Send Orders for Reprnts to reprnts@benthamscence.ae he Open Automaton and Control Systems Journal, 205, 7, 23-236 23 Open Access Research on the Dynamc Vbraton Control of Underwater Robot L-ya Wang and Yang Zhao 2,* School of Mathematcs and Informaton Scence, Langfang eachers Unversty, Langfang 065000,Chna 2 Department of Electronc and Informaton echnology, Jangmen Polytechnc, Jangmen 529090, Chna Abstract: Wth manknd s ncreasng development and utlzaton of the ocean, underwater robot whch can detect underwater envronment and autonomously accomplsh specfc tasks has drawn great attenton of researchers. Research was conducted on the knematc and dynamcal control of the underwater robot wth sx degrees of freedom. Frstly, consderng the nfluences of gravty, buoyancy force, thrust and hydrodynamc force, the knematc model and dynamcal model of the underwater robot were bult to descrbe the complcated underwater behavors of the robot. Based on ths, the dynamc model was smplfed and model predcton control of the dynamcs was conducted. Comprehensve comparson was also conducted on the results, whch showed that model predcton control can reduce the vbraton response of the system effectvely. Keywords: Dynamcs, knematcs, model predcton control, underwater robot.. INRODUCION Ocean reserves abundant resources and energy sources. How to reasonably use and develop the ocean has become an urgent problem and s drawng wde attenton of the all countres. Underwater robot plays an mportant role n the ocean utlzaton and development process. Wth the acceleraton of ocean development process, researches on the underwater robots whch can explore unknown underwater envronment and perform specfc underwater tasks also draw the wde attenton of research nsttutons. As the carrer for workng n the complcated ocean envronment, autonomy and securty are the mportant features of underwater robots, and ntellgent control technology s the mportant foundaton and technology for ensurng ts autonomy and securty [, 2]. o cater to the demands of ocean development such as marne ol explotaton etc., the development and applcaton work for underwater robot have been actvely conducted n the recent years. Research nsttutons such as Shangha Jaotong Unversty have developed remotely operated vehcle, undersea vehcles and autonomous underwater vehcle etc. W.Naeem et al. [3, 4] conducted on lnear model dentfcaton for the headng moton of Hammer head underwater robot. hey changed the headng of the underwater robot by controllng the devaton of the rudder, and bult ARX model. hey conducted on model predcton control for QnetQ underwater robot wth ARX model as the predcton model. K.Ish et al. [5, 6] bult a dynamc model for the expermental platform of WIN- BUEGER underwater robot by regressng from the output layer to the nput layer of the neural network based on Elman neural network. he dynamc model of the neural network *Address correspondence to ths author at the School of Department of Electronc and Informaton echnology, Jangmen Polytechnc, Jangmen, 529090, Chna; Emal: zhaoyang978023@gmal.com for ths underwater robot s manly used for on-lne adjustng the parameters of the moton controller of the underwater robot. F.J.Song [7] controlled the ptch and degree of freedom drecton of OEX underwater robot, and proved the effectveness of fuzzy sldng mode controller. However, the vbraton of the mechancal system n the moton process reduces the moton precson of the mechansm or makes the mechansm devate from the expected moton. o reduce the vbraton characterstcs of the mechansm, t needs to compensate the vbraton error by a reasonable control system, so as to enable the mechansm to realze the maxmum expected moton track. 