COMPLEX MODAL ANALYSIS OF LOCOMOTIVE MOTIONS OF SOFT ROBOTIC FISH

Zuo Cui and Hongzhou Jiang

References

  1. [1] J. Yu, M. Wang, H. Dong, Y. Zhang, and Z Wu, Motion controland motion coordination of bionic robotic fish: A review,Journal of Bionic Engineering, 15(4), 2018, 579–598.
  2. [2] D. Scaradozzi, G. Palmieri, D. Costa, and A. Pinelli, BCFswimming locomotion for autonomous underwater robots: areview and a novel solution to improve control and efficiency,Ocean Engineering, 130, 2017, 437–453.
  3. [3] L.Z. Dai, G.W. He, X. Zhang, and X. Zhang, Biolocomotionfluid-structure interaction computational fluid dynamics fishschooling energy efficiency intermittent locomotion of a fish-likeswimmer driven by passive elastic mechanism, Bioinspiration& Biomimetics, 13, 2018, 056011.
  4. [4] A. Jusufi, D.M. Vogt, R.J. Wood, and G.V. Lauder, Undulatoryswimming performance and body stiffness modulation in a softrobotic fish-inspired physical model, Soft Robotics, 4(3), 2017,202–210.
  5. [5] S. Subramanian, T. George, and A. Thondiyath, Real-timeobstacle avoidance for an underactuated flat-fish type au-tonomous underwater vehicle in 3D space, International Jour-nal of Robotics and Automation, 29(4), 2014, 424–431.
  6. [6] R. A. Hooshmand, A. Akbar Nasiri, and M. Ataei, Trajectoryangle control of fish-like robot motion by using fuzzy-PIDcontroller, International Journal of Robotics and Automation,27(2), 2012, 163.
  7. [7] F. Sun, J. Yu, P. Zhao, and D. Xu, Tracking control for abiomimetic robotic fish guided by active vision, InternationalJournal of Robotics and Automation, 31(2), 2016, 137–145.
  8. [8] H. Banerjee, Z.T.H. Tse, and H. Ren, Soft robotics withcompliance and adaptation for biomedical applications andforthcoming challenges, International Journal of Robotics andAutomation, 33(1), 2018, 69–80.
  9. [9] G.V. Lauder and E.D. Tytell, Hydrodynamics of undulatorypropulsion, Fish Physiology, 23, 2006, 425.
  10. [10] R.E. Shadwick and G.V. Lauder, Fish physiology: Fish biome-chanics, Vol. 23 (NY: Academic Press, 2006).
  11. [12].When the driving frequency changed from 1 Hz to3.2 Hz, the swimming trajectories are recorded and anal-ysed in the same way. As shown in Fig. 9, the travellingindex of robotic fish motions fluctuates around 0.61, evenat different driving frequencies. This result is consistentwith the theoretical analysis of travelling index in the pre-vious study [12], i.e., the travelling index of fish motionsis independent of its tail-beat frequency.Moreover, the relations between the steering angleand the travelling index are studied to investigate theinfluences of tail’s amplitude, and the results are shown inFig. 10. In the experiments, the steering angle of motoris changed from 18◦to 54◦, and the maximum amplitudeFigure 8. The components of standing and travelling wavedecomposed from the midline motions of soft robotic fish.Figure 9. Relation between the driving frequency and thetravelling index of midline motions.of tail varies from 0.1 BL to 0.15 BL. The experimentalresults show that the travelling index increases with thesteering gear angle, and the variation range is 0.58–0.70.It is demonstrated that the travelling index is affected bythe tail-beat amplitude.For the soft robotic fish, the lowest point of body mo-tions is related to the installation position of the steeringgear. Therefore, the deformed fish body or the midlinemotions have a similar pattern. When the angle of steering5Figure 10. Relation between the travelling index and thesteering angle of robotic fish.gear increases, the tail’s amplitude of robotic fish increasescorrespondingly, but the lowest point remains at the sameposition. Therefore, it makes the travelling index to in-crease gradually. The experimental results agreed wellwith the theoretical analyses in our previous study [12],and they also verify that the travelling index can be usedas a parameter to evaluate the movements of robotic fish.5. ConclusionIn this paper, the COD method is employed to analyse themovements of robotic fish, and the travelling index of itsmidline motions is 0.6. Further, the relations between thetravelling index and the tail-beat frequency and the steer-ing angle (or the undulating amplitude) are investigatedin the experiments. The results show that the midlinemotions of robotic