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 control and 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, BCF swimming locomotion for autonomous underwater robots: a review 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, Biolocomotion fluid-structure interaction computational fluid dynamics fish schooling energy efficiency intermittent locomotion of a fish-like swimmer driven by passive elastic mechanism, Bioinspiration & Biomimetics, 13, 2018, 056011.
  4. [4] A. Jusufi, D.M. Vogt, R.J. Wood, and G.V. Lauder, Undulatory swimming performance and body stiffness modulation in a soft robotic fish-inspired physical model, Soft Robotics, 4(3), 2017, 202–210.
  5. [5] S. Subramanian, T. George, and A. Thondiyath, Real-time obstacle avoidance for an underactuated flat-fish type autonomous underwater vehicle in 3D space, International Journal of Robotics and Automation, 29(4), 2014, 424–431.
  6. [6] R. A. Hooshmand, A. Akbar Nasiri, and M. Ataei, Trajectory angle control of fish-like robot motion by using fuzzy-PID controller, International Journal of Robotics and Automation, 27(2), 2012, 163.
  7. [7] F. Sun, J. Yu, P. Zhao, and D. Xu, Tracking control for a biomimetic robotic fish guided by active vision, International Journal of Robotics and Automation, 31(2), 2016, 137–145.
  8. [8] H. Banerjee, Z.T.H. Tse, and H. Ren, Soft robotics with compliance and adaptation for biomedical applications and forthcoming challenges, International Journal of Robotics and Automation, 33(1), 2018, 69–80.
  9. [9] G.V. Lauder and E.D. Tytell, Hydrodynamics of undulatory propulsion, Fish Physiology, 23, 2006, 425.
  10. [10] R.E. Shadwick and G.V. Lauder, Fish physiology: Fish biomechanics, Vol. 23 (NY: Academic Press, 2006).
  11. [12]. When the driving frequency changed from 1 Hz to 3.2 Hz, the swimming trajectories are recorded and analysed in the same way. As shown in Fig. 9, the travelling index of robotic fish motions fluctuates around 0.61, even at different driving frequencies. This result is consistent with the theoretical analysis of travelling index in the previous study [12], i.e., the travelling index of fish motions is independent of its tail-beat frequency. Moreover, the relations between the steering angle and the travelling index are studied to investigate the influences of tail’s amplitude, and the results are shown in Fig. 10. In the experiments, the steering angle of motor is changed from 18◦ to 54◦ , and the maximum amplitude Figure 8. The components of standing and travelling wave decomposed from the midline motions of soft robotic fish. Figure 9. Relation between the driving frequency and the travelling index of midline motions. of tail varies from 0.1 BL to 0.15 BL. The experimental results show that the travelling index increases with the steering gear angle, and the variation range is 0.58–0.70. It is demonstrated that the travelling index is affected by the tail-beat amplitude. For the soft robotic fish, the lowest point of body motions is related to the installation position of the steering gear. Therefore, the deformed fish body or the midline motions have a similar pattern. When the angle of steering 79 Figure 10. Relation between the travelling index and the steering angle of robotic fish. gear increases, the tail’s amplitude of robotic fish increases correspondingly, but the lowest point remains at the same position. Therefore, it makes the travelling index to increase gradually. The experimental results agreed well with the theoretical analyses in our previous study [12], and they also verify that the travelling index can be used as a parameter to evaluate the movements of robotic fish. 5. Conclusion In this paper, the COD method is employed to analyse the movements of robotic fish, and the travelling index of its midline motions is 0.6. Further, the relations between the travelling index and the tail-beat frequency and the steering angle (or the undulating amplitude) are investigated in the experiments. The results show that the midline motions of robotic fish are composed of the pure travelling wave and the pure standing wave, and the travelling index of midline motions is independent of the tail-beat frequency, but increases with the tail’s amplitude. These experimental results agreed well with the theoretical