Modeling and Pose Control of Robotic Manipulators and Legs using Conformal Geometric Algebra

Authors

  • O. Carbajal-Espinosa Instituto Tecnológico y de Estudios Superiores de Monterrey
  • L. González-Jiménez Instituto Tecnológico y de Estudios Superiores de Occidente
  • J. Obviedo-Barriga Universidad Veracruzana,
  • B. Castillo-Toledo Centro de Investigación y De Estudios Avanzados del IPN
  • A. Loukianov Centro de Investigación y De Estudios Avanzados del IPN
  • E. Bayro-Corrochano Centro de Investigación y De Estudios Avanzados del IPN

DOI:

https://doi.org/10.13053/cys-19-3-2012

Keywords:

Serial manipulators, pose control, motors, conformal geometric algebra

Abstract

Controlling the pose of a manipulator involvesfinding the correct configuration of the robot’s elementsto move the end effector to a desired position and orientation.In order to find the geometric relationshipsbetween the elements of a robot manipulator, it is necessaryto define the kinematics of the robot. We presenta synthesis of the kinematical model of the pose forthis type of robot using the conformal geometric algebraframework. In addition, two controllers are developed,one for the position tracking problem and another for theorientation tracking problem, both using an error feedbackcontroller. The stability analysis is carried out forboth controllers, and their application to a 6-DOF serialmanipulator and the legs of a biped robot are presented.By proposing the error feedback and Lyapunov functionsin terms of geometric algebra, we are opening a newvenue of research in control of manipulators and robotlegs that involves the use of geometric primitives, suchas lines, circles, planes, spheres.

Author Biographies

O. Carbajal-Espinosa, Instituto Tecnológico y de Estudios Superiores de Monterrey

Received his B.Sc. degreein Electrical Engineering from the TechnologicalInstitute of Merida, Yucatan, Mexico, theM.Sc. and Ph.D. degrees in Electrical Engineeringfrom the Advanced Studies and Research Centerof the National Polytechnic Institute, CampusGuadalajara, Mexico, in 2008, 2011, and 2015,respectively. His research interests include nonlinearrobust control design, sliding modes andits applications to robotics, modeling and controlof humanoid robots, conformal geometric algebraand its applications to robotics and mechanicalsystems.

L. González-Jiménez, Instituto Tecnológico y de Estudios Superiores de Occidente

Was born in Guasave,Sinaloa, Mexico. He received his B.Sc. degreein Electronic Engineering from Instituto Tecnol´ogico de Sonora (ITSON) in 2005, his M.Sc.and Ph.D. degrees in Electrical Engineering fromthe Advanced Studies and Research Center of theNational Polytechnic Institute (CINVESTAV-IPN),Guadalajara Campus, Mexico, in 2007 and 2011,respectively. Since 2013, he is a Research Professorat ITESO. He has published more than 10refereed journal and conference papers. His researchinterests include robust automatic control,computer vision, and robotics.

J. Obviedo-Barriga, Universidad Veracruzana,

Received his B.Sc. degree inElectronic Engineering from the Technological Instituteof Orizaba, Veracruz, Mexico, the M.Sc. degreein Science of Electronic Engineering from theAutonomous University of Puebla, Puebla, Mexico,and Ph.D. degree in Electrical Engineering fromthe Advanced Studies and Research Center of theNational Polytechnic Institute, Campus Guadalajara,Mexico, in 2004, 2007, and 2013, respectively.His research interests include non-linear robustcontrol design, sliding modes and its applicationsto robotics, modeling control of humanoid robots.Now he works at the Veracurzana University as fulltime Professor at the Faculty of Engineering.

B. Castillo-Toledo, Centro de Investigación y De Estudios Avanzados del IPN

Received his B.Sc.degree in Electrical Engineering from the NationalPolytechnic Institute (IPN), Mexico City, Mexico,the M.Sc. degree from the Research and AdvancedStudies Center of the National PolytechnicInstitute (CINVESTAV-IPN), Guadalajara, Mexico,and the Ph.D. degree from the University of Rome“La Sapienza,” Rome, Italy, in 1981, 1985, and1992, respectively. He was a Lecturer with theSchool of Electrical and Mechanics Engineering,IPN, from 1985 to 1989. From 1985 to 1995, hewas with the Automatic Control Section, Departmentof Electrical Engineering, CINVESTAV-IPN,and since 1995, with CINVESTAV-IPN, GuadalajaraCampus. He held several research stayswith the University of Rome “La Sapienza” andUniversity of L’Aquila, L’Aquila, Italy, and was avisiting Professor at the Laboratoire d’Automatiqueet d’Analyse des Systemes, French Council forScientific Research, from 1999 to 2000, and at theUniversity of Compi`egne, Compiegne, France, duringthe first semester of 2002. His main researchinterests include nonlinear control design, the robustregulation problem, and application of artificialneural networks and fuzzy logic techniques to controland fault diagnosis of dynamical systems.

