The Nonlinear Accuracy Model of Electro-Hydrostatic Actuator

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1 The Nonlinear Auray Model of Eletro-Hydrostati Atuator Rongjie Kang, Zongxia Jiao, Shuai Wu Yaoxing Shang Shool of Automation Siene and Eletrial Engineering Beihang University Beijing, China Jean-Charles Mare Department of Mehanial Engineering Institut National des Sienes Appliquees Toulouse, Frane Abstrat The Eletro-Hydrostati Atuator (EHA) is a ind of Power-By-Wire (PBW) atuator. In this paper, a typial arhiteture of EHA is desribed and the blo diagram model of EHA is established diretly from the mathemati equations without transfer funtions, whih is a nonlinear auray model. The influene of refeeding iruit on the EHA system is disussed based on the omparison with onventional linear model. A gainvariable PID ontroller is introdued to this model to ompensate the frition. motor speed (VPVM). Nowadays, the FPVM-EHA (Fig. 1) is more popular for its simple struture and effiieny. In this system, a bi-diretional pump rotates in variable speed and diretions given by eletri motor. As a result, the oil flow and supply pressure is variable to drive the symmetrial atuator [3]. Keywords EHA, blo diagram, modeling, ontrol I. INTRODUCTION The demand for onventional hydrauli atuation is gradually dereasing due to its limitations suh as: low energy effiieny, leaage, noise, low maintainability. The Power-By- Wire (PBW) tehnology is beoming an attrative diretion of future airborne atuation system. A PBW flight ontrol system would simplify the seondary power generation, eliminate the need for a entral hydrauli power supply, and replae the hydrauli pipes by eletri power ables. As a result, the reliability, survivability, effiieny and maintainability of the airraft would be greatly improved. The Eletro-Hydrostati Atuator (EHA) is one ind of PBW atuator whih uses a hydrauli pump to transfer the rotational motion of eletri motor to the atuator output. EHA is based on the priniple of losed-iruit hydrostati transmission, so that there are no requirements for oil reservoir and eletro-hydrauli servo-valves. A lot of researh papers have modeled the EHA system by transfer funtion in the past, however, this linear modeling method has some disadvantages: negleting the refeeding iruit whih ontains some nonlinearity; simplifying the frition, espeially the stati frition; supposing that all the initial onditions are zero [1,2]. To solve these problems, the blo diagram modeling method is presented in this paper. II. ARCHITECTURE OF FPVM-EHA There are several arhitetures of EHA: EHA with fixed pump displaement and variable motor speed (FPVM), EHA with variable pump displaement and fixed motor speed (VPFM), EHA with variable pump displaement and variable Fig.1. Arhiteture of FPVM-EHA III. EHA MODELING A. Modeling of DC Motor A Brushless DC Motor (BLDCM) is used as the driving motor. The maximum speed is 12000r/min, nominal voltage is 270V, and nominal power is 10W. The mathemati equations of BLDCM are: di U = E+ L + Ri dt E = Kω (1) Te = Kti Te = J ω+ friω+ Tl Where E is the ba eletromotive fore, L is the motor winding indutane, R is the motor winding resistane, T e is the eletromagnetism torque, J is the sum of inertia of motor and pump, K is the sum of visous oeffiient of motor and fri pump, T is the load torque ating on the motor shaft, and ω is l the rotational speed of motor /08 /$ IEEE RAM 2008

