literature (Johnson, Withrow-Maser, Young, Malpica, Koning, WJF, Fehler,
Tuano, Chan, Datta et al., 2020), however it lacks a dedicated advanced form
of control architecture for achieving higher levels of autonomy. In order to
address the shortcomings of the state of the art vehicle, a specially designed
Coaxial Quadrotor for Martian conditions and having a nonlinear model
predictive controller, is utilized for simulating a fully autonomous
exploration mission as it is proposed in this article. Moreover, a PID based
control might yield acceptable performance in position control in open areas
however, the targeted application for future aerial scouts are subterranean
lava tubes. Thus, advanced trajectory tracking, reactive navigation and
control architecture is absolute necessity for an autonomous vehicle to be
deployed in completely unknown areas. In response we propose a complete
exploration-planning-control architecture that considers energy efficiency
along with safety of the vehicle as ultimate constraints to simulate a
realistic lava tube exploration mission in Martian conditions.

In the context of subterranean exploration on another planet, there are
multiple driving factors to deploy autonomous robots to complete the task. The
factors include achieving a completely autonomous exploration and returning
with accurate 3D reconstructed map of the explored area while still
efficiently utilizing the available flight time by rapidly exploring unknowns.
Since robotic exploration of unknown area is a research direction that has
been pursued with many different approaches, a frontiers based approach has
been fundamental. The main idea of autonomous robotic exploration is to solve
the Next Best View (NBV) problem subject to adequate constraints that focus on
certain flight behaviour. For example, in this work the motive of the
exploration and planning architecture is to maximize exploration volume while
efficiently utilizing the energy resource. The exploration approaches in
literature can be classified in two high level strategies named as frontiers
based and sampling based approaches. In robotics, a frontier is defined as the
boundary between unknown and free space around the robot as it was initially
presented in (Yamauchi, 1997) for single robot and in (Yamauchi, 1998) for
multiple robots. In the initial work of frontiers based exploration, a closest
frontier from the mobile robot was chosen to navigate to followed by a 360∘
sensor sweep to detect further frontiers. Most of the modern frontiers based
approaches rely on similar approach (closest frontier) to select next goal for
the robot to visit with some additional modifications for example frontiers
clustering. The frontiers based approaches are further studies in (Juliá, Gil
& Reinoso, 2012) and (Holz, Basilico, Amigoni & Behnke, 2010) for comparisons
against various other exploration approaches. A frontiers clustering based 3D
exploration tool was proposed in (Zhu, Ding, Lin & Wu, 2015) that uses a
quadrotor equipped with a stereo visual camera for exploration. The approach
proposed in (Zhu, Ding, Lin & Wu, 2015) is based on selecting a closest
frontiers cluster from robots position for incremental exploration. However
selecting a closest frontier based approach in not optimal for the application
discussed in this work due to the fact that the closest frontier cluster might
not always lie in front of the aerial vehicle and therefore requiring
additional energy usage for subsequent yaw movements to navigate towards
closest frontier cluster. In subterranean exploration local way point
selection planner is presented in (Patel, Lindqvist, Kanellakis &
Nikolakopoulos, 2022b) where authors propose a local planner based on
computing safe look ahead poses for aerial robot while reactively navigating
in obstructed areas. More works in this direction also include (Fraundorfer,
Heng, Honegger, Lee, Meier, Tanskanen & Pollefeys, 2012) which is based on
vision-only based exploration-mapping scheme that also utilize Micro Aerial
Vehicle (MAV) to explore using continuously updating frontiers. A generic
exploration approach for UAVs is also discussed in (Cieslewski, Kaufmann &
Scaramuzza, 2017) that uses rapid frontiers selection approach in which
frontiers corresponding to high flight velocities are selected ahead of the
UAV. Such rapid frontier selection approach has shown relatively superior
exploration gain compared to the receding horizon next best view planner
(Bircher, Kamel, Alexis, Oleynikova & Siegwart, 2016). However, when there
exist no frontier ahead of the UAV, the method in (Cieslewski, Kaufmann &
Scaramuzza, 2017) switches to classical frontiers based approach which again
disregard the constraint of energy efficiency. Further studies in (Zhou,
Zhang, Chen & Shen, 2021) shows that rapid flight driven frontier selection
approach often gets stuck in maze or multi branches structure scenarios due to
switching to classical frontiers based approach at multiple instances. More
theoretical ideas of exploration including Stochastic Differential Equation
(SDE) based exploration approach is presented in (Shen, Michael & Kumar, 2012)
where expansion of system of particles with Newtonian dynamics for evolution
of SDE is simulated. The regions with high particle expansion are portrayed to
be unknown areas for the MAV to explore.

The second mainly researched direction in exploration is sampling based
approaches that often integrate exploration and path planning as coupled
problem to be solved. Such methods are often based on computing local goals
for exploration while maximizing information gain and at the same time
planning paths in order to minimize distance travelled, actuation cost or
similar metrics(Lindqvist, Agha-Mohammadi & Nikolakopoulos, 2021). These
sampling based approaches based on Rapidly-exploring Random Trees (RRT)
structures are extensively studied in both simulations and real life
experiments as presented in (Bircher, Kamel, Alexis, Oleynikova & Siegwart,
2016), (Pito, 1999) and (Dang, Mascarich, Khattak, Nguyen, Nguyen, Hirsh,
Reinhart, Papachristos & Alexis, 2020a). In (Sun, Wu, Xu & Kong, 2022) a
hybrid approach is discussed where, an adaptive frontier detector for rapid
exploration based on local sampling in RRT is presented. The approach has
bench marked the exploration in indoor, forest as well as multi story building
for improved exploration gain and sampling rate which is a challenge in RRT
based methods. However, multiple instances have been seen where the UAV goes
back and forth in local space while exploring and this is common challenge in
any sampling based approach. In (Dharmadhikari, Dang, Solanka, Loje, Nguyen,
Khedekar & Alexis, 2020) a motion primitives based exploration-planning
approach is presented where next exploration goal and subsequent path is
planned such that trajectory tracking can be improved for local exploration.
The work of (Dharmadhikari, Dang, Solanka, Loje, Nguyen, Khedekar & Alexis,
2020) is excellent candidate to be inline with the focus of energy efficient
exploration however, the method often gets stuck in local minima when
exploration gain is low and also due to the lack of dedicated global re-
positioning scheme that could be triggered automatically when local
exploration is insufficient as it will be presented in this work. Apart from
local exploration-planning methods, a global graph based planner is also
presented in (Dang, Mascarich, Khattak, Papachristos & Alexis, 2019), (Dang,
Tranzatto, Khattak, Mascarich, Alexis & Hutter, 2020b). So far the graph based
planner is the state of the art method in subterranean exploration of unknown
environments which has dedicated global re-positioning strategy. Therefore in
this work we have presented the comparisons of our work with (Dang, Tranzatto,
Khattak, Mascarich, Alexis & Hutter, 2020b) to evaluate exploration metric.
Apart from sampling and frontiers based approaches, there exist also hybrid
approaches. An information driven frontier exploration method for MAVs, which
uses a hybrid approach between control sampling and frontier based is
presented in (Dai, Papatheodorou, Funk, Tzoumanikas & Leutenegger, 2020). The
graph based strategy is also extended in terms of multi robot teamed
exploration of subterranean caves and tunnels as part of DARPA subterranean
challenge is presented in (Kulkarni, Dharmadhikari, Tranzatto, Zimmermann,
Reijgwart, De Petris, Nguyen, Khedekar, Papachristos, Ott et al., 2022a).
Exploration of unknown environments are also extended to legged or ground
robots. Probabilistic Local and Global Reasoning on Information roadMaps
(PLGRIM) as presented in (Kim, Bouman, Salhotra, Fan, Otsu, Burdick & Agha-
mohammadi, 2021), discusses a hierarchical value learning strategy for
autonomous exploration of large subterranean environments. The methodology
presented in (Kim, Bouman, Salhotra, Fan, Otsu, Burdick & Agha-mohammadi,
2021) uses a hierarchical learning to address local and global exploration of
large scale environments while focusing on near optimal coverage plans. A
Frontloaded Information Gain Orienteering Problem (FIG-OP) based strategy is
presented in (Peltzer, Bouman, Kim, Senanayake, Ott, Delecki, Sobue,
Kochenderfer, Schwager, Burdick et al., 2022) that uses topological maps to
plan exploration paths in fixed time budget exploration scenario. The method
presented in (Peltzer, Bouman, Kim, Senanayake, Ott, Delecki, Sobue,
Kochenderfer, Schwager, Burdick et al., 2022) is tested with ground robots in
multi-kilometers subterranean environment targeted at time constrained
exploration missions.

