Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T12:23:45.927Z Has data issue: false hasContentIssue false
  • English
  • Français

Newton–Euler modeling and Hamiltonians for robot control in the geometric algebra

Published online by Cambridge University Press:  22 July 2022

Eduardo Bayro-Corrochano Open the ORCID record for Eduardo Bayro-Corrochano [Opens in a new window] ,
Jesus Medrano-Hermosillo Open the ORCID record for Jesus Medrano-Hermosillo [Opens in a new window] ,
Guillermo Osuna-González  and
Ulises Uriostegui-Legorreta
Eduardo Bayro-Corrochano*
Affiliation:
Centro de Investigaciones y Estudios Avanzados, CINVESTAV, Department of Electrical Engineering and Computer Science, Campus Guadalajara, 1145 Del Bosque Ave., El Bajío, 45019, Zapopan, México
Jesus Medrano-Hermosillo
Affiliation:
Centro de Investigaciones y Estudios Avanzados, CINVESTAV, Department of Electrical Engineering and Computer Science, Campus Guadalajara, 1145 Del Bosque Ave., El Bajío, 45019, Zapopan, México
Guillermo Osuna-González
Affiliation:
Centro de Investigaciones y Estudios Avanzados, CINVESTAV, Department of Electrical Engineering and Computer Science, Campus Guadalajara, 1145 Del Bosque Ave., El Bajío, 45019, Zapopan, México
Ulises Uriostegui-Legorreta
Affiliation:
Centro de Investigaciones y Estudios Avanzados, CINVESTAV, Department of Electrical Engineering and Computer Science, Campus Guadalajara, 1145 Del Bosque Ave., El Bajío, 45019, Zapopan, México
*
*Corresponding author. E-mail: eduardo.bayro@cinvestav.mx
Get access Rights & Permissions [Opens in a new window]

Abstract

The principal objective of the paper is to show the importance of the Hamiltonian in control theory. Instead of using the Lagrangian formulation of electromechanical or robotic systems, our work is focused on robot dynamics by its Hamiltonian. Using the iterative Newton–Euler, we generate the local Hamiltonians and the derivative of the moments at each joint of the robot manipulator. Thus, we can apply decentralized controllers at each joint. We compare and discuss the efficiency of the controllers. We show that the performance of the sliding modes controller is more robust than that of the PD or Bang–Bang controllers.

