#  Spherical Rolling Robots Design, Modeling, and Control: A Systematic
Literature Review

Aminata Diouf
Corresponding author  Lab. INIT Robots
Department of Mechanical Engineering
École de technologie supérieure
Montréal, Canada
Email<EMAIL_ADDRESS>Bruno Belzile
Lab. INIT Robots
Department of Mechanical Engineering
École de technologie supérieure
Montréal, Canada
Email<EMAIL_ADDRESS>Maarouf Saad
Department of Electrical Engineering
École de technologie supérieure
Montréal, Canada
Email<EMAIL_ADDRESS>David St-Onge
Lab. INIT Robots
Department of Mechanical Engineering
École de technologie supérieure
Montréal, Canada
Email<EMAIL_ADDRESS>
###### Abstract

Spherical robots have garnered increasing interest for their applications in
exploration, tunnel inspection, and extraterrestrial missions. Diverse designs
have emerged, including barycentric configurations, pendulum-based mechanisms,
etc. In addition, a wide spectrum of control strategies has been proposed,
ranging from traditional PID approaches to cutting-edge neural networks. Our
systematic review aims to comprehensively identify and categorize locomotion
systems and control schemes employed by spherical robots, spanning the years
1996 to 2023. A meticulous search across five databases yielded a dataset of
3189 records. As a result of our exhaustive analysis, we identified a
collection of novel designs and control strategies. Leveraging the insights
garnered, we provide valuable recommendations for optimizing the design and
control aspects of spherical robots, supporting both novel design endeavors
and the advancement of field deployments. Furthermore, we illuminate key
research directions that hold the potential to unlock the full capabilities of
spherical robots.

##  1 Introduction

Spherical rolling robots (SRRs) are a fascinating category of robots
characterized by their ability to move by rolling on themselves, owing to
their unique spherical shape. However, beneath this seemingly simple concept
lies a plethora of sophisticated mechanisms and control strategies that enable
such motion. Nearly three decades ago, NASA introduced the idea of “Beach-
Ball” Robotic Rovers111https://ntrs.nasa.gov/citations/19950070425 for
planetary exploration, igniting the exploration of various systems in this
domain. Notably, the Rollo, designed in 1996 at Finland’s Helsinki University
of Technology [1], stands as one of the pioneering spherical robots aimed at
operating in hostile environments. The inherent protective nature of their
spherical shell renders them well-suited for challenging terrains,
safeguarding sensitive mechatronics, including sensors and actuators.
Consequently, their applications extend to underwater exploration [2], surveys
of dusty construction sites, tracking crop yields in muddy fields, and even
missions in extreme environments such as the moon, Mars, and beyond [3, 4].
Furthermore, spherical robots have also found utility in educational and
therapeutic contexts, particularly for children with developmental disorders
[5]; a market targetted by one of the only companies manufacturing primarily
spherical robots, Sphero.

Over the years, numerous researchers and companies have proposed diverse
designs, dynamic models, and control strategies for spherical rolling robots.
While previous reviews exist in the literature, they fail to encapsulate the
latest advancements in this field. For instance, a comprehensive examination
of rolling in robotics [6] delves into earlier designs of SRRs, while another
review [7] focuses primarily on the different actuation mechanisms specific to
spherical robots. However, since the publication of these reviews, several
novel designs have emerged, necessitating an updated analysis (see Fig. 1).
Although a recent review covered control algorithms [8], it provided limited
detail on the employed control strategies due to its concise nature as a
conference paper and failed to provide a holistic understanding of both
mechanical and control aspects of these robots.

Figure 1: Number of publications each year on the design of SRR (blue) and
their control strategies (red). The vertical lines highlight key events in the
recent SRR history: the previous literature reviews from Armour (2006) and
Chase (2012), the release of Sphero’s first product (2011), and the first
appearance of BB-8 in Star Wars movies (2015).

Figure 1 shows the trend in publication over the period covered in this review
(1996-2023) for both mechanical design and control strategies. Since 2010, an
increase in interest in SRR is observable. A common misconception is to grant
this popularity to the public apparition of BB-8, a fictional spherical robot
from the Star Wars movie franchise. However, we observe that the trend had
started before that, potentially inspiring the filmmakers. We can also observe
that Sphero’s first product release is right at the start of that new trend,
around 2011.

Spherical robots feature a range of internal mechanisms that enable their
movement, broadly categorized into three main groups: barycenter offset (BCO),
shell transformation, and conservation of angular momentum (CoAM). Barycentric
spherical robots (BSRs) manipulate the center of mass to achieve desired
motion, exemplified by wheeled mechanisms within a spherical shell or popular
pendulum-driven spherical robots. However, the torque capability of BSRs is
constrained as the center of gravity cannot be shifted beyond the shell. This
limitation can be circumvented through the utilization of control moment
gyroscopes (CoAM). Conversely, shell transformation, a less prevalent method,
involves deforming the outer body of the robot using wind, air, or water to
induce movement [7].

The highly nonlinear and non-holonomic nature of spherical robots presents
significant challenges in their control. Consequently, various control methods
have been proposed in the literature. They most often are designed to be fully
autonomous with only a few designed for teleoperation, to the extent of using
brain interfaces [9, 10]. When designed for autonomy, the majority of control
strategies can be achieved through nonlinear methods or local linear
approximations of the system. This review will show the proportion, advantages
and disadvantages of the control strategies found in the literature of SRR,
after assessing the underlying dynamic model differences.

