Robust Tracking Control of Robot Manipulator Using Dissipativity Theory

The robust H controller is designed for the problem of rigid robot tracking, which based on the dissipativity theory. The quadratic form dissipative feedback control law was given for interference suppression under the condition of existing model error and external disturbance. The scheme improved the robustness of the system. The simulation results show that the algorithm can achieve rapid tracking of the robot system.


Introduction
Robot system is a very complex multi-input multi-output nonlinear system with time-varying, and the strong coupling of nonlinear dynamics.dissipativity theory has been put forward by the people's concern and reached a wide range of research results (Van der Schaft A.J., 1999;Feng, 1998) in the 1970s.Its substance is that the internal energy system loss is always less than the external energy supply rate.H control (Mei,2003)and passive control are a special cases of dissipative control.The H control is a kind of interference suppression control, which can not only guarantee the stability of the system, but also achieve smallest degree requested by the interference to system output.If the supply rate is the product of input and output, the state will be passivity problem.Passivity theory has been widely used in many engineering problems, such as electrical systems and thermal power systems.A strict passive dynamic system generally has excellent dynamic characteristics and satisfactory robustness.(Feng, 1999,pp. 577-582) Stabilization controller is designed by using passivity theory on the definite part of the robot system, and then uses the dissipativity theory for interference suppression under the condition of existing model error and external disturbance.The scheme enhances the robustness of the robot tracking, at the same time, and improves the tracking accuracy and speed.

Model Control Law Design
Considering the following n-degree of robot, the dynamic equation is as follows (1) Where the vector ( ) q t is the 1 n joint angle; ( ) M q is the n n symmetric positive definite inertia matrix; ( , ) C q q q is the 1 n vector of coriolis and centrifugal torques; ( ) G q is the 1 n vector of gravitational torques; is the 1 n vector of actuator joint torques.
The description of robot systems in Eq (1), has the following characteristics: Property 1: ( ) M q is a bounded, symmetric positive definite matrix, its inverse matrix is a bounded.There are positive number 1 and 2 , is skew symmetric matrix, both M an C in Eq (1) satisfy the following equations Suppose that the desired trajectory of system is described by d q , d q and d q , then the corresponding error is defined as d e q q , d e q q , d e q q The can be given as u is a control input signal.
For system in Eq(1), we can give non-linear compensation in Eq(2), receive error dynamic equation: The state vector is defined: (5) In the new coordinates, and Eq(3) can be translated into the following equation of state: The output signal is defined: Choose the form of Lyapunov function as follows: Along the state trajectory, and its time derivative is as follows: The feedback control law is as follows T T T

V t x x x y v y v v
This shows that the closed-loop system is passive from the input, viz.v , to the output, viz.y .According to relations of passivity and asymptotic stability, let 2 v y x , the closed-loop system is gradual and stable.
The equation of state ( 6) is replaced by: If formula ( 10) is passive, which can be written for the general form: According to a KYP lemma, then:

Robust Controller Design
Consider the model error and external disturbances, the robot model: ( ) ( , ) ( ) M q q C q q q G q (13) The equation of state: Combining ( 9) and ( 14), we have

y y y y y L V y y y y y L V y y
So the closed-loop system is dissipative on supply rate 2 2 2 ( , ) s w y y .( ) V x is the storage function.