Spherical Robots: An Up-to-Date Overview of Designs and Features Текст научной статьи по специальности «Медицинские технологии»

Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, pp. 709-750. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221207

NONLINEAR ENGINEERING AND ROBOTICS

MSC 2010: 70Q05

Spherical Robots: An Up-to-Date Overview of Designs and Features

Yu. L. Karavaev

This paper describes the existing designs of spherical robots and reviews studies devoted to investigating their dynamics and to developing algorithms for controlling them. An analysis is also made of the key features and the historical aspects of the development of their designs, in particular, taking into account various areas of application.

Keywords: spherical robot, rolling, design, modeling

1. Introduction

The active development of the results of theoretical research into the dynamics of the rolling motion of bodies and the emergence of modern technologies of creation of actuating mechanisms and control systems have stimulated the development of a new class of mobile robotic systems possessing quite a number of advantages over the most widespread wheeled robots. Their key difference is that they have a spherical shell in which actuating mechanisms, a system for controlling the actuating mechanisms, and sensors for performing various tasks are installed. The spherical shell designed to be hermetic protects the structural elements and the control system against the action of corrosive and other media such as moisture, dust etc., and gives the robot a compact and symmetric form, which ensures high maneuverability.

Despite the fact that the first prototype of the self-propelled toy in the form of a sphere actuated by an electric motor installed inside a spherical shell was registered in 1957 [1], it was not until the 2000s, after publication of several studies devoted to developing the first spherical

Received September 19, 2022 Accepted November 21, 2022

The work was carried out within the framework of the state assignment of the Ministry of Education and Science of Russia FZZN-2020-0011.

Yury L. Karavaev [email protected]

Kalashnikov State Technical University ul. Studencheskaya 7, Izhevsk, 426069 Russia I.N.Ulianov Chuvash State University Moskovskii prosp. 15, Cheboksary, 428015 Russia

robots [2-4] that real impetus was given to this line of research in science and engineering. The review papers [5, 6] published in 2006 describe more than 10 various designs.

The development of spherical robots involves not only engineering activities, but also fundamental research in several subject areas such as theoretical mechanics, dynamical systems theory, and robotics. Most of the existing designs of spherical robots are based on the results of fundamental research on the problems concerning the rolling motion of a body on a solid surface. These results have been obtained by quite a number of renowned scientists in this field, in particular, by A.P. Markeev, whose monograph presents not only fundamental foundations, but also a detailed literature overview and references [7]. At present, active research is underway on quite a number of related problems. This includes investigation and development of models of contact interaction of a spherical body with the plane (surface) [8-19], explanation of the dynamics of motion of inhomogeneous spherical bodies, in particular, those with internal mechanisms changing the position of the center of mass and the angular momentum of the system [4, 20-25]. The problems of planning the motion of spherical robots along a trajectory [26-36] are no less popular.

It should be noted that the spherical robots include not only mobile robots whose actuating mechanisms are placed inside the spherical shell, but also robots in which external actuating mechanisms placed outside the sphere are used to set it in motion, or the motion is implemented by means of external forces, for example, wind or fluid flows. Such robots are not discussed in the literature review that follows,but their detailed description can be found in the dissertation [37], which is concerned with the dynamics of a spherical robot designed to move on the surface of Mars using wind energy, and in [38].

From the viewpoint of modeling and control, highly maneuverable robots on a spherical wheel [39-41] are also rather similar in design. Such robots are upgraded versions of the wheeled balancing devices (among which the segway is the most well-known design), but the latter have a spherical wheel which is rotated by external actuating mechanisms (usually by rollers, usual wheels or omniwheels), see Fig. 1.

(a) (b)

Fig. 1. Photographs of balancing robots on one spherical wheel (Ballbot): (a) a design proposed by Prof. Ralph Holli at the Carnegie Mellon University, USA; (b) photograph of the prototype of a balancing robot developed under the guidance of Prof. Masaaki Kumagai at the Tohoku Gakuin University, Japan

(a)

(b)

Fig. 2. (a) Gyrover created at the Carnegie Melon University, USA; (b) photograph of a full-scale specimen of the gyrowheel of Yu. G. Martynenko [44]

The work preceding the development of various designs of spherical robots includes activities aimed at creating and investigating mobile robots in the form of a wheel. For example, the gyrover [42, 43] is a one-wheeled robot actuated by an internal one-degree-of-freedom pendulum, and turns are performed by means of a servomotor, which tilts the axis of the rotating flywheel (gyroscope) (see Fig. 2a). Also, the gyroscope stabilizes the motion of the gyrover.

It should be noted that the principles of propulsion of a one-wheeled mobile robot "Gyrowheel" (see the photograph in Fig. 2b) were first set forth in 2001 in Perm at the All-Russian Congress on Theoretical and Applied Mechanics in a presentation by Yu. G. Martynenko, A. V. Lenskii and A. I. Kobrin, and control problems were addressed in [45, 46].

There is a toy that may be regarded as the prototype of modern spherical robots — the so-called hamsterball. Using this analogy, the first mechanical self-propelled toys in the form of a sphere were created. In 1917, A.D.McFaul (U.S. Patent 1,263,262) suggested using inside a ball a mechanism with rollers whose rotational energy was ensured by a spring. The rotation of the rollers due to friction forces between them and the internal surface of the spherical shell set the ball in motion. Earlier, the first self-propelled spherical ball was created in 1893 in the USA (J. L. Tate U.S. patent 508,588). The displacement of its center of mass was also performed by means of a spring, but the trajectory of its motion was chaotic. In 1906, B. Shorthouse (U.S. Patent 819,609) upgraded the design so that the ball began to move along a prescribed trajectory. It should be noted that still earlier, in 1889, W.Henry (U.S. Patent 396,486) suggested using a special vehicle in the form of a sphere for transportation of people by water by actuating the external spherical shell using motion relative to its internal surface. This idea was developed further in more than ten patents.