2. KINEMAIC AND DYNAMIC EQUAION ANAL- YSIS 2.. Knematc Equaton Analyss Underwater robot s shown n Fg. (). o descrbe the moton of the underwater robot, the knematc model of the underwater robot needs to be bult accordng to ts moton tracks and moton speeds planned for dfferent tasks. In the fxed coordnate system, the poston and posture of the underwater robot can be shown by the coordnates of the underwater robot s gravty n the fxed coordnate system ( xyzand,, ) Euleran angles ( αβγ,, ) between the movng coordnate system and the fxed coordnate system. he nstantaneous lnear velocty and angular velocty of the underwater robot are ( pqr,, ) and ( µν,, ω) respectvely, and components of the forces and moments appled on each axs of the underwater robot are ( X,Y, Z) and (K, M, Z) respectvely. he drecton of the velocty and the force s consstent wth the coordnate axs, and the angular velocty and moment drecton are judged accordng to rght-hand screw rule. he pathway and velocty of the underwater robot planned n the fxed coordnate system needs to be converted 874-4443/5 205 Bentham Open
232 he Open Automaton and Control Systems Journal, 205, Volume 7 Wang and Zhao Fg. (). Underwater robot and the knematc dagram of ts mechansm. to the movng coordnate system. he velocty generated by the force (moment) appled on the underwater robot needs to be converted to the fxed coordnate system. Only by dong so can make the current poston and posture of the robot be calculated. herefore, the vector converson between the two coordnate systems s necessary. For lnear velocty, converson matrx Jv from the movng coordnate system to the fxed coordnate system s as follows: J v γ c sαβγ s c cαγ s cαβγ s c + sαγ s = cθγ s sαβγ s s cαγ c cαβγ s s sαγ c + sβ sα cα For angular velocty, the converson matrx J from the w movng coordnate system to the fxed coordnate system s as follows: J w sαsβ cαsβ = 0 cα sα sα cα 0 he converson from the fxed coordnate system to the movng coordnate system s as follows:! # # "!! 2 Where $! & & = # # % " J v O O J w ' 2 $! & # & # % "!"!" 2 $ & & % η = [, xyz,], η = [ α, β, γ], ν [ p q r], ν [ u v w] = = 2 () (2) (3) 2.2. Dynamc Equaton Analyss Accordng to Newton Euler equatons for rgd body n flud, the dynamc model of the underwater robot wth sx degrees of freedom n the movng coordnate system can be descrbed as follows[8]: M!! + D(!)! + g(") = # (4)!! = J(!)" (5) 3. MODEL SIMPLIFICAION AND CONROL MODEL BUILDING he orgn pont of the coordnate system s consstent wth the center of gravty for the underwater robot, and the buoyancy force equals to the gravty. Only the elements on the dagonal lne between the mass nerta matrx and the dampng matrx of the hydrodynamc force are consdered, the rollng and tumblng motons are passve and controllable,.e., they can be neglected. he ptchng and rollng motons are controlled by the rghtng moment generated by the gravty and the buoyancy force to mnmze the ptchng and rollng angles. he underwater robot s symmetrcal about three tangent planes. he moton of underwater robot n sdeway degree of freedom drecton s often neglected. Corols and centrpetal force can be neglected under low speed. herefore, the dynamc model of the underwater robot can be further smplfed as follows: M!! + D(v)v + g(") = # (6) Smplfcaton of the mass and nerta matrx as follows: the orgn pont of the movng coordnate system s located at the center of gravty for the underwater robot, and the underwater robot s symmetrcal about three tangent planes. Smplfcaton for the vector of gravty and buoyancy force as follows: n the movng coordnate system, the coordnate of the center of gravty for the underwater robot s r [ 0 0 0] G =, and the coordnate of ts buoyancy force s r = 0 0 0.. herefore, the buoyant center s also on B [ ]
Research on the Dynamc Vbraton Control of Underwater Robot he Open Automaton and Control Systems Journal, 205, Volume 7 233 Fg. (2). Dagram of nput shapng. oz coordnate axs. he gravty of the underwater robot s 66N, and the buoyancy force of the water appled on the underwater robot s 82N. herefore, the resultant force of the gravty and the buoyancy force appled on the underwater robot wll only affect ts moton n heavng degree of freedom drecton. Moreover, when the underwater robots s mmersed n the water completely, the buoyancy force of the water appled on the underwater robot s 6N larger than ts gravty, whch can ensure that the underwater robot can float upwards by relyng on ts buoyancy force n case of falure. 