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Tool position tracking control of a nonlinear uncertain flexible robot manipulator by using robust H2/H∞ controller via T–S fuzzy model

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Abstract

In this paper, a robust H2/H∞ control with regional Pole-Placement is considered for tool position control of a nonlinear uncertain flexible robot manipulator. The uncertain nonlinear system is first approximated by Takagi and Sugeno’s (T–S) fuzzy model. To achieve a better tracking, an extra state (error of tracking) is then augmented to the T–S model. Based on each local linear subsystem with augmented state, a regional pole-placement state feedback H2/H∞ controller is properly designed via linear matrix inequality (LMI) approach. Parallel Distributed Compensation (PDC) is also used to establish the whole controller for the overall system and the total linear system is obtained by using the weighted sum of the local linear systems. A fuzzy weighted online computation (FWOC) component is employed to update fuzzy weights in real time for different operating points of the system. Simulation results are presented to validate the effectiveness of the proposed controller like robustness and good load disturbance attenuation and accurate tracking, even in the presence of parameter variations and also load disturbances on the motor and the tool. The superiority of the proposed control scheme is finally highlighted in comparison with the Quantitative feedback theory (QFT) controller, the QFT controller of order 13, a polynomial controller and the so-called linear sliding-mode controller methods.

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Cleveland State University, Cleveland, OH, USA

    VAHID AZIMI

  2. Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran

    MOHAMMAD BAGHER MENHAJ

  3. Department of Electrical, Biomedical and Mechatronic Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran

    AHMAD FAKHARIAN

Authors
  1. VAHID AZIMI
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  2. MOHAMMAD BAGHER MENHAJ
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  3. AHMAD FAKHARIAN
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Corresponding author

Correspondence to VAHID AZIMI.

Appendix

Appendix

1.1 Quasi-linear affine system presentation

The detailed quasi-linear affine model of original manipulator dynamics in (1) is presented in appendix.

By substituting (3) into (1), one can obtain

$$\begin{array}{@{}rcl@{}}{J}_{{m}}\ddot{{q}}_{{m}}&{=}&{u}_{{m}}{+\mathit{w}-} {f}_{{m}}\dot{{q}}_{{m}} {-[7500}({q}_{{m}}{-}{q}_{{a}_{{1}}} )^{{3}}{+10}({q}_{{m}}{-}{q}_{{a}_{{1}}}) {]-}{d}_{{1}}(\dot{q}_{{m}}{-}\dot{q}_{{a}_{{1}}})\\ {J}_{{a}_{{1}}}\ddot{q}_{{a}_{{1}}}&=&-{f}_{{a}_{{1}}}\dot{q}_{{a}_{{1}}}{+[7500}({q}_{{m}}{-}{q}_{{a}_{{1}}} )^{{3}}{+10}({q}_{{m}}{-}{q}_{{a}_{{1}}}){]+}{d}_{{1}}(\dot{q}_{{m}}{-}\dot{q}_{{a}_{{1}}} )\\ &&{- }{k}_{{2}}({q}_{{a}_{{1}}}{-}{q}_{{a}_{{2}}} ){-}{d}_{{2}}(\dot{q}_{{a}_{{1}}}{-}\dot{q}_{{a}_{{2}}})\\ {J}_{{a}_{{2}}}\ddot{q}_{{a}_{{2}}}&=&-{f}_{{a}_{{2}}}\dot{q}_{{a}_{{2}}}{+}{k}_{{2}}({q}_{{a}_{{1}}}{-}{q}_{{a}_{{2}}} ){+}{d}_{{2}}(\dot{q}_{{a}_{{1}}}{-}\dot{q}_{{a}_{{2}}} ){-}{k}_{{3}}({q}_{{a}_{{2}}}{-}{q}_{{a}_{{3}}} ){-}{d}_{{3}}(\dot{q}_{{a}_{{2}}}{-}\dot{q}_{{a}_{{3}}} )\\ {J}_{{a}_{{3}}}\ddot{q}_{{a}_{{3}}}&=&v-{f}_{{a}_{{3}}}\dot{q}_{{a}_{{3}}}{+}{k}_{{3}}({q}_{{a}_{{2}}}{-}{q}_{{a}_{{3}}} ){+}{d}_{{3}}(\dot{q}_{{a}_{{2}}}{-}\dot{q}_{{a}_{{3}}} ) \end{array} $$
(19)

