Multi-Layer Feedforward Neural Network Modelling of a Kinematics Solution of A 3-DoF Manipulator Robot
Received: 10 July 2025 | Revised: 4 August 2025 | Accepted: 22 August 2025 | Online: 8 December 2025
Corresponding author: Samara Munaem Naeem
Abstract
Modeling forward kinematics with neural networks allows for efficient handling of nonlinear relationships and realistic error correction in time-critical applications by relying on accurate training data. This paper presents a Multi-Layer Feed-Forward Neural Network (MLFFNN) to solve the forward kinematics of a 3-DOF robot. The proposed MLFFNN consists of 50 hidden neurons and was trained using 628319 samples to find only the position (x, y, z) of the end-effector. Data were generated by MATLAB, assuming an incremental motion of joints. The joint variables ( , , and ) are the inputs of the NN, which outputs the positions of the end effector (x, y, z) calculated using the Denavit-Hartenberg (DH) method. The results demonstrate that the proposed MLFFNN has high performance and is efficient for solving the forward kinematics, with a Mean Squared Error (MSE) between the desired and estimated position of 4.3881×10-11. This performance clearly demonstrates that, despite the large size of the dataset, it can be effectively mastered with only a small number of neurons. The simplicity of the network allows it to learn a compact and efficient representation of the data. This improves the reliability of using the proposed network for similar applications in other robotic systems.
Keywords:
multilayer neural network, forward kinematics, 3 DOF robot, serial robot, Denavit-Hartenberg method, incremental motionDownloads
References
L. Nguyen, R. V. Patel, and K. Khorasani, "Neural network architectures for the forward kinematics problem in robotics," in 1990 IJCNN International Joint Conference on Neural Networks, San Diego, CA, USA, 1990, pp. 393–399 vol.3. DOI: https://doi.org/10.1109/IJCNN.1990.137874
K. H. Mahmoud, A. N. Sharkawy, and G. T. Abdel-Jaber, "Development of safety method for a 3-DOF industrial robot based on recurrent neural network," Journal of Engineering and Applied Science, vol. 70, no. 1, Dec. 2023, Art. no. 44. DOI: https://doi.org/10.1186/s44147-023-00214-8
S. K. R. Moosavi, M. H. Zafar, and F. Sanfilippo, "Forward Kinematic Modelling with Radial Basis Function Neural Network Tuned with a Novel Meta-Heuristic Algorithm for Robotic Manipulators," Robotics, vol. 11, no. 2, Apr. 2022, Art. no. 43. DOI: https://doi.org/10.3390/robotics11020043
T. Tegegne, "Artificial Neural Network Based Position Control of Three-Degree of Freedom (3-DOF) Serial Robot," International Journal of Scientific Research, vol. 6, no. 3, pp. 1415–1424, 2020.
C. Cristoiu, M. Ivan, I. G. Ghionea, and C. Pupăză, "The Importance of Embedding a General forward Kinematic Model for Industrial Robots with Serial Architecture in Order to Compensate for Positioning Errors," Mathematics, vol. 11, no. 10, May 2023, Art. no. 2306. DOI: https://doi.org/10.3390/math11102306
E. Jiménez-López, D. Servín De La Mora-Pulido, L. A. Reyes-Ávila, R. Servín De La Mora-Pulido, J. Melendez-Campos, and A. A. López-Martínez, "Modeling of Inverse Kinematic of 3-DoF Robot, Using Unit Quaternions and Artificial Neural Network," Robotica, vol. 39, no. 7, pp. 1230–1250, Jul. 2021. DOI: https://doi.org/10.1017/S0263574720001071
M. Fouz, A. Bayoumy, and S. Rezeka, "Neural-Networks-Based Inverse Kinematics for a Robotic Manipulator," International Conference on Aerospace Sciences and Aviation Technology, vol. 15, pp. 1–18, May 2013. DOI: https://doi.org/10.21608/asat.2013.22084
