ResearchActivities

Geometric analysis in control

Geometric analysis in control and robotics

We are interested in

Linear Algebra – Geometric (Clifford) Algebras – Quaternions – Lie Groups and Algebras – Representation Theory – Mathematical (Geometric) Control Theory – Geometric Structures – Symmetry – Differential Geometry – Sub Riemannian Geometry

with applications in

Robotics – Mechanics – Nonholonomic systems – Inverse problems – Control theory – Route planning – Binocular vision – Quantum mechanics – Quantum computing – Theoretical physics

Currently realized topics

Control of a planar mechanism using symmetries

[1] J. Hrdina, A. Návrat, P. Vašík, L. Zalabová, Note on geometric algebras and control problems with SO(3) – symmetries, Mathematical Methods in the Applied Sciences (2022)
[2] J. Hrdina, A. Návrat, L. Zalabová, On symmetries of Sub--Riemannian structure with growth vector (4,7), Annali di Matematica Pura ed Applicata (2022)

[3] Hrdina, J., Návrat, A., Zalabová, L., Symmetries in geometric control theory using Maple, Mathematics and Computers in Simulation, 2021, 190, pp. 474–493

[DP 1] FROLÍK, S., Geometrická teorie řízení na nilpotentních Lieových grupách. Brno University of technology, Master's thesis.

Use of real geometric algebras in robotics

[4] Hrdina, J., Návrat, A., Vašík, P., Dorst, L., Projective Geometric Algebra as a Subalgebra of Conformal Geometric algebra, Advances in Applied Clifford Algebras, 2021, 31(2), 18
[5] Hrdina, J., Návrat, A. Binocular Computer Vision Based on Conformal Geometric Algebra. Adv. Appl. Clifford Algebras 27, 1945–1959 (2017)
[6] Hildenbrand, D., Hrdina, J., Návrat, A. et al. Local Controllability of Snake Robots Based on CRA, Theory and Practice. Adv. Appl. Clifford Algebras 30, 2 (2020).

[DP 2] STODOLA, M., Robotický manipulátor prostředky CGA Brno University of technology, 2019, Master's thesis.

Quantum computing (quantum game theory) using complex geometric algebras

[7] Hrdina J., Návrat A., Vašík P., Quantum computing based on complex Clifford algebras, Quantum Information Processing (2022)
[8] Alves, R., D. Hildenbrand, J. Hrdina, and C. Lavor, An Online Calculator for Quantum Computing Operations Based on Geometric Algebra, Advances in Applied Clifford Algebras 32 (1). 2022.
[9] Eryganov, I. Hrdina, J., Clifford algebra in repeated quantum prisoner's dilemma. Math Meth Appl Sci. 2022

[DP 3] KATABIRA, J.Groverův algoritmus v kvantovém počítání a jeho aplikace. Brno University of technology, 2021. Master's thesis.

Projects:

  • OC-2021-1-25132 Cost project, Cartan geometry, Lie, Integrable Systems, quantum group Theories for Applications – funding of participation in seminars and workshops of the network. 2023-2025

  • Cambridge University – Memorandum of Understanding – collaboration on topics related to geometric algebras

  • University of Defense – collaboration on the development of autonomous control.

  • OPVVV MSM EF16 026/0008404, Machine Tools and Precision Engineering, 2019–2022

Members:
Jaroslav Hrdina, Petr Vašik, Aleš Návrat, Ivan Eryganov
PhD students: Roman Byrtus, Marek Stodola, Johanka Brdečková

Quaternions in industrial engineering and robotics

The geometrical analysis research team within the Mathematics Department focuses on applying quaternions and dual quaternions in industrial machinery and robotics. These mathematical constructs are used to model closed and open kinematic chains, offering significant computational speed and accuracy advantages. Quaternions simplify the representation of three-dimensional rotations and translations, leading to faster and more efficient algorithms. Additionally, their structure is well-suited for parallel processing, enhancing performance in real-time applications. Our research demonstrates that quaternions improve the robustness and stability of control algorithms, which are crucial for precision tasks in robotics and automation.

Contact:
Doc. Mgr. Jaroslav Hrdina, Ph.D.