撓性等力機構可以在一定輸入位移下，輸出幾乎相同的力量，可以不使用力量回饋系統，而且能夠經由一體式的構造變形達到等力的效果，沒有複雜的零件組成；撓性等力機構可以用剛性疊加法進行設計，將負剛性機構與正剛性機構結合形成等力機構，同時用這種方法設計的等力機構能夠運用預壓，預先施加位移，使機構具有可變等力功能。本研究希望使用拓樸最佳化方法先設計負剛性機構，再設計對應的正剛性機構並加以結合以達成等力目標，為此本研究在非線性拓樸最佳化的基礎上，加入新的目標函數與限制式，讓此方法能夠用於設計特定剛性之機構。本研究以反向機構的最佳化作為範例，分別以三種目標剛性做測試，展示本研究之拓樸最佳化流程能應於不同目標。並透過此方法以雙穩態樑的邊界條件設計出一個負剛性機構，並依此設計對應的正剛性機構，將兩者結合成等力機構，後續將拓樸最佳化結果進行邊緣平滑化，以有限元素分析確認機構符合設計需求，最後使用軟性材料進行3D列印製作，對負剛性機構與正剛性機構進行輸入位移與輸出力量實驗，結果為負剛性機構於輸入位移28mm至58mm之剛性值為-0.246 N/mm，與模擬的剛性值-0.258 N/mm間相差約4.65%；正剛性機構於輸入位移30mm至54mm的剛性為0.2483 N/mm，與模擬的正剛性值0.2588 N/mm誤差約為4.06%；之後對等力機構做輸入位移與輸出力量實驗，進一步驗證本研究設計的等力機構效果，結果於輸入位移為22mm至64mm的區間中具有等力效果，等力值為18.75N，於區間中最大等力誤差為2.42%，觀察等力機構在不同預壓下對等力值大小的影響，結果為在預壓位移-15mm至15mm時，力量變化為-4.43N至3.32N。

The compliant constant-force mechanism outputs nearly identical forces over a range of input displacements without requiring a force feedback system. It achieves constant force through deformation. The mechanism is designed using the principle of stiffness combination, where a negative stiffness mechanism (NSM) is combined with a positive stiffness mechanism (PSM). This approach allows for preload application, enabling variable constant-force through pre-applied displacement. This research first designs an NSM using topology optimization, then a corresponding PSM, combining them to achieve the constant-force objective. New objective functions and constraints are introduced into nonlinear topology optimization to design mechanisms with specific stiffnesses. As an example, an inverse mechanism is optimized for three target stiffnesses, demonstrating the effectiveness of the proposed method. Furthermore, a combined NSM and PSM are designed and evaluated via finite element analysis, then fabricated using 3D printing with flexible materials. Experiments are conducted to evaluate the input displacement and output force of the designed mechanisms. The results show that the NSM exhibits a stiffness value of -0.246 N/mm within an input range of 28mm to 58mm, with a deviation of approximately 4.65% from the simulation The PSM exhibits a stiffness of 0.2483 N/mm within an input range of 30mm to 54mm, with a deviation of approximately 4.057% from the simulation. The results show that within the input range of 22mm to 64mm, a constant force value of 18.75N and a maximum error of 2.42%. Additionally, the study observes the impact of different preload on the constant force value with preload displacements ranging from -15mm to 15mm, the force variation is between -4.43N and 3.32N.

[1] J. A. Gallego and J. Herder, "Synthesis methods in compliant mechanisms: An overview," in International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2009, vol. 49040, pp. 193-214.
[2] L. L. Howell and A. Midha, "A method for the design of compliant mechanisms with small-length flexural pivots," Journal of Mechanical Design, vol. 116, no. 1, pp. 280-290, Mar 1994.
[3] O. Sigmund, "Topology optimization: a tool for the tailoring of structures and materials," Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 358, no. 1765, pp. 211-227, 2000.
[4] S. Perai, "Methodology of compliant mechanisms and its current developments in applications: a review," American Journal of Applied Sciences, vol. 4, no. 3, pp. 160-167, 2007.
