Robotic applications, especially robotic arm, become widely used in industries due to their simplicity and the ability to do multi-task/multi-function with few numbers of settings and/or arrangements. The problem with such applications is the necessary to apply nonlinear control signals to achieve the desired trajectories. The latter is not easy to be implemented and has several limitations [1] -[3] . For example, Sliding Mode Control (SMC) [1] is one of the robust control approaches. However, it suffers from chattering. Although several researches have proposed to eliminate the chattering, the problem is still not fully solved. The limitation of such controllers increases as uncertainties present, i.e. modeling uncertainties and noise. This becomes worse when the number of measurement is less than the number of states.

Filters, especially model based filters [2] - [7] , have been used to remove some of those constrains. It is a cheap method that could be used to obtain the unmeasured-hidden-states, and/or it could be used to reduce the noise effect. The optimal solution for such applications in their linear case is the Kalman Filter (KF) [7] - [12] . When the system is nonlinear, the KF is modified to be applicable for such applications. Several researches have been developed to overcome this limitation. Those include linearizing the system by Taylor Series Approximation (TSA) up to the first order such as the Perturbation Kalman Filter [9] [13] [14] , the Extended Kalman filter (EKF) [8] [15] - [17] , and the Iterated Extended Kalman filter (IEKF) [7] [15] [18] - [20] , or up to higher order such as the Higher Order Extended Kalman Filter (HOEKF) [15] [21] - [23] . The later shows that in order to increase the accuracy of high nonlinear application, TSA is not a suitable approach as it takes long computation time with complicated structure [24] . Therefore, different approaches were developed including the combination of KF with intelligent techniques such as [25] - [29] , or finding different approaches to approximate the nonlinearity such as the Sigma-Point Kalman Filter (SPKF) [2] [4] [5] and the Particle Filter (PF) [30] . The rest of the paper will be divided as the following: Section two includes an introduction to the SPKF including the algorithms used in this paper, UKF, CKF and CDKF. The mathematical model of the PRRR robotic arm application is showed in Section three. Results, discussion and conclusion are listed and discussed in Sections four and five.

The SPKFs linearize the nonlinear models statistically using weighted linear regression method. This is done by obtaining a certain number of points, referred to as sigma points, from the state neighborhood using the probability distribution function as shown in Figure 1. Those points are projected through the system model, and then combined together using appropriate weights as shown in Figure 2. This provides with a mechanism that covers

the actual mean and covariance without the need to linearize the model by TSA and calculate the Jacobian matrices. Moreover, it accommodates noise disturbances that are not Gaussian [4] [5] [15] [31] - [34] .

Several algorithms have been created using the above principle. Although, different approaches were used to derive those algorithms, the general outline remain the same as will be proven in the next subsections. The major differences between those methods could be summarized to the number of the sigma points, how to choose them, and what are the appropriate weights for the combining step. Moreover, they may differ on calculating the covariance matrices [35] . Some SPKFs algorithm will be described on the next subsections.

The algorithms in section two are applied to a four DOF robotic arm that consists of one prismatic joint and three revolute joints (PRRR) that is presented by Figure 3 and Figure 4. The model has been derived in [1] and [2] , and is summarized as follow.

where;

The system is discretized using the following definition

where is the sampling time and it is equal to 0.001 sec. If the states defined as the following.

And knowing that

Then the overall state space could be defined as

Equations (3.3)-(3.9) have several parameters. Those are summarized by Table 5.

The system in section 3 was simulated several time -for each filter including UKF, CKF and CDKF-. Four cases were obtained as follows:

1. Assuming all the states were measured.

2. Assuming that the position and angles were measured while their derivatives were not measured.

3. Similar to the first case. However, modeling uncertainties were injected; e.g. the masses were multiplied by 1.5.

4. Similar to the second case. However, modeling uncertainties were injected; e.g. the masses were multiplied by 1.5.

This work discussed the benefits of using Sigma-Point Kalman Filters in nonlinear application, i.e. PRRR robotic arm. Three types of SPKFs were used, namely Unscented, Cubature, and Central difference Kalman Filters. Four cases were used: the first and the second cases involved with system with no modeling errors; the third and the fourth cases involved with system injected with uncertainties. The first and the third cases assumed all the states were measured which was not the case in the other cases. The results showed that the filters gave good performance when all the states were measured. Reducing the number of measurements affected the results a little bit. The errors became larger than 10 times of those obtained in case 1 when modeling errors were presented and not all the states were measured. However, the CDKF showed stable performance in all cases. The latter gave an indication to use the CDKF in such applications.

MohammadAl-Shabi, (2015) Sigma-Point Filters in Robotic Applications. Intelligent Control and Automation,06,168-183. doi: 10.4236/ica.2015.63017