Motion control of an electrohydraulic actuated toggle mechanism

Mechatronics 11 (2001) 939±946

Technical Note

Motion control of an electrohydraulic actuated toggle mechanism Rong-Fong Fung a,*, Rong-Tai Yang b a

Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science & Technology, Kaohsiung 811, Taiwan, ROC b Department of Mechanical Engineering, Ming-Hsin Institute of Technology, Hsinchu 30404, Taiwan, ROC Received 1 April 1998; received in revised form 31 March 2000; accepted 26 June 2000

Abstract The main objective of this paper is to study the motion control of a toggle mechanism actuated by an electrohydraulic system. This control system is a nonlinear time-varying one due to the internal leakage of the electrohydraulic system as well as the nonlinear dynamics of the toggle mechanism. The speci®ed position±velocity pro®le of the output slider of the toggle mechanism is considered as the desired trajectory. The numerical results via the inverse dynamics control (IDC) and variable structure control (VSC) are compared for the control system with external disturbances. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Motion control; Toggle mechanism

1. Introduction Hydraulic servomechanisms are widely used in manufacturing machinery, heavyduty machines and the automobile industry because hydraulic actuators can provide a comparatively small device with high torque, high response speed, high sti€ness [1], very large force capability and high ratio of force to weight [2]. However, there are many disadvantages in the hydraulic control system, such as the limit of the upper temperature, ¯uid compressibility, ¯ow-pressure relationship deadband due to the internal leakage and hysteresis, and many uncertainties of a hydraulic system due to

*

Corresponding author. Tel.: +886-7-6011-000 Ext. 2221; +886-7-601-1066. E-mail address: r€[email protected] (R.-F. Fung).

0957-4158/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 7 - 4 1 5 8 ( 0 0 ) 0 0 0 5 4 - 4

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linearization. These drawbacks cause diculties in the control of a hydraulic system [3]. In this paper, two types of control laws are employed to compare their performances. The ®rst one is called the IDC method. This control method is usually used in robot control and is a special application of feedback linearization of a nonlinear system [4]. Experimental and theoretical studies on rapid position control based on inverse dynamics have been reported [5]. Matsui and Mochizuki [6] investigated the e€ect of positive angular velocity feedback on the torque control of a rotary hydraulic actuator. The second control method is the VSC method, which is one of the major approaches to deal with the nonlinear system with parameter uncertainties [7]. Gao and Hung [8] investigated a new approach for the design of the VSC of a nonlinear system. This approach was called the reaching law method, and was complemented by a sliding mode equivalence technique. Chern and Wu [9] designed an integral variable structure controller and applied it to the velocity control of an electrohydraulic servomechanism system. Hwang and Lan [3] developed a new VSC law for the electrohydraulic servomechanism with time-varying, parameter uncertainties and external disturbances. Recently, Vossoughi and Donath [10] have developed a dynamic feedback linearization for an electrohydraulic servo system, in which the controller was designed to eliminate the e€ects of the nonlinearities. Tunay and Kaynak [11] have proposed a new VSC method, which combined the VSC with variable structure adaptation and performs switching with hysteresis between the structures in order to avoid a sliding mode. To the authors' knowledge, there are little papers about the electrohydraulicmechanism control system. In this paper, the IDC and VSC methods are applied to an electrohydraulic-mechanism control system. Numerical results are provided for comparisons.

2. Coupled system equations The physical model of the electrohydraulic actuated toggle mechanism is shown in Fig. 1. This toggle mechanism is composed of two slider-crank mechanisms. PC is the driving force applied by the electrohydraulic actuator and PB is the force acting on slider B, which can be treated as disturbance force. The coupled system equations [12,13] have matrix form as follows: ^A ‡ Q ^ D; ^ v ‡ N…v; ^ v_ † ˆ Q M…v† where ^ ˆ Mvv M

M Uq 1 Uv

h ^ ˆ Nv N

UTv …Uq 1 † N

h uv UTv …Uq 1 †T M

vq

T

…1†

q

i

h vq ‡ M Uq 1

i uu M Uq 1 Uv ;

i qq T UTv …Uq 1 † M Uq 1 c;

Fig. 1. The physical model of electrohydraulic actuated toggle mechanism.

