1. Progress on the Control of Nonholonomic Systems
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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

2. 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)

3. 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

4. Examples
Mobile Robots
Fig. 1. A two-wheeled mobile robot.

5. Examples (cont’d) Mathematical Model and Constrained Equations
Non-integrable equations
Underactuated Structure:
Number of control variables (=2)
< Number of state variables (=3)

6. Examples (cont’d)
Ships and dynamic model

7. 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:

8. 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

9. Research Problems Fast tracking and regulation problems. Robustness.
Sensor based control: controllers design using the image sensor and GPS, e.t.c..

10. An Illustrated Example: a General Tracking Problem for Mobile Robots
Problems Statement:

11. Error Model: Coordinate transformation and new input: Error model:
Now, tracking problem is transformed into
stability problem, i.e.,

12. Tracking Controllers Design
A smooth function:
Lyapunov function:
Controllers:
Energy dissipation:

13. 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).

14. Fast Parking Problem Assumption on tracking trajectories:
Modified tracking trajectories:
where is a continuous periodic function with
and
Verifying (C2):

15. 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:

16. SIMULATIONS AND EXPERIMENTAL RESULTS
Trajectory for parallel-parking :
Trajectory for back-into-garage :

17. A Comparison for Simulations and Experimental Results

18. A Comparison for Different Controllers
Proposed Saturation Feedback Controller
Saturation Feedback Controller

19. A Comparison for Different Choices of tuning functions
: representing
: representing

20. Experimental Video : parallel-parking

21. Experimental Video : back-into-garage

22. 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 , June, 2003.