Development of Piezoelectric Actuator Control System using FEM*Daigoro Isobe1), Hiroshi Nakamura2), Ryuta Shimizu2) andHisashi Nakagawa3)
Institute of Engineering Mechanics, University of Tsukuba, Ibaraki 305-8573, JapanABSTRACT
In this paper, the Finite Element Method (FEM) is proposed for application to a real-time
control system of connected piezoelectric actuators, assuming an actuator as finite elements,
which are mainly used in the field of computational mechanics. The FEM is capable ofexpressing the state of an entire system by stiffness equations, and can cope flexibly with lack or
disability of constituent elements of the system by controlling the stiffness matrices. An inverse
analysis program using the FEM, combined with the empirical formulae considering timedependent characteristics of the piezoelectric material, is implemented into a parallel control
system. The system is verified by the experiments on quasi-static displacement control ofconnected piezoelectric actuators. The results show the possibility of highly accurate, real-time
control of the actuators using the FEM.
INTRODUCTION
Parallel control of robotic systems, which is an adaptive control scheme involving thereassembly of hardware systems into different configurations while still using the same software,
has been progressively developed in recent years. The concept of robotic systems with highadaptability to their environment is called Parallel Robotics, and is expected to be applied inspace, under the sea, mining, and construction, where environments tend to change rapidly.
Hamlin and Sanderson (1997), for example, had made a prototype of robotic architecture whichis capable of reassembling into many different patterns of platforms consisting of linear
actuators and multi-link spherical joints.
In this paper, a parallel control system for a continuous architecture formed by bimorph1) Assistant Professor
2) Graduate Student
3) Kajima Construction Co. (Ex-undergraduate Student)piezoelectric actuators has been developed and verified, as a part of the research project by Fujii
et al. (1997), with the goal of constructing more human-like robots from a biological point ofview (see Figure 1). The conventional control system necessitates the change of state equations
depending on the shape of a system or the quantity of the linked members. On the other hand,
the Finite Element Method (FEM), a generally used computational tool in analyzing structures,
fluids, etc., is capable of expressing the behaviors of each discrete element, as well as the wholecontinuous system, by evaluating the stiffness equations derived from the energy principle.
Flexible control can be achieved simply by changing the material properties of the finiteelements expressing the actuators, in case of disorder. Therefore, the FEM, when used as acontrol scheme, may achieve a higher redundant parallel control system.
Several steps are taken to enable real-time control by FEM. First, a simple inverse theory,
which is obtained by considering the linearity of voltage distribution in bimorph piezoelectric
actuators, is applied for calculating the control voltage. The algorithm using this theory
consumes less computational memory. Second, a noncompatible four-node element isimplemented into the FEM program. The finite element allows the in-plane bending mode byconsidering a noncompatible mode in shape functions. Although the continuity of displacement
along the elemental boundary is not satisfied in the element, practical accuracy can be achievedby a minimum number of elements. Third, to minimize the effect of time-dependent
characteristics of a piezoelectric material, empirical formulae for creep deformation and residualstrain are evaluated, and are combined with the FEM inverse analytical algorithm.
In this paper, the inverse analysis program using the FEM combined with the empiricalformulae, is implemented into a parallel control system. The system, which consists of a PC, abipolar DC unit and a signal-distributing controller, is examined and verified by experiments on
quasi-static displacement control of connected piezoelectric actuators.

(a) Conventional robotic system(b)Piezoelectric actuator robot

Fig.1 Robotic systems

INVERSE THEORY USED IN PIEZOELECTRIC ACTUATOR CONTROL
When no external force is applied to the material, the stiffness equation under static
condition considering piezoelectricity (Allik and Hughes, 1970) is:   where [Kuu] is the stiffness matrix, [Ku] the piezoelectric stiffness matrix, {u} the displacement
vector and { } the voltage vector. Displacements are calculated by multiplying the inverse
matrix of the [Kuu] matrix from the left on both sides of the equation above. On the contrary, thevoltage required for achieving target displacement can not be simply calculated from theequation, since the [Ku] matrix has no inverse matrix. Thus a simplified inverse theory using
the linearity of voltage distribution in a bimorph piezoelectric actuator is employed (Isobe andNakagawa, 1999).
By assuming the thickness of bimorph piezoelectric actuator to be t mm, the voltage at topand bottom planes to be 0V, the voltage at the central plane to be 0 V, and the distance of anode point (No. i) from the central plane to be di mm, the voltage applied at the node point iVis:      





By using the above equation, the right hand side of Eq.(1) can be expressed as:     When target displacement is given to the tip of an actuator by the pseudo-force scheme, acertain reaction force is calculated at the freedom. By comparing the reaction force fb (>0) andthe corresponding components of Eq.(3), the relation between control voltage 0 and reaction
force fb is obtained as:                  

where kbi(i=1,…,n) are the (freedom No. of target displacement, node point No.) components ofthe [Ku] matrix.
ne is a coefficient connecting the mechanical and electrical quantities, and it
shows a dependence on elemental size. It is calculated by carrying out a direct analysis and aninverse analysis. From Eq.(4), we can derive      and the control voltage of the actuator can be obtained. Furthermore, the control voltage,
considering the reaction force from other objects in the case of contact, can be evaluated byadding the corresponding value to the equation.

