A Review of Mathematical Models in Robotics
Source abstract
Abstract
In robotics, much emphasis is placed on mathematical modeling, as the creation, control, and optimization of robots for a wide field of work must be achieved precisely and adaptively. The aim of this paper is to present a systematic and structured approach to the literature review of mathematical models in robotics, critically considering mathematical frameworks that influence and shape robotics in light of current and prevailing trends. The paper underlines the complexities of maintaining accurate dynamic representations in robotic systems, revealing the challenges that arise from numerical simplifications. The study outlines the development of efficient remote-control systems that consider dynamic relationships among the components comprising the robot. The findings of the recent simulation prove that the developed mathematical model effectively supports designing an adaptive control system with artificial intelligence features, especially for autonomous mobile robotics with manipulators that are inherently complex and networked systems. If models are to accelerate robotics progress toward increasingly intelligent, adaptive, and efficient systems, they must learn to overcome some of the computational challenges while leveraging disciplinary synergies.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Robot Manipulation and Learning · Soft Robotics and Applications · Robotic Path Planning Algorithms
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
Affiliations
Department of Automation, Faculty of Electrical and Electronics Engineering, Kaunas University of Technology, Studentu St. 48-320, LT-51367 Kaunas, Lithuania
References (60)
Show source-provided references
1. A Geometrically Exact Model for Soft Continuum Robots: The Finite Element Deformation Space Formulation10.1089/soro.2018.0047 ↗
2. Mathematical modelling for a class of flexible robot10.1016/0307-904X(95)00036-J ↗
3. Mathematical modeling and analysis of the Delta robot with flexible links10.1016/j.camwa.2016.03.018 ↗
4. DQ Robotics: A Library for Robot Modeling and Control10.1109/MRA.2020.2997920 ↗
5. Mathematical deficiencies of numerically simplified dynamic robot models10.1109/9.35289 ↗
6. Bąkała, M., Duch, P., and Ostalczyk, P. (2023). New Approach of the Variable Fractional Order Model of a Robot Arm. Appl. Sci., 13.10.3390/app13053304 ↗
7. Fractional Modeling of Robotic Systems10.1007/978-3-030-58247-0_2 ↗
8. Coupled Longitudinal and Lateral Control for Autonomous Vehicle Dynamics Modeled Using a Robotics Formalism10.1016/j.ifacol.2017.08.2190 ↗
9. On Mathematical Modeling in Robotics
10. Construction of mathematical model for the controlled dynamics of connected bodies system in robotics10.1088/1757-899X/1030/1/012055 ↗
11. BoundMPC: Cartesian path following with error bounds based on model predictive control in the joint space10.1177/02783649241309354 ↗
12. Dynamic movement primitives in robotics: A tutorial survey10.1177/02783649231201196 ↗
13. Nonlinear model predictive dynamic positioning of a remotely operated vehicle with wave disturbance preview10.1177/02783649241286909 ↗
14. Rolling based locomotion on rough terrain for a wheeled quadruped using centroidal dynamics10.1016/j.mechmachtheory.2020.103984 ↗
15. Horizon-stability control for wheel-legged robot driving over unknow, rough terrain10.1016/j.mechmachtheory.2024.105887 ↗
16. Deep learning can accelerate grasp-optimized motion planning10.1126/scirobotics.abd7710 ↗
17. A memristor-based hybrid analog-digital computing platform for mobile robotics10.1126/scirobotics.abb6938 ↗
18. Nguyen, T.D.L., and Lu, J. (2023). A Review of Mathematical Methods for Flexible Robot Dynamics Modeling and Simulation. Machine Learning and Mechanics Based Soft Computing Applications, Springer. Studies in Computational Intelligence.10.1007/978-981-19-6450-3 ↗
19. Modeling and Simulation of Dynamics in Soft Robotics: A Review of Numerical Approaches10.1007/s43154-023-00105-z ↗
20. Mathematical modeling, analysis and control in soft robotics: A survey10.1007/s40324-023-00334-4 ↗
21. Soft robotics: Biological inspiration, state of the art, and future research10.1080/11762320802557865 ↗
22. Soft Robotics: A Perspective-Current Trends and Prospects for the Future10.1089/soro.2013.0001 ↗
23. Continuum Robots for Medical Applications: A Survey10.1109/TRO.2015.2489500 ↗
24. Park, Y.L., Chen, B., Perez-Arancibia, N., Young, D., Stirling, L., Wood, R.J., Goldfield, E.C., and Nagpal, R. (2014). Design and control of a bio-inspired soft wearable robotic device for ankle–foot rehabilitation. Bioinspir. Biomim., 9.10.1088/1748-3182/9/1/016007 ↗
25. Soft robotic glove for combined assistance and at-home rehabilitation10.1016/j.robot.2014.08.014 ↗
26. Soft Robots Modeling: A Structured Overview10.1109/TRO.2022.3231360 ↗
27. Ménager, E., Navez, T., Chaillou, P., Goury, O., Kruszewski, A., and Duriez, C. (2025). Modeling, Embedded Control and Design of Soft Robots using a Learned Condensed FEM Model. arXiv.
28. Boër, C.R., Molinari-Tosatti, L., and Smith, K.S. (2011). Parallel Kinematic Machines: Theoretical Aspects and Industrial Requirements, Springer.
