#### Abstract

An adaptive robust quadratic stabilization tracking controller with hybrid scheme is proposed for robotic system with uncertainties and external disturbances. The hybrid scheme combines computed torque controller (CTC) with an adaptive robust compensator, in which variable structure control (VSC) and optimal control approaches are adopted. The uncertain robot manipulator is mainly controlled by CTC, the VSC is used to eliminate the effect of the uncertainties and ensure global stability, and approach is designed to achieve a certain tracking performance of closed-loop system. A quadratic stability approach, which allows separate treatment of parametric uncertainties, is used to reduce the conservatism of the conventional robust control approach. It can be also guaranteed that all signals in closed-loop system are bounded. The validity of the proposed control scheme is shown by computer simulation of a two-link robotic manipulator.

#### 1. Introduction

Due to the uncertainties, disturbance, and nonlinear system dynamics, tracking control for robot manipulator always is a challenging problem [1, 2]. Therefore, in the past decades, many control approaches have been proposed and applied on controlling the robot manipulator, such as PID control [3], computed torque control method (CTC) [4–6], adaptive control [7, 8], variable structure control (VSC) [9, 10], robust control [11–16], fuzzy control [17–19], and neural networks control [20–24].

Varieties of hybrid control systems have been designed for controlling the complex robotic systems. Chang [25] and Wai [26] utilized neural networks to entirely approximate the equivalent control of VSC and then applied the technique to achieve a certain tracking performance. In these controllers, the robotic system nominal model is not included. Actually, the robotic nominal model could be known provided that the uncertainties are all considered to be reasonably modeled. For this reason, CTC could not be neglected in designing controller for complex robotic system due to its good performances [5], even though uncertainties exist in robotic system which would degrade the tracking performance. In order to eliminate the effect of the uncertainties, Song et al. [5] proposed an approach of CTC plus fuzzy compensator, the nominal system was controlled by using CTC method and for uncertain system, and a fuzzy controller acts as compensator. In [6], CTC plus a neural network compensator was proposed and simulations were conducted on a two-link robotic manipulator; furthermore, an experimental example was tested on PUMA560. However, they assumed that system actual accelerations are measurable. Although the accelerations can be obtained through installing accelerometers on the robotic systems, the measurement noises and weight of these extra utilities would both sacrifice the tracking performance of robotic systems [21]. Peng et al. [22] proposed a robust hybrid tracking control for robotic system, which combines computed torque control with a neural network based robust compensator.

In this paper, an adaptive robust quadratic stabilization tracking control scheme, which combines CTC, nonlinear control, and VSC for robotic manipulator, is proposed. The constant parameters of nominal system matrix are assumed to be known, which is controlled by CTC method. The uncertainties in robotic system are considered to be unknown, where VSC and adaptive controller are designed to eliminate the effect of the uncertainties and achieve tracking performance of closed-loop system. A quadratic stability approach, which allows separate treatment of parametric uncertainties, is used to reduce the conservatism of the conventional robust control approach. Based on Lyapunov stability theorem, the proposed adaptive robust controller can guarantee stability of closed-loop system and a certain tracking performance. Finally, simulation examples on a two-link robotic manipulator are presented to show the efficiency of the proposed method.

The rest of this paper is organized as follows. In Section 2, some preliminaries are addressed, which consist of mathematical notations, dynamical models of robotic manipulators with uncertainties, and detailed explanation related to CTC for robotic manipulators. The design of adaptive robust quadratic stabilization tracking control scheme is given in Section 3 and the robust stability is analyzed. The simulation results are given in Section 4, and the conclusions are drawn in Section 5.

#### 2. Preliminaries

Let be the real number set, let be the* n*-dimensional vector space, and let be the real matrix space. The norm of vector and that of matrix are defined, respectively, as and . If is a scalar, then denotes the absolute value. and are the minimum and the maximum eigenvalues of matrix , respectively. is identity matrix. And is a standard sign function.

Consider a general -link rigid robot, which takes into account the external disturbances, with the equation of motion given by [2, 7, 14–16]: where , , and are the joint angular position, velocity, and acceleration vectors of the robot, respectively; is the symmetric and positive definite inertia matrix; is the effect of Coriolis and centrifugal forces; is the gravity vector; is the torque input vector. denotes unknown external disturbance.

