Research Article | Open Access

# Adaptive Fuzzy Dynamic Surface Sliding Mode Position Control for a Robot Manipulator with Friction and Deadzone

**Academic Editor:**Xudong Zhao

#### Abstract

Precise tracking positioning performance in the presence of both the deadzone and friction of a robot manipulator actuator is difficult to achieve by traditional control methodology without proper nonlinear compensation schemes. In this paper, we present a dynamic surface sliding mode control scheme combined with an adaptive fuzzy system, state observer, and parameter estimator to estimate the uncertainty, friction, and deadzone nonlinearities of a robot manipulator system. We design a dynamic surface sliding mode basic controller by systematic recursive design steps that yields several adaptive laws for the compensation of nonlinear friction, deadzone, and other unknown nonlinear dynamics. The boundedness and convergence of this closed-loop system are guaranteed by the Lyapunov stability theorem. Experiments on the Scorbot robot manipulator demonstrate the validity and effectiveness of the proposed control scheme.

#### 1. Introduction

In recent decades, several advanced control approaches have been developed to solve complex control problems as industrial machines and devices have rapidly progressed, requiring higher performance control. Among these, a breakthrough nonlinear control method, adaptive backstepping, [1, 2] achieved stabilizing controllers for nonlinear system and guaranteed global or regional regulation and tracking properties. The cancellation of useful nonlinearities that occur with the feedback linearization techniques can be also avoided by using a step-by-step recursive algorithm. However, the application of the backstepping design method requires that nonlinear dynamic models be known either exactly or linearly parameterized with respect to known nonlinear functions. In real situations, this requirement is frequently difficult to accomplish since most uncertainty in a nonlinear system is unknown. To solve this problem, adaptive backstepping methods combined with fuzzy methods [3, 4] and neural networks (NNs) [5, 6] have been developed to approximate these unknown uncertainties. Thus, recently, this approximator-based backstepping method has become a very popular control scheme for dealing with a large class of nonlinear systems. However, repeated differentiation of the virtual control functions [7] gives rise to an explosion of complexity in the controller terms of the complex nonlinear system. Dynamic surface control (DSC) [7, 8] was developed to help a nonlinear systems overcome this “explosion of terms” by using a first-order filter of the synthetic input at each step of the backstepping design procedure. Thus, several adaptive DSCs combined with fuzzy methods [9] and NNs have been developed [10–12] because these controllers are relatively much simpler than backstepping-based ones. Another option is a model-free approach such as fuzzy methods, which is synthesized.

In a dynamic system consisting of the actuator of a robot manipulator, friction and deadzones are frequently encountered and are the main obstacles to high-performance positioning and tracking control. Friction between a moving part and a guide surface gives rise to problems such as stick slip, limit cycle, and steady-state error. Deadzone nonlinearity also causes inaccuracy in a control system [13]. A controller designed to compensate for friction or deadzone independently may perform poorly in the friction/deadzone overlap. Thus, compensation for both nonlinearities should be taken into consideration together. However, with the exception of [12], most DSC applications have focused largely on the compensation of linear or smooth nonlinear system.

The effect of nonlinear friction appears most strongly in a low-velocity regime, especially during velocity reversals. The LuGre [14] and Elastoplastic [15] models can construct a friction estimator relatively easily by virtue of their more systematic structure and lower complexity compared to other available modes. Lin and Chen [16], Yau and Yan [17], and Han et al. [18, 19] developed a sliding mode control and fuzzy logic scheme with the LuGre and Elastoplastic friction model to compensate for the nonlinear friction of a ball-screw and robot systems. For deadzone, several control schemes [20–23] have been developed. However, compensations for both deadzones and friction together have not often been considered until now.

Fuzzy technology [24, 25] has replaced many complex nonlinear control applications. One major feature of fuzzy logic is its ability to express an amount of ambiguity, similar to judgments based on human experiences or expert opinion. Thus, fuzzy logic is an alternative way to deal with the unknown mathematical model of a complex system due to its universal approximation property [24]. A fuzzy controller depends on the experience of experts to create a fuzzy rule base and parameters that are adjusted by adaptive laws for a specified control performance. Hence, adaptive fuzzy controls have been applied successfully in many nonlinear control systems and guarantee improved system performance and stability in the Lyapunov sense [26–28]. However, a specific performance decision table, complicated learning mechanisms, and/or a large amount of fuzzy rules require design by trial-and-error and make practical application difficult.