Apart from the exploration and planning modules, the third necessary component
of the autonomy architecture is a control scheme for the vehicle. The most
basic form of position control utilized by autonomous aerial vehicles is PID
control. The Mars helicopter also has nested low level control that is based
on a simple PID architecture. However, the position control using a PID method
does not always yield accurate tracking due to the overshoots in non smooth
trajectories. Other commonly used form of control strategies include Linear
Quadratic Regulator (LQR) and Model Predictive Control (MPC) based position
and trajectory tracking controls. Initial works on model predictive control
for quadrotor class of vehicle include (Bangura & Mahony, 2014),
(Abdolhosseini, Zhang & Rabbath, 2013), (Alexis, Nikolakopoulos & Tzes, 2014),
(Chikasha & Dube, 2017). More MPC based navigation scheme in subterranean
environments are proposed in (Mansouri, Kanellakis, Fresk, Lindqvist,
Kominiak, Koval, Sopasakis & Nikolakopoulos, 2020) and (Lindqvist, Mansouri &
Nikolakopoulos, 2020b). Separated from quad rotor vehicle control, recent
studies have presented more advanced form of control for space vehicles as
presented by authors in (Alsaade, Yao, Al-zahrani, Alzahrani & Jahanshahi,
2023b) for fixed-time attitude tracking control that utilizes berrier Lyapunov
functions with neural network. The extended form of neural adaptive fault-
tolerant control strategy is presented in (Alsaade, Yao, Al-Zahrani, Alzahrani
& Jahanshahi, 2022). The authors validate the proposed indirect neural
approximation based position and attitude tracking in presence of unknown
disturbances through extensive simulation studies to track freely tumbling
target in an orbit. Spacecraft control under uncertainties is a challenging
research direction pursued by novel control applications such as attitude
stabilization and vibration suppression for a flexible spacecraft as described
by authors in (Yao, Jahanshahi, Moroz, Alotaibi & Bekiros, 2022b). A neural
integral sliding mode based control scheme is proposed by authors in (Yao,
Jahanshahi, Bekiros, Mihalache & Alotaibi, 2022a) that establishes a finite
time attitude tracking of a spacecraft in presence of unknown uncertainties
and disturbances. The control method presented in (Yao, Jahanshahi, Bekiros,
Mihalache & Alotaibi, 2022a) also shows reduced computational complexity of
the control scheme due reduced number of learning parameters to be tuned
online. In (Alsaade, Jahanshahi, Yao, Al-zahrani & Alzahrani, 2023a) a neural
network based optimal mixed control scheme (targeting unknown uncertainties
and disturbances) for a custom modified UAV is proposed.

The related works in the field of autonomously exploring aerial vehicles have
laid the foundation for future aerial exploration of Mars. However, the
methods somehow lack the idea of preserving energy or efficient energy
utilization while exploring. Thus, through this work we present novel autonomy
framework to contribute towards solving challenge of energy efficient
autonomous exploration of potential lava tube on Mars. Based on the related
works, the key contributions of this work are presented in section 4.

##  3 Motivation

Martian terrain exploration with aerial robots is becoming emerging research
direction because it addressed the limitations of rovers in terms of
traversability and navigation speed, demanding novel solutions by
subdisciplines of Artificial Intelligence including navigation, exploration,
mapping, perception, planning and control. The aerial surveying further
complements the idea of exploration by not only being the aerial scouts for
rovers but also have the ability to enter potential subsurface voids or lava
tubes on Mars which are difficult or untraversable terrains for rovers. This
work is inspired from the successful flight attempts of Ingenuity, the Mars
helicopter which was sent to Mars along with the Perseverance rover. The
valuable research data collected from multiple flights of Ingenuity on Mars
have proved that powered flight in thin Martian atmosphere is possible.
However, the lessons learned from Ingenuity also show that flight time of such
vehicle is extremely limited and making the best out of limited flight time
directly relates with how efficiently the vehicle can utilize the available
energy from power source.

It is a major challenge for multi rotor vehicles to produce lift in thin
Martian atmospheric conditions. In design of such vehicles, this challenge is
addressed with combination of two factors that directly affect the lift
capacity and flight time of vehicle. The optimal trade-off between how fast
the rotors should spin and how big the rotors should be to generate enough
lift derives the flight time for such vehicles. The first factor (rotor
speeds) directly relates to the energy utilization from batteries and thus
proportionally the exploration volume and distance covered by a vehicle in the
limited flight time. The state of the art methods discussed in section 2 are
mainly classical frontiers or sampling based. The classical frontiers based
methods utilize a frontier selection based on minimum distance function which
result in multiple yaw movements for the aerial vehicle because the closest or
minimum distance frontier might not always lie in front of the vehicle and
therefore, exploration based on classical frontiers based method result in
multiple instances where aerial vehicle utilizes more energy for yaw
correction and resulting in low exploration volume gain in fixed time budged
based missions. Moreover in the classical approaches, the vehicle back tracks
previously visited positions in cases where the vehicle reaches dead end in
order to globally reposition to a partially seen area to continue exploration
in different branch of cave or lava tube. This often result in longer paths to
go back to junction and therefore proportionally utilizing more energy to
achieve same task. In contrast the proposed scheme utilizes a risk aware
expendable grid based planning method that computes shortest-collision-free
paths to navigate towards partially seen area in case of dead end. On the
other hand, sampling based approaches such as RRT inspired methods often lack
the ability of rapidly computing the future exploration goals and therefore,
often require the vehicle to hover at one place for a fraction of time while
the algorithm computes next goal and path. This result in explore-stop-plan-
explore flight behaviour which again disregard the idea of efficiently
utilizing the available flight time to maximize exploration.

The future Martian rotor crafts such as Mars Coaxial Quadrotor (MCQ) requires
intelligent rapid exploration strategies that prevents the vehicle to hover at
one place while planning future frontiers and path to the frontiers as well as
preventing motor saturation by minimizing yaw torque along vertical axis.
Therefore, the main motivation of this research work is to present a novel
exploration-planning autonomy architecture for MCQ that addresses the
efficient energy utilization for MCQ vehicle in a simulated Martian lava tube
exploration scenario. As it will be presented in next sections, we propose
energy preserving frontiers based exploration approach along with risk aware
planning, reactive navigation and control scheme for aerial exploration of a
potential lava tube on Mars.

##  4 Contributions

The exploration-planning architecture of this work is part of the development
efforts within the COSTAR team (Agha, Otsu, Morrell, Fan, Thakker, Santamaria-
Navarro, Kim, Bouman, Lei, Edlund et al., 2021), related with the DARPA Sub-T
competition (DARPA, 2020), while the applicability of the proposed scheme is
extended for planetary caves and lava tubes as subterranean environments. The
simulated exploration mission presented in this article is motivated from our
previous work on design and model predictive control of a coaxial quadrotor to
operate in Mars conditions. The MCQ used for simulation is presented in
(Patel, 2021). Based on the related works and existing approaches on aerial
robots for planetary exploration, the novel contributions of this work are
listed below.

The first contribution of this work is in relation with addressing the main
challenge on efficiently utilizing the energy resource of Mars Coaxial
Quadrotor (MCQ) while exploring. In response, this work establishes an
exploration and planning scheme for the MCQ that takes into consideration
minimizing the yaw movement of MCQ as well as computing next frontier and
planning the path to next frontier on the go. The back bone of the proposed
exploration-planning architecture is derived from the Look-Ahead Move-Forward
nature of flight. In order to rapidly move forward in an unknown environment,
the proposed work also establishes a two layer safe frontier detection and
selection approach. The first layer of exploration computes the occupancy
information of the surrounding, stored in an octree data structure format and
detects safe frontier points for the MCQ such that the MCQ can navigate to
such frontier in nearly straight path in a collision free manner. The second
layer deals with selecting an optimal frontier subject to constraints related
with preventing unnecessary iterations for fast processing and forcing the MCQ
to move forward as long as there is unknown space ahead of it by bifurcating
the frontiers into directly and indirectly accessible frontiers. As long as
there isn’t a dead end ahead of the MCQ, it prioritizes a frontier from
directly accessible frontiers set that require minimum yaw movement
corrections. In multi rotor systems, yaw movement tend to saturate the motors
by dawning excess energy from batteries and and therefore, exploring with
minimal yaw movements considerably helps the MCQ in efficient energy
utilization. The state of the art sampling based approaches for planning, work
in Explore-Stop-Plan-Explore manner. Moreover the multi rotor vehicle’s energy
is best utilized while flying at nearly constant speeds while avoiding
hovering in one place. In contrast, the in this work we propose an
exploration-planning architecture that rapidly computes frontiers and plans
paths while exploring forward. In simple words, avoiding the hovering in one
place to plan next steps.

There are moments in exploration of unknown subterranean environments, when
MCQ might reach a dead end of a lava tube. In such scenario the MCQ iterates
through the set of Indirectly accessible frontiers and computes for an optimal
frontier subject to minimizing energy usage to navigate to such frontier. We
propose a cost based indirectly accessible frontier that takes into
consideration yaw correction cost, euclidean distance as well as height
difference to avoid saturating the motors and to help in efficiently utilizing
energy resource.

Apart from the exploration algorithm, the proposed work also utilizes a risk
aware expendable grid based planning $D^{*}_{+}$ method along with Artificial
Potential Fields (APF) for redundant collision avoidance as well as a
Nonlinear Model Predictive control path following. We integrate the
exploration, planning as well as reactive navigation and control modules into
one autonomy framework designed specifically to address challenges of
autonomously exploring a lava tube structure beneath Martian surface. The
novelty of the proposed autonomy architecture stems from the redundancy and
dependency less overall work flow within the systems. The framework is
evaluated through intensive simulation studies and comparisons with state of
the art sampling based approach. The simulations are carried at true Mars
atmospheric conditions such as Martian air density, surface pressure and
gravity to evaluate the method in a truly realistic martian lava tube
environment. We have also constructed a custom Martian lava tube environment
which represent potential martian underground terrain for simulation purpose.
We have made the environment publicly available for fellow researchers to use.

The rest of the article is structured as follows. section 5 provides details
on the design and torque based dynamic model of a custom Martian Coaxial
Quadrotor used for simulation purposed. section 6 presents the proposed
frontier based exploration approach with focus on energy preserving frontier
generation. In section 7, simulation results are presented that prove the
efficacy of the proposed scheme. Finally section 8 provides a discussion with
concluding remarks and future directions of the proposed work.

##  5 Mars Coaxial Quadrotor (MCQ)

The Mars coaxial Quadrotor design optimizations are specifically done to
operate in the Mars thin atmosphere. The detailed analysis and modelling of
the Mars coaxial Quadrotor is presented in our previous work (Patel, 2021).
Due to the reason that the Mars coaxial Quadrotor is designed to operate in
thin atmosphere, it’s rotors have to spin at considerably high RPM in order to
produce the required thrust. Since the high RPM of the rotors by default
consume considerably high power from the batteries, it is important to
optimally utilize resources in terms of electrical power in an exploration
scenario. The model of the Mars coaxial Quadrotor is presented in Figure 4.