It was not until the 2000s that the first review papers appeared on the designs of spherical robots, including the description of the principles of actuating them. Among the most significant and the most cited reviews are [5, 6, 47-51]. But the most complete and detailed overview of all existing designs, their historical development before 2017, and examples of their practical use can be found in the dissertation by Tomi Ylikorpi [52]. The most relevant studies with focus on the existing designs and algorithms for controlling spherical robots have been published in recent years [53-55]. Also informative is the website [56], which contains photographs and references to primary sources that publish materials on various designs of spherical robots created before 2017.

Let us consider the main areas of application of the existing models of spherical robots.

2. Areas of application of spherical robots

Spherical robots have already gained widespread use [56, 57]. They are applied for monitoring of the flow of liquids and gases in pipes, underwater investigations of pipes in nuclear power plants, remote monitoring and exploration, and are used as service robots and toys. In the opinion of many authors, the use of spherical robots for investigating the surfaces of planets holds very much promise.

2.1. Robot toys and robots for entertainment

In 2000, Michaud developed a spherical robot toy whose outward appearance is attractive for children because they are used to playing with spherical objects such as rubber, plastic or inflatable balls. In addition, the round shape of the robot has no protrudent elements that could injure a child or could be easily broken. One of the first examples is the robot named Roball (see Fig. 3), which uses a pendulum propulsion system [58]. The motion of the Roball is determined by an operating mode providing four types of motion: "critical situation", "spinning", "straight-line motion" and "cruise", which can be activated or deactivated in accordance with events. These simplified motions are sufficient for the robot toy. The authors have actively conducted experiments involving children. The tests showed the interest of children in the Roball and they immediately began to play with them. These tests also showed that the durability of the robot is a very important parameter because children often throw toys on the floor [59].

Fig. 3. Roball (left); A boy playing with the Roball (right)

In 2002, Sony Corporation developed a spherical robot for entertainment purposes, called Q-taro, which is an abbreviation of the "Quasi-stable Traveling and Action Robot" [60]. The main purpose of the Q-taro is to create emotional contact between people and robotic devices (see Fig. 4). The Q-taro is a completely autonomous robot that has a total of 36 various sensors. Using temperature sensors, the robot is capable of detecting the presence of a human nearby, which actuates the robot. The audio sensors allow the robot to move in tune with music, and the light indication shows various "emotions" of the robot. Also, the Q-taro is capable of detecting obstacles and avoiding collisions with external objects, and the speech recognition system allows the robot to distinguish up to ten different words.

A spherical robot for observation of children and entertainment (voice and visual interaction) was presented by the company Panasonic in the IFA exhibition in Berlin in 2016 [61]. The main purpose of the robot is to entertain children using sound, and visualization of emotions on the internal displays of the robot (see Fig. 5).

Fig. 4. Entertainment robot Q-taro

Fig. 5. Robot for interactive interaction with children, manufactured by Panasonic

The robot LEKA [62], which has a similar functional, but has been supplemented with the ability to teach children and to perform remote control, has been developed by the company Toy Design. The spherical robot has a more modern design, an extended set of sensors, both tactile and optical ones, and a more functional application for interaction with the robot using a tablet PC or a smart phone. A photograph of the robot is shown in Fig. 6.

Two models of spherical robots, namely, Sphero and BB-8 (see Fig. 7), developed by the company Sphero, Inc [63] (the previous name of the company was Orbotix) have gained the most widespread use and become the most widely known. The first of them appeared in 2011. It was a remotely controlled toy in the form of a sphere 74 mm in diameter, inside which a two-wheeled platform was placed, and a sliding spring-loaded arm prevented the platform from turning. The spherical robot was controlled remotely from a smart phone and had induction charging with a special station. Later, the design had no essential changes, but various functional possibilities appeared for interaction with the robot, as well as the possibility of programming it to organize the educational process.

Fig. 6. Robot for entertainment and teaching, manufactured by the company Toy Design

Fig. 7. Robots of the company Sphero, Inc: Sphero SPRK+ and Sphero Bot BB-8

2.2. Robots for monitoring and reconnaissance

Monitoring and reconnaissance are obviously the most important and promising applications of spherical robots. Rotundus, a Swedish company founded in 2004, has developed a spherical robot named GroundBot for monitoring and security functions. The robot is moved by means of an internal pendulum and motors actuating it [57]. The outward appearance of the robot is presented in Fig. 8. The robot has a radar, sensors for detecting violators, and an alarm signal. Among the advantages of this robot the developers mention the simplicity and stability of the robot and the ability to move across various terrains, including snow, sand and dirty surfaces.

Researchers from the Carnegie-Mellon University developed in 1999 a prototype of the spherical robot "Cyclop" for military applications, namely, for intelligence activities. "Cyclop" is a miniaturized mobile spherical robot intended for remote monitoring and reconnaissance in urban areas [64]. The motivation for creation of such miniaturized spherical robots is the necessity of such a device for police or military operations under urban conditions. The small size makes it possible to perform covert observation in closed rooms. In adition, remote observation allows people to perform surveillance and identify dangerous situations from a distance. The small size and light weight allow the robot to be carried by hand.

The motor system of the robot has two degrees of freedom and consists of two motors which produce torques along the vertical and horizontal axes of the robot. The motor along the vertical axis ensures the change in the direction of the robot's motion. When the motor rotates the mass

Fig. 8. Robot "GroundBot" for monitoring, manufactured by Rotundus. Left: the image of the first prototype 2004. Right: the modern prototype

Fig. 9. The spherical robot "Cyclop" for reconnaissance and monitoring (the picture is taken from [64])

built in the robot, the spherical casing of the robot turns and the orientation of the robot changes. The motor along the horizontal axis ensures, in fact, the motion of the device.

The control and feeding system consists of microcontrollers and miniaturized transceivers. The microcontrollers perform the control of peripheral elements such as gyroscope, compass and the feeding system. The transceiver ensures the communication of the robot with a man-operated ground station. The feeding system includes batteries that are completely built in the spherical casing of the robot, which allows it to move freely. In this context, the sensor system means a sensor system for observation except for any sensors for control and motion. The main sensor of such kind that is installed in "Cyclop" is a miniaturized camera. The image from the camera is transmitted to the ground station, allowing the operator to control the robot. Recently, the created prototype has been updated many times. For example, a miniaturized omnidirectional high-resolution camera with a high-visibility video transmitter was developed for this prototype. In addition, it is planned to improve the size, power and reliability of the robot.

