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

I. INTRODUCTION

• Different interaction configurations favor different directional stiffness profiles, such as greater rigidity along motion or load-transmission directions and compliance along accommodation directions.Peg-in-hole and gluing are examples of tasks with different directional requirements.Source: §I · p. 1
• Existing topology-optimization approaches can incorporate prescribed stiffness, but attainable designs remain restricted by geometry constraints and changing requirements often demand time-consuming redesign.Composition-based building-block methods enable rapid design but remain constrained by the mechanism's geometric restrictions.Source: §I · pp. 1–2
• Stackable planar modules add a design dimension that enlarges the attainable stiffness range and alleviates single-domain geometric constraints.Module selection, geometric parameters, orientation, and parallel composition configure task-specific stiffness; a flexible wrist demonstrates the approach.Source: §I · p. 2
• The paper develops a unified stiffness model for parameterized modules and their parallel compositions, alongside a two-stage inverse-design method for prescribed stiffness requirements.The framework is experimentally validated and demonstrated in a flexible wrist.Source: §I · p. 2

A. Lego-Like Planar Compliant Module Design

• Symmetrically distributed compliant chains are designed to make off-diagonal stiffness terms ideally vanish, while mirrored half-modules can be stacked in separate planes to preserve symmetry.Out-of-plane translational stiffness is treated as load support rather than a principal configuration objective and must remain within a task-dependent range.Source: A. Lego-Like Planar Compliant Module… · p. 2
• Three module geometries provide complementary combinations of the five principal stiffness components, based on distinct curved-beam and wavy-chain designs.Type-I uses relatively small beam curvature, Type-II introduces curvature reversal, and Type-III uses higher-order wavy chains.Source: Figure 2 · p. 2, A. Lego-Like Planar Compliant Module… · p. 3
• Beam width and plate thickness tune relative stiffness-component ratios, while a 90° module rotation swaps stiffness directions between the two in-plane axes.Module geometry, these parameters, orientation, and parallel composition jointly tailor stiffness toward task requirements.Source: A. Lego-Like Planar Compliant Module… · p. 3
• Predetermined module geometries separate topology design from stiffness configuration, replacing time-consuming topology optimization with lower-dimensional parameter optimization.Figure 2 also depicts a discretization-based stiffness-modeling method for one chain.Source: Figure 2 · p. 2, A. Lego-Like Planar Compliant Module… · p. 3

B. Stiffness Modeling of Planar Compliant Modules

• Thickness varies krx, kdy, and kdz approximately linearly and kdx, kry, and krz approximately cubically; width gives the first group near-cubic and the remainder linear to quadratic trends.Source: Figure 3 · p. 3, B. Stiffness Modeling of Planar… · p. 4
• Each curved chain is discretized into equal-arc segments represented by equivalent six-DoF serial mechanisms, whose kinetostatic equilibrium is established algebraically.Source: B. Stiffness Modeling of Planar… · p. 3
• Increment analysis derives spatial stiffness while accounting for configuration and relative-position effects on force transformation; chain stiffness matrices are summed in a common frame.Source: B. Stiffness Modeling of Planar… · pp. 3–4
• Symmetry makes module stiffness diagonal, so six principal components characterize behavior; rotational components are expressed as kri R^-2, with R the outer-frame radius.Source: B. Stiffness Modeling of Planar… · p. 4
• Because components with similar deformation modes vary together, one geometry cannot tune all six independently; parallel composition of Type-II and Type-III modules broadens attainable stiffness profiles.Source: B. Stiffness Modeling of Planar… · p. 4

C. Optimization for Stiffness Configuration

• The design searches module types, dimensions, orientations, and stack count while minimizing stiffness deviation subject to geometric and stiffness constraints.The examples use at most six modules, a 30 mm maximum stack thickness, thicknesses of 1–10 mm, and widths of 1–3 mm.Source: C. Optimization for Stiffness… · pp. 4–5
• Errors were [0.7, 1.1, 1.4, 2.5, 0.2, 1.7]% for [krx, kry, krz, kdx, kdy, kdz] in the prescribed-target example.The solution stacks two Type-I modules and one Type-III module; Fig. 4B(i) shows the resulting stiffness profile.Source: Figure 4 · p. 4, C. Optimization for Stiffness… · p. 5
• A genetic algorithm selects integer configuration choices, then sequential quadratic programming refines continuous thickness and width parameters.The problem includes module type, dimensions, orientation, and stack count.Source: C. Optimization for Stiffness… · pp. 4–5
• Thicker, narrower modules favor higher rotational and lower translational stiffness, while wider, thinner parallel compositions favor higher kdy and lower remaining components.The examples include two additional qualitative requirements, represented in Fig. 4B(ii–iii).Source: Figure 4 · p. 4, C. Optimization for Stiffness… · p. 5
• Optimal solutions are obtained on a hundred-second-level time scale.Source: C. Optimization for Stiffness… · p. 5