By expanding above equations, it is concluded that

$$\begin{array}{@{}rcl@{}} {J}_{{m}}\ddot{q}_{{m}}&=&-\left({7500}{q}_{{m}}^{{3}}{+22500}{q}_{{m}}{q}_{{a}_{{1}}}^{{2}}{+10}{q}_{{m}} \right){+}\left({7500}{q}_{{a}_{{1}}}^{{3}}{+22500}{q}_{{a}_{{1}}}{q}_{{m}}^{{2}}{+10}{q}_{{a}_{{1}}} \right)\\ &&{-}\left({f}_{{m}}\dot{q}_{{m}}{+}{d}_{{1}}\dot{q}_{{m}} \right){+}{d}_{{1}}\dot{q}_{{a}_{{1}}}{+w+}{u}_{{m}}\\ {J}_{{a}_{{1}}}\ddot{q}_{{a}_{{1}}}&{=}&\left({7500}{q}_{{m}}^{{3}}{+22500}{q}_{{m}}{q}_{{a}_{{1}}}^{{2}}{+10}{q}_{{m}} \right){-}\left({7500}{q}_{{a}_{{1}}}^{{3}}{+22500}{q}_{{a}_{{1}}}{q}_{{m}}^{{2}}{+10}{q}_{{a}_{{1}}}{+}{k}_{{2}}{q}_{{a}_{{1}}} \right)\\ &&{+}{k}_{{2}}{q}_{{a}_{{2}}}{+}{d}_{{1}}\dot{q}_{{m}}{-}({f}_{{a}_{{1}}}\dot{q}_{{a}_{{1}}}{+}{d}_{{1}}\dot{q}_{{a}_{{1}}}{+}{d}_{{2}}\dot{q}_{{a}_{{1}}} ){+}{d}_{{2}}\dot{q}_{{a}_{{2}}}\\ {J}_{{a}_{{2}}}\ddot{q}_{{a}_{{2}}}&=&-{f}_{{a}_{{2}}}\dot{q}_{{a}_{{2}}}{+}{k}_{{2}}({q}_{{a}_{{1}}}{-}{q}_{{a}_{{2}}} ){+}{d}_{{2}}(\dot{q}_{{a}_{{1}}}{-}\dot{q}_{{a}_{{2}}} ){-}{k}_{{3}}({q}_{{a}_{{2}}}{-}{q}_{{a}_{{3}}} ){-}{d}_{{3}}(\dot{q}_{{a}_{{2}}}{-}\dot{q}_{{a}_{{3}}} ) \\ {J}_{{a}_{{3}}}\ddot{q}_{{a}_{{3}}}&=&v-{f}_{{a}_{{3}}}\dot{q}_{{a}_{{3}}}{+}{k}_{{3}}({q}_{{a}_{{2}}}{-}{q}_{{a}_{{3}}} ){+}{d}_{{3}}(\dot{q}_{{a}_{{2}}}{-}\dot{q}_{{a}_{{3}}} ) \end{array} $$
(20)