fish are composed of the pure travel-ling wave and the pure standing wave, and the travellingindex of midline motions is independent of the tail-beatfrequency, but increases with the tail’s amplitude. Theseexperimental results agreed well with the theoretical anal-yses of travelling index, which have been published in ourprevious study [12].Overall, the main contribution of this paper is that thetravelling index can be used to evaluate the movementsof robotic fish. It also provides an important backgroundfor expanding the complex modal analyse to evaluate theswimming abilities and the dynamic characteristics of fishbody in the future. The discussions are listed as follows:(1) In the present study, we analysed the influences of trav-elling index partly, because of the limited motions ofrobotic fish. In our previous study [17], a self-propelledCFD model of carangiform fish was developed, and thenumerical results showed that the travelling index hada close relationship with the swimming performance,including the thrust and the forward speed. Therefore,we predict that the travelling index can be regardedas a new parameter to evaluate the swimming per-formance. It is totally different from the traditionalparameters, such as the tail-beat frequency and theamplitude.(2) In swimming fish, the flexible body can be regarded asa viscoelastic beam, deformed in the fluid flow. Ac-cording to (11), the deformed motions are determinedby the viscoelastic properties of fish body, which is alsodemonstrated in reference [25]. However, the problemthat how these dynamic properties affect the travellingindex of the deformed patterns in fish body is still un-solved. Therefore, it is quite necessary to establish aningenious dynamic model of swimming fish and analysethe complex modal characteristics from the dynamicalaspect in the future.AcknowledgementThis work was supported by the Scientific Start-upProject of GuiZhou Institute of Technology [grant numberXJGC20190956]; and the fund of the Research Cultivationand Technology Exploration Program of GuiZhou Instituteof Technology [grant number [2017]5789-20].References[1] J. Yu, M. Wang, H. Dong, Y. Zhang, and Z Wu, Motion controland motion coordination of bionic robotic fish: A review,Journal of Bionic Engineering, 15(4), 2018, 579–598.[2] D. Scaradozzi, G. Palmieri, D. Costa, and A. Pinelli, BCFswimming locomotion for autonomous underwater robots: areview and a novel solution to improve control and efficiency,Ocean Engineering, 130, 2017, 437–453.[3] L.Z. Dai, G.W. He, X. Zhang, and X. Zhang, Biolocomotionfluid-structure interaction computational fluid dynamics fishschooling energy efficiency intermittent locomotion of a fish-likeswimmer driven by passive elastic mechanism, Bioinspiration& Biomimetics, 13, 2018, 056011.[4] A. Jusufi, D.M. Vogt, R.J. Wood, and G.V. Lauder, Undulatoryswimming performance and body stiffness modulation in a softrobotic fish-inspired physical model, Soft Robotics, 4(3), 2017,202–210.[5] S. Subramanian, T. George, and A. Thondiyath, Real-timeobstacle avoidance for an underactuated flat-fish type au-tonomous underwater vehicle in 3D space, International Jour-nal of Robotics and Automation, 29(4), 2014, 424–431.[6] R. A. Hooshmand, A. Akbar Nasiri, and M. Ataei, Trajectoryangle control of fish-like robot motion by using fuzzy-PIDcontroller, International Journal of Robotics and Automation,27(2), 2012, 163.[7] F. Sun, J. Yu, P. Zhao, and D. Xu, Tracking control for abiomimetic robotic fish guided by active vision, InternationalJournal of Robotics and Automation, 31(2), 2016, 137–145.[8] H. Banerjee, Z.T.H. Tse, and H. Ren, Soft robotics withcompliance and adaptation for biomedical applications andforthcoming challenges, International Journal of Robotics andAutomation, 33(1), 2018, 69–80.[9] G.V. Lauder and E.D. Tytell, Hydrodynamics of undulatorypropulsion, Fish Physiology, 23, 2006, 425.[10] R.E. Shadwick and G.V. Lauder, Fish physiology: Fish biome-chanics, Vol. 23 (NY: Academic Press, 2006).[11] G.V. Lauder, Locomotion, in D.H. Evans and J.B. Claiborne(eds.), The physiology of fishes, 3rd edn. (Boca Raton, FL:CRC Press, 2005), 3–46.[12] Z. Cui, L. Shen, Z.X. Yang, and H.Z. Jiang, Complex modalanalysis of midline motions of swimming fish propelled bybody/caudal fin, Wave Motion, 78, 2018, 83–97.
