非完整約束輪型機器人之建模與控制

Modeling and Control for Wheeled Mobile Robots with Nonholonomic Constraints

蔡樸生(Pu-Sheng Tsai)

指導教授 ： 張帆人

共同指導教授 ： 王立昇(Li-Sheng Wang)

國立臺灣大學/電機資訊學院/電機工程學研究所/博士(2006年)

https://doi.org/10.6342/NTU.2006.02062

摘要

本論文針對非完整約束系統提出一套階層化設計的架構。文中探討了三個實用的輪型機器人系統包括︰三輪式自走車、車型機器人以及具有方向盤的四輪車輛系統，解決平面上軌跡追蹤的問題。對於一個具有非完整約束的多剛体系統而言，如何建立一組適當的系統模型，幫助控制器的設計以及發揮控制器的效能，為本論文的研究動機。基於 Appell 方程式所蘊涵的解耦特性，整個系統可以分解成動力模型與運動模型兩部份，針對這兩個模型進行控制器的設計可減化控制器結構，在實際應用中更具彈性且易於實現。本文首先建立矩陣形式的 Appell 簡化模型，透過一個有系統以及結構化的程序來處理非完整約束系統模型化的問題。在建模過程中，適當地選擇一組所謂的特權變數使得動力模型與運動模型互相解耦，整個系統區分為路徑規劃、中間運動層以及低階動力層個別設計。在路徑規劃的部份，有鑑於B-樣條函數所建立的曲線具有快速收斂、局部學習與分段調整之特點。本文推導分段型B-樣條曲線來取代傳統的遞迴表示式，有效地整合約束最佳化之演算法，建構一條連接起始點與終點之最佳化的平滑軌跡，除了滿足轉彎時車輛的曲率限制以及車輪滾動時之運動約束外並可成功地避開障礙物。在中間運動層方面，審慎選擇特權變數的結果，可產生特殊結構的運動模型。針對上述不同型態之輪型機器人，本文提出三種不同的運動模型控制器，分別為模糊推論引擎、鏈狀模型之後退式策略以及 Lyapunov直接法，產生特權變數的補償信號回饋至低階層作為為動力模型之參考信號。低階動力層乃針對 Ap-pell 簡化模型進行控制器之設計，動力模型的推導是屬於解析力學的範疇，可以真實地描述輸入力矩與系統變數的關係，並且滿足反對稱的本質特性。本文採用適應性滑動模式控制器來保證動力模型之追蹤性能，所有系統變數均可收斂至追蹤信號。此外，運動模型與動力模型之間存在交互作用誤差，加上外界的干擾將直接影響追蹤性性能。本文在控制法則中引入強健控制補償項，企圖涵蓋不確定模型與雜訊干擾的成分，有效地將交互作用與外來干擾衰減至容忍的範圍之內。透過適應性與強健性控制理論的緊密結合，動力模型的控制應可獲得完美的效果。本論文探索了 Appell 簡化模型的解耦特性，成功地發展一套階層式控制架構。經由約束最佳化的演算法建構一條B-樣條參考曲線，結合運動模型的補償信號以及動力模型控制器的設計，提供輪型機器人軌跡追蹤的有效方法。藉由每一章節的模擬結果證實了本文所構思的方法極具可行性，其追蹤效能與預期相吻合。

關鍵字

非完整約束系統 ； 階層式追蹤控制 ； 路徑規劃 ； 動力模型 ； 運動模型 ； B樣條曲線 ； 特權變數 ； 鏈狀模型 ； 後退式控制器 ； 滑動模式控制器

並列摘要

Based on the decoupling feature arises from the Appell equations, this dissertation fo-cuses on the design of the following three hierarchical control architectures: fuzzy com-pensator for tri-wheeled mobile robots, backstepping controller for car-like mobile robots, and Lyapunov synthesis method for four-wheeled vehicle with steering wheel system. The problem to be attacked here is the tracking of a desired trajectory for three types of mobile robots moving on a horizontal plane. The reduced Appell equation in matrix form is first established to deal with the modeling problem of nonholonomic systems by a systematic and structural procedure. By appropriately choosing a set of privileged variable, the re-duced Appell equations are decoupled from the kinematics such that the system design may be separated into three levels: motion planning, kinematic, and dynamic. In motion planning level, the proposed algorithm is an integrated application of the constrained opti-mization method and B-spline function to generate a feasible path that the kinematic con-straints and curvature restrictions are satisfied. In addition, the matrix form of divided B-spline curve is derived, which makes it possible to realize the constrained optimization programming. Three types of the compensator, fuzzy inference engine, backstepping in chained form and Lyapunov synthesis method, in the kinematic level are respectively de-signed to update the desired trajectory computed in the motion planning level. The updated privileged variables are then fed into the dynamic controller in the bottom level. To deal with the uncertainties in the system parameters, an adaptive sliding mode controller is then adopted to track the new reference values of privileged variables assured by the skew-symmetric property in the dynamical level, which subsequently drives all system variables to the desired values by natural mechanism. The decoupling paves the way for the structure of hierarchical control which simultaneously takes kinematic equation and dynamic equation into consideration indeed gives rise to an effective methodology for the trajectory tracking problem. Furthermore, the effects of external disturbances and interac-tions between kinematics and dynamics will appear gradually while the initial condition is significantly away from the desired configuration. Two robust control compensations, switching function and saturation function, are investigated to guarantee the performance of developed hierarchical tracking control systems. As to what we expect, simulation re-sults demonstrate the success of the proposed systematic design of the hierarchical tracking control scheme.

並列關鍵字

nonholonomic constrained systems ； hierarchical tracking control ； path planning ； kinematics ； dynamics ； B-spline curve ； privileged variable ； chained form system ； backstepping controller ； reduced Appell equation ； sliding mode controller

參考文獻

[1].Ackerman, J., 1992, “Robust Yaw Damping of Cars with Front and Rear Wheel Steering,” Proc. 31th IEEE Conf. on Decision and Control, pp. 2586-2590.

[2].Anthony, L., 1993, Modern Inertial Technology: Navigation, Guidance, and Control, New York : Springer-Verlag.

[3].Astolfi, A., 1995, “Exponential Stabilization of a Car-Like Vehicle,” IEEE Int. Conf. on Ro-botics and Automation, pp. 1391-1396.

[4].Bloch, A. and Drakunov, S., 1996, “Stabilization and Tracking in the Nonholonomic Integra-tor via Sliding Modes,” Systems & Control Letters, vol. 29, pp. 91-99.

[6].Bushnell, L. G., Tilury, D. M. and Sastry, S. S., 1995, “Steering Three-Input Nonholonomic Systems: The Fire Track Example,” Int. J. Robot. Res., vol. 14, no. 4, pp. 366-381.