By factorization of system states (5), it yields

$$\begin{array}{@{}rcl@{}}\ddot{q}_{{m}}&=&-\left[ \frac{{7500}\left({q}_{{m}}^{{2}}{+3}{q}_{{a}_{{1}}}^{{2}} \right){+10}}{{J}_{{m}}} \right]{q}_{{m}}{+}\left[ \frac{{7500}\left({q}_{{a}_{{1}}}^{{2}}{+3}{q}_{{m}}^{{2}} \right){+10}}{{J}_{{m}}} \right]{q}_{{a}_{{1}}}\\ &&{-}\left[ \frac{{f}_{{m}}{+}{d}_{{1}}}{{J}_{{m}}} \right]\dot{q}_{{m}}{+}{d}_{{1}}\dot{q}_{{a}_{{1}}}{+\mathit{w}+}{u}_{{m}} \end{array} $$
$$\begin{array}{@{}rcl@{}} \ddot{q}_{{a}_{{1}}}&=&\left[ \frac{{7500}\left({q}_{{m}}^{{2}}{+3}{q}_{{a}_{{1}}}^{{2}} \right){+10}}{{J}_{{a}_{{1}}}} \right]{q}_{{m}}{-}\left[ \frac{{7500}\left({q}_{{a}_{{1}}}^{{2}}{+3}{q}_{{m}}^{{2}} \right){+10+}{k}_{{2}}}{{J}_{{a}_{{1}}}} \right]{q}_{{a}_{{1}}}\\ &&{+}\frac{{k}_{{2}}}{{J}_{{a}_{{1}}}}{q}_{{a}_{{2}}}{++}\frac{{d}_{{1}}}{{J}_{{a}_{{1}}}}\dot{q}_{{m}}{-}\left[ \frac{{f}_{{a}_{{1}}}{+}{d}_{{1}}{+}{d}_{{2}}}{{J}_{{a}_{{1}}}} \right]\dot{q}_{{a}_{{1}}}{+}\frac{{d}_{{2}}}{{J}_{{a}_{{1}}}}\dot{q}_{{a}_{{2}}}\\ \ddot{q}_{{a}_{{2}}}&{=}&\frac{{k}_{{2}}}{{J}_{{a}_{{2}}}}{q}_{{a}_{{1}}}{-}\frac{\left({k}_{{2}}{+}{k}_{{3}} \right)}{{J}_{{a}_{{2}}}}{q}_{{a}_{{2}}}{+}\frac{{k}_{{3}}}{{J}_{{a}_{{2}}}}{q}_{{a}_{{3}}}{+}\frac{{d}_{{2}}}{{J}_{{a}_{{2}}}}\dot{q}_{{a}_{{1}}}{-}\frac{({d}_{{2}}{+}{d}_{{3}}{+}{f}_{{a}_{{2}}} )}{{J}_{{a}_{{2}}}}\dot{q}_{{a}_{{2}}}{+}\frac{{d}_{{3}}}{{J}_{{a}_{{2}}}}\dot{q}_{{a}_{{3}}} \\ \ddot{q}_{{a}_{{3}}}&{=}&\frac{{k}_{{3}}}{{J}_{{a}_{{3}}}}{q}_{{a}_{{2}}}{-}\frac{{k}_{{3}}}{{J}_{{a}_{{3}}}}{q}_{{a}_{{3}}}{+}{d}_{{3}}\dot{q}_{{a}_{{2}}}{-}\frac{({d}_{{3}}{+}{f}_{{a}_{{3}}})}{{J}_{{a}_{{3}}}}\dot{q}_{{a}_{{3}}}{+\mathit{v}} \end{array} $$
(21)

States of system \(\underline {x}\) and nonlinearity terms R 1 R 2 can be expressed as follows:

$$\begin{array}{@{}rcl@{}} \underline{{x}}&{=}&[ {\begin{array}{*{20}c} {q}_{{m}} & {q}_{{a}_{{1}}} & {q}_{{a}_{{2}}} & {q}_{{a}_{{3}}} & \dot{q}_{{m}} & \dot{q}_{{a}_{{1}}} & \dot{q}_{{a}_{{2}}} & \dot{q}_{{a}_{{3}}} \end{array} } ]^{{T}}\\ {R}_{{1}}&{=}&{q}_{{m}}^{{2}}{+3}{q}_{{a}_{{1}}}^{{2}}{=}{x}_{{1}}^{{2}}{+3}{x}_{{2}}^{{2}}\\ {R}_{{2}}&{=}&3{q}_{{m}}^{{2}}{+}{q}_{{a}_{{1}}}^{{2}}{=}{{3}{x}_{{1}}}^{{2}}{+}{x}_{{2}}^{{2}} \end{array} $$
(22)

