An improved adaptive online neural control for robot manipulator systems using integral Barrier Lyapunov functions

ABSTRACT Conventional Neural Network (NN) control for robots uses radial basis function (RBF) and for n-link robot with online control, the number of nodes and weighting matrix increases exponentially, which requires a number of calculations to be performed within a very short duration of time. This consumes a large amount of computational memory and may subsequently result in system failure. To avoid this problem, this paper proposes an innovative NN robot control using a dimension compressed RBF (DCRBF) for a class of n-degree of freedom (DOF) robot with full-state constraints. The proposed DCRBF NN control scheme can compress the nodes and weighting matrix greatly and provide an output that meets the prescribed tracking performance. Additionally, adaption laws are designed to compensate for the internal and external uncertainties. Finally, the effectiveness of the proposed method has been verified by simulations. The results indicate that the proposed method, integral Barrier Lyapunov Functions (iBLF), avoids the existing defects of Barrier Lyapunov Functions (BLF) and prevents the constraint violations.


Introduction
With the widely use of complex robot manipulators which are nonlinear systems in our modern society and industry, research into robot technologies has attracted enormous attention Alford and Belyeu (1984); Cheng, Hou, Tan, and Zhang (2012); Gueaieb, Karray, and Al-Sharhan (2007); G.-W. Lee and Cheng (1996); T. Li, Duan, Liu, Wang, and Huang (2016) ;Na, Mahyuddin, Herrmann, Ren, and Barber (2015); Namvar and Aghili (2005) Alford and Belyeu (1984); Cheng et al. (2012); Gueaieb et al. (2007); G.-W. Lee and Cheng (1996); T. ; Na et al. (2015); Namvar and Aghili (2005). Meanwhile, the control of robot system in the presence of uncertain parameters and motion constraints were extensively studied (Z. Li, Ge, & Ming, 2007;Tang, Ge, Tee, & He, 2016a). In recent decades, adaptive control of complex nonlinear systems such as robot manipulators with full-state constraints and uncertainties has been developed to deal with theoretical challenges and practical needs (Cheng, Cheng, Yu, Deng, & Hou, 2016;Y. Huang, Na, Wu, Liu, & Guo, 2015; T. Lee, Koh, & Loh, 1996;G.-H. Yang & Ye, 2006). In the field of adaptive control, neural networks (NNs) are always considered as an efficient way to handle the uncertain or poorly known dynamics due to their universal approximation capabilities (Cheng, Liu, Hou, Yu, & Tan, 2015;Hou, 2001;Kennedy & Chua, 1988;Z. Li, Li, & Feng, 2016). It is very difficult to establish exact mathematical dynamics models with various uncertainties e.g., unknown payloads (Arefi & Jahed-Motlagh, 2013;Arefi, Jahed-Motlagh, & Karimi, 2015;G.-H. Yang & Wang, 2001;Zhang & Ge, 2009). However, by exploiting NN approximation, many complex and challenging models can be established easier but not sacrificing many characteristics of accurate models (Chen & Ge, 2013;Dai, Wang, & Wang, 2014;Gao & Selmic, 2006;T. H. Lee & Harris, 1998;Z. Li, Ge, Adams, & Wijesoma, 2008). In (C. Yang, Wang, Cheng, & Ma, 2016), a direct adaptive NN scheme is presented for a class of uncertain nonlinear strict-feedback systems. By utilizing a special property of the affine term, the developed scheme can avoid the controller singularity problem completely. The adaptive control of a strict feedback nonlinear systems using multilayer neural network was studied in (Zhang, Ge, & Hang, 2000) so as to guarantee the uniform ultimate boundedness of the closed-loop adaptive systems. In (C. Yang, Jiang, Li, He, & Su, 2017), RBF NN based control for coordinated dual arms robots have been proposed to settle the uncertainties. (He, Chen, & Yin, 2016) utilized NN of the conventional RBF NN structure for n-link robot but with lots of nodes. In this paper, an innovative DCRBF NN is proposed. The n − DOF input is split into n 1−DOF inputs and built a conventional RBF NN for each 1−DOF input. Subsequent mathematical manipulations provide the output of every node in each NN (by adding two layers) and it ensures that the tracking performance of the controller output does not deteriorate. In this way, we can compress the number of nodes and the weights in an extreme degree with the performance similar to the traditional one, which might be important for particular practical applications by saving time and energy. The approximation error between the output of DCRBF and conventional RBF is proved to be bounded.
