Adaptive block compensation trajectory tracking control based on LuGre friction model

Abstract

Aiming at the problems of modeling error and uncertain external disturbance in the multi-joint robot control model, an adaptive block compensation trajectory tracking controller based on LuGre friction model is proposed. Firstly, the algorithm divides the interference term of LuGre friction model into three parts with different physical quantities. Secondly, an adaptive neural network compensator is designed to assess the three parts of the LuGre friction model. Thirdly, a robust sliding mode controller is developed to reduce the influence of these estimation errors of neural network compensator and other uncertain disturbances and ensure that the system converges in a finite time at the same time. Finally, numerical simulations under different input and disturbance signals for the planar multi-joint robot and the inverted pendulum are conducted to validate the effectiveness of the proposed controller, and the performance of the proposed controller is compared with conventional sliding mode controller to illustrate the usefulness and efficiency of the proposed controller.

Keywords

Multi-joint robot LuGre friction model block compensation adaptive neural network robust sliding mode control

Introduction

The uncertainty of parameters and complex friction types of multi-joint robot are the main factors that make it difficult to establish the complete and accurate dynamic model of multi-joint robot. Some scholars have studied the sliding mode control method based on neural network.^{1

–15} For example, Vikas¹ studied wavelet neural networks controller based on fast terminal sliding mode control, where the wavelet neural network is used to compensate the unknown robot dynamics. Wai and Muthusamy² proposed a sliding mode controller based on fuzzy neural network genetic algorithm, where a projection algorithm is used to derive the adaptive adjustment algorithm of network parameters. Pham and Yaonan³ proposed a robust compensator based on radial basis function neural networks (RBFNNs), which implemented high-precision position tracking control and ensured the stability and robustness of the controller. Van and Wang⁴ proposed a robust adaptive control method based on RBFNNs for the joint position control and predefined trajectory tracking control of two link cleaning and detecting robot manipulators, where a three-layer RBFNN is used to approximate nonlinear robot dynamics. Considering the trajectory tracking of robotic manipulators with structured and unstructured uncertainties, Liu and Zhang⁵ proposed a neural network-based robust finite-time control strategy, which possesses finite-time convergence and strong robustness. Boudjedir et al.⁶ proposed an adaptive neural network control with neural state’s observer for quadrotor, where a new neural observer based on the structure of sliding mode observer is used to estimate the states. Chiang et al.⁷ proposed a sliding mode angle controller with neural network estimator for a fan-plate system, where the neural network estimator is used to estimate the unknown lumped bounded uncertainty of parameter variations and external disturbances. Tuan et al.⁸ proposed a robust adaptive system for a ship-mounted container crane using second-order sliding mode control and designed a modeling estimator on the basis of RBF network, which approximates almost all the structure of a crane model. Rahmani et al.⁹ proposed an adaptive neural network integral sliding mode controller to control a biped robot, where the adaptive neural network is applied to estimate the unknown disturbances. Vijay and Jena¹⁰ developed an adaptive observer backstepping terminal sliding mode control for three degrees of freedom overhead transmission line deicing robot manipulator, where the neural network observer is used to estimate tracking position and velocity vectors of the controller. Jung¹¹ presented a neural network control technique to improve the tracking performance of a three-link rotary robot manipulator, where a neural network compensator is used to deal with the stability and performance more intelligently. Toshani and Farrokhi¹² developed an optimal sliding mode control method based on the projection recurrent neural networks for a class of linear systems. Nguyen et al.¹³ proposed a neural network-based adaptive sliding mode control method for tracking of a nonholonomic wheeled mobile robot subject to unknown wheel slips, model uncertainties, and unknown bounded disturbances, where self-recurrent wavelet neural networks are employed to approximate unknown nonlinear functions. Mien¹⁴ proposed an adaptive neural integrated sliding mode control method, which integrates the advantages of neural network approximation ability and the robustness of integral sliding mode control. Pavol and Yuri¹⁵ established the multi-fingered manipulator model and conducted target recognition, check and attitude tracking based on the deep learning model of convolutional neural network. However, these aforementioned neural network sliding mode controllers have just provided an overall compensation control for structured and unstructured uncertainties. Most of them have paid no attention to the separate compensation control for these uncertainties. As a matter of fact, there are great differences between structured and unstructured uncertainties in many ways, for example, structured uncertainties are characterized by the existence of an upper bound, but unstructured uncertainties may be unbounded. These characteristics should be taken into account in the exploration of the control scheme for a robot manipulator with uncertain dynamics. Therefore, this article proposes an adaptive block compensation trajectory tracking controller.