In light of the aforementioned landscape, this paper aims to present a
systematic review of spherical rolling robots, encompassing the wide range of
actuation mechanisms and control strategies implemented in this domain.
Through comprehensive data gathering and analysis, the review will shed light
on research opportunities and identify blind spots. It is important to note
that the scope of this review is limited to robotic systems utilizing their
spherical shell for locomotion, as opposed to robots with a spherical shape
but employing limbs. Notably, this distinction arises in numerous publications
concerning amphibious and underwater spherical robots, where water jets or
propellers are utilized in aquatic environments while limbs are employed on
land [11]. Furthermore, this review focuses solely on active locomotion,
excluding spherical robots that rely on external forces, exemplified by the
NASA/JPL Tumbleweed polar rover [12].

The paper is structured as follows: Section 2 details our methodology,
inspired by PRISMA, then Section 3 presents the extracted designs, categorized
and compared, followed by Section 4 describing all the control strategies
found in our analysis. Finally, in Section 6, we shed light on potential
research opportunities and challenges that may unlock the full potential of
SRR.

##  2 Methodology

Table 1: Keywords prompt designed for each database of this literature review. Database |  Keywords prompt
---|---
Compendex |  (((((design OR control OR command) WN ALL) AND ((”spherical Robot*” OR “spherical roll* robot*”) WN ALL))) AND (english WN LA))
Web of Science |  (( ( design OR control OR command ) AND ( “spheric* Robot*” OR “spheric* roll* robot*” ) )) Timespan: All years. Databases: WOS, CCC, DIIDW, KJD, MEDLINE, RSCI, SCIELO. Search language=Auto
Science Direct |  (Design OR Control OR Command) AND (“spherical Robot” OR “spherical Robots” OR “spherical rolling robot” OR “spherical rolling robots”)
Scopus |  TITLE-ABS-KEY ( ( ( design OR control OR command ) AND ( “spheric* Robot*” OR “spheric* roll* robot*” ) ) ) AND ( LIMIT-TO ( LANGUAGE , “English” ) )
ProQuest |  ( ( design OR control OR command ) AND ( “spheric* Robot*” OR “spheric* roll* robot*” ) )

To ensure the utmost rigor and credibility of our analysis and results, we
conducted a literature review following the guidelines of the Preferred
Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) framework
[13]. Although initially developed for medical and pharmaceutical meta-
studies, PRISMA has gained traction and served as a valuable reference for
various fields in recent years. In adherence to this framework, our
methodology adhered to a comprehensive 27-item checklist encompassing key
aspects such as methods, results, and discussions.

###  2.1 Publication search and screening

Our study encompassed all research pertaining to spherical robots,
irrespective of publication year, with the caveat that the database content
before 1995 is limited. Additionally, we limited our search to English and
French publications. To compile our dataset, we performed thorough searches
across five information sources: ProQuest, Science Direct, Web of Science,
Engineering Village, and Scopus. Employing keywords such as ”design,”
”control,” ”spherical robots,” and ”rolling,” we identified a total of 3189
records. Subsequently, through a stringent screening process, we excluded
records that did not meet the predefined inclusion criteria. As a result, our
study included 126 papers concerning control strategies and 280 papers
addressing locomotion mechanism design.

During the eligibility assessment, we considered whether the spherical form
was an integral aspect of the movement generation, thereby excluding papers
where robots relied on external forces for propulsion. This criterion ensured
a focused review solely on spherical robots employing their spherical shell
for locomotion. To maintain an up-to-date and comprehensive analysis, we
encompassed publications from 1996 (no records available before that) up until
April 2023.

For each information source, the datasets utilized are presented in Tab. 1,
providing transparency and facilitating the replicability of our search
process. The study included 126 papers on control strategies for spherical
robots after screening and sorting through 3189 papers obtained from various
databases. The process from the start of the research to the inclusion of the
126 records is summarized in Fig. 2. An identical process was applied to the
mechanism design topics.

Figure 2: PRISMA 2020 flow diagram for new systematic reviews which included
searches of databases [13]

###  2.2 Data extraction

The included records were exported into a software for systematic literature
review, which allowed for screening and data extraction. A 7-item form was
created for data extraction, including questions about the control strategy
used, the objectives of the control method, the outcomes of the method, the
driving mechanism of the spherical robot, and whether a simulation or
experiment was performed to verify the control strategy:

  1. 1.

Which control strategy is used in the paper?

  2. 2.

Which dynamic modeling technique is used in the paper?

  3. 3.

What are the objectives of the control method?

  4. 4.

What are the outcomes of the method?

  5. 5.

What is the driving mechanism of the spherical robot in the study?

  6. 6.

What sensors have been used or considered?

  7. 7.

Is a simulation or an experiment performed to verify the control strategy?

Data extraction has been achieved in a redundant manner, in parallel by the
first two authors. The presentation and analysis of the content extracted are
covered in sections 3 and 4.

###  2.3 Bibliometric analysis of the keywords

To analyze the relationship between keywords, references from various
databases were imported into VosViewer [14], which generated a keyword
network. The final analysis included keywords that occurred more than 20 times
in the papers, and 57 keywords met this threshold. Keywords with a total link
strength of less than 600 were excluded from the analysis. The link strength
is the strength of the relationship between the different nodes. The most
frequent keywords were ”design,” ”spherical robots,” and ”control,” with total
occurrences of 2220, 1841, and 1643, respectively. The size of the node
indicated the frequency of a keyword, while the curves represented the co-
occurrence of keywords in the same publication. The distance between two nodes
determined the number of co-occurrences. The shorter that distance is, the
larger the number of co-occurrence will be [14]. Figure 3 shows the different
interactions between keywords. The analysis showed that there were more papers
on the design of a spherical robot than on its control, and sliding mode
control and PID were the most commonly used control strategies. The distance
from the spherical robot node to the control and from the spherical robot to
the design is the same as from control to design. Therefore, we can conclude
that several papers address both control and design aspects, which reinforces
the importance of conducting a holistic literature review like this one.

