A Derivation of the Equilibrium Point for a Controller of a Wheeled Inverted Pendulum with Changing Its Center of Gravity

무게중심이 변동되는 차륜형 역진자의 평형점 상태에 관한 연구

  • Lee, Se-Han (Division of Mechanical Engineering, Kyungnam University)
  • Received : 2011.06.27
  • Accepted : 2012.04.22
  • Published : 2012.05.01


An equilibrium point of a WIP (Wheeled Inverted Pendulum) with changing its center of gravity is derived and validated by various numerical simulations. Generally, the WIP has two equilibrium points which are unstable and stable one. The unstable one is interested in this study. To keep the WIP over the unstable equilibrium point, the WIP is consistently being adjusted. A state feedback controller for the WIP needs a control reference for the equilibrium point. The control reference can be obtained by studying an equilibrium point of the WIP based on statics. By using Lagrange method, this study is deriving dynamic equations of the WIP both with and without changing its center of gravity. Various numerical simulations are carried out to show the validation of the equilibrium point.


Supported by : 경남대학교


  1. P. L. Kapitza, in Collected Paper of P. L. Kapitza, edited by D. Ter Haar (Pergamon, London), pp. 174, 1965.
  2. S.-H. Jung, J.-N. Choi, and S.-K., "Design of optimized fuzzy controller by means of HFC-based genetic algorithms for rotary inverted pendulum system," Journal of Korean Institute of Intelligent Systems (in Korean), vol. 18, no. 2, pp. 236-242, 2008.
  3. D. Park, M. Park, D. Chwa, and S.-K. Hong, "An observer design and compensation of the friction on an inverted pendulum using adaptive fuzzy basis functions expansion," Journal of Control, Automation and Systems Engineering (in Korean), vol. 13, no. 4, pp. 335-343.
  4. S.-H. Lee and S.-Y. Rhee, "A mixed $H_{2}/H_{\infty}$ state feedback controller based on LMI scheme for a wheeled inverted pendulum running on the inclined road," Journal of Korean Institute of Intelligent Systems (in Korean), vol. 20, no. 5, pp. 617-623, 2010.
  5. G. Grasser, A. D'Arrigo, S. Colombi, and A. C. Rufer, "JOE a mobile, inverted pendulum," IEEE Transactions on Industrial Electronics, vol. 49, no. 1, pp. 107-114, 2002.
  9. H. U. Ha and J.M. Lee, "A control of mobile inverted pendulum using single accelerometer," Journal of Institute of Control, Robotics and Systems (in Korean), vol. 16, no. 5, pp. 440-445, 2010.
  10. S. H. Kim, J. O. Lee, J. M. Hwang, B. H. Ahn, and J. M. Lee, "Dynamic modeling and performance improvement of a unicycle robot," Journal of Institute of Control, Robotics and Systems (in Korean), vol. 16, no. 11, pp. 1074-1081, 2010.
  11. H. W. Kim and S. Jung, "Experimental studies of controller design for a car-like balancing robot with a variable mass," Journal of Korean Institute of Intelligent Systems (in Korean), vol. 20, no. 4, pp. 469-475, 2010.
  12. S. Matsumoto, S. Kajita, and L. Tani, "Estimation and control of the attitude of a dynamic mobile robot using internal sensors," Journal of Robotic Society of Japan (in Japanese), vol. 8, no. 5, pp. 37-46, 1990.
  13. K. Furuta, H. Kajiwara, and K. Kosuge, "Digital control of a double inverted pendulum on an inclined rail," International Journal of Control, vol. 32, no. 5, pp. 907-924, 1980.

Cited by

  1. A Study on the Visual Servoing of Autonomous Mobile Inverted Pendulum vol.19, pp.3, 2013,
  2. Balancing Control of a Ball Robot Based on an Inverted Pendulum vol.19, pp.9, 2013,
  3. LQR Controller Design for Balancing and Driving Control of a Bicycle Robot vol.20, pp.5, 2014,
  4. Travel Control of a Spherical Wheeled Robot (Ball-Bot) with Mecanum Wheel vol.20, pp.7, 2014,
  5. Balancing and Driving Control of a Mecanum Wheel Ball Robot vol.21, pp.4, 2015,