Kinematic Singularity Mitigation in Serial Manipulators Using Auxiliary Virtual Axis Insertion

The Denavit-Hartenberg (DH) convention, while foundational in robot kinematics, contains a critical representational flaw in a specific class of anthropomorphic manipulators. This paper identifies and formalizes the "vanishing length problem", a singularity arising in kinematic chains featuring three consecutive joints where the z (yaw), x (roll), and y (pitch) axes form a mutually orthogonal triad. This configuration, characteristic of sequences like Roll-Yaw-Pitch or Pitch-Yaw-Roll, causes the origin of the middle (yaw) joint’s frame to become coincident with the origin of the subsequent joint’s frame. Consequently, the two physical link lengths preceding and following the yaw joint collapse into a single, combined length in the kinematic model, thereby losing a critical degree of freedom in the representation. We introduce the Auxiliary Virtual Joint (AVJ), a passive joint with a fixed constraint, to decouple this collapsed chain. The AVJ reintroduces the missing frame origin, restoring the two distinct link lengths and ensuring kinematic integrity. We provide a complete kinematic and dynamic formulation of the AVJ-augmented system, proving its equivalence to the physical manipulator. Validation on a 6-DoF humanoid arm demonstrates kinematic consistency to machine precision and, through dynamic simulations in MATLAB, confirms the feasibility of computing physically consistent joint torques—a computation rendered impossible for the middle yaw joint in the degenerate model due to the indeterminate inertial properties of the collapsed link.

1. Denavit J., Hartenberg R. S. A kinematic notation for lower-pair mechanisms based on matrices, Journal of Applied Mechanics, 1955,vol. 22, no. 2, pp. 215—221.

2. Craig J. J. Introduction to Robotics: Mechanics and Control, Pearson, Prentice Hall, 2005.

3. Csiszar A. A combinatorial optimization approach to the Denavit-Hartenberg parameter assignment, IEEE International Conference on Advanced Intelligent Mechatronics, 2015, pp. 1451—1456.

4. Faria C., Vilaca J. L., Monteiro S., Erlhagen W., Bicho E. Automatic Denavit-Hartenberg parameter identification for serial manipulators, IECON 2019 — 45th Annual Conference of the IEEE Industrial Electronics Society, 2019, pp. 610—617.

5. Singh R., Kukshal V., Yadav V. S. A review on forward and inverse kinematics of classical serial manipulators, Advances in Engineering Design, P. K. Rakesh, A. K. Sharma, and I. Singh, Eds. Singapore, Springer, 2021, pp. 417—428.

6. Rocha C. R., Tonetto C. P., Dias A. A comparison between the Denavit-Hartenberg and the screw-based methods used in kinematic modeling of robot manipulators, Robotics and Computer-Integrated Manufacturing, 2011, vol. 27, no. 4, pp. 723—728.

7. Sheth P. N., Uicker J. J. A generalized symbolic notation for mechanisms, Journal of Engineering for Industry, 1971, vol. 93, no. 1, pp. 102—112.

8. Gupta K. C. Kinematic analysis of manipulators using the zero reference position description, The International Journal of Robotics Research, 1986, vol. 5, no. 2, pp. 5—13.

9. Yang X., Wu L., Li J., Chen K. A minimal kinematic model for serial robot calibration using POE formula, Robotics and Computer-Integrated Manufacturing, 2014,vol. 30, no. 3, pp. 326—334.

10. Qiao Y., Chen Y., Chen B., Xie J. A novel calibration method for multi-robots system utilizing calibration model without nominal kinematic parameters, Precision Engineering, 2017, vol. 50, pp. 211—221.

11. Qi Y. et al. An explicit solution of forward and inverse kinematics for a serial manipulator with insufficient intersection joints based on finite and instantaneous screw theory, International Journal of Advanced Robotic Systems, 2024, vol. 21.

12. Singh A., Singla A., Soni S. D-H parameters augmented with dummy frames for serial manipulators containing spatial links, In the 23rd IEEE International Symposium on Robot and Human Interactive Communication, 2014, pp. 975—980.