  12. [13] I. Borazjani and F. Sotiropoulos, On the role of form andkinematics on the hydrodynamics of self-propelled body/caudalfin swimming, Journal of Experimental Biology, 213, 2010,89–107.6
  13. [14] E.D. Tytell, J.A. Carr, D. Nicole, et al., Body stiffness anddamping depend sensitively on the timing of muscle activationin lampreys, Integrative and Comparative Biology, 58, 2018,860–873.
  14. [15] T.L. Williams, A new model for force generation by skeletalmuscle, incorporating work-dependent deactivation, Journalof Experimental Biology, 213, 2010, 643–650.
  15. [17], a self-propelledCFD model of carangiform fish was developed, and thenumerical results showed that the travelling index hada close relationship with the swimming performance,including the thrust and the forward speed. Therefore,we predict that the travelling index can be regardedas a new parameter to evaluate the swimming per-formance. It is totally different from the traditionalparameters, such as the tail-beat frequency and theamplitude.(2) In swimming fish, the flexible body can be regarded asa viscoelastic beam, deformed in the fluid flow. Ac-cording to (11), the deformed motions are determinedby the viscoelastic properties of fish body, which is alsodemonstrated in reference [25]. However, the problemthat how these dynamic properties affect the travellingindex of the deformed patterns in fish body is still un-solved. Therefore, it is quite necessary to establish aningenious dynamic model of swimming fish and analysethe complex modal characteristics from the dynamicalaspect in the future.AcknowledgementThis work was supported by the Scientific Start-upProject of GuiZhou Institute of Technology [grant numberXJGC20190956]; and the fund of the Research Cultivationand Technology Exploration Program of GuiZhou Instituteof Technology [grant number [2017]5789-20].References[1] J. Yu, M. Wang, H. Dong, Y. Zhang, and Z Wu, Motion controland motion coordination of bionic robotic fish: A review,Journal of Bionic Engineering, 15(4), 2018, 579–598.[2] D. Scaradozzi, G. Palmieri, D. Costa, and A. Pinelli, BCFswimming locomotion for autonomous underwater robots: areview and a novel solution to improve control and efficiency,Ocean Engineering, 130, 2017, 437–453.[3] L.Z. Dai, G.W. He, X. Zhang, and X. Zhang, Biolocomotionfluid-structure interaction computational fluid dynamics fishschooling energy efficiency intermittent locomotion of a fish-likeswimmer driven by passive elastic mechanism, Bioinspiration& Biomimetics, 13, 2018, 056011.[4] A. Jusufi, D.M. Vogt, R.J. Wood, and G.V. Lauder, Undulatoryswimming performance and body stiffness modulation in a softrobotic fish-inspired physical model, Soft Robotics, 4(3), 2017,202–210.[5] S. Subramanian, T. George, and A. Thondiyath, Real-timeobstacle avoidance for an underactuated flat-fish type au-tonomous underwater vehicle in 3D space, International Jour-nal of Robotics and Automation, 29(4), 2014, 424–431.