In 2015 the company "SET-1" [67] presented in the exhibition "Interpolytech" the reconnaissance and survey device "Sfera". This device is intended to accomplish tasks such as efficient gathering and transmission of audio and video information from hard-to-reach and dangerous zones. The device can perform all-round observation using 4 monochrome video cameras (CMOS) developed by SET-1; in addition, "Sfera" is equipped with a vertical positioning system. The device has the form of a sphere 91 mm in diameter. The upper part of the sphere contains 4 cam-

eras, and the lower part contains a storage battery. The device has no actuating mechanisms and is, in fact, a self-orienting camera in an impact-resistant casing (see Fig. 10a) and can be thrown into zones to be observed.

(a) (b)

Fig. 10. (a) Reconnaissance and survey device "Sfera", (b) tactical camera in the form of a sphere, manufactured by the company Bounce Imaging (USA)

The tactical camera with all-round view (see Fig. 10b), manufactured by Bounce Imaging [68], USA, is a foreign analog of the domestic development. Like "Sfera", the device has no actuating mechanisms, but is a multiobjective camera. Six cameras built in a spherical impact-resistant casing make all-round observation possible, and the stabilization algorithms allow one to transmit high-quality video in motion, for example, when the device falls and rolls after being thrown. The appearance of such devices allows one to draw a conclusion on their importance for observation, reconnaissance and monitoring.

2.3. Investigation of other planets

The Massachusetts Institute of Technology (MIT) in the USA investigated spherical robots which can move by jumping up and rolling [69]. It is assumed that these robots will be sent to other planets in a specialized container resembling an eggshell (illustrated in Fig. 11). The advantage of using dozens of microrobots is that they can cooperate to achieve their common goal. For example, when they are arranged correctly, they can transmit messages to the central unit even from the depth of a cavern.

In [70], a description is given of the design of a spherical robot named SphereX for investigation of terrains difficult to traverse for other known modifications of Mars rovers. In addition, when using several robots one can create a communication infrastructure for widening the survey zone.

The design of the spherical robot is shown in Fig. 12. To ensure passability, the robot has an internal hermetic spherical shell containing a control system, sensors and actuating mechanisms which rotate two external spherical segments with special edges ensuring passability. Also, to overcome more essential obstacles, the lower part of the robot between the external hemispheres has a mechanism for negociating obstacles in the form of an extendable arm.

The most illustrative examples of the application of spherical robots for exploration of other planets are the projects of the European Space Agency (ESA). The first project was called "Biologically Inspired Solutions for Robotic Surface Mobility" No. AO4532-03/6201 within the framework of the program ARIADNA and was implemented by the team of the Automation

Fig. 11. Spherical microrobots for investigation of Mars (the picture is taken from [69])

Fig. 12. Spherical robots SphereX for investigation of Mars (the picture is taken from [70])

Technology Laboratory of the University of Technology in Helsinki under the direction of Prof. A. Halme. The results of work on this project are presented in the dissertation of Tomi Ylikorpi (2005) [37]. The dissertation describes in detail the designs of robots that existed at that time for exploration under conditions of other planets, and focuses on robots that are planned to be exploited under conditions of Mars, with its heavy winds and specific soil [38]. The team of authors of this project proposed a design of the spherical robot shown in Fig. 13.

The robot proposed by them consists of tangential blades, but in the future all internal mechanisms can be placed and insulated in the internal sphere. For a more efficient transformation of wind energy, each arc forming a spherical shell has a special form improving the windage. If there is no wind, the spherical robot is actuated by a two-degree-of-freedom internal pendulum, which also ensures changes in the direction of motion.

A similar design of the spherical robot and its purpose are described in [71], in which the authors suggest using a wind-driven spherical shell in which sensors necessary for investigation are installed. In the absence of wind, the shell is capable of being set in motion by displacing the center of mass using permanent magnets placed in special tubes — solenoids which, with the robot being driven by wind, can be used to generate energy and to feed board equipment.

Fig. 13. Prototype of a spherical robot actuated by wind (the picture is taken from [37])

The second ongoing project of the European Space Agency is the project of a speleologist robot named DAEDALUS (Descent And Exploration in Deep Autonomy of Lava Underground Structures), which will be used for exploration of lunar caves [72, 73] in the near future.

Descent Rolling Scanning Negotiation

of obstacles

Fig. 14. 3D model of the prototype of the spherical robot DAEDALUS (the picture is taken from [73])

Figure 14 shows 3D models of the above-mentioned prototype under different operation conditions. The work on the project is carried out by a group under the direction of the Julius Maximilian University of Wurzburg in Germany together with several universities of Germany and Italy. It is planned that the sphere DAEDALUS with a diameter of 46 cm will carry an immersive stereoscopic camera, a lidar system for three-dimensional mapping of caverns, temperature sensors and a radiation dosimeter, and extendable arms that help overcome obstacles and check the properties of rocks. At first DAEDALUS will be lowered into the mouth of the cavern by means of a long rope, and then disconnected in order to move on its own inside the cavern or crater. The rope that was used for lowering will later be used as a Wi-Fi receiver.

In addition to engineering work, research was conducted aimed at developing the control and planning of the robot's motion in the form of a sphere on surfaces similar in geometry to the reliefs of other planets [74, 75]. Using the results of modeling, it is shown what possibilities exist for spherical robots to overcome craters and hills specific of the terrains of the planets under exploration.

2.4. Service robots

The first attempts to use spherical robots for cleaning were made in 2009 after the emergence of the robot Robomop (see Fig. 15). In reality, this spherical robot was placed into a special platform which the robot could move across the room, and the platform with the devices fastened on it performed cleaning. But as a commercial product such an application did not received wide acceptance, so nowadays it is rather difficult to find information on it. Nevertheless, after several large world companies had announced the creation of mobile service robots in the form of a sphere, a lot of effort went into developing this line of research.