analyses of travelling index, which have been published in our previous study [12]. Overall, the main contribution of this paper is that the travelling index can be used to evaluate the movements of robotic fish. It also provides an important background for expanding the complex modal analyse to evaluate the swimming abilities and the dynamic characteristics of fish body in the future. The discussions are listed as follows: (1) In the present study, we analysed the influences of travelling index partly, because of the limited motions of robotic fish. In our previous study [17], a self-propelled CFD model of carangiform fish was developed, and the numerical results showed that the travelling index had a close relationship with the swimming performance, including the thrust and the forward speed. Therefore, we predict that the travelling index can be regarded as a new parameter to evaluate the swimming performance. It is totally different from the traditional parameters, such as the tail-beat frequency and the amplitude. (2) In swimming fish, the flexible body can be regarded as a viscoelastic beam, deformed in the fluid flow. According to (11), the deformed motions are determined by the viscoelastic properties of fish body, which is also demonstrated in reference [25]. However, the problem that how these dynamic properties affect the travelling index of the deformed patterns in fish body is still unsolved. Therefore, it is quite necessary to establish an ingenious dynamic model of swimming fish and analyse the complex modal characteristics from the dynamical aspect in the future. Acknowledgement This work was supported by the Scientific Start-up Project of GuiZhou Institute of Technology [grant number XJGC20190956]; and the fund of the Research Cultivation and Technology Exploration Program of GuiZhou Institute of Technology [grant number [2017]5789-20]. References [1] J. Yu, M. Wang, H. Dong, Y. Zhang, and Z Wu, Motion control and 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, BCF swimming locomotion for autonomous underwater robots: a review 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, Biolocomotion fluid-structure interaction computational fluid dynamics fish schooling energy efficiency intermittent locomotion of a fish-like swimmer driven by passive elastic mechanism, Bioinspiration & Biomimetics, 13, 2018, 056011. [4] A. Jusufi, D.M. Vogt, R.J. Wood, and G.V. Lauder, Undulatory swimming performance and body stiffness modulation in a soft robotic fish-inspired physical model, Soft Robotics, 4(3), 2017, 202–210. [5] S. Subramanian, T. George, and A. Thondiyath, Real-time obstacle avoidance for an underactuated flat-fish type autonomous underwater vehicle in 3D space, International Journal of Robotics and Automation, 29(4), 2014, 424–431. [6] R. A. Hooshmand, A. Akbar Nasiri, and M. Ataei, Trajectory angle control of fish-like robot motion by using fuzzy-PID controller, International Journal of Robotics and Automation, 27(2), 2012, 163. [7] F. Sun, J. Yu, P. Zhao, and D. Xu, Tracking control for a biomimetic robotic fish guided by active vision, International Journal of Robotics and Automation, 31(2), 2016, 137–145. [8] H. Banerjee, Z.T.H. Tse, and H. Ren, Soft robotics with compliance and adaptation for biomedical applications and forthcoming challenges, International Journal of Robotics and Automation, 33(1), 2018, 69–80. [9] G.V. Lauder and E.D. Tytell, Hydrodynamics of undulatory propulsion, Fish Physiology, 23, 2006, 425. [10] R.E. Shadwick and G.V. Lauder, Fish physiology: Fish biomechanics, 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 modal analysis of midline motions of swimming fish propelled by body/caudal fin, Wave Motion, 78, 2018, 83–97.
  12. [13] I. Borazjani and F. Sotiropoulos, On the role of form and kinematics on the hydrodynamics of self-propelled body/caudal fin swimming, Journal of Experimental Biology, 213, 2010, 89–107. 80
  13. [14] E.D. Tytell, J.A. Carr, D. Nicole, et al., Body stiffness and damping depend sensitively on the timing of muscle activation in lampreys, Integrative and Comparative Biology, 58, 2018, 860–873.
  14. [15] T.L. Williams, A new model for force generation by skeletal muscle, incorporating work-dependent deactivation, Journal of Experimental Biology, 213, 2010, 643–650.
