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Mathematics
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Advanced Control of Complex Dynamical Systems and Robotics with Applications
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Advanced Control of Complex Dynamical Systems and Robotics with Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E2: Control Theory and Mechanics".

Deadline for manuscript submissions: 20 October 2025 | Viewed by 8027

Special Issue Editors

Dr. Gang Wang

E-Mail Website
Guest Editor
Institute of Machine Intelligence, University of Shanghai for Science and Technology, Shanghai 200093, China
Interests: distributed control; adaptive control; robotics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Chaoli Wang

E-Mail Website
Guest Editor
School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
Interests: nonlinear control; robot dynamic and control; visual servoing feedback control; pattern identification
Prof. Dr. Qingdu Li

E-Mail Website
Guest Editor
Institute of Machine Intelligence, University of Shanghai for Science and Technology, Shanghai 200093, China
Interests: biped robots; dynamic walking; nonlinear circuits; complex systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue showcases the latest research on the use of mathematical models and methods in the analysis and control of complex dynamical systems and robots.

These articles explore a range of topics, including nonlinear dynamics, adaptive control, optimal control, distributed control, and learning control. They present new theoretical results and practical applications that demonstrate the dynamics and control of complex systems and robots.

Specific applications of these techniques in different complex systems and robotic domains are addressed. The topics include the following:

  • The modelling and control of nonlinear systems using differential geometry and Lie group methods;
  • Designing adaptive controllers for uncertain systems using Lyapunov stability theory and sliding mode control techniques;
  • Robotic systems using optimal control theory and numerical optimization methods;
  • Learning control policies for autonomous robots using machine learning algorithms and reinforcement learning techniques;
  • Learning cross-domain knowledge for robotic interaction and comprehension using machine learning algorithms and large pretrained language models.

We hope this Special Issue will serve as a valuable resource for researchers and practitioners interested in this field. The articles provide a glimpse into the diverse range of mathematical techniques that can be applied to these fields while also highlighting the importance of collaboration between mathematicians and engineers in addressing real-world challenges.

Dr. Gang Wang
Prof. Dr. Chaoli Wang
Prof. Dr. Qingdu Li
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • modeling and control of robotics
  • machine learning algorithms
  • complex systems and stability analysis
  • distributed control of multiagent systems

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Published Papers (8 papers)