Fig. 15. Photograph of the cleaning robot Robomop

In 2013, in the annual competition of conceptual robots "Electrolux Design Lab 2013" [76], a prototype of the spherical robot which is an automated cleaning system (see Fig. 16) was presented. The robot's structure in the form of a sphere is designed not only to ensure motion, but also serves as a platform for placement, charging and cleaning of miniaturized flying robots, which filtrate the air and gather dust from surfaces. But this project still remains only a concept.

Fig. 16. Prototype of a vacuum cleaner robot, manufactured by Electrolux

In 2020, in the international exhibition CES 2020 in Los Angeles (see Fig. 17), the Samsung corporation presented a helper robot named Ballie. The robot has the form of a ball and, in addition to the ability to move indoors, is equipped with a functional for controlling a smart house.

Samsung Ballie contains a set of sensors intended for safe motion indoors, for determining the orientation, interaction with surrounding objects and voice interaction with humans. The camera of the robot records photographs and video and detects rubbish (the robot can call the

Fig. 17. Photograph of the helper robot Ballie, manufactured by Samsung (the picture is taken from [77])

vacuum cleaner to remove rubbish). Also, the robot, which is part of the "Smart House" can control electrical equipment and lighting. In terms of design, despite its spherical form, the robot is a model of the two-wheeled mobile robot with differential drive, each wheel of which is a segment of the sphere.

In 2016, young researchers from Kasakhstan proposed the design of a helper robot in the form of a sphere, which was presented at technological conferences of Kasakhstan and Singapur [78]. A photograph of the prototype is shown in Fig. 18.

Fig. 18. Photograph of the helper robot BalBo (the picture is taken from [78])

The spherical robot [79] can be used as a helper for remote observation, safeguarding premises, and voice and visual interaction with humans. In addition to these functions, the robot can be used as a high-tech toy for children.

In [80, 81], an example is described of using a spherical robot (see Fig. 19) in agriculture for monitoring the state of soils and plants. The authors of those papers show an example of using a simple design of the spherical robot with a two-degree-of-freedom pendulum for motion on cultivated soil between plants. In order that the sensors installed inside the spherical shell could be applied, the shell itself has been made nonhermetic and has special windows. The experiments described in [80, 81] have demonstrated the possibility of applying such robots for agricultural purposes. Also, the authors note that the robot in the form of a sphere causes less damage to topsoil when it moves. This conclusion was drawn by researchers and engineers from Saint Petersburg [82] after they had assessed the possibility of placing monitoring equipment on the platform of the spherical robot.

In the opinion of many authors, since the pressure of spherical robots on the underlying surface is smaller than that of wheeled or walking robots of similar sizes, the spherical robots

Fig. 19. Photograph of the robot Rosphere for monitoring the state of soils and plants (the picture is taken from [81])

can be applied for tasks in which damage to soil would be critical, for example, in agriculture. For example, in [65, 66], examples of the application of spherical robots for monitoring of the growth of plants are described and the results of experiments confirming the absence of compaction and erosion of soil after multiple travels of the robot are presented.

3. Principles of actuation of spherical robots

All well-known designs of spherical robots employing various principles of actuation can be presented in the form of a diagram in Fig. 20.

Fig. 20. Principles of actuation of spherical robots

They can be divided into the main four groups:

• control by producing variable gyrostatic momentum;

• control by displacing the center of mass;

• control by deforming the spherical shell; combination of several mechanisms.

The fourth group includes robotic systems in which mechanisms of the first three groups are combined. Combination of the above-mentioned mechanisms allows one to achieve a higher efficiency and functionality of spherical robots.

The robots belonging to the first group are set in motion by producing variable gyrostatic momentum, which is ensured by internal flywheels (gyrostats). This principle was studied by many researchers (in particular, the algebraic controllability of this system was shown and control laws were obtained), but in practice such robots possess low efficiency and are of interest primarily when it comes to combining them with other mechanisms.

Robots using the second control scheme are the most widespread. To displace the center of mass inside the sphere, use is made of a system of pendulum type, sliders, skids on which the mass moves, and various vehicles rolling inside the ball.

Robots that belong to the third group and have also found widespread application recently move by deforming the shell or by changing its shape. Having a soft shell, the spherical robots can move and even jump using its deformations.

We next consider each of the above-mentioned types of motion in more detail.

3.1. Robots with internal flywheels

The development of this principle of actuation of spherical robots is associated with the use of similar mechanisms for stabilization (adjustment of the trajectory and orientation) of the motion of satellites, vehicles and marine vessels. Handwheels were also used for nonbionic stabilization of two-legged walking robots [83].

Figure 21 shows a robot in the form of a cube with internal handwheels which produce a torque which allows it to turn on its edge or its top. The developers have called it M-block [84]. Owing to the magnets installed in the ribs of the cube, it can be positioned sufficiently accurately after turning on the faces of another identical cube. Thereby, in the presence of a large number of such cubes, one can create a spatial configuration. When the control of several cubes is coordinated, they can implement motion jointly, in the form of assemblies. The cube turns when the handwheel accelerated previously to a high angular velocity is abruptly decelerated using a special brake gear.

Fig. 21. Photograph of the robot M-block (the picture is taken from [84])

The first publications devoted to spherical robots with internal handwheels or flywheels have appeared relatively recently although they are based on the results of S. A. Chaplygin's work, who proved as early as 1903 the integrability of the problem of the motion of a dynamically asymmetric ball and showed a number of geometric properties of its trajectory [85]. A mechanical

system describing a spherical robot with internal flywheels is often called the Chaplygin ball with flywheels, although the first results on the dynamics of a ball with a gyroscope were obtained by Bobylev [87] and Zhukovskii [86].

The results of theoretical studies include investigations of integrability [88], qualitative analysis of motion [89], and the stability analysis of the motion of spherical bodies actuated by internal flywheels [90, 91]. More recent work is devoted to problems concerning the controllability and planning of the trajectory of motion [92-94]. Real prototypes and experiments with robots implementing this principle of actuation are described in [95, 96, 103].

