Dual quaternions for the kinematic description of a fish-like propulsion system

Kitowski, Zygmunt ; Piskur, Paweł ; Orłowski, Mateusz

International Journal of Applied Mathematics and Computer Science, Tome 33 (2023) no. 2, pp. 171-181.

Résumé

This study discusses the use of quaternions and dual quaternions in the description of artificial fish kinematics. The investigation offered here illustrates quaternion and dual quaternion algebra, as well as its implementation in the software chosen. When it comes to numerical stability, quaternions are better than matrices because a normalised quaternion always shows the correct rotation, while a matrix more easily loses its orthogonality due to rounding errors and oversizing. Although quaternions are more compact than rotation matrices, using quaternions does not always provide less numerical computation and the amount of memory needed. In this paper, an algebraic form of quaternion representation is provided which is less memory-demanding than the matrix representation. All the functions that were used to prepare this work are presented, and they can be employed to conduct more research on how well quaternions work in a specific assignment.

Détail

Keywords: dual quaternion, artificial fish, underwater vehicle, undulating propulsion system
Mots-clés : kwaternion dualny, sztuczna ryba, pojazd podwodny, układ napędowy

Bibliographie

[1] [1] Chen, L., Zielinska, T., Wang, J. and Ge, W. (2020). Solution of an inverse kinematics problem using dual quaternions, International Journal of Applied Mathematics and Computer Science 30(2): 351-361, DOI: 10.34768/amcs-2020-0027.

[2] [2] Denavit, J. and Hartenberg, R.S. (1955). A kinematic notation for lower-pair mechanisms based on matrices, American Society of Mechanical Engineers 22(2): 215-221, DOI: 10.1115/1.4011045.

[3] [3] Felski, A., Jaskólski, K., Zwolak, K. and Piskur, P. (2020). Analysis of satellite compass error’s spectrum, Sensors 20(15), Paper ID: 4067, DOI: 10.3390/s20154067.

[4] [4] Grzadziela, A., Szymak, P. and Piskur, P. (2020). Method for assessing the dynamics and efficiency of diving fins, Acta of Bioengineering Biomechanics 22(4): 139-150, DOI: 10.37190/ABB-01589-2020-06.

[5] [5] Horwitz, L.P. and Biedenharn, L.C. (1984). Quaternion quantum mechanics: Second quantization and gauge fields, Annals of Physics 157(2): 432-488.

[6] [6] Hożyń, S. and Zalewski, J. (2020). Shoreline detection and land segmentation for autonomous surface vehicle navigation with the use of an optical system, Sensors 20(10), Paper ID: 2799, DOI: 10.3390/s20102799.

[7] [7] Jarzebowska, E. and Klak, M. (2020). Quaternion-based spacecraft dynamic modeling and reorientation control using the dynamically equivalent manipulator approach, in T. Sands (Ed.), Advances in Spacecraft Attitude Control, InfoTech Open, Rijeka, Chapter 35, DOI: 10.5772/intechopen.88080.

[8] [8] Jaskólski, K. (2017). Two-dimensional coordinate estimation for missing automatic identification system (AIS) signals based on the discrete Kalman filter algorithm and universal transverse mercator (UTM) projection, Scientific Journals of the Maritime University of Szczecin 52 (124): 82-89.

[9] [9] Jaskólski, K., Felski, A. and Piskur, P. (2019). The compass error comparison of an onboard standard gyrocompass, fiber-optic gyrocompass (FOG) and satellite compass, Sensors 19(8), Paper ID: 1942, DOI: 10.3390/s19081942.

[10] [10] Jaskólski, K., Marchel, Ł., Felski, A., Jaskólski, M. and Specht, M. (2021). Automatic identification system (AIS) dynamic data integrity monitoring and trajectory tracking based on the simultaneous localization and mapping (SLAM) process model, Sensors 21(24), Paper ID: 8430.

[11] [11] Joldes¸, M. and Muller, J.-M. (2020). Algorithms for manipulating quaternions in floating-point arithmetic, 2020 IEEE 27th Symposium on Computer Arithmetic (ARITH), Portland, USA, pp. 48-55, DOI: 10.1109/ARITH48897.2020.00016.

[12] [12] Jurczyk, K., Piskur, P. and Szymak, P. (2020). Parameters identification of the flexible fin kinematics model using vision and genetic algorithms, Polish Maritime Research 27(2): 39-47, DOI: 10.2478/pomr-2020-0025.

