RBF Neural Network of Sliding Mode Control for Time-Varying 2-DOF Parallel Manipulator System

Abstract

This paper presents a radial basis function (RBF) neural network control scheme for manipulators with actuator nonlinearities. The control scheme consists of a time-varying sliding mode control (TVSMC) and an RBF neural network compensator. Since the actuator nonlinearities are usually included in the manipulator driving motor, a compensator using RBF network is proposed to estimate the actuator nonlinearities and their upper boundaries. Subsequently, an RBF neural network controller that requires neither the evaluation of off-line dynamical model nor the time-consuming training process is given. In addition, Barbalat Lemma is introduced to help prove the stability of the closed control system. Considering the SMC controller and the RBF network compensator as the whole control scheme, the closed-loop system is proved to be uniformly ultimately bounded. The whole scheme provides a general procedure to control the manipulators with actuator nonlinearities. Simulation results verify the effectiveness of the designed scheme and the theoretical discussion.

1. Introduction

The past several decades have seen a rapid increase in parallel manipulators connected to the control system. It is well known that parallel manipulators with high rigidity are of generally higher accuracy and of lower error accumulation than similarly sized serial manipulators. Their closed kinematics structure allows them to obtain high structural stiffness and perform high-speed motions. The inertia of its mobile parts is reduced, since the actuators of a parallel manipulator are often fixed to its base and the end effectors can perform movements with higher accelerations. Adaptive tracking control for a class of nonlinear systems is given in [1–3], and in [4] a design of the sliding mode surface integral algorithm is proposed to inhibit the steady-state error and enhance the robustness. In [5, 6] sliding mode control (SMC) with low-pass filter is used to keep trajectory tracking accurately.

It is known that SMC has the intrinsic nature of robustness, good transient fast response, and insensitivity to the variation of plant parameters and external disturbances in [7]. Thus, the SMC is considered as an effective approach for the control of many systems such as uncertain nonlinear systems in [8], discrete-time nonlinear systems in [9], and singular stochastic hybrid systems in [10–12]. The control process of SMC consists of two parts; one part is continuous and the other discontinuous. When the system reaches the sliding mode, the system with variable structure control is insensitive to the external disturbances and the variations of the plant parameters, and it has been widely applied to the manipulator system due to its operation characteristics for the sake of fastness, robustness, and stability in large load variations. All those merits are gotten at the cost of the chattering. Furthermore, in many time-variable systems, parameters and perturbation upper bounds are often uncertain, so switching control gain should be as high as possible to keep stable [13–16]. The control algorithm in [17–20] reduces the reaching phase extremely and achieves better robustness than SMC. However, in [17] with disturbance observer based on SMC, the algorithm assumes that the disturbance is produced by a linear exogenous system, but in fact it is difficult to accurately predict the uncertainty and disturbance of the time-variable system. In [18–20], the adaptive sliding mode control (ASMC) algorithm is proposed with the switching gain excessive the SMC values, which brings the more serious chattering problems. So the appropriate switching gain is the key to the accurate control in time-variable system.

Motivated by the above observations, we propose a new approach to using RBF neural network to estimate the actuator nonlinearities and their upper boundaries for the switching gain. In this paper, we aim to solve these problems by focusing on the accurate tracking problem of the uncertain mechanical system. Our research begins with a time-variable SMC (TVSMC) algorithm with the dynamics of the manipulator. The main characteristics of such a TVSMC algorithm are discussed. To deal with the finite static error brought by the continuous approximation of the TVSMC algorithm, the RBF neural network is utilized, a novel Lyapunov function is introduced, and then a new time-variable stability criterion is presented. In the numerical simulation, the RBF TVSMC algorithm and the TVSMC algorithm are compared after being tested.

2. System Model

The 2-degree-of-freedom parallel manipulator (2-DOF parallel manipulator) is made up of three groups of two links in one platform, in which one group has a base of open chain mechanism, respectively, installed by the AC servo motor and a speed reducer drive shown in Figure 1 and coordinate in Figure 2. The 2-DOF parallel manipulator system can be described by Laugrange’s equations [21, 22]: 𝐽𝜃⃛+𝐵𝜃̈+𝑊𝜃̇=𝑈+𝑈_𝑑. (1)

Figure 1 

Structure of 2-DOF parallel manipulator.