EMPIRICAL FORMULAE USED FOR TIME-DEPENDENT CHARACTERISTICS
A piezoelectric material shows time-dependent characteristics such as creep deformation
and residual strain (see Figure 2). Some empirical formulae are derived to consider such
nonlinearlity, which also has a strong dependence on applied voltage.

400

300



200
100

00100200300400

500

Fig.2 Creep deformation and residual strain of piezoelectric actuator

Empirical formulae considering creep deformation
As seen in Figure 2, the actuator tip displacement after a certain time lapse can beexpressed by the sum of the time-independent component ui and time-dependent component
u. Therefore, the displacement at time t can be determined by approximating the timedependency of u by the asymptotic exponential function:                  

where a and c are the constants for convergence. The function expressing dependence on theapplied voltage can be written in the same manner against ui and uc as:       where sinc and m are the constants of voltage dependence. Displacement including creep
deformation after convergence can be expressed as:   

where ccr is the mean ratio of uc and ui in a low-voltage area.
Empirical formulae considering residual strain
As also seen in Figure 2, a large amount of residual strain can be observed after theelectrical charge is reduced. The displacement at time t can be determined by approximating thetime-dependency of uR by the asymptotic exponential function:             where b and d are the constants for convergence. The same function finc from Eq.(7) is used asthe voltage dependency function for ui. Since crystal grain in a reduced electrical field acts
differently, another function for the second component is defined as:         

where sdec1 and sdec2 are the constants of voltage dependency, q and n the exponents, andPmax the maximum voltage applied in the control process. Displacement considering residualstrain after a complete discharge can be expressed as:     

where umax is the maximum target displacement in the process, and crs is the mean ratio of uR andumax in a low-voltage area.
DEVELOPMENT OF REAL-TIME CONTROL SYSTEM USING FEM
A GUI application using FEM inverse analysis combined with the empirical formulae is
implemented into a parallel control system. The system consists of a PC, a bipolar DC unit and asignal-distributing controller as shown in Figure 3. Figure 4 shows the algorithm used in theGUI application. The GUI application starts by initializing the element subdivision followed by
the calculation of a global stiffness matrix, to minimize the CPU time during the control process.
The application moves to the next step if target displacements are provided by the user, andstarts time loop calculation of the inverse analysis. One cycle of calculation by a 270 MHz clock
CPU takes only 30 ms. The output data of the control voltage is provided every 200 ms, and issent via a serial wire to the controller. The controller supplies independent DC power to eachactuator.
① SG tU aI ratpipnlgi c a t i o n
I con

Cl ick!

Initializing application

Co n tr ol l er・e l e m e n t s u b d i v i s i o n・c a l c u l at i n g s t i f f n e s s m a t r i x

D C s u pp ly

I n v e rs e a n a l y s i sC a l c u l a t i n g s ti f f n e s s e q .
C a l c u l a t i n g e m p ir i c a l f o r m u l aＰＣ
Se r ia l w i r e
A ct u at orFig.3 Control system using FEM

In p u t
ta r g e t d i s p .

②

Calculation ofcontrol voltage

200 permsoutputO u tp u t
c o nt r o l v o l .

Fig.4 Algorithm in GUI application

Four piezoelectric actuators (each with six finite elements) clumped at one end andconnected at the other end with elastic material (natural rubber, two finite elements), is selected
for the experimental model of quasi-static displacement control. Figure 5 shows the control
voltage applied to each actuator, which is obtained from the FEM control system. The control
voltage changes with time to minimize the effect of time-dependent characteristics. As seen inFigure 6, the creep deformation and residual strain are effectively reduced, and practicallyaccurate control is achieved. The experimental data suggest that the FEM control system
handles the effect of the material connecting the actuators, since the control voltage varies when
the property data of the connecting material are changed.

ch1: 0 50 100 150 100 50
ch2: 0 100 100 100 100 100
ch3: 0 50 150 50 150 50
ch4: 0 150 100 150 100 150

 








(unit:

ch1: 0 50 100 150 100 50
ch2: 0 100 100 100 100 100
ch3: 0 50 150 50 150 50
ch4: 0 150 100 150 100 150

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
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(unit: 

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Fig.5 Applied control voltage















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Fig.6 Attained displacement by FEM control

CONCLUSION
Although some steps must be taken to minimize the calculation time, there is a possibilityof using the FEM as an effective control scheme. Most control schemes handle an entire system
in series, which becomes a drawback when some elements lose their function. On the other hand,
the FEM can handle the system in parallel by subdividing it into discrete finite elements, andcan cope with flexibility in the change of form and material properties. This kind of approach
may be applied to other fields of robotic control.
REFERENCES
Allik, H. and Hughes, T.J.R. (1970), "Finite element method for piezoelectric vibration",
Int. J. Numer. Methods Eng., 2, 151-157.
Fujii, T., Isobe, D. and Kuroda, Y. (1997), "Finite Element Super Parallel Actuators: FESPA-Its Concept and Basic Analysis-", Proceedings of the JSME Annual Conference on
Robotics and Mechatronics ’97, B, 873-874, in Japanese.
Hamlin, G.J. and Sanderson, A.C. (1997), "TETROBOT: A Modular Approach to Parallel
Robotics", IEEE Robotics and Automation Magazine, 4(1), 42-49.
Isobe, D. and Nakagawa, H. (1999), “Real-Time Parallel Control of Connected PiezoelectricActuators by using FEM”, Transactions of the JSME, 65A(629), 120-126, in Japanese.