Fig. 4: Mars Coaxial Quadrotor (MCQ) used for the exploration mission.

The proposed exploration algorithm is based on minimizing the change in the
direction vector by forcing to select the frontier in the field of view. The
algorithm also implements a cost function based selection process when there
is no frontier lying in the field of view, which allows the MCQ to reject a
frontier for which the MCQ has to deviate significantly from the current
direction. Due to this novelty, the MCQ is able to maintain a stable flight,
avoiding aggressive maneuvers which eventually efficiently uses the limited
energy available from batteries. Due to the fact that Mars has thin atmosphere
and low surface pressure, the rotors of MCQ need to spin considerably faster
and thus, it is important to optimize the energy consumption by the Mars
Coaxial Quadrotor.

Fig. 5: Coaxial quadrotor system representation.

The torque based model of the coaxial Quadrotor is presented in Equation 1.
$T$ is the thrust force in the $Z$ direction, $\tau_{\phi}$, $\tau_{\theta}$
and $\tau_{\psi}$ are the roll, pitch and yaw torques respectively along the
$X_{\mathbb{B}},Y_{\mathbb{B}},Z_{\mathbb{B}}$ axis

$\begin{pmatrix}T\\\ \tau_{\phi}\\\ \tau_{\theta}\\\
\tau_{\psi}\end{pmatrix}=\begin{pmatrix}K_{T}&K_{T}&K_{T}&K_{T}&K_{T}&K_{T}&K_{T}&K_{T}\\\
0&0&dK_{T}&dK_{T}&0&0&-dK_{T}&-dK_{T}\\\
dK_{T}&dK_{T}&0&0&-dK_{T}&-dK_{T}&0&0\\\
-K_{D}&K_{D}&-K_{D}&K_{D}&-K_{D}&K_{D}&-K_{D}&K_{D}\end{pmatrix}\begin{pmatrix}\Omega_{1}^{2}\\\
\Omega_{2}^{2}\\\ \Omega_{3}^{2}\\\ \Omega_{4}^{2}\\\ \Omega_{5}^{2}\\\
\Omega_{6}^{2}\\\ \Omega_{7}^{2}\\\ \Omega_{8}^{2}\end{pmatrix}$

$=A\begin{pmatrix}\Omega_{1}^{2},\Omega_{2}^{2}...,\Omega_{8}^{2}\end{pmatrix}^{T}$
(1)

In Equation 1, the matrix $A\in\mathbb{R}^{4\times 8}$ is the allocation
matrix responsible for producing specific control inputs based on desired
state of the model, while $K_{T}$ and $K_{D}$ are the thrust and drag
coefficients and $\Omega_{i}^{2}$ is the rotor speed in $rad/s$. The
approximate values for $K_{T}$ and $K_{D}$ are presented in Table 1.

Table 1: Mars Coaxial Quadrotor and simulation model parameters Simulation Parameter | Value
---|---
Rotor inertia, $J$ ($Kg\cdot m^{3}$) |  4.240$e^{-4}$
Torque constant $k_{t}$ ($Nm/A$) |  0.010$e^{-3}$
Thrust coefficient $K_{T}$ | 0.60
Drag coefficient $K_{D}$ |  0.20$e^{-3}$
Density, $\rho$ (Kg/$m^{3}$) | 0.017
Static Pressure p (Pa) | 720
Operating Temperature T (K) | 223
Gas constant R ($m^{2}/s^{2}$/K) | 188.90
Dynamic viscosity $\mu$ (Ns/$m^{2}$) |  1.130$\cdot 10^{-5}$
Gamma $\gamma$ | 1.289

##  6 Incremental Frontiers Based Exploration

The proposed extended frontier based method uses occupancy grid maps as a
mapping algorithm, which can generate a 2D or 3D probabilistic map. A value of
occupancy is assigned to each cell that represents a cell to be either free or
occupied in the grid. Suppose $m_{x,y}$ is the occupancy of a cell at $(x,y)$
and $p(m_{x,y}\ |\ z^{t},x^{t})$ is the numerical probability, then by using a
Bayes filter (Chen et al., 2003) the odds of the cell to be occupied can be
denoted as:

$\frac{p(m_{x,y}\ |\ z^{t},s^{t})}{1-p(m_{x,y}\ |\
z^{t},s^{t})}=\frac{1-p(m_{x,y})}{p(m_{x,y}}\times\\\ \frac{p(m_{x,y}\ |\
z^{t-1},s^{t-1})}{1-p(m_{x,y}\ |\ z^{t-1},s^{t-1})}$

where $z^{t}$ represent a single measurement taken at the location $s^{t}$. In
order to construct a 3D occupancy grid an existing framework OctoMap (Hornung,
Wurm, Bennewitz, Stachniss & Burgard, 2013) based octree is formulated in this
work. In this case each Voxel is subdivided into eight Voxels until a minimum
volume defined as the resolution of the octree is reached. In the octree if a
certain volume of Voxel is measured and if it is occupied, then the node
containing that Voxel can be initialized and marked occupied. Similarly, using
the ray casting operation, the nodes between the occupied node and the sensor,
in the line of ray, can be initialized and marked as free. This leaves the
uninitialized nodes to be marked unknown until the next update in the octree.
The estimated value of the probability $P(n\ |\ z_{1:t})$ of the node $n$ to
be occupied for the sensor measurement $z_{1:t}$ is given by:

$P(n|z_{1:t})=[1+\frac{1-P(n|z_{t})}{P(n|z_{t})}\frac{1-P(n|z_{1:t-1})}{P(n|z_{1:t-1})}\frac{P(n)}{1-P(n)}]^{-1}$

where $P_{n}$ is the prior probability of node $n$ to be occupied. Let us
denote the occupancy probability as $P^{o}$.

$p^{Voxelstate}=\begin{cases}Free,&\text{if $P^{o}<P_{n}$}\\\
Occupied,&\text{if $P^{o}>P_{n}$}\\\ Unknown,&\text{if $P^{o}=P_{n}$}\\\
\end{cases}$

The proposed exploration algorithm is made up of three essential modules,
namely the frontier pose generation, accessible frontier selection and global
re positioning. The first module converts a point cloud into a Voxel grid $V$
defined as $V=\\{\vec{x}\\}$ based in the Octomap framework discussed earlier.
The next module generates frontiers based on the occupancy probability
function for each Voxel corresponding to each sensor measurement. The frontier
points are fed into the next module, which evaluates each frontier based on a
cost function and selects an optimal frontier point. This point is published
to the Mars Coaxial Quadrotor (MCQ) controller as a temporary goal point to
visit and the overall process continues for each sensor scan until no frontier
is remaining.

The classical definition of frontiers considers a cell in the Voxel grid to be
a frontier if at least one of it’s neighbour cell is marked as unknown. In the
real scenario, where the area that needs to be explored is vast (for example,
a lava tube stretching for multiple kilometers on Mars), it is practically
inefficient. Therefore in the proposed algorithm the number of unknown or free
neighbours can be set by user before the start of the exploration that reduces
the overall computational complexity by a significant margin. Another
modification compared to classical frontier approach is made by not allowing
any neighbour to be an occupied Voxel. This also allows constrained frontier
generation such that confined narrow passages (smaller than the safety margin
for the MCQ to traverse in) can be avoided and instead other areas can be
prioritised.

The third layer of frontier refinement is set as if a frontier is detected
within the sphere of radius $r$ (set by user at the start of exploration),
then the node is marked as free. The sphere of radius $r$ is assumed to be
known space around the drone such that $r<R$ where $R$ is the sensor range
configured in the octree formulation. This helps in limiting the undesired
change in vertical velocity of the MCQ if the frontier is detected above or
below the MCQ. The proposed algorithm uses a 3D Lidar to get the point cloud
and because of the limited vertical field of view of the 3D Lidar, the above
mentioned layer of refinement helps in an overall stable operation.

Algorithm 1 Frontier Point Detection

Input :  $Currentoctree,$
$n$ : Number of unknown or free neighbours required
$r$ : known boundary around MCQ
Output :  $\mathcal{F}$

1: for all $Cell:UpdatedCells()$ do

2: if $Cell.is.Free()$ then

3: if $Cell.distance()<r$ then

4: $i\leftarrow 0$

5: for all $Neighbours:Cell.Neighbour()$ do

6: if $Neighbour.is.Occupied()$ then

7: i = 0;

8: $break;$

9: else

10: $i\leftarrow i+1$

11: end if

12: end for

13: if $i>=n$ then

14: $\mathcal{F}.add(Cell);$

15: end if

16: end if

17: end if

18: end for

Fig. 6: Directly and Indirectly accessible frontiers classification. The
motive behind bifurcating frontiers in two sets is to prioritize unknown space
ahead of MCQ in a potential multi branched lava tube system.

The pseudo code for the frontier generation from octomap is depicted in the
Algorithm 1. $\mathcal{F}$ represent the set of frontiers generated in
Algorithm 1. In order to reduce the number of cells to be examined in each
update, only the cells whose state (free or occupied) is changed are examined.
For the selection of frontier, a directly accessible and indirectly accessible
set of frontiers are defined. For a large environment it is computationally
inefficient to go through each frontier, while evaluating the optimal frontier
to visit. Therefore, in the proposed method, the frontiers evaluation is done
in two stages. First the frontiers in the field of view in the direction of
the MCQ are added to the set which contain all frontiers that are directly
accessible from the MCQ’s current position and we define this set as
$\mathcal{D}$. The frontiers that are not in the field of view and the once
that are not directly accessible are then added top the set indirectly
accessible frontiers set and is denoted as $\mathcal{I}$. In the frontier
selection process as presented in algorithm 2, the frontiers in $\mathcal{D}$
are prioritized. When there is no frontiers in $\mathcal{D}$ frontier to
visit, a frontier from the $\mathcal{I}$ is selected based on the cost
function. The set of valid frontier which satisfy the requirements of having
$n$ unknown or free neighbours and which are out of sphere of radius $r$ as
formulated previously are then added to the set of valid frontiers denoted as
$\mathcal{V}$.