The stabilization of robot system is another important requirement of controller design. In practical systems, violation of constraints may cause degeneration of the control performance or even system failures (L. Huang, Ge, & Lee, 2006;Su, Leung, & Zhou, 1992;Tee, Ren, & Ge, 2011). To handle with the constraint problem, many methods have been proposed such as model predictive control (MPC), optimal control and reference governors so on. It is always necessary to know the exact model which is quiet complicated for complex robot when utilizing the method of MPC and optimal control (Berkovitz, 2013;Luo, Wu, & Li, 2015;Mayne, Rawlings, Rao, & Scokaert, 2000;Rubio, 2012). However, in the situation of uncertainties or some sophisticated model which cannot be calculated, these control methods can not be employed and it is necessary to find some alternatively solutions.
To solve the problem of system control in the presence of constraints and uncertainties, Barrier Lyapunov Functions (BLF) are popularly used, since they have the ability to shape the control performance (Liu & Tong, 2017). For example, the tracking control problem is studied in (He et al., 2016) for an uncertain n-link robot with full-state constraints and a BLF is designed to guarantee the uniform ultimate boundedness of the closed-loop system. In (Tee et al., 2011), BLF is employed at the outset to prevent the output from violating the time-varying constraint in strict feedback nonlinear systems. However, BLF-based controls have its limitations. One is that the feasibility conditions have a tendency to be conservative when ensuring constraint satisfaction, due to the original state are enforced indirectly by imposing transformed constraints on the errors. In (He, Zhang, Ge, & Liu, 2014) iBLF based boundary controls was proposed for a class of inhomogeneous Timoshenko beam satisfying the needs of suppressing the undesirable vibrations and preventing the constraint transgression. In (Tang, Ge, Tee, & He, 2016b), iBLFs are constructed to handle the unknown affine control gains with state constraints . In order to accomplish the prescribed tracking performance considering transient and steady states, the iBLF technique is exploited in this paper. The main contribution of this paper are as follows: • An innovative DCRBF NN is proposed and it can be employed to avoid the exponential growth of nodes and weights with an increase in the DOF. Additionally, it inherently takes care of the inevitable uncertainties in the dynamics of the robot. • The mathematical proof of the DCRBF NN is presented. A rigorous proof of the new algorithm is presented and its effectiveness is verified in theory and simulation. • In order to avoid the violation of constraints while using BLF on n-link robots, a novel iBLF is utilized to design the control strategy which incorporates the output constraints and provides an enhanced system stability.
The rest of the paper is organized as follows. Section II gives the problem formulation of the n-link robot manipulator and some useful preliminaries for deriving proof.
In Section III, The control design and the stability confirmation for the system are proposed. Besides, a comparison between conventional RBF NN control and DCRBF NN control is demonstrated and the mathematical proof of DCRBF NN is presented.
Simulation studies are carried out to testify the effectiveness of the designed control and DCRBF NN in Section IV.