Figure 3: Co-occurrence analysis of the main keywords found in the
publications. Made with VOSviewer [14].

##  3 Locomotion, Design and Actuation

Our first pass on the large literature gathered focuses on their different
actuation mechanisms, which will in turn define the type of system and the
control variables. As shown in Fig. 4, the number of publications, at least
partially addressing the design of the actuation mechanism of an SRR,
increased significantly over the last two decades.

Figure 4: Number of included publications addressing the design of a SRR (some
may appear in more than one category, having multiple driving systems)

As mentioned above, their locomotion systems can be summarized in three broad
categories:

  1. 1.

barycentric;

  2. 2.

conservation of the angular momentum;

  3. 3.

shell deformation.

Considering these categories, the papers identified and their corresponding
devices were classified in Fig. 5. Since an SRR can be actuated with more than
one mechanism or with a system fitting in more than one of the above three
categories, the sum of robots listed in this figure is higher than the total
number of papers addressing the design of the driving mechanism. It should be
noted that the BSR category was separated into subcategories in Fig. 5, since
it is the most common type, by far. Moreover, colors used in Fig. 4 correspond
to those of Fig. 5.

Figure 5: Number of design publications sorted by types of mechanism (robots
with more than one actuation mechanism are counted more than once), purple for
BSRs, blue for others

###  3.1 Nonholonomic motion with SRRs

Holonomic and nonholonomic motion are concepts that are frequently mentioned
when it comes to mobile robots. Basically, a nonholonomic mobile robot’s state
depends on the path taken to reach it. In this case, the velocity constraint
defining its motion is non-integral, for instance with “rolling without
slipping condition” with wheels as they cannot slide sideways. Fundamentally,
with SRRs, the spherical shape of the shell makes it capable of holonomic
motion, as it can roll in any direction. However, it is typically no longer
the case when we consider the internal driving mechanism. Indeed, most cannot
make their external shell roll in any direction, particularly about their
vertical axis. As stated by Chase and Pandya [7], for “true holonomy, the
research challenge becomes developing an internal driving mechanism that can
provide omnidirectional output torque to a sphere that can arbitrarily rotate
around it, regardless of the orientation of either the sphere or the drive
mechanism.” This thus requires an internal driving mechanism capable of moving
in three dimensions independently from the spherical shell. According to our
review, this is not the case in the overwhelming majority of SSRs, with rare
exceptions, which is probably motivated by the fact that holonomic motion is
not necessarily needed for most situations, as it is for a car.

###  3.2 Barycentric

By far the most common type of spherical robots according to our assessment,
as shown in Fig. 5, barycentric spherical robots (BSRs) are driven by a
displacement of their center of mass (CoM). Indeed, by destabilizing the
system by moving the CoM away from its point of lowest potential energy
(typically directly underneath the center of rotation (CoR) of the sphere),
the shell starts rolling. BSRs can be classified into several subcategories:

  1. 1.

pendulum-based;

  2. 2.

IDU;

  3. 3.

sliding masses.

####  3.2.1 Pendulum-based

Among BSRs, pendulum-based BSRs are the most popular designs. The pendulum
typically, but not necessarily, rotates about a shaft passing through the
center of the sphere. For a robot to be able to move in more than one
direction, at least two DoFs are needed. This can be done with a 2-DoF
pendulum, which can rotate in two directions. Otherwise, more than one
pendulum can be used. For instance, on the one hand, Li et al. [15] and Zhao
et al. [16] proposed BSRs with two, while on the other hand, Dejong et al.
[17], four, to increase their maneuverability. This is obviously done at the
expense of more complex control schemes. Moreover, multiple pendulums do not
need to be fully independent of each other. For instance, Asiri et al. [18]
added a second, smaller pendulum orthogonal to the main pendulum rotating
about their spherical shell’s main axis. Otherwise, to steer its single-
pendulum BSR, Schroll [19] designed and patented a differential mechanism to
tilt the bob in a direction orthogonal to the rolling motion. While the main
axle (passing through the center of the sphere with both extremities rigidly
attached to the spherical shell) is usually used as the axis of the
pendulum(s), it is not always the case, as demonstrated by Zhao et al. [16]
which used circular guides carved into the ellipsoid222For the sake of this
review, papers presenting rare rolling robots with slightly ellipsoid shapes
were considered, as they behave mostly as spherical robots and their actuation
and control scheme can be applied to a perfectly spherical robot shell of its
device. However, having a pendulum only attached to the shell at two places
makes it easier to use a flexible shell, as shown by Ylikorpi et al. [20].
This characteristic is valuable to overcome small obstacles or steps with a
reduced impact on the trajectory, while also attenuating the oscillation due
to the nature of a pendulum’s motion. Regardless of the number of pendulums
and their designs, the rolling torque of a pendulum-based spherical robot is
bounded by the diameter of its shell and the mass of its pendulum. Even if the
actuator inside the sphere is capable of generating a larger torque, it will
only result in the spinning of the pendulum about its axis, which is not the
desired behavior. Moreover, this torque limit is also proportional to the
gravitational acceleration. To increase the distance between the center of the
sphere and the CoM, many authors located the battery powering their robot in
the lowest part of their pendulum, as done by Landa and Pilat [21].

While most pendulum-driven robots are incapable of holonomic motion, some have
omnidirectional capabilities, notably, the four-pendulum BSR proposed by
DeJong et al. [17]. In this particular case, this is possible because the four
pendulums have four different skew axes intersecting at the center of the
sphere.

