Adaptive Fuzzy Control of Uncertain Robotic Manipulator

This paper designs a kind of adaptive fuzzy controller for robotic manipulator considering external disturbances and modeling errors. First, n-link uncertain robotic manipulator dynamics based on the Lagrange equation is changed into a two-order multipleinput multiple-output (MIMO) system via feedback technique. Then, an adaptive fuzzy logic control scheme is studied by using sliding theory, which adopts the adaptive fuzzy logic systems to estimate the uncertainties and employs a filtered error to make up for the approximation errors, hence enhancing the robust performance of robotic manipulator system uncertainties. It is proved that the tracking errors converge into zero asymptotically by using Lyapunov stability theory. Last, we take a two-link rigid robotic manipulator as an example and give its simulations. Compared with the existing results in the literature, the proposed controller shows higher precision and stronger robustness.


Introduction
The robotic control system is a kind of strongly coupling and highly nonlinear dynamical system.One of the main control aims is to guarantee the factual motion traces to track the given traces under the ideal dynamic quality.When the precise robotic mathematic model is established, this aim is achieved easily.However, at the practical operation scene, the robotic control systems is always influenced by the external stochastic disturbances, including the Coulomb force and the Friction force, and the inside parameter perturbations.Therefore, the mathematic model established under the face of ideal instance cannot work well and finding the robust controller which can compensate the uncertainties of practical mathematic model is very important [1][2][3][4][5].
With the development of cybernetics, more and more intelligent control methods such as neural control and fuzzy control are applied into some nonlinear dynamic systems.Among these intelligent control methods, the fuzzy logic is used more widely because of its nonlinear approximation ability.In [6], a fuzzy terminal sliding mode controller is developed for linear systems with mismatched timevarying uncertainties.In [7], an adaptive fuzzy tracking control is suggested for a class of MIMO nonlinear systems considering the system uncertainties, unknown nonsymmetric input saturation, and external disturbances.In [8], an adaptive fuzzy controller based on any observer for a class of affine nonlinear system is developed.And the performance of the developed controller is demonstrated in an inverted pendulum system and a chaotic system.In [9], a new adaptive fuzzy terminal sliding mode tracking controller is presented for a class of nonlinear systems.This control strategy is proved to achieve favourable control performance considering parameter variations and external disturbances.In [10], a fuzzy mixed  2 / ∞ sampled-data control scheme is proposed for nonlinear systems, and the effectiveness and feasibility of the proposed controller are illustrated by using a computer simulated truck-trailer system.In [11], an adaptive backstepping fuzzy controller for servo systems with unknown parameters and nonlinear backlash is developed.The developed controller shows that it has higher accuracy and robustness in performances than PID controller.In [12], a robust adaptive controller using a fuzzy compensator for MEMS triaxial gyroscope with nonlinearities, model uncertainties, and external disturbances is proposed.In [13], a robust adaptive fuzzy compensation based tracking controller including a second-order filler with physical constraints is proposed for nonlinear ship 2 Mathematical Problems in Engineering course-keeping system with modeling errors and external disturbances.
Of course, the application of the fuzzy control strategy in robotic manipulator systems' tracking problem has been a hot field.In [14], a robust fuzzy controller is designed for robotic manipulator to ensure both global stability and robust performance.In [15], a backstepping adaptive fuzzy control strategy is suggested to ensure the tracking errors convergence into zero.However, both [14,15] do not consider the effect of the approximation error.In [16], the approximation error is considered, but the proposed adaptive fuzzy backstepping controller can only achieve uniform ultimate boundness, rather than asymptotical stability.In addition, these references make little difference between modeling uncertainty and external disturbance or neglect the external disturbance.In fact, there is no relationship between external disturbances and system parameters.Moreover, it is difficult to guarantee the stability, error convergence, and robustness based on offline trained fuzzy logic because of the highly nonlinear of the fuzzy logic.
In this paper, an adaptive fuzzy logic based feedback control scheme is developed by using the sliding theory [17,18] to enhance the robustness of the closed-loop system.The adaptive fuzzy logic controller consists of the fuzzy logic approximation module and the sliding mode control module.The first module is used to model the dynamics of nonlinear systems and the second module is adopted to compensate for the approximation errors such that the presented control scheme can guarantee the asymptotic convergence of the trajectory tracking error and the global stability of the closedloop system.The main contribution of this scheme can be summarized as follows: (1) This paper adopts feedback control technique to transform the Lagrange equation into a concise state equation, which can be studied by adopting linear control technology.Compared with the nonlinear equations, there are a lot of achievements regarding linear equation: optimal control [19], fuzzy cerebellar model articulation control [20], and so on [2,21].(2) External disturbances and uncertain modeling errors are considered meanwhile for developing the weight updating algorithms which are updated online by employing Lyapunov stability theory.The Lyapunov stability theory is also employed to prove that the designed controller guarantees the stability of tracking.
(3) The two lumped modeling uncertainties are adopted by fuzzy logic system to reduce the number of fuzzy rules hereinbelow.
The organization of the paper is as follows.After a general description of the uncertain robotic system, the dynamical model based on the Lagrange equation is transformed into a more concise state equation via feedback control technique in Section 2. In Section 3, a lot of space is used to develop a kind of new controller that is adaptive fuzzy logic.The example simulations for a two-link manipulator are given in Section 4, and the paper has been concluded in Section 5.

