V.Kh. Ankudinov – Post-graduate Student, 

Department «Information Systems and Networks», Kaluga branch of the Bauman MSTU

E-mail: vladislav.ankudinov@gmail.com

A.V. Maksimov – Ph.D.(Eng.), Associate Professor, 

Department «Information Systems and Networks», Kaluga branch of the Bauman MSTU

E-mail: av_maximov@bk.ru

In the first part of the article, published in the journal «Electromagnetic waves and electronic systems» № 9 for 2016, a kinematic model of the robot hexapod «snowflake» based on the mathematical apparatus of homogeneous matrices was obtained. The equations of the model were derived on the basis of the solution of the direct kinematics problem of the considered robot. It is noted that the results can be successfully used in the future in solving the inverse problem of kinematics of this type of robot.

To improve the efficiency of hexapod motion control in this part of the article the derivation of the equations of its kinematics model based on dual quaternions is given. The features of dual quaternions, their unambiguous relationship with homogeneous matrices, which served as the basis for the derivation of kinematic equations in the quaternion basis are noted.

On the basis of the developed matrix and biquaternion models the problem of inverse kinematics was solved and the simplest statically stable gaits were synthesized: one-legged, two-legged and three-legged. To test the synthesized gait and further research, a hardware and software complex was developed and manufactured as part of the hexapod «snowflake» and its simulator «Robosim».

The experimental results have shown that dual quaternions are more computationally efficient than a homogeneous matrix. Thus, to store a dual quaternion 8 floating-point numbers are required, and to store all elements of a homogeneous matrix – 16. The multiplication of dual quaternions requires 42 multiplication operations and 38 addition operations, while the matrices require 64 multiplication operations and 48 addition operations. Using quaternions requires less time and memory when organizing calculations in the control controller.

1. Ankudinov V.KH., Maksimov A.V. Kinematicheskaya Model Geksapoda. Chast I. Matrichnye modeli. Elektromagnitnye volny i elektronnye sistemy. 2016. T. 21. № 9. S. 56−65.
2. Thomas F. Approaching dual quaternions from matrix algebra. IEEE Transactions on Robotics. 2014. 30(5). 1037−1048.
3. Branets N., Shmyglevskii I.P. Primenenie kvaternionov v zadachakh orientatsii tverdogo tela. M.: Nauka. 1973. 320 s.
4. Chelnokov Yu.N. Kvaternionnye i bikvaternionnye modeli i metody mekhaniki tverdogo tela i ikh prilozheniya. Geometriya dvizheniya. Saratov: Izd-vo Saratovskogo un-ta. 2006. 236 s.
5. Chelnokov Yu.N. Kvaternionnye i bikvaternionnye modeli i metody mekhaniki tverdogo tela i ikh prilozheniya. Geometriya i kinematika dvizheniya. M.: Fizmatlit. 2006. 512 s.
6. Chelnokov Yu.N. Bikvaternionnoe reshenie kinematicheskoй zadachi upravleniya dvizheniem tverdogo tela i ego prilozhenie k resheniyu obratnykh zadach kinematiki robotov-manipulyatorov. Izvestiya RAN. Mekhanika tverdogo tela. 2013. № 1. S. 38−58.
7. Ben Kenwright. A Beginners Guide to Dual-Quaternions: What They Are, How They Work, and How to Use Them for 3D Character Hierarchies. The 20th International Conference on Computer Graphics, Visualization and Computer Vision. 26−28 June 2012. P. 1−10.
8. Ben Keywright. Dual-quaternions: From classical mechanics to computer graphics and beyond. URL = www.xbdev.net: Technical Article. Oct 2012.
9. Ankudinov V.Kh., Maksimov A.V., Starkov S.O. Hardware-software complex of in-depth studies of hexapod walking robot. Journal of Physics: Conference Series. 2018. T. 945. № 1. S. 012030.
10. Ben Kenwright. Inverse Kinematics with Dual-Quaternions, Exponential-Maps, and Joint Limits. International Journal on Advances in Intelligent Systems. V. 6. № 1&2. 2013.
11. Ankudinov V.KH. Sintez prosteishego ALU dlya protsessinga kvaternionov s ispolzovaniem yazyka VHDL. Materialy regionalnoi nauchnotekhnich. konf. studentov, aspirantov i molodykh uchenykh «Naukoemkie tekhnologii v priboro- i mashinostroenii i razvitie innovatsionnoi deyatelnosti v vuze». M.: Izd-vo MGTU im. N.E. Baumana. 2012. T. 2. S. 259−262.
12. Maksimov A.V., Maksimova E.A., Frolkina A.A. Chetyrekhrazryadnyi summator chetyrekh slogaemykh. Elektromagnitnye volny i elektronnye sistemy. 2015. T. 20. № 7. S. 18−21.
13. Klimanova E.V., Maksimov A.V., Maksimova E.A., Stepanov S.E. Kompleksnoznachnye i giperkompleksnye modeli tipovykh zvenev sistem upravleniya. Chast 2. Matematicheskii apparat dlya chetyrekhmernykh sistem. Elektromagnitnye volny i elektronnye sistemy. 2017. T. 22. № 3. S. 22−29.
14. Konovalov I.V. i dr. Logicheskii sintez spetsializirovannykh vychislitelnykh ustroistv v SAPR «Decomposer». Elektromagnitnye volny i elektronnye sistemy. 2018. T. 23. № 3. S. 18−25.
15. Maksimov A.V. Optimalnoe proektirovanie assemblernykh programm matematicheskikh algoritmov: teoriya, inzhenernye metody: Ucheb. posobie. SPb.: Lan. 2016. 192 s.
16. Maksimov A.V., Maksimova E.A. Effektivnost metodov vosstanovleniya normy chislennogo resheniya kvaternionnykh kinematicheskikh uravnenii. Elektromagnitnye volny i elektronnye sistemy. 2014. T. 19. № 10. S. 18−24.