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Research

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17 pages, 2917 KiB  
Article
Combining Prior Knowledge and Reinforcement Learning for Parallel Telescopic-Legged Bipedal Robot Walking
by Jie Xue, Jiaqi Huangfu, Yunfeng Hou and Haiming Mou
Mathematics 2025, 13(6), 979; https://doi.org/10.3390/math13060979 - 16 Mar 2025
Viewed by 359
Abstract
The parallel dual-slider telescopic leg bipedal robot (L04) is characterized by its simple structure and low leg rotational inertia, which contribute to its walking efficiency. However, end-to-end methods often overlook the robot’s physical structure, leading to difficulties in maintaining the parallel alignment of [...] Read more.
The parallel dual-slider telescopic leg bipedal robot (L04) is characterized by its simple structure and low leg rotational inertia, which contribute to its walking efficiency. However, end-to-end methods often overlook the robot’s physical structure, leading to difficulties in maintaining the parallel alignment of the dual sliders, which in turn compromises walking stability. One potential solution to this issue involves utilizing imitation learning to replicate human motion data. However, the dual telescopic leg structure of the L04 robot makes it difficult to perform motion retargeting of human motion data. To enable L04 walking, we design a method that integrates prior feedforward with reinforcement learning (PFRL), specifically tailored for the parallel dual-slider structure. We utilize prior knowledge as a feedforward action to compensate for system nonlinearities; meanwhile, the feedback action generated by the policy network adaptively regulates dynamic balance and, combined with the feedforward action, jointly controls the robot’s walking. PFRL enforces constraints within the motion space to mitigate the chaotic behavior of the parallel dual sliders. Experimental results show that our method successfully achieves sim2real transfer on a real bipedal robot without the need for domain randomization techniques or intricate reward functions. L04 achieves omnidirectional walking with minimal energy consumption and exhibits robustness against external disturbances. Full article
(This article belongs to the Special Issue Advanced Control of Complex Dynamical Systems and Robotics with Applications)
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17 pages, 726 KiB  
Article
Optimal Control Problem and Its Solution in Class of Feasible Control Functions by Advanced Model of Control Object
by Askhat Diveev and Elena Sofronova
Mathematics 2025, 13(4), 674; https://doi.org/10.3390/math13040674 - 18 Feb 2025
Viewed by 392
Abstract
This paper is devoted to the solution of the optimal control problem. The obtained control should be optimal in terms of quality criteria and, at the same time, feasible when implemented in the control object. To solve the optimal control problem in the [...] Read more.
This paper is devoted to the solution of the optimal control problem. The obtained control should be optimal in terms of quality criteria and, at the same time, feasible when implemented in the control object. To solve the optimal control problem in the class of feasible control functions, an advanced mathematical model of the control object is used. Firstly, the universal stabilisation system of the motion along any trajectory from some class is developed via symbolic regression. Then, the obtained stabilisation system is inserted into the right part of the control object model instead of the control vector. A reference model with a free control vector in the right part is added to the model; thus, the advanced mathematical model of the control object is obtained. After this, the optimal control problem is solved with the advanced mathematical model of the control object. The optimal control problem is stated in the classical form when the control is a time function. Here, the control function is searched for the reference model. The preliminary design of the universal stabilisation system for some class of trajectories allows the solution of the optimal control problem via the control object in a reasonable time frame. The proposed methodology is computationally tested for a model of the spatial motion of a quadcopter and a group of two-wheeled mobile robots with a differential drive. The results of the experiments show that the universal stabilisation system ensures the stabilisation of the motion of the objects along optimal trajectories, which are not known beforehand but obtained as a result of solving the problem with an advanced model. Full article
(This article belongs to the Special Issue Advanced Control of Complex Dynamical Systems and Robotics with Applications)
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22 pages, 6331 KiB  
Article
A Target Domain-Specific Classifier Weight Partial Transfer Adversarial Network for Bearing Fault Diagnosis
by Yin Bai, Xiangdong Hu, Kai Zheng, Yunnong Chen and Yi Tang
Mathematics 2025, 13(2), 248; https://doi.org/10.3390/math13020248 - 13 Jan 2025
Cited by 1 | Viewed by 706
Abstract
In actual industry applications, the failure categories of practical equipment are usually a subset of laboratory conditions failure categories. Due to the strict constraints, partial transfer learning can address a more practical diagnostic scenario. In view of this, this paper proposes a target [...] Read more.
In actual industry applications, the failure categories of practical equipment are usually a subset of laboratory conditions failure categories. Due to the strict constraints, partial transfer learning can address a more practical diagnostic scenario. In view of this, this paper proposes a target domain-specific classifier weight partial transfer adversarial network. Initially, the 1-D convolutional neural network is employed as the basic architecture. By training the domain discriminator and feature generator with an adversarial strategy, the recognition ability of the domain discriminant network and the feature extraction ability of the feature generation network can be enhanced. After that, a weighted learning strategy is introduced to guide the model to learn the cross-domain invariant feature. Also, a specific target domain classifier is utilized to redivide the target domain decision boundary to accurately classify the unlabeled target domain samples. Finally, five mainstream deep neural network methods are taken for comparison using the data from Western Reserve University and the motor-magnetic brake test designed by us. The results show that the proposed method reaches 90.18% and 96.53% classification accuracy on two datasets, respectively, which demonstrates superior performance compared with the state-of-the-art methods. Full article
(This article belongs to the Special Issue Advanced Control of Complex Dynamical Systems and Robotics with Applications)
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15 pages, 1079 KiB  
Article
An Improved Hierarchical Optimization Framework for Walking Control of Underactuated Humanoid Robots Using Model Predictive Control and Whole Body Planner and Controller
by Yuanji Liu, Haiming Mou, Hao Jiang, Qingdu Li and Jianwei Zhang
Mathematics 2025, 13(1), 154; https://doi.org/10.3390/math13010154 - 3 Jan 2025
Viewed by 1223
Abstract
