Recurrent fuzzy wavelet neural network variable impedance control of robotic manipulators with fuzzy gain dynamic surface in an unknown varied environment

Abstract

In this paper, an intelligent variable impedance control combined with a fuzzy gain dynamic surface is proposed to improve the interaction of the robot manipulator with an unknown varied environment. The parameters of the proposed variable impedance are adapted by optimization an introduced cost function using a recurrent fuzzy wavelet network. The stability conditions for the varying inertial, stiffness and damping are presented to guarantee the stability of the variable impedance. Additionally, a fuzzy dynamic surface method is developed to tune the gains of the dynamic surface as a robust controller. The proposed fuzzy gain dynamic surface is used to force the end-effector of the manipulator to track the desired impedance profile in the presence of large disturbances. Using Lyapunov's method, the stability of the mentioned closed-loop system is proved. Finally, by using a designed simulator for IRB120 (ABB) robot, several simulations are carried out to verify the performance of the proposed method for the execution of various tasks in an unknown varied environment in the presence of large disturbances.

Introduction

Impedance control is a classical control method to realize compliant behavior in the robotic manipulators. Hogan in [1] introduced the main concept of the impedance controller that nowadays is considered a main compliant controller. The main statement of the impedance control is that the end-effector should modulate its behavior according to interaction with the environment. When a robot interacts with an unknown varied environment, the impedance parameters are necessary to be tuned in order to perform various tasks. In fact, the variable impedance control gives the robustness and the flexibility to change impedance and dynamic interaction of the end-effector in a continuous manner during task execution. In the following introduction section, we considered two procedures. First, we study the variable impedance control and its challenges. Additionally, we present the intelligent variable impedance method as one of the main continuations of our research. Then, we study the dynamic surface as a nonlinear controller used to force the robot manipulator to track the variable impedance profile. To present a robust controller in this work, we propose the fuzzy gain dynamic surface to observe and eliminate the uncertainties and disturbances.

There are several works that have addressed the variable impedance methods. In [2], [3], [4], variable impedance control is accomplished by using reinforcement learning methods. In [5], a novel design and fabrication methodologies are addressed to produce cost effective hardware prototypes for investigating the efficacy of the variable impedance approach. In [6], an adaptive impedance control approach is proposed to adjust the stiffness on-line using the tracking error of the motion trajectory. In [7], a fuzzy approach is used to generate the impedance gain to guarantee a certain contact force on the contour of the fixed environment. In [8], the impedance parameters are modulated on-line according to human behavior during the interaction. In [9], an iterative learning control method is proposed for impedance control robotics tasks when the endpoint of soft tool interacts with a rigid object or environment at a certain periodic force pattern. Seraji et al. [10] have used the direct and indirect adaptive method to generate the reference position and the estimated parameters of the unknown environment.

However, only a limited number of the works have addressed the stability conditions for the variable impedance profile. In fact, the impedance control with constant gains ensures the stability of the closed-loop system, and it makes a stable behavior of the system both in free motion and in case of interaction with passive environments [11], [12]. However, stability is lost if the impedance parameters are considered time varying. The authors in [13] have analyzed the stability conditions for varying stiffness and damping. In [14], a switched stiffness approach is presented for vibration control. Ferraguti et al. [15] presented the tank-based approach to guarantee the system passivity for time varying stiffness. Selvaggio et al. [16] proposed an online technique to adapt the pose and geometry of virtual fixtures and used the tank-theory to guarantee the stability of the systems. In [17], a multi-layer perceptron network was used to estimate and regulate the impedance parameters without considering the stability of the impedance profile. Therefore, we have to present a stability condition for the variable impedance control.

