Mathematics in Robot Control for Theoretical and Applied Problems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 29008

Special Issue Editors


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Guest Editor
Department of Applied Physics, Autonomous University of Baja California, Mexicali 21100, Mexico
Interests: automated metrology; 3D coordinates measurement; robotic navigation; machine vision; simulation of the robotic swarms behaviour
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Guest Editor
Department of Computer-Integrated Technologies, Automation and Mechatronics, Kharkiv National University of Radio Electronics, 61166 Kharkiv, Ukraine
Interests: control systems; mechatronics; robotics; decision-making; computer vision
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Technological development has not only boosted the use of mechanical systems for industrial uses but above all has made it possible for them to be used in areas and sectors unimaginable until a few years ago. Mechatronics is the neologism which now indicates in general modern robotic systems which are to be equipped with sophisticated electronic control devices. Such devices are capable of helping systems to achieve high performance and allowing their use and disparate aspects of our daily life. It is a synergy set which can radically change some aspects of the production world. A growing interest toward robots, a special class of mechanical systems, as well as fear and perplexity in relation to the impact that these systems have in the world of productivity, and then ultimately their social impact, has be witnessed in recent years. Future robotics represent a tremendous challenge in the field of mathematics because of the central role their control plays in the context of this field. In fact, robot control is one of the most important and challenging topics for mathematicians, engineers, physicians, and practitioners. Mathematical issues are the kernel of the design of control of movements and performance of robots. This Special Issue aims to collect the latest advancements of mathematical methods for solving not only theoretical but also applied problems of classical and also modern robot structures, such as robotic manipulators, walking robots, flexible robots, haptic robots, and any kind of old and new mechanisms with all possible tasks, in grasp, manipulation, and motion for any kind of their possible issues and applications.

Prof. Dr. Paolo Mercorelli
Dr. Oleg Sergiyenko
Dr. Oleksandr Tsymbal
Guest Editors

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Keywords

  • mathematics problems in robot motion and grasp
  • robot control
  • robot optimal control
  • robot stability and analysis of the dynamics

Published Papers (13 papers)