[6] R. A. Hooshmand, A. Akbar Nasiri, and M. Ataei, Trajectoryangle control of fish-like robot motion by using fuzzy-PIDcontroller, International Journal of Robotics and Automation,27(2), 2012, 163.[7] F. Sun, J. Yu, P. Zhao, and D. Xu, Tracking control for abiomimetic robotic fish guided by active vision, InternationalJournal of Robotics and Automation, 31(2), 2016, 137–145.[8] H. Banerjee, Z.T.H. Tse, and H. Ren, Soft robotics withcompliance and adaptation for biomedical applications andforthcoming challenges, International Journal of Robotics andAutomation, 33(1), 2018, 69–80.[9] G.V. Lauder and E.D. Tytell, Hydrodynamics of undulatorypropulsion, Fish Physiology, 23, 2006, 425.[10] R.E. Shadwick and G.V. Lauder, Fish physiology: Fish biome-chanics, Vol. 23 (NY: Academic Press, 2006).[11] G.V. Lauder, Locomotion, in D.H. Evans and J.B. Claiborne(eds.), The physiology of fishes, 3rd edn. (Boca Raton, FL:CRC Press, 2005), 3–46.[12] Z. Cui, L. Shen, Z.X. Yang, and H.Z. Jiang, Complex modalanalysis of midline motions of swimming fish propelled bybody/caudal fin, Wave Motion, 78, 2018, 83–97.[13] I. Borazjani and F. Sotiropoulos, On the role of form andkinematics on the hydrodynamics of self-propelled body/caudalfin swimming, Journal of Experimental Biology, 213, 2010,89–107.6[14] E.D. Tytell, J.A. Carr, D. Nicole, et al., Body stiffness anddamping depend sensitively on the timing of muscle activationin lampreys, Integrative and Comparative Biology, 58, 2018,860–873.[15] T.L. Williams, A new model for force generation by skeletalmuscle, incorporating work-dependent deactivation, Journalof Experimental Biology, 213, 2010, 643–650.[16] J.M. Donley and K.A. Dickson, Swimming kinematics of ju-venile kawakawa tuna (Euthynnus affinis) and chub mackerel(Scomber japonicus), Journal of Experimental Biology, 203,2000, 3103–3116.[17] Z. Cui, X.S. Gu, K.K. Li, and H.Z. Jiang, CFD studies of theeffects of waveform on swimming performance of carangiformfish, Applied Sciences-Basel, 7, 2017, 149.
  16. [18] B.F. Feeny and A.K. Feeny, Complex modal analysis of theswimming motion of a whiting, International Journal of Acous-tics and Vibration, 135, 2013, 021004.
  17. [19] M. Tanha and B.F. Feeny, Evaluation of traveling wave modelsfor carangiform swimming based on complex modes, ImacXXXVI Conf. & Exposition on Structural Dynamics, Orlando,FL, 2019, 335–341.
  18. [20] G.B. Gillis, Environmental effects on undulatory locomotionin the American eel Anguilla rostrata: kinematics in water andon land, Journal of Experimental Biology, 201, 1998, 949–961.
  19. [21] Z. Cui and H.Z. Jiang, Design and implementation of thun-niform robotic fish with variable body stiffness, InternationalJournal of Robotics & Automation, 32(2), 2017, 109–116.
