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Progress on the Control of Nonholonomic Systems

此簡報會與參加人員進行相關討論，所以你將需要加入執行項目來輔助討論。 請充份運用 PowerPoint 整合記錄這些執行項目於你簡報進行中，方法如下： 於投影片放映狀態按下滑鼠右鍵 選取 〔 會議記錄簿 〕 選取 〔 執行項目 〕 將意見記錄於其中 按下 〔 確定 〕 以結束此對話方塊 這個功能將隨著意見的加入，而自動於投影片下方產生一個執行項目. Progress on the Control of Nonholonomic Systems. Dr. Ti-Chung Lee

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Progress on the Control of Nonholonomic Systems

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  1. 此簡報會與參加人員進行相關討論，所以你將需要加入執行項目來輔助討論。此簡報會與參加人員進行相關討論，所以你將需要加入執行項目來輔助討論。 • 請充份運用 PowerPoint 整合記錄這些執行項目於你簡報進行中，方法如下： • 於投影片放映狀態按下滑鼠右鍵 • 選取〔會議記錄簿〕 • 選取〔執行項目〕 • 將意見記錄於其中 • 按下〔確定〕以結束此對話方塊 • 這個功能將隨著意見的加入，而自動於投影片下方產生一個執行項目 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 Experimental Simulation

  18. A Comparison for Different Controllers ProposedSaturation 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. 1045-1051, June, 2003.

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