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\title{Stabilizability analysis of a flexible manipulator model}
\author{A.L.~Zuyev}
\address{Institute of Applied Mathematics and Mechanics NASU,\\
	R.~Luxemburg str.~74, Donetsk  83114, Ukraine}
\email{al\_zv@mail.ru}

\begin{document}
\maketitle

Control issues for several models of flexible manipulators have been
intensively studied by many authors. A particular list of results and
references in this area can be found in \cite{LGM1999}.  In contrast to the
above publication, we study here a rotating beam that carries a payload
under the action of gravity, the control torque is applied at the hub, and the
longitudinal motion is taken into account. The motivation for this study is to
control the motion of a real flexible-link manipulator (turntable ladder).

In this talk, the Timoshenko beam rotating in the vertical plane
around the fixed point~$O$ is considered. We assume that the beam
is driven by a control torque at one its end (the hub at $O$), and
the other end carries a payload of mass $m$. Let $l$ be the length
of the beam. Consider a particle $P$ on the centerline and denote
by $x$ its coordinate in the reference configuration. At a given
time~$t$, let $\bigl(x+s(x,t),w(x,t)\bigr)$ be the coordinates of
the position vector for $P$ in the rotating Cartesian frame~$Oxy$.
We introduce the notation $\psi(x,t)$ for the rotation angle of
the cross section area at $P$ due to bending. By taking into
account the longitudinal, vertical, and shear motions, the
Lagrangian of the system considered takes the form: {\footnotesize
$$
2L = \int_0^l  \left\{ \left( (\dot w+x\dot
\varphi)^2+{\dot\varphi}^2 w^2\right)\rho-{\dot \varphi}^2
{w'}^2\rho_2+I_\rho (\dot \varphi+\dot\psi)^2-K
(\psi-w')^2-EI(\psi')^2 \right\}dx+
$$
$$
+m\left\{{\dot\varphi}^2 w^2(l,t)+\left(l\dot\varphi+\dot
w(l,t)\right)^2 \right\}+
J_c\left\{\dot\varphi+\dot\psi(l,t)\right\}^2+J_0{\dot\varphi}^2 -
$$
$$
-g\int_0^l \{(2\rho x-\rho_1{w'}^2)\sin\varphi+2\rho
w\cos\varphi\}dx -2mg\left\{l\sin\varphi+
w(l,t)\cos\varphi\right\},
$$}
where
$$
\rho_1(x) = \int_x^l \rho(\xi)\,d\xi+m,\quad \rho_2(x) = \int_x^l
\xi\rho(\xi)\,d\xi+ml.
$$
Here $\varphi(t)$ is the angle between the moving axis $Ox$ and the horizontal
direction, $\rho(x)$ is the mass per unit length of the beam, $I_\rho(x)$ is
the mass moment of inertia of the cross section, $J_0$ is the hub moment of
inertia, $E$ and $I$ are Young's modulus and the moment of inertia of the cross
section of the beam, respectively, $g$ is the acceleration of gravity. The
coefficient $K$ is equal to $kGA$, where $G$ is the modulus of elasticity in
shear, $A$ is the cross sectional area, and $k$ is a constant depending on the
shape of the cross section. The mass distribution for the payload is
characterized by the moment of inertia $J_c$ with respect to its center of
mass.

For the distributed parameter system corresponding to the Lagrangian $L$, we
construct a family of Galerkin approximations based on solutions of the
homogeneous Timoshenko beam equation. We derive sufficient conditions for
stabilizability of such finite dimensional system. In addition, the equilibrium
of the Galerkin approximation considered is proved to be stabilizable by an
observer-based feedback law, and an explicit control design is proposed.

A part of this presentation is based on paper \cite{ZS2007}.

\begin{thebibliography}{M}
\bibitem{LGM1999} \emph{Luo Z.-H., Guo B.-Z., Morgul O.}
	Stability and  Stabilization of Infinite Dimensional Systems.
	-- London: Springer--Verlag, 1999. -- 403~p.
\bibitem{ZS2007} \emph{Zuyev A., Sawodny O.}
	Stabilization and Observability of a Rotating Timoshenko Beam Model
	// Mathematical Problems in Engineering. -- 2007. -- No. 7. -- P.~1--19.
\end{thebibliography}

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