Topology-Optimized 4D Printing of a Soft Actuator


Soft robots and actuators are emerging devices providing more capabilities in the field of robotics. More flexibility and compliance attributing to soft functional materials used in the fabrication of these devices make them ideal for delivering delicate tasks in fragile environments, such as food and biomedical sectors. Yet, the intuitive nonlinearity of soft functional materials and their anisotropic actuation in compliant mechanisms constitute an existent challenge in improving their performance. Topology optimization (TO) along with four-dimensional (4D) printing is a powerful digital tool that can be used to obtain optimal internal architectures for the efficient performance of porous soft actuators. This paper employs TO analysis for achieving high bending deflection of a 3D printed polyelectrolyte actuator, which shows bending deformations in response to electrical stimuli in an electrolyte solution. The performance of the actuator is studied in terms of maximum bending and actuation rate compared with a solid, uniformly 3D printed and topology-optimized actuator. The experimental results proved the effectiveness of TO on achieving higher bending deformation and actuation rate against a uniformly 3D printed actuator.

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  1. 1.

    Cohen E, et al. Design methodologies for soft-material robots through additive manufacturing, from prototyping to locomotion. In: ASME 2015 international design engineering technical conferences and computers and information in engineering conference. American Society of Mechanical Engineers; 2015.

  2. 2.

    Koloor S, et al. FE model-based construction and progressive damage processes of FRP composite laminates with different manufacturing processes. Int J Mech Sci. 2018;141:223–35.

    Article  Google Scholar 

  3. 3.

    Zolfagharian A, et al. Polyelectrolyte soft actuators: 3D printed chitosan and cast gelatin. J 3D Print Addit Manuf. 2018;5(2):138–50.

    Article  Google Scholar 

  4. 4.

    Zolfagharian A, et al. Evolution of 3D printed soft actuators. J Sens Actuators A Phys. 2016;250:258–72.

    Article  Google Scholar 

  5. 5.

    Zolfagharian A, et al. Pattern-driven 4D printing. J Sens Actuators A Phys. 2018;274:231–43.

    Article  Google Scholar 

  6. 6.

    Bodaghi M, Liao W-H. 4D printed tunable mechanical metamaterials with shape memory operations. Smart Mater Struct. 2019;28:045019.

    Article  Google Scholar 

  7. 7.

    Bodaghi M, Damanpack A, Liao W. Self-expanding/shrinking structures by 4D printing. Smart Mater Struct. 2016;25(10):105034.

    Article  Google Scholar 

  8. 8.

    Bodaghi M, et al. 4D printing self-morphing structures. J Mater. 2019;12(8):1353.

    Article  Google Scholar 

  9. 9.

    Gao B, et al. 4D bioprinting for biomedical applications. Trends Biotechnol. 2016;34:746–56.

    Article  Google Scholar 

  10. 10.

    Zolfagharian A, et al. Rigid elements dynamics modeling of a 3D printed soft actuator. Smart Mater Struct. 2018;28(2):025003.

    Article  Google Scholar 

  11. 11.

    Zolfagharian A, et al. System identification and robust tracking of a 3D printed soft actuator. Smart Mater Struct. 2019;28:075025.

  12. 12.

    Ionov L. Biomimetic hydrogel-based actuating systems. Adv Funct Mater. 2013;23(36):4555–70.

    Article  Google Scholar 

  13. 13.

    O’Grady ML, Kuo P-L, Parker KK. Optimization of electroactive hydrogel actuators. ACS Appl Mater Interfaces. 2009;2(2):343–6.

    Article  Google Scholar 

  14. 14.

    Zolfagharian A, et al. Bending control of a 3D printed polyelectrolyte soft actuator with uncertain model. J Sens Actuators A Phys. 2019;288:134–43.

    Article  Google Scholar 

  15. 15.

    Shiga T, Kurauchi T. Deformation of polyelectrolyte gels under the influence of electric field. J Appl Polym Sci. 1990;39(11–12):2305–20.

    Article  Google Scholar 

  16. 16.

    Li Y, et al. Electric field actuation of tough electroactive hydrogels cross-linked by functional triblock copolymer micelles. ACS Appl Mater Interfaces. 2016;8(39):26326–31.

    Article  Google Scholar 

  17. 17.

    Sigmund O, Maute KJS, Optimization M. Topology optimization approaches. Struct Multidiscip Optim. 2013;48(6):1031–55.

    MathSciNet  Article  Google Scholar 

  18. 18.

    Huang X, Xie YM. Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem Anal Des. 2007;43(14):1039–49.