Description of Robotic System
The -link robotic manipulator dynamical model based on the Lagrange equation can be described by the following twoorder differential equation [22]: where , q , q ∈   are the joint displacement, the joint velocity, and the joint acceleration; () ∈  × is the inertia matrix of the robot; ℎ(, q ) ∈    is the coupled vector by the Coriolis, centrifugal, and gravitational force;  ∈    is the generalized control force;   ∈   stands for the external uncertain disturbance, which is bounded.The robotic model has the following property.
Property 1. ()  is symmetric and positive definite.For all  it is bounded; that is, there are two positive numbers   ≤   satisfying the following inequality: Besides external disturbances, the modeling uncertainties will be considered also in the actual robotic manipulator system.Then, ( 1) is rewritten as where  = −Δ() q − Δℎ(, q ) is the lumped modeling error, Δ() and Δℎ(, q ) are unknown parts, and  0 () and ℎ 0 (, q ) are known parts.Select the following generalized control force: From ( 3)∼(4), the following dynamical equation is obtained: Let  =  −1 0 ()  and (, q , q ) =  −1 0 (); then, the more concise state equation is obtained as follows: Then, the control aim of system ( 6) is to design  to make the actual trace follow the given   asymptotically.Here the given   is a two-order continuously bounded differentiable function.

Remark 2. Because of Property 1 and the boundness of
where  0 is a positive constant number.

Adaptive Fuzzy Controller
3.1.Nominal Model.When there is not modeling uncertainty and external uncertain disturbance   equals zero, system (1) is called the nominal model [22] of robotic manipulator.Then system (6) becomes q = .There exists a linear feedback control law: which makes the dynamics of system (6) become where  =  −   is the trajectory tracking error and ,  ∈  × are positive definite matrices.They are also usually specified to be diagonal matrices for decoupling.Obviously, ( 9) is stable; that is, the tracking error  asymptotically converges to zero.In fact, ( 9) can be rewritten as where And  is a positive definite matrix [23].Hence, there exists a positive definite matrix  meeting the following Lyapunov equation: where  is a positive definite matrix too.

Uncertainty Exists.
In actual system of robotic manipulator, there are always uncertainties including modeling errors and external disturbances.That is to say, (, q , q ) exits and  does not equal zero in (6).
In order to still use the feedback control technique, the uncertainty (, q , q ) needs to be estimated.Let's say the estimation is f(, q , q ).And then, in a relatively straightforward manner, we can get the following modified control law: Substituting ( 13) into ( 6), the following error dynamics will emerge: Due to (, q , q ) − f(, q , q ) ̸ = 0 and  ̸ = 0, the stability of error equation ( 14) cannot be guaranteed easily.
Therefore, in order to eliminate the effects of approximation error and outside disturbance , we need to redesign controller (13) by adding a compensation term   ; that is, the control input  has the following expression: Since it is proved that fuzzy logic can approximate a large range of nonlinear system to any given degree of accuracy, in this paper the estimation f(, q , q ) can be gotten by using the fuzzy logic [16,24,25].