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Table 1 The values of the electrohydraulic system parameters Parameter

Value

Unit 7

PS kq Clt Ap Mcy Bcy kv

N/m2 m2 /s m4 s=kg m2 kg N/m/s m/V

1:37  10 p 9:184  10 5 PS sgn…Xv †PL 1  10 13 3:14  10 4 0.28 4  10 5 1:7  10 4

^A ˆ Q



Ap kq kv r4 ‰ sin…h1 ‡ /† Clt

 cos…h1 ‡ /† tan h5 Šuc ;

^ D ˆ ‰PB r1 ‰ sin h1 ‡ cos h1 tan h2 ŠŠ: Q ^ is the mass matrix, N ^ the vector containing nonlinear displacement and velocity, M U the geometry constraint, Ap ; Clt ; kq and kv are the electrohydraulic system pa^ A , while the forcing rameters shown in Table 1. The control input uc is included in Q ^ term acted on slider B is casted into QD . 3. Inverse dynamics control method In this section, the IDC method is employed to construct a controller for the electrohydraulic actuated toggle mechanism system. The design of the IDC [4] includes (i) an inner nonlinear loop design and (ii) an outer loop control signal design. First, we design the inner nonlinear loop. De®ne the tracking error as e ˆ vd e ˆ vd

v; v;

e_ ˆ v_ d

v_ ;

…2†

where vd is the desired trajectory vector. From Eq. (1), we have   ^A ‡ Q ^D N ^ : ^ 1 Q v ˆ M

…3†

Substituting Eq. (3) into the third equation of Eq. (2) yields   ^A ^ D: ^ 1 N ^ Q ^ 1Q e ˆ vd ‡ M M

…4†

De®ne the control input function as   ^A ; ^ 1 N ^ Q u ˆ vd ‡ M

…5†

then inverting Eq. (5) yields the computed torque law as   ^A ˆ M ^ vd u ‡ N: ^ Q

…6†

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Secondly, we design the outer loop control signal. The PD control law is selected as a feedback control scheme. Let uˆ

Kv e_

Kp e;

…7†

where Kv and Kp are control gains. Substituting Eq. (7) into Eq. (6), the overall control force is obtained as ^ A ˆ M… ^ vd ‡ Kv e_ ‡ Kp e† ‡ N: ^ Q

…8†

4. Variable structure control method We adopt the reaching law method [8] of the VSC law for the electrohydraulic actuated toggle. De®ne the state vector h iT x ˆ vT ; v_ T : …9† Eq. (3) is rewritten as the form x_ ˆ a…x† ‡ b…x†u ‡ d…x†; where

…10†

"

#   v_ 0 a…x† ˆ ; d…x† ˆ ^D ; ^ ^ 1N ^ 1Q M M   0 ^ A: b…x† ˆ ^ 1 ; u ˆ Q M

De®ne the tracking error state vector as     e v v eˆ 1 ˆ d : e2 v_ d v_

…11†

The ®rst task of designing a VSC controller is to design a switching function. In general, the form of switching function is designed as s…e† ˆ Ce;

…12†

where C ˆ ‰c1 ; IŠ. Adopting the reaching law s_ ˆ

Q sgn…s†

Ks;

…13†

where Q and K are the gain matrices, and taking the time derivative of Eq. (11) and using Eq. (9) yields   ^D u : ^ 1 N ^ Q s_ ˆ c1 e2 ‡ vd ‡ M …14† Equating Eqs. (12) and (13) and solving for the control u yields i h ^ D: ^ c1 e2 ‡ vd ‡ Q sgn…s† ‡ Ks ‡ N ^ Q uˆM

