Elsevier

Neurocomputing

Volume 171, 1 January 2016, Pages 209-219
Neurocomputing

Neurofuzzy self-tuning of the dissipation rate gain for model-free force-position exponential tracking of robots

https://doi.org/10.1016/j.neucom.2015.06.034Get rights and content

Abstract

Simultaneous force and position control of robots interacting with a rigid environment has been broadly studied assuming several contact force models, being the differential algebraic – DAE – model the most complete one, however DAE robots show complex and strong nonlinear couplings that make difficult to achieve tracking, when dynamic model is not available. In this paper, considering the fundamental structural properties of DAE robots, in particular passivity and the orthogonalization of force and velocity vectors, it is proposed a model-free neurofuzzy-based self-tuning robot controller for exponential tracking, which is composed of orthogonalized PID-position plus an I-force (If) control terms, and a feed-forward desired force term. The salient feature of this proposal is a novel neurofuzzy self-tuning scheme aimed at tuning the dissipation rate gain (DRG) so as to enforce dissipativity in closed-loop, rather than the standard scheme of tuning the feedback control gains, or the control structure, which in our case stands for a simple constant feedback gain orthogonalized PID+If controller. In fact, in virtue of such orthogonalization, it emerges a simple and low cost parallel structure of the neuro-fuzzy network that targets solely the DRG so as to drive error dynamics to zero with exponential rate, without any knowledge of robot dynamics or carrying out any approximation of inverse dynamics. Thus, this technique can be applied to other class of systems and controllers that ensure passivity in closed-loop. Simulations show the validity and the feasibility of this new approach.

Introduction

Neurofuzzy systems (NFS) have proved effective for various applications, see [1] for a recent review, with versatile roles as approximator [2], classifier [3], [4], state estimation [4], system identification [5], to mention a few. For control purposes, it is used typically as a tuning scheme of feedback gains to stabilize the system with a simple regulator structure [6], assuming partial knowledge of a robot [7], or for multiple robots [8]. While NFS stands for an attractive control alternative that includes as principal feature the powerful learning tool to extract knowledge of the plant [9], typically, in synergy with an stabilizer regulator input [10], it outperforms fixed gain schemes due to its ability to introduce knowledge-based tuning with closed-loop stability analysis [11], [12].

Nevertheless, there exits three issues that, we believe, have limited the scope of NFS-based control for certain nonlinear plants, in particular for robots. Firstly, the control design is usually based on the state space representation, requiring stringent assumptions on the input matrix [13]; a problem that can be alleviated if the controller were designed for the Euler Lagrange model representation, where input matrix is the identity [14]. Secondly, Euler Lagrange robots exhibit some structural properties useful for control purposes, in particular passivity [15], which can be useful to improve the stability of the system if is included in the fuzzy-based design [16]. Thirdly, NFS-based control is usually aimed at a model-free control scheme where stability is generally taken for granted in most publications, either because of the approximation capabilities of inverse dynamics or by ignoring the flow of the state space representation [9].

Notwithstanding, closed-loop stability analysis is a requirement in any controller of a physical system, more importantly when the plant is subject to constraints, in particular for a robot constrained by a rigid environment. In such constrained robot, the characteristic of critically damped response is desirable because the system is governed by very stiff differential algebraic equations of index 2, or DAE [17]. Such response can be achieved if exponential convergence is assured, however exponential tracking without neglecting major dominant dynamic forces is difficult to obtain with NFS. In particular the control design of this type of highly nonlinear constrained plants is exacerbated if structural properties are not considered; indeed, the force-position exponential tracking control problem for a DAE robot using a model-free smooth controller remains an open problem in the literature.

In contrast to the usual notion of self-tuning of the feedback control gains of the controller [18], and/or the weights of the function approximation scheme [9], in this paper we propose a new NFS-based control design for synthesizing a self-tuning mechanism. The idea is to use the characteristic of NFS to extract knowledge of the plant that enables a self-tuning neurofuzzy mechanism, the so-called dissipation rate gain (DRG), through minimizing exponentially a quadratic cost function of the closed-loop error variable. To this end, we design a novel model-free orthogonalized PID-like controller with constant gains that renders a passive (stable) closed loop system. This result leads subsequently to a closed-loop energy balance that becomes dissipative (asymptotically stable). Thus, exponential stability is guaranteed, assuming only full access to the state of the robot as well as the environment kinematics, and without any inverse dynamics approximation. Notice that this approach extends the fundamental result of [19], based on [20], for free motion robots to constrained robots, which is formalized in [21] in terms of qISS (quasi Input to State Stable). This paper is organized as follows. Section 2 summarizes briefly the background on NFS-based robot force control. The robot dynamics is introduced in Section 3 together with the proposed open-loop error equation, and the robot controller design with constant feedback gains is addressed in Section 4. The self-tuning mechanism of the DRG is synthesized and its stability analysis is presented in Section 5, while the main result is given in Section 6, including its closed-loop stability analysis. Numerical validation and feasibility of our approach are discussed in Section 7, with final remarks addressed in Section 8.