The design of a spherical robot with two flywheels is described in [95], for which a comparison is made of the experimental trajectories with one and two constant control actions (angular velocities of the flywheels' rotation). Fairly short portions of the trajectory and a limited set of controls are considered, but even they allow the authors to draw conclusions on the complexity of the process of motion and the necessity of a more complex description of the system.

The paper [96] presents the design of a spherical robot with two flywheels (see Fig. 22). Two flywheels are installed inside a spherical shell with a radius of 30 cm, so that the axes of their rotation are perpendicular. To ensure that the center of mass lies symmetrically at the center of the sphere in the plane of their rotation axes, additional balance weights have been placed. The role of such weights is played, in particular, by power components.

Fig. 22. Photograph of a spherical robot with two internal flywheels (the picture is taken from [96])

The results of experimental research on this prototype are extremely few in number. The authors discuss only two maneuvers: rotation in place about the vertical axis and rolling motion on a plane along a straight line, but attempts to obtain and investigate typical trajectories have failed in practice.

Investigations of the controllability and planning of the trajectory of the motion of a spherical robot with internal rotors were conducted under the guidance of V. E. Pavlovskii [97] and were presented in the dissertation by G. P. Terekhov [98]. Despite the presence of the prototype shown in Fig. 23, they published only theoretical results on the modeling of motion and planning of maneuvers, taking into account the model of two-parameter friction [99], in particular, for a robot having a deviation of the center of mass from the geometrical center of the sphere.

Figure 24 shows photographs of the prototype of a spherical robot with internal flywheels 300 mm in diameter [100].

A demonstration of the motion of the developed prototype of the spherical robot is presented in the video [101]. The results of experiments, including those obtained by means of controls obtained using the algorithm proposed in [102-104] have shown the complexity of start from rest because friction plays an essential role, as well as the possibility of using rotors for changing the

Fig. 23. (a) photograph of a spherical robot with three internal flywheels (the picture is taken from [97]), (b) its 3D model

Fig. 24. Photograph of the prototype of a spherical robot with internal flywheels

trajectory of its motion, or stabilization. In general, the above-mentioned principle of propulsion is possible, but ineffective.

3.2. Robots actuated by changing the position of the center of mass

This principle of propulsion of spherical robots is the most widespread. As shown in Fig. 20, the position of the center of mass can be changed in various ways.

3.2.1. Internal wheeled vehicles

The idea of placing wheeled vehicles (or platforms, or cars) inside a sphere was motivated by the creation of the hamsterball (a ball with a hamster moving inside it). The dissertation [52] describes approximately ten patents of vehicles inside a spherical shell. Usually they are traditional radio-controlled models of wheeled and caterpillar vehicles, or real vehicles placed inside

a sphere. Since there are no descriptions of their real prototypes nor research papers devoted to their control, they do not require a more detailed analysis.

The first spherical robot created in 1996 by scientists and engineers from Finland under the direction of Prof. A. Halme [3] was also controlled by displacing the center of mass (see Fig. 25). The robot had one active wheel and one swivel wheel. The active wheel had a point of contact with the sphere in the lower part, above this wheel a control unit was installed, and the swivel wheel contacted with the sphere in the upper part. In the course of time this principle of controlling spherical robots became widespread, many robots with various internal wheeled vehicles were designed, and control schemes and algorithms were improved. A similar design was implemented in the robot ball developed by researchers from the Lomonosov Moscow State University. Its diagram and photograph are shown in Fig. 26. A control law is considered in [106] to prevent oscillations of the robot in longitudinal motion.

Fig. 25. Diagram and photograph of the first spherical robot with an internal wheeled vehicle

Fig. 26. Diagram and photograph of a robot ball with a one-wheeled internal vehicle (the picture is taken from [105])

Another modification that has gained widespread use as a robot for entertainment and teaching is a robot with an internal two-wheeled vehicle as shown in Fig. 27. This robot has the same shortcoming as the robot with one driving wheel, namely, the impossibility of the motion

of the spherical robot in the direction parallel to the axis of rotation of the wheels, which limits its maneuverability. Investigations of the dynamics and control problems are addressed in [107].

Fig. 27. Explosion model of a spherical robot with an internal two-wheeled vehicle

Robots with highly maneuverable vehicles exclude this shortcoming, although their designs are more sophisticated. For the first time, the design of a spherical robot with an internal vehicle with omniwheels was proposed in [47]. A photograph of a robot named Omnicron is shown in Fig. 28. In this design, use is made of standard omniwheels whose axes of the rollers' rotation lie in the plane of the wheel rim, and the control is performed using the kinematic model. But as the authors of [47] have noted, the agreement of the experimental trajectory of motion with the results of modeling using the kinematic model proposed by them did not exceed 50%.

Fig. 28. Diagram of the spherical robot "Omnicron" and photograph of the internal highly maneuverable vehicle with omniwheels (the picture is taken from [47])

The authors of [108-110] have been able to considerably improve, by changing the profile of the wheel roller, the accuracy of the motion of the spherical robot with an internal omniwheeled

platform along the trajectory within the framework of the kinematic model, and the construction of control using gaits (maneuvers connecting stable stationary states) allowed them to ensure a higher accuracy of positioning [111, 112].

Fig. 29. Design of a spherical robot with an internal omniwheeled platform

An interesting robot design was proposed by the authors of [113], when a homogeneous ball was set in motion inside a hollow sphere by means of special rollers. The design of the robot is presented in Fig. 30. A homogeneous ball placed inside a hollow sphere and ensuring that the position of the center of mass of the system changes during motion is used to set the spherical robot in motion.

Fig. 30. 3D model, photograph of a spherical robot actuated by an internal ball (the picture is taken from [113])

The rotation of the ball itself is ensured by two driving rollers whose axes of rotation are perpendicular. Mechanically they are fastened on a platform located above the ball, and to exclude sliding between the platform and the internal surface of the sphere, and between the platform and the ball, additional supports such as a spherical wheel are used.