This paper addresses the fundamental challenge of achieving stable and efficient walking in a lightweight, underactuated humanoid robot that lacks an ankle roll degree of freedom. To tackle this relevant critical problem, we present a hierarchical optimization framework that combines model predictive control [...] Read more.
This paper addresses the fundamental challenge of achieving stable and efficient walking in a lightweight, underactuated humanoid robot that lacks an ankle roll degree of freedom. To tackle this relevant critical problem, we present a hierarchical optimization framework that combines model predictive control (MPC) with a tailored whole body planner and controller (WBPC). At the high level, we employ a matrix exponential (ME)-based discretization of the MPC, ensuring numerical stability across a wide range of step sizes (5 to 100 ms), thereby reducing computational complexity without sacrificing control quality. At the low level, the WBPC is specifically designed to handle the unique kinematic constraints imposed by the missing ankle roll DOF, generating feasible joint trajectories for the swing foot phase. Meanwhile, a whole body control (WBC) strategy refines ground reaction forces and joint trajectories under full-body dynamics and contact wrench cone (CWC) constraints, guaranteeing physically realizable interactions with the environment. Finally, a position–velocity–torque (PVT) controller integrates feedforward torque commands with the desired trajectories for robust execution. Validated through walking experiments on the MuJoCo simulation platform using our custom-designed lightweight robot X02, this approach not only improves the numerical stability of MPC solutions, but also provides a scientifically sound and effective method for underactuated humanoid locomotion control. Full article
(This article belongs to the Special Issue Advanced Control of Complex Dynamical Systems and Robotics with Applications)
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17 pages, 753 KiB  
Article
Fixed-Time Event-Triggered Control of Nonholonomic Mobile Robots with Uncertain Dynamics and Preassigned Transient Performance
by Yong Wang, Yunfeng Ji, Wei Li and Xi Fang
Mathematics 2024, 12(22), 3544; https://doi.org/10.3390/math12223544 - 13 Nov 2024
Viewed by 870
Abstract
In this paper, a novel adaptive control scheme is proposed for the path-following problem of a nonholonomic mobile robot with uncertain dynamics based on barrier functions. To optimize communication resources, we integrate an event-triggered mechanism that avoids Zeno behavior and ensures that the [...] Read more.
In this paper, a novel adaptive control scheme is proposed for the path-following problem of a nonholonomic mobile robot with uncertain dynamics based on barrier functions. To optimize communication resources, we integrate an event-triggered mechanism that avoids Zeno behavior and ensures that the tracking error of the closed-loop system converges to a small neighborhood around zero within a fixed time, while consistently satisfying predefined transient performance requirements. Extensive simulation studies demonstrate the effectiveness of the proposed approach and validate the theoretical results. Full article
(This article belongs to the Special Issue Advanced Control of Complex Dynamical Systems and Robotics with Applications)
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19 pages, 2032 KiB  
Article
Path Planning Method and Control of Mobile Robot with Uncertain Dynamics Based on Improved Artificial Potential Field and Its Application in Health Monitoring
by Yuan Li, Hongkai Song, Yunfeng Ji and Lingling Zhang
Mathematics 2024, 12(19), 2965; https://doi.org/10.3390/math12192965 - 24 Sep 2024
Viewed by 1032
Abstract
To enhance the navigation and control efficiency of mobile robots in the field of health monitoring, a novel path planning and control strategy for mobile robots with uncertain dynamics based on improved artificial potential fields is proposed in this paper. Specifically, we propose [...] Read more.
To enhance the navigation and control efficiency of mobile robots in the field of health monitoring, a novel path planning and control strategy for mobile robots with uncertain dynamics based on improved artificial potential fields is proposed in this paper. Specifically, we propose an attractive potential field rotation method to overcome the limitation that traditional artificial potential fields tend to fall into local minima. Then, we define a new class of attractive potential fields to address the goals non-reachable with obstacles nearby (GNRON) and collisions caused by excessive attractive force at long distances from the target point. Furthermore, a control law is proposed for the mobile robot with uncertain dynamics, and the stability of the closed-loop system is rigorously proven using the Lyapunov method. Finally, the feasibility and effectiveness of the proposed method are verified by simulations and experiments. Full article
(This article belongs to the Special Issue Advanced Control of Complex Dynamical Systems and Robotics with Applications)
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21 pages, 842 KiB  
Article
Optimal Asymptotic Tracking Control for Nonzero-Sum Differential Game Systems with Unknown Drift Dynamics via Integral Reinforcement Learning
by Chonglin Jing, Chaoli Wang, Hongkai Song, Yibo Shi and Longyan Hao
Mathematics 2024, 12(16), 2555; https://doi.org/10.3390/math12162555 - 18 Aug 2024
Cited by 1 | Viewed by 1271
Abstract
This paper employs an integral reinforcement learning (IRL) method to investigate the optimal tracking control problem (OTCP) for nonlinear nonzero-sum (NZS) differential game systems with unknown drift dynamics. Unlike existing methods, which can only bound the tracking error, the proposed approach ensures that [...] Read more.
This paper employs an integral reinforcement learning (IRL) method to investigate the optimal tracking control problem (OTCP) for nonlinear nonzero-sum (NZS) differential game systems with unknown drift dynamics. Unlike existing methods, which can only bound the tracking error, the proposed approach ensures that the tracking error asymptotically converges to zero. This study begins by constructing an augmented system using the tracking error and reference signal, transforming the original OTCP into solving the coupled Hamilton–Jacobi (HJ) equation of the augmented system. Because the HJ equation contains unknown drift dynamics and cannot be directly solved, the IRL method is utilized to convert the HJ equation into an equivalent equation without unknown drift dynamics. To solve this equation, a critic neural network (NN) is employed to approximate the complex value function based on the tracking error and reference information data. For the unknown NN weights, the least squares (LS) method is used to design an estimation law, and the convergence of the weight estimation error is subsequently proven. The approximate solution of optimal control converges to the Nash equilibrium, and the tracking error asymptotically converges to zero in the closed system. Finally, we validate the effectiveness of the proposed method in this paper based on MATLAB using the ode45 method and least squares method to execute Algorithm 2. Full article
(This article belongs to the Special Issue Advanced Control of Complex Dynamical Systems and Robotics with Applications)
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Review