There are diverse frictional disturbances in the multi-joint robot control model. To reduce the negative effect of friction interference on robot’s control performance, some scholars have proposed compensation methods for friction interference.^16
–18 For example, Luo et al.¹⁶ designed a disturbance observer based on RBFNN to compensate for friction interference; however, the linear disturbance observer is only effective for a certain bandwidth signal but not enough for the friction signal acting on the whole bandwidth region. Ufnalski and Grzesiak¹⁷ designed a feed forward compensation method for friction interference based on neural network, but the speed tracking signal brought compensation error. Wang et al.¹⁸ designed two neural networks identifier based on different velocity directions to compensate for friction interference; however, the two neural networks identifier has the disadvantages of related nonlinearity and measurement limitations. In view of the abovementioned problems, this article proposes an adaptive block compensation trajectory tracking controller based on RBFNN, where the local approximation characteristics of the neural network are used to compensate for the three friction subfunctions with different physical quantities. A robust term is designed to eliminate the estimation errors and external disturbance of neural network.

This article is organized as follows. In the “Problem formulation and preliminaries” section, the robot control problem is formulated, and a brief description of the LuGre friction model is presented. In the “Design of controller” section, an adaptive block compensation trajectory tracking controller is developed. In the “Proof of convergence” section, the convergence of the proposed controller is analyzed. In the “Simulation results and discussion” section, the simulation results are presented and discussed. Conclusion is given in the last section.

Problem formulation and preliminaries

Control problem of multi-joint robot manipulator

According to the Lagrange–Euler approach, the dynamic model of the multi-joint robot manipulator with n serial links is expressed as follows

M (q) \ddot{q} + C (q, \dot{q}) \ddot{q} + G (q) + F (\dot{q}) + τ_{d} = τ

where $q \in R^{n}$ is the vector of joint displacement; $\dot{q} \in R^{n}$ represents the vector of joint velocity; $\ddot{q} \in R^{n}$ denotes the vector of joint acceleration; $M (q) \in R^{n \times n}$ is the inertia matrix bounded by $‖ M (q) ‖ \leq k_{m}$ with positive constant k_m; $C (q, \dot{q}) \dot{q} \in R^{n}$ is the vector representing centrifugal and Coriolis forces; $G (q) \in R^{n}$ denotes the vector of gravitational torques; $F (\dot{q}) \in R^{n}$ is the vector representing the dynamic effects of nonlinear frictions; $τ_{d} \in R^{n}$ is the vector of the external disturbances, τ_d is assumed to be bounded by $‖ τ_{d} ‖ \leq d_{m}$ with positive constant d_m; and $τ \in R^{n}$ is the vector of joint torques supplied by the actuators.

For the dynamic model of the multi-joint robot manipulator described in equation (1), it is generally of the following properties.

Property 1

$M (q)$ is a positive symmetric matrix, and its inverse matrix $M^{- 1} (q)$ exists. $M (q)$ and $M^{- 1} (q)$ as the function of q are uniformly bounded. That is, there are positive numbers $M_{min}$ and $M_{max}$ , which make the following formula holds: $0 < M_{min} \leq M (q) \leq M_{max}$ .

Property 2

$\dot{M} (q) - 2 C (q, \dot{q})$ is an oblique symmetric matrix, namely the following equation holds for any $x \in R^{n \times 1}$ : $x^{T} [\dot{M} (q) - 2 C (q, \dot{q})] x = 0$ .

Property 3

There is a parameter vector that depends on the robot parameters, so that $M (q)$ , $C (q, \dot{q})$ , $G (q)$ , and $F (\dot{q})$ satisfy the linear relationship

M (q) \ddot{q} + C (q, \dot{q}) \ddot{q} + G (q) + F (\dot{q}) = Φ (q, \dot{q}, \ddot{q}) θ

where $Φ (q, \dot{q}, \ddot{q}) \in R^{n \times m}$ is the regression matrix of the known joint variable function, and it is a known function matrix for the generalized coordinates of the robot and its derivatives. $θ \in R^{n}$ is an unknown constant parameter vector describing the quality characteristics of the robot.

LuGre friction model

In equation (1), the term $F (\dot{q})$ represents the nonlinear dynamic friction vector. The LuGre friction model adopts the idea of a bristle friction model and models it based on the average deformation of the bristles. The LuGre friction model has been used to describe most of the friction phenomena observed in experiments, such as Stribeck effect, preslip, and variable maximum static friction force.² Since the LuGre friction model can not only accurately predict the important characteristics of friction but also can better compensate the friction term dynamically, this article uses the LuGre friction model to conduct friction dynamic modeling for the multi-joint robot manipulator.