[5] S. Nishiwaki, M. I. Frecker, S. Min, and N. Kikuchi, "Topology optimization of compliant mechanisms using the homogenization method," International Journal for Numerical Methods in Engineering, vol. 42, no. 3, pp. 535-559, 1998.
[6] N. P. Van Dijk, K. Maute, M. Langelaar, and F. Van Keulen, "Level-set methods for structural topology optimization: a review," Structural and Multidisciplinary Optimization, vol. 48, pp. 437-472, 2013.
[7] M. P. Bendsøe and O. Sigmund, "Material interpolation schemes in topology optimization," Archive of Applied Mechanics, vol. 69, pp. 635-654, 1999.
[8] O. Sigmund, "A 99 line topology optimization code written in Matlab," Structural and Multidisciplinary Optimization, vol. 21, pp. 120-127, 2001.
[9] X. Huang and M. Xie, Evolutionary topology optimization of continuum structures: methods and applications. Chichester, West Sussex, U.K: John Wiley & Sons, 2010, pp. 1-221.
[10] K. Svanberg, "The method of moving asymptotes - a new method for structural optimization," International Journal for Numerical Methods in Engineering, vol. 24, no. 2, pp. 359-373, 1987.
[11] O. Sigmund and J. Petersson, "Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima," Structural Optimization, vol. 16, pp. 68-75, 1998.
[12] O. Sigmund, "On the design of compliant mechanisms using topology optimization," Journal of Structural Mechanics, vol. 25, no. 4, pp. 493-524, 1997.
[13] T. E. Bruns and D. A. Tortorelli, "Topology optimization of non-linear elastic structures and compliant mechanisms," Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 26-27, pp. 3443-3459, 2001.
[14] J. K. Guest, J. H. Prévost, and T. Belytschko, "Achieving minimum length scale in topology optimization using nodal design variables and projection functions," International Journal for Numerical Methods in Engineering, vol. 61, no. 2, pp. 238-254, Sep 2004.
[15] Q. Chen, X. M. Zhang, and B. L. Zhu, "A 213-line topology optimization code for geometrically nonlinear structures," Structural and Multidisciplinary Optimization, vol. 59, no. 5, pp. 1863-1879, May 2019.
[16] Y. J. Luo, M. Y. Wang, and Z. Kang, "Topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique," Computer Methods in Applied Mechanics and Engineering, vol. 286, pp. 422-441, Apr 2015.
[17] 鍾富名, "拓樸最佳化結合元圖策略於考慮真實輸出位移下之幾何非線性等力輸出撓性機構設計," 國立成功大學機械工程學系碩士論文, 2020.
[18] O. Sigmund, "Morphology-based black and white filters for topology optimization," Structural and Multidisciplinary Optimization, vol. 33, pp. 401-424, 2007.
[19] 何元平, "撓性等力端效器之拓樸最佳化設計," 國立成功大學機械工程學系碩士論文, 2022.
[20] H. T. Pham and D. A. Wang, "A constant-force bistable mechanism for force regulation and overload protection," Mechanism and Machine Theory, vol. 46, no. 7, pp. 899-909, Jul 2011.
[21] C.-C. Lan and J.-Y. Wang, "Design of adjustable constant-force forceps for robot-assisted surgical manipulation," in 2011 IEEE international conference on robotics and automation, 2011: IEEE, pp. 386-391.
[22] Y. H. Chen and C. C. Lan, "An Adjustable Constant-Force Mechanism for Adaptive End-Effector Operations," Journal of Mechanical Design, vol. 134, no. 3, Mar 2012.
[23] P. Wang and Q. Xu, "Design and modeling of constant-force mechanisms: A survey," Mechanism and Machine Theory, vol. 119, pp. 1-21, 2018.
[24] J. Ling, T. Ye, Z. Feng, Y. Zhu, Y. Li, and X. Xiao, "A survey on synthesis of compliant constant force/torque mechanisms," Mechanism and Machine Theory, vol. 176, p. 104970, 2022.
