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 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

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.