ementas

Geometric foundations of robotics

The course covers basic algebra and geometry (differential and algebraic) used for the fundamentals of robotics.

Program:

• Lie Groups. The group of rotations 𝑆𝑂(3,R), Euler’s theorem and Euler’s rotation axe. Hamilton’s quaternions and their representation by Euclidean rotations. Orthogonal group 𝑂(𝑛), special orthogonal group 𝑆𝑂(𝑛), symplectic group 𝑆𝑝(2𝑛, R), unitary group 𝑈(𝑛), special unitary group 𝑆𝑈(𝑛). Homomorphisms. Actions and Products. The Proper Euclidean Group 𝑆𝐸(3): isometries, Chasles’s theorem.
• Subgroups. The Homomorphism Theorems. Quotients and Normal Subgroups. Group Actions. Matrix Normal Forms. Subgroups of SE(3). Releaux’s Lower Pairs. Robot Kinematics.
• Lie Algebras. Tangent vectors. Adjoint and coadjoint representations. Commutators. Exponential Mapping. Robot Jacobians and Derivatives, Angular Velocity, Velocity Screw. Subalgebras, homomorphisms, ideals. The Killing Form. The Campbell–Baker–Hausdorff Formula.
• Kinematics of 3-R Wrists, 3-R Robots, Planar Motion. Singularities. Euler–Savaray equation. Inflection Circle. Ball’s Point. The Cubic of Stationary Curvature.
• Line Geometry. Plücker coordinates. Klein Quadric. Action of the Euclidean Group. Ruled Surfaces: the regulus, the cylindroid, curvature axes. Line complexes. Inverse Robot Jacobians. Grassmannians.
• Representation Theory. Representations of SO(3). Plethysm. Representations of SE(3).
• Screw Systems.
• Clifford Algebras. Geometric Algebra. Dual Quaternions. Geometry of Ruled Surfaces. Use of Clifford Algebra in Kinematics. Pieper’s Theorem.
• The Study Quadric. Linear Subspaces. Quadric Grassmannian. Partial Flags and Projections. Quadric Subspaces. Intersection Theory.
• Statics. Co-Screws. Forces, Torques and Wrenches. Gripping. Friction.
• Dynamics. Momentum and Inertia. Robot Equations of Motion. Lagrangian Dynamics of Robots. Hamiltonian Dynamics of Robots.
• Constrained Dynamics.
• Differential Geometry. Metrics, Connections, Geodesics. Mobility of Overconstrained Mechanisms. Hybrid Control.

Basic bibliography:

• J. M. Selig: Geometric Fundamentals of Robotics, Second Edition. 2004. Springer Monographs in Computer Science.

Complementary bibliography:

• Joe Harris: Algebraic geometry: a first course, Vol. 133. Springer Science and Business Media, 2013.
• Clifford Taubes: Differential geometry: bundles, connections, metrics and curvature, OUP Oxford, volume 23, 2011.
• Vladimir Arnold: Mathematical methods of classical mechanics, Vol. 60. Springer Science and Business Media, 2013.