Algorithm 2 Frontier Classification

Input :  $\mathcal{F}$, $n$, $r$
$\alpha$ : frontier vector angle
Output :  $\mathcal{D}$, $\mathcal{I}$

1: for all $Frontier:\mathcal{F}$ do

2: if $(Frontier.distance<r)\ or\ (Frontier.height>h_{r})$ then

3: $i\leftarrow 0;$

4: for all $Neighbours:Frontier.Neighbour()$ do

5: if $Neighbour.is.Occupied()$ then

6: $i\leftarrow 0;$

7: $break;$

8: else

9: $i\leftarrow i+1;$

10: end if

11: end for

12: if $i<n$ then

13: $\mathcal{F}.remove(Frontier);$

14: end if

15: for all $Frontier:\mathcal{V}$ do

16: if $\alpha$ $\leq$ $\theta/2$ then

17: $\mathcal{D}.add(Frontier)$

18: else

19: $\mathcal{I}.add(Frontier)$

20: end if

21: end for

22: end if

23: end for

The pseudo code for the frontier classification, for both directly and
indirectly accessible frontiers, is presented in Algorithm 2. $\theta$ is 2D
horizontal field of view considered to classify directly accessible frontiers.
In vertical direction each frontier height is checked as described in
algorithm 2 and if frontier height exceeds then maximum height difference
bound then such frontier is not considered directly accessible frontier. In
algorithm 2, $h_{r}$ is the maximum relative height at which frontiers are
consideref to be accessible. Such classification also limits how much the MCQ
would move in vertical direction while exploring forward. A frontier from the
$\mathcal{D}$ set is selected based on a minimum angle $\alpha\in[-\pi,\pi]$
between then frontier vector $\vec{p^{f}}$ and the MCQ’s direction of travel
$\vec{V_{fwd}}$. In Figure 6, $(X_{\mathbb{B}},Y_{\mathbb{B}},Z_{\mathbb{B}})$
represent the body fixed coordinate frame. The frontiers from Octomap are
generated in the world frame but the frontier vector $\vec{x_{f}}$ is
calculated relative to the position of the MCQ. Let us denote $P^{mcq}_{k}$ as
the position of the MCQ at time instance $k$, where
$P^{mcq}=\\{x,y,z,\psi\\}$. Therefore we can compute the vector defining the
direction of motion of the MCQ as,

$V_{fwd}=(P^{MCQ}_{k+1}-P^{MCQ}_{k}).normalize()$ (2)

As shown in Figure 6, the angle $\alpha$ with respect to $\mathbb{B}$ can be
defined as:

$\alpha_{i}=cos^{-1}({\frac{\vec{p^{f}_{i}}\cdotp\vec{V_{fwd}}}{|\vec{p^{f}_{i}}|\cdotp|{\vec{V_{fwd}}|}}})$

Thus, by selecting the frontier with a lowest value of the angle $\alpha$, the
exploration can be continued with a relatively higher speed as the MCQ does
not have to change the direction of motion as long as a selected frontier
regulates the heading of the MCQ as it explores. When there is no frontier in
the $\mathcal{D}$ set then a frontier from $\mathcal{I}$ set is selected. The
selection of a frontier from $\mathcal{I}$ set can be tuned based on a cost
function as shown in Equation 3

$\mathbb{C}=\left|\sum_{i=1}^{n}(\mathcal{W}_{\alpha}\ \alpha_{i},\
\mathcal{W}_{h}\ \Delta H_{i},\ \mathcal{W}_{d}\ d_{i})\right|$ (3)

where, $d$ is the distance between the MCQ’s current position and the frontier
point, $\Delta H$ is the height difference between MCQ and the frontier point,
and $\mathcal{W}_{\alpha}$, $\mathcal{W}_{h}$ and $\mathcal{W}_{d}$ are
weights on the angle, height and distance difference respectively. The cost
for each valid frontier in the global set is calculated and the frontier with
a minimum cost value $\mathbb{C}$ from Equation 3 is selected as the next
frontier in order to globally re position the MCQ in case of a dead end or
inaccessible frontier. As discussed previously, the weights can be set before
the start of the exploration, which allows where the frontier will be selected
relative to the MCQ’s current position in the 3D space. For example, if a high
value of $\mathcal{W}_{h}$ is set, then the cost concerning a height
difference is increased, resulting in a frontier higher or lower than the MCQ
position in the $Z$ direction that would be less likely to be selected.

###  6.1 Autonomy framework for Exploration

The autonomy framework is composed of exploration, planning and control
modules and is presented in Figure 7. Furthermore, the considered exploration
mission relies on a 3D lidar data for frontier generation, as well as for
collision avoidance. As presented in the previous section 5, the nonlinear
model predictive control of the MCQ is achieved by allocating rotor speeds
converted from the desired thrust, roll, pitch and yaw. In Figure 7 the
complete framework of exploration and risk aware path planning is shown for an
autonomous exploration mission in a mars like environment. The overall high-
level autonomy architecture of the MCQ is divided into two main modules namely
exploration and planning-control. As presented in section 6, the exploration
module takes point cloud and generates an octree based on occupancy
information. The octree provides the voxel states denoted as $V^{state}$ to
the frontier detection module. The process is followed by the frontier
classification module which classifies the frontiers in $\mathcal{D}$ and
$\mathcal{I}$ based on algorithm 2. Based on the minimum angle $\alpha$ for
$p^{f}\in\mathcal{D}$ and minimum cost $\mathbb{C}$ for $p^{f}\in\mathcal{I}$
a candidate frontier $p^{f}_{candidate}$ is selected. As presented in Figure
7, $D^{*}_{+}$ takes the $p^{f}_{candidate}$ and plans a risk-aware path to
$p^{f}_{candidate}$. All consecutive points from the path planned by
$D^{*}_{+}$ are sent to APF (Artificial Potential Fields) module which shift
the way point if it lies close to an occupied point resembling the low level
obstacle avoidance incorporated within the planning step. The shifted way
point is sent to the nMPC controller which is responsible for actuating the
MCQ in order to move to the first point in path. The loop runs at 20 $Hz$
while the exploration is in progress. Detailed explanation of the planing-
control-avoidance architecture is presented in subsections 6.1.1,6.1.2 and
6.2.

Fig. 7: High level autonomy framework composed of the proposed frontiers based
approach along with risk-aware planning and reactive navigation-control sub
modules.

####  6.1.1 Nonlinear Model Predictive Control of the Coaxial Quadrotor

While there are many different control approaches to a MAV system, we will
utilize a NMPC for reference tracking and stable flight behavior, as
successfully utilized in previous applications for MAVs as in (Lindqvist,
Mansouri, Agha-mohammadi & Nikolakopoulos, 2020a), and can be found in
Equation 4. This model considers eight states in a yaw-compensated body-frame
namely: positions coordinates
$p^{\mathcal{B}}=[p_{x}^{\mathcal{B}},p_{y}^{\mathcal{B}},p_{z}^{\mathcal{B}}]^{\top}$,
linear velocities
$v^{\mathcal{B}}=[v_{x}^{\mathcal{B}},v_{y}^{\mathcal{B}},v_{z}^{\mathcal{B}}]^{\top}$
as well as roll and pitch angles $\phi$ and $\theta\in[-\pi,\pi]$, and
computes control inputs as
$T,\phi_{\mathrm{ref}},\theta_{\mathrm{ref}}\in\mathbb{R}$ where $T\geq 0$ is
the total mass-less thrust produced by the motors. Furthermore, the model
simplifies the MCQ dynamics by assuming the existence of a low-level attitude
controller and modelling the roll/pitch angles as first-order systems (let
$K_{\phi},K_{\theta}\in\mathbb{R}$ and
$\tau_{\phi},\tau_{\theta}\in\mathbb{R}$ denote the gains and time constants)
as to approximate the closed-loop behavior of the attitude controller (here
utilizing the the RotorS framework modified to fit the co-axial Quadrotor
(Furrer, Burri, Achtelik & Siegwart, 2016)). Based on this model, let us
define the state vector as
$x=[p^{\mathcal{B}},v^{\mathcal{B}},\phi,\theta]^{\top}$, and denote the
control inputs as: $u=[T,\phi_{\mathrm{ref}},\theta_{\mathrm{ref}}]^{\top}$.
Thus, the dynamic model of the MCQ can be provided as:

$\displaystyle\dot{p}^{\mathcal{B}}(t)=v^{\mathcal{B}}(t)$ (4a)
$\displaystyle\dot{v}^{\mathcal{B}}(t)=R(\phi,\theta)\left[\begin{smallmatrix}0\\\
0\\\ T\end{smallmatrix}\right]+\left[\begin{smallmatrix}0\\\ 0\\\
-g\end{smallmatrix}\right]-\left[\begin{smallmatrix}A_{x}&0&0\\\ 0&A_{y}&0\\\
0&0&A_{z}\end{smallmatrix}\right]v^{\mathcal{B}}(t),$ (4b)
$\displaystyle\dot{\phi}(t)=\nicefrac{{1}}{{\tau_{\phi}}}(K_{\phi}\phi_{\mathrm{ref}}(t)-\phi(t)),$
(4c)
$\displaystyle\dot{\theta}(t)=\nicefrac{{1}}{{\tau_{\theta}}}(K_{\theta}\theta_{\mathrm{ref}}(t)-\theta(t)).$
(4d)

The model is then discretized by the forward Euler method to achieve the
predictive form $x_{k+1}=\zeta(x_{k},u_{k})$.