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Research

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18 pages, 622 KiB  
Article
Distributed Finite-Time Coverage Control of Multi-Quadrotor Systems with Switching Topology
by Hilton Tnunay, Kaouther Moussa, Ahmad Hably and Nicolas Marchand
Mathematics 2023, 11(12), 2621; https://doi.org/10.3390/math11122621 - 8 Jun 2023
Cited by 1 | Viewed by 983
Abstract
This paper studies the distributed coverage control problem of multi-quadcopter systems connected with fixed and switching network topologies to guarantee the finite-time convergence. The proposed method modifies the objective function originating from the locational optimization problem to accommodate the consensus constraint and solves [...] Read more.
This paper studies the distributed coverage control problem of multi-quadcopter systems connected with fixed and switching network topologies to guarantee the finite-time convergence. The proposed method modifies the objective function originating from the locational optimization problem to accommodate the consensus constraint and solves the problem within a given time limit. The coverage problem is solved by sending angular-rate and thrust commands to the quadcopters. By exploiting the finite-time stability theory, we ensure that the rotation and translation controllers of the quadcopters are finite-time stable both in fixed and switching communication topologies, able to be implemented distributively, and able to collaboratively drive the quadcopters towards the desired position and velocity of the Voronoi centroid independent of their initial states. After carefully designing and analyzing the performance, numerical simulations using a Robot Operating System (ROS) and Gazebo simulator are presented to validate the effectiveness of the proposed control protocols. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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19 pages, 3389 KiB  
Article
Mathematical Methods for an Accurate Navigation of the Robotic Telescopes
by Vadym Savanevych, Sergii Khlamov, Oleksandr Briukhovetskyi, Tetiana Trunova and Iryna Tabakova
Mathematics 2023, 11(10), 2246; https://doi.org/10.3390/math11102246 - 11 May 2023
Cited by 12 | Viewed by 1137
Abstract
Accurate sky identification is one of the most important functions of an automated telescope mount. The more accurately the robotic telescope is navigated to the investigated part of the sky, the better the observations and discoveries made. In this paper, we present mathematical [...] Read more.
Accurate sky identification is one of the most important functions of an automated telescope mount. The more accurately the robotic telescope is navigated to the investigated part of the sky, the better the observations and discoveries made. In this paper, we present mathematical methods for accurate sky identification (celestial coordinates determination). They include the automatic selection of the reference stars, preliminary and full sky identification, as well as an interaction with international databases, which are a part of the astrometric calibration. All described methods help to receive accurately calculated astrometric data and use it for the positional calibration and better navigation of the automated telescope mount. The developed methods were successfully implemented in the Collection Light Technology (CoLiTec) software. Through its use, more than 1600 small solar system objects were discovered. It has been used in more than 700,000 observations and successful sky identifications, during which, five comets were discovered. Additionally, the accuracy indicators of the processing results of the CoLiTec software are provided in the paper, which shows benefits of the CoLiTec software and lower standard deviation of the sky identification in the case of low signal-to-noise ratios. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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17 pages, 1729 KiB  
Article
Neural Network Based Control of Four-Bar Mechanism with Variable Input Velocity
by R. Peón-Escalante, Manuel Flota-Bañuelos, Roberto Quintal-Palomo, Luis J. Ricalde, F. Peñuñuri, B. Cruz Jiménez and J. Avilés Viñas
Mathematics 2023, 11(9), 2148; https://doi.org/10.3390/math11092148 - 4 May 2023
Viewed by 1395
Abstract
For control applications, the angular velocity of the drive crank of a four-bar mechanism is traditionally assumed to be constant. In this paper, we propose control of variable velocity of the drive crank to obtain the desired output motions for the coupler point. [...] Read more.
For control applications, the angular velocity of the drive crank of a four-bar mechanism is traditionally assumed to be constant. In this paper, we propose control of variable velocity of the drive crank to obtain the desired output motions for the coupler point. To estimate the reference trajectory for the crank velocity, a neural network is trained with data from the kinematic model. The control law is designed from feedback linearization of the tracking error dynamics and a Proportional–Integral–Derivative (PID) controller. The applicability of the proposed scheme is validated through simulations for three variable speed profiles, obtaining excellent results from the system. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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24 pages, 17366 KiB  
Article
CISA: Context Substitution for Image Semantics Augmentation
by Sergey Nesteruk, Ilya Zherebtsov, Svetlana Illarionova, Dmitrii Shadrin, Andrey Somov, Sergey V. Bezzateev, Tatiana Yelina, Vladimir Denisenko and Ivan Oseledets
Mathematics 2023, 11(8), 1818; https://doi.org/10.3390/math11081818 - 11 Apr 2023
Cited by 5 | Viewed by 1802
Abstract
Large datasets catalyze the rapid expansion of deep learning and computer vision. At the same time, in many domains, there is a lack of training data, which may become an obstacle for the practical application of deep computer vision models. To overcome this [...] Read more.
Large datasets catalyze the rapid expansion of deep learning and computer vision. At the same time, in many domains, there is a lack of training data, which may become an obstacle for the practical application of deep computer vision models. To overcome this problem, it is popular to apply image augmentation. When a dataset contains instance segmentation masks, it is possible to apply instance-level augmentation. It operates by cutting an instance from the original image and pasting to new backgrounds. This article challenges a dataset with the same objects present in various domains. We introduce the Context Substitution for Image Semantics Augmentation framework (CISA), which is focused on choosing good background images. We compare several ways to find backgrounds that match the context of the test set, including Contrastive Language–Image Pre-Training (CLIP) image retrieval and diffusion image generation. We prove that our augmentation method is effective for classification, segmentation, and object detection with different dataset complexity and different model types. The average percentage increase in accuracy across all the tasks on a fruits and vegetables recognition dataset is 4.95%. Moreover, we show that the Fréchet Inception Distance (FID) metrics has a strong correlation with model accuracy, and it can help to choose better backgrounds without model training. The average negative correlation between model accuracy and the FID between the augmented and test datasets is 0.55 in our experiments. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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21 pages, 5796 KiB  
