Course detail

Mathematical Robotics

FSI-SMR-A Acad. year: 2027/2028 Winter semester

The course focuses on developing an advanced mathematical framework for robotic applications (controllability algebra, the Chow–Rashevsky theorem, nilpotent approximations). It covers the theory of Lie groups and algebras (SO(3), SE(3), exponential maps, Lie brackets) for representing the motion of a rigid body, geometric algebra (multivectors, rotors, PGA/CGA) for the efficient formulation of kinematics, and Riemannian and sub- Riemannian geometry (Hamiltonian, Pontryagin’s maximization principle) for describing nonholonomic constraints and control. These findings find direct application in the analysis and control of robotic manipulators, the control of nonholonomic mechanisms, trajectory optimization, and
singularity avoidance.

Language of instruction

English

Number of ECTS credits

5

Supervisor

doc. Mgr. Jaroslav Hrdina, Ph.D.

Department

Institute of Mathematics

Entry knowledge

Fundamentals of Linear Algebra (ideally courses SLA and SLB)

Rules for evaluation and completion of the course

Active participation in class sessions, mastery of the entire course material, and a grade of “satisfactory” or better on all assignments. May be substituted with a semester project.

Exam: written and oral. The written portion consists of a series of problems covering the material presented in lectures and is graded on a scale of 0–80 points. The oral portion assesses theoretical knowledge and the ability to navigate the subject matter, and is graded on a scale of 0–20 points.

Aims

To familiarize students with the basic mathematical tools for the design and control of robotic systems, particularly robotic manipulators, but also nonholonomic mechanisms of various types. Graduates of this course should be able to solve forward and inverse kinematics problems, find optimal control solutions, and discuss the singularities of selected mechanisms using modern linear algebra and geometry.

Study aids

elearning

The study programmes with the given course

Programme N-MAI-A: Mathematical Engineering, Master's, compulsory

Type of course unit

 

Lecture

26 hours, optionally

Syllabus


  • Topic 1: Lie group of rigid-body motion. The SO(3) and SE(3) group apparatus for a nonsingular description of rotations and rigid transformations in space.

  • Topic 2: Geometry of helical motion of a body. The SO(3) Lie algebra, Rodrigues’ formula, and the concept of twists in the SE(3) Lie algebra.

  • Topic 3: Velocity kinematics, force application, and reciprocal screws. Spatial and body velocities of a rigid body related via the adjoint transformation Ad_g. The concept of a wrench for generalized force systems.

  • Topic 4: Forward and inverse kinematics of serial manipulators. Forward kinematics of open articulated chains using the product of exponentials (POE).

  • Topic 5: The manipulator’s Jacobian, singularities. The manipulator’s Jacobian derived from POE. Identification of kinematic singularities, measurement of manipulability, and extension to redundant and parallel mechanical structures.

  • Topic 6: Introduction to nonholonomic constraints. Nonholonomic (nonintegrable) Pfaff constraints; robotic systems involving roll-off without slip or momentum conservation. Conditions for testing integrability and configurability.

  • Topic 7: Nonholonomic systems and control using sinusoids. Drift-free canonical control systems (Brockett system, systems in chained form). Use of harmonic control signals (sinusoids) to generate motion in the direction of Lie brackets.

  • Topic 8: General methods for planning nonholonomic trajectories. Advanced algorithms for controlling nonholonomic systems using nilpotent approximations and conversion to chained form.

  • Topic 9: Hamilton quaternions and Clifford biquaternions. Representation of spatial rotations using unit Hamilton quaternions. Dual quaternions (Clifford biquaternions describing the full Euclidean motion SE(3))

  • Topic 10: Projective (PGA) and conformal (CGA) geometric algebra. Extension of Euclidean space by a point at infinity using PGA. Incorporation of spherical objects and circles using CGA.

  • Topic 11: Description and analysis of robotic systems. Construction of a kinematic chain using PGA; inverse kinematics using CGA.

  • Topic 12: Visual Servoing and Feedback Control Based on Camera Data. Controlling a robot’s motion based on visual information from cameras in a closed loop. Binocular vision.

  • Topic 13: Supplementary elective topic – reserve

Exercise

26 hours, compulsory

Syllabus

Copied the topics covered in lectures this week