The average deformation of the bristles is represented by a state variable z_i (joint i = 1, 2,…, n).¹⁹

d z_{i} / d t = {\dot{q}}_{i} - σ_{0 i} | {\dot{q}}_{i} | {(g ({\dot{q}}_{i}))}^{- 1} z_{i}

where ${\dot{q}}_{i}$ represents the relative velocity between the contact surfaces.

The LuGre friction model can be described as the function of the flexibility of the bristles, thus

F_{i} = σ_{0 i} z_{i} + σ_{1 i} {\dot{z}}_{i} + σ_{2 i} {\dot{q}}_{i}

where σ_0i represents the stiffness of the bristles, σ_1i denotes the microscopic damping coefficient, and σ_2i indicates the viscous friction coefficient.

The following function $g ({\dot{q}}_{i})$ is used to describe the Stribeck effect (i.e. in the case of lubrication conditions, as the speed increases, the frictional force first drops to a minimum value and then increases with increasing speed)

g ({\dot{q}}_{i}) = F_{c i} + (F_{s i} - F_{c i}) e^{{({\dot{q}}_{i} / {\dot{q}}_{s})}^{2}}

where F_ci and F_si represent the joint Coulomb friction and joint viscous friction, respectively; ${\dot{q}}_{s}$ denotes the Stribeck characteristic velocity.

To facilitate the implementation and programing of the model, it is necessary to discretize the continuous time-domain model of equations (3) to (5). Let $x_{1} = q$ , $x_{2} = \dot{q}$ ; and k represent the discrete time. Thus, the discretized model can be expressed as follows

z_{i} (k + 1) = x_{2} (k) - σ_{0 i} | x_{2} (k) | {(g (x_{2} (k)))}^{- 1} z_{i} (k)

F_{i} (k) = σ_{0 i} z_{i} (k) + σ_{1 i} z_{i} (k + 1) + σ_{2 i} x_{2} (k)

g (x_{2} (k)) = F_{c i} + (F_{s i} - F_{c i}) e^{- {(x_{2} (k) / {\dot{q}}_{s})}^{2}}

Substituting equation (6) into equation (7) yields the following discretized LuGre friction model

F (k) = σ_{0 i} z_{i} (k) - σ_{3 i} n_{i} (x_{2} (k)) z_{i} (k) + σ_{4 i} x_{2} (k)

where $σ_{3 i} = σ_{0 i} σ_{1 i}$ , $σ_{4 i} = σ_{1 i} + σ_{2 i}$ , $n_{i} (x_{2} (k)) = | x_{2} (k) | {(g (x_{2} (k)))}^{- 1}$ .

According to equation (9), we take $F_{0} = σ_{0 i} z_{i} (k)$ , $F_{3} = - σ_{3 i} n_{i} (x_{2} (k)) z_{i} (k)$ , $F_{4} = σ_{4 i} x_{2} (k)$ . Since the functions of the three friction parts in the model are different and represent different physical quantities, the friction model is decomposed into three parts of $F_{0} (z (\dot{q}))$ , $F_{3} (n (\dot{q}) z (\dot{q}))$ , and $F_{4} (\dot{q})$ . Then, the RBFNN is used to locally estimate the three friction subfunctions to reduce the error and further improve the control performance of the multi-joint robot manipulator. According to the three-part variable function, the input variables of the neural network are set to $z (\dot{q})$ , $n (\dot{q}) z (\dot{q})$ , and $\dot{q} (k)$ .

Design of controller

This article designs the controller block diagram as shown in Figure 1. The four RBFNNs in Figure 1 are used to estimate and compensate for the unknown model and friction model, of which three of them are used for the three subfunctions $F_{0} (z (\dot{q}))$ , $F_{3} (n (\dot{q}) z (\dot{q}))$ , and $F_{4} (\dot{q})$ of the LuGre friction model, and another neural network is used to estimate the sum of the three matrices $M (q)$ , $C (q, \dot{q})$ , and $G (q)$ in the dynamic model of the robot manipulator. The input variables of the four neural networks are $z (\dot{q})$ , $n (\dot{q}) z (\dot{q})$ , $\dot{q}$ , and q, respectively, and the output variables are the neural network estimates ${\hat{F}}_{0}$ , ${\hat{F}}_{3}$ , and ${\hat{F}}_{4}$ of the three friction components F₀, F₃, and F₄, and the estimates ${\hat{F}}_{f}$ of the sum of the three matrices in the dynamic model. The input of the robust sliding mode controller is e, $\dot{e}$ , q, and $\ddot{q}$ . The adaptive identification is used to identify and adjust the model parameters.

