Fuzzy Adaptive Switching Control for an Uncertain Robot Manipulators with Time-Varying Output Constraint

An adaptive control strategy based on a fuzzy logic system by introducing a nonzero time-varying parameter is studied for an n-link manipulator system with the condition of a complex environment. At the beginning, a universal approximator with a one timevarying parameter is proposed based on the analysis of the fuzzy logic system, which is utilized to equalize uncertainties in robot manipulators with time-varying output constraints. The novel design method is used to reduce greatly the online learning computation burden compared with traditional fuzzy adaptive control. The output and the position of robotic manipulators are constrained with time-varying, a good tracking performance can be guaranteed with the condition of unknown dynamics of robot manipulators, and the violation of constraints can be conquered by the analysis based on the barrier Lyapunov function. A switching adaptive control is proposed to extend the semiglobal stability to global stability. Effectiveness of the approach is demonstrated by simulation results.


Introduction
In recent decades, more challenges in an intelligent controller design or learning algorithms are urgent to be developed for the growing different applications in engineering environments.Intelligent technologies for manifold aims as a hot topic research and development emerged in many different regions or fields, like robotics and automation [1,2], manufacturing [3], process industries [4], and so on.Accordingly, the intelligent control has become a formidable problem owing to the different environment in complex systems.Especially in various sophisticated scenarios, intelligent control is a challenging issue because of the various changes of the target object or other complicated phenomenon.In [5], an intelligent shared control system combined with human-computer interaction and a brain computer interface was developed.Based on a spatiotemporal context learning algorithm, good tracking performance can be achieved for various challenging environments in [6].Under the condition of a sophisticated environment, a coupling interface proposed to learn human adaptive skills in [7].Extreme learning machine (ELM) was employed to compensate for the uncertain nonlinearities of manipulator systems or be used to the robot teaching systems such that satisfactory performance can be achieved in [8,9].
Adaptive technologies play a pivotal role in dealing with controlled systems in various complex environments and received more favor by scholars.Uncertain nonlinearities or unknown models commonly existed in systems with many complex environments; thus, fuzzy logic systems (FLSs) or neural networks (NNs) are employed to tackle with the performance indicators of the systems.Especially, FLSs and NNs are universal approximators [10][11][12][13][14], which can be used to approximate unknown terms by combining with an adaptive algorithm technique.In [15], a novel teleoperation control combining with NNs and wave variable is proposed to guarantee the stability of the closed-loop systems.For the unknown parameters of robotic systems, adaptive estimation design methods are developed in [16,17].For uncertain multiagent systems, adaptive NN synchronization controls by using the information of the neighboring agents are established in [18].A new idea of finite-time quantized feedback control with NNs is addressed for quantized nonlinear systems with unknown states in [19].Tracking controllers with prescribed performance are constructed for dynamical nonlinear systems in [20][21][22][23].A variety of different forms of control technologies are designed for robot manipulators with unknown dynamics based on NNs or FLSs, like output feedback control [24][25][26][27], robust position control [28], decentralized control [29,30], and so on.Nevertheless, it should be noted that the problem of constraint control was not considered in these literatures.
A barrier function as a useful tool was firstly proposed to protect the boundary on the feasible region in [31,32], and this pioneer work induced many excellent research results in preventing constraint violation.For example, constant output constraint can be ensured by the analysis of the barrier Lyapunov function (BLF) in [33][34][35].An adaptive neural tracking controller is designed by the stability of the integral barrier Lyapunov functions for robot manipulators in [36].Adaptive fuzzy output controllers united in wedlock backstepping are also be proposed in [37][38][39][40][41]. Adaptive controls by employing the BLF are developed for a class of nonlinear systems with full-state constraints in [40,41].However, it should be pointed out that the output constraint is assumed to be a constant value in these references [33][34][35][36][37][38][39][40][41], and thus, these methods will be invalided for time-varying output constraints.
In many special complex work environments, varying constraints are necessary to be taken into account for meeting specific predefined targets, so how to exploit some novelty controls is essential.In [42], an asymmetric bimanual coordinate control based on a novel BLF with the position and velocity constraints in tasks is developed, which can be used to make the robot behavior adaptive and flexible to detect human operator motion.In [11], time-varying constraints are considered, and an adaptive neural network control of robot manipulators is designed to satisfy the tracking performance.It should be needed to point out that these above-mentioned NN or FLS control methods only have been guaranteed for semiglobal uniformly ultimately bounded (SGUUB) because the approximation holds only on a certain compact domain; wherefore, the range of the domain must be ensured such that approximation accuracies can be obtained.However, it is impossible that the accurate domain can be chosen precisely in advance, especially in some complicated systems such as multiple input multiple output (MIMO) systems; hence, it is worth to exploit NN or FLS control that can be achieved by global uniformly ultimately bounded (GUUB).In [43], the GUUB stability can be ensured, but output constraint was not considered in the systems.A global adaptive NN control with presetting tracking performance is designed for a class hypersonic flight vehicle in [44].A switching adaptive NN controller was projected for the GUUB stability of bimanual robot systems in [45], and tailored transient performances can be achieved at the same time.Nevertheless, these global controllers are the main focus on the approximation of NNs to equalize the unknown nonlinearities or dynamic model in systems.From the viewpoint of engineering, FLSs have advantages of high interpretability of rich expert experiences over NNs, so it is important to design a fuzzy adaptive control of the GUUB stability in practical engineering.
Inspired by the above descriptions, we attempt to solve these problems by using FLSs with abundant experiences or languages of experts in this paper.A fuzzy adaptive control based on the BLF technique is proposed for an unknown robot manipulator with time-varying constraints.The stability of GUUB can be guaranteed for the closed-loop system by introducing switching mechanism.The main advantages of this design method are as follows: (1) A nonzero time-varying parameter is introduced in FLSs, and a fuzzy adaptive controller is designed to get the good tracking performance with time-varying output constraint.
(2) All the signals can be guaranteed GUUB in the closed-loop robot manipulator.