2 Aording to (1), a blo diagram model of motor is gotten in Fig. 2 by Simulin. The part in dashed is urrent protetion that ould be realized by software. C. Modeling of Refeeding Ciruit In order to eep the losed-iruit of EHA, a refeeding iruit omposed of aumulator and he valves is neessary. The shemati diagram is shown in Fig. 5. Fig.2. Blo diagram model of motor B. Modeling of Pump A shemati diagram of bi-diretional pump onsidering the internal and external leaage is shown in Fig. 3. Fig.3. Shemati diagram of pump The flow equation of node A is obtained as: Q = D ω Q Q = D ω K ( P P) K ( P P ) 1 il el1 ilp 1 2 elp 1 a (2) The flow equation of node B is obtained as: Q = D ω Q + Q = D ω K ( P P) + K ( P P ) 2 il el 2 ilp 1 2 elp 2 a (3) Where D is the displaement of pump, K is the internal ilp leaage oeffiient of pump, K elp is the external leaage oeffiient of pump, P a is the pressure of aumulator. The diagram model of pump is shown in Fig. 4. Fig.5. Shemati diagram of refeeding iruit The flow equations of refeeding iruit are: Q = Q Q Q a el 1 2 Q = Q + Q 1f 1 1 (4) Q = Q Q 2 f 2 2 Where Q is the external leaage of pump, Q el 1, Q are 2 the flow of he valves depending on ( P1 P a ) and ( ). P2 P a The relationship between Q a and P ould be desribed a as follows: Pa = PaiV /( V Qadt) (5) Where P is the initial pressure of aumulator, V is the ai initial volume of gas, is the polytropi exponent of gas within the range from 1.0 to 1.4. The blo diagram model of refeeding iruit and aumulator are shown in Fig. 6 and Fig. 7 respetively. The loo-up tables are used to desribe the flow harateristi of he valves. P V is defined as GasCons in Fig. 7. ai Fig.6. Blo diagram of refeeding iruit Fig.4. Blo diagram of pump

3 Fig.7. Blo diagram of aumulator D. Modeling of Hydrauli Atuator EHA requires a symmetrial atuator in order to ensure flow balane between the atuator and the pump. The hydrauli ja shown in Fig. 8 is divided into two woring hambers by the piston [5]. Fig.8. Shemati diagram of hydrauli ja The flow in Chamber 1 ould be desribed by the following equation: V10 + Axt Q1f = Ax t + P 1+ Kilj ( P1 P2) (6) B The flow in Chamber 2 ould be desribed by the following equation: V20 Axt Q = Ax P + K ( P P) (7) 2f t 2 ilj 1 2 B Where A is the ative area of piston, B is the bul modulus of oil, K ilj is the internal leaage oeffiient of hydrauli ja, V 10 and V20 are the initial volume of hamber 1 and hamber 2 respetively, whih are the same in this symmetrial atuator. The load fore balane equation of the piston is: AP ( 1 P2) = Mx t + Fex + Ffri (8) Where M is the total mass of piston and load, F ex is the external load fore, and F fri is the frition whih will be desribed in Setion E. The diagram model of hydrauli ja and piston are respetively shown in Fig. 9 and Fig. 10: Fig.10. Blo diagram of piston E. Modeling of Frition There are several solutions to desribe the frition as a funtion of veloity. The simplest one is: Ffri = Kvis x t, where K vis is the visous oeffiient of piston. For getting high auray, the frition model is given in the form: F ( x ) = [ F + ( F F) exp( x / α) + K x ] sign( x ) fri t s t vis t t (9) Where F is the Coulomb frition, ( Fs F) exp( x t / α) is the Stribe frition, and K x vis t is the visous frition [6]. For improvement, sign( x t ) ould be replaed by tanh( x t / β ) to mae the model ontinuous: F ( x ) = [ F + ( F F) exp( x / α) + K x ] tanh( x / β) (10) fri t s t vis t t Where α, β are the referene speeds approximately within the range from 0.001m/s to 0.01m/s.The diagram model of the frition is shown in Fig. 11. The relation between x t and frition is shown in Fig.12. Fig.11. Blo diagram of frition Fig.9. Blo diagram of hydrauli ja Fig.12. Frition model F. Blo Diagram of EHA Assembly Based on the above sub-models, a FPVM-EHA without ontroller ould be gotten. All the sub-models are enapsulated and onneted with eah other by the defined input and output ports. The blo diagram of the whole system is shown in Fig. 13.

