A Four-Quadrant Thrust Estimation Scheme Based on Chebyshev Fit and Experiment of Ship Model

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1 Send Orders of Rerints at 148 The Oen Mechanical Engineering Journal, 2012, 6, (Sul 2: M5) Oen Access A Four-uadrant Thrust Estimation Scheme Based on Chebyshe Fit and Exeriment of Shi Model Baoyu Ye 1, inruo Wang 2, Jiafu Wan *,3, Yi Peng 2 and Jianbin Xiong 2 1 Aircraft Maintenance Engineering College, Guangzhou Ciil Aiation College, China 2 Faculty of Automation, Guangdong Uniersity of Technology, Guangzhou, China 3 School of Comuter Science and Engineering, South China Uniersity of Technology, Guangzhou, China Abstract: This aer rooses a thrust estimation scheme for marine roellers in four-quadrant oerations. To calculate the thrust and torque coefficients of screw roeller in four-quadrant, a Chebyshe fit exression of the roeller roerties in four-quadrant for surface essel is gien, and then it is changed into an ordinary olynomial exression. These exressions are suitable for calculating the alue of the roeller thrust and conenient for studying the shi s maneuerability. On the basis of shi-roeller moement characteristics, the dynamical models of roeller in fourquadrant oerations are gien. The effectieness of the roosed thrust estimation scheme is alidated by exerimental results deried from an electrically drien fixed itch roeller, which roides a good reference for the essel oerations. Keywords: Proeller, Chebyshe olynomial, four-quadrant roerty, thrust estimation. INTRODUCTION In the design of essel control system, such as Dynamic Positioning (DP), thruster assisted osition mooring (PM), and autoilot, the high-leel control aroaches hae receied more attention comared to the roeller dynamics [1]. Howeer, the issue of local thruster dynamics and control has been fully studied in the last ten years (see [2-9] and the references therein). There are two difficulties in the design of effectie roeller controllers: 1) the roeller s dynamics modeling, and 2) measuring the enironmental state. In extreme conditions, thrusters on shis are frequently subject to thrust losses that are caused by entilation and in-and-out-of water effects. Moreoer, due to the interaction between essel hull and roeller, there are also losses of thrust. Industrial exerience showed that the roulsion unit would be seere mechanical wear and tear due to the high thrust losses [10]. Therefore, the knowledge of the roeller thrust, either measured or estimated, could be used to imroe the total erformance of the essel and reduce ower fluctuations as well as wear and tear of the roulsion in extreme conditions [1]. All these considerations motiate the deeloment of schemes to estimate the roeller thrust since in general it can t be measured directly. The incororation of the estimated thrust in a controller is essential for the total erformance of the essel. In addition, the erformance monitoring could imroe fault detection and thrust allocation from calm to extreme sea conditions [11-15]. Zhinkin [16] roosed a thrust estimation scheme for the roeller, where the thrust alues were comuted on the condition of ositie shaft seed and essel seed in calm sea. *Address corresondence to this author at School of Comuter Science and Engineering, South China Uniersity of Technology, Guangzhou, China; Tel: ; jiafu_wan@ieee.org Its estimation was based on the measurement of the roeller torque and a linear relation between thrust and torque. The toic of thrust estimation has also been inestigated in [17], the roeller thrust alues were roided by the roeller torque obtained with a Kalman filter where a linear shaft friction torque was considered. The relation between thrust and torque inoled an axial flow elocity model and a requirement of the knowledge of the adance seed, both of which are ery difficult to measure. Furthermore, simulation results showed that the scheme was highly sensitie to hydrodynamic and mechanical modeling errors. Another examle of thrust estimation scheme can be found in [1, 5, 18]. This scheme inoled a nonlinear obserer to estimate the roeller torque as well as shaft seed and comuted the thrust from the obserer estimates by a maing. In this aer, a thrust estimation scheme that works in fourquadrant lane is gien. Unlike the method in [1, 5, 18], our new scheme just need to know the roeller seed so as to comute the thrust by the relation between thrust and roeller seed, which makes it ery conenient for studying the shi s maneuerability. In order to simulate the dynamic roerties of the roeller load, this aer selects adance ratio of the bounded mode along with the test atlas of Nordstrom series and adots the fitting method of Chebyshe olynomial. The thrust and torque coefficients can also be adjusted according to different numbers of roeller blades and area ratio. The thrust estimation scheme resented in this aer is based on the results achieed in these work. This aer is organized as follows. Section 2 introduces the roeller model and the bounded form of adance ratio J. Section 3 resents the Chebyshe olynomial exression of roeller roerties and its ordinary olynomial exression. Section 4 is deoted to erifying the effectieness of our new thrust estimation scheme by exeriment results. Finally, the conclusions were drawn in Section X / Bentham Oen