Descriptions and Assumptions of Robot Manipulators.
The n-link robot manipulator is considered as the following dynamical equation: where q ∈ R n×1 denotes the joint angular positions, q ∈ R n×1 is the velocity, and q ∈ R n×1 denotes acceleration vectors; M q ∈ R n×n is a symmetric positive known inertia matrix, C q, q ∈ R n×n denotes the centripetal and Coriolis force vector, G q ∈ R n×1 denotes the gravity vector, τ t ∈ R n×1 represents the input torque, J q is the unknown reversible Jacobian matrix, and f t denotes the constrained force, which satisfies f t ≤ f for t > 0, f > 0 is a known constant.
In this paper, we assume the position of system (1) to be constrained in a time-varying compact set as where x 2n T , then the dynamical system (1) can be transformed into following dynamical equation: For a given reference signal vector the control aim is to design a fuzzy adaptive controller such that the output y = x 1 = q 1 , q 2 , … , q n T tracks this ideal reference signal vector with small errors, and all the signals in the closed-loop robot manipulators (1) are all be bounded.Meanwhile, the time-varying constrains must be guaranteed without damage.
Assumption 2 [11].There exist constants H cij and H cij satis- Assumption 3 [11].There exist two function vectors X 0 t < h c and X 0 > h c , ∀t > 0 and exist positive constants X i which satisfies Define If Assumptions 2 and 3 are hold, then there exist some positive constants h bi , h bi , h ai , and h ai , which can make the following inequalities true: The tracking error vector is defined as We introduce the following error coordinates that will be used in the procedure of the control design: where The following Lemmas 1 and 2 are introduced.

Descriptions of Fuzzy Logic Systems.
The following Mamdani type FLS with q fuzzy rules and the kth rule is to be considered: where A k h and B k denote fuzzy sets.A k h x h h = 1, 2, … , n is the membership function of x h .Now, a nonzero time-varying parameter ρ = ρ t is introduced in the fuzzy rule (8).According to the defuzzification in [12], the output of the fuzzy logic system (8) can be written as where y k = max y k ∈R μ B k y k .On a given compact set, we know that the FLS (8) can be utilized to approximate any given uncertain continuous function with any degree of accuracy.Therefore, based on this property of universal approximation, Lemma 3 can be easily obtained.
Lemma 3. If the uncertain continuous nonlinear function Δ i X satisfies the Lipschitz condition with coefficient l i and existing FLSs F i X make sup X∈U Δ i X − F i X ≤ ε i , where U = X | X ≤ α ρ and ε i is the approximate error, then the following inequality is hold: Proof.In terms of Δ i X , it satisfies the Lipschitz condition with coefficient l i , such that is true, then the following result is also true: This completed the proof for Lemma 1.
The values of ε and L are all unknown in this paper, so we introduce ε = ε t and L = L t to denote the estimation of ε and L, respectively.ε = ε − ε and L = L − L denote the estimate error.