  20. [23] and
  21. [24].Further, the midline motions of robotic fish in Fig. 7are decomposed by the COD method, and two parts ofstanding and travelling waves are shown in Fig. 8. Thecalculated travelling index is 0.60, which is also within thetravelling index range (0.52–0.78) of carangiform fish innature [12].When the driving frequency changed from 1 Hz to3.2 Hz, the swimming trajectories are recorded and anal-ysed in the same way. As shown in Fig. 9, the travellingindex of robotic fish motions fluctuates around 0.61, evenat different driving frequencies. This result is consistentwith the theoretical analysis of travelling index in the pre-vious study [12], i.e., the travelling index of fish motionsis independent of its tail-beat frequency.Moreover, the relations between the steering angleand the travelling index are studied to investigate theinfluences of tail’s amplitude, and the results are shown inFig. 10. In the experiments, the steering angle of motoris changed from 18◦to 54◦, and the maximum amplitudeFigure 8. The components of standing and travelling wavedecomposed from the midline motions of soft robotic fish.Figure 9. Relation between the driving frequency and thetravelling index of midline motions.of tail varies from 0.1 BL to 0.15 BL. The experimentalresults show that the travelling index increases with thesteering gear angle, and the variation range is 0.58–0.70.It is demonstrated that the travelling index is affected bythe tail-beat amplitude.For the soft robotic fish, the lowest point of body mo-tions is related to the installation position of the steeringgear. Therefore, the deformed fish body or the midlinemotions have a similar pattern. When the angle of steering5Figure 10. Relation between the travelling index and thesteering angle of robotic fish.gear increases, the tail’s amplitude of robotic fish increasescorrespondingly, but the lowest point remains at the sameposition. Therefore, it makes the travelling index to in-crease gradually. The experimental results agreed wellwith the theoretical analyses in our previous study [12],and they also verify that the travelling index can be usedas a parameter to evaluate the movements of robotic fish.5. ConclusionIn this paper, the COD method is employed to analyse themovements of robotic fish, and the travelling index of itsmidline motions is 0.6. Further, the relations between thetravelling index and the tail-beat frequency and the steer-ing angle (or the undulating amplitude) are investigatedin the experiments. The results show that the midlinemotions of robotic fish are composed of the pure travel-ling wave and the pure standing wave, and the travellingindex of midline motions is independent of the tail-beatfrequency, but increases with the tail’s amplitude. Theseexperimental results agreed well with the theoretical anal-yses of travelling index, which have been published in ourprevious study [12].Overall, the main contribution of this paper is that thetravelling index can be used to evaluate the movementsof robotic fish. It also provides an important backgroundfor expanding the complex modal analyse to evaluate theswimming abilities and the dynamic characteristics of fishbody in the future. The discussions are listed as follows:(1) In the present study, we analysed the influences of trav-elling index partly, because of the limited motions ofrobotic fish. In our previous study [17], a self-propelledCFD model of carangiform fish was developed, and thenumerical results showed that the travelling index hada close relationship with the swimming performance,including the thrust and the forward speed. Therefore,we predict that the travelling index can be regardedas a new parameter to evaluate the swimming per-formance. It is totally different from the traditionalparameters, such as the tail-beat frequency and theamplitude.(2) In swimming fish, the flexible body can be regarded asa viscoelastic beam, deformed in the fluid flow. Ac-cording to (11), the deformed motions are determinedby the viscoelastic properties of fish body, which is alsodemonstrated in reference