Figure 2 

Coordinate of 2-DOF parallel manipulator.

In (1),𝜃∈𝑅³stands for a displacement angle of generalized coordinates,𝐽for the symmetric and positive-definite inertia matrix,𝐵for a damping coefficient matrix,𝑊for a stiffness coefficient matrix,𝑈for a voltage vector of generalized control input, and𝑈_𝑑for a damping voltage vector of external disturbance.

Defining𝐽=𝐽̂+Δ𝐽,𝐵=𝐵+Δ𝐵, and𝑊=𝑊+Δ𝑊, the superscript(∧)stands for the nominal value, and the notation(Δ)represents the uncertainty. According to the structural feature, the effect caused by the model uncertainties can be merged into the disturbance term, which then can be regarded as the lumped disturbance in the following form: 𝑑=𝑈_𝑑−Δ𝑊𝜃̇−Δ𝐵𝜃̈−Δ𝐽𝜃⃛. (2)

From (2), it is assumed that the lumped disturbance is bounded by a upper bound; that is,‖𝑑‖_∞≤𝑑_max, where𝑑_max∈𝑅⁺is a constant scalar and‖⋅‖_∞is the infinite norm of a vector.𝐽̇̂−2𝐵is a skew-symmetric matrix (see details in [23, 24]).

So (1) can be rewritten as 𝐽𝜃̂⃛+𝐵𝜃̈+𝑊𝜃̇=𝑈+𝑑. (3)

3. TVSMC Design and Parameter Setting

In this paper, we will address the tracking control problem of the system in (3). For the desired trajectory with the system states of𝜃_𝑑,𝜃̇_𝑑, and𝜃̈_𝑑, a controller𝑢for the system is designed so that the system states𝜃_𝑑,𝜃̇_𝑑, and𝜃̈_𝑑can track the desired trajectory in the presence of parametric uncertainty and external disturbance.

For the given trajectory, the tracking error and tracking angle are defined as 𝐸=𝜃−𝜃_𝑑,𝐸̇=𝜃̇−𝜃̇_𝑑,𝐸̈=𝜃̈−𝜃̈_𝑑, (4) with the initial tracking error satisfying𝐸(0)≠0_3×1,𝐸̇(0)≠0_3×1, and𝐸̈(0)≠0_3×1, where the subscript denotes the appropriate dimensions of the matrix.

In order to derive the SMC algorithm, the switching surface can be chosen as 𝑆=𝑠₁𝑠₂𝑠₃^𝑇=𝐶𝐸, (5) where𝐶∈𝑅^3×3is the matrix with strictly positive every elements and𝐸=(𝐸𝐸̇𝐸̈)^𝑇. The switching surface is then determined by𝑆=0_3×1, which is the desired dynamics.

The input of SMC algorithm is set as 𝑈=𝑢_eq+𝑢_𝑛. (6)

In (6),𝑢_eqis the equivalent control. The equivalent control of the ideal sliding mode is obtained on𝑆=0with𝑆̇=0; it can be gotten as 𝑢_eq=−𝐽̂𝐶₁𝐸̇+𝐶₂𝐸̈−𝐵𝐶₁𝐸+𝐶₂𝐸̇+𝐽𝜃̂⃛_𝑑+𝐵𝜃̈_𝑑+𝑊𝜃̇_𝑑. (7)𝑢_𝑛is the switching control: 𝑢_𝑛=−𝛾⋅sign(𝑆). (8)𝛾=diag(𝛾₁,𝛾₂,𝛾₃)∈𝑅^3×3is the switching gain matrix with the elements𝛾_𝑖>𝑑_max, and thesign(𝑆)=(sign(𝑠₁)sign(𝑠₂)sign(𝑠₃))^𝑇 presents the sign function.