M. R. Diprasetya, J. Pöppelbaum, and A. Schwung, "KineNN: Kinematic Neural Network for inverse model policy based on homogeneous transformation matrix and dual quaternion," Robotics and Computer-Integrated Manufacturing, vol. 94, Aug. 2025, Art. no. 102945. DOI: https://doi.org/10.1016/j.rcim.2024.102945
F. Cursi, W. Bai, W. Li, E. M. Yeatman, and P. Kormushev, "Augmented Neural Network for Full Robot Kinematic Modelling in SE(3)," IEEE Robotics and Automation Letters, vol. 7, no. 3, pp. 7140–7147, Jul. 2022. DOI: https://doi.org/10.1109/LRA.2022.3180428
S. Alavandar and M. J. Nigam, "Inverse Kinematics Solution of 3DOF Planar Robot using ANFIS," International Journal of Computers, Communications & Control, vol. 3, pp. 150–155, 2008. DOI: https://doi.org/10.15837/ijccc.2008.3.2391
R. Bouzid, H. Gritli, and J. Narayan, "Investigating Feed-Forward Back-Propagation Neural Network with Different Hyperparameters for Inverse Kinematics of a 2-DoF Robotic Manipulator: A Comparative Study," Chaos Theory and Applications, vol. 6, no. 2, pp. 90–110, Jun. 2024. DOI: https://doi.org/10.51537/chaos.1375866
A. N. Sharkawy and S. S. Khairullah, "Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator Using Artificial Neural Network," International Journal of Robotics and Control Systems, vol. 3, no. 2, pp. 330–353, May 2023. DOI: https://doi.org/10.31763/ijrcs.v3i2.1017
P. Srisuk, A. Sento, and Y. Kitjaidure, "Forward kinematic-like neural network for solving the 3D reaching inverse kinematics problems," in 2017 14th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON), Phuket, Jun. 2017, pp. 214–217. DOI: https://doi.org/10.1109/ECTICon.2017.8096211
T. P. Valayil and R. S. Augustine, "Methods to solve forward kinematics of parallel and serial manipulators," presented at the 2nd International Conference on Energetics, Civil and Agricultural Engineering 2021 (ICECAE 2021), Tashkent, Uzbekistan, 2022, Art. no. 030003. DOI: https://doi.org/10.1063/5.0115314
A. Ghasemi, M. Eghtesad, and M. Farid, "Neural Network Solution for Forward Kinematics Problem of Cable Robots," Journal of Intelligent & Robotic Systems, vol. 60, no. 2, pp. 201–215, Nov. 2010. DOI: https://doi.org/10.1007/s10846-010-9421-z
U. A. Mishra and S. Caro, "Unsupervised Neural Network Based Forward Kinematics for Cable-Driven Parallel Robots with Elastic Cables," in Cable-Driven Parallel Robots, vol. 104, M. Gouttefarde, T. Bruckmann, and A. Pott, Eds. Springer International Publishing, 2021, pp. 63–76. DOI: https://doi.org/10.1007/978-3-030-75789-2_6
M. Dehghani, M. Ahmadi, A. Khayatian, M. Eghtesad, and M. Farid, "Neural network solution for forward kinematics problem of HEXA parallel robot," in 2008 American Control Conference, Seattle, WA, USA, Jun. 2008, pp. 4214–4219. DOI: https://doi.org/10.1109/ACC.2008.4587155
C. S. Yee and K. Lim, "Forward kinematics solution of Stewart platform using neural networks," Neurocomputing, vol. 16, no. 4, pp. 333–349, Sep. 1997. DOI: https://doi.org/10.1016/S0925-2312(97)00048-9
Z. Geng and L. Haynes, "Neural network solution for the forward kinematics problem of a Stewart platform," in Proceedings. 1991 IEEE International Conference on Robotics and Automation, Sacramento, CA, USA, 1991, pp. 2650–2655. DOI: https://doi.org/10.1109/ROBOT.1991.132029
S. N. Yurt, E. Anli, and I. Ozkol, "Forward kinematics analysis of the 6-3 SPM by using neural networks," Meccanica, vol. 42, no. 2, pp. 187–196, Mar. 2007. DOI: https://doi.org/10.1007/s11012-006-9037-3
