Adaptive Fuzzy Sliding Mode Control for a 3-DOF Parallel Manipulator with Parameters Uncertainties

Parallel manipulators possess the advantages of being compact structure, high stiﬀness, stability and high accuracy, so such parallel manipulators have been widely employed in application ﬁelds as diverse as parallel kinematic machine, motion simulator platform, medical rehabilitation device, and so on. Due to the complexity of the closed-loop structural system, an accurate dynamic model is very diﬃcult to be derived in the absence of some uncertainties parameters and external disturbances. In order to improve the trajectory tracking accuracy with time-varying and nonlinear parameters, this paper addresses the design and implement of adaptive fuzzy sliding mode control (AFSMC) for a three-degree-of-freedom (DOF) parallel manipulator, where internal force term can be linearly separated into a regression matrix and a parameter vector that contains the estimated errors. Furthermore, fuzzy inference unit is utilized to modify the gain parameters in real-time by using the state feedback from the task space and the adaptive law is performed to update uncertainties in dynamic parameters. The proposed controller is deduced in the sense of Lyapunov theory to guarantee the stability while improving the trajectory tracking performance. Finally, simulation experiment results demonstrate that the proposed control method is insensitive to uncertainties and disturbances and permits to decrease the requirement for the bound of these uncertainties, which validate the eﬀectiveness of the developed control method and exhibit good trajectory tracking performance compared with sliding mode control (SMC) and fuzzy sliding model control (FSMC).


Introduction
Parallel manipulators are widely believed to be an interdisciplinary research area which becomes an outstanding field attracting more and more attention from both industrial and academic domains. Some researchers began to focus on parallel manipulators because of the advantages they hold over their serial counterparts: high stiffness, high accuracy, quick response speed, and high payload to weight ratio. A large number of new architectures dealing with the kinematics and dynamics of parallel manipulators have been proposed in academic [1,2], while some of them have been implemented to practical applications, e.g., parallel kinematic machine [3], motion simulator platform [4], surgical medical equipment [5], dexterous hands [6], etc. However, in order to achieve potential advantages over serial manipulators, parallel manipulators still require improvement in the design, modeling, analysis, and optimization, while the development of control schemes also provides a field for improving the performance of the parallel manipulators to some extent [7].
Due to the complexity of the closed-loop structural system, the dynamic model of parallel manipulator is more complicated than it is in serial manipulator. In [8], the simplified model was obtained by neglecting the friction and their deformation, and based on an assumption that the rigidity of the moving platform is much greater than other components. However, the assumption cannot be generalized for all the parallel manipulators. Liang et al. [9] developed a nonlinear dynamic modeling of a redundantly actuated parallel manipulator based on the flexible multi-body dynamics theory. It is mentioned that the closed-loop dynamic control of the parallel mechanisms is a challenging domain due to the high complexity of their dynamic behavior [10][11][12].