[13] [13] Kiciński, R., Szturomski, B. and Marchel, Ł. (2021). A more reasonable model for submarines rescues seat strength analysis, Ocean Engineering 237, Paper ID: 109580.

[14] [14] Kluczyk, M. and Grzadziela, A. (2015). Simulation model of four stroke, six cylinder marine diesel engine, Solid State Phenomena 236: 113-118, DOI: 10.4028/www.scientific.net/SSP.236.113.

[15] [15] Leclercq, G., Lefèvre, P. and Blohm, G. (2013). 3D kinematics using dual quaternions: Theory and applications in neuroscience, Frontiers in Behavioral Neuroscience 7(7): 1-25, DOI: 10.3389/fnbeh.2013.00007.

[16] [16] Lighthill, M.J. (1971). Large-amplitude elongated-body theory of fish locomotion, Proceedings of the Royal Society of London B: Biological Sciences 179(1055): 125-138.

[17] [17] Ling, C., Qi, L. and Yan, H. (2022). Minimax principle for right eigenvalues of dual quaternion matrices and their generalized inverses, arXiv 2203.03161.

[18] [18] Morawski, M., Slota, A., Jerzy, Z. and Malec, M. (2020). Fish-like shaped robot for underwater surveillance and reconnaissance-Hull design and study of drag and noise, Ocean Engineering 217, Paper ID: 107889.

[19] [19] Mouton, H.D. (2021). Comparison of body rotations using Euler angles and quaternions, http://hdl.handle.net/11427/33222.

[20] [20] Naus, K., Szymak, P., Piskur, P., Niedziela, M. and Nowak, A. (2021). Methodology for the correction of the spatial orientation angles of the unmanned aerial vehicle using real time GNSS, a shoreline image and an electronic navigational chart, Energies 14(10), Paper ID: 2810, DOI: 10.3390/en14102810.

[21] [21] Pennestrí, E. and Valentini, P.P. (2010). Dual quaternions as a tool for rigid body motion analysis: A tutorial with an application to biomechanics, Archive of Mechanical Engineering 57(2): 187-205, DOI: 10.2478/v10180-010-0010-2.

[22] [22] Piórek, M. and Jabłoński, B. (2020). A quaternion clustering framework, International Journal of Applied Mathematics and Computer Science 30(1): 133-147, DOI: 10.34768/amcs-2020-0011.

[23] [23] Piskur, P. (2022). Strouhal number measurement for novel biomimetic folding fins using an image processing method, Journal of Marine Science and Engineering 10(4), Paper ID: 484, DOI: 10.3390/jmse10040484.

[24] [24] Piskur, P., Szymak, P., Flis, L. and Sznajder, J. (2020a). Analysis of a fin drag force in a biomimetic underwater vehicle, Naše more 67(3): 192-198.

[25] [25] Piskur, P., Szymak, P., Kitowski, Z. and Flis, L. (2020b). Influence of fin’s material capabilities on the propulsion system of biomimetic underwater vehicle, Polish Maritime Research 27(4): 179-185, DOI: 10.2478/pomr-2020-0078.

[26] [26] Piskur, P., Szymak, P., Przybylski, M., Naus, K., Jaskólski, K. and Żokowski, M. (2021). Innovative energy-saving propulsion system for low-speed biomimetic underwater vehicles, Energies 14(24), Paper ID: 8418, DOI: 10.3390/en14248418.

[27] [27] Radavelli, L., Simoni, R., De Pieri, E. and Martins, D. (2012). A comparative study of the kinematics of robots manipulators by Denavit-Hartenberg and dual quaternion, Mecánica Computacional 31(15): 2833-2848.

[28] [28] Sarabandi, S. and Thomas, F. (2019). A survey on the computation of quaternions from rotation matrices, Journal of Mechanisms and Robotics 11(2), Paper ID: 021006, DOI: 10.1115/1.4041889.

[29] [29] Sola, J. (2017). Quaternion kinematics for the error-state Kalman filter, arXiv 1711.02508, DOI: 10.48550/arXiv.1711.02508.

[30] [30] Wawrzyński,W., Zieja, M., Żokowski, M. and Sigiel, N. (2022). Optimization of autonomous underwater vehicle mission planning process, Bulletin of the Polish Academy of Sciences: Technical Sciences 70(2), Paper ID: e140371.