L. H. Sang and M. C. Han, "The estimation for forward kinematic solution of Stewart platform using the neural network," in Proceedings 1999 IEEE/RSJ International Conference on Intelligent Robots and Systems. Human and Environment Friendly Robots with High Intelligence and Emotional Quotients (Cat. No.99CH36289), Kyongju, South Korea, 1999, vol. 1, pp. 501–506. DOI: https://doi.org/10.1109/IROS.1999.813053
F. Tavassolian, H. Khotanlou, and P. Varshovi-Jaghargh, "Forward Kinematics Analysis of a 3-PRR Planer Parallel Robot Using a Combined Method Based on the Neural Network," in 2018 8th International Conference on Computer and Knowledge Engineering (ICCKE), Mashhad, Iran, Oct. 2018, pp. 320–325. DOI: https://doi.org/10.1109/ICCKE.2018.8566243
R. Bouzid, J. Narayan, and H. Gritli, "Exploring artificial neural networks for the forward kinematics of a SCARA robotic manipulator using varied datasets and training optimizers," Engineering Research Express, vol. 6, no. 4, Dec. 2024, Art. no. 045209. DOI: https://doi.org/10.1088/2631-8695/ad81cc
A. Ghobakhloo and M. Eghtesad, "Neural network solution for the forward kinematics problem of a redundant hydraulic shoulder," in 31st Annual Conference of IEEE Industrial Electronics Society, 2005. IECON 2005., Raleigh, NC, USA, 2005, Art. no. 6. DOI: https://doi.org/10.1109/IECON.2005.1569211
R. Kang, H. Chanal, T. Bonnemains, S. Pateloup, D. T. Branson, and P. Ray, "Learning the forward kinematics behavior of a hybrid robot employing artificial neural networks," Robotica, vol. 30, no. 5, pp. 847–855, Sep. 2012. DOI: https://doi.org/10.1017/S026357471100107X
C. Cristoiu and A. Nicolescu, "New Approach for Forward Kinematic Modeling of Industrial Robots," Research and Science Today, vol. 13, 2017, Art. no. 136.
H. Larochelle, Y. Bengio, J. Louradour, and P. Lamblin, "Exploring Strategies for Training Deep Neural Networks," Journal of Machine Learning Research, 2009.
F. Günther and S. Fritsch, "neuralnet: Training of Neural Networks," The R Journal, Jun. 2010. DOI: https://doi.org/10.32614/RJ-2010-006
M. Tiboni, G. Legnani, N. Pellegrini, and Università degli Studi di Brescia via Branze 38, Brescia 25123, Italy, "Study of Neural-Kinematics Architectures for Model-Less Calibration of Industrial Robots," Journal of Robotics and Mechatronics, vol. 33, no. 1, pp. 158–171, Feb. 2021. DOI: https://doi.org/10.20965/jrm.2021.p0158
A. Widyacandra, A. R. Al Tahtawi, and M. Martin, "Forward and inverse kinematics modeling of 3-DoF AX-12A robotic manipulator: Pemodelan kinematika maju dan terbalik dari manipulator robot 3-DoF AX-12A," JITEL (Jurnal Ilmiah Telekomunikasi, Elektronika, dan Listrik Tenaga), vol. 2, no. 2, pp. 139–150, Sep. 2022. DOI: https://doi.org/10.35313/jitel.v2.i2.2022.139-150
A. Ashagrie, A. O. Salau, and T. Weldcherkos, "Modeling and control of a 3-DOF articulated robotic manipulator using self-tuning fuzzy sliding mode controller," Cogent Engineering, vol. 8, no. 1, Jan. 2021, Art. no. 1950105. DOI: https://doi.org/10.1080/23311916.2021.1950105
T. T. K. Ly, N. T. Thanh, H. Thien, and T. Nguyen, "A Neural Network Controller Design for the Mecanum Wheel Mobile Robot," Engineering, Technology & Applied Science Research, vol. 13, no. 2, pp. 10541–10547, Apr. 2023. DOI: https://doi.org/10.48084/etasr.5761
Downloads
How to Cite
License
Copyright (c) 2025 Samara Munaem Naeem
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain the copyright and grant the journal the right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) after its publication in ETASR with an acknowledgement of its initial publication in this journal.