It is well known that the sliding mode control (SMC) technique is robust to system uncertainty due to its use of a sliding surface [29, 30]. To reduce the fuzzy rules in fuzzy control and significantly increase control performance, SMC is combined with fuzzy logic [31] and other control methods such as intelligent methods [32] and backstepping control [33]. The adaptive sliding mode backstepping control for a semistrict feedback system with unmatched uncertainty was proposed in [34, 35]. However, the backstepping control technique has an explosion of terms problem due to the repeated differentiation of the virtual control functions. This problem leads to a severe computational burden for real hardware implementations such as complex robotic systems. Thus, although the backstepping method is theoretically tractable, in real applications, its increasing complexity is an insurmountable obstacle that prevents its application to multiple state control systems.

We propose an adaptive fuzzy strict feedback positioning control for a robot manipulator based on a DSC design. SMC is applied to a DSC and FLC frame to enhance robustness for the compensation of uncertainty and an adaptive fuzzy system approximates the unknown nonlinear function. The main contributions of this paper are as follows. (1) The DSC scheme is introduced to overcome the drawback of backstepping control. (2) We show that both the deadzone and friction nonsmooth and nonlinear effects of a robot manipulator can be compensated for simultaneously. (3) We then detail and show how SMC is combined with an adaptive DSC and FLC system to enhance the performance robustness for lumped uncertainty and required fuzzy rules and can then be reduced to an approximation to reduce the controller complexity. (4) The proposed control approach is successfully applied to the problem of both reducing nonsmooth nonlinear effect and uncertainty of the robot manipulator in the presence of the friction and deadzone by experiment.

#### 2. Problem Formulations

##### 2.1. Description of the Nonlinear Plant

We consider a robot manipulator system in the presence of deadzone and friction including actuator dynamics whose dynamic equations [36, 37] are described by where denote the joint position, velocity, and acceleration vectors, respectively; the moment of inertia matrix is a positive definite symmetric matrix; is the centripetal Coriolis matrix; is a skew-symmetric matrix; is the gravity vector; is the nonlinear friction torque vector; is an external disturbance; is the deadzone control torque vector of the joint actuators; is the motor current vector; is the voltage vector applied to the motor drive; and are the inductance and resistance of the motor, respectively; and is the back electromotive-force (emf) constant of the motor.

Considering the modeling uncertainties and external disturbances, the robot system in (1) can be reformulated as where the subscript represents the system parameters in the nominal condition, , where ; is a lumped uncertainty defined as ; , , , and represent the unknown uncertainties of , , , and , respectively; and is the disturbance vector. The uncertainties of , , , and are bounded by some positive constants such that , , , and . For the disturbance, it is assumed that , for all , and is bounded by some positive constant :. Thus, the lumped uncertainty is assumed to be bounded by a finite value. To guarantee more improved control performance, an elaborate nonlinear friction model should be considered. The state equations for (1)–(4) are represented as where , , and .

The deadzone nonlinearities are shown in Figure 1(a) and their mathematical models are described by where and denote the slope of the deadzone and and stand for the deadzone width parameters. In the control problem, the practical assumptions of the deadzone are as follows.

**(a) Deadzone**

**(b) Inverse deadzone**

*Assumption 1. *The deadzone outputs are not available for measurement. Furthermore, the deadzone parameters , , , and are unknown but their signs are known, , , , and .

*Assumption 2. *The deadzone slopes are bounded by known constants , , , and such that and .

The deadzone inverse technique is a useful method to compensate for the deadzone effect [13]. Letting be the signal from the controller that does not take into account the deadzone, the following control signal is generated according to the certainty equivalence deadzone inverse described in Figure 1(b): where , , , and are the estimates of , , , and , respectively, and The resulting errors between and are given by where is known as the bounded function for all [13].

The nonlinear friction forces are assumed to be modeled as where is the stiffness of the elastic bristle, is the damping coefficients in the presliding range, is the viscous damping coefficients, and contains the bounded friction modeling errors. The presliding states are represented by the following Elastoplastic model [15]: where is the Coulomb friction, is the stiction level, is the relative velocity between two contact surfaces, is the Stribeck velocity, and is the unknown coefficient related to the presliding friction behavior. The function is positive and depends on many factors such as material properties, lubrication, and temperature. As the state variables cannot be measured directly, we use the friction state observers to estimate as follows [19]: where , , and are the estimations of , , and , respectively. The estimations of the friction can be expressed as where and are the estimations of and , respectively. From (10), (11), (13), and (14), we have where , , and .

We can transform the above state model into the following form: where , ,, , and , , and .

