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Chaos study of a vibratory micro-robot in hybrid motion

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Abstract

In this paper, motion of a micro-robot with two perpendicular vibrational actuators is studied. In this work, separation of the micro-robot body from the substrate of motion is modeled. If the applied vertical force to the micro-robot body is greater than its weight, body of the micro-robot jumps apart from the substrate and then returns to it. This motion will continue intermittently until it is damped. In this condition, the motion mechanism of “stick-slip” is not valid, and a hybrid motion according to “stick-slip-jump” mode is governed. By increasing the applied vertical force, the motion velocity of the micro-robot becomes disordered and unrepeatable. By numerical solving of the equation of motion and investigating the dynamic response, we find the micro-robot motion is chaotic, related to the restitution factor of the motion substrate and the ratio of vertical force to micro-robot weight.

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Abbreviations

\(A_{0}\) :

Non-dimensional mean velocity

\(A_{0}^{*}\) :

Non-dimensional mean velocity in another time duration

\(A_{n}\) :

Amplitude of the nth element of Fourier expansion of the non-dimensional velocity

\(a_{n}, b_{n}\) :

nth Coefficients of the cos and sin terms in Fourier expansion of the non-dimensional velocity

\(D_{{aV}}, D_{{aH}}\) :

Characteristics of the piezo-electric actuators

e :

Restitute factor of the motion substrate

\(\bar{{F}}\) :

Constant part of resultant applied horizontal force

\(F_{0}\) :

Amplitude of the horizontal harmonic actuator force

\(F_{s}\) :

Friction force

\(F_{x}, F_{y}\) :

Horizontal and vertical applied forces of the actuators

g :

Gravity acceleration

k :

Number of motion cycle for numerical calculating

M :

Micro-robot mass

m :

Order of Fourier expansion

\(m_{1}, m_{2} \) :

Vibrating masses of the horizontal and vertical actuators

N :

Substrate reaction force

\(\bar{{N}}\) :

Micro-robot weight

\(N_{0}\) :

Amplitude of vertical harmonic actuator force

n :

Subscribe of the number of the Fourier expansion

t :

time

\(V_{V}, V_{H}\) :

Vertical and horizontal applied voltage to the piezo-electric actuators

\(v_{x1}, v_{y1}\) :

Horizontal and Vertical velocities before impact

\(v_{x2}, v_{y2}\) :

Horizontal and Vertical velocities after impact

\(x, \dot{x}, \ddot{x}\) :

Horizontal displacement, velocity and acceleration

\({x}^{\prime }, {x}^{\prime \prime }\) :

Non-dimensional horizontal velocity and acceleration with respect to time

\(\bar{{{x}}}, \bar{{{x}}}^{\prime }, \bar{{{x}}}^{\prime \prime }\) :

Non-dimensional horizontal displacement, velocity and acceleration

\(\bar{{x}}_{i}, {\bar{{x}}}_{i}^{\prime }\) :

Non-dimensional x displacement and velocity before i’th impact

\(\bar{{x}}_{0i}, {\bar{{x}}}_{0i}^{\prime }\) :

Initial non-dimensional x displacement and velocity in i’th jumping (after i’th impact)

\(\Delta \bar{{x}}\) :

Non-dimensional jumping size (Non-dimensional horizontal displacement of the micro-robot between two successive jumps

\(y, \dot{y}, \ddot{y}\) :

Vertical displacement, velocity and acceleration

\({y}^{\prime }, {y}^{\prime \prime }\) :

Non-dimensional vertical velocity and acceleration with respect to time

\(\bar{y}, \bar{{{y}}}^{\prime }, \bar{{{y}}}^{\prime \prime }\) :

Non-dimensional vertical displacement, velocity and acceleration

\(\bar{{{y}}}_{i}^{\prime }\) :

Non-dimensional y velocity before i’th impact

\(\bar{{{y}}}_{0i}^{\prime }\) :

Initial non-dimensional y velocity in i’th jumping (after i’th impact)

\(\alpha \) :

The ratio of the amplitude of the vertical applied force to weight of the micro-robot

\(\beta \) :

Non-dimensional constant horizontal applied force

\(\delta _{i+1}\) :

Non-dimensional time between two impacts at \(\tau _{i}\) and \(\tau _{i+1}\)

\(\varphi \) :

Phase difference between horizontal and vertical applied forces

\(\varphi _{n}\) :

Phase of the nth term of Fourier expansion

\(\eta \) :

Performance coefficient of the micro-robot motion

\(\eta ^{*}\) :

