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| B3M33PKR |
Advanced robot kinematics |
| Roles: | PV |
Extent of teaching: | 2P+2C |
| Department: | 13133 |
Language of teaching: | CS |
| Guarantors: | Pajdla T. |
Completion: | Z,ZK |
| Lecturers: | Pajdla T. |
Credits: | 6 |
| Tutors: | Pajdla T. |
Semester: | Z |
Web page:
https://cw.fel.cvut.cz/wiki/courses/pkr
Anotation:
The course will explain and demonstrate theoretical and computational methods for describing and analyzing the kinematics of industrial robots, the principles of representing spatial motion—rotation matrices, quaternions, Euler vectors, and Cayley parametrization—and robot description using the Denavit–Hartenberg convention for the kinematic analysis of manipulators. The main topics will be: a) solving the inverse kinematics problem for a general 6-DOF serial manipulator, and b) analyzing its singularities. The fundamental theoretical and computational tools will be linear and polynomial algebra, as well as methods of computational algebraic geometry. The theoretical techniques will be verified through implementation tasks using simulations. The course is theoretical and suitable for students interested in mathematics and interested in pursuing an academic career.
Study targets:
The aim of the course is to introduce methods for analyzing and modeling robot kinematics based on algebraic geometry.
Content:
| 1. | | Introduction, algebraic equations, and eigenvalues of a matrix |
| 2. | | Motion as a coordinate transformation, rotation matrices, axis, and angle of rotation |
| 3. | | Parametrization of rotation: axis–angle, quaternions, Cayley parametrization, rational rotations |
| 4. | | Denavit–Hartenberg convention for a serial manipulator |
| 5. | | Axis of motion |
| 6. | | Rings, varieties, ideals |
| 7. | | Monomial ordering, polynomial “division” |
| 8. | | Gröbner bases |
| 9. | | Buchberger’s algorithm |
| 10. | | Algebraic-numerical solution of systems of polynomial equations |
| 11. | | Algebraic solution of the inverse kinematics problem for a general 6R serial manipulator I 12. Algebraic solution of the inverse kinematics problem for a general 6R serial manipulator II |
| 13. | | Kinematic singularities of a manipulator I 14. Kinematic singularities of a manipulator II |
Course outlines:
| 1. | | Algebraic equations and eigenvalues of a matrix, Test A 2. Motion as a coordinate transformation, rotation matrices, axis and angle of rotation |
| 3. | | Parametrization of rotation: axis–angle, quaternions, Cayley parametrization, rational rotations |
| 4. | | Denavit–Hartenberg convention for a serial manipulator |
| 5. | | Test 1 |
| 6. | | Rings, varieties, ideals |
| 7. | | Monomial ordering, polynomial “division” |
| 8. | | Gröbner bases |
| 9. | | Buchberger’s algorithm |
| 10. | | Test 2 |
| 11. | | Algebraic solution of the inverse kinematics problem for a general 6R serial manipulator I 12. Algebraic solution of the inverse kinematics problem for a general 6R serial manipulator II |
| 13. | | Kinematic singularities of a manipulator I 14. Test 3 |
Exercises outline:
Literature:
Reza N. Jazar: Theory of Applied Robotics: Kinematics, Dynamics, and Control. Springer, second edition, 2010. A textbook covering the geometry and kinematics of manipulators. Available in the CTU library.
| M. | | Meloun, T. Pajdla. Inverse Kinematics for a General 6R Manipulator. CTU-CMP 2013-29. 2013. An algebraic-numerical solution of the inverse kinematics problem for a 6R manipulator. |
ftp://cmp.felk.cvut.cz/pub/cmp/articles/meloun/Meloun-TR-2013-29.pdf
| T. | | Pajdla. Elements of Geometry for Robotics. 2025. Geometry and representation of motion. |
Available as a PDF:
https://cw.fel.cvut.cz/b251/_media/courses/pkr/pkr-lecture-2025-09-20.pdf
Requirements:
B0B01LAG – Linear algebra, B3B33ROB1 – Robotics
Keywords:
robotics, kinematics, algebraic geometry
Subject is included into these academic programs:
| Page updated 10.6.2026 17:52:01, semester: Z/2026-7, L/2029-30, L/2025-6, Z/2028-9, Z,L/2027-8, L/2028-9, L/2026-7, Send comments about the content to the Administrators of the Academic Programs |
Proposal and Realization: I. Halaška (K336), J. Novák (K336) |