## 4 Controller Design  
The LDR is a complex MIMO system with high nonlinearities and coupling states. In general, one can design a trajectory tracking controller for the end-effector through a kinetic model or a dynamic model. Thus, nonlinearities and time-varying characteristics increase when the PH-LDR carries heavyweight and high mass moment of interia, and it is so difficult to model the exact dynamics for overall control systems of the PH-LDR. Moreover, the complicated and nonlinear dynamic model make the PH-LDR auduous to design a controller to manipulate the trajectory tracking in real time, due to massive numerical computation to solve partial differential equations. The purpose of this study is to implement the PH-LDR based on the kinetic model with the intelligence based controllers, which has the self-adaptation ability to deal with system nonlinearities and parameter uncertainties.  
The dynamic for the PH-LDR has two parts: a rod-less PAs dynamic and a LDR motion dynamic. The rod-less PAs dynamic model can be separated into a pneumatic servo valve dynamic model and a pneumatic cylinder dynamic model, both of which have a significant effect on the stability of the PH-LDR, due to the highly nonlinear behavior caused by the airflow and air compression. In addition, without loss of generality, the motion dynamics of the LDR can be modeled in terms of the motion equations to configure the LDR in relation to the joint torque that is exerted by the three rod-less PAs. The inverse dynamics control (IDC), which is the so-called the computed torque control [23], is a well-known control design technique for conventional LDRs. It refers to the LDR motion model and directly predicts the desired output forces or torques for actuators using a feedforward method. However, the derived PH-LDR dynamic in this study is too complicated to obtain an accurate mathematical representation, so that the IDC method is difficult to be implemented in the developed PH-LDR.  
Hence, a hybrid controller is proposed, based on a simple and straightforward technique that accounts for nonlinearities and uncertainties of PH-LDR under environmental temperature changes. The designed hybrid controller consists of a fuzzy-PID controller and a NNPC. Figure 4 shows the overall control system for the PH-LDR, in which each of the three independent Fuzzy-PID + NNPC's regulates the position of a rod-less PA. The fuzzy-PID receives the feed-back measured position and sends a control voltage to the proportional servo valve to regulate the amount of air that flows into the rod-less PA. The NNPC compensates for turnaround delay due to variations in the target, \(\Delta Target\) and \(\Delta^{2}Target\), and variations in the temperature of the environment.  
The inverse position kinematic model calculates the reference positions \(\left(\rho_{1}(t),\rho_{2}(t),\rho_{3}(t)\right)\) for the three axes using a specific 3-DOF trajectory coordinate \(\left(x^{r}(t),y^{r}(t),z^{r}(t)\right).\) Accordingly, the feedback errors and the variations in the target are determined as:  
\[e_{i}=\rho_{i}-q_{i}\ \,\ \ \ \ \ (i=1,2,3) \tag{26}\]  
Additionally,  
\[\Delta Target=\rho_{i}(t)-\rho_{i}(t-T_{s}),\ \ \ i=1,2,3 \tag{27}\]  
where \(T_{s}\) is the sampling time in the control loop. The outputs of \(\left(x(t),\,y(t),\,z(t)\right)\) define the 3-DOF position of the moving platform.