##### 2.2. Function Approximation Using a Fuzzy Logic System

The basic configuration of a fuzzy system consists of the fuzzifier, fuzzy rule base, fuzzy inference engine, and defuzzifier. The fuzzy inference engine performs a mapping from an input linguistic vector to an output linguistic scalar variable . The fuzzy rule base consists of a collection of fuzzy IF-THEN rules. The th IF-THEN rules are described by where , , and are fuzzy sets characterized by the fuzzy membership functions and , respectively, and is the number of rules in the fuzzy rule base. The output of a fuzzy system with a center-average defuzzifier, product inference, and singleton fuzzifier is expressed as where is the point at which (its maximum value). This equation can be rewritten as where is a vector that groups all the consequence parameters, and is a set of fuzzy basis functions defined as It has been proven that a fuzzy logic system can approximate any nonlinear continuous function to an arbitrary degree accuracy if enough rules are provided [24]. Thus, a fuzzy logic system performs a universal approximation in the sense that, given any real continuous function on a sufficiently large compact set and an arbitrary , there exists a fuzzy logic system in the form of (21) such that Then the function can be expressed as where , is the error of the fuzzy approximation and is chosen to be the value of that minimizes the fuzzy approximation error ; that is, Since is unknown, it is replaced by , an estimation of . Adaptation laws are required to update the parameter and other related fuzzy parameters online to asymptotically minimize the reference tracking error. The optimal fuzzy output function can be rewritten as where .

#### 3. Design of Controller and Nonsmooth Nonlinear Compensator

In this section, the adaptive laws and controller are derived via recursive DSC design procedures. The control objective for a robot manipulator system is to determine a state feedback control system such that the system output can track a desired trajectory . We add a final assumption to the system.

*Assumption 3. *The desired trajectory vectors are continuous and available, and with the known compact set , where is a constant. The state feedback control system is designed step-by-step using a DSC technique as follows.

*Step 1. *We define the tracking error to be the first error
where time derivative of (27) is
We define the following Lyapunov function:
and its time derivative is given as
We choose a virtual control law to be
where is a design constant. We introduce the filtering virtual control and let pass through a first-order filter with a time constant as
Setting , from (25), it follows that
By using the definition of , (30) becomes
From (24) and (26), it follows that
where is a continuous function. From (35) and (36), we have

*Step 2. *We consider the following expression:
By defining , the time derivative of is given by
Next, we define the Lyapunov function candidate:
By differentiating (40) with respect to time, we obtain the following equation:
where ,, , is a design constant, and where . We specify a virtual control to be as follows:
where is a design constant and is the estimation of . We introduce a new filtering virtual control and let pass through a first-order filter with a time constant as
By setting , from (36), it follows that
We define . It then follows that
Substituting (45) into (41), we obtain the following expression:
From (36), it follows that
where is a continuous function. From (47), we obtain the following inequality:

*Step 3. *The final control law is derived in this step. Consider
From the third error surface , it follows that
where , is a positive constant, and is an estimation of .

The adaptive strict feedback dynamic surface control is modified to enforce robustness by adding a sliding mode control. The modification starts by defining the following sliding surface in terms of the error coordinates: We define the following Lyapunov function candidate: where are positive constants. The time derivative of is calculated as From (51), we obtain the following relation: By considering the previous results and (54), we obtain the following result: We choose the control input to be where and are design constants. By using Young’s inequality expressed as , (56) becomes We specify the adaptive laws as follows: where the projection mapping is defined as [35] Substituting (58) into (57), we can obtain the following relation: where and the positive definite matrix is described as Then (60) can be written as where , , and is a positive constant. By integrating both sides of (64) from to , we obtain the following inequality: for all with . This implies all the states and signals are bounded. Finally, we can conclude that , , , , , , and . Then, , , , , , and as by Barbalat’s Lemma [30].

#### 4. Experimental Example

The experiments to evaluate the proposed control scheme using the Scorbot robot system are described in this section. A photograph of the Scorbot robot is given in Figure 2, where the deadzone occurs in the timing belt. We select only two links (upper arm = link1 and forearm = link2) among the four links of the Scorbot robot manipulator to simplify the verification process of our position control. From (1) to (3), the dynamic equations for the two DOF (degree of freedom) links of the Scorbot robot manipulator are described as where

**(a)**

**(b)**

The parameter values chosen for each link and actuator are represented in Table 1. The sine wave joint and circle motions of the end effector are chosen to be the desired trajectory commands. The sine wave is chosen to be . The direct kinematics for a circle trajectory in a task space is given by Then, the desired end-effector trajectory of the manipulator becomes where , , and . This trajectory makes the manipulator tip trace a circle in the plane with a radius of . The desired trajectory was translated to the corresponding joint space desired position trajectory via the inverse kinematics of the simulated two DOF links manipulators: The design parameters of the controller are given in Table 2. The fuzzy membership functions for the link1 are chosen to be The second inputs of the fuzzy system in link1 are chosen to be and the membership functions are the same as the above expressions. The fuzzy membership functions for link2 are chosen to be