2 A Four-uadrant Thrust Estimation Scheme The Oen Mechanical Engineering Journal, 2012, Volume PROPELLER MODEL Proeller Mathematical Model As shown in Fig. (1), the roeller system comosed of an electric motor attached to the roeller through a shaft and a gear-box. = M /(n 2 D 5 ) (4) J = (5) n D where J is the adance ratio, is adance seed of the roeller relatie to water. Note that although in general is not aailable, it can be estimated by the following formula: = s (1 w) (6) Fig. (1). Proeller system. The function of motor is to roide ower, then with the form of torque outut, which is conerted to thrust through the roellers rotating to ush the shi forward. The roeller thrust and torque mathematical model can be written as [19]: F T = n 2 D 4 (1) M = n 2 D 5 (2) where F T is the roeller thrust (N), M is the roeller torque (Nm), n is the roeller seed(r/s), is water density (kg/m 3 ), D is roeller diameter(m), is the thrust coefficient, and is the torque coefficient. where s stands for essel seed (m/s); w stands for the wake fraction number. For a gien screw itch ratio H/D, are lotted as functions of J: = (J) (7) = (J) (8) Generally seaking, the and cures may stride across four quadrants. That is, when n 0, J,,. This exression is usually called the non-bounded form. Theoretically, the cures can be established een though J>1, but it s troublesome for both analog simulation and digital simulation when the ariable range is ery large. It will be more conenient to use the following form to analysis the roeller s dynamic roerty in four-quadrant. When n and are not equal to zero simultaneously, they are defined as = F T /[D 2 ( 2 + D 2 n 2 )] (9) = M /[D 3 ( 2 + D 2 n 2 )] (10) J = /[(Dn) 1+ ( / Dn) 2 ] (11) Fig. (2). Measurement system. In this aer, a suly shi model with two tunnel thrusters and two azimuth thrusters is resented. PLC analog outut module outut oltage (0~6) corresonding to the roeller seed n (0~3000rm) by controlling the motor sero drie. The azimuth thrusters orientation is controlled by the synchronous measurement system, which is comosed of the rotating transformer and the decoder board. A icture of the measurement system is resented in Fig. (2). This guarantee high reliability when it works in extreme sea states. Adance Ratio J with Bounded Form If the roeller seed n 0, the thrust coefficient, torque coefficient and adance ratio J can be written as: = F T /(n 2 D 4 ) (3) The following transformation relation holds: 1 n = 0, > 0 J / 1+ J 2 n > 0 J = J / 1+ J 2 n < 0 1 n = 0, < 0 (12) It can be seen that by (12), J (,+) is maed into J [1,1]. That is, the roeller s all-round dynamic is exressed by the bounded mode. The relationshis between and J, and J become: = ( n 2 D 2 )/( 2 + n 2 D 2 ) = (1 J 2 ) (13) = ( n 2 D 2 )/( 2 + n 2 D 2 ) = (1 J 2 ) (14) CHEBYSHEV POLYNOMIAL EXPRESSION OF PROPELLER PROPERTIES A continuous function oer the interal of [-1 1] can be exressed aroximately as an μ-th order Chebyshe olynomial:

3 150 The Oen Mechanical Engineering Journal, 2012, Volume 6 Ye et al. f (x) = 1 2 a g (x) + a g (x) ++ a g μt μ (x) (15) where g 0 (x) = 1, g 1 (x) = x, g 2 (x) = 2x 2 1,. The general recursie formula for g k (x) is: g k (x) 2xg k1 (x) + g k2 (x) = 0 ( k 2 ) (16) Note that g j and g k are orthogonal to each other, and the olynomial coefficients are indeendent of the order μ. Besides, the fitting error is small and the fitting result with finite order μ is the best aroximation in the sense of minimal square error. It s also conenient to change a gien Chebyshe olynomial exression into an ordinary olynomial. The Chebyshe olynomial coefficients a 0 ~ a μ are substituted into (15) and g 0 ~ g μ are extended by definition, and then the ordinary olynomial and its coefficients b 0 ~ b μ can be obtained. and show as functions of the bounded adance ratio J, which can be written as (J ) = 1 2 a g (J ) + a g (J ) ++ a g 0T 0 1T 1 μt μ (J ) (17) (J ) = 1 2 a g (J ) + a g (J ) ++ a g μ μ (J ) (18) where coefficients a 0 ~ a μ can be obtained from Tables 1 and 2 in [20]. Extending g 0 ~ g μ in (17) and (18) by definition, the ordinary olynomials corresonding to (17) and (18) can be written as: (J ) = b 0T + b 1T J ++ b μt (J ) μ (19) (J ) = b 0 + b 1 J ++ b μ (J ) μ (20) Trading off the accuracy