Stabilization of a 3-Link Pendulum in Vertical Position

The aim of the paper is to verify the linearizabilty conditions for the triple inverted pendulum driven by 2 inputs, and stabilize it in the upright position. Moreover, the zero dynamics is derived and illustrated graphically.

We express our thanks to Prof. W. Respondek for fruitful discussion and useful comments.

Authors and Affiliations

1. Poznan University of Technology, Poznañ, Poland

   Krzysztof Kozłowski, Dariusz Pazderski, Paweł Parulski & Patryk Bartkowiak

Corresponding author

Correspondence to Krzysztof Kozłowski .

Editors and Affiliations

1. Institute of Automatic Control, Lodz University of Technology, Lodz, Poland

   Andrzej Bartoszewicz

2. Institute of Automatic Control, Lodz University of Technology, Lodz, Poland

   Jacek Kabziński

3. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

   Janusz Kacprzyk

Appendix

In order to guarantee that the \(\bar{D}_2\) distribution is full rank some scalar functions from Sect. 2.2 need to be nonzero. The \(F_{95}\) scalar function is trivial and results in \(\theta _{2} \ne 2k \pi \). The scalar function \(F_{56}\) is complicated and hard to be written analytically. However one can depict it graphically (omitted here) assuming that \(\theta _{2}\) and \(\theta _{3}\) varies form \((-\frac{\pi }{2},\frac{\pi }{2})\) and thus their zeros can be observed for three link manipulator considered here. Calculations from Sect. 3.1 are valid when the system is not in its singularity, when determinants of matrices: \(\det \left[ \begin{array}{c} \bar{m}_{12} \; \bar{m}_{13} \\ \bar{m}_{22} \; \bar{m}_{23} \end{array} \right] ^{-1} = \frac{m_{13}}{\det M}, \) \( \det \left[ \begin{array}{c} \bar{m}_{11} \; \bar{m}_{13} \\ \bar{m}_{21} \; \bar{m}_{23} \end{array} \right] ^{-1} = \frac{m^2_{23}}{\det M} \) and \( \det \left[ \begin{array}{c} \bar{m}_{11} \; \bar{m}_{12} \\ \bar{m}_{21} \; \bar{m}_{22} \end{array} \right] ^{-1} = \frac{m_{33}}{\det M}, \) (here \(m_{33}>0\) by definition) must not be equal to zero, i.e. \(m_{13} \ne 0\), \(m_{23} \ne 0\), and moreover: \(J_1 \ne 0\) and \(J_2 \ne 0\), respectively, for two cases:

1. 1.

   \(a_3 = m_3 L_3 (L_1 + L_2)\) for \(\theta _2 = 0\), \(\theta _3 = \pi + 2k\pi \);

   \(a_3 < m_3 L_3 (L_1 + L_2)\) for solution of the following equation: \(a_3 = - r_1 - r_3\).

2. 2.

   \(a_3 = m_3 L_2 L_3\) for \(\theta _3 = \pi + 2k\pi \);

   \(a_3 < m_3 L_2 L_3\) for \(\theta _3 = -\arccos (\frac{a_3}{m_3 L_2 L_3})\).

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