According to (20) and (21) the state-space quasi-linear model can be concluded

$$\begin{array}{@{}rcl@{}} \dot{x}_{{1}}&{=}&{x}_{{5}}\\ \dot{x}_{{2}}&{=}&{x}_{{6}}\\ \dot{x}_{{3}}&{=}&{x}_{{7}}\\ \dot{x}_{{4}}&{=}&{x}_{{8}}\\\dot{x}_{{5}}&{=}&-\left[ \frac{{7500}{R}_{{1}}{+10}}{{J}_{{m}}} \right]{x}_{{1}}{+}\left[ \frac{{7500}{R}_{{2}}{+10}}{{J}_{{m}}} \right]{x}_{{2}}{-}\left[ \frac{{f}_{{m}}{+}{d}_{{1}}}{{J}_{{m}}} \right]{x}_{{5}}{+}{d}_{{1}}{x}_{{6}}{+w+}{u}_{{m}}\\ \dot{x}_{{6}}&{=}&\left[ \frac{{7500}{R}_{{1}}{+10}}{{J}_{{a}_{{1}}}} \right]{x}_{{1}}{-}\left[ \frac{{7500}{R}_{{2}}{+10+}{k}_{{2}}}{{J}_{{a}_{{1}}}} \right]{x}_{{2}}{+}\frac{{k}_{{2}}}{{J}_{{a}_{{1}}}}{x}_{{3}}{++}\frac{{d}_{{1}}}{{J}_{{a}_{{1}}}}{x}_{{5}}\\ &&{-}\left[ \frac{{f}_{{a}_{{1}}}{+}{d}_{{1}}{+}{d}_{{2}}}{{J}_{{a}_{{1}}}} \right]{x}_{{6}}{+}\frac{{d}_{{2}}}{{J}_{{a}_{{1}}}}{x}_{{7}}\\ \dot{x}_{{7}}&{=}&\frac{{k}_{{2}}}{{J}_{{a}_{{2}}}}{x}_{{2}}{-}\frac{\left({k}_{{2}}{+}{k}_{{3}} \right)}{{J}_{{a}_{{2}}}}{x}_{{3}}{+}\frac{{k}_{{3}}}{{J}_{{a}_{{2}}}}{x}_{{4}}{+}\frac{{d}_{{2}}}{{J}_{{a}_{{2}}}}{x}_{{6}}{-}\frac{({d}_{{2}}{+}{d}_{{3}}{+}{f}_{{a}_{{2}}} )}{{J}_{{a}_{{2}}}}{x}_{{7}}{+}\frac{{d}_{{3}}}{{J}_{{a}_{{2}}}}{x}_{{8}}\\ \dot{x}_{{8}}&{=}&\frac{{k}_{{3}}}{{J}_{{a}_{{3}}}}{x}_{{3}}{-}\frac{{k}_{{3}}}{{J}_{{a}_{{3}}}}{x}_{{4}}{+}{d}_{{3}}{x}_{{7}}{-}\frac{({d}_{{3}}{+}{f}_{{a}_{{3}}})}{{J}_{{a}_{{3}}}}{x}_{{8}}{+\mathit{v}} \end{array} $$
(23)