29. Analysis and dimensional synthesis of the DELTA robot for a prescribed workspace10.1016/j.mechmachtheory.2006.06.012 ↗
30. Zhang, L., and Song, Y. (2011, January 9–13). Optimal design of the Delta robot based on dynamics. Proceedings of the IEEE International Conference on Robotics and Automation, Shanghai, China.
31. Dynamic finite-element analysis of a planar high speed, high-precision parallel manipulator with flexible links10.1016/j.mechmachtheory.2004.12.007 ↗
32. Modeling and control of a rigid-flexible coupling robot for narrow space manipulations10.1007/s11431-024-2819-2 ↗
33. Adorno, B.V. (2011). Two-Arm Manipulation: From Manipulators to Enhanced Human-Robot Collaboration [Contribution à la Manipulation à Deux Bras: Des Manipulateurs à la Collaboration Homme-Robot]. [Ph.D. Dissertation, University of Montpellier]. Available online: https://theses.hal.science/tel-00641678/.
34. Spong, M.W., Hutchinson, S., and Vidyasagar, M. (2020). Robot Modeling and Control, Wiley. [2nd ed.]. Available online: https://www.perlego.com/book/2774514/robot-modeling-and-control-pdf.
35. Siciliano, B., Sciavicco, L., Villani, L., and Oriolo, G. (2008). Robotics: Modelling, Planning and Control, Springer. [1st ed.].10.1007/978-1-84628-642-1 ↗
36. Leclercq, G., Lefèvre, P., and Blohm, G. (2013). 3D kinematics using dual quaternions: Theory and applications in neuroscience. Front. Behav. Neurosci., 7.10.3389/fnbeh.2013.00007 ↗
37. An Improved Dual Quaternion Dynamic Movement Primitives-Based Algorithm for Robot-Agnostic Learning and Execution of Throwing Tasks10.1115/1.4068555 ↗
38. Govindarajan, V., Bhattacharya, S., and Kumar, V. (2016). Human-robot collaborative topological exploration for search and rescue applications. Distributed Autonomous Robotic Systems, Springer.10.1007/978-4-431-55879-8_2 ↗
39. A machine learning approach for collaborative robot smart manufacturing inspection for quality control systems10.1016/j.promfg.2020.10.003 ↗
40. Kilbas, A.A., Srivastava, H., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science Inc.. Available online: https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/204.
41. Fractional Order Systems. Applications in modelling, identification, and control
42. Existence uniqueness and stability analysis of solutions for the fractional fuzzy cellular neural networks with impulse and reaction diffusion10.1080/13873954.2025.2515875 ↗
43. Time-domain identification of One Noninteger Order Plus Time Delay models from step response measurements10.1142/S1793962319410113 ↗
44. Mobile ground-based robot systems for professional use10.31776/RTCJ.8101 ↗
45. The state and problems of standardization and unification of military ground robot systems10.31776/RTCJ.8102 ↗
46. Dynamics and control of space robot considering joint friction10.1016/j.actaastro.2015.02.010 ↗
47. Haptic based teleoperation with master slave motion mapping and haptic rendering for space exploration10.1016/j.cja.2018.07.009 ↗
48. Angeles, J. (1988). Rational Kinematics, Springer.10.1007/978-1-4612-3916-1 ↗
49. Some topological problems in robotics10.1007/BF03023989 ↗
50. Bochnak, J., Coste, M., and Roy, M.-F. (1987). Real Algebraic Geometry, Springer.
51. Boissonat, J.-D., and Laumond, J.-E. (1989). Geometry and Robotics, Springer. Lecture Notes in Computer Science.10.1007/3-540-51683-2 ↗
52. Lapshin, V.V. (2012). Mechanics and Motion Control Walking Machines, MSTU named after N.E. Bauman.
53. Zhao, C., Wei, Y., Xiao, J., Sun, Y., Zhang, D., Guo, Q., and Yang, J. (2024). Inverse kinematics solution and control method of 6-degree-of-freedom manipulator based on deep reinforcement learning. Sci. Rep., 14.10.1038/s41598-024-62948-6 ↗
54. Foundation models in robotics: Applications, challenges, and the future10.1177/02783649241281508 ↗
55. Shape modeling and experimental validation of continuum robots10.1016/j.apm.2024.06.013 ↗
56. Diveev, A., Sofronova, E., Konyrbaev, N., and Abdullayev, O. (2024). Advanced Model with a Trajectory Tracking Stabilisation System and Feasible Solution of the Optimal Control Problem. Mathematics, 12.10.3390/math12203193 ↗
57. Innovations in mathematical modeling, AI, and optimization techniques10.1007/s11227-024-06861-9 ↗
58. Optimized inverse kinematics modeling and joint angle prediction for six-degree-of-freedom anthropomorphic robots with Explainable AI10.1016/j.isatra.2024.12.008 ↗
59. LAMARS: Large Language Model-Based Anticipation Mechanism Acceleration in Real-Time Robotic Systems10.1109/ACCESS.2024.3524906 ↗
60. Autonomous Systems (2025, April 02). Design, Develop, and Test Autonomous Systems with MATLAB. Available online: https://blogs.mathworks.com/autonomous-systems/2025/02/27/let-robots-move-easy-the-power-of-generative-ai-for-robotics/.