[25] C. H. Liu, F. M. Chung, and Y. P. Ho, "Topology optimization for design of a 3D-printed constant-force compliant finger," IEEE-ASME Transactions on Mechatronics, vol. 26, no. 4, pp. 1828-1836, Aug 2021.
[26] J. W. Liang, X. M. Zhang, B. L. Zhu, H. C. Zhang, and R. X. Wang, "Topology optimization method for designing compliant mechanism with given constant force range," Journal of Mechanisms and Robotics-Transactions of the ASME, vol. 15, no. 6, 2023.
[27] G. Hao, J. Mullins, and K. Cronin, "Simplified modelling and development of a bi-directionally adjustable constant-force compliant gripper," Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 231, no. 11, pp. 2110-2123, 2017.
[28] T. Bruns and O. Sigmund, "Toward the topology design of mechanisms that exhibit snap-through behavior," Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 36-38, pp. 3973-4000, 2004.
[29] Q. Chen, X. M. Zhang, H. C. Zhang, B. L. Zhu, and B. C. Chen, "Topology optimization of bistable mechanisms with maximized differences between switching forces in forward and backward direction," Mechanism and Machine Theory, vol. 139, pp. 131-143, 2019.
[30] K. A. James and H. Waisman, "Layout design of a bi-stable cardiovascular stent using topology optimization," Computer Methods in Applied Mechanics and Engineering, vol. 305, pp. 869-890, Jun 2016.
[31] B. X. Ding, X. Li, and Y. M. Li, "Configuration design and experimental verification of a variable constant-force compliant mechanism," Robotica, vol. 40, no. 10, pp. 3463-3475, 2022.
[32] Y. Z. Wei and Q. S. Xu, "Design of a new passive end-effector based on constant-force mechanism for robotic polishing," Robotics and Computer-Integrated Manufacturing, vol. 74, Apr 2022.
[33] B. X. Ding, X. Li, and Y. M. Li, "FEA-based optimization and experimental verification of a typical flexure-based constant force module," Sensors and Actuators a-Physical, vol. 332, 2021.
[34] F. Wang, B. S. Lazarov, and O. Sigmund, "On projection methods, convergence and robust formulations in topology optimization," Structural and Multidisciplinary Optimization, vol. 43, pp. 767-784, 2011.
[35] O. Sigmund, "A 99 line topology optimization code written in Matlab," Structural and Multidisciplinary Optimization, vol. 21, no. 2, pp. 120-127, Apr 2001.
[36] B. Bourdin, "Filters in topology optimization," International Journal for Numerical Methods in Engineering, vol. 50, no. 9, pp. 2143-2158, Mar 2001.
[37] N.-H. Kim, Introduction to nonlinear finite element analysis. US: Springer Science & Business Media, 2014.
[38] C. B. W. Pedersen, T. Buhl, and O. Sigmund, "Topology synthesis of large-displacement compliant mechanisms," International Journal for Numerical Methods in Engineering, vol. 50, no. 12, pp. 2683-2705, Apr 2001.
[39] L. Y. Liu, J. Xing, Q. W. Yang, and Y. J. Luo, "Design of large-displacement compliant mechanisms by topology optimization incorporating modified additive hyperelasticity technique," Mathematical Problems in Engineering, vol. 2017, 2017.
[40] M. Kollmann, Sensitivity analysis: The direct and adjoint method. Linz: Johannes Kepler Universität Linz, 2010.
[41] K. Svanberg, "MMA and GCMMA-two methods for nonlinear optimization," Optimization and Systems Theory, vol. 1, pp. 1-15, 2007.
[42] S. Wang, K. Tai, and M. Y. Wang, "An enhanced genetic algorithm for structural topology optimization," International Journal for Numerical Methods in Engineering, vol. 65, no. 1, pp. 18-44, 2006.
[43] E. Dougherty, Mathematical morphology in image processing. Boca Raton: CRC press, 2018.
[44] C. H. Liu, Y. Chen, and S. Y. Yang, "Quantification of hyperelastic material parameters for a 3D-Printed thermoplastic elastomer with different infill percentages," Materials Today Communications, vol. 26, Mar 2021.