Improved Composite Nonlinear Feedback Control for Robot Manipulators

GONG Chenglong()， JIANG Yuan( )， 2， 3*， Lü Ke( )， 2

1 College of Information Engineering， Nanchang Hangkong University， Nanchang 330063， China2 School of Engineering Management and Information Technology， University of Chinese Academy of Sciences， Beijing 100049， China3 Non-destructive Testing Technology Ministry of Education Key Laboratory， Nanchang Hangkong University， Nanchang 330063， China

Abstract: External disturbances or inaccurate mathematical model built will inevitably impose a disadvantageous effect on the robot system， which generates positioning errors， vibrations， as well as weakening control performances of the system. The strategy of combining adaptive radial basis function (RBF) neural network control and composite nonlinear feedback (CNF) control is studied， and a robot CNF controller based on RBF neural network compensation is proposed. The core is to use RBF neural network control to approach the uncertainty of the system online， as the compensation term of the CNF controller， and make full use of the advantages of the two control methods to reduce the influence of uncertain factors on the performance of the system. The convergence of closed-loop system is proved. Simulation results demonstrate the effectiveness of this strategy.

Key words: robot; uncertainty; composite nonlinear feedback (CNF); adaptive RBF neural network; system convergence； trajecfory tracking

Introduction

Trajectory tracking is a basic task for robot control[1]. The main control objective is to design the controller， track the reference trajectory periodically， and make the robot manipulators track the reference trajectory asymptotically. There are numerous control methods for robot system[2-6].

Composite nonlinear feedback[7]is a computed torque control theory that can be used as an effective method to perform set point tracking tasks of saturated systems. Reference [8] verifies that the composite feedback control performance is better than the time optimal control in terming of set point tracking tasks. However， when there is disturbance in the system or the system model is not accurate， the system under the control of CNF would no longer be able to match the reference input accurately.

There are various solutions to the uncertainty of the robot system[9-12]. If the system uncertainty model is known， a model-based compensation method can be used[13]， but it is impossible to completely compensate the uncertainty effect according to the mathematical model. Although Ref.[14] cancompensate the impact of uncertainty on the system, the implementation algorithm is more complicated. Ref.[15] has a good inhibitory effect on external disturbances， but it only targets external disturbance factors. If other uncertain factors are considered， it will inevitably change the structure of the controller.

Radial basis function[16](RBF) neural network is a three-layer feedforward network with a single hidden layer. It has been proved that RBF network can approximate any continuous function to any degree of accuracy. Therefore， RBF neural networks control scheme can effectively improve the tracking precision， robustness and adaptability of the robot system.

1 Problem Formulation and RBF Neural Networks

The equations of motion of ann-link rigid robot can be expressed in the form of:

(1)

The robot dynamics equation has the following properties[17].

P1)M(q)∈Rn×nis symmetric， bounded， and positive definite.

In practice engineering， the precise mathematical model of robot system is difficult to be obtained， and the controller can only be designed based on the ideal robot nominal model[18]. The original CNF controller can be designed as follows.

(2)

Andkv=diag(2α， 2α) andkp=diag(α2，α2) are decoupling gain matrices designed based on the robot manipulator’s own parameters. When they are diagonally positive definite matrices， the robot system can be decoupled.

(3)

wheresatm(·)≥sath(·). This would allow for exemption of vibrations for a closed-loop system with large uncertainties. Then denoting

In this paper， we use the adaptive RBF networks[19]to approximate the uncertainties termf(x)， which can be characterized by

wherexis the input;yis the output of the network;φ=[φ1，φ2，…，φn] is the output of the Gaussian function; andθis the neural network’s weight matrix;ciandbiare the centre and width of theith kernel unit respectively.

It had been shown that under mild assumptions， RBF neural networks are capable of achieving universal approximations， meaning that it can approximate of any continuous function over a compact set to any degree of accuracy. Therefore， we can use an RBF neural network to approximate the nonlinear functionf(x). Based on the reported results， the following assumptions on the RBF neural networks are elaborated in order.

where

andθ*is then×norder matrix， which represents the best approximation parameter forf(x).

It is noted that with assumption A2)， an RBF neural network is guaranteed to exist which approximates the nonlinear functionf(x) to any degree of accuracy.

where

2 Improved CNF Control

The new control law can be designed as

(4)

According to Eqs. (1),(3)-(4)， we have

(5)

where

and

Assumption2If the robot closed-loop system satisfies the following conditions:

A3) Existingc>0， satisfied by:

wherekpiandkviare row vectors of matriceskpandkv， respectively;Lv(c) is the estimated ellipsoid invariant set of the system stability attraction domain[20];cis its radius;ui， maxis the maximum output amplitude of each manipulator.

Choosing a Lyapunov function candidate

where matrixPis symmetric and positive definite， satisfying the following Lyapunov equation

whereQis a positive definite matrix.

Combining Eqs. (2)，(3)，(5) with the feedback linearization function， the closed-loop system can be expressed as

(6)

Taking the time derivative ofValong the trajectories of Eq. (6)， we have

Considering the item∑vi[sath， i(ki-ρ(e)vi)-ki]， for each robot manipulator， ifki-ρ(e)vi≤ui， max， thenvi[sath， i(ki-ρi(e)vi)-ki]=-ρi(e)vi2≤0; ifki-ρ(e)vi>ui， max，by Assumption A1) givingki≤ui， max， we have

Therefore，viandsath， i(ki-ρ(e)vi)-kihave opposite signs， thenvi[sath， i(ki-ρ(e)vi)-ki]≤0.And we always have

which means

(7)

In the following section， we will choose two adaptive laws to design a new type of controller and perform simulation analysis on their respective control performance.