Article
Autonomous Navigation System of Indoor Mobile Robots Using 2D Lidar
by Jian Sun, Jie Zhao, Xiaoyang Hu, Hongwei Gao and Jiahui Yu
Mathematics 2023, 11(6), 1455; https://doi.org/10.3390/math11061455 - 17 Mar 2023
Cited by 5 | Viewed by 4159
Abstract
Significant developments have been made in the navigation of autonomous mobile robots within indoor environments; however, there still remain challenges in the face of poor map construction accuracy and suboptimal path planning, which limit the practical applications of such robots. To solve these [...] Read more.
Significant developments have been made in the navigation of autonomous mobile robots within indoor environments; however, there still remain challenges in the face of poor map construction accuracy and suboptimal path planning, which limit the practical applications of such robots. To solve these challenges, an enhanced Rao Blackwell Particle Filter (RBPF-SLAM) algorithm, called Lidar-based RBPF-SLAM (LRBPF-SLAM), is proposed. In LRBPF, the adjacent bit poses difference data from the 2D Lidar sensor which is used to replace the odometer data in the proposed distribution function, overcoming the vulnerability of the proposed distribution function to environmental disturbances, and thus enabling more accurate pose estimation of the robot. Additionally, a probabilistic guided search-based path planning algorithm, gravitation bidirectional rapidly exploring random tree (GBI-RRT), is also proposed, which incorporates a target bias sampling to efficiently guide nodes toward the goal and reduce ineffective searches. Finally, to further improve the efficiency of navigation, a path reorganization strategy aiming at eliminating low-quality nodes and improving the path curvature of the path is proposed. To validate the effectiveness of the proposed method, the improved algorithm is integrated into a mobile robot based on a ROS system and evaluated in simulations and field experiments. The results show that LRBPF-SLAM and GBI-RRT perform superior to the existing algorithms in various indoor environments. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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16 pages, 4065 KiB  
Article
Explicit Identification of Pointwise Terrain Gradients for Speed Compensation of Four Driving Tracks in Passively Articulated Tracked Mobile Robot
by Haneul Jeon and Donghun Lee
Mathematics 2023, 11(4), 905; https://doi.org/10.3390/math11040905 - 10 Feb 2023
Cited by 1 | Viewed by 1139
Abstract
Tracked mobile robots can overcome the limitations of wheeled and legged robots in environments, such as construction and mining, but there are still significant challenges to be addressed in terms of trajectory tracking. This study proposes a kinematic strategy to improve the trajectory-tracking [...] Read more.
Tracked mobile robots can overcome the limitations of wheeled and legged robots in environments, such as construction and mining, but there are still significant challenges to be addressed in terms of trajectory tracking. This study proposes a kinematic strategy to improve the trajectory-tracking performance of a PASTRo (Passively Articulated Suspension based Track-typed mobile robot), which comprises four tracks, two rockers, a differential gear, and a main body. Due to the difficulties in explicitly identifying track-terrain contact angles, suspension kinematics is used to identify track-terrain contact angles (TTCA) in arbitrarily rough terrains. Thus, the TTCA-based driving velocity projection method is proposed in this study to improve the maneuverability of PASTRo in arbitrarily rough terrains. The RecurDyn-Simulink co-simulator is used to examine the improvement of PASTRo compared to a tracked mobile robot non-suspension version. The results indicate that PASTRo has a 33.3% lower RMS(Root Mean Square) distance error, 56.3% lower RMS directional error, and 43.2% lower RMS offset error than the four-track skid-steer mobile robot (SSMR), even with planar SSMR kinematics. To improve the maneuverability of PASTRo without any information on the rough terrain, the TTCA is calculated from the suspension kinematics, and the TTCA obtained is used for both TTCA-based driving velocity projection methods. The results show that PASTRo, with the TTCA-based driving velocity projection method, has a 39.2% lower RMS distance error, 57.9% lower RMS directional error, and 51.9% lower RMS offset error than the four-track SSMR. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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18 pages, 8899 KiB  
Article
A Cooperative Control Algorithm for Line and Predecessor Following Platoons Subject to Unreliable Distance Measurements
by Carlos Escobar, Francisco J. Vargas, Andrés A. Peters and Gonzalo Carvajal
Mathematics 2023, 11(4), 801; https://doi.org/10.3390/math11040801 - 4 Feb 2023
Cited by 2 | Viewed by 1379
Abstract
This paper uses a line-following approach to study the longitudinal and lateral problems in vehicle platooning. Under this setup, we assume that inter-vehicle distance sensing is unreliable and propose a cooperative control strategy to render the platoon less vulnerable to these sensing difficulties. [...] Read more.
This paper uses a line-following approach to study the longitudinal and lateral problems in vehicle platooning. Under this setup, we assume that inter-vehicle distance sensing is unreliable and propose a cooperative control strategy to render the platoon less vulnerable to these sensing difficulties. The proposed control scheme uses the velocity of the predecessor vehicle, communicated through a Vehicle-to-Vehicle technology, to avoid significant oscillations in the local speed provoked by tracking using unreliable local distance measurements. We implement the proposed control algorithm in the RUPU platform, a low-cost experimental platform with wireless communication interfaces that enable the implementation of cooperative control schemes for mobile agent platooning. The experiments show the effectiveness of the proposed cooperative control scheme in maintaining a suitable performance even when subject to temporal distortions in local measurements, which, in the considered experimental setup, arise from losing the line-of-sight of the local sensors in paths with closed curves. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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19 pages, 6982 KiB  
Article
Active Disturbance Rejection Strategy for Distance and Formation Angle Decentralized Control in Differential-Drive Mobile Robots
by Mario Ramírez-Neria, Jaime González-Sierra, Alberto Luviano-Juárez, Norma Lozada-Castillo and Rafal Madonski
Mathematics 2022, 10(20), 3865; https://doi.org/10.3390/math10203865 - 18 Oct 2022
Cited by 5 | Viewed by 1638
Abstract
The important practical problem of robust synchronization in distance and orientation for a class of differential-drive mobile robots is tackled in this work as an active disturbance rejection control (ADRC) problem. To solve it, a kinematic model of the governed system is first [...] Read more.