Section snippets

Background of NFS-based robot control

It is a common practice to address the fuzzy controller based on its function approximation capabilities to compensate the dynamics, such that an approximated computed torque can be implemented without any knowledge of robot dynamics [13], neither assuming the regressor [22] nor partial knowledge of the plant, for inverse dynamic compensation. Thus, the residual unknown dynamics can be compensated using simple regulators for improved stability properties [18], or approximating the residual

Constrained dynamic model

Consider a robot end-effector exerting a given profile of force onto a rigid surface modeled by a holonomic constraint φ(q)=0:RnRr, for r independent contact points. The robot solution is constrained to evolve all the time in the implicit function φ(q)=0C2, and its derivatives, which describes the infinitely rigid surface of the environment. In this way, the constrained dynamic model of a robot manipulator can be represented by Differential Algebraic system Equations of index-2, DAE [17], [29]

Control design

Let the smooth model-free control law beτ=KdSJφT(q)(λdηΔF)where Kd,η are positive constant gains. Substituting (8) into (5), the following closed-loop equation becomes:H(q)Ṡ+C(q,q̇)S=KdS+JφT(q)(Δλ+ηΔF)YrΘWe have now the following result.

Proposition 1

Consider the model-free controller (8) into the constrained robot dynamics (5), then the closed-loop system (9) is locally stable for large enough Kd under small error on initial conditions, with all closed-loop signals bounded.

Proof

Consider the following

Fuzzy self-tuning scheme

Let a neuro-fuzzy algorithm be characterized in two stages, see Fig. 2. In the first stage, fuzzy logic rules are designed based on error manifold S(k)1 whose output is processed in the second stage with a neural network to obtain the corresponding value of K^d(k). The fuzzy inference system is represented by IF–THEN rules, based on the input signal S(k)Rn, with n inputs and r fuzzy

Main result

We have now the following result, see its structure in Fig. 4.

Theorem

Consider constrained robot dynamics (1), (2) in closed-loop with control (8) and Proposition 1. If K^d(k) is tuned according to (21), using (30), (33), then there exists a domain D1D0 such that local exponential convergence of S(k) is guaranteed. Consequently, the local tracking to desired trajectories is secured such that q(k)qd(k),q̇(k)q̇d(k),λ(k)λd(k) as k.

Proof

Let the following Lyapunov candidate function beV1(k)=12ST(k)S(k),

Simulations

Representative simulations for a 2-DOF nonlinear rigid arm in contact to a rigid environment are presented. The desired task is defined as a robot moving along the wall while a force is applied on it. As first case, we will assume that the force applied by the robot to the wall is constant while in the second case the force is a time-varying sinusoidal signal, see Fig. 5. The simulator is programmed in Matlab® using a stiff numerical solver tb1, with a sampling time of 1 ms, where the initial

Conclusions

A new model-free neurofuzzy controller aimed at tuning of the dissipation rate gain is proposed to enforce a dissipative mapping of the closed-loop dynamics that drives the surface error to zero. In this way, this scheme guarantees tracking of constrained robot dynamics by reshaping the closed-loop energy balance rather than tuning feedback control gains or inverse dynamics approximation, using constant feedback gains of an orthogonalized regulator PID+I. The rationale behind this self-tuning

Acknowledgments

This work was partially supported by CONICYT, Departamento de Relaciones Internacionales “Programa de Cooperación Científica Internacional” CONICYT/CONACYT Grant 2011-380, and by CONACYT of Mexico under Grants 133346 and 174597.

Vicente Parra-Vega received the B.Eng. degree in control and computing and the B.Eng. degree in electronics and communications, both from the Nuevo León University, Nuevo León, México, in 1987, the M.Sc. degree in automatic control from the Research Center for Advanced Studies (CINVESTAV), San Pedro Zacatenco, México, in 1989, and the Ph.D. degree in electrical engineering from the Mathematical Engineering and Information Physics Department of the University of Tokyo, Tokyo, Japan, in 1995,

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    Vicente Parra-Vega received the B.Eng. degree in control and computing and the B.Eng. degree in electronics and communications, both from the Nuevo León University, Nuevo León, México, in 1987, the M.Sc. degree in automatic control from the Research Center for Advanced Studies (CINVESTAV), San Pedro Zacatenco, México, in 1989, and the Ph.D. degree in electrical engineering from the Mathematical Engineering and Information Physics Department of the University of Tokyo, Tokyo, Japan, in 1995, under the supervision of Prof. S. Arimoto. Currently, he is a Full Professor at the CINVESTAV. His current research interests include collaborative multirobots, high-precision servosystems, haptic interfaces, control theory, visual servoing, teleoperators, high-speed CNC, and unmanned aerial and submarine robots. He is the leading cofounder of two new postgraduate Programmes in Mexico, the first one in Mechatronics and the second one in Robotics and Advanced Manufacturing. He is a Regular Member of the Mexican Academy of Science, the National Researcher System, and from 1997, he has participated in several committees of the Mexican Council for Science and Technology.

    Rodolfo García-Rodríguez has a B.S. degree in Industrial Electronic Engineering from the Instituto Tecnológico de Puebla in 1997. He received the Ph.D. and M.S. degrees from the Research Center for Advanced Studies (CINVESTAV) in 2005 and 2002, respectively. From 1998 to 2000 he was with Gates Rubber de México as an Engineer in the Engineering Department. Currently, he is a Professor at Universidad de los Andes, Chile in the Facultad de Ingeniería y Ciencias Aplicadas. His research interests include robotic hands, control theory and mechatronic systems.

    Jorge Armendariz received his B.Eng. degree in Advanced Manufacturing, from the Chihuahua Institute of Technology, Chihuahua, México, in 2008, and the M.Sc. degree in Robotics and Advanced Manufacturing from the Research Center for Advanced Studies (CINVESTAV), Coahuila, México, in 2010. Currently, he is working as specialist in Manufacturing Engineering at HELLA automotive. His current research interests include robot control, nonlinear and neuro-fuzzy control, fuzzy systems applied to robotics and automation, manufacturing, and automation technology for assembly systems.

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