In industrial applications, the parallel manipulators need to achieve trajectory tracking for control implementation, one of the most common control techniques applied to the parallel manipulators is the family of PID controllers (including PI and PD controllers) which are mainly designed in terms of kinematics [13][14][15]. In order to improve the control precision of the existing PID controller, a variable PID parameter controller optimized by a genetic algorithm was proposed by Sheng in [16]. Zhan and Kao [17] proposed a hybrid damped resolved-acceleration control (HDRAC) scheme, which considered both damped acceleration and velocity as a way to remove these unnecessary velocities, a modified particle swarm optimization (MPSO) was adopted to obtain simple and effective estimated damping value, and the illustrative examples of the 3RPS parallel manipulator are provided to verify the utility of the proposed control scheme. But as the parallel manipulator is multi-variable and multi-parameter coupling nonlinear system, its dynamic model is very complicated, and external disturbances such as friction, environment noise are always inevitable during the motion process. erefore, the dynamic model has some uncertainties and the traditional dynamic control generally cannot effectively solve the uncertainty problem. In this case, the advanced controllers can improve the trajectory tracking performance, high kinematic accuracy, and system stability [18]. Sliding mode control (SMC) is a robust method that has an appropriate performance to overcome the disturbance and uncertainties, which is extremely effective when the bounds of disturbances and uncertainties are known [19]. A sliding mode controller is designed using a presumed upper bound of uncertainties as a control parameter, which may generate large amplitude chattering in the control input. If the uncertainty is in the allowable range of the designed controller, the controller will perform well. But if the uncertainty exceeds the limits, the system will become unstable or the controller performance will be degraded greatly. To overcome the aforementioned problems in the design of SMC, fuzzy sliding mode control is proposed in [20] as a model-free controller scheme, which is an alternative effective approach to control such system for its advantage of independent on a precise mathematical model, as well as good robustness and nonlinear characteristics. A modified robust dynamic control (MRDC) method was proposed for a 2-DOF planar parallel robot to realize high precision and high speed cooperation motion. e designed control law was insensitive to the noise signal, which can help to improve error convergence ratio without causing undesirable chattering in [21]. Cazalilla et al. [22] developed an adaptive control of a 3-DOF parallel manipulator by considering the rigid parameters, the friction parameters, the actuator dynamics, and a combination of the former parameters. An adaptive controller was proposed on a planar parallel robot with uncertainties in dynamic and kinematic parameters and the simulation illustrated that the adaptation of the kinematic and dynamic parameters can improve the performance by adjusting the direction of resultant internal force [23,24]. Le et al. [25] presented a novel chattering free neuro-sliding mode controller for two degrees of planar parallel manipulator which had a complicated dynamic model, including frictional uncertainties, modeling uncertainties, and external disturbances. e simulation results showed the effectiveness for the tracking control of the 2-DOF parallel manipulator, and the control system can hold small errors and good robustness against uncertainties and external disturbances.
In addition, the combinations of fuzzy sliding mode controller and adaptive controller have also attracted great interest in the academic fields. Qi et al. [26] proposed a fuzzy adaptive supervisory controller (FASC) for the 4-DOF parallel manipulator to reduce the chattering, and the comparative analysis with the sliding surface control (SMC) indicated that the improved control can reduce greatly the chattering phenomenon and possess good robustness against the parameters uncertainties and external disturbances. Filabi et al. [27] proposed a fuzzy adaptive sliding mode controller for trajectory tracking with considering dynamics of electromechanical actuators, and several simulation cases demonstrated that the proposed control strategy can achieve favorable control performance with regard to uncertainties, nonlinearities, and external disturbances. An adaptive robust controller integrated with online dead-zone estimation was proposed in [28] and the nonlinear characteristics and unknown parameters in the dynamic model can be estimated and compensated by online estimation methods, and the unknown parameters converged to the actual values within finite time. A new extended adaptive fuzzy sliding mode controller and robust observer method was applied for position control of a Stewart manipulator whose parameters are strongly state-dependent and complex, so as to navigate toward the desired trajectory in the presence of uncertainty and unknown upper bound, the upper bound of the uncertainty required to be progressively estimated utilizing the designed adaptation rules [29]. A large number of literatures are presented, and the researches on the control of the parallel manipulators are most concentrated on the planar parallel manipulator and six degrees of freedom parallel manipulator. However, the control schemes on the spatial redundantly actuated parallel manipulator are still merely, furthermore, the conventional PID control is unavailable to be implemented on the designed parallel manipulator, so the advanced control algorithm is carried out on the redundantly actuated parallel manipulator to perform trajectory tracking in this paper. e main contribution of this paper is to propose an alternative adaptive fuzzy sliding mode control algorithm for a 3-DOF redundantly actuated parallel manipulator. e control algorithm combines the advantages of conventional sliding mode controller and fuzzy logic controller. e stability of the control system is theoretically proven in terms of Lyapunov method, and the tracking errors can converge to zero asymptotically. e parameters uncertainties of the control system are compensated by the adaptive control law and the unknown disturbances are rejected by the fuzzy self-tuning robust control law. e simulation experiment results demonstrate that the developed controller can exhibit excellent tracking performance compared with the two other controllers. Furthermore, the appearing characteristics of the proposed controller are fast convergence, high precision, and high robustness as compared with aforementioned controllers. e paper is organized as follows. In Section 2, a 3-DOF parallel manipulator is proposed as the research object. e kinematic and dynamic analysis is carried out, the mapping relation related to the velocity between actuator and endeffector is obtained, and the dynamics equation is established as well. In Section 3, the designed adaptive fuzzy sliding mode controller is introduced and derived in detail based on the sliding mode controller and fuzzy sliding mode controller. e simulation experiment results demonstrate the proposed adaptive fuzzy sliding mode controller can achieve the best trajectory tracking performance compared with other two controllers in Section 4. Finally in Section 5, the several remarks and further work are addressed.