In [114], the possibility of controlling a spherical robot using two omniwheels is considered. And in 2019 the authors of [115] suggested using a vehicle with one omniwheel; also, they

suggested using another drive for turnings, namely, a drive deflecting the vehicle relative to the sphere. However, this design received no further development.

3.2.2. Robots with a pendulum inside

Robots with a pendulum drive are no less widespread. There are modifications that use inside the robot several pendulums, and they are usually fastened at the geometrical center of the sphere and do not come in contact with it. In contrast to spherical robots inside which wheeled vehicle move, spherical robots with a pendulum drive are designed so that their internal mechanisms are fastened to the shell, which makes it possible, if necessary, to ensure its exact orientation and positioning. The main difference of the existing modifications from each other lies in the number of degrees of freedom of the pendulum and in the mechanisms of their fastening and actuation. The pendulum mechanism is the most widespread in the model problems of investigating the stability and planning of the trajectory of spherical robots.

The simplest example of a spherical robot with an internal pendulum is the toy "Squiggle Ball". It has inside it a mass (ballast or a pendulum) rotating about the axis connecting two hemispheres. The change in the position of the center of mass due to the rotation of the pendulum sets the ball in motion. The toy has no control, and the change of direction occurs after collision with obstacles in its way.

A robot in the form of a sphere with external pendulums placed outside its surface can be regarded as another prototype of a spherical robot with a pendulum drive. The design of this robot was developed by Koshiyama in 1992 [2]. His robot has external sensors which allow the robot to get its bearings in space. These sensors are installed on the opposite sides of the ball.

However, one of the first prototypes of spherical robots with an internal pendulum mechanism was developed under the direction of Prof. A. Halme in 2001 at the Technical University in Helsinki [116]. The prototype whose photograph is presented in Fig. 31 is a pendulum with two degrees of freedom. A special feature of this prototype is that the horizontal axis of rotation of the pendulum which is responsible for the rolling motion of the spherical robot is capable of turning relative to the vertical. This is made possible by fastening it and owing to the ability to move along a special rim in the diametral plane of the spherical shell.

Fig. 31. A spherical robot with a two-degree-of-freedom internal pendulum _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(4), 709-750

This modification was developed and manufactured (although this is mentioned very seldom in the publications of Finnish researchers) by the team of the closed joint-stock company "ROVER" in collaboration with the open joint-stock company "VNIITransmash" in Saint Petersburg [117]. In addition, it was developed further by adding another degree of freedom of the pendulum (see Fig. 32). The special features of the design are described in [82], and the control system for a spherical robot with an internal three-degree-of freedom pendulum mechanism was developed under the guidance of Prof. V. E. Pavlovskii [118, 119]. But this design had a number of shortcomings, first of all, the deformation of the spherical shell resulted in the elements of mechanisms being deformed, which often led to the loss of engagement in power transmissions, which partially could be eliminated in later inventions of the team [120], but the design turned out to be fairly sophisticated and demanding with respect to the accuracy of manufacture of the mechanical elements.

Fig. 32. A spherical robot with a three-degree-of-freedom internal pendulum

Modifications in the form of prototypes of spherical robots with a diameter of 350 mm, with a pendulum drive, created under the direction of Prof. A. Halme, have led to the creation of a prototype of a spherical robot of combined type intended for exploration of Mars and actuated by a two-degree-of-freedom pendulum and wind. Its design is shown in Fig. 13 and is described in detail in the dissertation of T. Ylikorpi [37]. The spherical robot had impressive dimensions (1.3 and 6 meters in diameter), which allowed it to easily negotiate considerable obstacles.

A similar design of the spherical robot with a two-degree-of-freedom pendulum was developed and manufactured at the South-West State University under the direction of Prof. S. F. Yat-sun in 2011 [121].

In [122], the motion the spherical robot is generated by a cylindrical actuated joint acting like a 2-DoF pendulum. This design allows to have a nearly empty upper hemisphere inside the spherical shell, which is dedicated to payloads adapted to the application.

In the opinion of the authors of [123-127], robots with two pendulums are more maneuverable and controllable. A design that captures all existing prototypes is shown in Fig. 33.

The authors of [128] proposed the design of a spherical robot with 4 internal pendulums. The arrangement of the pendulums is shown in Fig. 34. Examples of specifying the control actions for a simple rectilinear trajectory are given and the results of modeling for several sets of controls are shown.

Another modification of pendulum mechanisms that are used to actuate spherical robots is the gimbal mechanism. The designs and mathematical models describing their dynamics are

outer shell

linear

motor

bearing

guide pendulum

Fig. 33. A spherical robot with two internal pendulums (the picture is taken from [123])

Fig. 34. A spherical robot with four internal pendulums (the picture is taken from [128])

Fig. 35. Diagram of the pendulum principle of actuation with a gimbal mechanism (the picture is taken from [55])

considered in [129-134]. The general layout of these mechanisms is shown in Fig. 35, and Fig. 36 shows some full-scale specimens in which this propulsion method is employed.

L. I. Nadeina (Moscow State University of Technology "Stankin") [135] was one of the first to propose such a mechanism in 2005. The robot ball whose diagram is presented in Fig. 37 is set in motion by two motors, one of which is installed immediately on the internal surface of the sphere, and a sector in the form of a quarter of a circle is fastened to it; at the end of this sector the second motor is installed which causes another sector to rotate which plays the role of weight necessary for displacement of the center of mass of the system.

Fig. 36. Photographs of the prototypes of spherical robots actuated by a gimbal mechanism (the left picture is taken from [129], and the right picture is taken from [131])

Fig. 37. Diagram of the robot ball proposed by L. I. Nadeina (the picture is taken from [135])

3.2.3. Mechanisms of linear displacement for the offset of the center of mass

The decision to use the mechanisms of linear displacement for the offset of the center of mass of a spherical robot was first proposed in the work of researchers of the Michigan University [26, 136]. Although in their theoretical study [26] they proposed a kinematic scheme for radial displacement of masses (see Fig. 38), it was the mechanism of motion of masses along some mutually perpendicular axes that was applied in practice. Photographs of the developed prototype are shown in Fig. 39. Figure 40 shows a scheme with radial linear guides. It should be noted that most of the studies dealing with this principle of propulsion are only theoretical treatments describing the dynamics of motion, and their results have not been implemented in practice [137-140].