An example of a pendulum-based spherical robot, known as ARIES, is depicted in
Figure 6. The robot incorporates an actuated cylindrical joint that serves as
a pendulum. This novel design allows for simultaneous rolling and steering by
utilizing a continuous differential transmission.

Figure 6: ARIES: a pendulum-based spherical robot [22]

####  3.2.2 Internal drive unit

Another type of BSR is those with an internal drive unit. Here, a torque is
transmitted to the outer shell through wheels in contact with its internal
surface. Sometimes considered separately from IDUs, the hamster ball system
defines the combination of a spherical shell and a smaller wheeled robot
inside the sphere, such as the one proposed by Alves and Dias [23]. In fact,
this type of actuation is among the earliest example of SRR because of its
simplicity. Obviously, the potential holonomy of a hamster-ball SRR is linked
directly to the holonomy of the internal robot. Recently, Belskii et al. [24]
proposed an omnidirectional IDU to be used inside a spherical robot. This
concept was not fully tested however inside a shell.

Hamster balls suffer from a major issue, namely slipping between the shell and
the wheels. To overcome this limitation, IDUs generally incorporate a system
to apply a force on the wheel, reducing the risk of slipping. For example,
Zhan et al. [25] developed the concept of an omnidirectional IDU with two
wheels based on the concept of barycenter offset. Their device consists of a
smaller wheel used to orient a larger one in contact with the internal surface
of the shell, allowing holonomic motion. Yet, these systems are still known to
suffer from slipping between the internal drive and the spherical shell [7],
limiting their robustness.

####  3.2.3 Sliding masses

Finally, some BSRs use sliding masses to control the location of the CoM [26,
27]. For instance, Javadi and Mojabi [28] used masses moving along prismatic
joints to modify the location of the CoM.However, they are generally, more
difficult to control than the two other subcategories described above. Indeed,
in the case of masses moving linearly [28], a continuous variation of the
position of the masses is required, even moving in a single direction,
contrary to pendulum-driven robots. This can be overcome by masses moving
along circular guides, as Tafrishi et al. [29] did when they investigated the
potential of a fluid-based BSR. They used masses moving inside pipes to
control the location of the CoM thus generating motion. However, this system
was not experimentally validated by the authors. As motors can generally be
used as generators to accumulate energy, it is also true with spherical
robots. Indeed, in applications/environments where minimizing energy
consumption is critical, recharging the battery may be useful, as is the case
with the robot developed by Zhai et al. [30]. The latter harvests energy while
the robot is pushed by the wind with electromagnets moving along pipelines.
This system can be used to actuate the robot by controlling the position of
the magnets instead of letting them move according to the dynamics of the
system.

###  3.3 Conservation of the angular momentum

Instead of moving the CoM for the rolling motion, another spherical robot
design is to leverage the conservation of the angular momentum without having
to displace the CoM. The rolling motion is generated using reaction wheels
[31, 32, 33] or control moment gyroscopes (CGM) [34, 35]. The latter can
provide more torque than the former, taking advantage of gyroscopic
precession. However, the torque output is not continuous, contrary to the
former. Moreover, reaction wheels must be fixed to the shell, rendering the
mechanical design and the control challenging [33]. Chase [36] partially
overcame this limitation by using four CGMs mounted in dual-scissored pairs.

One of the main differences between BSRs and those based on the conservation
of the angular momentum is the non-continuous torque generated for the rolling
motion.

###  3.4 Shell deformation

Last but not least, other types of locomotion based on deformable shells are
also possible. As can be seen in Fig. 5, they are far less common than
barycentric and angular-momentum-based SRRs. This category includes shape-
memory alloys and pressurized air bladders [37]. Their deformable spherical
shell can increase maneuverability, as shown by Sugiyama and Hirai [38] with
their robot capable of crawling and jumping. Generally able of holonomic
motion with a basic controller, they can be complex to design and more prone
to failure (i.e. unprotected moving parts).

###  3.5 Types and number of actuators

Most of the mechanisms discussed before can be actuated by different means,
namely steppers, servos, brushless, etc. We tried to extract the most common
types of actuators from the papers studied, unfortunately, only a few papers
provide this level of detail on their prototype. Among them, Chowdhury et al.
[39] utilized two Faulhaber DC gear drive motors, each with 30mNm/3V, to
control a 90-gram spherical robot. Similarly, [40] used two Maxon EC 45 flat
30W brushless motors with corresponding controllers to actuate a 10kg
spherical robot. They justified this choice based on the motors’ lightweight
nature and their ability to fit within the limited available space. In
contrast, Azizi and Keighobadi [41] proposed a design with three motors
driving three independent rotors. Similarly, Jia et al. [42] proposed a design
with three independent actuators, including one drive motor, one steer motor,
and one flywheel. The spinning flywheel enabled increased angular momentum of
the spherical robot. In [43], two stepper motors were used as actuators, where
one rotated the main shaft and the other enabled the rotation of the pendulum.
Additionally, four gears were utilized for power transmission. In a different
approach, [44] employed three separate motors in the internal mechanism: a
flywheel motor for increasing angular momentum, a long-axis motor for
generating driving forces, and a short-axis motor for controlling the
counterweight’s angle relative to the shell’s axis.

###  3.6 Design scale

The papers examined in this study featured spherical robots of different
scales, e.g. weight and dimensions. Figure 7 visually represents the masses
and radii of the described robots, where such information was available. It is
worth noting that some authors, as observed in [45, 46], employed the same
spherical robot in multiple papers. To ensure accuracy, we considered each
robot only once, resulting in a total of 58 distinct robots represented in the
figure.