13. Chen Z., Wang H., Lu X., Pu Q., Li M., Yan J. Kinematics analysis and application of 5-DOF manipulator with special joint, In the Chinese Automation Congress, 2017, pp. 7421—7426.

14. Trullo H. D., Vivas Alb n O. A. A Systematic Review of Inverse Kinematics Methods for Fixed-Base Serial Manipulators: Analytical, Numerical, and Machine Learning Methods, International Journal of Robotics and Control Systems, 2025.

15. Parikh P. A. et al. Kinematics of an indigenous 3D printed 6 DoF feeding serial manipulator: Forward and inverse kinematics, singularity and workspace analysis, AIP Conference Proceedings, 2025.

16. Orbegoso Moreno L. A., Valverde Ram rez E. D. Automatic differential kinematics of serial manipulator robots through dual numbers, Transactions on Energy Systems and Engineering Applications, 2024.

17. Tarasova V. V. Reshenie Obratnoj Zadachi Kinematiki Manipuljatora Metodom Uslovnoj Minimizacii, V sbornike: Prikladnaja matematika i informatika: sovremennye issledovanija v oblasti estestvennyh i tehnicheskih nauk. Sbornik materialov IX Mezhdunarodnoj nauchno-prakticheskoj konferencii (shkoly-seminara) molodyh uchenyh. Tol’jatti, 2023, pp. 193—198 (in Russian)

18. Poezzhaeva E. V., Jaburov M. I., Reutov V. A., Russkih E. R. Reshenie V Zamknutoj Forme Obratnoj Zadachi Kinematiki Dlja Manipuljacionnyh Robotov S Izbytochnost’ju, Kachestvo i zhizn’, 2024, vol. 4, no 44, pp. 11—15 (in Russian)

19. Zakiev D. D., Krutskih N. A., Alibekov S. Ja., Zakiev N. D., Marjashev A. V., Kuzina N. A. Reshenie Trehmernoj Obratnoj Zadachi Kinematiki Chetyrehzvennogo Manipuljatora, Vestnik Tehnologicheskogo universiteta, 2025, vol. 28, no. 2, pp. 98—102 (in Russian).

20. Boshljakov A. A., Novak A. V. Ob Ogranichenijah Reshenija Obratnoj Zadachi Kinematiki Manipuljatora S Izbytochnymi Stepenjami Podvizhnosti S Obhodom Prepjatstvij, Tendencii razvitija nauki i obrazovanija, 2024, vol. 109, no. 13, pp. 27—31 (in Russian).

21. Dzhachaev A. Sh. Kombinirovannyj Podhod Dlja Reshenija Obratnoj Zadachi Kinematiki Manipuljatorov S Izbytochnym Chislom Stepenej Podvizhnosti, Studencheskij forum, 2025, vol. 329, no. 20-1, pp. 58—66 (in Russian).

22. Kolpashhikov D. Ju. Metod i Algoritmy Obratnoj Kinematiki i Planirovanija Dvizhenija Dlja Mnogosekcionnyh Nepreryvnyh Robotov, dissertacija na soiskanie uchenoj stepeni kandidata tehnicheskih nauk / federal’’noe gosudarstvennoe avtonomnoe obrazovatel’’noe uchrezhdenie vysshego obrazovanija " juzhnyj federal’’nyj universitet", 2023 (in Russian).

23. Kolpashhikov D. Ju., Gerget O. M. Reshenie Obratnoj Zadachi Kinematiki Dlja Nepreryvnogo Robota Cherez Razbienie Rabochej Oblasti, Mehatronika, avtomatizacija, upravlenie, 2025, vol. 26, no. 5, pp. 253—259 (in Russian).

24. Nguen H. T., Filipas A. A. Algoritm Reshenija Obratnoj Kinematiki Dlja Robototehnicheskogo Manipuljatora Na Mobil’noj Platforme S Uchetom Ogranichenijami, V sbornike: Molodezh’ i sovremennye informacionnye tehnologii. Sbornik trudov XXII Mezhdunarodnoj nauchno-prakticheskoj konferencii studentov, aspirantov i molodyh uchenyh. Tomsk, 2025, pp. 253—258 (in Russian).