System Description
The dynamics of an n-link rigid robotic system in the following Lagrange form (Craig, 2005): where q,q,q ∈ R n represent the position, velocity and acceleration respectively; M (q) ∈ R n×n denotes a symmetric positive definite inertia matrix; C(q,q) ∈ R n×n is the centripetal and Coriolis torques, which is hard to obtain; G(q) ∈ R n×n is the unknown gravitational force; J T is the Jacobian matrix for f (t); f (t) represents the unknown internal and external disturbances such as friction and so on; τ ∈ R n×n represents the input torques.
Property 1. The matrix M (q) is symmetric and positive definite and there exist positive constraints 0 < m 1 < m 2 so that M (q)satisfies m 1 I < M (q) < m 2 I.

Required Technology Lemmas and Definitions
Lemma 2.1 ( (He et al., 2016)). If there existe a Lyapunov function V (x) ,which is positive definite and continuous satisfying where c 1 , c 2 are the positive constants and ξ 1 , ξ 2 are the functions making R n → R, the parameters and states of the system will remain in a compact set and eventually converge to a specific compact sets.
Lemma 2.4 (RBF approximation (Wang & Yang, 2017)). If there allow sufficient nodes, under suitable width ∆ and node centersδ, RBF NN can approximate any smooth function F a (x) over a compact set x ∈ Ω x with convergent errors: Lemma 2.5 (RBF and optimal weights). According to (Haykin, Haykin, Haykin, & Haykin, 2009), without loss of generality, we can use least-square method and recursive least-square method to solve the optimal wights and there is equivalence between two methods in mathematical terms. For leat-square method, we have where n i is the number of training sample and g(n i ) is n i × 1 desired response. † is the symbol for generalized inverse. φ(Z) = [S(Z 1 ) T , S(Z 2 ) T , · · · , S(Z ni ) T ] T is a interpo-lating matrix. As for using recursive least-square method, we have
For the n − link robotic arm, consider an iBLF candidate where e i = x 1i − x di and x di are continuously differentiable functions satisfying |x di | < k ai for i = 1, 2, · · · , n. According to Property 3, we can see that V 1 is positive definite. Differentiating V 1 with respect to time, we havė where Then, a virtual controller α i can be designed as where k 1i is a positive control gain for i = 1, 2, · · · , n, we obtaiṅ Then, we design a positive Lyapunov candidate function as Then differentiating V 2 with respect to time leads tȯ According to the expression ofV 2 , we design the control law as τ = τ 1 + τ 2 , where according to lemma 2.4, τ 1 uses adaptive dimension compressed RBF neural network control described in subsection B and τ 2 is designed as where k 2 is the control gain, and k 2i , i = 1, 2, · · · , n are positive constants. Then substituting τ into (14), according to Moore-Penrose inverse, , when z = [0, 0, · · · , 0] T . According to the lemma 2.3, we can still draw the asymptotic stability of the system. Otherwise, in the case of z = [0, 0, · · · , 0] T , we obtaiṅ

Useful Property
Property 2. For any positive constant k ai , the following inequality holds for any e i and x di , in the interval |x di | < k ai ,|e i + x di | = |x 1i | < k ai , for i = 1, 2, · · · , n: for |e i + x di | < k ai , which leads to the (17) after substituting for p.
Property 3. By Assumption1, the V 1 is positive definite, continuously differentiable, and satisfies the decrescent condition in the set |x 1i | ≤ k c1i < k ai , for i = 1, 2, · · · , n: which is useful for establishing uniformly stability.
Property 4. Using L'Hôpital's rule, it can be shown that Since |x di | ≤ k c1i < k ai , for i = 1, 2, · · · , n, by Assumption 1, ρ i is bounded and well-defined in a neighborhood of e i = 0. Figure 1 shows the architecture of the inputs space of an adaptive neural network control with n−DOF arm. Z = [x 1 , x 2 , α,α] T are the inputs, which has 4n dimensions. If each dimension has m types of centres, there will be m 4n dots and nm 4n weights in neural network. Thus it can be seen that with the degree of freedom n increasing, the dots and weights increase exponential.

Dimension Split for Radial Basis Function
Without loss of generality, for n − link arms (n − DOF ), let us express the conventional RBF NN as the form below.

Compression Matrix A
For the better illustration of DCRBF, let us introduce an operator matrix A(m 4n × nm 4 )first, which could be used to compress the numbers of the weights. To construct compression matrix A, a series of m 4 × m 4 submatrices ψ i , for i = 1, 2, 3 · · · , m 4 is built as . . .
Then use ψ i and unit matrix E to design the compression matrix A: (27), a dimension-split RBF NN of n − DOF is built in Figure2, which only has nm 4 dots and nm 4 weights for each degree of freedom of the outputs, which avoids the exponential growth with the

