本論文之目的在於分析雙足機器人的平衡。本論文針對馬達扭力需求大小、ZMP值、摩擦係數的需求這三項為分析特徵以進行分析雙足機器人的平衡。而研究需要許多不同的運動軌跡，所以本論文提供了一些快速建構軌跡的方法。這些軌跡的資料是一組限制條件，由不同的限制條件建構不同的軌跡。

在馬達的驅動需要計算多連桿的動力方程式，本論文利用Newton-Euler method加上自由體圖的概念，建構出一個多連桿平面運動的動力學公式，利用這些公式將能夠快速建立動力學計算的矩陣資料。而ZMP 的計算需要各桿件質量中心的運動學資訊，本論文利用機器人學的理論，進而擷取質量中心的位置、加速度值，這兩項元素將決定ZMP 的位置。ZMP 將決定機器人是否穩定。最後透過動力學的公式，我們發現腳與地面的反作用力可藉由運動學的資訊取得，由於這些運動學的資訊計算本來就是必要的，所以將可以花費很少的時間計算出摩擦係數的需求量。藉以分析雙足機器人滑行的可能性。

The purpose of this thesis is to analyze the balance of a biped robot. We use three features including the torque, the ZMP (Zero-Moment-Point), and the coefficient of friction to analyze the balance of a biped robot. Many different trajectories are needed, therefore we need some methods for rapid construction of trajectories. In this thesis, trajectories are built from given constraints. We use different sets of constraints to construct different trajectories.

In order to ensure the feasibility of driven motors in a multi-link, we need to calculate the dynamics equations. We use a combination of the Newton-Euler method and the free body diagram to construct the general formula of a multi-link dynamics on the plane. The use of these formulas allows us to rapidly construct matrix data of the dynamics. The concept of dynamic equilibrium is usually based on judgement of the ZMP, which requires the computation of the center of mass kinematics for each rod. This thesis uses robot theory to compute the center of mass and the acceleration from which we derive the ZMP and in turn determine whether the robot is balanced or not. Finally, through the dynamics equations, we derive the action-reaction force between the foot and the ground. Since the information needed to obtain such reaction force is available, rapid computation of the coefficient of friction can be achieved, which enables us to analyze the possibility of any sliding of the biped robot.

References
[1] G. B. Christine Chevallereau, Cybernetics and G. A. Y. Aoustin, `` Bipedal Robots : Modeling, Design and Walking Synthesis,' Wiley-ISTE, JAN 2009.
[2] M. M. S. Jadran Lenarcic, ``Advances in robot kinematics : motion in man and machine,' Springer Science + Business Media B.V., 2010.
[3] O. Kurt and K. Erbatur, ``Biped robot reference generation with natural ZMP trajectories,' IEEE International on Workshop in Advanced Motion Control, pp. 403 --410, 2006.
[4] J. H. Park and H. Chung, ``ZMP compensation by online trajectory generation for biped robots,' IEEE International Conference on IEEE SMC'99 Conference Proceedings in Systems, Man, and Cybernetics, vol. 4, pp. 960 --965, 1999.
[5] T. Sugihara, Y. Nakamura, and H. Inoue, ``Real-time humanoid motion generation through ZMP manipulation based on inverted pendulum control,' ICRA IEEE International Conference on Robotics and Automation, vol. 2, pp. 1404 -- 1409, 2002.
[6] K. Erbatur and O. Kurt, ``Natural ZMP trajectories for biped robot reference generation,' IEEE Transactions on Industrial Electronics, vol. 56, pp. 835 --845, MAR 2009.
[7] S. Lim, S. Oh, and K. Kim, ``Balance control for biped walking robots using only Zero-Moment-Point position signal,' Electronics Letters, vol. 48, pp. 19 --20, MAY 2012.
[8] K. Akbari Hamed, N. Sadati, W. Gruver, and G. Dumont, ``Exponential stabilisation of periodic orbits for running of a three-dimensional monopedal robot,' Control Theory Applications, IET, vol. 5, pp. 1304 --1320, 2011.
[9] P. R. Vundavilli and D. K. Pratihar, ``Balanced gait generations of a two-legged robot on sloping surface,' Sadhana-Academy Proceedings in Engineering Sciences, vol. 36, pp. 525--550, AUG 2011.
[10] R. Gregg, Y. Dhaher, A. Degani, and K. Lynch, ``On the mechanics of functional asymmetry in bipedal walking,' IEEE Transactions on Biomedical Engineering, vol. 59, pp. 1310 --1318, MAY 2012.
[11] J.-Y. Kim and Y.-S. Kim, ``Development of motion capture system using dual video cameras for the gait design of a biped robot,' International Journal of Humanoid Robotics, vol. 8, pp. 275--299, JUN 2011.
[12] T. Sato, S. Sakaino, E. Ohashi, and K. Ohnishi, ``Walking trajectory planning on stairs using virtual slope for biped robots,' IEEE Transactions on Industrial Electronics, vol. 58, pp. 1385 --1396, APR 2011.
[13] Y. Hu, G. Yan, and Z. Lin, ``Feedback control of planar biped robot with regulable step length and walking speed,' IEEE Transactions on Robotics, vol. 27, pp. 162 --169, FEB 2011.
[14] M. T. Trojnacki and T. Zielinska, ``Motion synthesis and force distribution analysis for a biped robot,' Acta of Bioengineering and Biomechanics, vol. 13, no. 2, pp. 45--56, 2011.
[15] J.-Y. Kim and Y.-S. Kim, ``Human-like gait generation for biped android robot using motion capture and ZMP measurement system,' International Journal of Humanoid Robotics, vol. 7, pp. 511--534, DEC 2010.
[16] R. G. K.S. Fu and C. Lee, eds., ``Robotics : Control, Sensing, Vision, and Intelligence,' McGraw-Hill, 1987.
[17] S. Shah, S. Saha, and J. Dutt, ``Modular framework for dynamic modeling and analyses of legged robots,' Mechanism and Machine Theory, vol. 49, pp. 234 -- 255, 2012.
[18] P. Sardain and G. Bessonnet, ``Forces acting on a biped robot. center of pressure-zero moment point,' Part A: Systems and Humans, IEEE Transactions on Systems, Man and Cybernetics, vol. 34, pp. 630 -- 637, SEP 2004.
[19] D. E. Orin and W. W. Schrader, ``Efficient computation of the jacobian for robot manipulators,' The international of robotics research, vol. 3, pp. 66--75, DEC 1984.