Figure 1.

Block diagram of the proposed controller.

Design of sliding mode robust controller

The following sliding surface is designed to track the ideal position $q_{d} (t)$ of the multi-joint robot

r = \dot{e} + Λ e

where $Λ = diag (Λ_{1}, Λ_{2}, \dots, Λ_{n})$ is the sliding surface matrix, and $Λ = Λ^{T} > 0$ ²⁰; e is a single variable. For any initial state, the convergence point will be reached in finite time after the system state variables enter the sliding surface. At the same time, the convergence point is the terminal attractor, that is, the system can reach the convergence state and be stable in finite time.¹⁷

The position tracking errors, speed tracking errors, and acceleration tracking errors are, respectively, expressed as follows

{\begin{matrix} e = q_{d} - q \\ \dot{e} = {\dot{q}}_{d} - \dot{q} \\ \ddot{e} = {\ddot{q}}_{d} - \ddot{q} \end{matrix}

Differentiating r in equation (10) results in

\dot{r} = \ddot{e} + Λ \dot{e}

According to equations (10) to (12), the closed-loop error dynamic equation is expressed as follows

\begin{matrix} M \dot{r} = M ({\ddot{q}}_{d} - \ddot{q} + Λ \dot{e}) = M ({\ddot{q}}_{d} + Λ \dot{e}) - M \ddot{q} \\ = M ({\ddot{q}}_{d} + Λ \dot{e}) + C \dot{q} + G + F + τ_{d} - τ \\ = M ({\ddot{q}}_{d} + Λ \dot{e}) - C r + C ({\dot{q}}_{d} + Λ e) + G + F + τ_{d} - τ \\ = - C r - τ + f + τ_{d} \end{matrix}

where the uncertain nonlinear function f can be expressed as follows

f = M (q) ({\ddot{q}}_{d} + Λ \dot{e}) + C (q, \dot{q}) ({\dot{q}}_{d} + Λ e) + G (q) + F (\dot{q})

The control inputs of the sliding mode controller are defined as follows

τ_{0} = \hat{f} + K_{V} r

where $\hat{f}$ is the estimated term of f, K_V is a gain matrix, and $K_{V} = K_{V}^{T} > 0$ .

Design of RBFNN controller

RBFNN has good properties of approaching arbitrary nonlinear functions and expressing the inherent law of the system.²¹ Therefore, RBFNN is appropriate to compensate for the modeling errors of the LuGre friction model. The function of RBFNN chooses Gauss function as the basis function, thus the ideal algorithm of RBFNN can be expressed as follows²²

h_{i} = g ({‖ x - c_{i} ‖}^{2} / σ_{i}^{2}) i = (1, 2, \dots, n)

The output function of the neural network is described as follows

y = W^{T} h

where c_i is the center of the Gaussian function, σ_i is the width of the Gaussian function, $h = {[h_{1}, h_{2}, \dots, h_{n}]}^{T}$ is the output matrix of the Gaussian function, and W is the weight of the neural network.

Assuming that there is a constant weight value W, the uncertain nonlinear function f in equation (14) can be estimated using the following output function of the neural network

f = W^{T} h + ε (x)

where h denotes a suitable basis function, $ε (x)$ represents the estimation error of the neural network, ε(x) is a known bounded function, and $‖ ε (x) ‖ \leq ε_{N} (x)$ .

The estimated value of neural network in equation (18) is defined as follows

\hat{f} = {\hat{W}}^{T} h

where $\hat{W}$ is the estimated weight of the neural network and the estimated error is defined as $\tilde{W} = W - \hat{W}$ .

The RBFNN can be used not only to compensate for the uncertain nonlinear function but also to evaluate the three parts of the LuGre friction model. Thus, the RBFNNs are designed to evaluate the three components of friction F₀, F₃, and F₄ and the sum of four matrices F_f

{\begin{cases} {\hat{F}}_{0} = {\hat{W}}_{0}^{T} h_{0} \\ {\hat{F}}_{3} = {\hat{W}}_{3}^{T} h_{3} \\ {\hat{F}}_{4} = {\hat{W}}_{4}^{T} h_{4} \\ {\hat{F}}_{f} = {\hat{W}}_{f}^{T} h_{f} \end{cases}

According to equation (15), the control law of the neural network is designed as follows

\begin{matrix} τ = τ_{0} - v = {\hat{W}}^{T} h + K_{V} r - v \\ = {\hat{W}}_{f}^{T} h_{f} + {\hat{W}}_{0}^{T} h_{0} + {\hat{W}}_{3}^{T} h_{3} + {\hat{W}}_{4}^{T} h_{4} + K_{V} r - v \end{matrix}

where $v = - (ε_{N} (x) + b_{d}) sgn (r)$ is a robust term for compensating the estimation error $ε (x)$ of these neural networks.