Consequently, the matrices of system (22) can be represented as

$$\begin{array}{@{}rcl@{}} &&\underline{\dot{x}}=A\underline{{x}}{+}{B}_{\mathit{w}}{\mathit{w}+}{B}_{\mathit{v}}{\mathit{v}+}{B}_{{u}}{u} \\ &&A=\left[\!\!\!\!\! {\begin{array}{*{20}c} {0}_{{4\ast 4}} & {I}_{{4}}\\ {\begin{array}{*{20}c} {-}\frac{({7500}{R}_{{1}}{+10})}{{J}_{{m}}} & \frac{({7500}{R}_{{2}}{+10})}{{J}_{{m}}} & {0} & {0}\\ \frac{({7500}{R}_{{1}}{+10})}{{J}_{{a}_{{1}}}} & {-}\frac{({k}_{{2}}{+7500}{R}_{{2}}{+10})}{{J}_{{a}_{{1}}}} & \frac{{k}_{{2}}}{{J}_{{a}_{{1}}}} & {0}\\ {0} & \frac{{k}_{{2}}}{{J}_{{a}_{{2}}}} & {-}\frac{({k}_{{2}}{+}{k}_{{3}})}{{J}_{{a}_{{2}}}} & \frac{{k}_{{3}}}{{J}_{{a}_{{2}}}}\\ {0} & {0} & \frac{{k}_{{3}}}{{J}_{{a}_{{3}}}} & {-}\frac{{k}_{{3}}}{{J}_{{a}_{{3}}}} \end{array} } & {\begin{array}{*{20}c} {-}\frac{({f}_{{m}}{+}{d}_{{1}})}{{J}_{{m}}} & \frac{{d}_{{1}}}{{J}_{{m}}} & {0} & {0}\\ \frac{{d}_{{1}}}{{J}_{{a}_{{1}}}} & {-}\frac{({d}_{{1}}{+}{d}_{{2}}{+}{f}_{{a}_{{1}}})}{{J}_{{a}_{{1}}}} & \frac{{d}_{{2}}}{{J}_{{a}_{{1}}}} & {0}\\ {0} & \frac{{d}_{{2}}}{{J}_{{a}_{{2}}}} & {-}\frac{({d}_{{2}}{+}{d}_{{3}}{+}{f}_{{a}_{{2}}})}{{J}_{{a}_{{2}}}} & \frac{{d}_{{3}}}{{J}_{{a}_{{2}}}}\\ {0} & {0} & \frac{{d}_{{3}}}{{J}_{{a}_{{3}}}} & {-}\frac{({d}_{{3}}{+}{f}_{{a}_{{3}}})}{{J}_{{a}_{{3}}}} \end{array} } \end{array} } \!\!\!\!\!\!\right] \end{array} $$
$$\begin{array}{@{}rcl@{}} B&=&\left[ {\begin{array}{*{20}c} {B}_{\mathit{w}} & {B}_{\mathit{v}} & {B}_{{u}} \end{array} } \right]{=}\left[ {\begin{array}{*{20}c} {0} & {0} & {0}\\ {0} & {0} & {0}\\ {0} & {0} & {0}\\ {0} & {0} & {0}\\ \frac{{1}}{{J}_{{m}}} & \frac{{1}}{{J}_{{m}}} & {0} \\ {0} & {0} & {0}\\ {0} & {0} & {0}\\ {0} & {0} & \frac{{1}}{{J}_{{a}_{{1}}}} \end{array} } \right]\\ \end{array} $$
(24)

Referring above matrices, nonlinearity terms R 1 and R 2 and also uncertain parameters f m , \(f_{a_{1}}\), \(f_{a_{2}}\) and \(f_{a_{3}}\) appear only in matrix A. Define an affine parameter-dependent system is defined as:

