Disturbance Compensation Based Finite-Time Tracking Control of Rigid Manipulator

The finite-time tracking control problem of rigid manipulator systemwithmismatched disturbances is investigated via a composite control method. The proposed composite controller is based on finite-time disturbance observer and adding a power integrator technique. First, a finite-time disturbance observer is designed which guarantees that the disturbances can be estimated in a finite time.Then, a composite controller is developed based on adding a power integrator approach and the estimates of the disturbances. Under the proposed composite controller, themanipulator position can track the desired position in a finite time. Simulation results show the effectiveness of the proposed control scheme.


Introduction
Due to some superior capacities, such as high accuracy, high stiffness, and high load-carrying [1], manipulator systems have been widely used in robots [2], parallels [3], and mechanical systems [4].To achieve the control goals for manipulators and improve the performances of the closedloop manipulator systems, PID control laws were given in [5][6][7], sliding mode control techniques were adopted in [8,9], adaptive control laws were designed in [10][11][12][13], robust control algorithm was presented in [14], and neural network based control law was shown in [15].
Note that the aforementioned control schemes for manipulators with disturbances work in a robust way, which implies that the disturbances attenuation is at the price of sacrificing their nominal control performances.To improve this problem, a feasible way is to use feedforward-feedback composite control rather than pure feedback control to solve the control problems of manipulators with disturbances.Disturbance observer based control (DOBC) is an effective composite control method, which is composed of disturbance observer (DO) design and nominal feedback controller design [37][38][39][40].Compared with feedback control methods, DOBC has several superiorities, such as faster rejection of disturbances and recovery of the nominal performances.Due to such nice features, DOBC approaches have been adopted in [41][42][43] to reject disturbances for manipulator systems.However, the DOBC methods in these papers for the control of manipulators still have several aspects to be improved.On one hand, in the existing literatures, most DOBC techniques are only available for systems with matched disturbances: namely, the disturbances enter the system in the same channel with 2 Mathematical Problems in Engineering control inputs.On the other hand, the estimated disturbances converge to the real disturbances as time goes to infinity.To reject disturbances in a shorter time, a disturbance observer which provides a faster convergence rate is desired.The sliding mode differentiator in [44,45] is such an observer and it has been utilized by our research group to solve the control problem of manipulators with matched disturbances in [46].In practice, there are mismatched disturbances in the manipulators.However, up to now, there are still no finite-time control results through DOBC methods for manipulators in the presence of mismatched disturbances.
In this paper, a composite control scheme is developed for manipulator systems with mismatched disturbances.The composite control algorithm is designed based on finite-time disturbance observer (FTDO) and adding a power integrator control method.Under the proposed composite controller, in the presence of mismatched disturbances, the manipulator position can track the desired position in a finite time.
The remainder of this paper is organized as follows.Section 2 presents useful definitions and lemmas.The manipulator system model is given in Section 3. The control design is presented in Section 4. Section 5 shows the simulation results and the conclusions of this paper are drawn in Section 6.

Preliminaries
Consider the following nonlinear autonomous system: where  :  → R  satisfies the locally Lipschitz continuous condition.Under this condition, the definition of finite-time stability can be described as follows.
(ii) There exist real numbers  > 0,  ∈ (0, 1) and an open neighborhood  0 ⊂  of the origin such that Then the origin is a finite-time stable equilibrium of system (1).
If  =  0 = R  , the origin is a globally finite-time stable equilibrium of system (1).

Controller Design and Stability Analysis
Controller design is mainly composed of two parts, that is, disturbance observer design and composite controller design.

Finite-Time Controller Design
Theorem 8. Assume that the desired position vector   = [ 1 , . . .,   ]  is twice differentiable.For system (9), if Assumption 6 holds and controller  is designed as the manipulator position  1 will track the desired position   in a finite time, where Proof.Defining the tracking errors as . .,  2, ]  = q  −  2 , then the following tracking error system can be derived: Let ẽ1 =  1 , ẽ2 =  2 − d1 .Then system (13) can be written as The stability analysis for closed-loop system ( 12) and ( 14) is shown in the following.
The stability analysis can be divided into two steps.In Step 1, system states of ( 12) and ( 14) will not escape to infinity for  <  1 .In Step 2, system (12) and ( 14) is finite time stable when  ≥  1 .
Remark 9.Under the condition without disturbance observer, with adding a power integrator method, a controller for manipulator system (9) can be designed as where Under controller (32), manipulator position  can not track the desired trajectory   in any long time, which will be shown in simulations.

Numerical Simulations
Simulations are conducted on a two-link rigid robot manipulator.The cases without/with output noises are considered in the simulations.The manipulator model is shown in Figure 1.The dynamic of the manipulator is where The reference trajectories are [32] The system parameters and disturbances are selected as

The Case without Output Measurement Noises.
To validate the effectiveness of the proposed composite control algorithm, closed-loop system performances under composite controller (12) and under finite-time controller (32) will be compared in this part.
Taking the practical input saturation into consideration, the control inputs for both controller (12) and controller (32) are limited not to exceed 60 Nm.Under these limitations, efforts have been spent to make both closed-loop systems as good as possible.For composite controller (12) Simulation results are presented in Figures 2-4.Figures 2 and 3 show that observation errors of the disturbance observers converge to the origin in a finite time for both links of the manipulator.From Figure 4, it can be seen that manipulator positions can track the desired positions in a finite time under controller (12) while they cannot track the desired positions under controller (32).

The Case with Output Measurement
Noises.In this part, in the presence of output measurement noises, the closedloop system performances under composite controller (12) and under finite-time controller (32) are compared to validate the effectiveness of composite controller (12).Generally speaking, in practice, higher-frequency measurement noises can be filtered by some filters, for example, Kalman filters.In this way, the measured states used by the controllers are usually signals with only lower-frequency noises.Hence, in simulations, only lower-frequency noises are considered and the output measurement noises are assumed to be  = [0.01sin(5); 0.01 cos(5)].In other words, the measured output is  =  1 + .Differentiating  yields ẏ =  2 +  1 , where  1 =  1 + ṅ ( 1 is defined in system ( 9)).By an observer almost the same as (10) (the only difference is that the observer estimates  1 rather than  1 in system ( 9)), the mismatched disturbance  1 can still be estimated.Then based on the disturbances estimates, a composite controller almost the same as (12) (the only difference is the replacement  of  1 , d1 , ̂ḋ 1 by  =  1 +, d1 , ̂ḋ 1 , resp.) can be designed.Under the same input saturation as the case without output noises, the parameters for disturbance observer (10) and composite controller (12) and 6 show that, even in the presence of lower-frequency noises, observer (10) still works well and provides accurate disturbances estimates in a fast way.Moreover, manipulator positions can still track the desired positions in a finite time under controller (12) while controller (32) fails to do this.

Conclusions
This paper has studied the position tracking control problem of rigid manipulator system with mismatched disturbances.By using adding a power integrator technique and FTDO method, a composite control scheme has been developed.The proposed control method has realized that the manipulator positions tracked the desired positions in finite time and simulations have shown the effectiveness of the proposed composite control algorithm.

Figure 1 :
Figure 1: The two-link robot manipulator model.

Figure 7 :
Figure 7: Response curves of system (9) under controllers (12) and (32) in the presence of output measurement noises.(a) Positions of the 1st link (rad).(b) Position tracking errors of the 1st link (rad).(c) Positions of the 2nd link (rad).(d) Position tracking errors of the 2nd link (rad).(e) Control signals for the 1st link (Nm).(f) Control signals for the 2nd link (Nm).