The parameters , , and in the robot dynamic model (1) are functions of physical parameters of systems like links masses, links lengths, moments of inertial, and so on. The precise values of these parameters are difficult to acquire due to measuring errors, environment, and payloads variations. Therefore, here it is assumed that actual values , , and can be separated as nominal parts denoted by , , and and uncertain parts denoted by , , and , respectively. These variables satisfy the following relationships:

*Assumption 1. *The bound of uncertainty parameters is known, which can be expressed as
where , , and are positive constants.

Suppose that dynamical model of robotic system is known precisely and unmodeled dynamics and disturbances are excluded; that is, , , , and are all zeros. Dynamic model (1) can be converted into the following nominal model:

According to the CTC method, the control law for robot can be chosen as
where is the tracking error defined by and and are the actual and desired joint trajectories, respectively. The coefficients and should be chosen such that all the roots of the polynomial are in the open left-half plane.

*Assumption 2. *The desired trajectory is continuous and bounded known functions of time with bounded known derivatives up to the second order.

Substituting (5) into (4) yields

In practical robot systems, it is impossible to ignore the parameters uncertainties and external disturbances. Therefore, applying (5) to manipulator systems (1) yields

*Remark 3. *All the parameters in the proposed scheme may be uncertain, which is true in practical situation.

Our objective is to seek a robust control law such that joint motions of robotic system (1) can follow the desired trajectories despite the presence of the system uncertainties and external disturbances.

#### 3. Adaptive Robust Quadratic Stabilization Tracking Control Design

The robust tracking control law designed in this paper comprises of a main controller (CTC) and an adaptive robust compensator. This tracking controller is defined as where is the CTC defined like (5) and is the robust compensator to be determined later.

Let , and let ; then the robotic system error dynamic (8) can be written as

Define the joint variables and parameter variations to be . According to Assumption 1, we assume that is the bound value of the uncertain term so that ; that is,

*Remark 4. *According to Assumption 2, and are both bounded. Therefore, (10) is correct. However, the upper bound cannot be obtained directly, in this paper, an adaptive mechanism is adopted to estimate the upper bound .

Then, the error dynamic state-space equation has the following form: where , , and , with

*Assumption 5. * and represent the time-varying parametric uncertainties having the following structure:
where , , and are known constant matrix appropriate dimensions and is unknown Lebesgue measurable matrix which is bounded as follows:

Suppose there exists a matrix and positive numbers , , , and , such that the following matrix inequality holds: and , where is a prescribed weighting matrix and is some positive gains.

The robust compensator law in (8) is designed as

with where is an online estimated value of the upper bound . Since the positive constants , , and are unknown, the following adaptive algorithm is adopted to estimate the parameters uncertainty :

*Remark 6. *The quadratic stability constraint is obtained by the matrix inequality (15). Therefore, the design problem becomes an optimization problem that minimizes the upper bound of the performance indices subject to the matrix inequality constraint (15).

From the above considerations, the following theorem can be stated as follows.

Theorem 7. *Consider the robotic system dynamic (1) and suppose that Assumptions 1–5 are satisfied and the control law is provided by (8), where is CTC like (5) and is the robust compensator designed as (16) with VSC and optimal controller designing as (18) and (17), respectively. Then, the proposed control law can guarantee that (i) all the variables of the closed-loop system are bounded and (ii) the following tracking performance is achieved:
**
where and is an attenuation level and defined thereafter, .*

*Proof. *Let us select a Lyapunov function candidate:
where . Differentiating the above equation and substituting the state-space equation (9) yield
Substituting the matrix inequality (15) into (22), we obtain
Considering the controller in (18) and Assumption 2, we obtain
Considering the controller in (17) and substituting (24) into (23) yield
Then, according to Assumption 5, we have
where .

Integrating the above inequality from to yields
Since , the above inequality leads to the following criterion:
Because is a positive definite matrix, the fact that implies
Using Barbalat’s lemma [27], it can be concluded that the tracking error converges to zero as .

Since and is bounded, there is also . That means that there is a finite constant such that ; then we have
where and are the initial values of and , respectively. It can be concluded that all signals of the closed-loop system are bounded.

*Remark 8. *Since the VSC controller (18) contains the sign function, direct application of such control signals to the robotic system (1) may result in chattering caused by the signal discontinuity. To overcome this problem, the control law is smoothed out within a thin boundary layer [18, 28] by replacing the sign function by a saturation function defined as

*Remark 9. *In this paper, the VSC is designed to eliminate the effect of the uncertainties, and the robust controller is employed to achieve the desired tracking performance. In this way, global asymptotic stability and tracking performance of closed-loop systems are achieved.

