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CMPE-633C / PHYS-401: Topics in Robotics and Control
Spring 2012: Geometric Mechanics and Control


Dr Abubakr Muhammad, Assistant Professor of Electrical Engineering

Email: abubakr [at]

Office: Room 9-311A, 3rd Floor, SSE Bldg

Course Details

Year: 2011-12

Semester: Spring

Category: Grad

Credits: 3

Elective course for electrical engineering, computer engineering and physics majors

Course Website:

Course Description

The aim of this course is to provide a unified treatment of nonlinear control theory and mechanical systems subject to motion constraints. Both Lagrangian and Hamiltonian formulations will be developed with an emphasis on solving control problems for nonholonomic and under-actuated systems. Important topics covered in this course include the center manifold theorem for stability, Frobenius and Chow’s theorems, Poisson geometry of nonholonomic systems, symplectic geometry of Hamiltonian flows etc. The students will also learn basic concepts of differential forms, Lie algebras, distributions and Riemannian geometry to formulate these results. Applications of this formulation span controller design in robotics, quantum information, aerospace and other systems that do not yield to non-geometric methods in control.


  • Use of geometric frameworks to understand nonholonomic and under-actuated systems
  • Lagrangian and Hamiltonian formulation of analytical mechanics
  • Differential geometry methods in control problems
  • Applications in robotics, systems theory and mathematical physics

Learning Outcomes

  • Link control theory with a geometric view-point of classical mechanics.
  • Identify holonomic & nonholonomic constraints in physical systems.
  • Understand control limitations due to under-actuation and motion constraints.
  • Appreciate the value of abstract differential geometry methods in solving real-world problems.


  • CMPE-633b (Robot Dynamics & Control) OR
  • PHYS-310 (Classical Mechanics) OR
  • MATH-361 (Dynamical systems) OR
  • By Permission of Instructor

Self check. Before start of class you should be familiar with configuration spaces, Euler-Lagrange equations, phase-plane analysis and fluent with use of vector calculus, linear algebra and differential equations.

Who should take it? EE/CMPE majors seriously interested in robotics and control theory; SSE students interested in pursuing mathematical physics.

Text book

Main text book.

  • Nonholonomic Mechanics and Control by Anthony Bloch, P. Crouch, J. Baillieul, J. Marsden. Interdisciplinary Applied Mathematics Springer-Verlag NY, 2003.

Reference texts.

  • An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised, Volume 120, Second Edition (Pure and Applied Mathematics) by William M. Boothby.
  • Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems (Texts in Applied Mathematics) by Jerrold E. Marsden, Tudor S. Ratiu.
  • Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics) by V.I. Arnold [ geometric mechanics]
  • Riemannian Geometry by Manfredo P. do Carmo , Francis Flaherty.
  • Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems by Francesco Bullo, Andrew D. Lewis.

Grading Scheme

  • Home-works (weekly): 25%
  • Class Participation: 10%
  • Paper Project: 40%
  • Midterm Examination: 25%


Week 1. Jan 23 Lec 1. Configuration spaces; topology and geometry of important configuration spaces; visualization methods for S^n, RP^n, SO(n), T^n; Bloch Ch1;
Week 2. Jan 30 No classes
Week 3. Feb 6 Lec 2. Largrangian mechnaics; variational principles; Hamilton's principle; Euler-Lagrange Equations; Legendre transformation; Hamiltonian mechanics;

Lec 3. Holonomic and nonholonomic constraints; Lagrange D'Alembert's principle; external forces; dynamical and variational nonholonomic equations of motion; optimal control problem; vertical rolling disk example;

Bloch Ch1;
Week 4. Feb 13
Week 5. Feb 20

Lec 4.

Lec 5.

Week 6. Feb 27

Lec 6.

Lec 7.

Week 7. Mar 5 Lec 8.

Lec 9.

Week 8. Mar 12

Lec 10.

Lec 11.

Week 9. Mar 19

Lec 15.

Lec 16.

Week 10. Mar 26

Lec 18.

Week 11. April 2

Lec 19.

Lec 20.

Week 12. April 9

Lec 21.

Lec 22.

Week 13. April 16

Lec 23.

Lec 24.

Week 14. April 23 Lec 25.
Week 15. April 30

Lec 26.

Lec 27.

Week 16. May 7 Lec 28.
Week 17. May 14

Project Ideas

  • Parallel-parking a car: From Chow’s theorem to sub-Riemannian geometries.
  • Quantum feedback control: Steering trajectories over SU(2^n).
  • Discrete geometric mechanics for computer simulation.
  • Berry’s phase in Foucault’s pendulum, cyclic adiabatic processes and other applications.
  • Rigid body attitude stabilization: Time-optimal control on SO(n).
  • Robotic grasping: Contact kinematics in rolling bodies.
  • Cars, trailers and roller racers: Motion planning algorithms with nonholonomic constraints.
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