## 2 The Real 2-DOF CKCM
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**LVDT** & **Description** & **Value** & **Unit** \\ \hline Linear range & \(\pm\)7.5 & inch \\ Linearity & Best fit straight line & \\ Resolution & Infinite (theoretically) & \\ Input & \(\pm\)14.5 to \(\pm\)28 VDC, \(\pm\)100 mA & \\ Output & \(\pm\)5 & VDC \\ Operating temperature range & \(-\)67 to 257 & F \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The specifications of the LVDTs. Figure 1: The 2-DOF CKCM. The frame assignment of the manipulator is shown in Figure 3, where the lengths of its links, \(q_{1}\) and \(q_{2}\), are denoted as its joint variables. From Figure 3, we obtain the following:  
\[q_{1}=\sqrt{x^{2}+y^{2}} \tag{1}\]  
\[q_{2}=\sqrt{(d-x)^{2}+y^{2}} \tag{2}\]  
where d is the distance between the pin joints hanging the two actuators and \((x,y)\) represents the Cartesian position of the manipulator. Equations (1) and (2) provide a closed-form solution for the inverse kinematics, meaning that they can be used to calculate the link lengths \(q_{1}\) and \(q_{2}\) from the desired Cartesian position \((x,y)\). Additionally, from (1) and (2), the Cartesian variables \(x\) and \(y\) can be derived as follows. \[x=\frac{{q_{1}}^{2}-{q_{2}}^{2}+d^{2}}{2d} \tag{3}\]  
\[y=\frac{-\sqrt{4d^{2}{q_{1}}^{2}-({q_{1}}^{2}-{q_{2}}^{2}+d^{2})}}{2d} \tag{4}\]  
Equations (3) and (4) offer a closed-form solution for the forward kinematics, allowing the determination of a Cartesian position \((x,y)\) based on the link lengths \(q_{1}\) and \(q_{2}\). Figure 3: The frame assignment of the planar 2-DOF CKCM. Figure 2: The closed-loop control system of the manipulator. We proceed to derive the dynamic model the manipulator. Its Lagrangian dynamic equations can be obtained as a special case of Equation (5) [31] with n = 2, as follows:  
\[\mathbf{M(q)\ddot{q}(t)+C(q\dot{q})\dot{q}(t)+G(q)+F(q\dot{q})}=\boldsymbol{\tau} \tag{5}\]  
with  
\[\boldsymbol{\tau}(t)=(\tau_{1}\,\tau_{2})^{T};\,\mathbf{q}(t)=(q_{1}\,q_{2})^{T} \tag{6}\]  
where \(\tau_{i}\) denotes the joint force of the \(i\)th actuator for i = 1,2. The inertia matrix, the Centrifugal and Coriolis forces, and the friction and gravitational forces at two joints are given by the following [31]:  
\[\mathbf{M}= \begin{bmatrix}m_{1}&0\\ 0&m_{1}\end{bmatrix} ,\,\,\mathbf{C=} \begin{bmatrix}0&\frac{ml_{i}(q_{2}-q_{1} )}{3v}\\ \frac{ml_{i}(q_{2}-q_{1})}{3v}&0\end{bmatrix} ,\,\,\mathbf{G}=[G_{1}\,\,G_{2}]^{T} \tag{7}\]  
with  
\[\mathbf{G}_{1}=\left.(-m_{1}g[2v_{1}q_{1}^{\,2}(q_{1}l_{s}+q_{2}l_{s}+2q_{1}q_ {2})-q_{2}l_{s}v^{2}]-mgl_{s}[2q_{1}^{\,2}v_{1}(q_{1}+q_{2})-q_{2}v^{2}])\, \right/4 dq_{1}^{\,2}q_{2}v \tag{8}\]  
\[\mathbf{G}_{2}=\left.(-m_{1}g[2v_{1}q_{2}^{\,2}(q_{1}l_{s}+q_{2}l_{s}+2q_{1}q_ {2})-q_{2}l_{s}v^{2}]-mgl_{s}[2q_{2}^{\,2}v_{2}(q_{1}+q_{2})-q_{1}v^{2}])\, \right/4 dq_{2}^{\,2}q_{1}v \tag{9}\]  
where \(v_{1}\), \(v_{2}\), and \(v\) are actuator coupling constraints that can be expressed as follows:  
\[v_{1}(t)=v_{2}(t)=q_{2}^{\,2}-q_{1}^{\,2}+d^{\,2},\,v(t)=\sqrt{4d^{2}q_{1}^{ \,2}-v_{1}(t)} \tag{10}\]  
\[\mathbf{F}= \begin{bmatrix}\mathrm{F}_{V_{1}}\dot{q}_{1}+\mathrm{F}_{C_{1}} \mathrm{sgn}(\dot{q}_{1})\\ \mathrm{F}_{V_{2}}\dot{q}_{2}+\mathrm{F}_{C_{2}}\mathrm{sgn}(\dot{q}_{2}) \end{bmatrix} . \tag{11}\]