  22. [25]. However, the problemthat how these dynamic properties affect the travellingindex of the deformed patterns in fish body is still un-solved. Therefore, it is quite necessary to establish aningenious dynamic model of swimming fish and analysethe complex modal characteristics from the dynamicalaspect in the future.AcknowledgementThis work was supported by the Scientific Start-upProject of GuiZhou Institute of Technology [grant numberXJGC20190956]; and the fund of the Research Cultivationand Technology Exploration Program of GuiZhou Instituteof Technology [grant number [2017]5789-20].References[1] J. Yu, M. Wang, H. Dong, Y. Zhang, and Z Wu, Motion controland motion coordination of bionic robotic fish: A review,Journal of Bionic Engineering, 15(4), 2018, 579–598.[2] D. Scaradozzi, G. Palmieri, D. Costa, and A. Pinelli, BCFswimming locomotion for autonomous underwater robots: areview and a novel solution to improve control and efficiency,Ocean Engineering, 130, 2017, 437–453.[3] L.Z. Dai, G.W. He, X. Zhang, and X. Zhang, Biolocomotionfluid-structure interaction computational fluid dynamics fishschooling energy efficiency intermittent locomotion of a fish-likeswimmer driven by passive elastic mechanism, Bioinspiration& Biomimetics, 13, 2018, 056011.[4] A. Jusufi, D.M. Vogt, R.J. Wood, and G.V. Lauder, Undulatoryswimming performance and body stiffness modulation in a softrobotic fish-inspired physical model, Soft Robotics, 4(3), 2017,202–210.[5] S. Subramanian, T. George, and A. Thondiyath, Real-timeobstacle avoidance for an underactuated flat-fish type au-tonomous underwater vehicle in 3D space, International Jour-nal of Robotics and Automation, 29(4), 2014, 424–431.[6] R. A. Hooshmand, A. Akbar Nasiri, and M. Ataei, Trajectoryangle control of fish-like robot motion by using fuzzy-PIDcontroller, International Journal of Robotics and Automation,27(2), 2012, 163.[7] F. Sun, J. Yu, P. Zhao, and D. Xu, Tracking control for abiomimetic robotic fish guided by active vision, InternationalJournal of Robotics and Automation, 31(2), 2016, 137–145.[8] H. Banerjee, Z.T.H. Tse, and H. Ren, Soft robotics withcompliance and adaptation for biomedical applications andforthcoming challenges, International Journal of Robotics andAutomation, 33(1), 2018, 69–80.[9] G.V. Lauder and E.D. Tytell, Hydrodynamics of undulatorypropulsion, Fish Physiology, 23, 2006, 425.[10] R.E. Shadwick and G.V. Lauder, Fish physiology: Fish biome-chanics, Vol. 23 (NY: Academic Press, 2006).[11] G.V. Lauder, Locomotion, in D.H. Evans and J.B. Claiborne(eds.), The physiology of fishes, 3rd edn. (Boca Raton, FL:CRC Press, 2005), 3–46.[12] Z. Cui, L. Shen, Z.X. Yang, and H.Z. Jiang, Complex modalanalysis of midline motions of swimming fish propelled bybody/caudal fin, Wave Motion, 78, 2018, 83–97.[13] I. Borazjani and F. Sotiropoulos, On the role of form andkinematics on the hydrodynamics of self-propelled body/caudalfin swimming, Journal of Experimental Biology, 213, 2010,89–107.6[14] E.D. Tytell, J.A. Carr, D. Nicole, et al., Body stiffness anddamping depend sensitively on the timing of muscle activationin lampreys, Integrative and Comparative Biology, 58, 2018,860–873.[15] T.L. Williams, A new model for force generation by skeletalmuscle, incorporating work-dependent deactivation, Journalof Experimental Biology, 213, 2010, 643–650.[16] J.M. Donley and K.A. Dickson, Swimming kinematics of ju-venile kawakawa tuna (Euthynnus affinis) and chub mackerel(Scomber japonicus), Journal of Experimental Biology, 203,2000, 3103–3116.[17] Z. Cui, X.S. Gu, K.K. Li, and H.Z. Jiang, CFD studies of theeffects of waveform on swimming performance of carangiformfish, Applied Sciences-Basel, 7, 2017, 149.[18] B.F. Feeny and A.K. Feeny, Complex modal analysis of theswimming motion of a whiting, International Journal of Acous-tics and Vibration, 135, 2013, 021004.[19] M. Tanha and B.F. Feeny, Evaluation of traveling wave modelsfor carangiform swimming based on complex modes, ImacXXXVI Conf. & Exposition on Structural Dynamics, Orlando,FL, 2019, 335–341.[20] G.B. Gillis, Environmental effects on undulatory locomotionin the American eel Anguilla rostrata: kinematics in water andon land, Journal of Experimental Biology, 201, 1998, 949–961.[21] Z. Cui and H.Z. Jiang, Design and implementation of thun-niform robotic fish with variable body stiffness, InternationalJournal of Robotics & Automation, 32(2), 2017, 109–116.[22] M.J. Lighthill, Large-amplitude elongated-body theory of fishlocomotion, Proceedings of the Royal Society B: BiologicalSciences, 179(1055), 1971, 125–138.[23] B.P. Epps, P.V.Y Alvarado, K. Youcef-Toumi, and A.H. Techet,Swimming performance of a biomimetic compliant fish-likerobot, Experiments in Fluids, 47, 2009, 927–939.[24] J. Gray, Studies in animal locomotion III. the propulsivemechanism of the whiting (Gadus merlangus), Journal ofExperimental Biology, 10, 1993, 391–402.[25] S. Ramananarivo, R. Godoy-Diana, and B. Thiria, Propagatingwaves in bounded elastic media: transition from standingwaves to anguilliform kinematics, Europhysics Letters, 105,2014, 54003.

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