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**Composite control of the \(n\)-link chained mechanical systems.**
*(English)*
Zbl 1206.70015

Summary: In this paper, a new control concept for a class of underactuated mechanical system is introduced. Namely, the class of \(n\)-link chains, composed of rigid links, non actuated at the pivot point is considered. Underactuated mechanical systems are those having less actuators than degrees of freedom and thereby requiring more sophisticated nonlinear control methods. This class of systems includes among others frequently used for the modeling of walking planar structures. This paper presents the stabilization of the underactuated \(n\)-link chain systems with a wide basin of attraction. The equilibrium point to be stabilized is the upright inverted and unstable position.

The basic methodology of the proposed approach consists of various types of partial exact linearization of the model. Based on a suitable exact linearization combined with the so-called “composite principle”, the asymptotic stabilization of several underactuated systems is achieved, including a general \(n\)-link. The composite principle used herein is a novel idea combining certain fast and slow feedbacks in different coordinate systems to compensate the above mentioned lack of actuation.

Numerous experimental simulation results have been achieved confirming the success of the above design strategy. A proof of stability supports the presented approach.

The basic methodology of the proposed approach consists of various types of partial exact linearization of the model. Based on a suitable exact linearization combined with the so-called “composite principle”, the asymptotic stabilization of several underactuated systems is achieved, including a general \(n\)-link. The composite principle used herein is a novel idea combining certain fast and slow feedbacks in different coordinate systems to compensate the above mentioned lack of actuation.

Numerous experimental simulation results have been achieved confirming the success of the above design strategy. A proof of stability supports the presented approach.

### MSC:

70Q05 | Control of mechanical systems |

70E60 | Robot dynamics and control of rigid bodies |

93D15 | Stabilization of systems by feedback |

### References:

[1] | Bortoff S., Spong W.: Pseudolinearization of the acrobot using spline functions. 31st IEEE Conference on Decision and Control, Tucson 1992, pp. 593-598 |

[2] | Furuta K., Yamakita M.: Swing up control of inverted pendulum. Industrial Electronics - Control and Instrumentation, Japan 1991, pp. 2193-2198 |

[3] | Grizzle J., Moog, C., Chevallereau C.: Nonlinear control of mechanical systems with an unactuated cyclic variable. IEEE Trans. Automat. Control 50 (2005), 559-576 · Zbl 1365.70035 · doi:10.1109/TAC.2005.847057 |

[4] | Khalil H. K.: Nonlinear Systems. Prentice-Hall, Englewood Cliffs, N.J. 1996 · Zbl 1140.93456 · doi:10.1016/j.automatica.2006.08.010 |

[5] | Mahindrakar A. D., Rao, S., Banavar R. N.: Point-to-point control of a 2R planar horizontal underactuated manipulator. Mechanism and Machine Theory 41 (2006), 838-844 · Zbl 1101.70008 · doi:10.1016/j.mechmachtheory.2005.10.013 |

[6] | Martinez S., Cortes, J., Bullo F.: A catalog of inverse-kinematics planners for underactuated systems on matrix groups. IEEE Trans. Robotics and Automation 1 (2003), 625-630 |

[7] | Murray R., Hauser J.: A case study in approximate linearization: The acrobot example. Proc. American Control Conference, San Diego 1990 |

[8] | Spong M.: Control Problems in Robotics and Automation. Springer-Verlag, Berlin 1998 |

[9] | Stojic R., Chevallereau C.: On the Stability of biped with point food-ground contact. Proc. 2000 IEEE Internat. Conference Robotics and Automation ICRA, 2000, pp. 3340-3345 |

[10] | Stojic R., Timcenko O.: On control of a class of feedback nonlinearizable mechanical systems. WAG98, World Automation Congress, 1998 |

[11] | Wiklund M., Kristenson, A., Åström K.: A new strategy for swinging up an inverted pendulum. Proc. IFAC 12th World Congress, 1993, vol. 9, pp. 151-154 |

[12] | Zikmund J., Moog C.: The structure of 2-bodies mechanical systems. 45st IEEE Conference on Decision and Control, San Diego 2006, pp. 2248-2253 |

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