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Single-loop PID controller design for electrical flexible-joint robots

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Abstract

Most of previous approaches presented for electrical flexible-joint robots (EFJR) utilize back-stepping-based control strategy. In order to have a satisfactory performance in these approaches, the internal signals should converge to their desired trajectories defined by the designer. Usually, these desired trajectories are known as fictitious control signals. In EFJR, each joint is modeled by a fifth-order cascade differential equation. Thus, back-stepping-based approach seems complicated and time-consuming. Thus, the contribution of this paper is focusing on the convergence of the system output and meanwhile guaranteeing boundedness of other system’s states. As a result, the control law dimension and its implementation costs are reduced. In other words, a single-loop PID controller for EFJR has been presented, while previous approaches contain multi-loop controllers. The controller design strategy is based on the actuators’ electrical subsystem. Voltage saturation nonlinearity has been compensated in the control law. Hence, knowledge of the actuator/manipulator dynamics model is not required in the proposed method. The overall closed-loop controlled system is established as BIBO stable, and the link position-tracking errors are asymptotically stable based on the Lyapunov’s stability concept. Numerical and experimental implementations support the viability of the proposed theoretical results.

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Abbreviations

\( A \) :

State transition matrix

\( B \) :

Friction coefficient

\( C(q,\dot{q})\dot{q} \) :

Centrifugal and Coriolis torques

\( D(q) \) :

Inertia matrix

\( D_{t}^{\beta } \) :

Fractional derivative with order \( \beta \)

\( {\text{dzn}}() \) :

Dead-zone function

\( e(t) \) :

Tracking error

\( F_{s} \) :

Coulomb friction

\( F_{d} \) :

Viscous friction coefficient

\( g(q) \) :

Gravitational torques

\( I_{\text{a}} \) :

Armature current

\( I_{n} \) :

Identity matrix

\( J \) :

Actuator inertia

\( K \) :

Joint stiffness

\( K_{\text{b}} \) :

Back-emf coefficient

\( K_{\text{d}} \) :

Derivative gain

\( K_{\text{I}} \) :

Integrator gain

\( K_{m} \) :

Torque constant

\( \left\| \cdot \right\| \) :

Euclidian norm

\( K_{\text{p}} \) :

Proportional gain

L :

Inductance

\( n \) :

Manipulator degree of freedom

\( q \) :

Joint angular position

\( P, Q \) :

Positive definite matrix

\( R \) :

Electrical resistance

\( \Re^{n} \) :

n-dimensional real space

\( r \) :

Gear ratio

\( {\text{sat}}() \) :

Saturation function

\( u(t) \) :

Controller output

\( u_{\text{r}} (t) \) :

Robust control term

\( V_{1} \) :

Lyapunov candidate for integer-order controller

\( V_{2} \) :

Lyapunov candidate for fractional-order controller

\( v(t) \) :

Motor voltage

\( \theta_{\text{m}} \) :

Motor angular position

\( \rho \) :

Norm of uncertainty

\( \rho_{m} \) :

Uncertainty upper bound

\( \varGamma (n) \) :

Gamma function

\( \mu \) :

Convergence rate

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Authors and Affiliations

  1. Department of Electrical Engineering, Garmsar Branch, Islamic Azad University, Garmsar, Iran

    Alireza Izadbakhsh

  2. Department of Electrical and Computer Engineering, University of Birjand, Birjand, Iran

    Saeed Khorashadizadeh

Authors
  1. Alireza Izadbakhsh
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  2. Saeed Khorashadizadeh
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Correspondence to Saeed Khorashadizadeh.

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Technical Editor: Adriano Almeida Gonçalves Siqueira.

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Izadbakhsh, A., Khorashadizadeh, S. Single-loop PID controller design for electrical flexible-joint robots. J Braz. Soc. Mech. Sci. Eng. 42, 91 (2020). https://doi.org/10.1007/s40430-020-2172-2

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