Among the examined robots, the lightest variant, proposed in [47], weighed a
mere 55 grams, while the heaviest robot, documented in [48], weighed a
substantial 50 kilograms. The analysis revealed that the majority of the
examined robots weighed less than 10 kg, which can be attributed to their
primary usage in exploration scenarios. The benefits of lighter robots are
twofold: they require less energy to operate and are more manageable for
transportation purposes.

Furthermore, the investigation into the papers unveiled that the majority of
the robots possessed a radius ranging between 0.1 and 0.2 meters, with the
largest radius belonging to the device proposed by Rigatos et al. [49],
measuring 1.2 meters. In contrast, the smallest robot introduced by Nguyen et
al. [50], exhibited a radius of 30 cm. At this scale, the onboard computer and
sensing capacity are inherently limited, imposing constraints on the robot’s
capabilities.

Figure 7: Radius and mass of included mechanisms

##  4 Control and dynamics

During our thorough analysis of the wide array of publications encompassed in
this study, our subsequent phase focused on an examination the diverse range
of control strategies. In this context, Fig. 8 illuminates a significant
trend: the emergence of publications focusing on the control of spherical
robots began gaining prominence around 2012, culminating in a notable peak in
2019.

Figure 8: Number of papers addressing the control of a SRR following the most
used control strategies.

###  4.1 Dynamic modeling

Spherical robots possess a distinctive motion mechanism, resulting in
intricate kinematics and dynamics to model and analyze. The incorporation of
non-holonomic constraints, as discussed in Section 3.1, further amplifies the
complexity of these models. While certain papers, such as [51, 52, 53], rely
solely on the kinematic model for controlling spherical robots, most studies
employ a dynamic model due to the influence of conservative forces, such as
the pendulum gravity effect. However, when feasible, the kinematics-only
control approach offers a simplified depiction of the system’s behavior,
emphasizing motion or positional characteristics rather than intricate
dynamics.

The dynamics of a spherical robot can deviate from those of traditional robots
due to factors such as the shape of the spherical body, ground interactions,
and force or torque distributions, all of which impact its motion. Developing
a dynamic model is a crucial step prior to delving into the control of robotic
systems. In this research, various methods were employed to model spherical
robots.

Figure 9 illustrates the most commonly utilized techniques for modeling
spherical robots. The Euler-Lagrange method is extensively employed in
numerous papers for deriving the dynamic model. This method provides a
systematic approach by considering the system’s energy and serves as a
valuable tool in modeling mechanical systems. However, the presence of motion
constraints may require the use of Lagrange’s multipliers, which in turn adds
an additional computational burden for calculating the corresponding
constraint forces. The Newton-Euler method, widely utilized in robotics,
physics, control systems, and other fields requiring accurate dynamic
modeling, offers explicit equations of motion and a recursive formulation,
enabling detailed analysis and control of complex mechanical systems. Its
explicit representation of inter-body forces facilitates structural analysis
of all the components. Alternatively, system identification techniques can be
employed to create mathematical models describing a system’s behavior based on
input-output relationships, rather than relying on a physics-based
representation. This successful strategy has been implemented in [54, 55],
eliminating the need for explicit knowledge of the underlying physical
equations governing the system. However, system identification has its
limitations, such as the requirement for high-quality data, the potential for
modeling errors if the data is unrepresentative, and the need for careful
experimental design.

In [41], the Kane method was utilized to obtain the dynamic model of a
spherical robot. Although this method may be more complex to implement
compared to simpler approaches like the Newton-Euler method, it offers
significant advantages in terms of precision and representation, particularly
for complex mechanical systems or detailed analyses. In [56] a comparison of
the different methods for modeling a Small-size Helicopter has been done. The
results showed that Kane’s method offers unique advantages compared to
traditional approaches. It incorporates constraints using generalized
coordinates and speeds, resulting in concise equations. Constraint forces are
considered from the start, and the method is known for its computational
efficiency. However, implementing Kane’s method necessitates a solid
understanding of multibody system dynamics.

Figure 9: Dynamic modeling methods found in SRR papers from 1996 to 2023. The
Euler-Lagrange method largely predominates the others.

###  4.2 Control strategies

A total of 33 control strategies were identified in this study, with sliding
mode-based control, PID, PD, fuzzy control, and feedback linearization being
the most prevalent. However, these strategies are often enhanced through the
incorporation of additional methods, such as fuzzy control or neural network
algorithms, into traditional control approaches. Other strategies, such as PD,
feedback linearization, and backstepping, are also reported.

This section provides a comprehensive listing and detailed explanation of the
various control strategies. Some strategies combine two or three control
methods, while other papers independently investigate multiple control
strategies for the purpose of comparison. Figure 10 provides a summary of the
different control strategies documented in the papers.

Among the designs used, the barycentric spherical robot design was the most
commonly employed, with sliding mode-based control being the preferred control
strategy. However, no direct correlation between the design of the spherical
robot and the chosen control strategy was observed.

For an overview of the diverse control strategies included in this study,
please refer to Fig. 11.

Figure 10: Most prominent identified control strategies and their relative
importance from the number of related publications. ’others control’ are
detailed in Fig. 11. Figure 11: Different control strategies included in the
study

####  4.2.1 Sliding mode control

Sliding mode control is a widely employed technique for controlling various
types of robotic systems, including spherical robots and manipulators. Its
effectiveness in handling uncertainties and unmodeled disturbances, along with
its robust performance, has made it a popular choice. Numerous studies have
proposed different SMC-based control strategies, such as cascade sliding mode
control (CSMC) [57], high-order sliding mode control (HSMC) [58], and adaptive
hierarchical sliding mode control (AHSMC) [46], to enhance the robustness and
stability of spherical robots. Researchers have also explored the combination
of SMC with adaptive control laws, neural networks, and fuzzy control to
address uncertainties and reduce chattering. These hybrid approaches have
demonstrated improved tracking accuracy, convergence speed, and robustness
compared to traditional SMC. Fuzzy control, in particular, is well-suited for
complex robotic systems with challenging modeling requirements.

