# Topics in Physical Computing:

Control Theory

### Instructors

**Mark Moll**, mmoll AT rice DOT edu, x8-3834, DH 3093

**Andrew Ladd**, aladd AT rice DOT edu, x8-3889, DH 3119

**Time:** Tuesdays & Thursdays, 10:50–12:05

**Place:** Duncan Hall 3110

## Introduction

This is an advanced graduate level class on control theory for motion planning applications. With motion planning the goal is to find a path for a robot that connects a given start and goal pose such that the robot does not collide with any obstacles. For simplicity it is often assumed that a robot can move around in any direction at any speed. In practice this is usually not true. By using results from control theory we can find feasible solutions to the motion planning problem. This course will be of interested to students working in robotics, control theory, and computer graphics.

The first half of the course will consist of lectures by Mark Moll and Andrew Ladd. The students are expected to complete a project and give presentations during the second half of the semester. Final grades are based on the project report and the project presentation. There will be a few voluntary homework assignments.

## Prerequisites

- Advanced calculus
- Linear algebra
- COMP 360, COMP 450, MECH 498, or permission of the instructor
- Real analysis, algebra, topology or differential geometry is useful.

## Text

Lecture notes plus hand-outs.

Some reference material:

- Spivak,
*A Comprehensive Introduction to Differential Geometry,*Vol. 1, Publish or Perish, 1999. - Murray, Li, & Sastry,
*A Mathematical Introduction to Robotic Manipulation,*CRC Press, 1994. - Latombe,
*Robot Motion Planning,*Kluwer, 1991. - Ogata,
*Modern Control Engineering,*Prentice Hall, 2002. - Luenberger,
*Introduction to Dynamic Systems,*Wiley & Sons, 1979. - Isidori,
*Nonlinear Control Systems,*Springer Verlag, 1995. - Messner and Tilbury,
*Control Tutorials for Matlab*, http://www.engin.umich.edu/group/ctm/, Addison-Wesley, 1998. - Laumond (ed.),
*Robot Motion Planning and Control,*http://www.laas.fr/~jpl/book-toc.html, Springer Verlag, 1998. - Q. Zhang, B. Delyon, A new approach to adaptive observer design for MIMO systems, in: 2001 American Control Conference (ACC), Arlington, American Automatic Control Council, pages 1545-1550, June 2001 PostScript

## Schedule

Manifold theory (2 lectures)

- basics of point-set topology
- topology and differentiability of R^n
- multi-variable derivatives (jacobians)
- function classes C^n and C^infty
- representation: charts and atlases
- manifolds and C^n manifolds
- submanifolds and products
- diffeomorphisms
- vectors
- tangent spaces
- tangents of mappings
- vector bundles
- paths and lifting
- configuration spaces R^2, R^3, SE(2), SE(3)

Vector Fields and Lie Algebra (1 lecture)

- vector fields
- integral curves
- the Lie Bracket
- Lie Bracket
- completions

Differential 1-forms (.5 lecture)

- linearity
- forms
- singularities
- vector fields
- holonomic and non-holonomic constraints
- inequality constraints

Examples of Non-Holonomic Systems (.5 lecture)

- car-like robot
- tractor trailer robot
- differential drive robot
- diff. drive robot
- snakeboard

The Laplace Transform and State Space model(1 lecture)

- Laplace in control theory
- block diagrams
- state space
- error dynamics
- basic controllers

Controllability Properties (1 lecture)

- controllability
- via Lie Algebra analysis
- small-time locally controllable
- small-time locally configuration controllable
- reachability under kinematic constraints (OPEN)

Stability and Damping (1 lecture)

- different notions of stability
- Lyapunov’s direct and indirect method
- convergence analysis
- stability analysis
- damping

Observability Properties (1 lecture)

- observability
- state space linearization
- canonical forms
- reduced order observer

Lie Groups (1/2 lecture)

- definition
- exponentiation
- examples
- actions
- Harr measure and sampling (what is uniform?)
- applications

Symmetry and Maneuver Automata (1 1/2 lecture)

- symmetry
- primitives, trims and maneuvers
- maneuver automata
- Frazzoli’s Theorem
- further discussion of reachability
- System identification / adaptive observers

Complete Solutions (1/2 lecture)

- Reed-Sheps curves for car-like robots
- Differential drive robots
- Time rescaling results and applications to second order systems
- GENERALIZATION: Kinematic reduceability

Applications (1/2 lecture)

- steering or tracking
- parking
- trajectory following

## Project Presentation Schedule

Date | Speaker |
---|---|

3/9 | Kostas |

3/11 | Erion |

3/16 | Derek |

3/18 | Amarda |

3/23 | Jan |

3/25 | Evan |

3/30 | Allison |

4/1 | Kostas |

4/6 | Erion |

4/8 | Derek |

4/13 | Amarda |

4/15 | Jan |

4/20 | Evan |

4/22 | Allison |