25. Tam Bui et al. DE algorithm method for Solving the inverse kinematics, Journal of Physics: Conference Series, 2021, vol. 1922, 012008.

26. Shastri S., Parvez Y., Chauhan N. R. Inverse Kinematics for A 3-R Robot Using Artificial Neural Network and Modified Particle Swarm Optimization, Journal of The Institution of Engineers (India): Series C, 2020, vol. 101, pp. 355—363.

27. Cimen M. E. Q Learning Based PSO Algorithm Application for Inverse Kinematics of 7-DOF Robot Manipulator, Bitlis Eren Universitesi Fen Bilimleri Dergisi, 2024.

28. Sheik Masthan S. A. R., Kanagaraj G., Yu V. F. Gravitation search-based hybrid algorithm for solving inverse kinematics of an n-link redundant manipulator, Journal of Computational Design and Engineering, 2023, vol. 10, no. 5, pp. 2019—2035.

29. Singh N., Tewari V. K., Biswas P. K., Dhruw L. K., Ranjan R., Ranjan A. Optimizing cotton-picking robotic manipulator and inverse kinematics modeling using evolutionary algorithmassisted artificial neural network, Journal of Field Robotics, 2024, vol. 41, pp. 2322—2342.

30. Zhao C., Wei Y., Xiao J. et al. Inverse kinematics solution and control method of 6-degree-of-freedom manipulator based on deep reinforcement learning, Scientific Reports, 2024, vol. 14, pp. 12467.

31. Bai Y., Hsieh S. J. A hybrid method using FABRIK and custom ANN in solving inverse kinematic for generic serial robot manipulator, The International Journal of Advanced Manufacturing Technology, 2024, vol. 130, pp. 4883—4904.

32. Ruch’ev I. A., Shimanchuk D. V. Metody Mashinnogo Obuchenija Dlja Reshenija Obratnoj Zadachi Kinematiki RobotaManipuljatora S Izbytochnym Chislom Stepenej Svobody, Processy upravlenija i ustojchivost’, 2025, vol. 12, no. 1, pp. 130—138.

33. L pez-Franco C., Hern ndez-Barragan J., Alanis A. Y., Arana-Daniel N. A soft computing approach for inverse kinematics of robot manipulators, Engineering Applications of Artificial Intelligence,2018,vol.74,pp. 104—120.

34. Dzhachaev A. Sh., Vorotnikov S. A. Metod Roja Chastic Dlja Reshenija Obratnoj Zadachi Kinematiki Manipuljatorov S Izbytochnym Chislom Stepenej Podvizhnosti, Izvestija Volgogradskogo gosudarstvennogo tehnicheskogo universiteta, 2025, vol. 299, no. 4, pp. 35—43 (in Russian).

35. Macheha V. M., Ljubin V. A. Predstavlenie Denavita — Hartenberga Dlja Mnogozven’evogo Robota. Reshenie Obratnoj Zadachi Kinematiki, V sbornike: Algebra i prilozhenija. sbornik nauchnyh trudov. Krasnodar, 2025, pp. 71—78 (in Russian).

36. Stebuljanin M. M., Pimushkin Ja. I. Metod Reshenija Obratnoj Zadachi Kinematiki Mehatronnogo Ob#ekta V Ramkah Koncepcii Razdelenija Izmeritel’nogo I Fizicheskogo Prostranstv Dvizhenija, Izmeritel’naja tehnika, 2025, vol. 74, no. 4, pp. 36—46 (in Russian).

37. Hamouda A. M., Abdellatif E. M., Al Akkad M. A. Design and Implementation of Human-like Ambidextrous Robotic Arms for Cooperative Tasks. Mekhatronica, Avtomatizatsiya, Upravlenie, 2024, vol. 25, no. 12, pp. 615—623, DOI 10.17587/mau.25.615-623.