The selection of SMC in the included papers can be attributed to several
reasons, primarily the desire to achieve enhanced robustness and cope with
uncertainties and unmodeled disturbances. This rationale is discussed in [59,
60, 61, 62, 63, 41, 64, 65, 66, 50].

One promising implementation strategy involves decomposing a complex robotic
system into simpler subsystems, where a combination of the dynamic model
states and sliding surfaces, known as cascade sliding mode control (CSMC), can
be applied as demonstrated in [57]. However, conventional CSMC faced
challenges in achieving the desired position control performance due to
switching constants. To overcome this, the authors introduced positive
constants and referred to this modified control method as CSMC1.

Typically, the dynamic model of robots is necessary for designing a sliding
mode controller. However, in [52], the authors introduced a kinematic-based
SMC that can simultaneously track the position and attitude states of a
spherical robot. To reduce chattering, a saturation function was added to the
traditional SMC approach.

In the pursuit of stability in robot control, [58] proposed both first and
second-order SMC methods. They demonstrated that high-order sliding mode
control (HSMC) outperformed the first-order approach, a finding corroborated
in [67]. The control strategy presented in the latter paper reduced chattering
and enhanced robust operation in the presence of disturbances.

For improved performance in the face of uncertainties, in [68] the authors
introduced a fractional-order adaptive integral hierarchical sliding mode
control. This approach combines adaptive control laws, hierarchical sliding
mode control, and an integrator in the control loop. Comparative analysis with
adaptive hierarchical sliding mode control and HSMC revealed that the proposed
method exhibited superior convergence speed, stability, and robustness.
Adaptive hierarchical sliding mode control was initially proposed in [45] and
[69], while HSMC was presented by authors such as [70] and [71].

In [72], the authors introduced a hybrid super-twisting fractional-order
terminal sliding mode control for a rolling spherical robot, which
demonstrated higher accuracy and reduced chattering compared to traditional
SMC and other SMC-based controls. Hybrid super-twisting fractional-order
terminal sliding mode control is an advanced control technique that combines
super-twisting control, fractional-order control, and terminal sliding mode
control. It aims to achieve robust and accurate control of dynamic systems,
especially nonlinear and uncertain systems.

To stabilize a spherical robot moving on an inclined plane, in [73] the
authors presented a terminal sliding mode control. They showed that this
control approach effectively reduced chattering compared to SMC and
facilitated rapid convergence of the tracking error to zero in less than one
second, outperforming SMC for their specific spherical robot.

Some authors have integrated adaptive laws with SMC to enhance results.
Adaptive control enables the modeling of uncertainties, thereby reducing
chattering [74]. In [75], an adaptive method was employed to identify the
exact model. Similarly, in [76] the authors employed the same approach to
improve robustness and suppress vibrations in the inner suspension platform of
a spherical robot. Hierarchical sliding mode control with adaptive methods,
referred to as adaptive hierarchical sliding mode control (AHSMC), not only
offers improved robustness and parameter insensitivity characteristics but
also estimates dynamic disturbances [46]. In [69], AHSMC was utilized for
estimating the moment and rolling friction of a spherical robot.

Authors of [77] combined SMC with neural networks to achieve better
performance. The former proposed a radius-based function for controlling the
forward movement of the robot, demonstrating good convergence. The latter
employed recurrent neural networks in conjunction with SMC (RNNSMC) for
stabilization and tracking control. This approach improved the accuracy and
performance of the model, with comparisons against fuzzy PID indicating its
effectiveness in reducing chattering. Fuzzy PID was chosen as a baseline due
to the performance improvements offered by neural network-based control and
the suitability of fuzzy control for complex robot modeling.

We note from the above studies that SMC alone often falls short in adequately
addressing uncertainties and modeling errors in SRR systems. To tackle this
limitation, researchers have also explored the integration of neural networks
and fuzzy control with SMC to achieve improved robustness and faster
convergence. For instance, in [78], the authors incorporated neural networks
and fuzzy control into the SMC framework, yielding higher robustness and
faster convergence in the presence of uncertainties. Similarly, in [79], fuzzy
control was combined with SMC to effectively handle parameter uncertainties.
This integration of fuzzy control with SMC proved instrumental in enhancing
the overall performance of SRR systems.

####  4.2.2 PID, PI, and PD control strategies

PID control is widely employed by researchers in their studies, either in its
traditional form or in combination with other control laws such as fuzzy
control, neural networks, Cerebellar-model articulation control, or geometric
and adaptive laws. It is probably the most intuitive and less demanding, in
terms of processing power, controller for its performance. It is thus common
to select this strategy for a fast deployment. For instance, the traditional
PID control is utilized in [80] to demonstrate its functionalities. In the
work by [43], PID control is proposed, following Ziegler-Nichols rules, to
stabilize a spherical robot on an inclined plane. The Ziegler-Nichols rules
are a set of heuristic guidelines used in control theory to tune the
parameters of a proportional-integral-derivative (PID) controller. The Authors
in [66] apply PID control for pitch angle control. Furthermore, in [81], PID
control is combined with neural networks to compensate for actuators’
nonlinearity when single input multiple output PID control fails. This
integrated approach exhibits fast convergence and proves to be an efficient
method for controlling SRRs.