An interesting example of practical implementation is that of an art object (see Fig. 41) in the form of a sphere 3 m in diameter [141], with plants situated inside it, which is capable of moving by changing the position of the center of mass by means of linear guides.

In [142], the linear change in the position of weights (permanent magnets) is performed by means of solenoids. Also, the control system allows a recovery of energy which, in the authors' opinion, is generated when the robot comes to a stop or moves under the action of external forces, for example, wind. A diagram of this spherical robot is presented in Fig. 42.

Fig. 39. Photograph of a spherical robot with masses moving in perpendicular directions (the picture is taken from [136])

Fig. 40. A spherical robot with radial linear guides

The patent of G. A. Prokopovich [143] presents the design of a spherical robot with an internal mechanism using parallel kinematics, namely, four linear drives moving in a coordinated

Fig. 41. Designer's art object — a spherical robot created at the Bartlett School of Architecture

Batten-reinforced Mylar exterior

Fig. 42. The model of a spherical robot with linear guides in the form of a solenoid (the picture is taken from [142])

Fig. 43. Diagram of a robot ball with four linear guides (the picture is taken from [142])

fashion the mass inside the spherical shell. A diagram of the mechanism is shown in Fig. 43, and a description and approaches to control are presented in [144].

A similar design was proposed in [145], in which the change in the position of the center of mass is ensured by four cord mechanisms.

In addition to the well-known methods and mechanisms of changing the position of the center of mass which are described in this chapter, new ones appear, albeit mainly only in concepts. For example, the authors of [146] describe the design of a spherical robot in which the position of the center of mass is changed by means of a special hydraulic system consisting of several closed channels along which the working fluid displaces some mass. A diagram of the prototype is presented in Fig. 44.

Fig. 44. Diagram of a robot ball with a hydraulic system for changing the position of the center of mass (the picture is taken from [146])

3.3. Robots moving by deforming their form

Another method of actuating the spherical robot is to change the form of its shell. This method is also fairly widespread because it allows the spherical robot to negotiate obstacles, and some modifications based on this principle of propulsion enable a loss of contact with the surface, i.e., they allow a jump.

For the first time, a similar design was proposed by the authors of [147, 148], researchers from the Ritsumeikan University in Japan. Photographs of the prototypes developed by them are shown in Fig. 45. The motion is implemented by changing the length of drives with memory of the form; the drives are arranged radially and connect the center of the sphere (or the cylinder) and the shell. It should be noted that external cable power supply was used for these prototypes, and the robots had no payload during motion.

In [149] the authors have developed a prototype of the spherical robot consisting of 20 silicone segments which form a spherical shell. Each segment is made of silicone and has a cavity which is filled with a special suspension; changing its pressure leads to deformation of the shell. An image of the prototype is shown in Fig. 46.

This approach was developed further in [150]. The shell of their spherical robot also consisted of segments, but in contrast to the preceding modifications, the valve and the electronics that controlled the change in pressure were built immediately into the segment, which made it possible

Fig. 45. Prototypes of a robot wheel and a spherical robot moving by deformation

Fig. 46. A spherical robot with varying form with low internal pressure

Robot Hemispherical Superstructure

Fig. 47. A spherical robot with varying form and a cross-section of its 3D model (the picture is taken from [150])

to achieve larger autonomy. The motion of the sphere was implemented by widening the segments that surround the segment which is in contact with the surface. An image of the prototype and a cross-section of its 3D model are presented in Fig. 47. Other studies dealing with designs of spherical robots with a pneumatically deformable shell include [151-153].

Another modification of spherical robots is a rolling robot which interacts discretely with the surface. Photographs of prototypes of such robots are shown in Fig. 48. Their motion is implemented by changing the length of the rod, a "leg" contacting with the surface.

(a) (b) (c)

Fig. 48. Discretely interacting rolling robots a) Dodecahedral Robot [154] b) "Spike" Robot [155] c) Robotic All-Terrain Surveyor (RATS)

The active development of robots discretely interacting with the surface, especially in recent years, is aimed at creating and investigating robots that have the structure of tensegrity [156,157]. A photograph of one of the prototypes is presented in Fig. 49.

Fig. 49. A tensegrity robot during tests (the picture is taken from [156])

3.4. Combination of several methods of actuation

Most of the modern designs of spherical robots involve a combination of the actuation methods described above. This makes it possible, on the one hand, to simplify the design of the spherical robot and, on the other hand, to enhance the efficiency of propulsion for each particular case of application of the mechanisms presented above, while using the advantages of each of them. Such a combination of mechanisms is often an interesting motion control and planning problem. In what follows, we consider the most important and interesting papers and designs of combined mechanisms of spherical robots.

One of the most popular ways of combination of the above-mentioned mechanisms is the combination of mechanisms of changing the position of the center of mass with mechanisms of changing the angular momentum. These include mainly vehicles and pendulums with flywheels (handwheels) installed on them.

One of the first proposed designs of combined type applied as a real prototype is the design of the robot Volvot described in [158] and shown in Fig. 50. A special feature of the design is that the internal mechanisms ensuring the change in the position of the center of mass and the change in the internal angular momentum are rigidly fastened on the internal surface of the spherical shell. The rolling motion of the spherical robot is implemented by means of two eccentric flywheels, and the rotation about the vertical is achieved by rotating the platform with mechanisms about the shell.

Planar Field

Fig. 50. Photograph of the prototype and the layout of the spherical robot Volvot (the picture is taken from [158])

In [159], a description is given of the design of a spherical robot with an internal pendulum mechanism containing three one-degree-of-freedom pendulums fastened on the same axis (see Fig. 51). On one of the pendulums, a flywheel is fastened whose rotation axis is parallel to the rotation axis of the pendulum. The maximal angular velocity of the flywheel is 1600 rev/s, and the mechanism of fast braking allows one to ensure a high absolute value of the angular acceleration of the flywheel, which produces an angular momentum sufficient to negociate obstacles and even a jump of the robot.