Kinematic and Dynamic Model of the Parallel Manipulator
In this paper, a serial-parallel hybrid kinematic machine tool has been presented, which is a combination of 3-DOF parallel manipulator and two long X-Y tracks. And it can be applied to high speed machining for a large heterogeneous complex freedom surface in the aerospace field, as shown in Figure 1. e main unit module is a 2RPU-2SPR parallel manipulator, and its architectural diagram is depicted in Figure 2, which consists of an upper platform, a fixed base platform, and four actuated limbs connecting the base and upper platform. e parallel manipulator can achieve three degrees of freedom on the basis of the simultaneous movement of the four prismatic actuators. It is a remarkable fact that the proposed parallel manipulator belongs to parallel manipulator with redundant actuation due to the number of actuated joints is greater than that of the degree of freedom. e geometric model of the parallel manipulator and its vector diagram can be seen in Figure 2, where the radius of the upper platform is defined as ra, likewise, the parameter of the fixed base platform is defined as rb, respectively. For purpose of analysis, Figure 2 shows the coordinates of the parallel manipulator; positions of the upper platform and base can be uniquely defined by the coordinates of the four kinematic joints. e vector a i describes the positions of four vertices with respect to the coordinate system A-uvw which is attached on the upper platform. Vector b i describes the positions of four vertices with respect to the coordinate system B-xyz, which is attached on the base platform. Let the parameters 푧, 훼, 훽 be as the three independent variables to describe the translation and rotations of the upper platform, i.e., the three independent degrees of freedom. e mobility analysis of the parallel manipulator including initial configuration and general configuration has been addressed in detail in our previous work [30] by resorting to the screw theory and modified Grubler-Kutzbach (G-K) criterion. To avoid repetition, herein, the mobility regarding the parallel manipulator is not described in this paper. e kinematics problems of parallel manipulator can be divided into the inverse kinematics and the forward kinematics, where the inverse one means to compute the distance of prismatic actuators when the position and orientations are given, while the forward one, vice versa. In general, if given the desired trajectory of the upper platform, the inverse kinematics could be employed to transform the trajectory into the displacement of each actuator, then to formulate the dynamic equation in the joint space. e other way is that the measured displacement of each prismatic joint can be transformed into task space by utilizing the forward kinematics, and then the dynamic equation can be developed in the task operational space. However, there is another simpler approach that the dynamic equation can be developed in task space by means of the camera technique avoiding the complicated forward kinematic solutions [31,32]. e position and orientation of the upper platform with respect to the coordinate frame B-xyz can be described by a matrix T that contains the rotational transformation matrix R and the translation vector p. (1) and the condition J 푇 + J 푇 = E should be satisfied.
In general, due to the presence of structured and unstructured uncertainties and the external disturbance, such as mass, moment of inertia, and friction, it is very difficult to establish an accurate dynamic model, but if the modeling errors caused by the uncertainties are bounded with known functions, then we can utilize modeling errors between the measured value and the real values. erefore, the dynamic equation can be derived by combining the estimated values with parameters errors.
where M (X), Ĉ X,Ẋ and Ĝ (X) are the estimated parameter value of M(X), C X,Ẋ , and G(X), respectively.
Equation (9) subtracting Equation (10), yields In which Here τ is the uncertainty term caused by the parameters errors and external disturbance. For brevity, we denote M(X) as M, C X,Ẋ as C, and G(X) as G.