3 Simulation and Results

3.1 Case 1: adaptive law 1

We choose the adaptive law 1 as

(8)

It is known that ‖η‖≤‖η0‖， ‖B‖=1. Assume thatλmin(Q) is the minimum value of the eigenvalue of matrixQandλmax(P) is the maximum value of the eigenvalue of matrixP， so

A simulation study is conducted to demonstrate the performance of our algorithms. A two-degree of freedom manipulator was used in the simulation.The manipulator’s dynamic model is described in Ref.[21].

External disturbances are selected as

and the desired position is

q1d=1+0.2sin(0.5πt)，q2d=1-0.2cos(0.5πt), consideringΔM，ΔCandΔGas their total’s 20%， respectively; and choosingα=3， the nonlinear feedback portion is

The adaptive law parameters areγ=20 andk1=1×10-3; the initial values of the Gaussian function parameters are [-2，-1， 0， 1， 2] and 3.0， respectively.

Then

AssumingQ=50I，c=50.26 can be calculated from the level set estimate in Ref. [21].

The simulation results are shown in Figs. 1-3. Figures 1 and 2 show the trend of the trajectory tracking error of the two robot joints under the control of different controller. When the system is disturbed， the system trajectory tracking under the original CNF control has a large overshoot and steady state error， the control performance is poor， which cannot be accepted in the real project. Although the system trajectory tracking under the adaptive RBF neural controller greatly reduces the overshoot and steady state errors， the adjustment time is still long and the control performance needs to be improved. Under the new controller， the system tracking response is rapid， the adjustment time is short， the overshoot and steady state error are small， and the control performance is greatly improved.

Fig.1 Case 1: position tracking error for link 1

Fig.2 Case 1: position tracking error for link 2

Figure 3 shows the estimation and compensation of robot joint disturbances under the new controller. WhenT>3， the adaptive RBF compensator can accurately approach the uncertainty of the system online and can ensure the system to obtain better control performance. The effect at this time can also be seen from Figs. 1-2， the system have a good track tracking results whenT>3.

Fig.3 Case 1: f(x) estimation(f1, f2) and compensation(fn1, fn2)

3.2 Case 2: adaptive law 2

We can choose the RBF neural networks adaptive law 2 as

(9)

whereλ>0，k>0.

Combining Es. (7) with Eq. (9)， we have

Suggesting that the closed-loop system’s convergence condition is

Obviously， a smaller convergence value ofxcan be secured by using a small eigenvalue of matrixP， a smaller upper boundη0of the neural network modeling error， and a smallerθmax， leading to a better tracking effect.

The simulation and analysis of the new controller 2 also use the robot model of section 3.1. The simulation results are shown in Figs. 4-6. Compared with adaptive law 1， the new controller designed according to adaptive law 2 has better effect on the influence of suppressing uncertainty factors on the system. It can fully retain the characteristics of the fast response and overshoot of the original CNF control method and the adaptive RBF neural network for the uncertainty of the online approximation ability.

Fig.4 Case 2: position tracking error for link 1

Fig.5 Case 2: position tracking error for link 2

Fig.6 Case 2: f(x) estimation and compensation

Compared with Fig. 3， the adaptive RBF control under the new adaptive law has a better approximation effect on the uncertainties. The results are shown in Fig. 6. The new controller at this time can achieve a better online approximation effect whenT>2， and the effect can also be reflected in Figs. 4-5.

Figures 7-8 show the total control input(torque) of different controllers to the system. Obviously， the control input of the original CNF controller is unstable， which will inevitably cause the system to produce large jitter and error during the trajectory tracking process. Although the both control methods can provide stable control input for the system， the control input of the new controller 2 has a shorter stabilization time， the effect can be shown in Figs. 4-5.

From the control performance of the controller designed by the above two adaptive laws， the effect produced by the adaptive law 2 is better， and the boundedness of the parameters can be guaranteed.

Fig.7 Control input of joint 1

Fig.8 Control input of joint 2

4 Conclusions

In this paper， a new type of controller is designed by combining CNF control with an adaptive RBF neural network control that can be switched in. The RBF adaptive controller is independent of model-based CNF controller， so that its entry does not affect the original control system design. The working principle of the new controller in the system is described as follows. At the initial stage， the system is controlled by a model-based CNF controller. Because there exists uncertainties， the tracking error of the robot system is large. At the same time， the RBF adaptive neural networks perform online learning on the uncertainty of the system. After a certain period of time， the RBF neural network can better estimate the system’s uncertainty. Att=T(Tshould be greater than the learning time of the RBF neural networks)， an adaptive RBF neural network controller derived from the Lyapunov method is switched in and coordinated with the CNF control. This new controller not only retains the advantages of fast response， small overshoot， and effective suppression of system jitter of the CNF controller， but also preserves the advantages of the adaptive RBF neural networks controller for effective estimation of system uncertainties and rapid convergence.