The application of fuzzy PID control is proposed in [43] for the motion of a
spherical robot, while a comparison with PID control in [82] demonstrates that
fuzzy PID control outperforms PID in terms of performance and stabilization
time. In [83], fuzzy PID control is combined with feedback linearization and a
grey predictor, with the output of the grey predictor utilized instead of the
current system outputs. This method exhibits superior performance compared to
PID and fuzzy PID controls in terms of overshoot and settling time. Moreover,
some studies employ the proportional-integral (PI) control strategy. For
example, [84] proposes PI control for controlling the angular velocity of a
spherical robot, and [85] employs PI control for velocity control of a two-
wheel differential spherical robot. In [86], PI control is employed for real-
time control of a spherical robot, with the addition of a genetic algorithm to
enhance stability.

PD control is also utilized in certain papers, such as [87] and [42], where it
is proposed for velocity control of an omnidirectional robot with flywheels.
In [50] and [47], PD control is employed for controlling the yaw orientation
of the spherical robot, while no other strategies are presented for
controlling the remaining degrees of freedom.

####  4.2.3 Other control strategies

In addition to sliding mode control (SMC) and proportional-integral-derivative
(PID) control, various other control strategies have been employed to control
robotic systems. For instance, backstepping control combined with neural
networks was used in [88] to achieve high-performance cooperative control of a
group of mobile robots. The linear quadratic regulator (LQR), dynamic
programming, and model predictive control have also been utilized to control
spherical robots in some of the included papers. For example, LQR has been
used to control robots or stabilize motion, as in [89], while dynamic
programming has enabled optimal motion planning and control of spherical
robots in [90]. Feedback linearization has been proposed in [91, 92] to
achieve control and stabilization objectives, and in [87], the linear
quadratic regulator is used in combination with percentage derivative control
for better performance. Some control methods have been used only once, such as
the active disturbance rejection control in [93] to achieve trajectory
tracking control for biomimetic spherical robots without model parameter
information, energy-based control in [94] for a spherical robot to track high
moving speeds, and H-infinity control in [49] to provide a solution to the
robot’s optimal control problem under model uncertainty and external
perturbation.

##  5 Experiments and sensing

###  5.1 Simulations vs experimental validation

In all included papers, authors performed either experiments or simulations in
order to validate or verify the feasibility of the proposed control
strategies. From the control aspect, 32 % of the papers performed experiments
to validate their control strategies while 78 % realized only simulations.
These validations were done in numerical simulation software such as Matlab
Simulink [70] or by implementing the control strategy on a prototype [42]. If
we look only at BSRs, the broad mechanism category with the largest number of
publications over the years, a significant portion of them included a physical
prototype, namely 50 % for pendulum-based BSRs, 38 % with sliding masses and
82 % for IDU-driven BSRs. These numbers are not unexpected, as robots with an
IDU are generally cheaper to prototype. On the opposite end, using sliding
masses increases the mechanical complexity of the system, as well as requiring
more complex control schemes, the sliding masses needing to continuously
change direction to provide a uniform and continuous motion of the sphere.

Beyond the presentation of a prototype, experimental validation with different
tests and scenarios is less common. For instance, Chowdhury et al. [39]
conducted experiments in both indoor (smooth surface) and outdoor environments
(rough surfaces) to validate their model and their control strategy as well.
In the paper by Alves and Dias [23], a path curvature-based control was
presented. Some authors used motion capture systems able to track markers
inside the shell in order to validate the accuracy of robot localization [95].
When conducting experiments, different metrics were selected for the
validation of the results. The most common were the overshoot, the steady
state response, the rise time, the convergence time, and the maximum absolute
value of motion trajectory error.

###  5.2 Embedded sensing

The few reviewed papers that progressed to prototype validation necessitated
the incorporation of sensing components to close the control loop. The
majority of these papers employed commonly used sensors, including
accelerometers, cameras, and encoders. Accelerometers, typically housed within
an inertial measurement unit (IMU), were utilized in some papers to acquire
feedback on the robot’s state. Notably, [23] proposed to use a three-axis
accelerometer combined with a three-axis rate gyroscope to obtain information
about the robot’s acceleration, angular velocity, and position. In [16], the
feedback is obtained from a module of sensors that consists of encoders of the
servomotors, angular rate sensors, angular acceleration sensors, and
directional gyro, which are all microelectromechanical systems (MEMS) devices
However, IMUs were predominantly deployed to provide orientation information
in studies such as [23, 47, 48, 96, 97, 44]. To mitigate noise and drift
issues, an extended Kalman filter was applied to the IMUs in [98].

Some researchers incorporated additional sensors into their systems. Encoders
were commonly used in conjunction with IMUs [99, 100, 101, 95, 102, 103, 104,
105, 106, 107], while cameras were employed for real-time position tracking in
[108, 69]. In [87], an IMU, gyroscope, and inclinometer were combined to
acquire the pitch, roll, and yaw angles, as well as the lean angles and spin
rate of the robot. To improve the accuracy of position measurement considering
slipping, [70] proposed a vision system in addition to an IMU. Vision systems
were utilized by other authors for obtaining feedback information as well. For
example, [109] proposed a system comprising four infrared cameras along with
dedicated software to determine the position of the robot in space. In [110],
an RGB camera is used to track the spherical robot pose at 30Hz (each frame).
Similarly, in [90, 111], a camera system was utilized to acquire the robot’s
position. Furthermore, in [40], the combination of a depth camera and a LIDAR
was used to map the environment and infer the robot’s position in the map
(Simultaneous Localization and Mapping - SLAM). It should be noted that
employing camera systems either requires a transparent shell or a turret to
house the camera, resembling the popular design of BB-8.

An overview of the sensors used in the reviewed papers is depicted in Fig. 12.