Substituting equation (21) into equation (13) yields the following closed-loop error dynamic equation

\begin{matrix} M \dot{r} = - (K_{V} + C) r + {\tilde{W}}^{T} h + (ε (x) + τ_{d}) + v \\ = - (K_{V} + C) r + ξ_{1} \end{matrix}

where $ξ_{1} = {\tilde{W}}^{T} h + (ε (x) + τ_{d}) + v$ , $ε (x) = ε_{f} + ε_{0} + ε_{3} + ε_{4}$ , and ${\tilde{W}}^{T} h = {\tilde{W}}_{f}^{T} h_{f} + {\tilde{W}}_{0}^{T} h_{0} + {\tilde{W}}_{3}^{T} h_{3} + {\tilde{W}}_{4}^{T} h_{4}$ .

The adaptive laws of the four adaptive neural network controllers are designed as follows

{\begin{matrix} {\dot{\hat{W}}}_{f} = F_{f} h_{f} r^{T} - k_{f} F_{f} | | r | | {\hat{W}}_{f} \\ {\dot{\hat{W}}}_{0} = F_{0} h_{0} r^{T} - k_{0} F_{0} | | r | | {\hat{W}}_{0} \\ {\dot{\hat{W}}}_{3} = F_{3} h_{3} r^{T} - k_{3} F_{3} | | r | | {\hat{W}}_{3} \\ {\dot{\hat{W}}}_{4} = F_{4} h_{4} r^{T} - k_{4} F_{4} | | r | | {\hat{W}}_{4} \end{matrix}

Proof of convergence

Theorem 1

If the ideal trajectory q, $\dot{q}$ , and $\ddot{q}$ is bounded and neural network estimation error $ε_{N} (x)$ is continuous, then the tuning weight can be expressed as follows

\dot{\hat{W}} = V h r^{T}

where $V = V^{T} > 0$ represents a constant matrix. The neural network weight estimation value $\hat{W}$ is bounded, and the tracking error can be reduced as much as possible by increasing the gain K_V.

Proof

Lyapunov function is defined as

L = \frac{1}{2} r^{T} M r + \frac{1}{2} tr ({\tilde{W}}^{T} F^{- 1} \tilde{W})

where F represents a positive definite matrix.

Differentiating L of equation (25) results in

\dot{L} = r^{T} M \dot{r} + \frac{1}{2} r^{T} \dot{M} r + tr ({\tilde{W}}^{T} F^{- 1} \dot{\tilde{W}})

Substituting equation (22) into equation (26), one can get

\begin{matrix} \dot{L} = - r^{T} K_{V} r + \frac{1}{2} r^{T} (\dot{M} - 2 C) r (t) + tr {\tilde{W}}^{T} (F^{- 1} \dot{\tilde{W}} + h r^{T}) \\ + r^{T} (ε (x) + τ_{d}) \end{matrix}

Then, by considering property 2 and the fact that $\dot{\tilde{W}} = - \dot{\hat{W}}$ , equation (27) is written as

\begin{matrix} \dot{L} = - r^{T} K_{V} r + r^{T} (ε (x) + τ_{d}) \\ \leq - K_{V min} {‖ r ‖}^{2} + (ε_{N} + b_{d}) ‖ r ‖ \end{matrix}

where $K_{V min}$ is the minimum singular value of K_V.

Let $| | r | | > (ε_{N} (x) + b_{d}) / K_{V min}$ , where $ε_{N} (x) + b_{d}$ is a constant, we can get $\dot{L} \leq 0$ from equation (28). The right half of the inequality $| | r | | > (ε_{N} (x) + b_{d}) / K_{V min}$ can be used as a practical constraint on the tracking error function r, so r will never be larger than it. Therefore, as the upper bound of the neural network estimation error $ε_{N} (x)$ and the upper boundary of the external disturbance b_d increase, the tracking error r increases, but an arbitrarily small tracking error can be obtained by selecting a larger gain K_V. (If K_V is a diagonal matrix, $K_{V min}$ is the smallest gain function.)