The discrete model is used as the prediction model of the NMPC and the
prediction horizon is denoted by $N$. A cost is associated to the states and
control inputs at current time and using nonlinear optimizer, optimal set of
control actions can be optimized corresponding to the minimum cost. Let
$x_{k+i|k}$ represent the prediction state at time step $k+i$, produced at
time step $k$. The corresponding control action can be denoted as $u_{k+i|k}$.
Therefore, $\textbf{x}_{k}$ and $\textbf{u}_{k}$ can be denoted as the full
predicted states and control inputs. The cost function (Lindqvist, Mansouri,
Agha-mohammadi & Nikolakopoulos, 2020a) is defined as:

$\displaystyle\mathcal{C}(\textbf{x}_{k},\textbf{u}_{k},u_{k-1|k})=\sum_{i=0}^{N}\overbrace{\mathcal{W}_{x}{||x_{ref}-x_{k+i|k}||}^{2}}^{\text{state
cost}}+$
$\displaystyle\overbrace{\mathcal{W}_{u}{||u_{ref}-u_{k+i|k}||}^{2}}^{\text{input
cost}}+\overbrace{\mathcal{W}_{\Delta
u}{||u_{k+i|k}-u_{k+i-1|k}||}^{2}}^{\text{Input rate cost}}$

Where, $\mathcal{W}_{x}\in\mathbb{R}^{8\times 8}$ and
$\\{\mathcal{W}_{u},\mathcal{W}_{\Delta u}\\}\in\mathbb{R}^{3\times 3}$ are
adaptive weight matrices for the states, control inputs and input rates
respectively. In the above mentioned cost function, the state cost penalizes
the deviation from the reference trajectory, the input cost penalizes the
reference input needed for hovering and the input rate penalizes the control
input rates to provide a smooth trajectory to reach the desired state of the
system. The input rate cost allows the MCQ to avoid aggressive control
actions, which promotes stable operation while minimizing the change in
attitude of the MCQ. Additionally, we impose constraints on the inputs and
input-rates respectively. The constraints on control inputs are defined as
$u_{min}\leq u_{k+i|k}\leq u_{max}$ and the constraints on rate of control
input are defined as:

$\displaystyle|\phi_{(ref,k+i-1|k)}-\phi_{(ref,k+i|k)}|\leq\Delta\phi_{max}$
(5)
$\displaystyle|\theta_{(ref,k+i-1|k)}-\theta_{(ref,k+i|k)}|\leq\Delta\theta_{max}$
(6)

In Equation 6, $\Delta\phi_{max}$ and $\Delta\theta_{max}$ represent the
maximum allowed rate of change in roll and pitch for each time step. In order
to minimize the aforementioned cost function, the optimization problem is
solved using the Optimization Engine (Sopasakis, Fresk & Patrinos, 2020) where
a penalty method is used to compute solutions that satisfy the input rate
constraints.

####  6.1.2 Collision avoidance

While obstacle-free paths are provided by the $D^{*}_{+}$ module, the frontier
based exploration framework should still have decent level of local obstacle
avoidance in order to maintain a safe distance to obstacles and walls in the
environment, or as an additional safety layer in the case of error in
occupancy information within planning module. This work utilizes a purely
reactive 3D Artificial Potential Field (APF), described in more detail in
(lindqvist2021compra), relying only on the instantaneous pointcloud stream
from the 3D lidar. It is based on letting each point in the lidar pointcloud
within a specified volume defined by radius $r_{F}$ result in a repulsive
force, and then summing all such point-forces to get the total repulsive
force. Let us denote the local point cloud generated by the 3D lidar as
$\\{P\\}$, where all points are described by a relative position to the lidar
as $\rho=[\rho_{x},\rho_{y},\rho_{z}]$. Also denote the repulsive force as
$F^{r}=[F_{x}^{r},F_{y}^{r},F_{z}^{r}]$. Denote the list of points
$\boldsymbol{\rho}_{r}\in\\{P\\}$, where $\mid\mid\rho_{r}^{i}\mid\mid\leq
r_{F}$ and $i=0,1,\ldots,N_{\rho_{r}}$ (and as such $N_{\rho_{r}}$ is the
number of points to be considered for the repulsive force). The repulsive
force is
$F^{r}=\sum^{N_{\rho_{r}}}_{i=0}L(1-\frac{\mid\mid\rho_{r}^{i}\mid\mid}{r_{F}})^{2}\frac{-\rho_{r}^{i}}{\mid\mid\rho_{r}^{i}\mid\mid}$
where $L$ is the repulsive constant. The attractive force $F^{a}$ can be seen
as the vector from $\hat{p}^{\mathcal{B}}$ to the next way-point provided by
D∗-lite as $F^{a}=w_{1}^{\mathcal{B}}-\hat{p}^{\mathcal{B}}$. Additionally, we
impose saturation limits on the magnitude of forces, the rate of change of
forces and also normalizing the resulting total force $F$ as to always
generate forces of the same magnitude and to enforce stable and non-
oscillatory flight behavior. The output is the obstacle-free position
reference given to the NMPC as
$p_{ref}^{\mathcal{B}}=F+\hat{p}^{\mathcal{B}}$, where $p_{ref}^{\mathcal{B}}$
are the first three elements of $x_{\mathrm{ref}}$, such that
$x_{\mathrm{ref}}=[p_{ref,x}^{\mathcal{B}},p_{ref,y}^{\mathcal{B}},p_{ref,z}^{\mathcal{B}},0,0,0,0,0]^{\top}$.

###  6.2 Incrementally Built Map Based Planning

The path planning module is based on a modified version of D∗-lite (Sheckells,
2018) called D${}^{*}_{+}$ (DSP), and the occupancy mapper Octomap (Hornung,
Wurm, Bennewitz, Stachniss & Burgard, 2013). The modification to D∗-lite is to
its internal map representation ($\mathbb{G}$) where a representation for
unknown areas is added as well as a risk-cost associated to space close to
occupied spaces. By assigning different traversal costs $C$ to the risky
areas, DSP is made to plan the safest and shortest path ($P$) to the
destination, such that $P$ is the path in $\mathbb{G}$ where $\sum{\forall
C\in P}$ is minimized. Here DSP is used to plan the map-based obstacle-free
path from the agents current position to the position of the selected
frontier. $C$ is assigned to each Voxel sow that:

$C_{n}=\begin{cases}1,&\text{if $Free$}\\\ C_{o},&\text{if $Occupied$}\\\
C_{u},&\text{if $Unknown$}\\\ C_{n}+\frac{c_{u}}{d+1},&\text{if $d<r$ (only
added once per Voxel)}\end{cases}$

where $d$ is the distance in Voxels to the closes occupied Voxel, and $r$ is
the range that are consider risky from a occupied Voxel measured in Voxels.
Values of $C_{o}$ should be set to a significant higher value than $C_{u}$ is
set to. $C_{u}$ can in turn be set to a low value to encourage some
exploration during traversal of the path or to a higher to have safer paths
within known free space. In Figure 8 is an example path of how DSP plans $P$
in free space where it is possible and only enters a risky area when there is
no other option. This behavior is achieved with the gradient risk for
proximity to occupied Voxels. A frontier Voxel is marked inaccessible when the
size of the MCQ is higher than the distance between two occupied Voxels
through which a safe path to a frontier can be planned. Such formulation is
advantageous in discarding frontiers which require extremely risky path to
traverse to the point. The detailed study in risk aware occupancy information
based path planning is presented in our previous work (Karlsson, Koval,
Kanellakis, akbar Agha-mohammadi & Nikolakopoulos, 2021).

Fig. 8: A illustration of how DSP plans a path from the blue ball to the red
ball with regards to the risk layer, red voxels is higher risk, green voxels
is lower risk, and the black voxels are occupied.

##  7 Lava Tube Simulated Exploration Mission

A gazebo world is constructed to replicate true potential Mars lava tube
environment used in the simulated exploration mission in this work as
presented in Figure 9. We have made the Mars lava tube gazebo world available
for public use at: https://github.com/LTU-RAI/MarsLavaTubeWorld.git.

Fig. 9: Open sourced realistic lava tube environment created and used for the
Gazebo simulations.

The simulations are carried out using a true Mars atmospheric model in gazebo
in order to simulate exploration mission with utmost realistic scenario
corresponding to surface pressure and gravity. As described in section 5, we
use Mars Coaxial Quadrotor with a top mounted VLP-16 Lidar which has
$30^{\circ}$ vertical FOV and $360^{\circ}$ horizontal FOV. The simulations
are carried out on a Intel core i7 unit, with 32GB RAM. We use Rotor Simulator
(Furrer, Burri, Achtelik & Siegwart, 2016) with a custom model of Mars Coaxial
Quadrotor as a simulation framework. All visualization representations are
derived from Rviz. As presented in section 6, the proposed framework computes
for safe next best frontier in the field of view of the MCQ and the
corresponding local planning framework plans an obstacle free path to such
frontier pose. The same can be visualized in Figure 10. The green marker
represents the candidate frontier while the red path is the safe local path to
such $p^{candidate}$. When there exist no $safe$ directly accessible frontier
ahead of the MCQ, an indirectly accessible frontier is computed based on the
cost function described earlier in section 6. This also applies when
inaccessible frontiers are generated due to the uncertainty in octree
formation or simply because of the holes in Octomap. A global re-positioning
is triggered in such cases to re locate the MCQ to previously partially seen
areas of the lava tube. The same behaviour can be visualized in Figure 10.