Simultaneously, the parameters errors can be defined as

Controller Design
e purpose of the control is that the parallel manipulator is at the defined control law u, and the upper platform can still move according to the predicted trajectory and maintain the stability of the system when there are some uncertainties such as modeling errors, parameters errors, external disturbances, unknown loads, etc.
We proposed an adaptive fuzzy sliding mode controller (AFSMC), in order to validate the effectiveness, and comparisons with other control schemes including Sliding Mode Control (SMC) and Fuzzy Sliding Mode Control (FSMC) were conducted. In the following, three controllers: SMC, FSMC, and AFSMC, will be designed to implement the desired trajectory tracking.

Sliding Mode Control.
Define X , Ẋ is the desired state and velocity, X and Ẋ is the actual state and velocity, and e and ė are the tracking error and the derivate of the tracking error, respectively.
Define the sliding surface function s as where λ ∈ 푅 3×3 is the positive matrix.   where s and c are the abbreviation of sine and cosine, respectively, and h represents the motion distance in the direction of perpendicular to the upper platform. e translation vector p can be also written as So the coupling relationship in Equation (3) can be deduced from (2) e length q i of the i-th actuated limb can be derived as where ‖ ⋅ ‖ 2 corresponds to the standard Euclid norm, and s i is the unit vector along the i-th prismatic linear actuator. e derivative of Equation (4) with respect to time can be written in a matrix form as where the matrix J ∈ 푅 4×3 is the Jacobian matrix that indicates the velocity mapping relationship between actuated joints and the end-effector. e dynamic model of the parallel manipulator can be regarded as the motion equation of the closed-loop system. It represents the relationship between the driving force/torque and the variables of joints. e dynamic equations can be established by means of the principle of virtual work [33], the Lagrange method [34], the Newton-Euler method [35] and Kane formulation [36], and so on. Assuming the upper platform is a rigid body, and ignoring friction force of the kinematic joints as well as the uncertainties and external disturbances. e dynamic model can be established in terms of the principle of virtual work as follows [37] where X = [ 푧 훼 훽] represents the independent Cartesian task space vector, q = [ 푞 1 푞 2 푞 3 푞 4 ] 푇 is the joint space vector, F x is the force vector in the task space, is the force vector in joint space, M(X) and M(q) are the inertia matrix, C(X) and C(q) are the Coriolics and Centrifugal coefficient matrix, and G(X) and G(q) are the gravity vector. e dual relationship between the force in task space and in the joint space can be denoted by resorting to the Jacobain matrix J of the kinematics. erefore, the driving force τ can be expressed as where [⋅] + denotes the Moore-Penrose inverse matrix. e Moore-Penrose inverse of the matrix J can be denoted as where is the thickness of boundary switching layer, −휀 and form a continuous approximation region for the switching function, and interval −휀 , 휀 is called a continuous approximation interval or a boundary layer, where a linear variation with respect to sliding mode surface s exits. Hence, the saturation function 푠푎푡(푠/ε) can provide a very smooth control action and reduce the chattering phenomenon of the parallel manipulator.
So the sliding mode control law can be finally expressed as Using the control law Equation (25), the parallel manipulator can realize the trajectory tracking control with parameters uncertainties and external disturbances. But one of the main drawbacks is that the constant gain Λ has to be chosen very large, and the control effort provided by the actuators must also be very large, which may cause chattering to some extent. On account of this, the SMC needs to be improved for good trajectory tracking performance.

Fuzzy Sliding Mode Control.
In order to control the position and orientations of the parallel manipulator effectively, the Fuzzy sliding model control is proposed on the basis of the sliding model control mentioned above. e sliding mode surface function and the change rate ṡ of sliding surface are treated as the input variables, and is the output variable.
Referring to the sliding mode control with sliding mode surface s stated in Equation (25), the fuzzy sliding mode control law can be written as follows where 푢 = N푠푎푡 푠 /휀 , N = diag 푛 1 , 푛 2 , 푛 3 , and n 1 , n 2 , n 3 are adjustable constants.