Solution of W k and Approximation Error
Considering a training procedure for the weights, according to lemma 2.5, use a leastsquare method to solve the weights W and we can obtain the optimal weights W * γ , for γ = 1, 2, · · · , n.
where n i is the number of training sample and g γ (n i ) is T is a n i × m 4n interpolating matrix Considering an error constant m 4n × m 4n matrix κ: where E 1 is a m 4n × m 4n unit matrix. Considering (37) and (39), the least-square method solvation of W k can be derived of where A is the compress matrix derived above and φ k (Z) = φA = [S (Z 1 ), S (Z 2 ), · · · , S (Z ni )] T is a n i × nm 4 interpolating matrix for S (Z).The optimal solution of W * k,γ is The weights approaching error can be expressed as which is a constant for γ = 1, 2, · · · , n. Using the expression of γ , and the output approximation error µ γ (Z) can be expressed as According to lemma 2.2 whereμ γ = T γ s is a positive constant. It can be seen that the DCRBF can obtain the similar desired response as conventional RBF with much less weights and dots,and a transform from conventional RBF to DCRBF with a bounded errorμ γ for γ = 1, 2, · · · , n is accessible by using the operator matrix A.
Theorem 3.1. According to Property 2, we can know that x 1i = e i + x di < k ai is bounded. For the condition satisfies −k ai < −k c1i ≤ x 1i ≤ k c1i < k ai , according to (6) and (7), we have −(k ai − k c1i ) < e i < (k ai − k c1i ), which is bounded. Then considering the definition of α in (11) and Property 4, α is bounded too. According to (54), lemma 2.1, Property 3 and in terms of (7), (13) and (49), we can safely conclude the e, z and NN weight estimated error W k are bounded. In the term of the boundness of α and z, according to x 2 = z + α, x 2 is bounded. Thus, we can safely say that the signals of the closed-loop system are semiglobally uniformly bounded (SGUB). And the closed-loop error signals e and z will remain within the compact sets Ω e , Ω z , respectively, defined by where i = 1, 2, · · · , n and D = 2(V 3 (0) + C/p), p and C are two positive constants.
Remark 1. The designed parameter k ai in the controller can be chosen simply as positive and the matrix k 2 should satisfy the condition in (57). The gains in NN adaptive law Q γ and θ γ should be positive. According to (58), θ γ should also be smaller than √ 2. In terms of (55),(59),(60), if the parameters k ai , k 2 and θ γ are chosen to be relatively small, while Q γ chosen relatively large, then the amplitude of trucking error could be made smaller.

Simulation Studies
In order to test the validity of the control, we have done a simulation on 2-DOF robot manipulator which have two revolute joints in the vertical plane. In the model which is shown Figure 3, the manipulator material is uniform. We define m i , l i , l ki , I i as the mass, the length of link i, the center distance of link i and the inertia of link i, i = 1, 2. q = [q1, q2]. Other simulation parameters are shown in Table. 1.
The results of the simulation are shown in Figure 4- Figure 6. All the pictures in Figure 4 represent the results of DCRBF with iBLF. Pictures in Figure 5 denote the results of conventional RBF with iBLF and Figure 6 is graphed to display the approximating error for weights between DCRBF and conventional RBF. Specially, the small pictures inserted in Figure 4(c) and Figure 5(c) represent the magnified position tracking errors from time 5s -30s.
In these pictures, we know that the prescribed trajectory tracking performance of proposed DCRBF with the implementation of iBLF is satisfactory from Figure  4 Figure 5(e), we can conclude that the DCRBF can approximate the system uncertainties as well as the conventional RBF.
According to Figure 4(f), Figure 5(f), W k and W approach to be stable with the test time going. So we assume that neural network approaches to the ideal model W * k , W * when times comes to 30s. Then we graph Figure 6 to show the approaching error between the weights of DCRBF and the conventional RBF, which is formulated as ∆W γ = W * γ − AW * k,γ − γ , for γ = 1, 2, where γ = κW * γ and κ = E 1 − AA † . The results are very small values close to zero, which proves that the approaching error of weights is definitely as what we calculate in (44). Then according to (45) and (46), we know that the output approximation error W * S(Z) − W * S (Z) is bounded. All the arguments above show that in terms of fitting ability, DCRBF can obtain a similar performance to conventional RBF with a bounded approximation error.

Conclusion
This paper presents an innovative adaptive neural network control using DCRBF for n-DOF robot system with full-state constraints and unknown dynamics. By utilizing DCRBF, the problem of superfluous number of nodes and weights in conventional RBF is overcome without compromising the tracking performance. The rigorous mathematical proofs of the effectiveness of DCRBF have been denoted. And an adaptive control for the system is formulated based on the methods of iBLF and backstepping for tracking performance and stability of the system with constraints and unknown disturbance. Finally, the effectiveness of the proposed method we proposed has been demonstrated through the results presented in this paper. Figure 6. Weights approaching error between DCRBF and conventional RBF