4 Fig.13. EHA assembly IV. SIMULATION AND ANALYSIS A. Close-loop Simulation A PID ontroller is introdued to the EHA model for loseloop simulation. The system parameters are given in Table 1. The step input is 10mm, and the external fore of 10000N is loaded at 2.5s. The response shown in Fig. 14 proves the orretness of this model, indiates that the system has good harateristi in rapidity and loading. B. Effets of Refeeding Ciruit General speaing, the roles of EHA refeeding iruit are as follows: To mae up the losed-iruit To prevent avitation To prevent the exessive pressure build up in pump To ompensate the external leaage In the onventional linear model, the refeeding iruit is always negleted due to some nonlinearity suh as: the flow harateristi of he valves and the gas in aumulator. For improvement, the refeeding iruit is onsidered in this nonlinear auray model. Fig. 15 and Fig. 16 show the system pressure derived from these two models respetively. An external fore of 10000N is loaded at 2.5s. The simulation result indiates that the refeeding iruit has another effet on reduing the pressure ripple in EHA system. Fig.15. Pressure response without refeeding iruit Fig.14. Position response Table 1 EHA Parameters Symbol Value Symbol Value L [H] 2.5e-3 F [N] s 50 R [Ohm] 1.5 F [N] 15 K [N*m/A] 0.2 V, V t [ml] 152 K [V/(rad/s)] 0.2 B [N/m^2] 6.5e8 J [Kg*m^2] 1.2e-3 A [m^2] 19 K [N*m/(rad/s)] M [Kg] 2000 fri K [(m^3/s)/pa] 1e-13 V [ml] ilp 150 K [(m^3/s)/pa] 1e-13 P [MPa] elp ai 2.5 D [ml/r] 1.2 V [ml] oili 150 K [(m^3/s)/pa] ilj 1e K [N/(m/s)] 150 vis Fig.16. Pressure response with refeeding iruit C. Frition Problem For the EHA, frition is an unavoidable fore that exits in motor, pump and piston, whih always lead to traing errors and reeping in low speed. Fig. 17 shows the obvious osillations in EHA veloity response of very low value (0.075mm/s). Beause of the negative slope due to Stribe frition (Fig.12), the damping will derease while the veloity inreases, whih leads to the

5 instability of this system. Moreover, there is frition-indued sti when the veloity passes through zero. A lot of nonlinear and adaptive ontrol strategies have been developed in the past for frition ompensation [7]. Fig. 18 desribes a gain-variable PID ontroller. The gain K v is hanged aording to the veloity of piston: Txt Kv = K1 e (11) Where K1 is the variable part of K, T is the time onstant v to adjust the rate of deay. Fig. 19 shows the traing performane of system without ompensation, for omparison, Fig. 20 shows the improved traing performane with this gain-variable PID ontroller. Fig.20. Position traing and the error (with gain-variable ontroller) Fig.17. Frition-indued osillations in veloity Fig.18. Gain-variable PID ontroller V. CONCLUSION The nonlinear auray model of FPVM-EHA is established by blo diagrams in this paper. It ontains more information than onventional linear model. The omparison analysis indiates the effet of EHA refeeding iruit on reduing pressure ripple. A gain-variable PID ontroller is introdued to this model and effiiently ompensates the frition. All the simulation results prove the orretness and reliability of this model whih is helpful for the further study on EHA. REFERENCES [1] Saeid Habibi and Andrew Goldenberg. Design of a new high performane eletrohydrauli atuator, Proeeding of the 1999 IEEE/ASME International Conferene on Advaned Intelligent Mehatronis Proeedings, Septemper,1999, Atlanta,USA, pp [2] Ravi N. Banavar and Vineet Aggarwal, A loop transfer reovery approah to the ontrol of an eletro-hydraili atuator, Control Engineering Pratie, Volume 6, Issue 7, July 1998, pp [3] Gao Bo, Fu Yongling, Pei Zhongai and Ma jiming. Researh on dualvariable integrated eletro-hydrostati atuator, Chinese Journal of Aeronautis, 2006, Vol 19(1), pp [4] Stephen C J, Flight test experiene with an eletromehanial atuator on the F-18 systems researh airraft, NASA,1998. [5] Wang Z L, Modern Hydrauli Control. Beijing: China Mahine Press, (in Chinese) [6] Evangelos G. Papadopoulos and Georgios C. Chasparis. Analysis and model-based ontrol of servomehanisms with frition, Proeedings of the IEEE/RSJ International Conferene on Intelligent Robots and Systems, Otober, 2002, Lausnne,Switzerland, pp [7] S. S. Ge, T. H. Lee and S. X. Ren, Adaptive frition ompensation of servo mehanisms, Proeedings of the IEEE International Conferene on Control Appliations, August 1999, pp Fig.19. Position traing and the error (without ompensation)