Approximation of the performance coefficient

\(\mu \) :

Coefficient of friction

\(\mu _{0}\) :

Ratio of the horizontal force amplitude to the micro-robot weight

\(\tau \) :

Non-dimensional time

\(\tau _{1}, \tau _{i}\) :

Non-dimensional time corresponding to the 1st and i’th jumps

\(\omega \) :

Angular frequency of the applied forces

References

  1. Stemme, T.E.A.G.: Microrobotics. In: Gad-el Hak, M. (ed.) The MEMS Handbook, vol. 28, pp. 1–42. CRC Press, Inc., Boca Raton (2005)

  2. Driesen, W., Rida, A., Breguet, J.M., Clavel, R.: Friction based locomotion module for mobile MEMS robots. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, 2007. IROS 2007, October 29–November 2, pp. 3815–3820 (2007)

  3. Breguet, J.-M., Johansson, S., Driesen, W., Simu, U.: A review on actuation principles for few cubic millimeter sized mobile micro-robots. In: Proceedings of the 10th International Conference on New Actuators (Actuator 2006), Bremen, Germany, pp. 374-381 (2006)

  4. Zhang, Z.M., An, Q., Li, J.W., Zhang, W.J.: Piezoelectric friction-inertia actuator—a critical review and future perspective. Int. J. Adv. Manuf. Technol. 62(5–8), 669–685 (2012). doi:10.1007/s00170-011-3827-z

    Article  Google Scholar 

  5. Eigoli, A., Vossoughi, G.: Dynamic analysis of microrobots with Coulomb friction using harmonic balance method. Nonlinear Dyn. 67(2), 1357–1371 (2012). doi:10.1007/s11071-011-0073-8

    Article  MathSciNet  Google Scholar 

  6. Kim, B., Lee, M.G., Lee, Y.P., Kim, Y., Lee, G.: An earthworm-like micro robot using shape memory alloy actuator. Sensors Actuat. A Phys. 125(2), 429–437 (2006). doi:10.1016/j.sna.2005.05.004

    Article  Google Scholar 

  7. Eigoli, A.K., Vossoughi, G.R.: Locomotion modes of a novel piezo-driven microrobot: analytical modeling and performance evaluation. Mech. Mach. Theory 52, 248–266 (2012). doi:10.1016/j.mechmachtheory.2012.01.010

    Article  Google Scholar 

  8. Ikegami, T., Torii, A., Doki, K., Ueda, A.: Motion analysis of a micro-actuator using three piezoelectric actuators. In: 2006 International Symposium on Micro-Nano Mechatronics and Human Science, November 5–8, pp. 1–6 (2006)

  9. Donald, B.R., Levey, C.G., McGray, C.D., Paprotny, I., Rus, D.: An untethered, electrostatic, globally controllable MEMS micro-robot. J. Microelectromech. Syst. 15(1), 1–15 (2006). doi:10.1109/jmems.2005.863697

    Article  Google Scholar 

  10. Mohebbi, M.H., Terry, M.L., Bhringer, K.F., Kovacs, G.T.A., Suh, J.W.: Omnidirectional walking microrobot realized by thermal microactuator arrays. In: ASME International Mechanical Engineering Congress and Exposition, New York, November 11–16, 2001, pp. 1–7 (2001)

  11. Vartholomeos, P., Papadopoulos, E.: Analysis, design and control of a planar micro-robot driven by two centripetal-force actuators. In: Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006, May 15–19, pp. 649–654 (2006)

  12. Chatterjee, S., Chatterjee, Saikat, Singha, T.K.: On the generation of steady motion using fast-vibration. J. Sound Vib. 283, 1187–1204 (2005)

  13. Kardan, I., Kabganian, M., Abiri, R., Bagheri, M.: Stick-slip conditions in the general motion of a planar rigid body. J. Mech. Sci. Technol. 27(9), 2577–2583 (2013). doi:10.1007/s12206-013-0701-y

    Article  Google Scholar 

  14. Bowen, Z., Lining, S., Liguo, C., Zhenhua, W.: The dynamics study of the stick-slip driving system based on LuGre dynamic friction model. In: 2011 International Conference on Mechatronics and Automation (ICMA), August 7–10, pp. 584–589 (2011)

  15. Sun, L., Li, W., Li, M., Jiang, Z., Guo, W.: Experiments and resonant locomotion principle based on impact force. J. Mech. Eng. 46(9), 12–18 (2010)

    Article  Google Scholar 

  16. Bergbreiter, S., Pister, K.S.J.: Design of an autonomous jumping microrobot. In: 2007 IEEE International Conference on Robotics and Automation, April 10–14, 2007, pp. 447-453 (2007)