In fuzzy sliding mode control, the switching gain matrix N is used to compensate the uncertainties that easily arises the chattering and brings the strong oscillation. So in order to decrease the dash, N should be varied with the time according to the tracking errors through the fuzzy logic system. e gains parameters can self-tune based on the sliding surface and its differentiating, which is a big difference from the traditional SMC control method. e fuzzy tuning technique can modify the output variable gain for a continuous adjustment to reduce the errors without destroying the sliding mode control characteristics. e gain is no longer a fixed constant, but the adjustable constant. So according to the Lyapunov stability theory, the closed-loop system is still asymptotically stable and the trajectory tracking errors can converge to the sliding surface. e sliding switching condition can guarantee the desired trajectory to reach the sliding surface within finite time. Once stabilized, the trajectory will remain on the surface all the time, and thus the upper platform can follow the desired trajectory [39,40]. e fuzzy From Equation (15), we can see that if the designed control law is reasonable, and the sliding surface function s will be close to zero asymptotically, then the error e → 0, and ė → 0, and the closed-loop system can be globally stable. Differentiating both sides of Equation (15) Note that, with another particular property of the parallel manipulator that the matrix Ṁ − 2C is indeed skew-symmetric matrix [38], and substituting into Equation (19), differentiating of Lyapunov function V can be derived as, Define sliding mode control law u 1 where K is positive define feedback gain matrix, and Λ is a constant matrix.
Substituting Equation (20) into (19), yields If the condition 儨 儨 儨 儨 휏 儨 儨 儨 儨 ≤ |훬| is satisfied, then Equation (21) can be simplified to erefore, according to the Lyapunov stability theory, the desired trajectory of the closed-loop system will be eventually convergence to the sliding surface And therefore, the error e will evolve toward zero, then the derivative of the error ė will evolve zero as well if time is infinity, which means that the designed controller can guarantee the actual trajectory X to track the given X d , and the corresponding closed-loop system can be globally stable.
Since the range of the sgn function in (20) varies from −1 to 1, then chattering phenomenon is inevitable around the sliding mode surface. erefore, to further avoid the undesirable phenomenon of oscillations with finite frequency and amplitude, the discontinuous sgn function needs to be replaced by a continuous approximation saturation function in Equation (20), i.e., (16) s =ë + λė .   controller mainly consists of three steps: fuzzification, rules evaluation, and defuzzification. In the following, we will introduce in details.

Fuzzification.
In the fuzzy system, the sliding surface s and its derivative ṡ are taken as the input variables of the fuzzy system, and is chosen as the output variable of the fuzzy system. Fuzzy sets of input and output of the system are defined with the following symbols, i.e., where NB denotes negative big, NM denotes negative middle, and NS denotes negative small, ZO denotes zero, PS denotes positive small, PM denotes positive middle, and PB denotes positive big. e membership function of input for s and ṡ are depicted in Figures 3 and 4, respectively. Triangular shape and S-Z membership functions are utilized and the fuzzy interval is limited to −1 to1 and −10 to 10 for ṡ . e membership function of output is shown in Figure 5. Triangular shape and S-Z membership functions are employed and the fuzzy interval is treated as -to 1. e surface viewer is also depicted in Figure  6 about the parameters (s, ṡ , and ). Without loss of generality, there is a proportional coefficient (i.e., ks, kds, ku) between the fuzzy interval and the actual interval, as depicted in Figure 7. e coefficients are very important to a nonlinear uncertainties system. e coefficients can affect the dynamic control performance, and they are generally derived from experience.