  15. [17], a self-propelled CFD model of carangiform fish was developed, and the numerical results showed that the travelling index had a close relationship with the swimming performance, including the thrust and the forward speed. Therefore, we predict that the travelling index can be regarded as a new parameter to evaluate the swimming performance. It is totally different from the traditional parameters, such as the tail-beat frequency and the amplitude. (2) In swimming fish, the flexible body can be regarded as a viscoelastic beam, deformed in the fluid flow. According to (11), the deformed motions are determined by the viscoelastic properties of fish body, which is also demonstrated in reference [25]. However, the problem that how these dynamic properties affect the travelling index of the deformed patterns in fish body is still unsolved. Therefore, it is quite necessary to establish an ingenious dynamic model of swimming fish and analyse the complex modal characteristics from the dynamical aspect in the future. Acknowledgement This work was supported by the Scientific Start-up Project of GuiZhou Institute of Technology [grant number XJGC20190956]; and the fund of the Research Cultivation and Technology Exploration Program of GuiZhou Institute of Technology [grant number [2017]5789-20]. References [1] J. Yu, M. Wang, H. Dong, Y. Zhang, and Z Wu, Motion control and 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, BCF swimming locomotion for autonomous underwater robots: a review 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, Biolocomotion fluid-structure interaction computational fluid dynamics fish schooling energy efficiency intermittent locomotion of a fish-like swimmer driven by passive elastic mechanism, Bioinspiration & Biomimetics, 13, 2018, 056011. [4] A. Jusufi, D.M. Vogt, R.J. Wood, and G.V. Lauder, Undulatory swimming performance and body stiffness modulation in a soft robotic fish-inspired physical model, Soft Robotics, 4(3), 2017, 202–210. [5] S. Subramanian, T. George, and A. Thondiyath, Real-time obstacle avoidance for an underactuated flat-fish type autonomous underwater vehicle in 3D space, International Journal of Robotics and Automation, 29(4), 2014, 424–431. [6] R. A. Hooshmand, A. Akbar Nasiri, and M. Ataei, Trajectory angle control of fish-like robot motion by using fuzzy-PID controller, International Journal of Robotics and Automation, 27(2), 2012, 163. [7] F. Sun, J. Yu, P. Zhao, and D. Xu, Tracking control for a biomimetic robotic fish guided by active vision, International Journal of Robotics and Automation, 31(2), 2016, 137–145. [8] H. Banerjee, Z.T.H. Tse, and H. Ren, Soft robotics with compliance and adaptation for biomedical applications and forthcoming challenges, International Journal of Robotics and Automation, 33(1), 2018, 69–80. [9] G.V. Lauder and E.D. Tytell, Hydrodynamics of undulatory propulsion, Fish Physiology, 23, 2006, 425. [10] R.E. Shadwick and G.V. Lauder, Fish physiology: Fish biomechanics, 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 modal analysis of midline motions of swimming fish propelled by body/caudal fin, Wave Motion, 78, 2018, 83–97. [13] I. Borazjani and F. Sotiropoulos, On the role of form and kinematics on the hydrodynamics of self-propelled body/caudal fin swimming, Journal of Experimental Biology, 213, 2010, 89–107. 80 [14] E.D. Tytell, J.A. Carr, D. Nicole, et al., Body stiffness and damping depend sensitively on the timing of muscle activation in lampreys, Integrative and Comparative Biology, 58, 2018, 860–873. [15] T.L. Williams, A new model for force generation by skeletal muscle, incorporating work-dependent deactivation, Journal of Experimental Biology, 213, 2010, 643–650. [16] J.M. Donley and K.A. Dickson, Swimming kinematics of juvenile 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 the effects of waveform on swimming performance of carangiform fish, Applied Sciences-Basel, 7, 2017, 149.
  16. [18] B.F. Feeny and A.K. Feeny, Complex modal analysis of the swimming motion of a whiting, International Journal of Acoustics and Vibration, 135, 2013, 021004.
  17. [19] M. Tanha and B.F. Feeny, Evaluation of traveling wave models for carangiform swimming based on complex modes, Imac XXXVI Conf. & Exposition on Structural Dynamics, Orlando, FL, 2019, 335–341.
  18. [20] G.B. Gillis, Environmental effects on undulatory locomotion in the American eel Anguilla rostrata: kinematics in water and on land, Journal of Experimental Biology, 201, 1998, 949–961.