Theorem 1. Considering the system in (3) under the lumped disturbance with an upper boundary of𝑑_max, by adopting the time-varying sliding mode function in (5) and the corresponding input control in (6), if𝛾_𝑖>𝑑_max, then the controlled system is stable.

Proof. Consider the following Lyapunov function: 1𝑉=2𝑆^𝑇𝐽𝑆̂. (9)
Differentiating𝑉with respect to time yields 𝑉̇=𝑆^𝑇𝐽𝑆̂̇+12𝑆^𝑇𝐽̇𝑆̂=𝑆^𝑇𝐽̂𝐶₁𝐸̇+𝐶₂̈𝐸⃛+1𝐸+2𝑆^𝑇𝐽̇𝑆̂=𝑆^𝑇𝑈+𝑑+𝐽̂𝐶₁𝐸̇+𝐶₂𝐸̈−𝐵𝐸̈−𝑊𝐸̇−𝐽𝜃̂⃛_𝑑−𝐵𝜃̈_𝑑−𝑊𝜃̇_𝑑+12𝑆^𝑇𝐽̇̂𝐶₁𝐸+𝐶₂𝐸̇+𝐸̈=𝑆^𝑇𝑈+𝑑+𝐽̂𝐶₁𝐸̇+𝐶₂𝐸̈+𝐵𝐶₁𝐸+𝐶₂𝐸̇−𝐽𝜃̂⃛_𝑑+𝐵𝜃̈_𝑑+𝑊𝜃̇_𝑑. (10)
𝑈=𝑢_eq+𝑢_𝑛is substituted in Lyapunov function, and for𝐽̇−2𝐵is a skew-symmetric matrix such that(1/2)𝑆^𝑇(𝐽̇̂−2𝐵)𝑆=0. The time derivative of𝑉will be 𝑉̇=𝑆^𝑇(𝑆)=−𝛾sign+𝑑_3𝑖=1−𝛾_𝑖⋅𝑠_𝑖+𝑑_𝑖𝑠_𝑖≤−³_𝑖=1𝛾_𝑖−𝑑_max𝑠_𝑖<0. (11)
As𝑉is positively defined and𝑉̇<0, according to the Lyapunov stability theory, the TVSMC is stable. And Figure 3 is gotten.

(a) Phase locus

(b) Sliding mode function

Figure 3 

Phase locus and sliding mode function.

4. RBF SMC Algorithm Design

From Figure 3(b), the process of SMC action is divided into two states: one is the reaching state and the other is the sliding state. Then reaching state will be joined to the sliding state inside the sliding tranche with bandwidth2𝛾surrounding the sliding surface, in a finite time limited by the switching frequency. The system adopts the dynamic of the surface and reaches the equilibrium point. In a short phrase, the switching gain depends on𝛾. Consider the following saturation function to replace the sign function to decrease chattering: (𝑆)=𝑆sat|𝑆|_∞,𝑠+𝛿^𝑖(𝑆)≤𝛿,sign,otherwise. (12)

In (12)𝛿is tiny positive number and is also the boundary layer thickness, which can reduce the chattering if appropriately chosen.

On the other hand, radial basis function neural network based on controller design is one of the popular methods of high precision control. In the SMC based controller, the RBF NN is used to approximate the upper boundary of lumped disturbance (Figure 4).

Figure 4 

Architecture of the RBF neural network.

The upper boundary of the lumped disturbance can be designed as 𝛾_𝑖=⁴_𝑗=1𝑤_𝑖𝑗ℎ_𝑖𝑗, (13) whereℎ_𝑖𝑗is the Gaussian radial basis function.

Here, further consideration will be given about adjusting the network weights. The weight adjustment is𝐴_𝑑=(1/2)𝐸^𝑇𝐸.