Rule Evaluation.
e control rules must be designed such that the actual trajectory of the states always turns toward and does not cross the sliding surface on phase plane to satisfy existence condition. e min-max composition is chosen as a fuzzy Mamdani inference method. A fuzzy logic system determines the relationship between inputs and output. is database demonstrates the collection    en, we introduce some regression matrices for each inertial force term, And finally, the above regression matrices can be written into one big matrix form as, where Y is a known (3 × 36) regression matrix (whose symbolical expression of regression matrix is very complicated and the results document in MATLAB is available here) and Θ is a (36 × 1) parameter vector as follows e dynamics parameters which in the case of the parallel manipulator are implemented for simplicity only considering four parameters (i.e., the mass, , and inertia parameters, I xx, i , I yy, i , I zz, i , actually the dynamic parameters should contain ten estimated inertial parameters values as the form [XX XY XZ YY YZ ZZ MX MY MZ M] T and the coulomb frictions and viscous frictions of the joints, the details are addressed in [42,43]) for each of the nine bodies including the upper (28) ere are 49 fuzzy rules in total, which can be automatically adjusted to achieve satisfactory system response and described in more detail with Table 1.

Defuzzification.
e output of the fuzzy controller is fuzzy sets, so it is necessary to convert these fuzzy sets to numerical values. Here, center of gravity (COG) algorithm is used to perform the defuzzification process. According to the membership function, above mentioned rules can be utilized to complete the individual parameter configuration of the input variables (s, ṡ ) and output variable in their respective domains. e fuzzy control technique can automatically adjust the output variable value online against the uncertainties parameters and external disturbances.

Adaptive Fuzzy Sliding Mode Control.
e aforementioned two controllers need to determine the maximum boundary of the uncertainties and external disturbances, but in fact we have difficulty in obtaining the extreme values. If the value of gain matrix is too large, the controller cannot be employed due to the chattering phenomenon. It is with consideration of its complexity that we proposed an adaptive control scheme based on the previous aforementioned approaches to implement the trajectory tracking in this paper.
Adaptive control can be employed to compensate for parametric uncertainties, constraint uncertainties, and bounded external disturbances. at is to say that adaptive controller is insensitive to uncertainties and permits to decrease the requirement for the bound of these uncertainties, and the controller can hold reliability and appropriate stability in the presence of uncertainties and disturbances, which is the main advantage in actual implementation of this method.
en, one can obtain e adaptive law of the unknown dynamic parameters, can be estimated with Equation (49) is ultimately simplified to As K and N are a positive definite matrix, 푉 (0) = 0, and 푉 (푡) ≤ 0 which is proved to be a semi-negative definite matrix. e sliding surface s will asymptotically converge to zero, and 푡 → ∞, s = 0. From Equation (15), we can see that will asymptotically be zero, ė also converge to zero, and the upper platform can asymptotically track a desired trajectory in the task space, i.e., X → X , the closed-loop system will be globally stable in finite time.
e block diagram of the proposed adaptive fuzzy sliding mode controller is illustrated in Figure 8.

Simulation and Discussion
e parallel manipulator in real environment is fabricated as shown in Figure 9. e implementation of the proposed controller requires controller hardware. erefore, the computer workstation is based on a frequency 2.8 GHz Intel Core i7-7700HQ processor. It should be mentioned that the closedloop system consists of a six-axis motion controller unit, the four prismatic actuators which are driven by four Delta servo motors ECMA-C20604SS with the output power of 0.4 KiloWatts and rated speed of 3000r/min and maximum speed 5000 r/min, and four servo drivers ADSA-B2-0421-B where the incremental photoelectric encoder sensors are mounted coaxially with servo motor to provide position and velocity feedback.
To demonstrate the performance of the proposed control scheme, the adaptive fuzzy sliding-mode controller is first implemented on integrated simulation environment based on platform and eight upper and lower limbs of the parallel manipulator [44], i.e., erefore, the dynamic Equation (6) can be written in a linear regression matrix form as Similarly, the following dynamic control equation can be written as where Θ is the estimate value of the uncertainty dynamics parameter Θ.
In order to obtain the control law and the adaptive law, the following Lyapunov function candidate is defined as It can be shown that the derivative of V with respect to time becomes Considering the relation in Equations (14) and (15) (41) X =Ẋ + λe =Ẋ + s,

Complexity
Meanwhile, the constant parameters employed in the three controllers are presented in Table 3.