The important practical problem of robust synchronization in distance and orientation for a class of differential-drive mobile robots is tackled in this work as an active disturbance rejection control (ADRC) problem. To solve it, a kinematic model of the governed system is first developed based on the distance and formation angle between the agents. Then, a special high-order extended state observer is designed to collectively estimate the perturbations (formed by longitudinal and lateral slipping parameters) that affect the kinematic model. Finally, a custom error-based ADRC approach is designed and applied assuming that the distance and orientation between the agents are the only available measurements. The proposed control strategy does not need time-derivatives of the reference trajectory, which increases the practical appeal of the proposed solution. The experimental results, obtained in laboratory conditions with a set of differential-drive mobile robots operating in a leader–follower configuration, show the effectiveness of the proposed governing scheme in terms of trajectory tracking and disturbance rejection. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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40 pages, 7697 KiB  
Article
Kinematic Modelling for Hyper-Redundant Robots—A Structured Guide
by Diego Cerrillo, Antonio Barrientos and Jaime Del Cerro
Mathematics 2022, 10(16), 2891; https://doi.org/10.3390/math10162891 - 12 Aug 2022
Cited by 4 | Viewed by 1816
Abstract
Obtaining mathematical equations to model the kinematics of a hyper-redundant robot is not intuitive and of greater difficulty than for traditional robots. Depending on the characteristics of the robot, the most appropriate methodology to approach the modelling may be one or another. This [...] Read more.
Obtaining mathematical equations to model the kinematics of a hyper-redundant robot is not intuitive and of greater difficulty than for traditional robots. Depending on the characteristics of the robot, the most appropriate methodology to approach the modelling may be one or another. This article provides a general overview of the different approaches there are when modelling a hyper-redundant cable-driven robot, while proposing a guide to help the novel researcher that approaches this field decide which methodology to apply when modelling a robot. After providing some definitions, a simple framework to understand all the underlying models is presented. Afterwards, the mathematical equations for the most important methods of modelling are developed. Finally, the proposal for a step-by-step tutorial is included, and it is exemplified by applying it to three real robots. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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22 pages, 591 KiB  
Article
Towards the Sign Function Best Approximation for Secure Outsourced Computations and Control
by Mikhail Babenko, Andrei Tchernykh, Bernardo Pulido-Gaytan, Arutyun Avetisyan, Sergio Nesmachnow, Xinheng Wang and Fabrizio Granelli
Mathematics 2022, 10(12), 2006; https://doi.org/10.3390/math10122006 - 10 Jun 2022
Cited by 2 | Viewed by 1570
Abstract
Homomorphic encryption with the ability to compute over encrypted data without access to the secret key provides benefits for the secure and powerful computation, storage, and communication of resources in the cloud. One of its important applications is fast-growing robot control systems for [...] Read more.
Homomorphic encryption with the ability to compute over encrypted data without access to the secret key provides benefits for the secure and powerful computation, storage, and communication of resources in the cloud. One of its important applications is fast-growing robot control systems for building lightweight, low-cost, smarter robots with intelligent brains consisting of data centers, knowledge bases, task planners, deep learning, information processing, environment models, communication support, synchronous map construction and positioning, etc. It enables robots to be endowed with secure, powerful capabilities while reducing sizes and costs. Processing encrypted information using homomorphic ciphers uses the sign function polynomial approximation, which is a widely studied research field with many practical results. State-of-the-art works are mainly focused on finding the polynomial of best approximation of the sign function (PBAS) with the improved errors on the union of the intervals [1,ϵ][ϵ,1]. However, even though the existence of the single PBAS with the minimum deviation is well known, its construction method on the complete interval [1,1] is still an open problem. In this paper, we provide the PBAS construction method on the interval [1,1], using as a norm the area between the sign function and the polynomial and showing that for a polynomial degree n1, there is (1) unique PBAS of the odd sign function, (2) no PBAS of the general form sign function if n is odd, and (3) an uncountable set of PBAS, if n is even. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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13 pages, 1623 KiB  
Article
Predicate-Based Model of Problem-Solving for Robotic Actions Planning
by Oleksandr Tsymbal, Paolo Mercorelli and Oleg Sergiyenko
Mathematics 2021, 9(23), 3044; https://doi.org/10.3390/math9233044 - 26 Nov 2021
Cited by 8 | Viewed by 1800
Abstract
The aim of the article is to describe a predicate-based logical model for the problem-solving of robots. The proposed article deals with analyses of trends of problem-solving robotic applications for manufacturing, especially for transportations and manipulations. Intelligent agent-based manufacturing systems with robotic agents [...] Read more.
The aim of the article is to describe a predicate-based logical model for the problem-solving of robots. The proposed article deals with analyses of trends of problem-solving robotic applications for manufacturing, especially for transportations and manipulations. Intelligent agent-based manufacturing systems with robotic agents are observed. The intelligent cores of them are considered from point of view of ability to propose the plans of problem-solving in the form of strategies. The logical model of adaptive strategies planning for the intelligent robotic system is composed in the form of predicates with a presentation of data processing on a base of set theory. The dynamic structures of workspaces, and a possible change of goals are considered as reasons for functional strategies adaptation. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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20 pages, 4100 KiB  
Article
Adaptive Active Disturbance Rejection Control of Solar Tracking Systems with Partially Known Model
by Sergio Isai Palomino-Resendiz, Norma Beatriz Lozada-Castillo, Diego Alonso Flores-Hernández, Oscar Octavio Gutiérrez-Frías and Alberto Luviano-Juárez
Mathematics 2021, 9(22), 2871; https://doi.org/10.3390/math9222871 - 11 Nov 2021
Cited by 4 | Viewed by 1824
Abstract
In this article, the trajectory tracking control of a solar tracking system is tackled by means of an adaptive active disturbance rejection control scheme. The state and disturbance estimation system is based on the combination of a time varying identification system and an [...] Read more.
In this article, the trajectory tracking control of a solar tracking system is tackled by means of an adaptive active disturbance rejection control scheme. The state and disturbance estimation system is based on the combination of a time varying identification system and an adaptive observer. The stability and robustness of the controller is mathematically tested by means of the second method of Lyapunov, and its effectiveness is experimentally tested in a robotic test bed, achieving both lower energy consumption and better tracking results with respect to a PID-based controller. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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Review