and efficiency in terms of actual simulation of the shi, the 8-th order Chebyshe olynomial is adoted in this study. Therefore, a 0 ~ a 8 can be obtained from Tables 1 and 2 in the aendix of [20]. With the itch ratio H/D=1.0, the coefficient are listed in Table 1 for ease of reference. The test atlas of Nordstorm series in [21] can estimate the roeller characteristics cure in four-quadrant. The rincial arameters of this series of roellers are: 1) disc square ratio A / A d = 0.45, 2) number of blades Z=4, and 3) screw itch ratio H/D=0~1.6. In case the actual arameters are different from the test atlas of Nordstorm series, that is, A / A d 0.45, and/or Z 4, the following formulas can be used [21]: " = (21) " = (22) where, are the modified coefficients, The conersion factor = 3 Z 0 (A / A d ) 0 /[Z(A / A d )], where Z 0 = 4, (A / A d ) 0 = 0.45, Z is the actual number of blades, A / A d is the actual disc square ratio. The characteristics of = (J ) and = (J ) are shown in Fig. (3a, b). For the conenience of simulation, the Chebyshe olynomial is often changed into an ordinary olynomial, namely, (17) and (18) are changed into (19) and (20). Resectiely, the results are shown in Table 2 with resect to the method mentioned aboe EXPERIMENTAL RESULTS A. Shi-Proeller model According to theorem of momentum and Newtons second law, the motion equation of the shi-roeller system is established as follows 2 In = R ab m f (23) (m + m) d s = F TE R (24) dt where I is the total moment of inertia including additional moment of inertia, m is the motor torque, P is the roeller torque, R ab is the gear ratio, f is the static friction torque, m is the quality of the shi (kg), m is additional quality of the shi (kg), and R is total resisting force, which can be written as [22]: R = r s 2 (25) with r being the total resistance coefficient of the shi and constant at gien condition. Here, the gear ratio and the static friction torque are chosen as R ab = 3, f = (0.02 ~ 0.04) m (26) According to the analyses in reious sections, a combination of (1) (2) (9) (10) (21) and (22) yields: F T = D 2 ( 2 + n 2 D 2 ) = D 2 2 / J 2 (27) Table 1. Chebyshe Polynomial Coefficients of Thrust and Torque Proerties a a 0 a 2 a 2 a 3 a 4 a 5 a 6 a 7 a 8 n > n < n > n <

4 A Four-uadrant Thrust Estimation Scheme The Oen Mechanical Engineering Journal, 2012, Volume a) Thrust characteristics b)torque characteristics Fig. (3). Four-quadrant roeller characteristics. Table 2. Ordinary Polynomial Coefficients of Thrust and Torque Proerties b b 0 b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 n> n< n> n< M = D 3 ( 2 + n 2 D 2 ) = D 3 2 / J 2 (28) For thrust deduction coefficient t: t = (F T F TE )/F T, the roeller effectie thrust F TE (N) is gien by F TE = (1 t)f T = (1 t) D 2 2 / J 2 (29) In the shi simulation, due to lack of the measured data, esecially the data of backward naigation, t and w are determined as [22]: t = 0.33 n < n e t = 0.33n / n e n e n < 0 (30) t = 0.13n / n e 0 n < n e t = 0.13 n n e w = 0 s 0 w = 0.22 s / se 0 s < se w = 0.22 s se (31) where se stands for shi rated seed. Fig. (4) summarizes the motion equation of the shi-roeller system. B. Result Analysis The main urose of this aer is to estimate the thrust from the relation between the thrust and the roeller seed. A shi-roeller model is gien for the test of electric roulsion system with roellers. The test were erformed at the ocean engineering Dynamic Positioning System lab in Guangdong uniersity, where has a ool with length L=11m, wide H=6m. A 2.8m shi model of an offshore suly essel with a scale of 1:26 has been conducted (See Fig. 5). Fig. (6) shows design of the thrust estimate exeriment scheme. The main arameters are shown in Table 3. At the condition of the gien initial shi seed s, J is comuted from (6) and (11),, can be obtained by (17) and (18). Then the roeller torque and effectie thrust can be calculated by (27) and (28). Shi seed can be obtained by (24) (27) and (29). The rated essel seed of 2.8m shi model is se =1m/s by test. This measuring exeriment record a roeller thrust of the corresonding roeller seed eery 20 RPM from-1000 ~ 1000 RPM. Fig. (7) gies the exeriment and theoretical alues of roeller thrust. Using the model gien in (27) and (28), combined with the, alues obtained by (17) and (18), the roeller thrust can be estimated under the gien roeller seed. Fig. (7) shows the measured and estimated thrust from the static water. It can be seen from Fig. (7) that the roeller thrust increases with the increase of the roeller seed n; and for the same roeller seed n, the thrust in the case of the roeller reerse is less than that when the roeller is in ositie rotation. That is, for ositie alues of n, the efficiency is higher because the roeller was designed to work mainly at forward essel seed.

