Stratis A. Kanarachos, Konstantinos Spentzas

Nonlinear vibration control of a flexible-link robot arm based on a Taylor series expansion of the control law

STRATIS A. KANARACHOS, KONSTANTINOS SPENTZAS

Mechanical Engineering Department

National Technical University of Athens

Polytechnioupolis Zografou, 15780 Athens

GREECE

Abstract: - This paper discusses a methodology for improving the dynamics of a flexible-link robot arm based on a Taylor series expansion of the control law, using time simulation and parameter optimization. The method can be interpreted as a special neural network (NN) technique with predefined structure (Taylor series expansion of the control law) and weights (parameters) to be optimized. As such, the paper focuses on the performance of the nonlinear control related to actuators with u-limits and to time optimal control. The above method is presented and discussed in the context of a typical flexible-link robot arm. Finally, test cases are used to illustrate the performance of the proposed method.

Key-Words: - Flexible robot arm; Control; Neural networks; Parameter optimization; Taylor series expansion;

1 Introduction

Over the last decade, the modeling and control of lightweight flexible-link manipulators have been challenging research areas with the objective of improving robot performance and a number of control methods have been proposed. Latest contributions for modeling the dynamics of a flexible robot or mechanism are given in [1], while methods for control, including command shaping techniques, feedback control and inverse dynamics are given in [2]-[8].

This paper discusses the nonlinear vibration control of a flexible-link robot arm, based on a Taylor series expansion of the control law, using time simulation and parameter optimization methods. The method as such can be interpreted as a special neural network (NN) technique with predefined structure (Taylor expansion of the control law) and weights (parameters) to be optimized. In this context, the paper focuses on special features related to control using actuators with u-limits, and to time optimal control.

This is considered to be an essential part of the necessary controller synthesis, because:

1.  The actuator’s weight, a function of the actuator’s power, influences the masses and the overall weight to be controlled, and

2.  Time optimal control is desirable in order to minimize the duration of the vibration.

The above methodology is presented and discussed using a flexible-link robot arm [2]. Then results for a linear controller using a u-limited actuator are presented. Finally a nonlinear controller is designed using the proposed methodology to achieve the same tasks. The following test cases illustrate the performance of the proposed method.

2 Modeling of the robot arm

This section briefly describes modeling of the flexible robot as one beam-type finite element with nodes 1 and 2, used as a simulation environment for the application of the proposed method (see also [3]). For a small angular displacement θ(t) and a small elastic deflection w(x,t), the FE method results in:

(1)

In eq. (1) u denotes the control vector, x the nodal DOFs

(2)

and M and K the mass matrix and stiffness matrix respectively:

(3a) (3b)

(4)

In (3) M1 and M2 are discrete masses and Θ1 and Θ2 discrete mass moments associated with the the nodes 1 and 2. For the case under consideration w1=0, while the rigid body motion θ is defined as follows:

(5)

Equations (1)-(5) describe the motion of the flexible robot arm with satisfactory accuracy.

They above equations are transformed in the state-space form using the vector z:

(6)

In (6) ua is the actuator’s output acting on the θ1-DOF. The actuator is modelled as a first order transfer function:

(7)

with Ta denoting the time constant, τ the applied torque and ua,lim the u- or torque-limit of the actuator.

3 Robot arm control

The state-space formulation of (1)-(7) is described by the following equation:

(8)

The torque τ is generally a nonlinear function of the state vector z

(9)

with zD denoting the desired state. It can be expanded in a Taylor series according to (10):

(10)

In case of the linear control law with no full state feedback, τ can be set equal to

(11)

In (11) the states and result from the desired states and, therefore, they are not considered in the control law (11).

In case of the nonlinear control law, only second order terms of the Taylor series expansion are considered in this paper. A principle component analysis showed that the following nonlinear form is sufficient with respect to second order approximation:

(12)

Equation (12) is a p-control with variable p-coefficients pk , k=1-4.

The control laws (11) and (12) depend now on a number of constants (weights or parameters) c

(13)

that have to be computed, so that an appropriate performance index J is minimized.

(14)

Equations (11)-(14) constitute a parameter optimization or a neural network problem with a given nonlinear controller structure (constant or variable pk), which has to be solved.

The performance index J is defined using the time response characteristics of each variable zi as follows:

1.  The maximum overshooting Sij for a given desired state zDj and initial conditions

(15)

2.  The zero Zij and first order Vij dynamic characteristics of zi (t) at time t:

(16)

This way, the following time dependent performance index Jij referring to the variable zi, to the desired state zDj and to the initial conditions is built up:

(17)

In (17) the index “lim” denotes limits that have to be defined by the user, λ weighting factors and the bracket :

(18)

If within a given observation period To the performance indices Jij are for a time t=Ti equal to zero, then the time simulation is ended ant Tij is considered to represent the response time of the system with respect to the variable zi and the initial conditions . Thus Jij is equal to:

(19)

The total performance index of the system within the observation time period of To is then equal to:

(20)

For the solution of (20) the Nelder-Mead algorithm (FMINSEARCH of MATLAB), has been successfully applied.