Fuzzy Logic System
. A fuzzy logic system contains four parts [24], which are the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier, respectively.The primal fuzzy system is fixed and uniformity.To hold the consistent performance of the fuzzy system under this circumstance where there is a lot of uncertainty or unknown development tendency in system parameters and structures, the fuzzy system should possess adaptive performance.Without loss of generality, the output of an uncertain system can be assumed as ().The fuzzy inference engine adopts the fuzzy if-then rules to form a mapping from an input vector  = [ 1 , . . .,   ]  ∈   to an output variable  ∈ .The th fuzzy rule is of the form   : If  1 is   1 and . . .  is    , then  is   , where   1 , . . .,    and   are fuzzy sets characterized by fuzzy membership functions, for example, the Gaussian type.
Take the singleton fuzzifier, the product inference engine, and the center-average defuzzifier; then, the output variable  can be described as where  is the number of fuzzy rules,  = [ ỹ1 , . . ., ỹ ] T with ỹ being an adjustable value where the fuzzy membership function    ( ỹ ) has the maximum value, choosing    ( ỹ ) = 1 usually, and () = [ 1 , . . .,   ] T is a fuzzy basis vector, whose element is of the form ,  = 1, . . ., .
And the adaptive fuzzy system can be considered as the type of a neural network, which is shown in Figure 1 [24-26].

Fuzzy Approximators.
In this subsection we propose the use of fuzzy system to estimate the unknown function (, q , q ).Then, the approximation is sued to develop a welldefined adaptive controller with its adaptation law in order to satisfy control objective.
First, for reducing the number of fuzzy rules hereinbelow, the uncertainty (, q , q ) can be represented as an addition of two functions: where And each function will be replaced by the fuzzy logic system f1 (, q |  1 ) and f2 (, q |  2 ), respectively, which are of the form of ( 16) and ( 17): Thus, there is Letting the optimal parameter matrices of the fuzzy logic system be  1 * and  2 * , respectively, the minimum approximation error vectors can be defined as follows: And they are bounded according to the approximation principle of the fuzzy logic; that is to say, there are very small positive constants  1 and  2 satisfying where  1  and  2  denote the th element of the vector  1 and  2 .Define the approximation error of weight matrices as (24)

Two Lemmas
Lemma 3. If ,  ∈   are row vectors,  ∈  × is a square matrix; then there is T , and On the left, On the right, first Then so is equal to the left.

Main Result.
In order to eliminate the effects of the approximation error and the external disturbance, we design the second part   in (15) as follows: where  = [ And   is the control gain given by the following expression: where sgn() ∈   is a switching function vector, whose common element is a switching function in scalar case: where   ∈  ( = 1, 2, . . ., ).
The block diagram of the control scheme is shown in Figure 2.
Proof.Construct a Lyapunov function as follows: where  is a solution of (12).Differentiate  with respect to the state trajectories of (35), under (32) and (36), yielding + tr ( θ1T  −1 Further, substitute the adaptive updating rules (37) and (38) into the above expression, and use Lemma 3, obtaining Then, substitute (31) and (33) into V, displaying Considering (34), no matter   > 0,   = 0 or   < 0, there are always In fact, take (23) and Remark 2 into consideration; when  ≥ 0, the latter part of (42) satisfies And when   < 0, the latter part of (42) satisfies Now, we have proved that the Lyapunov function  decreases monotonically.Thus, it can be concluded that the close loop system is globally stable and , θ1 , θ2 are uniformly bounded.
Furthermore, it is easy to show that  1 and  2 are uniformly bounded too.With boundedness of (, q , q ), we can say that the estimations f1 (, q |  1 ) and f2 (, q |  2 ) are also bounded.Thus, it can be concluded that   in (13) and   in (31) are uniformly bounded.That is to say, all the right parts of (36) are bounded, and hence  is bounded.Therefore, (35) implies that Ṡ is uniformly bounded, which tells us that  will be uniformly continuous. Let Because of the existence of V ≤ −1/2 T , then  1 () ≥ 0, so we can say that  1 () has below boundedness.Further, as 1 is semi-negativedefinite.Finally, the uniform continuousness of  will make V 1 uniformly continuous too.Hence, using Barbalat's lemma, we can deduce that lim  1 = 0; that is, when  → ∞, we have  → 0. Then the tracking error  and its derivative ė converge to zero asymptotically.