Therefore, we can draw a conclusion that the tracking error r and the tuning weight error $\tilde{W}$ are bounded. The boundedness of r guarantees the continuity of e and $\dot{e}$ . The boundedness of the ideal trajectory indicates that q and $\dot{q}$ are also bounded.

Theorem 2

The control input is designed as follows

τ = {\hat{W}}^{T} h + K_{V} r

Then, the sliding mode controller in equation (15) and the adaptive laws in equation (23) can guarantee a consistent final boundedness of the control system.

Proof

Lyapunov function is defined as follows

\begin{matrix} L = \frac{1}{2} r^{T} M r + \frac{1}{2} tr ({\tilde{W}}_{f}^{T} F_{f}^{- 1} {\tilde{W}}_{f}) + \frac{1}{2} tr ({\tilde{W}}_{0}^{T} F_{0}^{- 1} {\tilde{W}}_{0}) \\ + \frac{1}{2} tr ({\tilde{W}}_{3}^{T} F_{3}^{- 1} {\tilde{W}}_{3}) + \frac{1}{2} tr ({\tilde{W}}_{4}^{T} F_{4}^{- 1} {\tilde{W}}_{4}) \end{matrix}

Differentiating L along the tracking error equation (13), the following equation can be obtained as follows

\begin{array}{l} \dot{L} = r^{T} M \dot{r} + \frac{1}{2} r^{T} \dot{M} r + tr ({\tilde{W}}_{f}^{T} F_{f}^{- 1} {\dot{\tilde{W}}}_{f}) + tr ({\tilde{W}}_{0}^{T} F_{0}^{- 1} {\dot{\tilde{W}}}_{0}) \\ + tr ({\tilde{W}}_{3}^{T} F_{3}^{- 1} {\dot{\tilde{W}}}_{3}) + tr ({\tilde{W}}_{4}^{T} F_{4}^{- 1} {\dot{\tilde{W}}}_{4}) \end{array}

Substituting equation (13) into equation (31) yields

\begin{array}{l} \dot{L} = - r^{T} K_{V} r + \frac{1}{2} r^{T} (\dot{M} - 2 C) r + r^{T} (ε (x) + τ_{d}) \\ + r^{T} v + tr {\tilde{W}}_{f}^{T} (F_{f}^{- 1} {\dot{\tilde{W}}}_{f} + h_{f} r^{T}) + tr {\tilde{W}}_{0}^{T} (F_{0}^{- 1} {\dot{\tilde{W}}}_{0} + h_{0} r^{T}) \\ + tr {\tilde{W}}_{3}^{T} (F_{3}^{- 1} {\dot{\tilde{W}}}_{3} + h_{3} r^{T}) + tr {\tilde{W}}_{4}^{T} (F_{4}^{- 1} {\dot{\tilde{W}}}_{4} + h_{4} r^{T}) \end{array}

Substituting the adaptive laws (23) into equation (32) can obtain the following equation

\begin{array}{l} \dot{L} = - r^{T} K_{V} r + k_{f} | | r | | tr ({\tilde{W}}_{f}^{T} (W_{f} - {\tilde{W}}_{f})) \\ + k_{0} | | r | | tr ({\tilde{W}}_{0}^{T} (W_{0} - {\tilde{W}}_{0})) + k_{3} | | r | | tr ({\tilde{W}}_{3}^{T} (W_{3} - {\tilde{W}}_{3})) \\ + k_{4} | | r | | tr ({\tilde{W}}_{4}^{T} (W_{4} - {\tilde{W}}_{4})) + r^{T} (ε (x) + {τ_{d}}^{d}) + r^{T} v \end{array}

Then, by considering the fact that $tr ({\tilde{W}}^{T} (W - \tilde{W})) = {(\tilde{W}, W)}_{F} - | | \tilde{W} | |_{F}^{2} \leq | | \tilde{W} | |_{F} | | W | |_{F} - | | \tilde{W} | |_{F}^{2}$ , equation (33) is written as

\begin{array}{l} \dot{L} \leq - | | r | | [K_{V max} | | r | | + k_{f} {‖ {\tilde{W}}_{f} ‖}_{F} ({‖ {\tilde{W}}_{f} ‖}_{F} - W_{f max}) \\ + k_{0} {‖ {\tilde{W}}_{0} ‖}_{F} ({‖ {\tilde{W}}_{0} ‖}_{F} - W_{0 max}) + k_{3} {‖ {\tilde{W}}_{3} ‖}_{F} ({‖ {\tilde{W}}_{3} ‖}_{F} - W_{3 max}) \\ + k_{4} | | {\tilde{W}}_{4} | |_{F} (| | {\tilde{W}}_{4} | |_{F} - W_{4 max})] \end{array}

