Type-1 and Type-2 Fuzzy Techniques: Application to Robotic Systems

Abstract

Autonomous mobile robot navigation is a very important task in mobile robotic field. The navigation process is defined by the task to define the values of motion permitting the robot to displace from the start situation to the final destination without human actions and safely. If the mobile robot environment is free of obstacles, the problem is easy to handle. But as the environment is more complex, the robot control system needs more treatments, intelligence and capacities to handle the interactive uncertainties, imprecision that have affect each application of such robotic system. Behaviour-based navigation approaches are successful tools of subdivides the global navigation process into small systems in order to simplify the global task. These basic behaviours are: goal seeking, obstacle avoidance, wall-following, moving target pursuing, avoiding dynamic objects... Mobile robot navigation is one of the popular tasks encountered by imprecision and uncertainties and it has been studied using fuzzy logic systems. Controllers based on fuzzy theory have been applied and implemented in various types of engineering problems, especially in control and robotics. Type-1 fuzzy system is based on two-dimensional fuzzy membership functions. In some applications, this function cannot deal with the resulted uncertainty. Type-2 fuzzy membership function with their imprecise boundaries can give fuzzy sets with membership values instead of a crisp number. Type-1 fuzzy logic systems (T1FLSs) use precise and crisp type-1 fuzzy sets. The modelling process is based on the behaviour of the operator under specific conditions. However, this type of control system presents some limitations in treatment of uncertainties. Type-2 fuzzy logic systems (T2FLSs) introduce important tools to improve the robot performances. Type-2 fuzzy sets and their theoretical concepts create a new version of fuzzy logic controllers in robotic field. Type-2 fuzzy logic systems offer a good tool to model some levels of uncertainties which type-1 fuzzy logic system cannot do. The additional dimension of T2MF may give a better representation of uncertainty than T1MF. The T1FLS produces a T1FM and the result is a real value, whereas a Type-2 FLS produces a T2FM, and using the type-reducer, this fuzzy set will be converted to a T1FM. The type-reduced gives decision-making flexibilities. The objective of this chapter is to control a wheeled mobile robot using type-1 and type-2 Takagi-Sugeno (TS) fuzzy logic controllers. The proposed TS fuzzy controllers are used to infer actions for autonomous robot motion based-behaviors: goal seeking, obstacle avoidance, and wall-following. The simulation results are discussed and compared to show the effectiveness of the designed fuzzy logic controllers for automatic robot motion.

Appendix 1

• Calculation steps of the Takagi-Sugeno Type-2 FLC output.

  For a first order type-2 TSKFLS with M rules, N inputs $\left( x_{1} \in X_{1}, \ldots , x_{N} \in X_{N}\right)$ and one output ($y \in Y$). The r th rule can be expressed by:

  ${\text {IF}} x_{1}$ is $\widetilde{F}_{1}^{r}$ and $x_{2}$ is $\widetilde{F}_{2}^{r}$ and...and $x_{p}$ is $\widetilde{F}_{i}^{r}$, THEN $y^{r}=c_{0}^{r}+c_{1}^{r} x_{1}+\cdots +c_{p}^{r} x_{p}$

  • The firing degree of the i th fuzzy rule is calculated y the following equations:

    $W^{i}\left( x^{\prime }\right) =\left[ \underline{w}^{i}\left( x^{\prime }\right) , \bar{w}^{i}\left( x^{\prime }\right) \right]$

    $\underline{w}^{i}=\underline{\mu }_{\widehat{F}_{i}^{i}}\left( x_{1}\right) * \ldots * \underline{\mu }_{\widehat{F}_{p}^{i}}\left( x_{p}\right)$

    $\bar{w}^{i}=\bar{\mu }_{\widehat{F}_{i}^{i}}\left( x_{1}\right) * \ldots * \bar{\mu }_{\widehat{F}_{p}^{i}}\left( x_{p}\right)$

  • The final output is calculated by:

    $Y\left( Y^{1}, \ldots , Y^{M}, W^{1}, \ldots , W^{M}\right) =\left[ y_{l}, y_{r}\right]$

    $=\int _{y^{1}} \ldots \int _{y^{^{ }}} \int _{w^{U}} \ldots 1 / \frac{\sum _{i=1}^{M} w^{i} y^{i}}{\sum _{i=1}^{M} w^{i}}$

    Where $y_{i} \in Y^{i}$, and $Y^{i}=\left[ y_{l}^{i}, y_{r}^{i}\right] ,(i=1 \ldots M), y_{l}$ and $y_{r}$ are calculated as follows:

    $y_{l}=\frac{\sum _{i=1}^{M} w_{l}^{i} y_{l}^{i}}{\sum _{i=1}^{M} w_{l}^{i}}, y_{r}=\frac{\sum _{i=1}^{M} w_{r}^{i} y_{r}^{i}}{\sum _{i=1}^{M} w_{r}^{i}}$

  • The overall output is:

    $y=\frac{y_{l}+y_{r}}{2}$

Appendix 2

Table 6.

Table 6 Nomenclatures, symbols and abbreviations