Irregular-Shaped Torsion Spring Design for Gravity Compensation Mechanism Using Chain Algorithm and DIRECT Optimization

Shan Zexin, Endo Mitsuru, Nakamura Hiroshi, and Yukio Tsutsui

The 5th International Conference of IFToMM Italy IFIT2024

pp. 152–161, 2024

背景 – Background

Gravity compensation mechanisms reduce actuator power by counteracting gravitational torques, with counterweight-based and spring-based methods being the two primary approaches. While counterweights handle large torques effectively, spring-based methods offer weight reduction but often require bulky structures or fail to cancel nonlinear gravitational torques fully. To address this, the authors previously proposed an irregular-shaped torsion spring, designed using a modified Chain-Pseudo-Rigid-Body Model (CPRBM) combined with a Genetic Algorithm (GA). This method improved design flexibility but suffered from computational deformation approximation and optimization inefficiencies. This study aims to enhance efficiency by incorporating a chain algorithm for deformation approximation and the DIRECT (DIviding RECTangles) optimization algorithm for improved spring design.

手法 – Method

1. Chain Algorithm with PRBM

The Chain Algorithm discretizes a flexible beam into multiple PRBM segments, each modeled as a cantilevered element connected in sequence. The deformation is calculated iteratively from the base to the free end, accumulating the effects of previous segments based on static equilibrium, as shown in Figure 2. The deformation of each segment is governed by force and moment balance equations, with PRBM stiffness coefficients used to compute deflection due to external forces and moments. The actual deformation angles and displacements are obtained by superposing these effects and transforming them into the global coordinate system. This method provides an efficient approximation of beam deformation for irregular-shaped torsion springs.

Fig.2 The Chain Algorithm for approximating the deformation of the flexible beam.

2. DIRECT Optimization Algorithm

The DIRECT (DIviding RECTangles) algorithm optimizes the torsion spring design, minimizing the difference between the desired and approximated endpoint positions. Design variables include segment length, initial orientation angle, and thickness. The algorithm systematically divides the search space into hyper-rectangles, evaluating objective function values at their centers and refining promising regions iteratively. This approach balances global and local search, improving optimization efficiency compared to previous GA-based methods.

3. Optimization Constraints and Objective Function

The optimization process includes constraints on stress and self-intersection. The stress constraint ensures that the maximum bending stress in each PRBM segment does not exceed the material’s allowable limit, penalizing violations. The self-intersection constraint prevents overlapping segments by detecting intersections using a line-segment intersection algorithm. The objective function is defined as the average squared difference between the desired and approximated paths of the free end, with penalty terms for stress and self-intersection violations:

\[f(x) = \frac{1}{N}\sum^N_{j=1} | \boldsymbol{p}_j - \boldsymbol{p}_{j, target} | + C_\mathrm{stress} + C_\mathrm{intersect}\]

This ensures the optimized spring closely follows the target deformation path while maintaining structural feasibility.

検証結果 – Result

1. Case Study Conditions

A case study was conducted to design an irregular-shaped torsion spring achieving a sinusoidal moment-deflection relationship with an output torque of 0.1 Nm and a 90° rotation range. The spring consists of 10 PRBM segments with a radius of 0.04 m. The rotation range was discretized into 90 steps, simulating gradual force changes. The flexible member has a uniform rectangular cross-section and is made of silicone rubber (Young’s modulus: 0.0621 GPa, yield strength: 145 MPa).

2. Results

A C++ program was developed to implement the Chain Algorithm for deformation approximation and the DIRECT optimization algorithm for design optimization. The optimized spring shape was obtained after 500 iterations (~6 minutes). The average square difference between the desired and approximated endpoint paths was 0.002 m. An FEA simulation in Abaqus CAE confirmed a sinusoidal moment-deflection relationship to verify the output torque, though the reaction torque was slightly higher than the target. The RMSE over the 90° rotation range was 0.0082 Nm.

3. Discussion

The results confirm that the proposed method can design an irregular-shaped torsion spring with a sinusoidal moment-deflection relationship. The small deviation in the endpoint path (0.002 m) suggests feasibility, but challenges remain in optimizing the high-dimensional design space. Future improvements may include refining the optimization strategy, allowing variable segment thickness, and increasing the number of PRBM segments for better deformation accuracy.

The FEA results confirm the expected torque behavior, though minor discrepancies exist (RMSE: 0.0082 Nm). The limited 10-segment model may reduce accuracy, suggesting a need for higher segment resolution in future designs.

Since this study focused on a small target torque (0.1 Nm) for feasibility testing, future studies should evaluate larger torque levels and explore different materials for structural robustness in real-world applications.

結論 – Conclusion

This paper presents a method that combines the Chain Algorithm for approximating the deformation of an irregular-shaped torsion spring with the DIRECT optimization algorithm for design optimization. A case study was conducted to verify its effectiveness, and the results confirm that the proposed method is feasible for designing an irregular-shaped torsion spring that achieves a prescribed sinusoidal moment-deflection relationship.

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