$$\begin{array}{@{}rcl@{}} &~&\left\{ {\begin{array}{*{20}c} {E}\left({\rho } \right)\dot{x}{=A}\left({\rho } \right){x+}{B}_{{1}}\left({\rho } \right){\mathit{w}+}{B}_{{2}}\left({\rho } \right){u}\\ {z}_{{\infty }}{=}{C}_{{1}}\left({\rho } \right){x+}{d}_{{11}}\left({\rho } \right){\mathit{w}+}{d}_{{12}}\left({\rho } \right){u}\\ {z}_{{2}}{=}{C}_{{2}}\left({\rho } \right){x+}{d}_{{12}}\left({\rho } \right){\mathit{w}+}{d}_{{22}}\left({\rho } \right){u} \end{array} } \right.\\ {S}({\rho })&{=}&{S}_{{0}}{+}{\rho }_{{1}}{S}_{{1}}{+\mathellipsis +}{\rho }_{{n}}{S}_{{n}}\\ &~&\left[ {\begin{array}{*{20}c} {A}\left({\rho } \right){+jE}\left({\rho } \right) & {B}\left({\rho } \right)\\ {C}\left({\rho } \right) & {D}\left({\rho } \right) \end{array} } \right]{=}\left[ {\begin{array}{*{20}c} {A}_{{0}}{+j}{E}_{{0}} & {B}_{{0}}\\ {C}_{{0}} & {D}_{{0}} \end{array} } \right]{+}{\rho }_{{1}}\left[ {\begin{array}{*{20}c} {A}_{{\rho }_{{1}}}{+j}{E}_{{\rho }_{{1}}} & {B}_{{\rho }_{{1}}}\\ {C}_{{\rho }_{{1}}} & {D}_{{\rho }_{{1}}} \end{array} } \right]\\&&{+{\ldots} +}{\rho }_{{n}}\left[ {\begin{array}{*{20}c} {A}_{{\rho }_{{n}}}{+j}{E}_{{\rho }_{{n}}} & {B}_{{\rho }_{{n}}}\\ {C}_{{\rho }_{{n}}} & {D}_{{\rho }_{{n}}} \end{array} } \right]\\ {B}\left({\rho } \right)\!\!&{=}&\!\!\left[ {\begin{array}{*{20}c} {B}_{{1}}\left({\rho } \right)\! & \!{B}_{{2}}\left({\rho } \right) \end{array} } \right],C\left({\rho } \right)\!{=}\!\left[ {\begin{array}{*{20}c} {C}_{{1}}\left({\rho } \right) & {C}_{{2}}\left({\rho } \right) \end{array} } \right]^{{T}}\!, D\left({\rho } \right){\,=\,}\left[ \!{\begin{array}{*{20}c} {d}_{{11}}\left({\rho } \right) & \!{d}_{{12}}\left({\rho } \right)\\ {d}_{{12}}\left({\rho } \right) & \!{d}_{{22}}\left({\rho } \right) \end{array} }\! \right] \end{array} $$
(25)

where S 0,S 1,…,S n are given system matrices; A(.), B(.), C(.), D(.) and E(.) are fixed affine functions of some vector ρ=(ρ 1,…,ρ n ) The parameters p i are uncertain parameters. In this paper, uncertain parameters are given by following vector:

$$ {\rho =}\left({\rho }_{{1}}{,}{\rho }_{{2}}{,}{\rho }_{{3}}{,}{\rho }_{{4}} \right){=}({f}_{{m}}{f}_{{a}_{{1}}}{f}_{{a}_{{2}}}{f}_{{a}_{{3}}}) $$
(26)

As a result, according to definition of (24) and uncertain parameters of (25), the matrix A can be decomposed as