In this work, an adaptive control method is proposed to optimize a cost function using Recurrent Fuzzy Wavelet Neural Network (RFWNN). In fact, the presented method is an online learning approach to tune the impedance parameters of the robot manipulator. According to the task requirements such as convergence to the desired environment without large overshooting or with an acceptable desired force tracking, the cost function is defined to improve the interaction of the robot manipulator. It is worth to mention that, RFWNNs have both the properties of the fuzzy logic and wavelet neural networks. In fact, the combination of the fuzzy mechanism as an effective solution for dealing with a complex nonlinear process and wavelet neural network as an approach for estimation of the local changes makes a powerful adaptive method to model the nonlinear dynamics [18], [19], [20]. Additionally, the recurrent property of the RFWNN is able to store the past information of the network and use it to estimate the model of sudden jumps of the dynamical system. In addition, a stabilizer coefficient is proposed to guarantee the stability of the variable impedance. Therefore, the presented approach leads to improve the stability and robustness of the variable impedance profile while the robot interacts with a varied and unknown environment.

Because of the nonlinear dynamics of the robot manipulators, it is necessary to use an effective nonlinear control such as sliding mode control [21], feedback linearization [22] or the backstepping control [23], [24] to force the robot manipulator to track the generated impedance profile. In [25], an event-triggered control method is developed using backstepping approach and according to a nonlinear decomposition of input quantization. Additionally, Two methods can be considered including force impedance control or position-based impedance control [26], [27]. In our work, we present a fuzzy approach to tune the gains of the dynamic surface as a robust force impedance control.

There are several works addressed disturbances and saturation problems in the control systems [28], [29], [30], [31]. In [32], an interval type-2 fuzzy approach is used to detect the fault in the sensor saturation for continues-time fuzzy semi-Markov jump [33] systems. With respect to the previous works, the presented fuzzy approach, as a simple structure, is able to remove the external disturbances to handle saturation and to improve the response of the actuators. Furthermore, the dynamic surface control technique can solve the problem of the “explosion of complexity” caused by repeated differentiation of virtual controllers during the recursive procedures in the back-stepping method [34], [35]. The dynamic surface solves this problem by using a low-pass filter of the synthetic virtual control law at each step of the back-stepping procedure [34]. Several works have addressed the dynamic surface control using adaptive methods in the robotic systems [36], [37], [38], [39]. However, in our work, the gains of the dynamic surface are adapted with respect to the presence of uncertainty and disturbances using the fuzzy approach. Therefore, the simple presented method leads to improve the robustness of the robot manipulators. Additionally, we used the dynamic surface methods as a novel application of this approach to obtain a simple structure, the target impedance is formulated in the state space, and effective dynamic surfaces are defined to track the desired impedance behavior. In fact, dynamic surface control is used to force the robot manipulator to track the desired impedance, while the robot interacts with an environment.

According to the above discussion, this paper presents an intelligent and robust impedance control of the robot manipulators to interact with the unknown varied environment. According to Fig. 1, the proposed controller includes two closed loops. The outer loop is related to the intelligent variable impedance and the inner loop is considered for the fuzzy gain dynamic surface. Using the Lyapunov method, the stability conditions are induced to guarantee the bounded signals in the closed-loop system. The contributions of this paper are summarized as follows:

• An adaptive control method is proposed to optimize a cost function based on the recurrent fuzzy wavelet neural network as an online learning approach to tune the impedance parameters of the robot manipulator. In the proposed approach the impedance parameters including inertial, damping and stiffens are considered as only one parameter and are estimated by the RFWNN. It leads to decrease the computational complexity. Additionally, the introduced online learning approach improves the interaction of the robot with the environment and effectively reduces the contact force error with an acceptable transient response.

• To solve the large external disturbances, we have designed an effective impedance controller based on the Fuzzy Gain Dynamic Surface (FGDS). As a novel application of the dynamic surface method to obtain a simple structure, which the target impedance is formulated in the state space, and effective dynamic surfaces are defined to track the desired impedance behavior. Additionally, FGDS is able to estimate the unknown uncertainties and disturbances of the robotic system. In fact, introduced dynamic surface control in combination with the fuzzy rules is able to force the end-effector of the robot manipulator to track the desired impedance model.

The rest of this paper is organized as follows: In section 2, the system dynamics of the robot manipulators and the variable impedance are presented. Section 3 proposes the intelligent variable impedance control and fuzzy gain dynamic surface. Additionally, the stability conditions are analyzed by Lyapunov stability theory. Section 4 shows the performance of the proposed approach using the designed simulator for the IRB120 ABB robot, and the last section concludes with a short summary of the paper.