To compare the performance of three controllers in the presence of uncertainties and disturbances, some assuming conditions are considered, i.e., the mass and the moment of inertia in dynamic parameters Θ are 15% lower than the real values located in Table 2. Additionally, external disturbances, ὔ , applied to the moving platform and they are modeled with the function as follows: modular Matlab/Simulink, and real prototype experiments will be conducted in the near future. In addition, for the purpose of comparison, SMC and FSMC control techniques are also implemented. A series of simulation experiments are conducted to illustrate the tracking performance of the adaptive fuzzy sliding mode controller. e geometric parameters used for the parallel mechanism and the location of the gravity center with regard to the world coordinate system B-xyz obtained by the SolidWorks are listed in Table 2.   where j represents the j-th point of tracking trajectory, and m is the total number of data point. e tracking error of the i-th limb is similar to Equation (55), and the obtained results are summarized in Table 4.
As observed in Table 4, the different parameters cases are presented. In comparison with MSC and FMSC, the results illustrate that the FSMC controller has much better root square mean error performance than the SMC controller regardless of which parameter listed above. However, the proposed adaptive controller possesses the best performance compared with other two controllers. Obviously, the displacement parameters (i.e., , ) can follow the desired trajectory in a better performance, and the rotational angle also has good performance compared with the rotational angle who still need to be improved to some extent. e trajectory tracking performance of three controllers with the parameters uncertainties and external disturbances are shown in Figures 10-12.
In comparison with Figures 10 and 11, the trajectory tracking errors applied the adaptive fuzzy sliding model controller are a little smaller than the errors associated with sliding mode controller and fuzzy sliding mode controller. In other words, the trajectory tracking performance employed adaptive fuzzy sliding mode controller is better than the sliding mode controller and fuzzy sliding mode controller. As shown in Figure 12, there is a big difference in the response speed and time among three controllers, which indicates the adaptive fuzzy sliding mode controller has better transient response characteristic than other two controllers. It is also concluded that the tracking performance is affected with different controllers, and the proposed adaptive fuzzy sliding mode controller possesses superior trajectory tracking performance to others.
Analogously, the tracking performances of the four actuated limbs with three types of controllers are also depicted in Figures 13-16.
As can be seen from the above figures, the proposed adaptive sliding mode controller combining the sliding mode and fuzzy logic can perform better tracking performance than the two other controllers. For all the trajectories of the actuated limbs, the adaptive fuzzy sliding mode controller shows a good tracking performance. With consideration of uncertainties and disturbances, the adaptive fuzzy sliding mode controller is more robust and insensitive to irregular parametric variation compared with the two other controllers. It is obvious of the second limb and the fourth limb in Figures 14 and 16, the proposed adaptive fuzzy sliding mode controller has a much faster response speed and follows the desired trajectory much better as well.
For verifying the superiority of the proposed adaptive fuzzy sliding mode controller, the tracking error curves of the (56) Specifically, the desired trajectory functions of the upper platform in the workspace are defined to achieve zero velocity and acceleration initially and are as follows Furthermore, to quantify the trajectory tracking performance of three controllers, the root square mean error (RSME) is introduced as evaluation index, where the translational displacements and rotational angles are described in separated equation as follows (53) 휏 ὔ = 2 × 푟푎푛푑표푚(−1, 1). (55) trajectory tracking can be stable in 0.4 s, which illustrates the adaptive fuzzy sliding mode controller designed in this paper can quickly and accurately estimate the actual value of the parameters to trace the desired trajectory. In particular, the proposed controller can bring about the smallest tracking errors, i.e., approximately converging to zero, compared with sliding mode controller and fuzzy sliding mode controller.
In addition, the trajectory tracking errors under three controllers can be also drawn with box plot, which can display the full range of variation from minimum to maximum, and parameters (푧 훼 훽) are drawn, respectively, in  rough comparative analysis, the tracking error curves based on the sliding mode controller and fuzzy sliding mode controller can be close to zero asymptotically, especially the rotation angle . However, the tracking error curves applied the proposed adaptive fuzzy sliding mode controller are asymptotically stable and eventually converge to zero, indicating the closed-loop system is globally stable within finite time. From the trends of the points in simulation process, the     13 Complexity substantially more variation in the SMC controller which ranges approximately from −0.0018 to 0 whereas the one in FSMC controller ranges approximately from −0.001 to 0. e other Figures 21 and 22 are similar to Figure 20, and we can see that the proposed adaptive controller is superior to the two other controllers in terms of tracking accuracy and stability through comparative analysis, respectively. e trajectory tracking errors of the proposed adaptive fuzzy model controller tend to be zero asymptotically, which illustrate that the proposed controller can effectively compensate the uncertainties caused by parameters variation to deduce the tracking errors.