$$\begin{array}{@{}rcl@{}} &&{A}\left({\rho } \right){=}{A}_{{0}}{+}{\rho }_{{1}}{A}_{{1}}{+}{\rho }_{{2}}{A}_{{2}}{+}{\rho }_{{3}}{A}_{{3}}{+}{\rho }_{{4}}{A}_{{4}}{=}{A}_{{0}}{+}{f}_{{m}}{A}_{{f}_{{m}}}{+}{f}_{{a}_{{1}}}{A}_{{a}_{{1}}}{+}{f}_{{a}_{{2}}}{A}_{{a}_{{2}}}{+}{f}_{{a}_{{3}}}{A}_{{a}_{{3}}}\\ &&A=\left[\!\! {\begin{array}{*{20}c} {0}_{{4\ast 4}} & {I}_{{4}}\\ {\begin{array}{*{20}c} {-}\frac{({7500}{R}_{{1}}{+10})}{{J}_{{m}}} & \frac{({7500}{R}_{{2}}{+10})}{{J}_{{m}}} & {0} & {0}\\ \frac{({7500}{R}_{{1}}{+10})}{{J}_{{a}_{{1}}}} & {-}\frac{({k}_{{2}}{+7500}{R}_{{2}}{+10})}{{J}_{{a}_{{1}}}} & \frac{{k}_{{2}}}{{J}_{{a}_{{1}}}} & {0}\\ {0} & \frac{{k}_{{2}}}{{J}_{{a}_{{2}}}} & {-}\frac{({k}_{{2}}{+}{k}_{{3}})}{{J}_{{a}_{{2}}}} & \frac{{k}_{{3}}}{{J}_{{a}_{{2}}}}\\ {0} & {0} & \frac{{k}_{{3}}}{{J}_{{a}_{{3}}}} & {-}\frac{{k}_{{3}}}{{J}_{{a}_{{3}}}} \end{array} } & {\begin{array}{*{20}c} {-}\frac{{d}_{{1}}}{{J}_{{m}}} & \frac{{d}_{{1}}}{{J}_{{m}}} & {0} & {0}\\ \frac{{d}_{{1}}}{{J}_{{a}_{{1}}}} & {-}\frac{({d}_{{1}}{+}{d}_{{2}})}{{J}_{{a}_{{1}}}} & \frac{{d}_{{2}}}{{J}_{{a}_{{1}}}} & {0}\\ {0} & \frac{{d}_{{2}}}{{J}_{{a}_{{2}}}} & {-}\frac{({d}_{{2}}{+}{d}_{{3}})}{{J}_{{a}_{{2}}}} & \frac{{d}_{{3}}}{{J}_{{a}_{{2}}}}\\ {0} & {0} & \frac{{d}_{{3}}}{{J}_{{a}_{{3}}}} & {-}\frac{{d}_{{3}}}{{J}_{{a}_{{3}}}} \end{array} } \end{array} }\!\! \right]\\ &&{A}_{{f}_{{m}}}{=}\left[ {\begin{array}{*{20}c} {0}_{{4\ast 8}}\\ {\begin{array}{*{20}c} {0} & {0} & {0} & {0} & {-}\frac{{1}}{{J}_{{m}}} & {0} & {0} & {0} \end{array} }\\ {0}_{{3\ast 8}} \end{array} } \right],\qquad{A}_{{a}_{{1}}}{=}\left[ {\begin{array}{*{20}c} {0}_{{5\ast 8}}\\ {\begin{array}{*{20}c} {0} & {0} & {0} & {0} & {0} & {-}\frac{{1}}{{J}_{{a}_{{1}}}} & {0} & {0} \end{array} }\\ {0}_{{2\ast 8}} \end{array} } \right]\\ &&{A}_{{a}_{{2}}}{=}\left[ {\begin{array}{*{20}c} {0}_{{6\ast 8}}\\ {\begin{array}{*{20}c} {0} & {0} & {0} & {0} & {0} & {0} & {-}\frac{{1}}{{J}_{{a}_{{2}}}} & {0} \end{array} }\\ {0}_{{1\ast 8}} \end{array} } \right]{,}\qquad{A}_{{a}_{{3}}}{=}\left[ {\begin{array}{*{20}c} {0}_{{7\ast 8}}\\ {\begin{array}{*{20}c} {0} & {0} & {0} & {0} & {0} & {0} & {0} & {-}\frac{{1}}{{J}_{{a}_{{3}}}} \end{array} } \end{array} } \right] \end{array} $$
(27)

This completes the subsection 2.2 (Proposed T-S fuzzy model) of section 2 (Model Description of ABB manipulator IRB6600).

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AZIMI, V., MENHAJ, M.B. & FAKHARIAN, A. Tool position tracking control of a nonlinear uncertain flexible robot manipulator by using robust H2/H∞ controller via T–S fuzzy model. Sadhana 40, 307–333 (2015). https://doi.org/10.1007/s12046-015-0354-x

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