cs.ROOct 5, 2026

Lego-Like Stiffness Configuration of Planar Compliant Modules for Task-Specific Flexible Interfaces

Authors: Siyue Yao, Xiaochi Xie, Shixuan Zhao, Yutong Li, Hao Li, Mark R. Cutkosky, Genliang Chen

Organizations: State Key Laboratory of Mechanical Systems and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China · Department of Mechanical Engineering, Stanford University, USA · Shanghai Key Laboratory of Intelligent Robotics; META Robotics Institute, Shanghai Jiao Tong University, Shanghai 200240, China · META Robotics Institute, Shanghai 200240, China

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Abstract

Compliant mechanisms provide compact and intrinsic structural compliance for regulating physical interactions between mechanisms and environments. However, different tasks demand distinct stiffness characteristics, often requiring task-specific optimization and redesign due to limited geometric design space and inherent coupling among multiple stiffness components. This paper presents a Lego-like stiffness configuration approach using stackable planar compliant modules. Three complementary module geometries are introduced, with their stiffness characteristics further regulated through beam width, plate thickness, and module orientation. A unified stiffness model is established for quantitative analysis of individual and composed modules. Further, a two-stage optimization method is presented to achieve desired stiffness profiles, combining a genetic algorithm for configuration and sequential quadratic programming for parameter refinement. Experimental verification shows deviations below 6.5% for simulated stiffness. A flexible wrist is further developed as a representative implementation, exhibiting distinct compliant and dynamic responses under different stiffness characteristics. An optimized modular composition realizes prescribed stiffness values and maintains compliant obstacle interaction during high-speed motion at 1 m/s, with a maximum tested angular compliance of approximately 15∘15^\circ. The proposed framework provides a systematic approach for constructing flexible interfaces with task-specific stiffness characteristics.

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Fig. 1: Overview of the proposed Lego-like stiffness configuration framework. (A) Different tasks demand distinct stiffness requirements, especially differing along task and other directions. (B) Stiffness configuration through optimization using parallel composition of planar compliant modules. (C) Demonstrative flexible wrist implementation with desired stiffness.
Fig. 2: Planar compliant module design. (A) Several types of modules with different geometries and complementary stiffness characteristics. (B) Radar plot of normalized stiffness characteristics ( kri∗=5⋅kri⋅R−2,i∈{x,y,z}k^{*}_{ri}=5\cdot k_{ri}\cdot R^{-2},i\in\{x,y,z\} ) of modules with different geometries, widths, and thicknesses. (C) Discretization-based stiffness modeling method for one chain.
Fig. 3: Variance of principal stiffness components with respect to the (A) plate thickness tt and (B) chain width ww for modules with type-I geometry.
Fig. 4: Optimization of module combination for stiffness configuration. (A) Flowchart of the two-stage strategy for solving the optimization design problem. (B) Stiffness radar plots of optimal results under different requirements using a same scale. (i) Given target stiffness components. (ii) Requiring higher rotational stiffness and lower translational stiffness. (iii) Requiring a higher kdyk_{dy} and lower the remaining components other than kdxk_{dx} .
Fig. 5: Stiffness validation using a Type-I module. (A) Experimental setup. (B) Measurement configurations for the six principal stiffness components.
Fig. 6: Experimental stiffness results and comparison with the simulated model. (A) Force-displacement curves and linear fitting near the unloaded configuration. (B) Comparison between experimentally identified and simulated stiffness components. (C) Radar plot of relative errors of the six principal stiffness components.
Fig. 7: Demonstration of task-dependent contact responses using two compliant wrist configurations. (A) Flexible wrist setup and two compliant module configurations with their corresponding stiffness characteristics. (B) Low rotational stiffness enables the wrist to accommodate the obstacle after contact. (C) Low translational stiffness enables initial compliance after contact, whereas high rotational stiffness eventually tips the obstacle.
Fig. 8: Stiffness configuration and experimental evaluation of the optimized flexible wrist. (A) Optimized stiffness characteristics and corresponding modular composition. (B) Compliance and dynamic performance when interacting with an obstacle at high-speed.

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