# Computationally Efficient Adaptive Type-2 Fuzzy Control of Flexible-Joint Manipulators

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## Abstract

**:**

## 1. Introduction

## 2. Flexible-Joint Manipulator Dynamics

#### 2.1. Modeling of a Flexible-Joint Manipulator

**Property 1**

- (1)
- Positive Definite Symmetric (PDS), i.e., ${M}^{T}\left(q\right)=M\left(q\right)$ and ${x}^{T}M\left(q\right)\phantom{\rule{3.33333pt}{0ex}}x>0$ for any non-null vector x.
- (2)
- Upper and lower bounded, i.e., there exist two scalars ${\alpha}_{1}\left(q\right)$ and ${\alpha}_{2}\left(q\right)$ such that ${\alpha}_{1}\left(q\right)I\le M\left(q\right)\le {\alpha}_{2}\left(q\right)I$, where I is the identity matrix.

**Property 2**

- (1)
- Matrix $\dot{M}\left(q\right)-2C(q,\dot{q})$ is skew symmetric, i.e., $\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{x}^{T}(\dot{M}\left(q\right)-2C(q,\dot{q}))\phantom{\rule{3.33333pt}{0ex}}x=0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}x\in {\mathbb{R}}^{n}$
- (2)
- $C(q,\dot{q})\dot{q}$ is quadratic in $\dot{q}$ and bounded, i.e., there exists a scalar ${\alpha}_{3}\left(q\right)$ such that $\parallel C(q,\dot{q})\dot{q}\parallel \le {\alpha}_{3}\left(q\right){\parallel \dot{q}\parallel}^{2}$.

**Property 3**

**Assumption 1**

#### 2.2. Friction Modeling

#### 2.3. Problem Statement

## 3. Type-2 FLSs

## 4. Interval Type-2 FLSs

- the firing strength of the lth fuzzy rule is an interval type-1 fuzzy set defined as$${F}^{l}\left({x}^{\prime}\right)\equiv {\sqcap}_{i=1}^{p}{\mu}_{\tilde{{F}_{i}^{l}}}\left({x}_{i}^{\prime}\right)=[{\underline{f}}^{l}\left({x}^{\prime}\right),{\overline{f}}^{l}\left({x}^{\prime}\right)]\equiv [{\underline{f}}^{l},{\overline{f}}^{l}]$$$$\begin{array}{cc}\hfill {\underline{f}}^{l}\left({x}^{\prime}\right)& ={\underline{\mu}}_{\tilde{{F}_{1}^{l}}}\left({x}_{1}^{\prime}\right)\star \cdots \star {\underline{\mu}}_{\tilde{{F}_{p}^{l}}}\left({x}_{p}^{\prime}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {\overline{f}}^{l}\left({x}^{\prime}\right)& ={\overline{\mu}}_{\tilde{{F}_{1}^{l}}}\left({x}_{1}^{\prime}\right)\star \cdots \star {\overline{\mu}}_{\tilde{{F}_{p}^{l}}}\left({x}_{p}^{\prime}\right)\hfill \end{array}$$
- the fired output consequent set of the lth rule is a type-1 fuzzy set characterized by a membership function$${\mu}_{\tilde{{B}^{l}}}\left(y\right)={\int}_{{b}^{l}\in [{\underline{f}}^{l}\star {\underline{\mu}}_{\tilde{{G}^{l}}}\left(y\right),{\overline{f}}^{l}\star {\overline{\mu}}_{\tilde{{G}^{l}}}\left(y\right)]}1/{b}^{l}\phantom{\rule{2.em}{0ex}}\forall y\in Y$$
- if N out of a total of L fuzzy rules in the FLS fire, where $N\le L$, then the overall aggregated output fuzzy set is defined by a type-1 membership function ${\mu}_{\tilde{B}}\left(y\right)$ obtained by combining the fired output consequent sets into one. In other words, ${\mu}_{\tilde{B}}\left(y\right)={\bigsqcup}_{l=1}^{N}{\mu}_{\tilde{{B}^{l}}}\left(y\right)$, where ${\mu}_{\tilde{{B}^{l}}}\left(y\right)$ is defined in Equation (5).