Fig. 10: (a) Next Best frontier with respect to MCQ’s current position (b)
Inaccessible Frontier detection and and triggered homing maneuver.

The full (99$\%$) exploration of the Virtual Mars lava tube is achieved by the
proposed framework in a 1700 second exploration mission. As it is in all
exploration mission, at the beginning of exploration the MCQ has $zero$
information available about the environment. Using the proposed Incremental
Frontier Exploration approach with a global planing and collision avoidance
modules, the MCQ explores the lava tube environment. in Figure 11, a snap
shots of incremental map building is presented with the interval of 300
seconds. In Figure 11, at 10 seconds into the mission, the MCQ has just
started exploring the environment while at 1700 seconds the MCQ has almost
(99$\%$) explored the full lava tube.

Fig. 11: Incremental exploration of Mars lava tube using proposed autonomy
architecture.

Due to the complexity of the lava tube environment, there exist some parts of
the tube where the MCQ meets dead ends while exploring locally. At such points
it is essential that the MCQ does not spend more time hovering around a dead
end and autonomously re-position itself to a different part of the tube where
more information gain is believed. The proposed framework emphasise on global
re-positioning of the MCQ in such scenarios by computing a frontier from
$\mathcal{I}$ based on the traverse cost as well as actuation cost with
respect to the current location of the MCQ and potential global re positioning
poses. This behaviour is clearly visualized in Figure 6 in which as soon as
the MCQ meets a dead-end/inaccessible frontier and a global re-positioning
path (green) is computed which will re locate the MCQ to a previously
partially seen (while exploring) part of the lava tube. The proposed framework
also keeps track of globally leftover frontiers and at some point in the
exploration mission if the global leftover frontiers (previously partially
seen areas) as found in the field of view and if such frontiers are accessible
then they are merged with the current $\mathcal{D}$ frontiers set and the map
is merged at such points. This behaviour is presented in visualization form in
Figure 13. In such scenario after merging the frontiers, the MCQ re positions
itself to another part of the tube where it continues exploring the tube so
that rapid, energy efficient yet safe exploration mission behaviour can be
maintained.

Fig. 12: Global re-positioning due to inaccessible frontier (dead end) ahead
of the MCQ. Fig. 13: Global frontiers are merged when previously partially
seen area is discovered again.

In order to validate and benchmark the proposed rapid yet safe exploration
framework, we also compare our exploration results with the State-of-the-Art
Graph Based Planner (Dang, Tranzatto, Khattak, Mascarich, Alexis & Hutter,
2020b) which is revised and modified to use in real subterranean worlds by the
winning team at DARPA Sub-T Challenge (Kulkarni, Dharmadhikari, Tranzatto,
Zimmermann, Reijgwart, Petris, Nguyen, Khedekar, Papachristos, Ott, Siegwart,
Hutter & Alexis, 2022b). In order to make fair comparison we simulate multiple
Mars lava tube exploration mission using the both methods with exactly the
same parameters in terms of maximum forward velocity, sensor parameters,
mission time, simulation set up etc. Since the proposed work is addresses an
utmost autonomous exploration of Mars lava tube, we cap the exploration time
limit at 10 minutes and 15 minutes in two exploration scenarios for both
methods. This relates to a realistic MCQ flight duration and it yields real
exploration volume in constrained mission time. For both methods, autonomous
home return is triggered as soon as times exceeds the mission duration (10 and
15 minutes). In Figure 14, the part of the explored lava tube by the two
methods is shown where the mission duration is 10 minutes. The MCQ start and
return at the same position in both methods however, the path taken by the MCQ
differs due to the computed information gain or directly accessible frontier
by the two method differs. The similar mission is also carried out and the MCQ
trajectory and the explored lava tube is presented in Figure 15. It is evident
that the using the proposed method the MCQ covers more ground and also
explores more unexplored part of the lava tube in the same time as compared to
the graph based planner. In Figure 14 and Figure 15 it is also evident that
the MCQ trajectory tracked is also different due to the fact that the proposed
method implements a risk aware global planning method to next best view points
where as the graph based planner plans a safe trajectory to maximum
information gain nodes which often leaves the MCQ to go back and forth before
the global planner is triggered when local information gain is low.

Fig. 14: Mission duration : 10 min — Mars lava tube exploration comparison
against Graph Based Planner2. Fig. 15: Mission duration : 15 min — Mars lava
tube exploration comparison against Graph Based Planner2.

We also compute the volume of explored lava tube by our methods as well as
graph based planner. As it is presented in Figure 18, the exploration volume
is increased as the MCQ traverses through more unexplored areas in given time.
The proposed framework is developed with the focus of efficiently utilizing
the resource constrained MCQ’s energy while in flight by rapidly traversing
through unexplored areas. The expected behaviour is evident in Figure 18 as
using the proposed method the MCQ explores more volume of the environment. In
Figure 20 exploration metrics were derived based on the explored volume by the
two methods. Multiple simulation runs in different environment are evaluated
to make comparison show the higher exploration gain by proposed autonomy
framework compared to the graph based exploration planner. In given fixed time
budget based missions the proposed framework equipped MCQ also covers more
ground as it is evident in Figure 21. Moreover, the maximum velocity for
exploration corresponding to Figure 18 is 0.8 $m/s$ and the maximum allowed
velocity corresponding to the exploration metrics in Figure 20 are 1.5 $m/s$.
Which also evidently explains the change in explored volume over time. In the
distance metrics, the distance is calculated from the base. The decreasing
distance for gbplanner in Figure 21 correspond to looping behaviour of the
sampling based approach which further disregard the idea of efficiently
utilizing available energy to maximize exploration. In contrast, as presented
in section 6 the proposed approach is based on rapidly selecting future way
points and planning paths to such points on the go in order to avoid hovering
at same place. The effects on this on exploration volume gain and covered
ground is evident in Figure 20 and Figure 21. The graph based planner also
performs very well in terms of exploration volume in given time due to the
ability to re locate the MCQ to global frontier of interest when low local
information gain is encountered. Nearly same exploration volume gain observed
in the last 150 to 200 seconds because once the maximum exploration mission
reaches either 10 or 15 minutes, homing is triggered in both cases to bring
the MCQ back to start position. In Figure 16 we have also considered an
exploration mission in unconstrained time budged based scenarios to see the
evaluate energy efficient exploration by the two framework. It is evident that
even in longer missions, the proposed autonomy architecture enabled MCQ shows
higher exploration gain as well as covers more distance from the base as shown
in Figure 16. The simulations were carried out with same configurations
multiple times to validate the redundancy of the proposed system and the
explored parts of lava tube in four different runs is presented in Figure 17.

Fig. 16: Exploration mission comparisons in a non fixed time budget mission.
(a) Explored volume (b) Traveled distance from the base. Fig. 17: The explored
parts of lava tube and different paths planned by the MCQ in four different
exploration runs.

During the autonomous homing the MCQ travels back using the path computed in
map which was built while exploring thus, not leading to more new unexplored
places. We also compute the distance travelled by the MCQ in both methods in a
10 minute and 15 minute exploration missions. As described earlier, the
proposed framework is designed to rapidly explore Martian lava tube in order
to maximize the ground covered in given time. As shown in Figure 19 the
proposed framework allows the MCQ to cover more ground in a time constraint
based exploration scenario as compared to the graph based planner. The graph
based planner also shows a nearly linear increase in distance travelled by the
MCQ. However, the difference in covered distance in comparison is mainly due
to the $Plan\rightarrow\ Execute\rightarrow\ Stop\rightarrow\ Plan\rightarrow\
Execute...$ nature of the graph based planner. Since the gbplanner is designed
and tested intensively in field experiments, such planning nature has turned
out to be considerably safe for MAVs in subterranean exploration scenarios.
For fair comparison we also keep the maximum velocity of the MCQ to be exactly
the same ($1.5m/s$) as it plays big role in exploration volume gain. In order
to efficiently use the available energy of the MCQ it is important to maximize
the forward flight velocity while exploring. In order to demonstrate rapid
exploration behaviour the forward velocities of MCQ are plotted against the
simulation mission time using the data acquired from recorded rosbags using
both methods in 15 minute exploration. In Figure 24 and Figure 25 it is
evident that using the proposed method the MCQ aims at maintaining similar
forward velocity in order to efficiently cover more ground using available
energy. The state of the art method uses the RRT based nodes to compute
exploration gain thus the MCQ hovers at same place in between planning
iterations. The result of which is seen as high fluctuation in forward
velocity as presented in Figure 24. Moreover such behaviour is expected given
that the planner iteratively plans short yet safe segmented paths based on
local information gain as mentioned in (Dang, Tranzatto, Khattak, Mascarich,
Alexis & Hutter, 2020b). The negative values of forward velocity simply
indicate that the MCQ pitched backward to travel in reverse direction with
minimal change in heading. As discussed previously, the MAV’s energy is
consumed optimally by continuously flying forward at high speeds such that
more ground can be covered in given time. We demonstrate this by comparing the
accelerations samples and velocity samples during the exploration flight.
Figure 22 shows the velocity sampled during exploration. From Figure 22 it is
evident that the proposed framework shows more number of samples which
correspond to high forward velocities as compared to the graph based planner.
The proposed method also shows low samples which correspond to approximately
zero forward velocity (near hover point in flight) due to the compute-
frontier-plan-path while exploring nature of exploration of the proposed
approach. In addition to that Figure 23 shows the acceleration samples for
both methods. From Figure 23 is is clear that using proposed approach the MCQ
continuously flies forward as it explores exhibiting higher samples that
correspond to nearly zero accelerations. This clearly shows the exploration
performance of the proposed method in terms of efficiently utilizing the
energy of MCQ to explore the large lava tube environment on Mars.