A. Experiment Setup

• A Type-I module with {t,w} = {2,1.5} mm and R = 50 mm was tested across six principal stiffness components.Different fixture arrangements and module orientations were used for the component measurements.Source: A. Experiment Setup · p. 5
• A motorized stage applied displacements while displacement was recorded at 30 Hz and reaction force at 200 Hz.A ball joint at the loading interface allowed force-direction self-alignment and reduced friction and parasitic constraints.Source: A. Experiment Setup · p. 5
• The imposed motion was limited to 4 mm, and the load cell’s compliance was accounted for because it acted in parallel with the tested module.Source: A. Experiment Setup · p. 6

B. Stiffness Characterization

• Near-unloaded linear fits had R2 ≥ 0.99, although instantaneous stiffness varied by approximately 20% across the full motion range.Repeated measurements had a maximum force spread below 0.1 N at the same displacement.Source: B. Stiffness Characterization · p. 6
• Module stiffness was calculated from the measured equivalent stiffness of the load cell and module in parallel.The relation used was kM = kC·kE/(kC−kE).Source: B. Stiffness Characterization · p. 6
• The wrist with low rotational stiffness accommodated the obstacle after contact, whereas low translational stiffness enabled initial compliance before high rotational stiffness tipped it.Fig. 7 depicts these two distinct contact responses.Source: Figure 7 · p. 6
• All six simulated stiffness components deviated from experiment by less than 6.5%, with an average deviation below 5%.The comparison calibrated the material’s elastic properties to account for possible differences in actual and nominal Young’s modulus; Fig. 6B–C presents the comparison and errors.Source: B. Stiffness Characterization · p. 6

• The compliant module sits between the robotic arm and gripper, transmitting tool-environment interactions through the wrist.Source: A. Flexible Wrist Design · p. 6
• Fig. 8A pairs optimized stiffness characteristics with module composition, while Fig. 8B covers high-speed obstacle compliance and dynamics.Source: Figure 8 · p. 7
• Type III uses {t,w} = {4.5,1.8} mm for low kry and kdz; Type I uses {10,1} mm for lower kdz and higher kry.Source: A. Flexible Wrist Design · p. 7
• Under the same obstacle interaction, the Type-III wrist reorients after contact, whereas the Type-I wrist's higher kry limits rotation and eventually tips the obstacle.Source: A. Flexible Wrist Design · p. 7

• Large rotational stiffness restricts wrist reorientation, while excessively low rotational stiffness provides greater accommodation but introduces noticeable oscillation.Source: B. Wrist Modular Design for Stiffness… · p. 7
• The target kry = 5 N·m/rad and kdz = 6000 N/m was realized using mirrored Type-II modules and a Type-I module, all oriented at 0°.Source: B. Wrist Modular Design for Stiffness… · p. 7
• The optimized wrist passed the obstacle without tipping and showed higher oscillation frequency, consistent with its increased prescribed stiffness.Source: B. Wrist Modular Design for Stiffness… · p. 7
• At 1 m/s, the wrist accommodated each obstacle contact with a maximum tested rotation of approximately 15°, while further angular compliance remained available.Source: B. Wrist Modular Design for Stiffness… · p. 7

V. DISCUSSION AND FUTURE WORK

• The approach transforms planar geometry optimization into three-dimensional modular composition, with rapid inverse design, low-cost manufacturing, compact stacking, and extensibility.Source: §V · p. 7
• The framework assumes unloaded stiffness evaluation and a diagonal target stiffness matrix; partial coupling means arbitrary configurations generally remain unattainable.Source: §V · p. 7
• Stack thickness can alter equivalent rotational stiffness through module offsets, while out-of-plane deformation can cause interference and change effective stiffness.Source: §V · p. 8
• Future extensions propose spatial compliant chains, asymmetric layouts for non-diagonal stiffness, and task-specific regulation of nonlinear stiffness.Source: §V · p. 8

VI. CONCLUSION

• Experiments validated simulated results, and an optimized wrist demonstrated desired compliance and dynamic performance during high-speed interaction.Source: §VI · p. 8