Figure 12: Embedded sensors distribution and combination in the publications
covered. The vast majority combine IMUs with encoders.

##  6 Discussion

After a comprehensive analysis of the papers, several challenges associated
with spherical robots have been identified. While some of these challenges
have been addressed in the literature, others remain unexplored. In this
section, we will delve into these challenges and discuss potential avenues for
further research.

One significant challenge arises from the limited number of contact points
that spherical robots have with the ground. This constraint makes it
challenging to employ traditional sensors similar to those used in wheeled or
legged robots for measuring motion and orientation. Although some papers
suggest using IMUs for measurements, controlling the robot becomes problematic
due to drift errors inherent in IMUs. Consequently, there is a research
opportunity to explore robust control methods that can effectively model
disturbances and uncertainties. However, relying solely on proprioceptive
measurements may prove insufficient for most realistic missions such as
exploration and inspection.

To overcome the limitations of proprioceptive measurements, additional sensors
like cameras, LIDAR, or ultrasonic sensors can be integrated to complement IMU
data and provide more accurate information about the robot’s surroundings and
motion. An example of such integration can be found in [40], where a camera
and a LIDAR are utilized. However, the use of cameras and LIDARs in spherical
robots introduces challenges related to refraction and distortion. These
phenomena can alter distance and position measurements captured by the
sensors, leading to perceptual errors. Moreover, inadequate perception of the
environment’s geometry can result in errors in motion planning and
localization. Thus, investigating the impact of refraction and distortion
becomes crucial in enhancing perception precision, motion planning,
localization, and sensor calibration. Such research efforts would ultimately
improve the reliability and efficiency of navigation for robots equipped with
cameras and LIDARs.

When considering the integration of cameras and LIDARs, it is imperative to
ensure a clear line of sight. Alternatively, cameras can be positioned outside
the shell on both sides, as demonstrated in [112]. However, integrating
cameras outside a non-transparent shell may impose limitations on the robot’s
omnidirectional movements and increase the risk of camera damage.

The unique shape and locomotion mechanism of spherical robots poses
significant challenges in obstacle avoidance. These robots rely on changing
their center of gravity or rolling in different directions to move. However,
the lack of independent directional control makes efficient navigation around
obstacles challenging. Additionally, SRRs maintain contact with the ground or
surfaces using a single point or a limited set of points, which can affect
their stability and maneuverability, especially when encountering uneven
terrain or obstacles with varying heights. Maintaining balance and stability
while avoiding obstacles becomes increasingly challenging in such scenarios.
One potential solution to address this challenge is incorporating jumping
mechanisms in spherical robots, enabling them to overcome obstacles
vertically. By introducing a jumping mechanism, spherical robots gain the
ability to leap over barriers, gaps, or rough terrain that would otherwise be
challenging or impossible to traverse with rolling or sliding locomotion
alone. However, achieving controlled leaps while managing the position of the
center of mass and the point of application of the propulsion force remains an
open challenge due to the mechanical complexity involved.

Moreover, two distinct approaches can be employed to tackle the intricate
dynamics of spherical robots: the decoupled approach and the complete
approach. The decoupled dynamic approach involves separating the transversal
and longitudinal motions of the robot and studying them independently. This
approach simplifies the dynamics and reduces computational effort. On the
other hand, the complete dynamic approach involves modeling the robot without
decoupling the motions, which adds complexity to the system. Comparing these
two approaches can be valuable in understanding their respective advantages
and drawbacks, enabling individuals to determine the most suitable approach
for their needs.

In addition to the aforementioned challenges, controlling a spherical robot
can be particularly demanding, especially for non-holonomic systems such as
pendulum-driven spherical robots. These non-holonomic systems possess motion
constraints that cannot be easily integrated into their configuration space.
As a result, designing control laws capable of steering the system along a
desired trajectory becomes difficult. Unlike holonomic systems, non-holonomic
systems require careful consideration of their constraints and dynamics to
develop control laws that can achieve the desired motion. Therefore, further
research is needed to explore effective control strategies tailored to non-
holonomic spherical robots.

##  7 Conclusion

The paper provides a comprehensive analysis of spherical robots, covering
various aspects such as design, locomotion, control, and embedded sensing. It
reviews multiple research papers to gain insights into the state of the art in
spherical robot development. The design of spherical robots involves
considerations such as the mechanisms for locomotion. Different locomotion
mechanisms are discussed, including pendulum-driven systems, internal mass
shifting, COAMs, and shell transformation. The paper explores various control
strategies for spherical robots, with a focus on efficiency, stability,
robustness, and feasibility. A comprehensive analysis identified a total of 33
control strategies used for controlling spherical robots. Among these
strategies, the findings indicate that sliding mode-based control is the most
prevalent approach employed in the control of spherical robots. The challenges
associated with spherical robots encompass the limitations posed by their
contact points, the integration of additional sensors, obstacle avoidance,
controlled leaps, and control of non-holonomic systems. Addressing these
challenges through targeted research efforts would drive advancements in
spherical robotics, paving the way for more capable and versatile robotic
systems in various applications.

This review emphasizes the importance of ongoing research and innovation in
spherical robot design, locomotion, control, and embedded sensing. It
emphasizes the potential of spherical robots in various applications and
identifies areas for future exploration, such as refining control strategies,
addressing challenges in obstacle avoidance, and comparing different
approaches to modeling their dynamics.

Overall, the paper provides a comprehensive overview of spherical robot
research, highlighting the current state of the field, key challenges, and
potential avenues for further advancement.

This work was supported by NSERC Discovery Grant (RGPIN-2020-06121). We would
like to express our gratitude to Simon Bonnaud for his valuable support for
the redaction of this manuscript.

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