PKR − Advanced Robot Kinematics

,,Drahá slečno Gloryová, Roboti nejsou lidé. Jsou mechanicky dokonalejší než my, mají úžasnou rozumovou inteligenci, ale nemají duši. Ó, slečno Gloryová, výrobek inženýra je technicky vytříbenější než výrobek přírody.“ - Karel Čapek, R.U.R.

[“Miss Glory, robots are not people. They are mechanically much better than we are, they have an amazing ability to understand things, but they don't have a soul. Young Rossum created something much more sophisticated than Nature ever did - technically at least!”]

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.

Course material: pkr-lecture-2025-09-20.pdf.

Lecturers: Tomáš Pajdla

Teachers: Tomáš Pajdla See Labs for details.

1. All homework must be submitted via BRUTE and accepted.
2. At least 50% of points in total for the homework, i.e. $$p_{\mathrm{homework}} = \frac{1}{H} \sum_{j=1}^H \frac{h_j}{H_j} \geq 0.5$$ where $h_j$ (resp. $H_j$) is the student's (resp. maximum) number of points for the $j$-th homework assignment and $H$ is the number of homework assignments.
3. At least 50% of points in total from the tests, i.e. $$p_{\mathrm{tests}} = \frac{1}{T} \sum_{j=1}^T \frac{t_j}{T_j} \geq 0.5$$ where $t_j$ (resp. $T_j$) is the student's (resp. maximum) number of points for the $j$-th test and $T$ is the number of tests.
4. Regular submission of homework ends on January 12, 2025. Later submissions are possible only by an agreement with the assistants.

The exam consists of a written and an oral part. Having an assessment is not required for the written part, but is required for the oral part. You may skip the oral exam if you are satisfied with the result after the written exam. Exam content:

1. Linear algebra (linear space, basis, coordinates, linear dependence/independence, matrices, rank, determinant, eigenvalues, and eigenvectors, solving systems of linear equations, linear mapping, and its matrix), computing roots of a polynomial via eigenvalues of its companion matrix, motion as transformation of coordinates, rotation matrix, eigenvalues and eigenvectors of rotation matrices, rotation angle and rotation axis, axis of motion, Rodriguez parametrization of rotations, quaternions, quaternion parametrization of rotations, multivariate polynomials, monomial ordering, multivariate polynomial division algorithm, Groebner basis, Buchberger's algorithm, DH notation, forward and inverse kinematics.
2. - Course material: PKR-Lecture-2021.pdf.

Written In-Person exam organization:

1. The written exam is taken in a classroom.
2. You may NOT use any prepared material.

Oral exam organization:

1. The oral exam will be organized in groups of 3-5 students.
2. We will go over the written exam and resolve problematic answers.
3. Understanding of theoretical concepts will be examined.

The grade depends on the exam (40%), tests (30%), and homework (30%). This means that we compute the relative points for the exam, tests and homework as $$ p_{\mathrm{exam}} = \frac{e}{E} \in [0,1], \quad p_{\mathrm{tests}} = \frac{1}{T}\sum_{j=1}^T\frac{t_j}{T_j} \in [0,1], \quad p_{\mathrm{homework}} = \frac{1}{H}\sum_{j=1}^H\frac{h_j}{H_j} \in [0,1] $$ where $e$ (resp. $E$) is the student's (resp. maximum) number of points for the exam and $p_{\mathrm{tests}}, p_{\mathrm{homework}}$ are as described in the “assessment” part above. The total relative points that define the grade are computed as $$ p_{\mathrm{total}} = 0.4\cdot p_{\mathrm{exam}} + 0.3\cdot p_{\mathrm{tests}} + 0.3\cdot p_{\mathrm{homework}} \in [0,1]. $$

1. Lecture: It is very difficult to pass the course without attending lectures.
2. Labs: It is impossible to pass the course without attending labs.
3. Homework: Homework is assigned at a lab where it can be discussed with teaching assistants. Students work out homework individually (rulesin Czech). The deadline for submitting homework via BRUTE is on Monday two weeks after the assignment. Late submissions are penalized (10% for each commenced day of delay but not more than 50% of points).
4. Assessment: see above.
5. Tests: Students work out test independently.

1. PKR-Lecture-2021.pdf

1. Basic algebra (Groups, Rings, Ideals, Fields, Vector spaces) in 33 simple videos @ Socratica
2. Northwestern University Coursera Course Modern Robotics
3. Stanford cs223a Introduction to Robotics
4. Math Doctor Bob. Math Instruction Online. In Plain Language.
5. Lung-Wen Tsai. Robot Analysis And Design: The Mechanics of Serial And Parallel Manipulators, John Wiley and Sons, 1999.
6. G Sanderson Essence of Linear Algebra from 3Blue1Brown
7. J Strom, K Astrom, T Akenine-Moller Interactive Linear Algebra Course
8. P Pták. Introduction to Linear Algebra. Vydavatelství ČVUT, Praha, 2007.
9. E Krajník. Maticový počet. Vydavatelství ČVUT, Praha, 2000.
10. D Cox, J Little, D O'Shea. Ideals, Varieties, and Algorithms. 2nd edition, Springer, 1998.
11. B Sturmfels. Polynomials, Ideals, and Grobner Bases
12. M Michalek, B Sturmfels. Invitation to Nonlinear Algebra
13. B Sturmfels. Inroduction to Groebner bases youtube

1. A0B01LAG Linear Algebra (must have)
2. A3B33ROB Robotics (nice to have)