本論文之研究在於完成基於三維空間動態系統的倒單擺系統的控制器分析與設計，進而建構一個由多維倒單擺系統集合而成的實驗環境以進行平衡與彈跳控制研究。在彈跳方面，利用角動量守恆及能量守恆進行方塊彈跳控制。而在平衡方面，透過六軸感測器(MPU6050)偵測系統的加速度與角速度後，再以姿態演算法計算出方塊體當前的角度與角速度並結合雙環平行比例-積分-微分控制演算法以達到平衡控制。
本論文目前建構了多維方塊機器人的系統設計整合方案，包含硬體機構設計與韌體架構設計還有軟體開發等等…. 其中也包含設計實驗量測系統模型中的重要參數。在硬體方面煞車系統使用3D列印的PLA材質做出煞車機構外，慣性輪是以不銹鋼所製造，而帶動慣性輪為無刷馬達，並搭配馬達內建的光電編碼器進行轉速偵測。在韌體設計方面，以STM32嵌入式控制板及嵌套式定時中斷系統為基礎，使用IAR為編譯器建構控制器及資料收集，而在軟體方面是使用Microsoft Visual Studio Professional 2013平台進行開發及程式語言C語言進行撰寫，目前可實現一維平面彈跳控制與平衡控制還有方塊單邊彈跳控制與平衡控制。

The thesis aims at investigating the challenging problem of self-balancing and self-jumping control of a cubical robot. By using the structure that the thesis builds as the research object, the dynamic model and state-space representation can be derived with the conservation of angular momentum theory, the torque equilibrium theory on the coefficient related to the cubical robot. In self-balancing, the thesis uses the attitude and heading reference system algorithm to compute the system’s Euler Angle and angular velocity. The controller designed for the cubical robot’s balance is based on the PID control algorithm, and with the brushless motor generating the reaction torque, the cubical robot can readjust to the balance position after encountering an external force.
In self-jumping, the thesis uses the conservation of energy theory to compute the needed velocity of the reaction wheel and designs a braking system that can effectively stop the rotation of the reaction wheel. The experimental results show that the cubical robot can jump and balance by itself.

[1] HOFFMANN, Christopher J.; OZRELIC, Anthony J. “Electric-powered self-balancing unicycle.” U.S. Patent No 9, 2015.
[2] “2018 Segway-Ninebot”, https://www.kocpc.com.tw/archives/297802
[3] “MOTOROiD”, https://www.supermoto8.com/articles/2143
[4] MIRZAEI, Faraz M.; ROUMELIOTIS, Stergios I. “A Kalman filter-based algorithm for IMU-camera calibration: Observability analysis and performance evaluation.” IEEE transactions on robotics, pp. 1143-1156, 2008.
[5] MohanarajahGajamohan, Michael Muehlebach, Tobias Widmer, and Raffaello D’Andrea, “The Cubli: A Reaction Wheel Based 3D Inverted Pendulum”, IEEE Control Conference (ECC), Zurich, Switzerland, July 17-19, 2013.
[6] LI, Weiwei; TODOROV, Emanuel. “Iterative linear quadratic regulator design for nonlinear biological movement systems.” In: ICINCO (1), pp. 222-229, 2004.
[7] CHEN, Zhigang, et al. “Dynamic modeling of a self-balancing cubical robot balancing on its edge.” In: 2017 2nd International Conference on Robotics and Automation Engineering (ICRAE). IEEE, pp. 11-15, 2017.
[8] Che-Yu Lin “Realization of one-dimen sional Cube Robot Capable of Bounce and Self-balancing based on Linear Quadratic Regulator Control”, Cheng Kung University Master thesis, July 15, 2018.
[9] ABDULGHANY, A. R. “Generalization of parallel axis theorem for rotational inertia.” American Journal of Physics, pp. 791-795, 2017.
[10] SHU, Guanghui; ZHAN, Qiang; CAI, Yao. “Motion control of spherical robot based on conservation of angular momentum.” In: 2009 International Conference on Mechatronics and Automation. IEEE, pp. 599-604, 2009.
[11] ARONS, Arnold B.; REDISH, Edward F. “Teaching introductory physics.” New York: Wiley, 1997.
[12] LAWLOR, Gary R. “A L'hospital's rule for multivariable functions.” arXiv preprint, pp.1209-1363, 2012.
[13] FOY, Wade H. “Position-location solutions by Taylor-series estimation.” IEEE Transactions on Aerospace and Electronic Systems, pp. 187-194, 1976.
[14] SHIELDS, Robert; PEARSON, J. “Structural controllability of multi-input linear systems.” IEEE Transactions on Automatic control, pp. 203-212, 1976.
[15] LEE, Kok-Meng; DALEY, Wayne; MCKLIN, Tom. “An interactive learning tool for dynamic systems and control.” In: Proc. of International Mechanical Engineering Congress & Exposition, 1998.
[16] GENTLE, James E. “Numerical linear algebra for applications in statistics.” Springer Science & Business Media, 2012.
[17] HERMANN, Robert; KRENER, Arthur. “Nonlinear controllability and observability.” IEEE Transactions on automatic control, pp. 728-740, 1977.
[18] VLASE, Sorin; MARIN, Marin; ÖCHSNER, Andreas. “Eigenvalue and Eigenvector Problems in Applied Mechanics.” Springer International Publishing, 2019.