5 152 The Oen Mechanical Engineering Journal, 2012, Volume 6 Ye et al. n J n D K K ( J) T T K K ( J) F TE M ( 1 t) KT D / J K D 3 2 / J 2 T P 1 m m dt s 2 R r s ( 1 w) s Fig. (4). Shi-roeller system. Fig. (5). Shi model: scale 1:26 of a suly essel. a) ool b) Shi model a) Exerimental schemes b) Exerimental hysical ma Fig. (6). Thrust estimate exeriment scheme design.

6 A Four-uadrant Thrust Estimation Scheme The Oen Mechanical Engineering Journal, 2012, Volume Table 3. Parameters of Vessel Length Oerall (m) Oerall Width (m) Mean Draught (m) H/D (Pitch Ratio) M (uality) (kg) D (Proeller Diameter) (m) A/A d (Disc Square Ratio) (Water Density) (kg/m 3 ) F(N) theoretical alue exerimental alue1 exerimental alue n(r/s) Fig. (7). Thrust exeriment and theoretical alues. CONCLUSIONS In this aer, a thrust estimation scheme for marine roellers across the full four-quadrant range is resented. First, the roeller roerties across four quadrants described with Chebyshe olynomial and its ordinary olynomial are gien. Based on such olynomial, the simulation model of roeller in four-quadrant oerations is established. The exeriment results show that the erformance of the thrust estimation scheme using the algorithm mentioned aboe is satisfactory. The roosed thrust estimation scheme will be used in high erformance roeller controllers and thrust allocation for shi control. ABOUT THE AUTHORS Baoyu Ye is now an instructor at Guangzhou Ciil Aiation College, Guangzhou, China. Her current research interests include shi motion control, automation equiment and control technology. inruo Wang is now with School of Automation Science and Engineering, Guangdong Uniersity of Technology, China as Prof. and instructor of Ph.D. His current research interests include integratie technology of machine and electricity and automatic network control. Jiafu Wan is currently an associate research fellow in School of Comuter Science and Engineering of South China Uniersity of Technology. His current research interests are cyber-hysical systems, embedded systems, internet of things, and machine- to-machine communications. Yi Peng is now a teaching assistant at Guangdong Uniersity of Technology, Guangzhou, China, His current research is automation equiment and control technology. Jianbing Xiong is now a Ph.D. candidate at Guangdong Uniersity of Technology, Guangzhou, China. His current research interests include data mining, information security, and comuter alication technology. CONFLICT OF INTEREST The authors confirm that this article content has no conflict of interest. ACKNOWLEDGEMENTS This work is suorted by the National Natural Science Foundation of China No ; the Natural Science Foundation of Guangdong Proincial of China under Grant No ; the Key Item in the Guangdong Proince Science & Technology Secial Program under Grant No. 2009A ; the Ministry of Education of Guangdong Proince Secial Fund funded rojects through Cooeratie under Grant No. 2009B REFERENCES [1] L. Piano, T.A. Johansen, and Ø.N. Smogeli, "A Four-quadrant thrust estimation scheme for marine roellers: theory and exeriments", IEEE Transactions on Control Systems Technology, ol. 17, no. 1, , [2] L.L. Whitcomb, and D. Yoerger, "Deeloment comarison and reliminary exerimental alidation of nonlinear dynamic thruster models", IEEE Journal of Oceanic Engineering, ol. 24, no. 4, , [3] R. Bachmayer, L.L.Whitcomb, and M.A. Grosenbaugh, "An accurate four-quadrant nonlinear dynamical model for marine thrusters: Theory and exerimental alidation", IEEE Journal of Oceanic Engineering., ol. 25, no. 1, , [4] M. Blanke, K. Lindegaard, and T.I. Fossen, "Dynamic model for thrust generation of