Uniform ultimate boundedness for underactuated mechanical systems as mismatched uncertainty disappeared

  • Rongrong Yu
  • Ye-Hwa Chen
  • Han Zhao
  • Hao SunEmail author
Original Paper


We propose to design control for uncertain underactuated mechanical systems. The underactuated mechanical system is to follow prescribed holonomic or nonholonomic constraints. The uncertainty in the system does not in general fall within the range space of the input matrix, which is a major obstacle for control design. To resolve this difficulty, we decompose the uncertainty into matched uncertainty and mismatched uncertainty in a unique manner using the geometric structural characteristics of the system. A control scheme is designed to guarantee uniform boundedness and uniform ultimate boundedness of a constraint-following performance measure. The control is based solely on the matched uncertainty. The mismatched uncertainty turns out to be disappeared, as far as the performance analysis is concerned, since it is orthogonal to the geometric space of interest. For demonstrations, a vehicle/inverted pendulum platform is selected. We charge the system to follow either holonomic or nonholonomic constraint. The simulation shows the system performance, in following the prescribed constraint, is superior.


Underactuated system Mechanical system Control Nonholonomic constraint Uniform ultimate boundedness 



The research is supported by the China Scholarship Council (No. 201606690017). This research is also supported by the National Natural Science Foundation of China (No. 51505116) and the army aviation equipment “13th Five-Year” special research project (No. 30103090201).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mechanical EngineeringHefei Universityof TechnologyHefeiPeople’s Republic of China
  2. 2.College of Mechanical and Electronic EngineeringShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  3. 3.The George W. Woodruff School of Mechanical Engineering, Georgia Institute of TechnologyAtlantaUSA

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