Fig. 18: Exploration Volume gain for fixed-time budget based missions (a) 10
minute exploration mission (b) 15 minute exploration mission.

Fig. 19: Travelled Distance from the base during exploration (a) 10 minute
mission (b) 15 minute mission.

Fig. 20: Exploration Volume gain for 4 runs (a) Proposed approach (b) GB
planner.

Fig. 21: Travelled Distance from base during exploration for 4 runs (a)
Proposed approach (b) GB planner.

Fig. 22: (a) Comparisons based on Samples (Ns) corresponding to rapid
exploration behaviour at continuous forward velocity with minimal hovering.

Fig. 23: (a) Comparisons based on Samples (Ns) corresponding to change in
forward velocity over $\delta T$.

Fig. 24: (a) Forward Velocity of the MCQ (Proposed framework) (b) Forward
Velocity of the MCQ (GB Planner).

Fig. 25: (a) Average Velocity of the MCQ (Proposed framework) (b) Average
Velocity of the MCQ (GB Planner).

In order to show the nearly complete (99$\%$) exploration of the Mars lava
tube using the proposed method, the Octomap built during the exploration
mission is presented in Figure 26. The nearly complete exploration mission is
achieved in 1700 seconds using the proposed approach.

Fig. 26: nstructed occupancy based map of the explored area by the proposed
autonomy architecture on MCQ.  Fig. 27: Point cloud map and MCQ’s tracked
trajectory after 99$\%$ exploration of Mars lava tube environment using the
presented method.

##  8 Concluding Remarks and Future Directions

Subsurface voids and lava tubes are prime candidates for search of life on
Mars. Such sites are also areas of interest for potential habitats for future
colonization of Mars. Robotic exploration and surveying of planetary bodies
further strengthen our understandings of terrain in order to prepare humanity
for future Mars missions. The Mars helicopter surpassed the expectations by
proving the first powered flight in Martian thin atmosphere. The Mars
helicopter also proved to be an excellent candidate to be aerial scout for the
ground rovers on Mars in order to assist in planning safe paths. However, in
regard to the application of exploring subsurface voids and lava tube channels
on Mars, the technology demonstrator vehicle lacks autonomy in certain aspects
of independent decision making while exploring. In response, inspired from the
Ingenuity helicopter, this work focuses on the application of energy efficient
exploration of subterranean areas on Mars. The importance of efficiently
utilizing the energy from batteries to make the most out of the flight is
huge, considering the thin atmosphere and low surface pressure on Mars. This
work proposed an energy preserving incremental frontiers based exploration
approach coupled with risk-aware planning and advanced control scheme to
complement overall autonomy architecture of the Mars Coaxial Quadrotor (MCQ).
The proposed exploration approach is formulated with the goal of rapidly
exploring forward while minimizing motor saturating yaw movements. The
proposed approach considers directly accessible frontiers ahead of the MCQ to
prioritize forward exploration and also consist of a dedicated cost based
global re positioning scheme in order to navigate the MCQ to partially seen
area in case of dead end in exploration. The proposed scheme was further
validated through extensive simulations to evaluate the energy efficient
exploration and redundancy of the autonomy architecture. We also compare our
method with state of the art graph based exploration planner (designed for
subterranean exploration) in order to show the exploration gain and covered
ground in fixed time budged based exploration missions. The comparisons
clearly show the shortcomings of sampling based approach when energy efficient
exploration is targeted due to the lack of focused exploration goals
generation in one directions. Through comparisons, the proposed exploration
method also showed the impact on increased exploration gain by computing
future exploration goals and plan paths on the go. In conclusion, we propose
the autonomy architecture for a potential future Mars Coaxial Quadrotor that
considers energy efficient exploration with redundant systems and completely
autonomous operations in which mission plan could be loaded from Earth and the
MCQ autonomously executes exploration mission on Mars and returns with
accurate 3D map of the lava tube or subsurface void.

The design of the MCQ was proposed in our previous work which is still at
early stage in design for a space craft class vehicle to operate on Mars.
Therefore as part of future direction, the Design of vehicle to operate in
thin Martian atmosphere could be further improved with focus on increasing
payload size to equip more sophesticated sensor suite. Moreover, experimental
analysis of the vehicle to operate in controlled atmospheric chambers could be
performed to further optimize the design. The planned future directions also
include evaluating modern control frameworks by dropping the vehicle from
stratospheric balloons to evaluate hover and trajectory tracking of Mars
coaxial quadrotor at high altitudes that somewhat resemble actual flight
conditions on Mars. On the navigation side, the autonomy components could be
further evaluated in presence of external disturbances such as wind gusts etc.
The idea of swarms of Mars coaxial quadrotors but with smaller size vehicles
to perform multiple autonomy tasks to assist the rovers and other robots in
high level decision making process is one of the challenging yet highly
rewarding research direction that could benefit from the proposed autonomy
architecture.

##  9 Acknowledgement

This work has been partially funded by the European Unions Horizon 2020
Research and Innovation Programme under the Grant Agreement No. 869379
illuMINEation.

