The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS8.002

A Novel Fuzzy Optimal Controller on Motion and Vibration Coordination Control

for a Multi-flexible Link Manipulator

Ji Wang Yangjun Pi Yumei Hu Xiansheng Gong

State Key Laboratory State Key Laboratory State Key Laboratory College of Mechanical

of Mechanical Transmission, of Mechanical Transmission, of Mechanical Transmission, Engineering,

Chongqing University Chongqing University Chongqing University Chongqing University

Chongqing, China Chongqing, China Chongqing, China Chongqing, China

wangji@cqu.edu.cn cqpp@ cqu.edu.cn cdrhym@163.com cqxsgong@cqu.edu.cn

Abstract

In this paper, a new motion and vibration coordination

controller - a Fuzzy Linear Quadratic Regulator (FLQR)

controller is proposed for the control of a multi-flexible link

manipulator. The FLQR controller bases on the dynamic model

of multi-flexible link manipulators, which is based on the finite

element method(FEM) and the Lagrange equation. Then a

synthesized LQR controller in which both rigid-body motion

and elastic deformation are considered, is combined with a

fuzzy controller to adjust weight ratio between motion state

variables and vibration state variables in weight matrix of LQR

online, according to motion and vibration feedback information.

The performance of the FLQR controller is compared with that

of previous typical motion and vibration controllers based on a

flexible four-bar linkage in simulation. Results show that the

proposed controller is more effective on motion and vibration

coordination control.

Key words: LQR, fuzzy controller, flexible manipulators,

trajectory tracking , vibration control

1 Introduction

The important trends in the development of

industrial robots are light weight and high speed, leading

to advantages such as lower energy consumption, faster

response and lower overall cost[1]. At the same time, the

flexibility of modern machines brings challenges into

modeling and control accordingly. For modeling of

flexible manipulators, the popular methods are the finite

element method (FEM) and the assumed-mode method

based on either the Lagrange equation or the

Newton–Euler recursive formulation. For control of

flexible manipulators, the main difficulty is the

interaction between the objective motion and undesired

vibration. In other words, motion control and vibration

control should be considered simultaneously in controller

design. It is the research hotspot and the great challenge

for researchers in this field.

The model of flexible manipulators is a distributed

parameter system, which is built by partial differential

equations (PDEs). In general, PDEs are divided into a

finite number of ordinary differential equations (ODEs)

by the FEM [2,3] or the assumed-mode method [4,5]

with linearization, in order to apply existing control

techniques such as PD control [6], model predictive

control [7], adaptive control [8], sliding control[9],

singular perturbation approach [10] et al. These control

methods have been applied in single flexible link

manipulators successfully. However, control of

multi-flexible link manipulators is a more difficult task

than that of single flexible link manipulators, since it is a

more complex nonlinear model. Some researchers

proposed the control method based on smart materials

featuring distributed actuators, such as piezoelectric

patches (PZTs)[11,12]. However this is not economical

from the practical point of view due to the requirement of

additional actuators and their accessories. Therefore

research focused on collected actuator control and it is

classified into feedforward type and feedback type. The

typical one in the first type is the input shaping

technique[13]. Whereas the drawback of this method is

that the expected trajectory may be affected or even

altered by shaping process. Hence the feedback type has

been paid more attention in recent years. In this field, the

most representative control methods are PID+SRF(strain

rate feedback)[14]，sliding control[15], LQR[16] and so

on. In these control methods, the common problem

which needs to be solved is that motion control and

vibration suppression affect each other. Specially, in

PID+SRF controller，PID is responsible for motion

control and SRF is used to reduce vibration

independently at the same time. Unfortunately, the

vibration control force from SRF would degrade the

motion control precision of PID. Reciprocally, the

differential function in PID would excite further

vibration of flexible linkages. In LQR, the weight

coefficient ratio between motion state variables and

vibration state variables is difficult to be defined. It

means a hard balance between motion control and

vibration suppression. Overall, the common exiting

problem of aforementioned control methods is that

motion control and vibration suppression can not be

coordinated sufficiently .

In this paper, a fuzzy linear quadratic regulator

(FLQR) is proposed for the first time in our knowledge.

A LQR controller in which both rigid-body motion and

elastic deformation are considered, is combined with a

fuzzy controller to adjust the weight ratio between

motion and vibration control to different situations,

according to the feedback information on vibration

amplitude. The relation of motion control and vibration

suppression can be coordinated well without sudden

change and high frequency changes during process of

real-time adjustment. It guarantees high-precision of

motion and vibration control in the process of

manipulator movement.