The RBF NN earning algorithm is Δ𝑤=−𝜕𝐴_𝑑𝜕𝑤=−𝐸^𝑇𝜕𝐸𝜕𝑤=−𝐸^𝑇𝜕𝐸𝜕𝑢𝜕𝑢𝜕𝛾𝜕𝛾𝜕𝑊≈−𝐸^𝑇sign𝜕𝜃𝜕𝑢𝜕𝑢𝜕𝛾𝜕𝛾,𝜕𝑊 (14) where the value of(𝜕𝜃/𝜕𝑢)can be substituted by coefficient of learning rate,𝜃is in direct proportion to𝑢in step response, sign(𝜕𝜃/𝜕𝑢)=1,(𝜕𝑢/𝜕𝛾)=−sign(𝑆), and(𝜕𝛾/𝜕𝑊)=𝜙.

SoΔ𝑤can be described as follows: Δ𝑤≈𝐸^𝑇sign(𝑆)𝜙=𝐶^−1𝑇𝑆^𝑇⋅𝜙. (15)

Theorem 2. Considering the system in (3) under the lumped disturbance with an upper boundary of𝑑_max, by adopting the time-varying sliding mode function in (5) and the lumped disturbance identified by using the RBF neural network in (13), the controlled system is global asymptotic stable.

Proof. Let optimal network weights be𝑤^∗and the estimated upper boundary value of RBF NN be𝛾; then we get ³_𝑗=1𝑤^∗_𝑗𝑖ℎ_𝑗−𝛾_𝑖=𝜀,|𝜀|<𝜀₀, (16) where𝜀is a tiny number and𝜀₀is tiny positive number, and let𝛾_𝑖−|𝑑_𝑖|>𝜀₁>𝜀₀,𝑤̇=(𝐶⁻¹)^𝑇|𝑆^𝑇|𝜙. A neural Lyapunov function can be designed as 𝑉₁=12𝑆^𝑇̂1𝐽𝑆+2𝑤^∗−𝑤^𝑇𝐶^𝑇𝑤^∗−𝑤. (17)
Differentiating𝑉₁with respect to time yields, we get 𝑉̇=𝑆^𝑇𝐽𝑆̂̇+12𝑆^𝑇𝐽̇𝑆̂−𝑤^∗−𝑤^𝑇𝐶^𝑇𝑤̇, (18)
with𝑈=𝑢_eq+𝑢_𝑛and simplifying 𝑉̇₁=𝑆^𝑇−𝛾sign(𝑆)+𝑑−𝑤^∗−𝑤^𝑇𝐶^𝑇𝑤̇=𝑆^𝑇(𝑆)+(𝑆)−(𝑆)−𝛾sign𝛾sign𝛾sign+𝑑−𝐶^𝑇𝑤^∗−𝑤^𝑇𝑤̇≤−𝑆^𝑇𝑤^𝑇𝜙−𝑤^∗𝑇−𝑆𝜙+𝜀_𝑇𝛾−|𝑑|−𝑆^𝑇𝑤^∗−𝑤^𝑇𝜙=−³_𝑖=1𝑠_𝑖⋅⁶_𝑗=1𝑤_𝑖𝑗−𝑤^∗_𝑖𝑗ℎ_𝑗−𝑆^𝑇𝜀−𝑆^𝑇|𝑑|−𝛾−_3𝑖=1𝑠_𝑖⁶_𝑗=1𝑤_𝑖𝑗−𝑤^∗_𝑖𝑗ℎ_𝑗=−𝑆^𝑇𝜀−𝑆^𝑇|𝑑|≤𝑆𝛾−_𝑇𝜀₀−𝑆^𝑇𝜀₁=−𝑆^𝑇𝜀₁−𝜀₀≤0. (19) Using integral transform in (19), the neural Lyapunov function can be gotten as 𝑉₁(𝑡)=𝑉₁(0)+^𝑡₀𝑉̇₁𝑑𝜏≤𝑉₁(0)+^𝑡₀−𝑆^𝑇𝜀₁−𝜀₀𝑑𝜏. (20) By the Barbalat Lemma, the system is asymptotically stable.