marine roellers", In: Proc. 5th IFAC Conf. Manoeuring Control Marine Craft (MCMC), Aalborg, Denmark, , [5] L. Piano, T.I. Fossen, and T.A. Johansen, "Exerimental alidation of a marine roeller thrust estimation scheme", Is resented at the 7th IFAC conf. Manoeuring control marine craft (MCMC), Lisbon, Portugal, [6] L. Piano, T.I. Fossen, and T.A. Johansen, "Nonlinear model identification of a marine roeller oer four-quadrant oerations", In the 14th IFAC Sym. Syst. Identification (SYSID), Newcastle, Australia, [7] L. Piano, N. Smogeli, T.A. Johansen, T.I. Fossen, "Marine roeller thrust estimation in four-quadrant oerations", In the 45th IEEE Conf. Dec. Control, San Diego, CA, [8] R. Bachmayer, L.L. Whitcomb, and A. Grosenbaugh, "An accurate four-quadrant nonlinear dynamical model for marine thrusters: theory and exerimental alidation", IEEE Journal of Oceanic Engineering, ol. 25, no. 1, , [9] K. Koushan, "Enironmental and interaction effects on roulsion systems used in dynamic ositioning, an oeriew", In roceedings of the ninth international symosium on ractical design of shis and other floating structures (PRADS 04), Lubeck-Tracemunde, Germany, [10] Ø.N. Smogeli, A.J. Sørensen, and K.J. Minsaas, "The concet of anti-sin thruster control", Control Engineering Practice, ol. 16, no. 2008, , [11] M. Blanke, R. Izadi-Zamanabadi, and T.F. Lootsma, "Fault monitoring and reconfigurable control for shis roulsion lant", International Journal of Adatatie Control and Signal Processing, , December [12] Ø.N. Smogeli, "Control of marine roellers: from normal to extreme conditions, in Marine Technol", Norwegian Uni. Sci.

7 154 The Oen Mechanical Engineering Journal, 2012, Volume 6 Ye et al. Technol.(NTNU): Trondheim, Norway, [13] Ø.N. Smogeli, A.J. Sørensen, and T.I. Fossen, "Design of a hybrid ower/torque thruster controller with loss estimation", Is Presented at the IFAC Conf. Control Al.Marine Syst. (CAMS), Ancona, Italy, [14] L. Piano, T. A. Johansen, Ø. N. Smogeli, and T. I. Fossen, "Nonlinear thrust controller for marine roellers in four-quadrant oerations", In the 26th Amer. Conf.(ACC), New York, July [15] T.F. Lootsma, "Obserer-based fault detection and isolation for non-linear systems", In Control Eng., Aalborg Uni.: Aalborg, Eenmark, [16] V.B. Zhinkin, "Determination of the screw roeller thrust when the torque or shaft ower is known", In the 4th International Symosium on Practical Design of Shis and Mobile Units, Bulgaria, , [17] C. Guibert, E. Foulon, N. Ait-Ahmed, L. Loron, "Thrust control of electric marine thrusters", In the 31st Annu. Conf. IEEE Ind. Electron. Soc.(IECON), Raleigh, NC, [18] L. Piano, T.A. Johansen, Ø.N. Smogeli, and T.I. Fossen, "Nonlinear thrust controller for marine roellers in four-quadrant oerations", In American Control Conference, New York, , [19] J.S. Carlton, "Marine roellers and roulsion", Butterworth- Heinemann Ltd: Oxford, UK, [20] L. Dian-u, W. Zone-yi, and C. Hai-hong, "Chebyshe fitting way and error analysis for roeller atlas four quadrants", Journal of Marine Science and Alication, ol. 1, no. 1, , [21] J. Yuh, "Underwater robotic ehicles: design and control", Albuquerque TSI Press: USA, [22] F. Shangyong, "Vehicle ower setting and simulation technique", Shanghai: Shanghai Jiao Tong Uniersity, Receied: June 11, 2012 Reised: July 18, 2012 Acceted: July 19, 2012 Ye et al.; Licensee Bentham Oen. This is an oen access article licensed under the terms of the Creatie Commons Attribution Non-Commercial License (htt://creatiecommons.org/licenses/by-nc/3.0/) which ermits unrestricted, non-commercial use, distribution and reroduction in any medium, roided the work is roerly cited.