本文首先係藉由自行車的系統動態分析，進而設計適當之控制器，以實現自行車在無人操控的狀態下能平衡行駛。在自行車的動態模型中，可以發現系統的參數會受到車速的影響，若以一般的方式設計控制器，很難能夠應付系統的參數變異。因此，本研究將模型分解為兩個有時變係數的線性子系統的凸集，所提出的控制器也包括兩個線性全狀態回授控制器的凸集。經證明若全狀態回授控制器滿足一套線性矩陣不等式，便能夠保證在所設的行駛速度範圍內，自行車能夠保持平衡。
確保了平衡前進，接著將探討如何實現自行車的軌跡追蹤。首先，當自行車行駛時，龍頭的角度輸出以及後輪無刷直流馬達的速度均為已知。因此能透過這兩資訊可間接地知道當下自行車之位置以及方向角速度。有了位置以及方向角的資訊，便可與指定之直線軌跡做比較，得知當下自行車質心與直線軌跡之距離，以及方向角與直線軌跡的角度差。當兩誤差值收斂至零，便能確保自行車在指定的路徑上行駛。

關鍵字：
腳踏車動態、參數變異、平衡控制、軌跡追蹤

By performing dynamic analysis of the bicycle, a novel controller is designed to balance and steer the bicycle autonomously along the road surface. According to the dynamic model of the bicycle, the system parameters depend on the bicycle speed. Therefore, to design a controller which can deal with the speed-dependent characteristics, the dynamic model is firstly decomposed into a convex combination of two linear subsystems with time-varying coefficients. The proposed controller is then expressed as a convex combination of two linear, full-state feedback controllers. It can be proved that if the two full-state feedback controllers satisfy a set of linear matrix inequalities, then it can be ensured that the bicycle can maintain balance under a range of travel speed.
After considering the balancing problem, this thesis will investigate how to control the bicycle to achieve the trajectory tracking. By using steering angle and the measured bicycle speed, one can compute the bicycle's position and the orientation change which can be subsequently used to calculate, for feedback control purposes, the distance and orientation deviation from the COG of bicycle to the given trajectory. The balancing, steering as well as the tracking performance of the autonomous bicycle are all verified through simulations and experiments.

Keyword: Bicycle dynamic, parameter variation, balancing control, trajectory tracking

[1] “雲馬 C1, ” 雲造科技股份有限公司, [Online]. Available: https://zhixingche.com/product/c1.

[2] C. F. Huang, Bang-Hao Dai and T. J. Yeh, “Determination of motor torque for power-assist electric bicycles using observer-based sensor fusion” J. Dyn. Sys., Meas., Control. 2018; 140(7):071019-071019-11.

[3] F. J. W. Wipple. “The stability of motion of a bicycle”. In: Quarterly Journal of Pure and Applied Mathematics 30 (1899), pp. 312-348.

[4] K. J. Åström, R. E. Klein, and K. Lennartsson. “Bicycle dynamics and control: adapted bicycles for education and research”. In: IEEE Control Systems 25.4 (Aug. 2005), pp. 26-47.

[5] C. F. Huang, Y. C. Tung and T. J. Yeh, "Balancing control of a robot bicycle with uncertain center of gravity," 2017 IEEE International Conference on Robotics and Automation (ICRA), Singapore, 2017, pp. 5858-5863.

[6] J. Lowell, H. D. McKell. “The stability of bicycles”. In: America Journal of Physics 50.12 (1982), pp. 1106-1112.

[7] Brad Lignoski. “Bicycle stability, is the steering angle proportional to the lean?” In: Physics Department, The College of Wooster, Wooster, Ohio 44691 (2002)

[8] J. Yi, Y. Zhang, and D. Song. “Autonomous motorcycles for agile maneuvers, part I: Dynamic modeling”. In: Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on. Dec. 2009, pp.4613-618.

[9] J. H. Ginsberg. Advanced Engineering Dynamics. Cambridge University Press, 1995.

[10] Arend L. Schwab, Jaap P. Meijaard, Jim M. Papadopoulos, “Benchmark result on the linearized equations of motion of an uncontrolled bicycle”. In: Journal of Mechanical Science and Technology 19.1 (2005), pp. 292-304.

[11] Arend L. Schwab, Jaap P. Meijaard, Jim M. Papadopoulos, “A multibody dynamics benchmark on the equations of motion of an uncontrolled bicycle” in Proc.5th EUROMECH Nonlinear Dynamics Conf. The Netherlands: Eindhoven Univ. Technol., Aug. 7–12, 2005, pp. 511–521.

[12] J. Meijaard, J. Papadopoulos, A. Ruina, A. Schwab, “Linearized dynamics equations for the balance and steer of a bicycle: A benchmark and review,” Proc. R. Soc., vol. A463, pp. 1955–1982, 2007.

[13] D. Andreo, M. Larsson, V. Cerone, D. Legruto, “Stabilization of a riderless bicycle: a linear parameter varying approach”. IEEE Control Syst. Mag., vol. 30, no. 5, pp. 23-32, 2010.

[14] S. H. Zak, Systems and control. Oxford University Press New York, 2003, vol. 174.