###### Abstract  
Robotic manipulators have been widely used in industries, mainly to move tools into different specific positions. Thus, it has become necessary to have accurate knowledge about the tool position using forward kinematics after accessing the angular locations of limbs. This paper presents a simulation study in which an encoder attached to the limbs gathers information about the angular positions. The measured angles are applied to the Kalman Filter (KF) and its variants for state estimation. This work focuses on the use of fractional order controllers with a Two Degree of Freedom Serial Flexible Links (2 DSFL) and Two Degree of Freedom Serial Flexible Joint (2 DSFI) and undertakes simulations with noise and a square wave as input. The fractional order controllers fit better with the system properties than integer order controllers. The KF and its variants use an unknown and assumed process and measurement noise matrices to predict the actual data. An optimisation problem is proposed to achieve reasonable estimations with the updated covariance matrices. Serial Flexible Link; Serial Flexible Joint; integer order PID; fractional order PID state estimation; Kalman Filter; Unscented Kalman Filter; Extended Kalman Filter; particle filter +
Footnote †: journal: Medical  
and tear of the physical model. Thus, it is quite impractical to determine system states, making state estimation methods necessary [4]. Figure 1 presents the general analogy for the state estimation techniques used in MIMO systems. Usually, linear state estimations are performed through the implementation of Kalman Filters, but this may not be a good option in the case of dynamic models, measurements that are highly nonlinear and non-Gaussian noise. Filters such as the particle filter (PF) [5], Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) [6; 7] give a generalised solution for such systems, where low performance is caused due to intractable linearisation and Gaussian approximations. Traditionally, these Bayesian methods are significantly used in moving robot scenarios [8]. As an example, [PERSON] et al. presented the application of the EKF to a rigid 2 DOF robotic arm using measurements of tool acceleration and arm angle to improve trajectory tracking [9]. A similar extension of this work has been presented by [PERSON] et al. [10], where the EKF was used for a 7 DOF robotic manipulator. In [11], the authors used the Kalman filter for single flexible links, where they used joint angle and tool acceleration measurements for estimation. Similarly, ref. [12] used an EKF for a manipulator with two links, with tool positions and joint action as the measurements. There are few robotic applications in which the applications have been embedded with sensor fusion techniques using particle filters [13; 14] and the simulated data. The particle filter is additionally promoted as one can fully utilize the probability density function (PDF), which makes it easier to design control laws and therefore obtain more significant performance improvements. This also allows hard limits on system parameters and gives a benchmark for an easier solution, as in the case of the EKF. Modern manufacturing requires robots that are capable of functioning at accuracies beyond those of present-day industrial robots. Accurately moving a tool on a predefined path is the aim. However, many modern industrial robot control approaches involve joint angle measurements of the manipulator [15]. [PERSON] et al. [16] described a method to achieve improved accuracy through the identification of uncertain parameters in robot system by the use of the iterative learning control (ILC) method. Precise modelling has been a significant part of gaining information about systems [17]. The more accurate the modelling of the physical plant, the better the possible control of the system [18]. As the mathematical models have always been important, the use of fractional calculus gives users an upper hand when determining the model.