The paper is organized as follows: Section 2 briefly

outlines the general dynamic model of the planar

multi-flexible link manipulators derived by the finite

element method and the Lagrange equation. In Section 3

the FLQR controller is described. In Section4, the

numerical simulation on a flexible four-bar linkage is

carried out. Finally, the conclusions of this work are

provided in Section 5.

2. Multi-flexible link manipulator dynamic model

The combination of FEM and Lagrange equation is

a general method[17-19] in modeling of planar

multi-flexible link manipulator, shown in Fig 1. Firstly,

the multi-flexible link manipulator is divided into several

elastic beams by the FEM. Define as

elastic generalized coordinates in an elastic element,

representing nodal elastic displacements of the elastic

beam element in the local reference. The elastic

displacements can indicate the amplitude of vibration.

Define as system’s elastic generalized

coordinates, representing all nodal elastic displacements

of the multi-flexible link manipulator in the fixed

reference. Let f be the number of the system elastic

generalized coordinates. Define as the

rigid-body generalized coordinates, representing joint

angular displacements in accordance with rigid-body

degrees of freedom. d is the number of the rigid-body

generalized coordinates. The Rayleigh damping

coefficients are introduced into the system. And these

hypotheses [20,21] are adopted to simplify model: 1) the

instantaneous structure hypothesis; 2) total motion

composites of the large rigid-body motion and the small

elastic deflection; 3) the high level minim in the

elastodynamic equation can be neglected. Consequently,

a suitable dynamic model meeting the need of online

control of multi-flexible link manipulator can be written

as:

(1)

Where and are an elastic mass matrix and

rigid mass matrix respectively, and are

rigid-elastic coupling mass matrices. , , ,

are elements in the damping matrix and is a flexural

structure stiffness matrix. is a vector of the driving

forces of actuators. is a vector of the generalized

forces corresponding to the system’s elastic generalized

coordinates, only including external forces because the

interaction forces between different elastic elements offset

each other. is a vector of the generalized forces

corresponding to the rigid-body generalized coordinates.

Defining , a compact form of Eq. (1) can be

expressed as

(2)

Where

The above dynamic equation also can be rearranged as

(3)

Taking as the augmented

state vector, Eq. (3) can be expressed as

(4)

Where

Therefore the state equation of the multi- flexible

link manipulator can be given by

(5)

Where , , is the input variable

vector for the multi-flexible link manipulator. H is

temporary identity matrix. If the measured state variable

is taken into consideration, H should be changed

accordingly and the appropriate observer should be

added into system to estimate state variables.

Fig 1. General planar multi-flexible link manipulators. Detailed view of a deformed beam element (Inset).

3. FLQR system

A fuzzy linear quadratic regulator(FLQR) is proposed

in this paper for the first time. In this controller, the

weight ratio L between motion state variables and

vibration state variables can be adjusted online by fuzzy

method according to feedback information on vibration

amplitude. The traditional LQR for motion and vibration

control adopts constant weight ratio of above two types

of state variables. It is unable to get good results for both

motion and vibration control due to the phenomenon that

which state variable weight is high, the corresponding

error with the desired value is small. Therefore it is a

hard choice between motion control and vibration

suppression. The proposed fuzzy linear quadratic

regulator can distribute the weights of motion control and

vibration control to adapt to different situations online.

The diagram of FLQR controller can be seen in Fig 2

Fig.2 The diagram of FLQR system

The goal of the synthesized LQR controller is to

seek the optimal control force ( )t to minimize the

performance index J ,defined as:

(6)

Where is the feedback state variable

vector,

is

the objective state variable vector. and

stand for the joint objective angular velocities and the

joint objective angular displacements. It is worthwhile to

note that minimizes the absolute values of nodal

elastic displacements, nodal elastic velocities, errors of

joint angular displacements and errors of joint angular

velocities. The performance index reaches a minimum

value as long as the control force is defined as

(7)

Where the optimal control gain matrix is

defined as

(8)

And is the solution of the Riccati equation as follow

(9)

Where , are time-varying coefficient matrices of the

state - space Eq.(5). Hence the Riccati equation must be

solved online within a certain control period. According

to the characteristic of the infinite time LQR, the Riccati

Eq.(9) is an algebraic equation which can satisfy the

demand for online computing. The matrix in Eq.(9)

contains the weights of the control motor torques. The

matrix which contains the weights of all state

variables in the state - space Eq. (5) can be defined as

Q .

represent the

weights of elements in , ,

represent the weights of elements in and

represent the weights of

tracking errors of . L is the ratio of weights of

tracking errors to the weights of elastic displacements in

. L is an important factor for the motion control and

vibration suppression performance of the synthesized

LQR controller. The precision of motion control shows a

positive correlation with L and the effect of vibration

suppression shows a negative correlation with L in its

valid range.