*Remark 10. *From (29), it is clear that as ; that is, is satisfied. The boundedness of in (30) implies that . From the closed-loop dynamic equation (15) and boundedness of and , one can get . It implies that and ; thus, is achieved [23]. Therefore, the closed-loop system is globally asymptotically stable.

*Remark 11. *From (28), if the system start with the initial condition , that is, and , the gain must satisfy
where and . Inequality (29) indicates that when smaller attenuation levels are specified, a better tracking performance can be achieved.

#### 4. Simulation Example

To verify the theoretical results, simulations were carried out in two degrees of freedom robotic manipulator as shown in Figure 1 described by [22, 24]: where and are the mass of link 1 and link 2, respectively; and are the length of link 1 and link 2, respectively; denotes , denotes , and denotes , for and . is acceleration of gravity.

##### 4.1. Design Procedure

To summarize the analysis in Section 3, the step-by-step procedures of the adaptive robust quadratic stabilization tracking control for robotic system are outlined as follows.

*Step 1. *Select controller gains and such as
is a Hurwitz matrix.

*Step 2. *Choose proper parameters,
Solve from matrix inequality (15),

*Step 3. *Select the appropriate positive constants; then the robust quadratic stabilization tracking control law can be obtained from Theorem 7.

##### 4.2. Simulation Results

In this section, the proposed control approach is applied to control a two-link manipulator. The dynamic equation and parameters of the manipulator are similar to those in [22]. The nominal parameters of the robot used for simulation are and , , while actual parameters of robot are chosen as and to introduce the parameters uncertainties. The desired trajectories to be tracked are , . The initial conditions are and . For the purpose of comparison, simulation studies in three cases are conducted. To show the robustness of the controller, we choose the exogenous disturbances .

*Case 1. *Firstly, the conventional CTC (6) for controlling robotic system without model uncertainties and disturbances is demonstrated; that is, the nominal parameters are used to design the CTC controller and obtain the robotic manipulator model. Figure 2 shows the results of tracking performance. It is observed that CTC can track the given trajectories perfectly under precise known model parameters.

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*Case 2. *The conventional CTC (5) for controlling robotic system under model uncertainties and disturbances is demonstrated. To introduce the system uncertainties, the nominal parameters are used to design the CTC controller, while the actual parameters are used to obtain the robotic manipulator model. Furthermore, the exogenous disturbances are added. Figure 3 shows the simulation results of CTC. It can be seen that the controller cannot drive the joints to reach the desired positions and steady-state tracking error exist.

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*Case 3. *Under the same conditions as in Case 2, the proposed adaptive robust quadratic stabilization tracking controller is used to control robotic manipulator. The control procedure is described in foregoing subsection. Figure 4 shows the tracking results. It is observed that the tracking error decreases quickly, and the effects of uncertainties are successfully compensated by the adaptive robust control term.

To quantify the control performance, the root-mean square average of tracking error (based on the norm of the tracking errors ) is given as follows [14, 22, 24, 29, 30]:
where represents the total simulation time. Table 1 shows the error norms for CTC method (Case 2) and the proposed adaptive robust quadratic stabilization tracking control method (Case 3). Note that a smaller norm represents a better performance.

From Figures 3 and 4 and Table 1, the results thus demonstrate that the proposed robust quadratic stabilization tracking control can effectively control the robotic system in the presence of uncertainties and disturbances and present the better transient response and the smaller tracking error norm in comparison to the CTC method.

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#### 5. Conclusions

This paper presents an adaptive robust quadratic stabilization tracking control scheme for robotic system with mathematical derivations of global stability. The idea is based on combining a CTC controller plus an adaptive robust compensator. The CTC controller acts as the main controller, and the adaptive robust compensator is used to handle system uncertainties and external disturbances. In addition, an controller is used to achieve a certain tracking performance. The adaptive hybrid control method demonstrated robust and effective control performance on robotic system having uncertainties with good disturbance rejection. As a future work, the proposed adaptive robust controller will be proven experimentally and applied to other kinds of electromechanical systems.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20124101120001), China Postdoctoral Science Foundation (no. 2013M541992), National Natural Science Foundation of China (no. 61104022 and no. 61374128), and Key Project for Science and Technology of the Education Department of Henan Province (no. 14A413009 and no. 14A410001).