## References

  * Abdolhosseini et al. (2013) Abdolhosseini, M., Zhang, Y. M., & Rabbath, C. A. (2013).  An efficient model predictive control scheme for an unmanned quadrotor helicopter.  Journal of intelligent & robotic systems, 70(1), 27–38.
  * Agha et al. (2021) Agha, A., Otsu, K., Morrell, B. et al. (2021).  Nebula: Quest for robotic autonomy in challenging environments; team costar at the darpa subterranean challenge.  arXiv preprint arXiv:2103.11470, .
  * Alexis et al. (2014) Alexis, K., Nikolakopoulos, G., & Tzes, A. (2014).  On trajectory tracking model predictive control of an unmanned quadrotor helicopter subject to aerodynamic disturbances.  Asian Journal of Control, 16(1), 209–224.
  * Alibay et al. (2022) Alibay, F., Koch, J., Verma, V. et al. (2022).  On the operational challenges of coordinating a helicopter and rover mission on mars.  In 2022 IEEE Aerospace Conference (AERO) (pp. 1–17).  IEEE.
  * Alsaade et al. (2023a) Alsaade, F. W., Jahanshahi, H., Yao, Q. et al. (2023a).  A new neural network-based optimal mixed h2/h$\infty$ control for a modified unmanned aerial vehicle subject to control input constraints.  Advances in Space Research, .
  * Alsaade et al. (2022) Alsaade, F. W., Yao, Q., Al-Zahrani, M. S. et al. (2022).  Indirect-neural-approximation-based fault-tolerant integrated attitude and position control of spacecraft proximity operations.  Sensors, 22(5), 1726.
  * Alsaade et al. (2023b) Alsaade, F. W., Yao, Q., Al-zahrani, M. S. et al. (2023b).  Neural-based fixed-time attitude tracking control for space vehicle subject to constrained outputs.  Advances in Space Research, .
  * Balaram et al. (2018) Balaram, B., Canham, T., Duncan, C. et al. (2018).  Mars helicopter technology demonstrator.  In 2018 AIAA Atmospheric Flight Mechanics Conference (p. 0023).
  * Bangura & Mahony (2014) Bangura, M., & Mahony, R. (2014).  Real-time model predictive control for quadrotors.  IFAC Proceedings Volumes, 47(3), 11773–11780.
  * Bayard et al. (2019) Bayard, D. S., Conway, D. T., Brockers, R. et al. (2019).  Vision-based navigation for the nasa mars helicopter.  In AIAA Scitech 2019 Forum (p. 1411).
  * Bircher et al. (2016) Bircher, A., Kamel, M., Alexis, K. et al. (2016).  Receding horizon” next-best-view” planner for 3d exploration.  In 2016 IEEE international conference on robotics and automation (ICRA) (pp. 1462–1468).  IEEE.
  * Chen et al. (2003) Chen, Z. et al. (2003).  Bayesian filtering: From kalman filters to particle filters, and beyond.  Statistics, 182(1), 1–69.
  * Chikasha & Dube (2017) Chikasha, P. N., & Dube, C. (2017).  Adaptive model predictive control of a quadrotor.  IFAC-PapersOnLine, 50(2), 157–162.
  * Cieslewski et al. (2017) Cieslewski, T., Kaufmann, E., & Scaramuzza, D. (2017).  Rapid exploration with multi-rotors: A frontier selection method for high speed flight.  In 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (pp. 2135–2142).  IEEE.
  * Cushing (2012) Cushing, G. E. (2012).  Candidate cave entrances on mars.  Journal of Cave and Karst Studies, 74(1), 33–47.
  * (16) Cushing, G. E., Titus, T. N., Wynne, J. J. et al. ().  Themis observes possible cave skylights on mars.  Geophysical Research Letters, 34(17). URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2007GL030709. doi:https://doi.org/10.1029/2007GL030709. arXiv:https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2007GL030709.
  * Dai et al. (2020) Dai, A., Papatheodorou, S., Funk, N. et al. (2020).  Fast frontier-based information-driven autonomous exploration with an mav.  In 2020 IEEE International Conference on Robotics and Automation (ICRA) (pp. 9570--9576).  IEEE.
  * Dang et al. (2020a) Dang, T., Mascarich, F., Khattak, S. et al. (2020a).  Autonomous search for underground mine rescue using aerial robots.  In 2020 IEEE Aerospace Conference (pp. 1--8).  IEEE.
  * Dang et al. (2019) Dang, T., Mascarich, F., Khattak, S. et al. (2019).  Graph-based path planning for autonomous robotic exploration in subterranean environments.  In 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (pp. 3105--3112).  IEEE.
  * Dang et al. (2020b) Dang, T., Tranzatto, M., Khattak, S. et al. (2020b).  Graph-based subterranean exploration path planning using aerial and legged robots.  Journal of Field Robotics, 37(8), 1363--1388.  Wiley Online Library.
  * DARPA (2020) DARPA (2020).  DARPA Subterranean (SubT) challenge.  URL: https://www.darpa.mil/program/darpa-subterranean-challenge accessed: February 2021.
  * Dharmadhikari et al. (2020) Dharmadhikari, M., Dang, T., Solanka, L. et al. (2020).  Motion primitives-based path planning for fast and agile exploration using aerial robots.  In 2020 IEEE International Conference on Robotics and Automation (ICRA) (pp. 179--185).  IEEE.
  * von Ehrenfried (2022) von Ehrenfried, M. (2022).  Perseverance and the Mars 2020 Mission.  Jersey City: Springer.
  * Farley et al. (2020) Farley, K. A., Williford, K. H., Stack, K. M. et al. (2020).  Mars 2020 mission overview.  Space Science Reviews, 216(8), 1--41.
  * Fraundorfer et al. (2012) Fraundorfer, F., Heng, L., Honegger, D. et al. (2012).  Vision-based autonomous mapping and exploration using a quadrotor mav.  In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems (pp. 4557--4564).  IEEE.
  * Furrer et al. (2016) Furrer, F., Burri, M., Achtelik, M. et al. (2016).  Robot operating system (ros): The complete reference (volume 1).  chapter RotorS---A Modular Gazebo MAV Simulator Framework. (pp. 595--625).  Cham: Springer International Publishing.  URL: http://dx.doi.org/10.1007/978-3-319-26054-9_23. doi:10.1007/978-3-319-26054-9_23.
  * Grip et al. (2022) Grip, H. F., Conway, D., Lam, J. et al. (2022).  Flying a helicopter on mars: How ingenuity’s flights were planned, executed, and analyzed.  In 2022 IEEE Aerospace Conference (AERO) (pp. 1--17).  IEEE.
  * Holz et al. (2010) Holz, D., Basilico, N., Amigoni, F. et al. (2010).  Evaluating the efficiency of frontier-based exploration strategies.  In ISR 2010 (41st International Symposium on Robotics) and ROBOTIK 2010 (6th German Conference on Robotics) (pp. 1--8).  VDE.
  * Hornung et al. (2013) Hornung, A., Wurm, K. M., Bennewitz, M. et al. (2013).  Octomap: An efficient probabilistic 3d mapping framework based on octrees.  Autonomous robots, 34(3), 189--206.
  * Johnson et al. (2020) Johnson, W., Withrow-Maser, S., Young, L. et al. (2020).  Mars Science Helicopter Conceptual Design.  Technical Report 20200002139.
  * Juliá et al. (2012) Juliá, M., Gil, A., & Reinoso, O. (2012).  A comparison of path planning strategies for autonomous exploration and mapping of unknown environments.  Autonomous Robots, 33(4), 427--444.
  * Karlsson et al. (2021) Karlsson, S., Koval, A., Kanellakis, C. et al. (2021).  $d^{*}_{+}$: A generic platform-agnostic and risk-aware path planing framework with an expandable grid.  arXiv:2112.05563.
  * Kim et al. (2021) Kim, S.-K., Bouman, A., Salhotra, G. et al. (2021).  Plgrim: Hierarchical value learning for large-scale exploration in unknown environments.  In Proceedings of the International Conference on Automated Planning and Scheduling (pp. 652--662).  volume 31.
  * Kulkarni et al. (2022a) Kulkarni, M., Dharmadhikari, M., Tranzatto, M. et al. (2022a).  Autonomous teamed exploration of subterranean environments using legged and aerial robots.  In 2022 International Conference on Robotics and Automation (ICRA) (pp. 3306--3313).  IEEE.
  * Kulkarni et al. (2022b) Kulkarni, M., Dharmadhikari, M., Tranzatto, M. et al. (2022b).  Autonomous teamed exploration of subterranean environments using legged and aerial robots.  In 2022 IEEE International Conference on Robotics and Automation (ICRA).  Philadelphia (PA), USA: IEEE.
  * Lindqvist et al. (2021) Lindqvist, B., Agha-Mohammadi, A.-A., & Nikolakopoulos, G. (2021).  Exploration-rrt: A multi-objective path planning and exploration framework for unknown and unstructured environments.  In 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (pp. 3429--3435).  IEEE.
  * Lindqvist et al. (2020a) Lindqvist, B., Mansouri, S. S., Agha-mohammadi, A.-a. et al. (2020a).  Nonlinear mpc for collision avoidance and control of uavs with dynamic obstacles.  IEEE Robotics and Automation Letters, 5(4), 6001--6008.
  * Lindqvist et al. (2020b) Lindqvist, B., Mansouri, S. S., & Nikolakopoulos, G. (2020b).  Non-linear mpc based navigation for micro aerial vehicles in constrained environments.  In 2020 European Control Conference (ECC) (pp. 837--842).  IEEE.
  * Lorenz (2022) Lorenz, R. D. (2022).  Planetary Exploration with Ingenuity and Dragonfly: Rotary-Wing Flight on Mars and Titan.  Reston: American Institute of Aeronautics and Astronautics, Inc.
  * Mansouri et al. (2020) Mansouri, S. S., Kanellakis, C., Fresk, E. et al. (2020).  Subterranean mav navigation based on nonlinear mpc with collision avoidance constraints.  IFAC-PapersOnLine, 53(2), 9650--9657.
  * Patel (2021) Patel, A. (2021).  Design, Modelling and Control of a Space UAV for Mars Exploration.  Technical Report 82441 Lulea University of technology.
  * Patel et al. (2022a) Patel, A., Banerjee, A., Lindqvist, B. et al. (2022a).  Design and model predictive control of a mars coaxial quadrotor.  In 2022 IEEE Aerospace Conference (AERO) (pp. 1--11).  IEEE.
  * Patel et al. (2022b) Patel, A., Lindqvist, B., Kanellakis, C. et al. (2022b).  Fast planner for mav navigation in unknown environments based on adaptive search of safe look-ahead poses.  In 2022 30th Mediterranean Conference on Control and Automation (MED) (pp. 545--550).  IEEE.
  * Peltzer et al. (2022) Peltzer, O., Bouman, A., Kim, S.-K. et al. (2022).  Fig-op: Exploring large-scale unknown environments on a fixed time budget.  arXiv preprint arXiv:2203.06316, .
  * Pipenberg et al. (2022) Pipenberg, B. T., Langberg, S. A., Tyler, J. D. et al. (2022).  Conceptual design of a mars rotorcraft for future sample fetch missions.  In 2022 IEEE Aerospace Conference (AERO) (pp. 01--14).  IEEE.
  * Pito (1999) Pito, R. (1999).  A solution to the next best view problem for automated surface acquisition.  IEEE Transactions on pattern analysis and machine intelligence, 21(10), 1016--1030.
  * Sheckells (2018) Sheckells, M. (2018).  DSL Gridsearch: D*-lite on a uniformly spaced 3D or 2D grid.  URL: https://github.com/jhu-asco/dsl_gridsearch accessed: February 2021.
  * Shen et al. (2012) Shen, S., Michael, N., & Kumar, V. (2012).  Autonomous indoor 3d exploration with a micro-aerial vehicle.  In 2012 IEEE international conference on robotics and automation (pp. 9--15).  IEEE.
  * Sopasakis et al. (2020) Sopasakis, P., Fresk, E., & Patrinos, P. (2020).  Open: Code generation for embedded nonconvex optimization.  IFAC-PapersOnLine, 53(2), 6548--6554.
  * Sun et al. (2022) Sun, Z., Wu, B., Xu, C. et al. (2022).  Ada-detector: Adaptive frontier detector for rapid exploration.  arXiv preprint arXiv:2204.06237, .
  * Tzanetos et al. (2022a) Tzanetos, T., Aung, M., Balaram, J. et al. (2022a).  Ingenuity mars helicopter: From technology demonstration to extraterrestrial scout.  In 2022 IEEE Aerospace Conference (AERO) (pp. 01--19).  IEEE.
  * Tzanetos et al. (2022b) Tzanetos, T., Bapst, J., Kubiak, G. et al. (2022b).  Future of mars rotorcraft-mars science helicopter.  In 2022 IEEE Aerospace Conference (AERO) (pp. 1--16).  IEEE.
  * Yamauchi (1997) Yamauchi, B. (1997).  A frontier-based approach for autonomous exploration.  In Proceedings 1997 IEEE International Symposium on Computational Intelligence in Robotics and Automation CIRA’97.’Towards New Computational Principles for Robotics and Automation’ (pp. 146--151).  IEEE.
  * Yamauchi (1998) Yamauchi, B. (1998).  Frontier-based exploration using multiple robots.  In Proceedings of the second international conference on Autonomous agents (pp. 47--53).
  * Yao et al. (2022a) Yao, Q., Jahanshahi, H., Bekiros, S. et al. (2022a).  Indirect neural-enhanced integral sliding mode control for finite-time fault-tolerant attitude tracking of spacecraft.  Mathematics, 10(14), 2467.
  * Yao et al. (2022b) Yao, Q., Jahanshahi, H., Moroz, I. et al. (2022b).  Neural adaptive fixed-time attitude stabilization and vibration suppression of flexible spacecraft.  Mathematics, 10(10), 1667.
  * Zhou et al. (2021) Zhou, B., Zhang, Y., Chen, X. et al. (2021).  Fuel: Fast uav exploration using incremental frontier structure and hierarchical planning.  IEEE Robotics and Automation Letters, 6(2), 779--786.
  * Zhu et al. (2015) Zhu, C., Ding, R., Lin, M. et al. (2015).  A 3d frontier-based exploration tool for mavs.  In 2015 IEEE 27th International Conference on Tools with Artificial Intelligence (ICTAI) (pp. 348--352).  IEEE.