[19] SILVERMAN, Leonard M.; MEADOWS, H. E. “Controllability and observability in time-variable linear systems.” SIAM Journal on Control, 1967.
[20] YAO-YONG, D.U.A.N. “The History of Reduction to Absurdity.” College Mathematics, 2006.
[21] STEWART, Doug. “A platform with six degrees of freedom.” Proceedings of the institution of mechanical engineers, pp. 371-386, 1965.
[22] SHOEMAKE, Ken. “Animating rotation with quaternion curves.” In: Proceedings of the 12th annual conference on Computer graphics and interactive techniques, pp. 245-254, 1985.
[23] SHEPPERD, Stanley W. “Quaternion from rotation matrix.” Journal of Guidance and Control, pp. 223-224, 1978.
[24] DORMAND, John R.; PRINCE, Peter J. “A family of embedded Runge-Kutta formulae.” Journal of computational and applied mathematics, pp.19-26, 1980.
[25] ZUPAN, Eva; SAJE, Miran; ZUPAN, Dejan. “Quaternion-based dynamics of geometrically nonlinear spatial beams using the Runge–Kutta method.” Finite Elements in Analysis and Design, pp. 48-60, 2012.
[26] BJERKE, E. R. I. K.; PEHRSSON, BJÖRN. “Development of a Nonlinear Mechatronic Cube.” Department of Signal and Systems, Division of Automatic control, Automation and Mechatronics, Chalmers University of Technology, 2016.
[27] BITTANTI, Sergio; LAUB, Alan J.; WILLEMS, Jan C. (ed.). “The Riccati Equation.” Springer Science & Business Media, 2012.
[28] TAGUCHI, Hidefumi; ARAKI, Mituhiko. “Two-degree-of-freedom PID controllers—their functions and optimal tuning.” IFAC Proceedings Volumes, pp. 91-96, 2000.
[29] SONG, Eugene Y.; LEE, Kang B. “Sensor Network based on IEEE 1451.0 and IEEE p1451. 2-RS232.” In: 2008 IEEE Instrumentation and Measurement Technology Conference. IEEE, pp. 1728-1733, 2008.
[30] KALLMAN, Kurt Albert; TURCOTTE, Randy Lee. “Crosspole interference canceling receiver for signals with unrelated baud rates.” U.S. Patent No 5,838,740, 1998.
[31] BUCZ, Štefan; KOZÁKOVÁ, Alena. “Advanced methods of PID controller tuning for specified performance.” PID Control for Industrial Processes, pp. 73-119, 2018.
[32] SILVA, Guillermo J.; DATTA, Aniruddha; BHATTACHARYYA, Shankar P. “PID controllers for time-delay systems.” Springer Science & Business Media, 2007.
[33] HOLTZ, Joachim. “Pulsewidth modulation-a survey.” IEEE transactions on Industrial Electronics, pp.410-420, 1992.
[34] O'DWYER, Aidan. “Handbook of PI and PID controller tuning rules.” Imperial college press, 2009.
[35] LATTIMER, James M.; SCHUTZ, Bernard F. “Constraining the equation of state with moment of inertia measurements.” The Astrophysical Journal, pp. 629- 979, 2005.
[36] “Gate count”, https://en.wikipedia.org/wiki/Gate_count
[37] LEENS, Frédéric. “An introduction to I^2C and SPI protocols.” IEEE Instrumentation & Measurement Magazine, pp. 8-13, 2009.
[38] Venema, Steven C. “A kalman filter calibration method for analog quadrature position encoders.” MS thesis. University of Washington, 1994.
[39] “分電板”,
https://shopee.tw/%E7%A2%B3%E7%BA%96%E6%A9%9F%E6%9E%B6-DIY-i.5198380.33728350
[40] “BEC與UBEC”,
https://codertw.com/%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80/633086/
[41] FEDOROV, D. S., et al. “Using of measuring system MPU6050 for the determination of the angular velocities and linear accelerations.” Automatics & Software Enginery, 2015.
[42] “EMAX ES08MA II 12g Mini PWM servo”, https://emaxmodel.com/es08ma-ii.html
[43] ZHANG, Yongchang; YANG, Haitao. “Model predictive torque control of induction motor drives with optimal duty cycle control.” IEEE Transactions on Power Electronics, pp. 6593-6603, 2014.
[44] Mubeen, Muhammad. “Brushless DC motor primer.” Motion Tech Trends, 2008.
[45] CAO, Bin-qian, et al. “Design of the Stepper Motor Subdivision Control Based on the STM32” Science Technology and Engineering 23, 2013.
[46] Jaouadi, Aymen, et al. “Adaptive STM32F4 microcontrollers.” Proc. 3rd Int. Conf. Smart Grids and Green IT Syst, 2014.
[47] KARABOGA, Nurhan; CETINKAYA, Bahadir. “Design of digital FIR filters using differential evolution algorithm.” Circuits, Systems and Signal Processing, pp. 649-660, 2006.
[48] PROAKIS, John G. “Digital signal processing: principles algorithms and applications.” Pearson Education India, 2001.
[49] CHEN, Zhigang; RUAN, Xiaogang; LI, Yuan. “Dynamic modeling of a cubical robot balancing on its corner.” In: MATEC Web of Conferences. EDP Sciences, 2017.