Fig. 51. Combined spherical robot with a flywheel (the picture is taken from [159])

Figure 52 shows photographs of the prototypes of spherical robots of combined type whose center of mass is changed by means of various mechanisms: by means of a one-degree-of-freedom

pendulum [160] (Fig. 52a) and by means of a wheeled vehicle moving along the rim [161] (Fig. 52b). The flywheel installed on the pendulum or the wheeled vehicle in the lowest possible position so that its axis of rotation passes through the center of the spherical shell ensures turnings of the robot. The algorithms for controlling such robots are based on the investigation of the kinematics and dynamics of the motion of the model of a spherical robot with the Lagrange pendulum [162]. An analysis is also made of the control algorithm using gaits which takes into account viscous rolling friction [163], and the stabilization of its motion [164, 165] is investigated.

(a) (b)

Fig. 52. Photographs of spherical robots of combined type a) with a pendulum mechanism b) with a moving platform

A high-tech design of combined type is that of the prototype of a gyrostabilized spherical vehicle [166].

In this modification, the center of mass changes by rolling along the rim of the spherical shell of the wheeled vehicle. Turnings are performed using the mechanism of linear displacement of the center of mass in the direction perpendicular to that of the rolling motion of the sphere. The motion is stabilized using a pair of flywheels. The angles of inclination of the rotation axes of the flywheels are changed in opposite directions by means of a special power transmission from one electric motor. A photograph of the prototype and the layout of the internal mechanisms are shown in Fig. 53. Although experiments demonstrating the functioning of the stabilization system [166] had been conducted for the developed prototype, in particular, for the case where it moves on a rough surface and along a stretched rope, no published research describing the dynamics and adjustment of the stabilization system had been found by the time when this paper was written.

The authors of [167] present the design of the spherical robot "Spring Pendulum" with a pendulum drive having a soft deformable shell, which is assumed to be nondeformable in the process of simulation (after equivalent transformations), but the pendulum has no rigid point of attachment to the shell. The connection of the pendulum with the shell is modeled by a spring.

Figure 54 shows photographs of the prototype of a spherical robot with four pendulums. The pendulums can move outside beyond the spherical shell through special holes in the shell owing to the kinematics of the actuating mechanisms. This can be used as a means to negociate obstacles and as arresters if it is necessary to fix the spherical robot in an unstable position. A detailed description of the kinematics and dynamics of the robot and various schemes and regimes of motion is given in [168].

Fig. 53. A gyrostabilized spherical robot with two flywheels [166] a) photograph of the prototype b) 3D model of internal mechanisms

Fig. 54. A spherical robot with four pendulums enabling the motion that makes repulsion from the surface possible (the picture is taken from [168])

Fig. 55. A spherical robot with two extendable arms for negociating obstacles (the picture is taken from [169])

To ensure the possibility of negociating obstacles, the simply designed spherical robot consisting of two hemispheres each of which is rotated by a separate drive was equipped by the authors of [169] with extendable arms. The three-dimensional model of this robot, KisBot, which explains the design, is presented in Fig. 55. The experiments conducted by the authors

have shown that such a mechanism of extendable arms can be used both for rolling and for negociating obstacles.

The designs no less promising from the viewpoint of application and the ability to negociate obstacles are those of spherical robots in which the shell can be transformed, for example, to a walking robot, as in [170, 171]. An example of the design of such robots is presented in Fig. 56.

> -LL

Fig. 56. A combined spherical robot with a transformable shell (the picture is taken from [170])

A spherical robot named TumbelBot [172] was presented at the international conference devoted to intelligent robots and systems (IROS 2022) in Japan. This robot is a spherical millipede which moves using growing tentacle legs, into which sensors have been built to obtain information on the environment and the surface on which the robot moves. A photograph of the robot is shown in Fig. 57.

Fig. 57. A spherical robot presented in the exhibition IROS2022 (the picture is taken from [172])

Some designs involve combining the mechanisms which enable not only motion on the surface, but also motion in various media, for example, on a solid surface, in the air or in a fluid. In [173], a description is given of the design of a spherical robot which uses a pendulum mechanism for rolling and in which a flywheel is used for rotation about the vertical and for turnings. Special ribs on the external surface of the sphere and the hermetic shell allow the spherical robot to move in a fluid. The three-dimensional model and a photograph of its motion in a fluid are shown in Fig. 58.

Figure 59 shows a spherical robot designed to move in a fluid [174] by means of three pumps which form fluid flows. The drive system allows each pump (water-jet propulsion device) not only to set the required propulsion force, but also direction.

Fig. 58. A combined spherical robot for motion in a fluid (the picture is taken from [173])

In recent years, a lot of effort has gone into developing flying robots, including those in the form of a sphere [175-177]. The shell of the robot often has the form of a sphere and is used to protect internal screws and elements of the control system against collisions. It also allows the robot to roll, which is achieved by means of the aerodynamic thrust of the screws or by displacing the center of mass. Examples of the structure of spherical robots are shown in Fig. 60.

Fig. 60. Examples of flying robots in the form of a sphere (the left representation is taken from [175], and the right one from [176])

4. Conclusions

The increasing number of publications devoted to spherical robots, new designs and areas of application show a great importance of studies aimed at developing new designs of spherical robots and at controlling them. The results of analysis of the investigations presented in this work allow conclusions on a number of important problems whose solution involves the development of new propulsion mechanisms of spherical robots which ensure their maneuverability, simplicity and accuracy of control, and on problems aimed at investigating and describing the contact interaction of the spherical shell with the surface on which the spherical robot moves.

At the same time, examples can be given of the application of robots in the form of a sphere in practice, namely, as service robots and robots for teaching and entertainment. Further development and introduction of this kind of robotic systems involves, first of all, engineering development and research work, tests of designs under real conditions of their functioning, and development of specialized drives and payloads.

Conflict of interest

The author declares that he has no conflicts of interest.

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