Fuzzy Adaptive Control Design
In this section, the fuzzy adaptive controller is designed based on backstepping technology and the asymmetric BLF.
From, an asymmetric TVBLF in [11], the following is considered: where r ≥ 1 is an integer of differentiability of 13) can be rewritten as and with k 1i t = h ai /h ai 2 + h ai /h ai 2 + o i ; positive parameters k 1i and o i are to be given suitable positive constants by the designer.
Based on ( 6), ( 14) becomes where Theorem 1.Consider the robot system described by (1) under Assumptions 1-3, with the condition of z 2 > α ρ .The following control scheme design as shown in (18) can make sure that the state vector T reaches the sliding model surface s as shown in (19) in limited time period.
where the parameters λ and α are known positive constants.
Proof.Since z 2 > α ρ and X ≥ z 2 ≥ α ρ is satisfied, open control is adopted with this condition; choose the sliding model surface as Define the positive function V = 1/2 s 2 , and according to ( 17) and ( 18), the derivative of V can be obtained as We denote the function vector Δ X = J T x 1 f + C x 1 , x 2 x 2 + G x 1 + M x 1 β and satisfy Assumption 1; we have According to [46], (21) means that Theorem 1 can be guaranteed.
Theorem 2. If Assumptions 1-3 are hold and z 2 ≤ α ρ can be satisfied, then the proposed fuzzy adaptive control (22) can make the tracking errors converge to a small zero field.All the signals in the closed-loop system can be ensured to be uniformly ultimately bounded.
Proof.Because of z 2 ≤ α ρ and X ≥ z 2 , so if X > α ρ is satisfied, the control scheme will go back to Theorem 1.But if X > α ρ , we consider the following candidate Lyapunov function: In differentiation V 2 , then We know T , so the following inequality is true: Due to the function vector Δ = Δ 1 , Δ 2 , … Δ n T which is unknown, so the approximation of the fuzzy logic systems (9) can be used to deal with the unknown function; then Define the candidate Lyapunov function.
Based on the analysis of Theorem 1 and Theorem 2, we can draw the following Theorem 3 about global stability.Theorem 3.For the robot manipulator (1) with Assumptions 1-3 and with the initial output, h c 0 < y 0 < h c 0 .When the tracking run out of the domain U = X | X ≤ α ρ , the updated laws in control (18) will pull the states back to the sliding surface s, and then the proposed switching controls (18) and (22) can guarantee that all the signals in the closedloop systems are to be GUUB, and all the tracking errors z 1i will went to a small field Remark 1.According to the analysis in Theorem 3, we know that the approximation accuracies only updated the online time-varying parameter in a fuzzy logic system, which broke the limitation of exponential growth with the linguistic variables in traditional fuzzy rules such as [24,27,28,33].This advantage can make designers focus on his/her energy to convey information from human experience to controller action in a complex environment and neglect the computational burden because of the adaptive laws independent with fuzzy rules.
We choose domain , and fuzzy linguistic variables selected as negative big (NB), negative middle (NM), negative small (NS), positive small (PS), positive middle (PM), and positive big (PB) with the fuzzy membership functions are defined as i = 1, 2 ; j = 1, 2 : , , The fuzzy logic system F 1 X is the six rules in Table 1; The simulation results of tracking performance for the robot system are shown as in Figures 1-7.
From the simulation results, Figure 1 shows the output of the first manipulator and the desired trajectory with    7 Complexity time-varying constraint, and Figure 2 indicates the output of the second manipulator and the reference signal with timevarying constraint.The tracking errors are shown in Figure 3.The time response of velocity of the first manipulator and that of the second manipulator are depicted in Figures 4 and 5, respectively, and the speed of tracking errors is shown in Figure 6.All the adaptive parameters of online time response in controls (18) and (22) are shown in Figure 7, which can be achieved GUUB successfully, and a good tracking performance can be obtained with timevarying constraints from these simulation results.

Conclusions
This paper presented a novel fuzzy adaptive switching control based on a time-varying BLF for a robot manipulator with uncertain models; the FLSs combining with the timevarying parameter are used to compensate for the uncertain model system, which extended the traditional fuzzy control that is only emphasized on approximation accuracies and     18) and (22).8 Complexity neglected the interpretability of expert language in the procedure of design controllers.By introducing the time-varying parameter in the FLS, global stability can be guaranteed by the switching fuzzy control scheme with neglecting the outside or the inside of the compact domain of the universal approximator.

Figure 2 :
Figure 2: The time response of the output x 12 of Link 2 and ideal reference y d 2 .

Figure 1 :
Figure 1: The time response of the output x 11 of Link 1 and ideal reference y d 1 .

Figure 3 :
Figure 3: The time response of tracking error z 11 and z 12 .

Figure 6 :Figure 5 :
Figure 6: The time response of tracking error z 21 and z 22 .

Figure 4 :
Figure 4: The time response of velocity tracking trajectory of Link 1.

Table 1 :
Fuzzy rules of the FLS F 1 X .