Likely modified to get to geometric mechanics and control earlier.

1. Non-Euclidean spaces, robotic examples, course overview, class introductions
2. Configuration space, degrees-of-freedom, constraints (holonomic and nonholonomic)
3. Metric spaces, neighborhoods, open/closed spaces, boundary/interior, topology induced by a metric
4. Connected and compact spaces, C^n mappings, injection, surjection, bijection
5. Morphisms (iso, homeo, diffeo), charts and atlases, differentiable manifold
6. Paper discussion, "The Construction of Analytic Diffeomorphisms for Exact Robot Navigation on Star Worlds," E. Rimon and D. Koditschek, 1989 IEEE International Conference on Robotics and Automation, pp. 21-26.
7. Differentiable manifolds, groups, Lie groups
8. Lie groups, matrix Lie groups
9. Tangent vector, tangent space, tangent bundle (cotangent vector)
10. Vector fields, integral curves, Lie derivative, Lie bracket
11. Lie bracket, examples
12. Distributions, involutivity, integral manifolds, Frobenius theorem, dynamical polysystem, control affine nonlinear systems, controllability definitions (accessibility, controllability, STLA, STLC)
13. Lie algebra of vector fields, Lie algebra rank condition (LARC)
14. Paper discussion, "Controllability of a multibody mobile robot," J.-P. Laumond, IEEE Transactions on Robotics and Automation, 9(6):755-763, December 1993.
15. STLC, LARC, dynamic systems with drift, "bad" Lie brackets
16. Neutralizing bad brackets, Sussmann's theorem on local controllability, planar body example
17. Vector fields on Lie groups: left and right translations, left and right invariant vector fields
18. Left and right invariant vector fields (cont.), matrix Lie groups and their Lie algebras
19. Invariant vector fields (cont.), Jurdjevic and Sussmann theorem on global controllability of right-invariant systems on compact connected Lie groups
20. Exponential map, log (cotangent vectors)
21. Paper discussion, "Control synthesis and adaptation for an underactuated autonomous underwater vehicle," N. E. Leonard, IEEE Journal of Oceanic Engineering, 20(3):211-220, July 1995.
22. Riemannian metrics, right and left invariant metrics
23. Riemannian metrics and examples from Park's paper, geodesic
24. Parallel transport, affine connection, covariant derivative
25. Levi-Civita (Riemannian) connection, Christoffel symbols, dynamics, geodesics
26. Dynamics (cont.)
27. Paper discussion, "Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems," F. Bullo and K. M. Lynch, IEEE Transactions on Robotics and Automation, 17(4):402-412, August 2001.
28. Wrap-up

Readings

• Weeks 1 and 2: Chapter 3 and Appendices B, C from Principles of Robot Motion, Choset, Lynch, et al., MIT Press, to appear. Supplementary reading: Point set topology: Chapter 1 of Lecture Notes on Elementary Topology and Geometry, M. Singer and J. A. Thorpe, Springer-Verlag, 1967 (3rd printing 1987); Differentiable manifolds: pp. 23-31 of Geometrical Methods of Mathematical Physics, B. Schutz, Cambridge University Press, 1980.