Moreover, in order to validate the behavior of the mentioned controllers, the input of driving force of the each limb is also very important compared with the desired driving force in Cartesian joint workspace. Figures 23-26 illustrate the tracking errors of four driving force parameters.
As observed in Figures 24-26, the sliding model controller and fuzzy controllers fail to cope with the external disturbances successfully. However, the proposed adaptive fuzzy model controller can eliminate the parameters uncertainties and disturbances, and can be also effectively implemented a typical median value. From Figure 20, the datasets are approximately balanced around zero; evidently the median value in the three cases is near to zero. However, there is    regarding parameters uncertainties and irregular external disturbances. In addition, it can be observed that the parameters uncertainties and external disturbances are tremendously compensated for utilizing the proposed adaptive controller. at is to say, the proposed controller can effectively prevent high frequency fluctuation resulting in a more stable trajectory tracking performance, which further demonstrates that the proposed adaptive fuzzy sliding mode controller is superior against the uncertainties and external disturbances compared with the other two controllers mentioned above.

Conclusions
In this paper, a 3-DOF 2RPU-2SPR parallel manipulator has been presented, which can be applied to high speed machining in serial X-Y linear rail for milling some large heterogeneous complex freedom surfaces in the aerospace field. According to the investigation of this paper, the following contributions can be drawn: (1) e dynamic model for the 3-DOF parallel manipulator is formulated in the task space utilizing the principle of the virtual work, and the generalized coordinates can be separated into three independent parameters to significantly reduce the closed-loop system complexity in terms of kinematics coupling characteristics.
(2) To achieve high accuracy and stable cooperation, an adaptive fuzzy sliding mode controller for a 3-DOF parallel manipulator is developed in detail. e proposed controller is designed based the sliding mode control and fuzzy logic rules for self-tuning the gains.
In the presence of parameters uncertainties and external disturbances with unknown upper bound, a linear regression dynamic equation and adaptive update law are developed for the proposed controller. Besides, the stability of the proposed controller is proven in terms of the Lyapunov theorem. (3) e simulation experiment results demonstrate that the proposed controller can exhibit accurate and robust trajectory tracking performance without some chattering, which are superior to the results obtained with sliding mode control and fuzzy sliding mode control. erefore, the proposed adaptive fuzzy sliding mode controller is theoretically effective and practical for the experimental prototype. Further works would be devoted to extending the adaptive controller to unknown parameters identification in practical experiment applications with due consideration of coulomb frictions and viscous frictions term.
Data Availability e data used to support the findings of this study are available from the corresponding author upon request. online to deal with uncertainties and external disturbances in the closed-loop system. From the enlargements of the localized regions, it can be observed that the chattering or oscillation phenomenon still remains with the sliding mode controller and fuzzy sliding mode controller. If we increase the thickness of boundary layer or decrease the switching gains for reducing the fluctuation of the chattering, then the tracking errors will be increased and the robustness of the closed-loop system will not be guaranteed. Especially from Figures 24 and 26, we can see that the amplitude of the driving force fluctuation of the second and fourth limbs applied the adaptive fuzzy sliding mode controller, which are low in comparison with the other two controllers. Furthermore, the sliding mode controller vibrates more seriously than the fuzzy sliding mode controller which indicates the two controllers presented chattering problem, while adaptive fuzzy sliding mode controller results in a relative smooth driving force. erefore, the simulation results show that, in comparison with other two controllers, adaptive fuzzy sliding mode controller can deduce the undesirable chattering and improve the robustness of the parallel manipulator