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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