4. RESEARCH ON HE MEHODS OF ACIVE CONROL he elastc deformaton of the components wll cause dynamc error for ts output moton; therefore, t cannot accurately realze the expected track. he dynamc error of the underwater robot s manly caused by the vbraton of the components durng the vbraton of the system. Vbraton wll dstort the moton track of the system and devates the nput and output relaton. How to effectvely control the vbraton of the mechansm to ncrease the moton precson of the system and reduce the ndustral nose caused by vbraton s of great practcal sgnfcance [9]. For actve vbraton control, model reducton s conducted by modal truncaton durng controller desgn. Neglectng hgh-frequency dynamc problems may cause falure to observaton and control, and there s external perturbaton and uncertantes to the actual system, so t needs to ncrease the robustness of the control system. 4.. Dynamc Vbraton Control Based on PZ Intellgent Control Materal Vbratons wll occur durng the moton of robots. he vbratons not only cause nose, but also affect the moton precson. So t needs to reduce the vbratons. here are many factors causng system vbraton. herefore, t needs a real-tme vbraton control method to reduce the vbraton. In recent several decades, attenton has been pad to retranng the vbraton by ntellgent structure. An ntellgent structure ncludes four factors: actuator, sensor, control strategy and dynamc control devce. Pezoelectrc materals can be used as the actuator and sensor of an ntellgent structure. PZ ntellgent pezoelectrc materal requres lower drvng voltage and can be appled n larger frequency range. herefore, t s wdely used [0]. PZ sensor and actuator coordnated wth multple control strateges can realze dynamc vbraton control for robots, of whch stran rate feedback control strategy has wder dynamc dampng frequency area, and can realze dynamc restran n larger area. Because of pezoelectrc characterstcs, PZ ntellgent materal can be used as a sensor to measure the vbraton and as an actuator to restran the vbraton. he control methods for restranng vbraton are model predcton control. 4.2. heory and Methods of Input Shapng Input shapng s to convolve the nput sgnals and a seres of pulses, and then nput the convolved results to the actuator, thus reducng the vbraton of the system. he ampltude and work tme of the pulses can be calculated by the resonant frequency and dampng rato of the system. When the last pulse work ends, the vbraton of the system shall be the mnmum. As shown n Fg. (2), t s the executon process of the convoluton between a two-pulse nput shaper and the nput sgnals. 4.3. ZVD Input Shaper When the accurate resonant frequency and dampng rato of the system s obtaned n advance, ZV nput shaper wll be able to reduce the resdual vbraton of the system. However, n case that there are some errors or fluctuaton n buldng model, the resonant frequency and dampng rato of the system, ZV nput shaper wll not be able to effectvely restran the resdual vbraton. So the robustness of the nput shaper must be mproved. In case of dynamc parameter change or error n the system, Zero Vbraton and Dervatve (ZVD) method wll become a method whch can effectvely mprove the robustness of the shaper. he resdual vbraton of the system can be ndcated by V ( ωζ, ). = n ζωtn ζω(t t ) 2 2 = V ( ωζ, ) = e ( [A e cos( ω ζ t )] + n ζω(t t ) 2 2 2 ω ζ [A e sn( t )] ) o mprove the robustness of the shaper, a constrant equaton s added as follows. V ( ωζ, ) = 0 ω (8) (7)
234 he Open Automaton and Control Systems Journal, 205, Volume 7 Wang and Zhao Fg. (3). Comparson of the vbratons wthout nput shapng and wth ZVD nput shapng. hs constrant equaton shows that the dervatve of constrant equaton of resdual vbraton to the frequency s made to be zero. All the constrant equatons of ZVD method are as follows: V ( ωζ, ) = 0, n A =, V ( ωζ, ) = 0 (9) = ω Where V ( ωζ, ) = 0 s to ensure that the vbraton ampltude of the resdual vbraton s zero when the last pulse work ends. ZVD nput shaper not only requres to lmt the vbraton modal of the system, but also requres that the change rate of the system vbraton equals to zero, so make! V (!," ) = 0 means that t requres zero dervatve. 