L would be adjusted to different situations online in

the proposed controller. It should be noted that high

frequency changes and steps of weights may cause

system parameter perturbation which would excite high

frequency vibration in turn. Therefore, we use the fuzzy

algorithm to adjust the weight ratio L, which can avoid

the high frequency changes and steps with an appropriate

set of membership functions and fuzzy inference rules.

The input of the fuzzy controller is the equal envelope

curve (EEC) of the elastic displacement of the midpoint

in the longest flexible link. The EEC is constituted of two

stages. Stage1 is the constant compensation and stage 2

is the envelope curve gained by real-time rectification

and filtering of elastic displacement of the midpoint in

the longest flexible link. At the beginning, the curve

value is a constant KC , which compensates the lag of

envelope detection. The switch index from stage1 to

stage2 is:

(10)

Where are values of the envelope curve at

adjacent sampling points respectively. So Eq. (10) means

the switch point of EEC is the first peak point of

envelope curve.

The EEC is converted into the input variable fuzzy

field through scaling factor 1/Kf . There are two input

variable membership functions, and they should be

located in proper location with relatively large overlap to

avoid mutation. The membership functions used in

simulation for a flexible four-bar linkage is shown in

Fig.3.

Fig.3(a) Membership functions for input of the fuzzy controller

Fig.3(b) Membership functions for output of the fuzzy controller

The fuzzy rule is adopted to avoid high frequent changes

of weight ratio L .The fuzzy rule is shown in Tab1. The

Output of fuzzy controller is the weight ratio L, which is

sent to the LQR controller to renew the weight

distribution of motion control and vibration suppression

online.

Tab.1 The fuzzy inference rule in FLQR

fuzzy variables fuzzy inference rule

input B S

output S B

4. Simulation on a flexible four-bar linkage

4.1 Model of the flexible four-bar linkage

To prove the effectiveness of the proposed FLQR

controller, it is applied on a flexible four-bar linkage,

which is shown in Fig.1 with N=4. So the system elastic

generalized coordinates are and the

rigid-body generalized coordinate is . The system

parameters of the flexible four-bar linkage are given in

Tab 2.

deg

ree

of

mem

ber

ship

d

egre

e o

f m

emb

ersh

ip

Tab.2 Kinematic and dynamic features of the flexible four-bar linkage under investigation.

Items Parameter values of the model

Link length Link1

0.358m

Link2

0.530m

Link3

0.639m

Fixed link

0.329m

Concentrated masses

and inertias

Joint 2(mass)

0.030 kg

Joint 3(mass)

0.030 kg

Joint4 (inertia)

1.156×10-4kg m2

Square cross-section 6 mm×6 mm

Density 7.5×10-3kg/m3

Modulus of elasticity 193×109N/m2

The concrete form of matrices in model of the

flexible four-bar linkage according to Eq. (1) is written as

follow:

Elastic mass matrix are:

(11)

Where , are the density and the sectional area of the

beam element; is the shape function matrix of the ith

elastic beam element in the local coordinate system

shown in Fig.1 (insert):

is the coordinate transformation matrix, the

transformation from the global coordinate system to the

ith element local coordinate system:

(12)

Where represent respectively. is

the angle between the local coordinate system of the ith

element and the global coordinate system. which

consists of 1 or 0 is a matrix which represents the

transformation from the system elastic generalized

coordinates to the ith element elastic generalized

coordinates .

Rigid-elastic coupling mass matrix are:

(13)

is a sensitivity matrix [3] which represents the

relationship between the joint angular velocity of

link 1 and the rigid-body nodal velocities.

Rigid mass matrix are:

(14)

Where are the moments of inertia of link 1,

link2 and link3, respectively. is mass of the link 2.

are elements of which

respectively represent the gain coefficient from the joint

angular velocity of link 1 to the velocity in the

X-direction ,the velocity in the Y-direction, the angular

velocity of the midpoint of link 2 and the angular

velocity of the joint 4 (the last joint).

Damping matrix are:

(15)

(16)

(17)

Stiffness matrix are:

(18)

Where is the material’s elastic modulus and is the

cross sectional moment of inertia.

Generalized forces corresponding to the system

elastic generalized coordinates are:

(19)

Where is the ith element generalized forces

vector which corresponds to the elastic generalized

coordinates of the ith element.

