Lumped-parameter model of dual–drive gantry stages

The lumped-parameter model in the attached images can be described using eleven physical parameters.

1. Four masses $m_b,m_k,m_1$ and $m_2$ corresponding to the mass of the beam, the moving head and actuators $X_1$ and $X_2$.
2. Five friction coefficients $C_{g1}, C_{g2}, C_y, C_{b1}$ and $C_{b2}$, corresponding to the viscous friction coefficients of $X_1, X_2$, and $Y$ actuators, and to the damping coefficients of the flexible joints.
3. Two stiffness coefficients $k_{b1}$ and $k_{b2}$ of the flexible joints of the beam to actuator junctions.

The beam and the head have lengths $L_b$ and $L_h$. The head's position is measured from the center of mass of the beam and is denoted by $Y$ and $d$, In the attached image, $X$ denotes the linear position of the center of mass of the beam and $\Theta$ is the yaw angle of the beam. They are defined as

$\begin{bmatrix} X \\ \Theta \\ Y \end{bmatrix} = \begin{bmatrix} 1/2 &1/2 & 0 \\ 1/L_{b} & -1/L_{b} & 0 \\ 0 & 0 &1 \end{bmatrix}\begin{bmatrix} X_1 \\ X_2 \\ Y \end{bmatrix}$

The Lagrange-Euler formalism is applied to derive the motion equations of the system.

$$M\ddot{q}+H\dot{q}+C\dot{q}+Kq=F,$$

where $M, C$ and $K$ are the inertia, viscous damping and stiffness matrices, $H$is the coriolis and centripetal acceleration matrix, $F$ is the vector of forces and $q$ is the vector of coordinates composed of $X, \Theta$ and $Y$ (8).

$\begin{align*} M &=\begin{bmatrix} M_{11} & M_{12} & m_{h}\sin(\Theta) \\ M_{12} & J_{t}+m_{h}Y^{2} &-m_{h}d \\ m_{h}\sin(\Theta) & -m_{h}d & m_{h} \end{bmatrix}, \\ H &=\begin{bmatrix} 0 & If_{12}\dot{\Theta} & -2m_{h}\dot{\Theta} & \cos(\Theta) \\ 0 & 2m_{h}Y\dot{Y} & 0 \\ 0 & -m_{h}Y\Theta & 0 \end{bmatrix}, \\ C &= \begin{bmatrix} C_{11} & C_{12} & 0 \\ C_{12} & C_{22} & 0 \\ 0 & 0 & c_{y} \end{bmatrix},\ K=\begin{bmatrix} 0 & 0 & 0 \\ 0 & K_{22} & 0 \\ 0 & 0 & 0 \end{bmatrix}, \\ F &=[F_{c}\ M_{c}\ F_{y}]^{\mathrm T},\ q=[X \ \Theta \ Y]^{\mathrm T }. \end{align*}$