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21 pages, 1125 KiB  
Review
A Review on the Recent Development of Planar Snake Robot Control and Guidance
by Ningwei Li, Fei Wang, Shanjun Ren, Xin Cheng, Gang Wang and Peng Li
Mathematics 2025, 13(2), 189; https://doi.org/10.3390/math13020189 - 8 Jan 2025
Viewed by 1415
Abstract
Snake robots, inspired by the biology of snakes, are bionic robots with multiple degrees of freedom and strong robustness. These robots represent a current area of significant research interest within the field of robotics. Snake robots have a wide range of applications in [...] Read more.
Snake robots, inspired by the biology of snakes, are bionic robots with multiple degrees of freedom and strong robustness. These robots represent a current area of significant research interest within the field of robotics. Snake robots have a wide range of applications in many fields, advancing the integration of bionics, robotics, and cybernetics, while playing a crucial role in performing survey and rescue missions. This survey presents the latest technological advancements in modeling, motion control, and guidance laws for planar snake robots, and provides a unified perspective based on the existing results. To achieve target-tracking control of robots in complex environments, we present a feasible approach that integrates guided vector field technology and transforms the target-tracking and obstacle avoidance problem into a reference angle tracking issue. Finally, this paper analyzes and summarizes the development process and key technologies of snake robot control and provides an outlook on future development trends. Full article
(This article belongs to the Special Issue Advanced Control of Complex Dynamical Systems and Robotics with Applications)
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