A. Loukianov, Centro de Investigación y De Estudios Avanzados del IPN

Received his B.Sc. degreefrom the Polytechnic Institute, Moscow, Russia, in1975 and the Ph.D. degree in Automatic Controlfrom the Institute of Control Sciences, RussianAcademy of Sciences, Moscow, Russia, in 1985.He was with the Institute of Control Sciences in1978 and was the Head of the Discontinuous ControlSystems Laboratory from 1994 to 1995. From1995 to 1997, he held a visiting position at theUniversity of East London, London, U.K., and sinceApril 1997, he has been with the Research andAdvanced Studies Center of the National PolytechnicInstitute, Guadalajara Campus, Mexico, as aProfessor of Electrical Engineering graduate programs.From 1992 to 1995, he was in charge ofan industrial project between the institute and thelargest Russian car plant and also several internationalprojects supported by the International Associationand Specific International Scientific CooperationActivities-COPERNICUS, Brussels. Hehas published more than 100 technical papers ininternational journals, books, and conferences andhas been a reviewer for various international journalsand conferences. His research interests includenonlinear system robust control and variablestructure systems with sliding modes as appliedto dynamical plants with delay, electric drives, andpower systems control, robotics, spice, and automotivecontrol.

E. Bayro-Corrochano, Centro de Investigación y De Estudios Avanzados del IPN

Gained his Ph.D. inCognitive Computer Science in 1993 from the Universityof Wales at Cardiff. From 1995 to 1999he was Researcher and Lecturer at the Institutefor Computer Science, Christian Albrechts University,Kiel, Germany, working on applications ofgeometric Clifford algebra to cognitive systems.At present he is a full Professor at CINVESTAV,Guadalajara campus, Mexico, Department of ElectricalEngineering and Computer Science. Since2006 he has been a fellow of the IAPR society.His current research interest focuses on geometricmethods for artificial perception and action systems.It includes geometric neural networks, visuallyguided robotics, humanoids, color image processing,Lie bivector algebras for early vision androbot maneuvering. He developed the quaternionwavelet transform for quaternion multi-resolutionanalysis using the phase concept. He conductsresearch in robot medical image processing aswell. He is associate editor of Journal Robotica,Journal of Advanced Robotic Systems, Journal ofPattern Recognition, and Journal of MathematicalImaging. He is editor and author of the followingbooks: Geometric Computing for Perception ActionSystems, E. Bayro-Corrochano, Springer Verlag,2001; Geometric Algebra with Applications in Scienceand Engineering, E. Bayro-Corrochano andG. Sobczyk (Eds.), Birkhauser 2001; Handbookof Geometric Computing for Pattern Recognition,Computer Vision, Neurocomputing and Robotics,E. Bayro-Corrochano, Springer Verlag, 2005, GeometricComputing for Wavelet Transforms, RobotVision, Learning, Control and Action. E. Bayro-Corrochano, Springer Verlag, London, 2009; GeometricAlgebra Computing for Engineering andComputer Science, E. Bayro-Corrochano and G.Sheuermann, Springer Verlag, London, 2009. Hehas published over 150 refereed journal and conferencepapers and book chapters.

Downloads

Additional Files

Published

2015-09-30

Issue

Vol. 19 No. 3 (2015): 19(3) 2015 Thematic issue: Vision and Robotics (Guest editors: S. Ortega, A. Pedroza, O. Carbajal, B. Castillo Toledo)

Section

Articles

License 

Hereby I transfer exclusively to the Journal "Computación y 
Sistemas", published by the Computing Research Center (CIC-IPN),
the Copyright of the aforementioned paper. I also accept that these 
rights will not be transferred to any other publication, in any other 
format, language or other existing means of developing.
I certify that the paper has not been previously disclosed or simultaneo
usly submitted to any other publication, and that it does not contain 
material whose publication would violate the Copyright or other 
proprietary rights of any person, company or institution. I certify that 
I have the permission from the institution or company where I work or 
study to publish this work.
The representative author accepts the responsibility for the publication
of this paper on behalf of each and every one of the authors. 
This transfer is subject to the following conditions:
  • The authors retain all ownership rights (such as patent rights) of this work, except for the publishing rights transferred to the CIC, through this document.
  • Authors retain the right to publish the work in whole or in part in any book they are the authors or publishers. They can also make use of this work in conferences, courses, personal web pages, and so on.
  • Authors may include working as part of his thesis, for non-profit distribution only.