AA 546
Manifolds and Geometry for Systems and Control

Winter 2006

 

 

Instructor

Prof. Kristi A. Morgansen
morgansen@aa.washington.edu
Gug. 310

 

Office Hours

M 1-2pm and by appointment

 

Lectures: T/TH 12:30-1:50pm, LOW 219

Course Description

This course provides an introduction to the fundamentals of calculus on manifolds and group theory focusing on applications in robotics and control theory. We will begin with an overview of the use of differential geometry in control theory relative to other techniques and build a rigorous foundation from which current literature can be understood. Topics to be covered include: manifolds, tangent spaces and bundles, Lie algebras, groups and semi-groups, and coordinate versus coordinate-free representations. Applications that will be addressed are modeling of mechanical systems, potential fields, nonholonomic systems, and self-assembling systems.

Suggested prerequisites: EE510

Homework and Exam Policy

Collaboration on homework assignments is allowed. You may consult outside reference materials, other students, or the instructor. All solutions that are handed in should reflect your understanding of the subject matter at the time of writring. No collaboration is allowed on the midterm or the final exam.

Grading

The final grade will be based on homework, a midterm, a project and a final exam.

·         Homework: 40%.  There will be 8 homework assignments due at the beginning of class on Tuesdays. Late homework will not be accepted without prior permission from the instructor.

·         Midterm: 25%.  The midterm will be on Thursday, Feb. 9 and will be a take-home exam.

·         Project:  10%.  The project will involve reviewing a published paper and presenting the material to the class.  Presentations will likely be scheduled for Friday March 10.

·         Final exam: 25%.  The final exam will be a take-home exam.

 

Homework Section:  There will be a weekly homework solving section (time and date TBA).  In these sections, students will present the solutions to homework problems.  This activity will help with technical presentation skills as well as problem solving techniques. 

Course Text

The required reading source for the course is

F. Bullo and A.D. Lewis, Geometric Control of Mechanical Systems, Springer, 2005.

 

Additional references that may be useful:

·         J. E. Marsden, T. Ratiu, and R. Abraham, Manifolds, Tensor Analysis, and Applications, 3ed., Springer-Verlag, 2003.

·         W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. Academic Press, 1986.

·         Anthony Bloch, Nonholonomic Mechanics and Control, Springer Verlag.

·         V. Guilleman and A. Pollak, Differential Topology, Prentice-Hall, 1974.

·         J. Milnor, Topology from the Differentiable Viewpoint, University Press of Virginia, 1965.

·         B. Schutz, Geometrical Methods of Mathematical Physics, Cambridge University Press, 1980.

·         M. Spivak, A Comprehensive Introduction to Differentiable Geometry, v. 1. Publish or Perish, 1970.

·         F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, 1983.

 

Schedule

Date

Topics

Reading

Assignments

Jan 3

Introduction

BL 1

 Homework #1 Assignment

(Solutions)

Jan 5

Sets and sequences; Vector spaces

BL 2.1-2.2

 

Jan 10

Inner products and bilinear maps;

Tensors

BL 2.3-2.5

 Homework #2 Assignment

(Solutions)

Jan 12

Topology; manifolds

BL 3.1-3.2

 

Jan 17

Tangent bundles

BL 3.3

Homework #3 Assignment

(Solutions)

Jan 19

Vector bundles

BL 3.4

 

Jan 24

Vector fields

BL 3.5

Homework #4 Assignment

(Solutions)

Jan 26

Vector fields

BL 3.5

 

Jan 31

Tensor fields

BL 3.6

Homework #5 Assignment

(Solutions)

Feb 2

Tensor fields

BL 3.6

 

Feb 7

Distributions and codistributions

BL 3.7

Midterm

(Solutions)

Feb 9

Distributions and codistributions

BL 3.7

 

Feb 14

Affine differential geometry

BL 3.8

 

Homework #6 Assignment

(Solutions)

Feb 16

Affine differential geometry

BL 3.8

 

Feb 21

Rigid body kinematics

BL 5.1

Homework #7 Assignment

(Solutions)

Feb 23

Lie groups and Lie algebras

BL 5.2

 

Feb 28

Metrics, connections and systems on Lie groups

BL 5.3

Homework #8 Assignment

(Solutions)

Mar 2

Metrics, connections and systems on Lie groups

BL 5.3

 

Mar 7

Group actions, isometries, and symmetries

BL 5.4

 

Mar 9

Group actions, isometries, and symmetries

BL 5.4

 

 

FINAL EXAMINATION

 

FINAL (Solutions)

 

Web page maintained by K. Morgansen (morgansn@u.washington.edu)
Last updated: 22 March 2006