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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