Keywords

Newton–Euler modelingmotor algebra G+3,0,1dynamic modelHamiltoniansnonlinear controlSE(3) PD controlsliding mode controllocalized controlrobot armstracking
Type
Research Article
Information
Robotica , Volume 40 , Issue 11 , November 2022 , pp. 4031 - 4055
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aguilar-Ibáñez, C., Moreno-Valenzuela, J., García - Alarcón, O., Martinez-Lopez, M., Acosta, J. A. and Suarez-Castanon., M., “PI-type controllers and $\Sigma$ - $\Delta$ modulation for saturated DC-DC buck power converters,” IEEE Access 9, 20346–20357 (2021).10.1109/ACCESS.2021.3054600CrossRefGoogle Scholar
Aguilar-Ibáñez, J., Orozco, E., Cordova, D., Islas, M., Pacheco, J., Gutierrez, G., Zacarias, A., Soriano, L., Meda-Campaña, J., Mujica-Vargas, D., “Modified linear technique for the controllability and observability of robotic arms”,” IEEE Access 10, 3366–3377 (2022).Google Scholar
Ortega, R., van der Schaft, A. J., Maschke, B. and Escobar, G., “Energy-shaping of port-controlled Hamiltonian systems by interconnection,” In: Proc. IEEE Conf. on Decision and Control, vol. 2 (1999) pp. 1646–1651.Google Scholar
Záda, V. and Belda, K., “Robot Control in Terms of Hamiltonian Mechanics,” In: Proc. of the 22nd International Conference Engineering Mechanics, Svratka, Czech Republic, 9–12 May (2016) pp. 627–630Google Scholar
Wanxie, Z., Zhigan, W. and Shujun, T.. Theory and Computation of State Space Control (Science Publishing House, Beijing, 2006).Google Scholar
Ortega, R., van der Schaft, A., Castanos, F. and Astolfi, A., “Control by interconnection and standard passivity-based control of port-Hamiltonian systems,” IEEE Trans. Autom. Cont. 53(11), 2527–2542 (2008).10.1109/TAC.2008.2006930CrossRefGoogle Scholar
Hestenes, D., “Hamiltonian mechanics with geometric calculus,” In: Spinors, Twistors, Clifford Algebras, and Quantum Deformations (Z. Oziewicz et al., eds.) (Springer Science+Bisiness Media, B.V., 1993) pp. 203–214.10.1007/978-94-011-1719-7_25CrossRefGoogle Scholar
Abou El Dahab, E. T., “A formulation of Hamiltonian mechanics using geometric algebra,” Adv. Appl. Clifford Al. 10(2), 217–223 (1972).10.1007/s00006-000-0004-0CrossRefGoogle Scholar
Pappas, R., “A Formulation of Hamiltonian Mechanics Using Geometric Calculus,” In: Clifford Algebras and Their Applications in Mathematical Physics (eds.), F. Brackx et al. (Kluwer Academic Publishers, 1993) pp. 251–258.CrossRefGoogle Scholar
Hestenes, D.. New Foundation of Classical Mechanics (D. Reidel Publishing Co., Dordrecht/Boston, 1986).10.1007/978-94-009-4802-0CrossRefGoogle Scholar
Bayro-Corrochano, E. and Osuna-Gonzáles, G., “Modeling, control and tracking in robotics using screw theory in geometric algebra,” J. Robot. (to appear) (2022).Google Scholar
Romero, J. G., Ortega, R. and Sarras, I., “A globally exponentially stable tracking controller for mechanical systems using position feedback,” IEEE Trans. Automat. Contr. 60(3), 818–823 (2015).CrossRefGoogle Scholar
Yaghmeaei, A. and Yazdanpanah, M. J., “Trajectory tracking for a class of contractive port hamiltonian systems,” Automatica 83(September), 331–336 (2017).10.1016/j.automatica.2017.06.039CrossRefGoogle Scholar
Reyes-Baez, R., Van der Schaft, A. J. and Jayawardhana, B., “Tracking Control of Fully Actuated Port-Hamiltonian Mechanical Systems via Sliding Manifolds and Contraction Analysis,” In: IFAC Papers On-Line (2017) pp. 50–51.Google Scholar
Kelly, R., Santibánez, V. and Loriá, A., Control of Robot Manipulators in Joint Space Advanced Textbooks in Control and Signal Processing (Springer, 2005).Google Scholar
Ortega, R., Loria, A., Nicklasson, P. J. and Sira-Ramirez, H., “Passivity based Control of Euler Lagrange Systems: Mechanical, Electrical and Electromechanical Applications,” In: Communication and Control Systems (Springer, 1998).Google Scholar
Fujimoto, K., Sakurama, K. and Sugie, T., “Trajectory tracking control of port-controlled hamiltonian systems via generalized canonical transformations,” Automatica 39(12), 2059–2069 (2003).10.1016/j.automatica.2003.07.005CrossRefGoogle Scholar
Ortega, R., Van Der Schaft, A., Maschke, B. and Escobar, G., “Interconnection and damping assignment passivity-based control of port-controlled hamiltonian systems,” Automatica 38(4), 585–596 (2002).10.1016/S0005-1098(01)00278-3CrossRefGoogle Scholar
Fujimoto, K. and Sugie, T., “Time-varying stabilization of hamiltonian systems via generalized canonical transformations,” IFAC Proc. Vol. 33(2), 63-68 (2000). D. Hestenes, New Foundation of Classical Mechanics (D. Reidel Publishing Co., Dordrecht/Boston, 1986)Google Scholar
Mulero-Martinez, J. I., “Canonical transformations used to derive robot control laws from a port-controlled Hamiltonian system perspective,” Automatica 44(9), 2435–2440 (2008).CrossRefGoogle Scholar
Dirksz, D. and Scherpen, J. M. A., “Structure preserving adaptive control of port-hamiltonian systems”,” IEEE Trans. Automat. Contr. 57(11), 2880–2885 (2021).10.1109/TAC.2012.2192359CrossRefGoogle Scholar
Donaire, A., Perez, T. and Bartlett, N., “Tracking control of a class of hamiltonian mechanical systems with disturbances,” In: Proceedings of Australasian Conference on Robotics and Automation. Australian Robotics and Automation Association ARAA (2014) pp. 1–7.Google Scholar
Donaire, A. and Junco, S., “On the addition of integral action to port-controlled hamiltonian systems,” Automatica 45(8), 1910–1916 (2009).10.1016/j.automatica.2009.04.006CrossRefGoogle Scholar
Bayro-Corrochano, E., Geometric Algebra Applications. vol. I, Computer Vision, Graphics and Neurocomputing (Springer Verlag, Heidelberg, 2018).Google Scholar
Clifford, W. K., “Preliminary sketch of bi-quaternions”,” Proc. London Math. Soc. 4, 381–395 (1873).Google Scholar
Craig, J.. Introduction to Robotics 2nd edition (Mechanics and Control, Pearson, 1989).Google Scholar
Selig, J., Introductory Robotics (Prentice-Hall International, Hertfordshire, UK, 1992).Google Scholar
Utkin, V., Guldner, J. and Shi, J.. Sliding Mode Control in Electro-Mechanical Systems, 2nd edition. Automation and Control Engineering (Taylor & Francis, London, UK, 2009).Google Scholar
Spong, M. W., Hutchinson, S. and Vidyasagar, M.. Robot Modeling and Control (John Wiley & Sons, 2006).Google Scholar
Moreno, J. A. and Osorio, M., “A Lyapunov approach to second-order sliding mode controllers and observers,” In: 47th IEEE Conference on Decision and Control (December 2008) pp. 2856–2861.CrossRefGoogle Scholar