We can also observe that $\dot{L}$ is negative as long as the sliding surface r is bounded by $| | r | | > (0.25 k_{f} W_{f max}^{2} + 0.25 k_{0} W_{0 max}^{2} + 0.25 k_{3} W_{3 max}^{2} + 0.25 k_{4} W_{4 max}^{2}) / K_{V max}$ , and these vectors of tuning parameter errors are bounded by $| | {\tilde{W}}_{f} | |_{F} > W_{f max}$ , $| | {\tilde{W}}_{3} | |_{F} > W_{3 max}$ , $| | {\tilde{W}}_{0} | |_{F} > W_{0 max}$ , and $| | {\tilde{W}}_{4} | |_{F} > W_{4 max}$ .

Using Theorem 2, it can be seen that the tracking error converges, and all the solutions are also bounded.

Simulation results and discussion

A two-link planar robot manipulator and a two-stage inverted pendulum are applied to validate the effectiveness of the adaptive block compensation trajectory tracking controller.

Control of two-link planar robot manipulator

The dynamic equation of the two-link planar robot manipulator is given as follows

M (q) = [\begin{matrix} p_{1} + p_{2} + 2 p_{3} cos x_{3} & p_{2} + p_{3} cos x_{3} \\ p_{2} + p_{3} cos x_{3} & p_{2} \end{matrix}]

C (q, \dot{q}) = [\begin{matrix} - p_{3} x_{4} sin x_{3} & - p_{3} (x_{2} + x_{4} sin x_{3}) \\ p_{3} x_{2} sin x_{3} & 0 \end{matrix}]

G (q) = [\begin{matrix} p_{4} g cos x_{1} + p_{5} g cos (x_{1} + x_{3}) \\ p_{5} g cos (x_{1} + x_{3}) \end{matrix}]

Numerical simulations are performed to evaluate the trajectory tracking control performance of the proposed controller. The idea behind these simulations is to force the proposed controller to track a desired trajectory sin(t) and t². The parameters in the dynamic model are picked as $p = {[p_{1}, p_{2}, p_{3}, p_{4}, p_{5}]}^{T} = {[3.1, 2.3, 2.3, 3.1, 2.3]}^{T}$ . The initial average change value in the pair of the LuGre friction model is set as z(0) = [0, 0]^T, and the parameter b of Gaussian function in the RBFNN is designed as 10. The two parameters ε_N(x) and b_d in the robust term v = −(ε_N(x) + b_d)sgn(r) are picked as ε_N(x) = [3, 0, 3, 0]^T and b_d = 0.05. The sliding surface matrix in the sliding mode surface of equation (10) is taken as L = diag{5, 5}. In the sliding mode controller of equation (15), the gain matrix K_v is given as K_v = diag{20, 20}, and these parameters in the adaptive laws (23) are picked as follows: k_f = k₀ = k₃ = k₄ = 0.01. The initial position of the two-link planar robot manipulator is selected as q_d(0) = [−0.03, 0.03]^T.

Case 1

The traditional proportional derivative (PD) controller is used to control the two-link planar robot manipulator. The tracking positions illustrated in Figures 2 and 3 show that large errors exist between the desired trajectory and the actual datum. The actual position has noticeable ups and downs that are mainly caused by model errors and friction terms. The total simulation time is designed as 10 s. The position tracking errors displayed the first 2 s and the total 10 s to effectively show the transient and steady-state responses. Figures 2 and 3 show that the tracking position errors of the traditional PD controller have a tendency to deteriorate. Thus, the traditional PD controller is not suitable for the robot manipulator with uncertain dynamics.

Figure 2.

Position tracking control for two-link robot manipulator when input function is sin(t): (a) first joint and (b) second joint.

Figure 3.

Position tracking control for two-link manipulator when input function is t²: (a) first joint and (b) second joint.

Case 2

Adaptive block compensation trajectory tracking control with model error compensation is applied to the two-link planar robot manipulator. The model errors and friction terms are the same as in case 1. The green curves in Figures 2 and 3 illustrate the tracking position of the proposed controller including the model error compensation. Compared with the traditional PD controller in case 1, the tracking position errors in transient and steady state in case 2 are remarkably reduced. The reason is that the model errors are reduced by the model error compensator. However, it is noteworthy that unavoidable large error between the actual and desired trajectories in case 2 still exists. Thus, many additional compensators should be appropriately designed and embedded into the proposed controller to eliminate these unknown uncertainties.

