Lecture: MWF 11-11:50, Tech L158
Prerequisites: Graduate standing in engineering or permission of instructor

Most engineers have studied calculus and dynamics on real vector spaces, such as the plane R^2 or three-dimensional space R^3. However, the configuration (position) of a robotic system can rarely be described by a vector space such as R^n. Instead, the configuration space of a robot system is often "curved." For example, the configuration space of a two joint robot arm (with revolute joints) is properly described as a torus, not a plane. The geometry of the configuration space (and more generally the state space) plays a large role in the dynamic behavior of the system.

In this course we introduce some mathematical tools from differential geometry and Lie groups that allow us to study different robotic systems in a more unified way. Since many important results in mathematical robotics and control theory are derived and presented using these tools, a major goal of this course is to make these works accessible to students conducting robotics research.

This course is a math course, but geared specifically toward robotics researchers who do not have a strong background in these fields of mathematics. In a ten week quarter, it is impossible to cover the range of topics we cover with great depth. Instead, we will focus on applications of the concepts to robotic systems. For example, instead of discussing Lie groups in full generality, we will focus on SE(3) and its subgroups, since these are most relevant to robotics. When a new topic is introduced, a relevant robotic example will be given along with it.

There is no single book for the course; readings will be taken from different sources. To supplement the text readings, we will read a few robotics papers which apply the concepts we are currently studying.

Assignments
Students will be asked to prepare and present one lecture. There will be periodic assignments during the course. Students will also be graded on participation in class discussion of research papers. There are no exams.

Topics to be covered include:

• Mathematical preliminaries from topology: manifolds, mappings, morphisms
• Vector fields, tangent bundles, cotangent bundles, fiber bundles, natural projection
• Lie groups, Lie derivatives, Lie algebras, exponential map; distributions, Frobenius theorem, invariant vector fields, controllability
• Riemannian manifolds and metrics, Levi-Civita connection, covariant derivative, Christoffel symbols
• Geometric mechanics and control

Approximate syllabus

Code for calculating the Levi-Civita connection and the covariant derivative.

Readings will be taken from several references, including the following:

An Introduction to Differentiable Manifolds and Riemannian Geometry, W. M. Boothby, Academic Press, 1986. ISBN 0-12-116053-X.

Geometrical Methods of Mathematical Physics, B. Schutz, Cambridge University Press, 1980. ISBN 0-521-29887-3.

A Mathematical Introduction to Robotic Manipulation, R. M. Murray, Z. Li, and S. Sastry, CRC Press, 1994. ISBN 0-8493-7981-4.

Riemannian Geometry, M. P. do Carmo, Birkhauser, Boston, MA, 1992.

Lecture Notes on Elementary Topology and Geometry, M. Singer and J. A. Thorpe, Springer-Verlag, 1967, 3rd printing 1987. ISBN 0-387-90202-3. Chapter 1.

"Distance Metrics on the Rigid-Body Motions with Applications to Mechanism Design," F. C. Park, ASME Journal of Mechanical Design, v. 117, pp. 48-54, March 1995.

"Metrics and Connections for Rigid-Body Kinematics," M. Zefran, V. Kumar, and C. Croke, International Journal of Robotics Research, 18(2):243-258, February 1999.

Readings from "Principles of Robot Motion" (book in preparation), H. Choset, K. Lynch, et al., MIT Press 2005.