## 2 LI Device Design for Fast Motion
### Design of Actuation System for Desired Performance
As the designed ODT requires (1) a power transmission mechanism for driving the 3-row GOP shafts simultaneously, (2) a chain-sprocket mechanism for constraining and carrying the segments, and (3) a frame stiffness suitable for high velocity and acceleration, the interior of the ODT has very limited space. Therefore, motion actuation along each axis is generated using two synchronized motors, as shown in Figure 7. Under this distributed actuation, the motors' power can be distributed uniformly to all the power transmission components without excessive stress. Moreover, this actuation design provides comparatively faster command response than a single motor design [23]. Since the system uses a distributed power scheme, it is important to achieve precise speed control of both actuators to prevent damage to the power transmission components [17]. To simultaneously minimize the differences in speed and torque of both actuators, a cross-couple control scheme [24] was incorporated into the low-level control of the proposed ODT, as shown in Figure 8.  
Figure 7: Mechanically coupled actuation system design: power transmission mechanisms of (**a**) the Y-axis and (**b**) the X-axis. The low-level controller synchronizes the motor speeds according to the desired speed command (\(v_{c}\)), where \(\mathbf{v_{c}}=[v_{c}{}^{x},\,v_{c}{}^{y}]\mathbf{T}\) is the desired speed of the active surface, and \(v_{1}{}^{i}\) and \(v_{2}{}^{i}\) correspond to the velocity of each motor, determined via encoder feedback with respect to the X or Y-axis, \(T_{c}{}_{1}{}^{i}\) and \(T_{c}{}_{2}{}^{i}\) are the torque values. Synchronization for reducing the difference of each motor's speed (\(v_{v}{}^{i}\)) and torque (\(v_{T}{}^{i}\)) in each axis was achieved by proportional-derivative (PD) control, as follow:  
\[v_{v}{}^{i}=C_{v}{}^{i}\Big{(}v_{1}{}^{i},v_{2}{}^{i}\Big{)}=k_{Pv}{}^{i} \Big{(}v_{1}{}^{i}-v_{2}{}^{i}\Big{)}+k_{Dv}{}^{i}\frac{d\big{(}v_{1}{}^{i}-v_ {2}{}^{i}\big{)}}{dt},\,\,i=X,Y, \tag{4}\]  
where \(k_{Pv}\) and \(k_{Dv}\) are the positive gains of the speed controller. The torque synchronization controller is also implemented using PD control, as follows:  
\[v_{T}{}^{i}=C_{t}{}^{i}\Big{(}T_{c}{}_{1}{}^{i},T_{c}{}_{2}{}^{i} \Big{)}=k_{Pt}{}^{i}\Big{(}T_{c}{}_{1}{}^{i}-T_{c}{}_{2}{}^{i}\Big{)}+k_{Dt} \frac{d\big{(}T_{c}{}_{1}{}^{i}-T_{c}{}_{2}{}^{i}\big{)}}{dt},\,\,i=X,Y, \tag{5}\]  
where \(k_{Pt}{}^{i}\) and \(k_{Dt}{}^{i}\) are the positive gains of the torque synchronization controller, and \(T_{c}{}_{1}{}^{i}\) and \(T_{c}{}_{2}{}^{i}\) are the torque values for the control input of \(C_{t}{}^{i}\). To secure control stability of the low-level controller, suitable gains were selected based on the Nyquist criterion by considering the backlash model [19; 25]. In terms of the Nyquist criterion, the intersection shows a marginally stable condition, i.e., within 0.4 Hz. Regarding the gain of the controller, the X and Y values were set as; \(k_{Pv}{}^{i}=0.1\) and 0.5, \(k_{Dv}{}^{i}=0.02\) and 0.04, \(k_{Pt}{}^{i}=0.2\) and 0.4, and \(k_{Dt}{}^{i}=0.08\) and 0.1, respectively.