4.2 simulation result

The step and sine object trajectories are used to test

the performance of the FLQR controller. The Final

position of step is 0.5 (rad), the sine object trajectory is

defined as

(rad) (20)

The control parameters of FLQR, LQR and PD+SRF in

two simulation cases are shown below.

Tab 3. Control parameters of controllers.

Step(position control)

Sine(trajectory tracking)

Controller Control parameters Control parameters

FLQR Kf=1.6×10-3 KC=1.3×10-3 Kf=1.2×105 KC=1.2×10-5

LQR L=80 L=80

PD+SRF kp=21

The input and output of FLQR controller, EEC

multiplied by scaling factor and weight ratio L adjusted

online, shown in Fig.4 and Fig.5.

Fig.4 The curve of equal envelope curve(EEC) multiplied

by scaling factor

Fig.5 The curve of weight ratio L

With the proposed appropriate fuzzy controller,

online adjusting of the weight ratio L is achieved without

sudden changes and high frequency changes, which

avoid further vibration stimulation and achieve

coordinating motion control and vibration control well.

4.2.1 Step object trajectory

Under the objective position 0.5rad for joint space,

the response of joint 1 angular displacement and the

elastic displacement of the midpoint in the longest

flexible are shown in Fig.6 and Fig.7 respectively.

Fig.6 The elastic displacement of the midpoint in link3

Fig.7 The response of joint1 angular displacement

It can be seen from Fig.6 that the vibration

amplitude and decay time of the FLQR controller are

smaller than other two controllers. It means the proposed

controller possesses better vibration suppression,

compared to previous LQR and PID+SRF. From Fig.7, it

can be concluded that motion control of the PID+SRF

controller is worst among above controllers. The undue

overshoot appeared in the response of joint 1 angular

displacement, because differential coefficient in PID is

subject to vibration. The effect of motion control in

FLQR and traditional LQR is similar. The response time

of FLQR is slightly longer than that of LQR, because the

FLQR pay more attention on vibration suppression in the

start-up stage when the vibration is violent. The

difference in response time of FLQR and LQR is small

enough to be ignored, as opposed to the advantage in

vibration suppression of FLQR.

4.2.2 Sine object trajectories

Under the sine object trajectory for joint space, the

effects of vibration suppression and motion control are

shown in Fig.8 and Fig.9 respectively.

Fig.8 The elastic displacement of the midpoint in link3

Fig.9 The tracking error of joint1 angular displacements

The elastic displacements of the midpoint of the 3rd

flexible link and tracking errors of joint1 under the sine

objective trajectory Eq.(20) are shown in Fig.8 and Fig.9.

The similar result with previous simulation is that the

vibration suppression of FLQR controller is better than

that of LQR and PID+SRF. The tracking error of FLQR

in the stable stage is much smaller than others, although

its tracking error at the start-up stage is larger than other

two controllers. It needs to be emphasized that the more

important index of motion control is errors in the stable

stage rather than start-up stage, because people pay more

attention on motion trail in work stage of manipulators.

5 Conclusion

In this paper, a new motion and vibration

coordination controller (FLQR) is proposed for a

multi-flexible link manipulator. The FLQR controller can

adjust the weight ratio L of the motion state variables to

the vibration state variables in weight matrix of LQR

online by a fuzzy controller. It can redistribute the

weights of motion control and vibration control

according to different feedback information on vibration

state of manipulators, without sudden changes and high

frequency changes which may excite high frequency

vibration. Hence the FLQR controller can coordinate the

motion control and vibration suppression well. Both step

and sine object trajectories were taken to test the

performance of the FLQR controller and previous two

typical motion and vibration controllers. The results

show that the FLQR controller has better behavior on

trajectory tracking and vibration suppression than

traditional LQR and PID+SRF and its position control

effect is very close to that of LQR with ignorable

difference, which is the best position control among these

controllers. It has successfully been verified that the

FLQR controller can guarantee high-precision motion

and vibration control in process of manipulator

movement. Without loss of generality, the proposed

controller can be applied to the control for general

flexible manipulators.

However, the differences between the above

dynamic model of the multi-flexible link manipulator and

the actual multi-flexible link manipulator often appear in

engineering practice. Therefore, our future work would

add robust into proposed control system to compensate

the uncertainty between the dynamic model and the

actual multi-flexible link manipulator.

Acknowledgement

The authors gratefully acknowledge the financial support

from the National Basic Research Program of China(973)

[2014CB049403], the National Basic Research Program

of China(973) [2014CB049404] and the Special

Research Fund for the Doctoral Program of Higher

Education [20130191110004].

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