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Course Descriptions

MECH_ENG 450: Geometry in Robotics

Prerequisites

Description

Application of tools from differential geometry and Lie groups to problems in dynamics, controllability, and motion planning for mechanical systems, particularly with non-Euclidean configuration spaces.

Who Takes It

This course is taken by graduate students interested in robotics.

What It's About

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 is a math course, but geared specifically toward robotics researchers who do not have a strong background in these fields of mathematics. In a 10-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.

Lectures

Topics 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

Textbook

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. Students may be asked to present lectures. The final project will be to study a paper in the mathematical robotics literature and write a summary explaining the methods and results.

Contact

Professor: Kevin Lynch
Email: kmlynch@northwestern.edu