#### 4.1. Type-2 Fuzzification

#### 4.2. Type-2 Fuzzy Rule Base

#### 4.3. Type-2 Fuzzy Inference Engine

#### 4.4. Type Reduction

**Calculation of the Rule Consequents Centroids**The centroid of the ${t}^{th}$ output fuzzy set ${y}_{k}^{t}$ is a type-1 interval set determined by its left and right most points, ${y}_{lk}^{t}$ and ${y}_{rk}^{t}$, respectively, which are expressed by [57]:

#### 4.5. Calculation of the Type-Reduced Set

#### 4.6. Type-2 Defuzzification

## 5. Control Strategy

$\Delta q$ | |||||

$\mathbf{\Delta}{\dot{\u03f5}}_{\mathbf{r}}$ | NL | NS | Z | PS | PL |

PL | Z | PL | PL | PL | PL |

PS | NS | Z | PS | PS | PL |

Z | NL | NS | Z | PS | PL |

NS | NL | NS | NS | Z | PS |

NL | NL | NL | NL | NL | Z |

#### 5.1. Adaptive Type-2 FLC

**Theorem 1**

**Proof:**

## 6. Simulation Results and Discussion

#### 6.1. Simulation Setup

Parameter | Link | Motor |
---|---|---|

rotational inertia (kg·m${}^{2}$) | $I=5.05\times {10}^{-2}$ | ${J}_{m}=4\times {10}^{-3}$ |

viscous friction coefficient (N·m·s/rad) | ${F}_{vl}=4\times {10}^{-3}$ | ${F}_{vm}=3\times {10}^{-3}$ |

Coulomb friction coefficient (N·m) | ${F}_{cl}=1\times {10}^{-2}$ | ${F}_{cm}=4\times {10}^{-3}$ |

static friction coefficient (N·m) | ${F}_{sl}=2\times {10}^{-3}$ | ${F}_{sm}=2\times {10}^{-3}$ |

static friction decreasing rate (rad/s) | ${\eta}_{sl}=7\times {10}^{-2}$ | ${\eta}_{sm}=5\times {10}^{-2}$ |

#### 6.2. Numerical Simulations and Results

**Figure 10.**System’s response to varying load’s mass and inertia: (

**a**), (

**b**) manipulator’s position error; (

**c**), (

**d**) manipulator’s velocity error; (

**e**), (

**f**) motor’s velocity vs. manipulator’s velocity; and (

**g**), (

**h**) controller’s output torque (${\tau}_{m}$).

**Figure 11.**System’s response to varying stiffness coefficient: (

**a**), (

**b**) manipulator’s position error; (

**c**), (

**d**) manipulator’s velocity error; (

**e**), (

**f**) motor’s velocity vs. manipulator’s velocity; and (

**g**), (

**h**) controller’s output torque (${\tau}_{m}$).

## 7. Conclusions

## Acknowledgements

## Conflict of Interest

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**MDPI and ACS Style**

Chaoui, H.; Gueaieb, W.; Biglarbegian, M.; Yagoub, M.C.E.
Computationally Efficient Adaptive Type-2 Fuzzy Control of Flexible-Joint Manipulators. *Robotics* **2013**, *2*, 66-91.
https://doi.org/10.3390/robotics2020066

**AMA Style**

Chaoui H, Gueaieb W, Biglarbegian M, Yagoub MCE.
Computationally Efficient Adaptive Type-2 Fuzzy Control of Flexible-Joint Manipulators. *Robotics*. 2013; 2(2):66-91.
https://doi.org/10.3390/robotics2020066

**Chicago/Turabian Style**

Chaoui, Hicham, Wail Gueaieb, Mohammad Biglarbegian, and Mustapha C. E. Yagoub.
2013. "Computationally Efficient Adaptive Type-2 Fuzzy Control of Flexible-Joint Manipulators" *Robotics* 2, no. 2: 66-91.
https://doi.org/10.3390/robotics2020066