Simulations Results
In this section, we will take a two-link manipulator to verify the feasibility of the suggested controller.The configuration of the manipulator and its parameters are shown in Figure 3 [27][28][29].
In simulations, we choose five fuzzy levels, that is, NB, NS, ZO, PS, and PB on the universe of each input variable and we use the following Gaussian membership function [24]: where    are −/6, −/12, 0, /12, and /6, respectively.When the modeling uncertainties Δ() and Δℎ(, q ) change by 20%, the corresponding simulation results are shown in Figures 4(a) and 4(b).From these simulation results, we can see that the designed controller guarantees actual trajectory tracking of the desired one completely.Meanwhile, when the uncertainties Δ() and Δℎ(, q ) change in a greater range, for example 50%, the corresponding simulation results are listed in Figures 4(c) and 4(d).Figure 4(c) tells us that in this case there are tracking trajectory errors.However, if we make parameters  and  larger, for example, let  =  = 500 2 , then there are no tracking trajectory errors again.This is shown in Figures 4(e) and 4(f), while Figure 4(f) indicates that when these uncertainties change in a bigger width and parameters ,  are larger, the control inputs tend to be more bad.
In order to investigate the performance of the proposed controllers, we will give a simulated comparison of the robust control scheme [23,30] in the end of this chapter.In this scheme, (4) becomes  = ℎ 0 (, q )+ 0 ()+ 0 ( q  − ė −) and the controller  = − T  2 /(| T | + ), where  is selected as  = [0, 0; 0, 0; 1, 0; 0, 1], that is, the same as [23,30],  is a solution of the Lyapunov equation (12),  is a very small positive number chosen as  = 0.001, and  describes the upper bound of modeling uncertainties Δ() and Δℎ(, q ), which can be chosen with a greater value than the upper bound of the modeling uncertainties, such as  = 10.All of the other used expressions and parameters do not change.
The simulation results adopting robust control scheme are shown in Figure 5, where Figures 5(a From these simulations results, we can see that the proposed controller in this paper is very effective and shows higher precision.In fact, [23,30] tell us that the robust controller  = − T  2 /(| T | + ) can only guarantee global uniform ultimate boundedness rather than global asymptotic stability.

Conclusion
This paper proposes an adaptive fuzzy control scheme for uncertain robotic system.Although the control scheme is proposed for uncertain robotic system, it can be suitable for a kind of MIMO system.First, we prove our control scheme effectiveness based on Lyapunov method.Then, we take a two-link manipulator to verify the feasibility of the suggested controller.The simulation results tells us that what we do is very meaningful.
However, it should be noted that the variable structure term sgn() is adopted to increase the robust performance, but it will generate "chattering" phenomenon, which may irritate unmolded high-frequency dynamics and even destroy the physical device.In order to weaken or eliminate the "chattering," there are some good directions stated as follows.
(1) The saturation function method, which is described below, can be used when frequency is too high, but using the saturation may degrade the robustness.It is very important to choose an appropriate boundary layer thickness .
(2) Adjust the term gain switching online, study a new sliding mode control method, and so on.

Figure 1 :
Figure 1: Structure of a fuzzy logic system.