  19. [21] Z. Cui and H.Z. Jiang, Design and implementation of thunniform robotic fish with variable body stiffness, International Journal of Robotics & Automation, 32(2), 2017, 109–116.
  20. [23] and
  21. [24]. Further, the midline motions of robotic fish in Fig. 7 are decomposed by the COD method, and two parts of standing and travelling waves are shown in Fig. 8. The calculated travelling index is 0.60, which is also within the travelling index range (0.52–0.78) of carangiform fish in nature [12]. When the driving frequency changed from 1 Hz to 3.2 Hz, the swimming trajectories are recorded and analysed in the same way. As shown in Fig. 9, the travelling index of robotic fish motions fluctuates around 0.61, even at different driving frequencies. This result is consistent with the theoretical analysis of travelling index in the previous study [12], i.e., the travelling index of fish motions is independent of its tail-beat frequency. Moreover, the relations between the steering angle and the travelling index are studied to investigate the influences of tail’s amplitude, and the results are shown in Fig. 10. In the experiments, the steering angle of motor is changed from 18◦ to 54◦ , and the maximum amplitude Figure 8. The components of standing and travelling wave decomposed from the midline motions of soft robotic fish. Figure 9. Relation between the driving frequency and the travelling index of midline motions. of tail varies from 0.1 BL to 0.15 BL. The experimental results show that the travelling index increases with the steering gear angle, and the variation range is 0.58–0.70. It is demonstrated that the travelling index is affected by the tail-beat amplitude. For the soft robotic fish, the lowest point of body motions is related to the installation position of the steering gear. Therefore, the deformed fish body or the midline motions have a similar pattern. When the angle of steering 79 Figure 10. Relation between the travelling index and the steering angle of robotic fish. gear increases, the tail’s amplitude of robotic fish increases correspondingly, but the lowest point remains at the same position. Therefore, it makes the travelling index to increase gradually. The experimental results agreed well with the theoretical analyses in our previous study [12], and they also verify that the travelling index can be used as a parameter to evaluate the movements of robotic fish. 5. Conclusion In this paper, the COD method is employed to analyse the movements of robotic fish, and the travelling index of its midline motions is 0.6. Further, the relations between the travelling index and the tail-beat frequency and the steering angle (or the undulating amplitude) are investigated in the experiments. The results show that the midline motions of robotic fish are composed of the pure travelling wave and the pure standing wave, and the travelling index of midline motions is independent of the tail-beat frequency, but increases with the tail’s amplitude. These experimental results agreed well with the theoretical analyses of travelling index, which have been published in our previous study [12]. Overall, the main contribution of this paper is that the travelling index can be used to evaluate the movements of robotic fish. It also provides an important background for expanding the complex modal analyse to evaluate the swimming abilities and the dynamic characteristics of fish body in the future. The discussions are listed as follows: (1) In the present study, we analysed the influences of travelling index partly, because of the limited motions of robotic fish. In our previous study [17], a self-propelled CFD model of carangiform fish was developed, and the numerical results showed that the travelling index had a close relationship with the swimming performance, including the thrust and the forward speed. Therefore, we predict that the travelling index can be regarded as a new parameter to evaluate the swimming performance. It is totally different from the traditional parameters, such as the tail-beat frequency and the amplitude. (2) In swimming fish, the flexible body can be regarded as a viscoelastic beam, deformed in the fluid flow. According to (11), the deformed motions are determined by the viscoelastic properties of fish body, which is also demonstrated in reference