4.4. Smulaton Analyss o enhance the robustness of the nput shaper, the reducton vbraton effect of ZVD nput shapng method can be analyzed by smulaton. he frst order dampng rato and resonant frequency areζ = 0.057 and f = 76.6Hzrespec- tvely. he nput sgnals of the system are unt step sgnals. he parameters of the three-pulse ZVD nput shaper for the robot are as follows: A 0.2967 0.496 0.2072 t = 0 0.0065 0.03 (0) he smulaton analyss s conducted and the smulaton results are shown n Fg. (3), showng the vbraton responses of the connectng equpments for the robot wth or wthout ZVD nput shaper. he vbraton s restraned by 97%, and the delay tme of ts response s 0.0074s. It s dffcult to accurately buld a systematc mathematcal model due to nfluences of the non-lnear factors and the couplng among varous factors, whch brng dffculty n the actve vbraton control of the elastc mechansm. Model predcton control method whch has flexble forms and requres less for the system model can compensate the control errors caused by the tme varaton of the system parameters and non-lnear factors, etc. n a real-tme manner, and can effectvely overcome the nfluences of the uncertantes of the system, wth good control effect and strong robustness [0]. he control structure dagram s shown n Fg. (4). In ths secton, ts space model n dscrete state s bult. he vbraton response of the mechansm s restrcted by means of model predcton control. Its schematc dagram s shown n Fg. (4). he predcton model for predctng the dynamc deformaton response of the mechansm was bult by makng the nfluences of non-lnear factors, couplng factors as well as the hgh order mode of the system n the dynamc model of the complcated mechansm as the perturbaton. he predcton model for the dynamc response of the system was bult by regardng the modal force as uncertan perturbaton and by takng nto account the nfluence of the output nose on the system. he state varables of the system were estmated by Kalman flterng estmator. he controller was obtaned by solvng quadratc programmng optmzaton problem. 5. SIMULAION ANALYSIS ON NUMERICAL CAL- CULAION Wth underwater robot as the research object, ts controller was desgned by applyng H norm and u comprehensve robustness control theory. Accordng to the structure characterstcs of the transmsson moton, the harmonc gear s featured by the flexble jonts; the output end of the harmonc reducer s connected to the pezoelectrc ntellgent flexble beam. And the partal connectng equpments are the appled the ZVD nput shaper to reduce the vbraton response effectvely. hs platform s a rgd-flexble couplng system platform ntegratng sensor, pezoelectrc ntellgent materal sensor and actuator as well as harmonc gear reducer drven by AC servo motor. Comparatve study was conducted on the actve vbraton control methods. he geometrc dmenson of the flexble lnk s as follows: length s 0.65m, wdth s 0.m, thckness s.78mm, structure dampng rato s 0.3%. he mechancal property of the beam s as follows: elastcty modulus E=34.64Gpa, Posson's rato u=0.33, densty ρ =840kg/m 3. he dscretely dstrbuted pezoelectrc transducer (PZ) sensors are pasted at the root
Research on the Dynamc Vbraton Control of Underwater Robot he Open Automaton and Control Systems Journal, 205, Volume 7 235 Fg. (4). Schematc dagram of model predcton control. Fg. (5). Smulaton and experment results of model predcton control. and the mddle of the beam. he geometrc dmenson of the pezoelectrc transducer (PZ) s 50mm!5mm!mm, ts elastcty modulus, Posson's rato, densty and pezoelectrc stran constant are Ep 0 = 6.5 0 pa, 0.35 p 2 3 µ =, ρ = 7650 kg / m and d = 66 0 m/ Vrespectvely. o verfy the effectveness of the controller, numercal smulaton analyss was conducted []. he sold lne n Fg.5 s predcton control, and the dotted lne s the uncontrolled model. Fg. (5) shows that the response of the uncontrolled system s bascally consstent wth that of the predcton control system. It can well reflect the real characterstcs of the uncontrolled model, thus carryng out