…15†

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Secondly, following the standard process of designing the control input, we have i h ^ DC : ^ c1 e2 ‡ vd ‡ Q sgn…s† ‡ Ks ‡ N ^ Q uˆM …16† ^ DC is the bounded disturbance with a known quantity. The chattering where Q phenomenon due to the sign function including the control input is undesirable for a mechanical system. To eliminate the chattering phenomenon, a smoothing function Md …s† ˆ s=…jsj ‡ d† [14] is introduced to replace the sign function in Eq. (15), and d is a positive constant value chosen small enough to avoid the chattering phenomenon. 5. Simulation results and discussions The elements of toggle mechanism have the following dimensions: r1 ˆ 145 mm, r2 ˆ 70 mm, r3 ˆ 190 mm, r4 ˆ 100 mm, r5 ˆ 55 mm, f ˆ 25 mm and h ˆ 80 mm.

Fig. 2. Comparing the tracking performance between the IDC and VSC without disturbance ((. . .) IDC, …І VSC). (2a) Tracking trajectory, (2b) P±V diagram, (2c) position error, (2d) velocity error, (2e) acceleration error and (2f) control input voltage.

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The masses are as follows: m2 ˆ 25 kg, m3 ˆ 20 kg, m5 ˆ 2:5 kg, mB ˆ 80 kg and mC ˆ 5 kg. The parameter d of the smoothing function is taken as 0.01. A commercial product of servovalve is used in this paper and the parameters of the electrohydraulic system are given in Table 1. The tracking problems via the IDC and VSC methods are compared in Figs. 2(a)± (f) without the external disturbance. The parameters are Kp ˆ 400; Kv ˆ 40 for the IDC, and Q ˆ ‰10Š; K ˆ ‰40Š and c1 ˆ ‰30Š for the VSC. In these ®gures, the initial position error is 0.02 m and the initial velocity and acceleration errors are zero. Both the IDC and VSC, shown in Fig. 2(a), can track the desired trajectory well. The comparison of the position±velocity via the IDC and VSC methods is shown in Fig. 2(b). In Figs. 2(c)±(e), it is found that the errors of position, velocity and acceleration via the VSC are larger than those via the IDC at the beginning. Fig. 2(f) shows the comparison of the control voltages. To avoid the input voltage with sharp peak and oscillatory valley due to changing the states of the desired trajectory, a modi®ed trajectory shown in Fig. 3(b) is designed by smoothing the original P±V diagram shown in Fig. 3(a). The constant disturbance is 1 N and initial position error is 0.02 m. The parameters are Kp ˆ 100 and Kv ˆ 20 for the IDC, and Q ˆ ‰10Š; K ˆ ‰40Š and c1 ˆ ‰30Š for the VSC. The control voltages by both the IDC and VSC methods are compared in Fig. 3(c) for tracking the original trajectory, and in Fig. 3(d) for tracking the modi®ed trajectory. It can be seen that the voltages of tracking the modi®ed trajectory are smoother.

Fig. 3. Comparing the input voltages between tracking the original and modi®ed trajectories with constant disturbance ((. . .) IDC, …І VSC). (3a) Original position±velocity diagram, (3b) modi®ed position± velocity diagram, (3c) control input voltages by the IDC and VSC with tracking original trajectory, (3d) control input voltages by the IDC and VSC with tracking modi®ed trajectory.

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6. Conclusion The purpose of this paper is to apply the IDC and VSC methods to an electrohydraulic-mechanism coupled system. By using the IDC, the desired output response can be speci®ed by the outer loop control law directly from the behavior of a standard second-order system with the desired damping ratio and natural frequency. The IDC method converts a complex nonlinear control problem into a simple one. The VSC system could be adjusted to have fast response without overshooting during the control process. While the state is in the sliding mode, the system responses are insensitive to the parameter variations and external disturbances. In the tracking performance, the VSC is better than the IDC. Especially, when the initial error is not zero and disturbances exist during the control procedure.

Acknowledgements Support of this work by the National Science Council of the Republic of China under Contract NSC-88-2212-E033-002 is gratefully acknowledged.

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