  22. [25]. However, the problem that how these dynamic properties affect the travelling index of the deformed patterns in fish body is still unsolved. Therefore, it is quite necessary to establish an ingenious dynamic model of swimming fish and analyse the complex modal characteristics from the dynamical aspect in the future. Acknowledgement This work was supported by the Scientific Start-up Project of GuiZhou Institute of Technology [grant number XJGC20190956]; and the fund of the Research Cultivation and Technology Exploration Program of GuiZhou Institute of Technology [grant number [2017]5789-20]. References [1] J. Yu, M. Wang, H. Dong, Y. Zhang, and Z Wu, Motion control and 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, BCF swimming locomotion for autonomous underwater robots: a review 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, Biolocomotion fluid-structure interaction computational fluid dynamics fish schooling energy efficiency intermittent locomotion of a fish-like swimmer driven by passive elastic mechanism, Bioinspiration & Biomimetics, 13, 2018, 056011. [4] A. Jusufi, D.M. Vogt, R.J. Wood, and G.V. Lauder, Undulatory swimming performance and body stiffness modulation in a soft robotic fish-inspired physical model, Soft Robotics, 4(3), 2017, 202–210. [5] S. Subramanian, T. George, and A. Thondiyath, Real-time obstacle avoidance for an underactuated flat-fish type autonomous underwater vehicle in 3D space, International Journal of Robotics and Automation, 29(4), 2014, 424–431. [6] R. A. Hooshmand, A. Akbar Nasiri, and M. Ataei, Trajectory angle control of fish-like robot motion by using fuzzy-PID controller, International Journal of Robotics and Automation, 27(2), 2012, 163. [7] F. Sun, J. Yu, P. Zhao, and D. Xu, Tracking control for a biomimetic robotic fish guided by active vision, International Journal of Robotics and Automation, 31(2), 2016, 137–145. [8] H. Banerjee, Z.T.H. Tse, and H. Ren, Soft robotics with compliance and adaptation for biomedical applications and forthcoming challenges, International Journal of Robotics and Automation, 33(1), 2018, 69–80. [9] G.V. Lauder and E.D. Tytell, Hydrodynamics of undulatory propulsion, Fish Physiology, 23, 2006, 425. [10] R.E. Shadwick and G.V. Lauder, Fish physiology: Fish biomechanics, 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 modal analysis of midline motions of swimming fish propelled by body/caudal fin, Wave Motion, 78, 2018, 83–97. [13] I. Borazjani and F. Sotiropoulos, On the role of form and kinematics on the hydrodynamics of self-propelled body/caudal fin swimming, Journal of Experimental Biology, 213, 2010, 89–107. 80 [14] E.D. Tytell, J.A. Carr, D. Nicole, et al., Body stiffness and damping depend sensitively on the timing of muscle activation in lampreys, Integrative and Comparative Biology, 58, 2018, 860–873. [15] T.L. Williams, A new model for force generation by skeletal muscle, incorporating work-dependent deactivation, Journal of Experimental Biology, 213, 2010, 643–650. [16] J.M. Donley and K.A. Dickson, Swimming kinematics of juvenile 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 the effects of waveform on swimming performance of carangiform fish, Applied Sciences-Basel, 7, 2017, 149. [18] B.F. Feeny and A.K. Feeny, Complex modal analysis of the swimming motion of a whiting, International Journal of Acoustics and Vibration, 135, 2013, 021004. [19] M. Tanha and B.F. Feeny, Evaluation of traveling wave models for carangiform swimming based on complex modes, Imac XXXVI Conf. & Exposition on Structural Dynamics, Orlando, FL, 2019, 335–341. [20] G.B. Gillis, Environmental effects on undulatory locomotion in the American eel Anguilla rostrata: kinematics in water and on land, Journal of Experimental Biology, 201, 1998, 949–961. [21] Z. Cui and H.Z. Jiang, Design and implementation of thunniform robotic fish with variable body stiffness, International Journal of Robotics & Automation, 32(2), 2017, 109–116. [22] M.J. Lighthill, Large-amplitude elongated-body theory of fish locomotion, Proceedings of the Royal Society B: Biological Sciences, 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-like robot, Experiments in Fluids, 47, 2009, 927–939. [24] J. Gray, Studies in animal locomotion III. the propulsive mechanism of the whiting (Gadus merlangus), Journal of Experimental Biology, 10, 1993, 391–402. [25] S. Ramananarivo, R. Godoy-Diana, and B. Thiria, Propagating waves in bounded elastic media: transition from standing waves to anguilliform kinematics, Europhysics Letters, 105, 2014, 54003.

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