# 家用機器人之影像追蹤控制系統研發（13） by yurtgc548

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```									Progress on the Control of
Nonholonomic Systems

Dr. Ti-Chung Lee
Department of Electrical Engineering
Ming Hsin University of
Science and Technology
Outline
¨ Introduction to nonholonomic
Systems (3-7)
¨ Research problems in the control
of nonholonomic systems (8-10)
¨ An illustrated example: a general
tracking  problem for mobile robots (11-
16)
¨  Simulations and experimental results
(17-22)
¨  Conclusion (23)
Basic Concept
¨ Holonomic Systems : having
integrable constrained equations

¨ Nonholonomic Systems : having non-
integrable constrained equations

¨ Underactuated Systems : number of
control variables less than number
of state variables
Examples
¨ Mobile Robots

Fig. 1. A two-wheeled mobile robot.
Examples (cont’d)
¨ Mathematical Model and Constrained Equations

Non-integrable equations

¨ Underactuated Structure:
Number of control variables (=2)
<    Number of state variables    (=3)
Examples (cont’d)
¨ Ships and dynamic model
Limitation of Continuous Static
feedback
¨ Necessary Condition of Brackett

¨ Mobile Robots, underacuated ships,   spacecraft
and induction motors do not satisfy the
necessary condition of Brackett!
¨ An example – Mobile Robots:
Overcoming the Limitation
¨ Method 1: Time-varying smooth feedback : with a

slow convergence rate

¨ Method 2: Discontinuous feedback : with a

singular surface

¨ Method 3: Homogeneous feedback: Non-robustness

¨ Other Methods: Hybrid feedback, switch control,
practical stabilization and Motion planning
approaches
Research Problems
¨ Fast tracking and regulation problems.

¨ Robustness.

¨ Sensor based control: controllers design using
the image sensor and GPS, e.t.c..
An Illustrated Example: a General
Tracking  Problem for Mobile Robots
¨ Problems Statement:
Error Model:
¨ Coordinate transformation and new input:

¨ Error model:

¨ Now, tracking problem is transformed into
stability problem, i.e.,
Tracking Controllers Design
¨ A smooth function:

¨ Lyapunov function:
¨ Controllers:

¨ Energy dissipation:
Assumptions and Theorem
¨ Assumptions on tracking trajectories:

¨ Theorem 1: Consider the system (2) with controllers chosen
as (3). Then, the origin is uniformly globally asymptotically
stable and locally exponentially stable under condition (C2).
In addition to                               , the same result holds
under the weaker condition (C1).
Fast Parking Problem
¨ Assumption on tracking trajectories:

¨ Modified tracking trajectories:

where          is a continuous periodic function with
and
¨ Verifying (C2):
Fast Parking Control : From
tracking to parking control
¨ Given constants and function:

¨ The PTCP is solvable by the following
controllers:

¨         can be described in the following
expression:
SIMULATIONS AND
EXPERIMENTAL RESULTS
¨  Trajectory for parallel-parking :

¨ Trajectory for back-into-garage :
A Comparison for Simulations
and Experimental Results

Experimental       Simulation
A Comparison for Different Controllers

Proposed Saturation Feedback Controller
Saturation Feedback Controller
A Comparison for Different Choices
of tuning functions

:  representing

:  representing
Experimental Video :   parallel-parking
Experimental Video : back-into-garage
Conclusion
¨     Nonholonomic systems are very interesting and deserve more
deeper study.
¨    They are also important due to many practical applications for
examples,  in  the  motion  control  of  home  robots  and    the
control of underactuated  mechanic systems.
¨    In present literature, it can be observed that a tool developed
in one system can be applied to another system usually.
¨   Thus,  it  may  be  asked  if  there  exists  a  unified  approach  or
guide-line to treat a class of nonholonomic systems.
¨   To  answer  this  question,  it  may  start  from  cases-study  and
observe  some  common  properties  for  the  investigated
nonholonomic systems. Audience are refered to the following
paper :
Lee, T. C.  Exponential stabilization for nonlinear systems with applications to
nonholonomic systems. Automatica,  Vol. 39, pp. 1045-1051, June, 2003.

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