## 5 Simulation and Results  
Figure 6 shows the basic Simulink model that has been designed for FOPID and IOPID controllers. The values of the constant chosen for the fractional order PID and integer order PID controllers are shown in Table 2, where \(F_{lower}\) and \(F_{upper}\) are the Oustaloup filter frequency range and the order is the order of approximation. The input was provided by a square wave generator and then fed to the state-space model with their respective controllers. From Table 2, it can be stated that a low value of actuator effort is required and better fits the system. However, from Table 3, it can also be concluded that lower actuator effort leads to a higher rise time. The parameters of the controllers are tuned by the tool available \begin{table}
\begin{tabular}{c l l} \hline \hline
**Symbol** & **Description** & **Units** \\ \hline \(I_{m}\) & Motor current for first link & A \\ \(K_{t}\) & Torque constant for first motor drive & Nm/A \\ \(T\) & Torque produced by Drive 1 & Nm \\ \(\theta\) & Angular position of Link 1 drive shaft & rad \\ \(\frac{d\theta}{dt}\) & Angular velocity of Link 1 drive shaft & rad/s \\ \(\frac{d\phi}{dt}\) & Link 1 Relative end-effector angular position & rad \\ \(\frac{d\phi}{dt}\) & Link 1 relative end-effector angular velocity & rad/s \\ \(J_{1}\) & First joint equivalent moment of inertia & kg m\({}^{2}\) \\ \(B_{1}\) & First joint equivalent viscous damping coefficient & Nm s/rad \\ \(J_{2}\) & Link 1 end-effector equivalent moment of inertia (combined with second stage) & kg m\({}^{2}\) \\ \(B_{2}\) & Link 1 end-effector equivalent damping coefficient (combined with second stage) & Nm s/rad \\ \(K_{s}\) & First link torsional stiffness constant & Nm/rad \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of robotic manipulator for link 1. Figure 6: Basic process flow for 2 DSFL and 2 DSFJ, where \(\theta\): angular position, \(\phi\): angular shift, \(\theta\): angular velocity, \(\phi\): rate of angular shift. Figure 8: Controller responses for angular shift (\(\phi\)) for Link 1. Figure 10: Controller responses for angular shift (\(\phi\)) for Joint 1. Figure 9: Controller responses for angular position (\(\theta\)) for Joint 1. After implementing the FOPID controller, a noise signal arbitrarily introduced in the range of \(10^{-4}\) helped to mimic real-world noise. Various types of Kalman fFlters have been applied and tested to ensure feedback with minimum noise. The filter parameters were chosen using the trial and error method to get the best responses. The figures below show the performance of the filters when applied. The responses are shown for both SFL and SJF, but only for Link 1 and Joint 1. The responses for Link 2 and Joint 2 are similar to their parent links and joint as the system considered is a cascaded system. Figures 11 and 12 present the main response and its enlarged image for better visibility. A square wave input is given to the model. Figures 11 and 12 show that the state estimation methods as applied to integer order and fractional order controller systems provide better performance than the PF. The fractional order controller systems for both the UKF and PF provide better adaptability and manoeuvering. These contribute to lower actuator effort with better performances. Figures 13 and 14 present the comparison of all the filters studied in this manuscript for Link 1 and Joint 1, and the performance of filters can be analyzed from these. The number of particles used in the PF is 100. The PF provides a significantly improved performance than the other approaches. As we know, the distribution considered in the EKF and UKF is Gaussian, and so the same is considered in the PF for a fair comparison. The filter matrices were chosen to be the same as the system state matrices without considering the noise. It can be concluded that the particle filter provides better estimations.