* Introductory discussion, "The Geometric Mechanics of Undulatory Robotic
Locomotion," Jim Ostrowski and Joel Burdick, International Journal of Robotics
Research, 17(7):683-701, July 1998.
* Non-Euclidean spaces, robotic examples, course overview
* Configuration space, degrees-of-freedom, constraints (holonomic and nonholonomic)
* Metric spaces, neighborhoods, open/closed spaces, boundary/interior, topology induced by a metric
* Connected and compact spaces, C^n mappings, injection, surjection, bijection
* Morphisms (iso, homeo, diffeo), charts and atlases, differentiable manifold
* Differentiable manifolds, groups, Lie groups
* Lie groups, matrix Lie groups
* Tangent vector, tangent space, tangent bundle (cotangent vector)
* Vector fields, integral curves, Lie derivative, Lie bracket
* Lie bracket, examples
* Distributions, involutivity, integral manifolds, Frobenius theorem, dynamical polysystem, control affine nonlinear systems, controllability definitions (accessibility, controllability, STLA, STLC)
* Lie algebra of vector fields, Lie algebra rank condition (LARC)
* STLC, LARC, dynamic systems with drift, "bad" Lie brackets
* Neutralizing bad brackets, Sussmann's theorem on local controllability, planar body example
* Vector fields on Lie groups: left and right translations, left and right invariant vector fields
* Left and right invariant vector fields (cont.), matrix Lie groups and their Lie algebras
* Invariant vector fields (cont.), Jurdjevic and Sussmann theorem on global controllability of right-invariant systems on compact connected Lie groups
* Exponential map, log (cotangent vectors)
* Riemannian metrics, right and left invariant metrics
* Riemannian metrics and examples from Park's paper, geodesic
* Parallel transport, affine connection, covariant derivative
* Levi-Civita (Riemannian) connection, Christoffel symbols, dynamics, geodesics
* Dynamics (cont.)
Readings