    Article  Google Scholar 

  19. 19.

    van Dijk NP, et al. Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim. 2013;48(3):437–72.

    MathSciNet  Article  Google Scholar 

  20. 20.

    Bendsøe MP, Sigmund O. Material interpolation schemes in topology optimization. J Arch Appl Mech. 1999;69(9–10):635–54.

    MATH  Google Scholar 

  21. 21.

    Yang R, Chen C. Stress-based topology optimization. J Struct Optim. 1996;12(2–3):98–105.

    Article  Google Scholar 

  22. 22.

    Hongying Z. Development of topology optimized 3D printed soft grippers and dielectric soft sensors. PhD diss. National University of Singapore (Singapore). 2018.

  23. 23.

    Schumacher A. Mathematische Grundlagen der Optimierung. In: Optimierung mechanischer Strukturen. Springer; 2013. p. 45–55.

  24. 24.

    Harzheim L. Der Natur in die Karten geschaut–Optimierungsverfahren aus dem Bereich der Bionik. In: Karosseriebautage Hamburg 2016. Springer; 2016. p. 3–16.

  25. 25.

    Huang X, Xie M. Evolutionary topology optimization of continuum structures: methods and applications. Hoboken: Wiley; 2010.

    MATH  Book  Google Scholar 

  26. 26.

    Zhang H, et al. Topology optimized multimaterial soft fingers for applications on grippers, rehabilitation, and artificial hands. J IEEE/ASME Trans Mechatron. 2019;24(1):120–31.

    Article  Google Scholar 

  27. 27.

    He D, Liu S. BESO method for topology optimization of structures with high efficiency of heat dissipation. Int J Simul Multidiscip Des Optim. 2008;2(1):43–8.

    Article  Google Scholar 

  28. 28.

    Kim M-G, Kim J-H, Cho S-H. Topology design optimization of heat conduction problems using adjoint sensitivity analysis method. J Comput Struct Eng Inst Korea. 2010;23(6):683–91.

    Google Scholar 

  29. 29.

    Gersborg-Hansen A. Topology optimization using the finite volume method. Proceedings of 6th World Congresses of Structural and Multidisciplinary Optimization. 2005.

  30. 30.

    Kaup IA. Frequency Selective Deblocking Filter for HEVC. Master diss. 2018.

  31. 31.

    Svanberg K. The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng. 1987;24(2):359–73.

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Wiedemann J. Leichtbau: Elemente und Konstruktion. Berlin: Springer; 2007.

    Google Scholar 

  33. 33.

    Tavakoli R, Mohseni SM. Alternating active-phase algorithm for multimaterial topology optimization problems: a 115-line MATLAB implementation. Struct Multidiscip Optim. 2014;49(4):621–42.

    MathSciNet  Article  Google Scholar 

  34. 34.

    Zhang H, et al. Topology optimized design, fabrication and evaluation of a multimaterial soft gripper. In: 2018 IEEE International Conference on Soft Robotics (RoboSoft). IEEE; 2018.

  35. 35.

    Stamm C. Hochgeschwindigkeitsvideoverarbeitung im Sport. Informatik-Spektrum. 2013;36(5):431–9.

    Article  Google Scholar 

  36. 36.

    Wüst S, et al. Tunable hydrogel composite with two-step processing in combination with innovative hardware upgrade for cell-based three-dimensional bioprinting. Acta Biomater. 2014;10(2):630–40.

    Article  Google Scholar 

  37. 37.

    Lee JM, et al. Design and printing strategies in 3D bioprinting of cell-hydrogels: a review. Adv Healthc Mater. 2016;5(22):2856–65.

    Article  Google Scholar 

  38. 38.

    Wu Q, et al. Solvent-cast 3D printing of chitosan hydrogel scaffolds for guided cell growth. 2016.

  39. 39.

    Hinton TJ, et al. Three-dimensional printing of complex biological structures by freeform reversible embedding of suspended hydrogels. Sci Adv. 2015;1(9):e1500758.

    Article  Google Scholar 

  40. 40.

    Zolfagharian A, et al. 3D printed soft parallel actuator. Smart Mater Struct. 2018;27(4):045019.

    Article  Google Scholar 

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Correspondence to Ali Zolfagharian.

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Zolfagharian, A., Denk, M., Bodaghi, M. et al. Topology-Optimized 4D Printing of a Soft Actuator. Acta Mech. Solida Sin. 33, 418–430 (2020).

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  • Topology optimization
  • 4D printing
  • 3D printing
  • Soft actuator