Complexity

Volume 2017 (2017), Article ID 7683785, 14 pages

https://doi.org/10.1155/2017/7683785

## Neural Learning Control of Flexible Joint Manipulator with Predefined Tracking Performance and Application to Baxter Robot

School of Automation Science and Engineering, Guangzhou Key Laboratory of Brain Computer Interaction and Applications, South China University of Technology, Guangzhou 510641, China

Correspondence should be addressed to Min Wang

Received 20 July 2017; Accepted 14 September 2017; Published 31 October 2017

Academic Editor: Yanan Li

Copyright © 2017 Min Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper focuses on neural learning from adaptive neural control (ANC) for a class of flexible joint manipulator under the output tracking constraint. To facilitate the design, a new transformed function is introduced to convert the constrained tracking error into unconstrained error variable. Then, a novel adaptive neural dynamic surface control scheme is proposed by combining the neural universal approximation. The proposed control scheme not only decreases the dimension of neural inputs but also reduces the number of neural approximators. Moreover, it can be verified that all the closed-loop signals are uniformly ultimately bounded and the constrained tracking error converges to a small neighborhood around zero in a finite time. Particularly, the reduction of the number of neural input variables simplifies the verification of persistent excitation (PE) condition for neural networks (NNs). Subsequently, the proposed ANC scheme is verified recursively to be capable of acquiring and storing knowledge of unknown system dynamics in constant neural weights. By reusing the stored knowledge, a neural learning controller is developed for better control performance. Simulation results on a single-link flexible joint manipulator and experiment results on Baxter robot are given to illustrate the effectiveness of the proposed scheme.

#### 1. Introduction

Due to the great demands in industrial applications, the tracking control problem for flexible joint robot (FJR) manipulator has attracted much attention in recent years. Unlike rigid joint robot, the joint flexibility of FJR results in complex control situation, so that the control problem of FJR becomes much more difficult. In the past few decades, lots of efforts have been made on the research of FJR systems. Based on the model of FJR presented in [1], multifarious nonlinear control methods are presented such as backstepping method [2–4], sliding-mode control [5–8], switching control [9], fuzzy control [10], and neural network control [11, 12]. In consideration of the problem caused by the inherent structure of FJR under practical circumstance, such as friction, time delay, and variable stiffness, some researchers proposed effective strategies to solve such problem [13–15]. Moreover, the teleoperation control method is also widely used in robot research [16, 17].

The backstepping control [18] is known as one of the popular method for designing the control scheme of FJR. Nevertheless, it should be pointed out that this method has a drawback called “explosion of complexity” [19]. This problem generally occurs in the design of neural networks (NNs) during backstepping procedure. To overcome this problem, some researchers used the intermediate variables as neural inputs to reduce the dimension of neural network input vector [20]. The method in [20] did work well but the problem remained unsolved. Then other researchers proposed a dynamic surface control (DSC) method by introducing a first-order filter at each step of the backstepping procedure [21]. Due to the property of DSC method, many researchers presented their control schemes combined with DSC method [22–27]. In [24], a new robust output feedback control approach for flexible joint electrically driven robots via the observer-based dynamic surface method was proposed, which only requires position measurement of the system.

Besides, the transient and steady-state tracking performance constraints of system’s output are an important issue that needs to be taken into consideration [28, 29]. According to the practical operating environment, the manipulator is not only demanded to trace the reference trajectory accurately but also required to keep the tracking error within a specified range. To satisfy this condition, a performance function transformation was used to convert the “constrained” system into the “unconstrained” one [30]. Based on the idea in [30], further researches on prescribed performance for a variety of systems are proposed [31–36]. Authors in [31, 32] presented novel controllers for FJRs to achieve tracking control of link angles with any prescribed performance requirements. By combining neural learning control scheme, further results are given in [33–35]. In [36], an adaptive prescribed performance tracking control scheme is investigated for a class of output feedback nonlinear systems with input unmodeled dynamics based on dynamic surface control method.

In addition, adaptive neural control of nonlinear system has been widely studied for decades, but most of the traditional works focus on the system stability through online adjustment of neural weights and less works discuss the knowledge acquisition, storage, and utilization of optimal neural weights. To achieve such learning ability, the key problem is to verify the persistent excitation (PE) condition. Thanks to the results in [37], a deterministic learning mechanism is proposed, which proved the satisfaction of PE condition for the localized radial basis function (RBF) NN centered in a neighborhood along recurrent orbits. The result was extended to nonlinear systems satisfying matching conditions [38–40]. By combining recursive design technologies such as backstepping control and the system decomposition strategy, the deterministic learning was also applied to solve learning problem of accurate identification of ocean surface ship and robot manipulation in uncertain dynamical environments [41–44]. However, due to the recursive property of backstepping control, the convergence of neural weights has to be recursively verified based on the system decomposition strategy. It would be a tedious and complex process since the intermediate variables grow drastically as the order of system increases. Therefore, it is difficult to prove all neural weights convergence for high-order system by existing works.