Jump to: Research

34 pages, 10502 KiB  
Review
Concrete 3D Printing: Process Parameters for Process Control, Monitoring and Diagnosis in Automation and Construction
by Tan Kai Noel Quah, Yi Wei Daniel Tay, Jian Hui Lim, Ming Jen Tan, Teck Neng Wong and King Ho Holden Li
Mathematics 2023, 11(6), 1499; https://doi.org/10.3390/math11061499 - 19 Mar 2023
Cited by 7 | Viewed by 4510
Abstract
In Singapore, there is an increasing need for independence from manpower within the Building and Construction (B&C) Industry. Prefabricated Prefinished Volumetric Construction (PPVC) production is mainly driven by benefits in environmental pollution reduction, improved productivity, quality control, and customizability. However, overall cost savings [...] Read more.
In Singapore, there is an increasing need for independence from manpower within the Building and Construction (B&C) Industry. Prefabricated Prefinished Volumetric Construction (PPVC) production is mainly driven by benefits in environmental pollution reduction, improved productivity, quality control, and customizability. However, overall cost savings have been counterbalanced by new cost drivers like modular precast moulds, transportation, hoisting, manufacturing & holding yards, and supervision costs. The highly modular requirements for PPVC places additive manufacturing in an advantageous position, due to its high customizability, low volume manufacturing capabilities for a faster manufacturing response time, faster production changeovers, and lower inventory requirements. However, C3DP has only just begun to move away from its early-stage development, where there is a need to closely evaluate the process parameters across buildability, extrudability, and pumpability aspects. As many parameters have been identified as having considerable influence on C3DP processes, monitoring systems for feedback applications seem to be an inevitable step forward to automation in construction. This paper has presented a broad analysis of the challenges posed to C3DP and feedback systems, stressing the admission of process parameters to correct multiple modes of failure. Full article
(This article belongs to the Special Issue Mathematics in Robot Control for Theoretical and Applied Problems)
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