  17. Yang, B.D., Chu, M.L., Menq, C.H.: Stick-slip-separation analysis and non-linear stiffness and damping characterization of friction contacts having variable normal load. J. Sound Vib. 210(4), 461–481 (1998). doi:10.1006/jsvi.1997.1305

    Article  Google Scholar 

  18. Nhleko, S., Williams, M.S., Blakeborough, A., Stebbins, J.: Horizontal dynamic forces generated by swaying and jumping. J. Sound Vib. 332(11), 2856–2871 (2013). doi:10.1016/j.jsv.2012.12.035

    Article  Google Scholar 

  19. Tavakoli, A., Hurmuzlu, Y.: Robotic locomotion of three generations of a family tree of dynamical systems. Part II: impulsive control of gait patterns. Nonlinear Dyn. 73(3), 1991–2012 (2013). doi:10.1007/s11071-013-0917-5

    Article  MathSciNet  Google Scholar 

  20. Tsobgni-Fozap, D.C., Kenfack-Jiotsa, A., Koumene-Taffo, G.I., Kofané, T.C.: Effect of coupling, synchronization of chaos and stick-slip motion in two mutually coupled dynamical systems. Nonlinear Dyn. 78(2), 1159–1177 (2014). doi:10.1007/s11071-014-1504-0

    Article  Google Scholar 

  21. Efimov, D., Perruquetti, W., Shiriaev, A.: On existence of oscillations in hybrid systems. Nonlinear Anal. Hybrid Syst. 12, 104–116 (2014). doi:10.1016/j.nahs.2013.11.005

    Article  MATH  MathSciNet  Google Scholar 

  22. Harata, Y., Asano, F., Taji, K., Uno, Y.: Efficient parametric excitation walking with delayed feedback control. Nonlinear Dyn. 67(2), 1327–1335 (2012). doi:10.1007/s11071-011-0071-x

  23. Siu, D.P., Ladde, G.S.: Stochastic hybrid system with non-homogeneous jumps. Nonlinear Anal. Hybrid Syst. 5(3), 591–602 (2011). doi:10.1016/j.nahs.2010.12.007

    Article  MATH  MathSciNet  Google Scholar 

  24. Hiskens, I., Reddy, P.: Switching-induced stable limit cycles. Nonlinear Dyn. 50(3), 575–585 (2007). doi:10.1007/s11071-006-9175-0

    Article  MATH  MathSciNet  Google Scholar 

  25. Nishimura, T., Hosaka, H., Morita, T.: Resonant-type smooth impact drive mechanism (SIDM) actuator using a bolt-clamped Langevin transducer. Ultrasonics 52(1), 75–80 (2012). doi:10.1016/j.ultras.2011.06.013

    Article  Google Scholar 

  26. Burden, R.L., Faires, J.D.: Numerical Analysis, 5th edn. PWS-Kent, Boston, MA (1993)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Sharif University of Technology, PO Box 11155-9567, Tehran, Iran

    Hadi Jalili, Hassan Salarieh & Gholamreza Vossoughi

Authors
  1. Hadi Jalili
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  2. Hassan Salarieh
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  3. Gholamreza Vossoughi
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Corresponding author

Correspondence to Hassan Salarieh.

Appendix

Appendix

In Fig. 21a, b, a micro-robot with two perpendicular piezoelectric actuators and its dimensions are shown. The motion capability of the micro-robot is tested for various slopes of the substrate and two magnitudes of applied voltages to its actuators [200 and 320 V (Vpp)]. The results which confirm the motion capability of micro-robot are shown in Fig. 21c. Based on the physical parameters of the micro-robot and applied voltages to its actuators, the motion is in hybrid mode. The obtained results confirm that the motion performances are less than the conditions which are achieved for the uniform motion. For monitoring the chaotic behavior and sequences between the hybrid dynamic modes, some precision instruments are needed.

Fig. 21
figure 21

a Micro-robot with two perpendicular piezoelectric actuators, b micro-robot configuration and dimensions (mm), and c micro-robot velocity with respect to applied voltages and surface slope in the first resonance frequency (glass surface)

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Jalili, H., Salarieh, H. & Vossoughi, G. Chaos study of a vibratory micro-robot in hybrid motion. Nonlinear Dyn 82, 1355–1378 (2015). https://doi.org/10.1007/s11071-015-2243-6

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  • DOI: https://doi.org/10.1007/s11071-015-2243-6

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