This paper focuses on learning from adaptive neural control of flexible joint manipulator with unknown dynamics under the prescribed constraints. A performance function is introduced to transform the constrained tracking error into the unconstrained variable. To avoid the curse of dimensionality of RBF NN, first-order filters are introduced to reduce the number of NN approximators and decrease the dimension of NN inputs. The control law is constructed based on Lyapunov stability, which guarantees the closed-loop stability and the tracking error satisfying the prescribed performance during the transient process. Subsequently, due to the property of DSC and structure features of the considered manipulator, a system decomposition strategy is employed to decompose the stable closed-loop system into two linear time-varying (LTV) perturbed subsystems on the basis of the number of NNs in the whole system. Through the recursive design, the recurrent properties of NN input variables are easily proven. Consequently, with the satisfaction of the PE condition of RBF NNs, the convergence of partial neural weights is verified, and the unknown dynamics of system are approximated accurately in a local region along recurrent orbits. By utilization of the constant neural weights stored, a neural learning controller is developed to achieve the closed-loop stability and better control performance under the prescribed constraints for the same or similar control task. Compared with the existing neural learning results, the proposed neural learning control scheme not only achieves better control performance with specified transient and steady-state constraints but also reduces the dimension of NN inputs and the number of NNs significantly.

This paper is organized as follows. In Section 2, the problem formulation and preliminaries are stated before the control scheme design. In Section 3, a novel adaptive neural dynamic surface control scheme is proposed to guarantee that the constrained tracking error converges to a small neighborhood around zero with the prescribed performance in a finite time, and all the signals in the closed-loop system are uniformly ultimately bounded. Section 4 shows that the knowledge acquisition, expression, storage, and utilization of the manipulator’s unknown dynamics can be achieved after the steady-state control process. To verify the effectiveness of the proposed control scheme, simulation results on a single-link flexible joint manipulator and experiment results on Baxter robot are given in Section 5. Last but not least, the conclusions are drawn in Section 6.

#### 2. Problem Formulation and Preliminaries

##### 2.1. System Formulation

In this paper, we consider an -link manipulator with flexible joints, whose model is described by [1]where is the vector of links’ angle positions and is the vector of motors’ angle positions. is the link inertia matrix and is the diagonal and positive definite motor inertia matrix. Moreover, denotes the Coriolis and centrifugal matrix and represents the gravitational terms. is a diagonal and positive definite matrix of joint spring constants; thus is also positive definite. Finally, is the control input of system (1), and the output of system (1) is .

*Property 1 (see [2]). *The inertia matrix is symmetric and positive definite; both and are uniformly bounded.

*Property 2 (see [2]). *The Coriolis and centrifugal matrix can be defined such that is skew symmetric; that is, .

The reference trajectory vector is generated by the following smooth and bounded reference model:where and is the system outputs vector. is a smooth known nonlinear function. Assume , are recurrent signals and the reference orbit (denoted as ) is a recurrent motion. Moreover, , , , , and are assumed as unknown terms.

Our goal is to design a neural learning controller, which forces the tracking error vector (i.e., ) converges to a small neighborhood around zero with prescribed performance in a finite time. Before the design of learning control (LC) scheme, a stable adaptive neural dynamic surface controller with prescribed performance is developed to verify the feasibility of ANC scheme. According to the deterministic learning theory, the unknown system dynamics are accurately approximated by localized RBF networks along the recurrent orbits of NN inputs. Then, based on the ANC scheme and the approximation of localized RBF networks, the knowledge on unknown system dynamics is stored in static neural weights, which is also reused to develop a neural learning controller. This neural learning controller is verified to achieve the closed-loop system stability and better control performance with prescribed constraints for the same or similar tasks.

##### 2.2. Prescribed Tracking Performance

In this paper, the output error vector of system (1) is defined as . To achieve the prescribed performance (i.e., overshoot, convergence rate, and convergence accuracy), each element in is constrained into the following prescribed region: where and are positive design constants. is a bounded, smooth, strictly positive, and decreasing performance function. In addition, is chosen as the following form by setting and : where , , and are positive constants. With (3) and (4), it can be concluded that the convergence rate of is constrained by the decreasing rate of , while its maximum bound of overshoot at initial moment is constrained by and , and its steady error is constrained within a range from to .

##### 2.3. RBF Neural Network

According to [45], RBF NN can approximate any continuous function vector over a compact set to any arbitrary accuracy as where is the ideal weights matrix, is the NN node number, is the basis function vector with chosen as the Gaussian function , and is the approximation error vector which satisfies , with constant .

On the other hand, it has been shown in [46] that, for any bounded trajectory over the compact set , the continuous and smooth function can be approximated to an arbitrary accuracy using the localized RBF NNs with a limited number of neurons located in a local region along the trajectory: where is the subvector of and with . is the approximation error which is close to .

Lemma 1 ((partial PE condition for RBF NNs) [46]). *Consider any continuous recurrent trajectory . Assume that is a continuous map from into , and remains in a bounded compact set with . Then, for the RBF NN with centers placed on a regular lattice (large enough to cover the compact set ), the regression subvector consisting of RBFs with centers located in a small neighborhood of is persistently exciting.*

#### 3. Adaptive Neural DSC Design with Predefined Tracking Performance

In this section, performance function is introduced for describing constraints of system (1). Then an adaptive neural DSC is developed, with the design of adaptive control law based on the transformed error. Meanwhile, RBF NN is used to approximate the unknown dynamics.

*Step 1. *Similarly to the traditional backstepping design, we set

It should be pointed out that any errors set in previously traditional design are under unrestricted condition [20], while is constrained to satisfy condition (3) in this paper, which is rewritten as

It implies that can not be used for design directly due to the limitation of the traditional design method. To solve the constrained tracking control problem, the constrained error should be transformed into the unconstrained one equivalently. Therefore, a new error vector is defined as called transformed error vector. Define a smooth transformed functions vector with being chosen as where , and . Moreover, for the symmetric tracking error constraints , the transformation function (9) can be constructed as . For clarity, Figure 1 illustrates the relationship between and .