Case 3

Adaptive block compensation trajectory tracking control with model error and friction compensation is applied to the two-link planar robot manipulator. As can be calculated from the data shown in Figure 2, the position errors of the two joints in case 2 are 0.0013 mm and 0.0042 mm, whereas their corresponding values in case 1 are 0.045 mm and 0.057 mm. The comparison results demonstrate that the position errors in case 2 are sharply smaller than those in case 1. The results confirm that the adaptive block compensation trajectory tracking controller with model error and friction compensation eliminates uncertainties.

To verify the adaptive ability of the algorithm, this article performs simulations about the approximation effect of the uncertainty term f under the action of RBFNN. As can be seen from Figure 4, the simulation data of the proposed controller with model error and friction compensation of fn can quickly estimate the desired term f.

Figure 4.

Approximation results of neural network when input function is (a) sin(t) and (b) t².

Control of a two-stage inverted pendulum

In the case of the small swing angle (<5°), the linearization model of the two-stage inverted pendulum can be described as follows^23

–27

{\begin{cases} \dot{x} = A x + B_{1} w + B_{2} u \\ Z = C_{1} x + D_{11} w + D_{12} u \\ y = x \end{cases}

where $A = [\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & - 3.7864 & 0.2009 & - 4.5480 & 0.0037 & - 0.0017 \\ 0 & 41.9965 & 9.3378 & 7.6261 & - 0.0570 & 0.0349 \\ 0 & - 25.0347 & - 29.5778 & 1.0850 & 0.0675 & - 0.0543 \end{matrix}]$ , $B_{1} = [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}]$ , $B_{2} = [\begin{matrix} - 1.1902 & 68.6019 \\ - 55.3119 & - 115.0316 \\ 175.2019 & - 16.3660 \end{matrix}]$ , $C_{1} = [\begin{matrix} q_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & q_{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & q_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & q_{4} & 0 & 0 \\ 0 & 0 & 0 & 0 & q_{5} & 0 \\ 0 & 0 & 0 & 0 & 0 & q_{6} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$ , $D_{12} = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & ρ \end{matrix}]}^{T}$ , and $D_{11} = 0$ , in which ρ = 1, q₁ = 1.69, q₂ = 1.0, q₃ = 0.01, q₄ = 0.3, q₅ = 0.1, and q₆ = 0.01.

The initial state of the two-link planar robot manipulator is selected as x(0) = [0, 0, 0, 0, 0, 0]^T. Numerical simulations are performed to evaluate the trajectory tracking control performance of the proposed controller and the traditional PD controller. The idea behind these simulations is to force the proposed controller to track a desired trajectory sin(t).

Case 1

The traditional PD controller is applied to the two-stage inverted pendulum. Numerical simulation results of the tracking position under the actions of the traditional PD controller are illustrated in Figure 5. It can be seen that the tracking position errors tend to deteriorate, and especially the positions of the y₄, y₅, and y₆ are off the track. Hence, some compensators should be appropriately designed and embodied into the traditional PD controllers to reduce the effects of uncertainties.

Figure 5.

Tracking position of traditional PD controller when the input signal is sin(t). PD: proportional derivative.

Case 2

The proposed controller with model error and friction compensation is applied to the two-stage inverted pendulum. Figure 6 illustrates the tracking position of the proposed controller with model error and friction compensation. The simulation results demonstrate that the proposed controller tracks the desired trajectory extremely well and also reaches convergence rapidly. Furthermore, the position tracking ability shown in Figure 6 clearly demonstrates the advantage of the proposed controller over the traditional PD controller. Therefore, a conclusion can be drawn that the proposed controller provides the best control efficiency and compensation ability compared with the traditional PD controller.

Figure 6.

Tracking position of the proposed controller when the input signal is sin(t).

Conclusion

An adaptive block compensation trajectory tracking controller based on LuGre friction model is proposed. An adaptive neural network compensator is designed to assess the three parts of the LuGre friction model. A robust sliding mode controller is developed to reduce the influence of these estimation errors of neural network compensator and other uncertain disturbances and ensure that the system converges in a finite time at the same time.

The proposed controller guarantees that all the involved control signals in the control system are bounded and the trajectory tracking errors converge asymptotically to zero. Finally, simulation studies are conducted to verify the effectiveness of the proposed controller. Simulation results confirm the superiority of the proposed adaptive block compensation trajectory tracking controller with model error and friction compensation. In the future, a two-link robot manipulator will be applied to verify the effectiveness of the proposed control algorithm.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Science and Technology Program of Shenzhen, China [grant no. JCYJ20170818114408837].

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