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Sigma-Point Kalman Filters (SPKFs) are popular estimation techniques for high nonlinear system applications. The benefits of using SPKFs include (but not limited to) the following: the easiness of linearizing the nonlinear matrices statistically without the need to use the Jacobian matrices, the ability to handle more uncertainties than the Extended Kalman Filter (EKF), the ability to handle different types of noise, having less computational time than the Particle Filter (PF) and most of the adaptive techniques which makes it suitable for online applications, and having acceptable performance compared to other nonlinear estimation techniques. Therefore, SPKFs are a strong candidate for nonlinear industrial applications, i.e. robotic arm. Controlling a robotic arm is hard and challenging due to the system nature, which includes sinusoidal functions, and the dependency on the sensors’ number, quality, accuracy and functionality. SPKFs provide with a mechanism that reduces the latter issue in terms of numbers of required sensors and their sensitivity. Moreover, they could handle the nonlinearity for a certain degree. This could be used to improve the controller quality while reducing the cost. In this paper, some SPKF algorithms are applied to 4-DOF robotic arm that consists of one prismatic joint and three revolute joints (PRRR). Those include the Unscented Kalman Filter (UKF), the Cubature Kalman Filter (CKF), and the Central Differences Kalman Filter (CDKF). This study gives a study of those filters and their responses, stability, robustness, computational time, complexity and convergences in order to obtain the suitable filter for an experimental setup.

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 [

Filters, especially model based filters [

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

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 [

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 [

The Unscented Kalman Filter is a SPKF that has been developed using the unscented transformations. The latter has several form including general unscented [

////Comments //// q is the number of the sigma point //// draw the sigma points and their weights using |
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Method | |||
---|---|---|---|

UKF | |||

Simplex UKF | |||

Spherical UKF | Similar to the simplex UKF except that |

The statistical regression used in unscented filters provides with better approximation that the Jacobian matrices. It has been proven that UKFs approximates up to a third order TSA for Gaussian distributions [

The Cubature Kalman filter (CKF) is derived by using the third-degree cubature rule to numerically approximate the Gaussian-weighted integrals defined as [

where

mean and standard deviation. Assuming that the states are Gaussian as well, a scheme similar to the UKF could be obtained. However, due to the Gaussian Nature, the covariance matrices will differ from those obtained from UKF. Those are illustrated by

The Central Difference Kalman Filter (CDKF), described in [

//// Comments //// q is the number of the sigma point //// draw the sigma points //// propagate the points through the filter //// combining the sigma points to obtain the a priori estimate //// calculating the a priori covariance matrix //// Redefine the sigma point to obtain their a priori measurements //// combining the sigma points’ measurements to obtain the a priori measurement //// Calculating the output's error covariance matrix //// The correction gain //// Updating the estimate and its covariance matrix //// Repeat Stages |
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The previous stages result on a scheme that is similar to the weighted regression of the UKF as shown in

other SPKFs [

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

//// Comments //// draw the sigma points //// propagate the points through the filter //// combining the sigma points to obtain the a priori estimate //// calculating the a priori covariance matrix //// Redefine the sigma point to obtain their a priori measurements //// combining the sigma points’ measurements to obtain the a priori measurement //// Calculating the output's error covariance matrix //// The correction gain //// Updating the estimate and its covariance matrix //// Repeat Stages |
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where;

The system is discretized using the following definition

where

And knowing that

Then the overall state space could be defined as

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

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.

The results of cases 1 and 2 were summarized by

When modeling errors presented, the RMSE increased as shown in

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

21.5 kg | 0.25 m | ||||

16 kg | 1.2 m | ||||

8.5 kg | 0.8m | ||||

7.9 kg | 1.2 | ||||

6.3 kg |

RMS in | UKF | CKF | CDKF |
---|---|---|---|

32.7 | 32.7 | 32.7 | |

29.2 | 29.2 | 29.2 | |

31 | 31 | 31 | |

35.8 | 35.8 | 35.8 | |

22.2 | 22.2 | 22 | |

46.1 | 46.1 | 46.1 | |

25.6 | 25.6 | 25.6 | |

44.1 | 44.1 | 44.1 |

RMS in | UKF | CKF | CDKF |
---|---|---|---|

35.9 | 35.9 | 35.9 | |

203.4 | 203.4 | 203.4 | |

41.4 | 41.4 | 41.3 | |

162.8 | 162.8 | 161.6 | |

30.6 | 30.6 | 31.9 | |

171.4 | 171.4 | 171.1 | |

49.2 | 49.2 | 49.1 | |

291.7 | 291.7 | 291.4 |

RMS in | UKF | CKF | CDKF |
---|---|---|---|

44.6 | 44.6 | 44.6 | |

375.8 | 375.8 | 375.8 | |

44.5 | 44.5 | 44.5 | |

367.4 | 367.4 | 367.4 | |

34.9 | 34.9 | 34.9 | |

365.1 | 365.1 | 365.1 | |

38.9 | 38.9 | 38.9 | |

366.5 | 366.5 | 366.5 |

RMS in | UKF | CKF | CDKF |
---|---|---|---|

146.9 | 146.9 | 146.8 | |

4588.5 | 4588.5 | 4588.9 | |

111.3 | 111.3 | 103.8 | |

2710.8 | 2710.8 | 1693.2 | |

120.3 | 120.3 | 112.2 | |

5590.7 | 5590.7 | 2447.4 | |

133.3 | 133.3 | 109.3 | |

4079.6 | 4079.6 | 1982.2 |

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