4 Numerical Implementation

Following [2], an aluminum flexible link-robot arm of length L=1.2 m, cross sectional area A=4·10-4 m2 and moment of Inertia I=1.33·10-8 m4 has been used for the numerical implementation. The arm carries at its end a (reduced) bulk mass of M2=10 Kg and Θ2=10 Kgm. Τhe eigenfrequencies of the link robot arm are in this case 0 Hz (rigid body motion) and 3.28Hz and 0.705Hz (elastic motion). The actuator’s characteristics are Ta=0.025 sec and ua,lim=10 Nm. The motion has to be controlled for the θ-range between -10 ο and 10ο degrees.

4.1 Linear controller

In the case of a linear controller, application of the proposed method for z0j=0 and ua,lim=10 Nm yields

(21)

The performance of the controller is shown in Fig. 1 and Table 1.

Fig. 1 Linear controller (results for ).

Table 1. Response times of the linear controller.

4.2 Nonlinear controller

In the case of a nonlinear controller, five desired states (θD=2o, 4o, 6o, 8o and 10o; z0j=0) have been simultaneously considered. For stability reasons, the variable pk-coefficients are constrained:

(23)

with θlim denoting the limit of the linear branch for a given ua,lim. With (23), the parameters to be optimized for a given ua,lim are:

(24)

For the case under consideration the following values are computed for θlim=10o.

(25)

The performance of the controller (24) is shown in Fig. 2 and Table 2. The above results (linear and nonlinear cases) were obtained for λVi=0 and Zi,lim equal to θD/400.

Table 2. Response times of the nonlinear controller.

Fig. 2 Nonlinear controller (results for ).

5 Stability

In the following simulation results for the linear and nonlinear controllers for a severe disturbance of the initial state with respect to the w2-deflection (approximately 49% of the maximum deflection for θD=10o)

(26)

are presented in Fig. 3 and 4.

Fig. 3 Stability of the linear controller (total response time ΣTi =18.38 sec).

Fig. 4 Stability of the nonlinear controller (total response time ΣTi =15.94 sec).

As it can be seen, stability is maintained, even for the above severe disturbance of the initial state.

6 Minimizing the actuator’s weight

The design of nonlinear controllers serves two important targets: Faster response of the dynamic system and minimization of the actuators u-limit and weight respectively. In this section three computational results (see Fig. 5-7) are presented.

Fig. 5 Linear controller. Results for .

Fig. 6 Nonlinear controller. Results for .

Fig. 2 Nonlinear controller. Results for .

These results refer to a linear and to two nonlinear controllers according to chapters 3 and 4, now for an actuator with half of the original u-limit (5 Nm): The corresponding p-controls are:

(27)

The results of chapter 4 and 6 are summarized in Table 3.

§  Within the nominal control range between -10 ο and 10ο the p|10Nm(10o) and p|5Nm(10o) controllers have a linear characteristic (the linear branch is extended up to 10ο), corresponding to a 1st order Taylor series expansion within the nominal control range.

§  The p|5Nm(5o) controller has within the nominal control range between -10 ο and 10ο a bilinear characteristic (the linear branch is extended up to 5ο), corresponding practically to a 2nd order Taylor series expansion within the nominal control range, with the two linear branches representing the two asymptotes of a quadratic function.

The advantages of the proposed Taylor series based nonlinear controllers, are obvious: they lead to total response times that are much smaller than the total response times of the linear controllers, even if the ua,lim is decreased tom 10 Nm to 5 Nm.

Table 3. Total response times of flexible-link robot arm controllers.

7 Conclusions

In this paper a method for the nonlinear vibration control of a flexible-link robot arm based on a Taylor series expansion of the control law has been presented. A principle component analysis was applied to select the members of the Taylor expansion and the resulting feedback coefficients have been computed using time simulation and the parameter optimization method of Nelder-Mead. The method can be interpreted also as a special neural network (NN) technique with predefined structure (Taylor expansion of the control law) and weights (parameters) to be optimized. The non-full- state feedback control of the flexible-link robot arm has been computed for five cases, including actuators with different u-limits, showing the superiority of the proposed nonlinear control. The results are at this stage encouraging for a further investigation of the proposed method with respect to the optimum structure of the Taylor series expansion and to the implementation of other parameter optimization methods.

References:

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[2]  Z. Mohamed, M.O. Tokhi. Command shaping techniques for vibration control of a flexible robot manipulator. Mechatronics (2003) (to be published)

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[8]  H.W. Park, H.S. Yang, Y.P. Park, S.H. Kim. Position and vibration control of a flexible robot manipulator using hybrid controller. Robotics and Autonomous Systems 28 (1999) 31-41

Acknowledgement: Work presented in this paper was developed in the European project “Herakleitos” and was supported by the European Union, Hellenic Ministry of Education. We are very grateful for their support.