dynamc characterstc analyss and actve control desgn. herefore, based on the model of predcton control, the control strategy can be desgned accordng to the optmum control theory, so as to restran the elastc vbraton of the flexble plate. p In concluson, wth the underwater robot wth sx degrees of freedom as the research object, theoretcal and expermental study s conducted on tme doman dynamc modelng and actve control based on the nput and output data of the system. he numercal smulaton results show that predcton model control can meet the requrements of actve control desgn, and can effectvely reduce the vbraton response of the mechansm system. It can reduce the vbraton response of the vbraton system and reduce the vbraton characterstcs to the maxmum extent, thus mprovng the moton precson of the mechansm. In concluson, the control methods proposed n the paper are feasble and effectve; the methods based on master control can effectvely restran the elastc vbraton of the system. CONCLUSION he knematc and dynamc model for the underwater robot wth sx degrees of freedom s bult. Based on ths, predcton model smulaton control was conducted. he smula-
236 he Open Automaton and Control Systems Journal, 205, Volume 7 Wang and Zhao ton results of the control strategy show that the control strategy proposed n ths paper can enable the underwater robot to have strong ant-nterference ablty and accurately trace the tme-varyng theoretcal trace. Model predcton control can reduce the vbraton response of the mechansm to the maxmum extent, thus ncreasng the moton precson of the mechancal system. CONFLIC OF INERES he authors confrm that ths artcle content has no conflct of nterest. ACKNOWLEDGEMENS he work was supported by the Langfang Scence and echnology Research and development Gudance Program 204 (204023095). REFERENCES [] N. Sakaqam, M. Inoue, and S. Kawamura, heoretcal and expermental studes on teratve learnng control for underwater robots, Internatonal Journal of Offshore and Polar Engneerng, vol. 3, no. 2, pp. 20-27, 2003. [2] J. Yuh, Desgn and control of autonomous underwater robots: a survey, Autonomous Robots, vol. 8, no., pp. 7-24, 2000. [3] W. Naeem, R. Sutton, and J. Chudley, System dentfcaton, modelng and control of an autonomous underwater vehcle, In: Proceedngs of MCMC 2003 Conference, Grona, Span, 2003, pp. 37-42. [4] W. Naeem, Model predctve control of an autonomous underwater vehcle, In: Proceedngs of UKACC 2002 Postgraduate Symposum, Sheffeld, UK0002, 2002, pp. 9-23. [5] K. Ish,. Um and. Vuj, A feed forward neural network for dentfcaton and adaptve control of autonomous underwater vehcles, In: IEEE ICNN, 994, pp. 326-322. [6] K. Ish,. Fuj and. Ura, An adaptve neural-net controller system for an underwater vehcle, Control Engneerng Practce vol. 8, pp. 77-84, 2000. [7] R. Crst, M. Cacca, and G. Veruggo, Moton estmaton and modelng of the envronment for underwater vehcle, Internatonal Journal of Systems Scence, vol. 29, no. 0, pp. 35-43, 998. [8] C. Puaut, Hydrodynamc Analyss of Autonomous Underwater Vehcles n Shallow Water Waves, MS hess, Department of Ocean Engneerng, Florda Atlantc Unversty, 200. [9] D. Cortes, N. Vazquez, and J. A. Gallegos, Dynamcal sldngmode control of the boost nverter, IEEE ransactons on Industral Electroncs, vol. 56, no. 9, pp. 3467-3476, 2009. [0] S. Lu, L. ong, and Z. Ln, Smultaneous optmzaton of control parameters and confguraton of PZ actuators for morphng structural shapes, Fnte Elements n Analyss and Desgn, vol. 44, pp. 47-424, 2008. [] K.Y. Pettersen, and H. Njmejer, Under actuated Shp rackng Control: heory and Experments, Internatonal Journal of Control, vol. 74, no. 4, pp. 435-46, 200. Receved: May 26, 205 Revsed: July 4, 205 Accepted: August 0, 205 Wang and Zhao; Lcensee Bentham Open. hs s an open access artcles lcensed under the terms of the Creatve Commons Attrbuton-Non-Commercal 4.0 Internatonal Publc Lcense (CC BY-NC 4.0) (https://creatvecommons.org/lcenses/by-nc/4.0/legalcode), whch permts unrestrcted, non-commercal use, dstrbuton and reproducton n any medum, provded that the work s properly cted.