5. Simulation Tests

The parameters of the 2-DOF parallel manipulator are listed as follows: joints reduction ratio is 40?:?1,𝑙₁₁=𝑙₁₂=𝑙₂₁=𝑙₂₂=𝑙₃₁=𝑙₃₂=244mm, back electromotive force constant is 0.04297?V/(s/rad), torque constant is 3.41?N·m/A, resistance factor in driving side is 8.1 × 10^-5?N·m/(rad/s), winding resistance of electrical machine is 1.025?O, winding inductance of electrical machine is 0.03837?H, rotational inertia𝐽is 0.39?kg·m², and the uncertain inertiaΔ𝐽≤0.1?kg·m². The desired manipulator trajectory is the right triangle shown in Figure 5, and the desired joints trajectories are shown in Figure 6.

Figure 5 

The desired manipulator trajectory.

Figure 6 

The desired joints trajectories.

The trajectory tracking processes controlled by the TVSMC algorithm and the RBF TVSMC algorithm are compared in Figures 7–12 with the corresponding angle displacement in Figure 6.

(a) Control voltage of joint𝐴1

(b) Control voltage of joint𝐴2

(c) Control voltage of joint𝐴3

Figure 7 

Control voltage generated by TVSMC.

(a) Control voltage of joint𝐴1

(b) Control voltage of joint𝐴2

(c) Control voltage of joint𝐴3

Figure 8 

Control voltage generated by RBF TVSMC.

(a) Trajectory tracking responses of joint𝐴1

(b) Trajectory tracking responses of joint𝐴2

(c) Trajectory tracking responses of joint𝐴3

Figure 9 

Trajectory tracking by TVSMC.

(a) Trajectory tracking responses of joint𝐴1

(b) Trajectory tracking responses of joint𝐴2

(c) Trajectory tracking responses of joint𝐴3

Figure 10 

Trajectory tracking by RBF TVSMC.

(a)

(b)

(c)

Figure 11 

Angle displacement error responses controlled by TVSMC.

(a)

(b)

(c)

Figure 12 

Angle displacement error responses controlled by RBF TVSMC.

According to the angle displacement responses in Figures 7 and 8, it can be seen that the maximum control voltage generated by TVSMC algorithms is 10?V, while that by RBF TVSMC algorithms is no more than 2.5?V. And the control voltage generated by the RBF TVSMC algorithms is more stable than the that by TVSMC algorithm. When we focus on the control accuracy, significant differences exist in steady-state regime of the closed-loop system as seen from the angle trajectory tracking responses in Figures 9 and 10. Figure 10 has more accurate trajectory tracking responses than Figure 9. According to angle displacement error responses of joint A1 in Figures 11 and 12, we can see that the angle error of joint stable at origin is within 40?s based on TVSMC algorithm, but based on RBF TVSMC algorithm it is not later than 3?s. The average error responses controlled by the TVSMC algorithm are about 0.3?rad, while for the RBF SMC algorithm, the average error is less than 0.03 rad, which proves the precision improvement of the RBF TVSMC algorithm.

6. Conclusions

In this paper, the accurate tracking control problem of 2-DOF parallel manipulator in the presence of parameter variation and uncertain disturbance is investigated via the TVSMC technique. An effective method is provided for the parameter selection in the TVSMC framework. RBF NN based time-varying sliding mode control algorithm is proposed to address the global chattering problem and increase the control accuracy. Simulation results verify the effectiveness of the proposed algorithm.

(1)This paper provides a trajectory algorithm of RBF neural network based time-varying sliding mode control for 2-DOF parallel manipulator system. And a compensator using RBF network is proposed to estimate the actuator nonlinearities and eliminate their upper boundaries. So that an RBF neural network controller can work properly requiring neither the evaluation of off-line dynamical model nor the time-consuming training process. The other article, “The Implementation to Servomotor Based on RBF Neural Network Equivalent to Sliding Mode Variable Structure Control,” provides a trajectory optimization algorithm for 2-DOF parallel manipulator system of servomotor which is assumed as one linear system. The algorithm of input control is divided into two parts: one is the sliding mode control with the linear control and the other is the nonlinear control of output of the RBF replacing the switching input.(2)This paper is further research based on the other article, and it involves the three joints’ trajectories but the other article just focuses on one joint.

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and constructive suggestions. This work is supported by the National Natural Science Foundation of China (61201248) and Youth Foundation of JUST (KYY11081).