# Double-spiral Spring Compliant Structure for a Passive Joint of a Legged Robot

Harn Sison, Mohsen Jafarpour, Stanislav Gorb, Poramate Manoonpong

► To cite this version:

Harn Sison, Mohsen Jafarpour, Stanislav Gorb, Poramate Manoonpong. Double-spiral Spring Compliant Structure for a Passive Joint of a Legged Robot. Multidisciplinary Biomechanics Journal, 2025, 50th congress of the., 10.46298/mbj.16242. hal-05191797v2

HAL Id: hal-05191797

https://hal.science/hal-05191797v2

Submitted on 23 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Double-spiral Spring Compliant Structure for a Passive Joint of a Legged Robot

Harn Sison<sup>a*</sup>, Mohsen Jafarpour<sup>b</sup>, Stanislav Gorb<sup>b</sup>,
Poramate Manoonpong<sup>a*</sup>

a Bio-inspired Robotics and Neural Engineering, School of Information Science and Technology, Vidyasirimedhi Institute of Science and Technology Rayong 21210, Thailand

b Functional Morphology and Biomechanics, Zoological Institute, Kiel University, 24118 Kiel, Germany

* Corresponding authors: harn.s@vistec.ac.th, poramate.m@vistec.ac.th

Received date: 05/04/2025

Accepted date: 27/06/2025

Publication date: 27/10/2025

**Keywords:** double spiral, compliant joint, multi-DOF joint

© 2025 The Authors

Licence CC-BY 4.0

Published by Société de Biomécanique

## 1. Introduction

In the past, most researched and commercially developed legged robots were typically designed based on engineering and mathematical principles, featuring rigid structures with inorganic shapes such as rectangles, triangles, spheres, and cylinders. These designs prioritized simple manufacturing and assembly solutions, leading to straightforward linear control systems that were easily modeled and functioned effectively in well-structured environments (Bjelonic, Kottege, & Beckerle, 2016). However, common human living spaces and non-factory environments are not designed with suitable locomotion paths for automated systems. The mismatch between robotic design and real-world settings creates challenges for robots with rigid structures and linear control systems, making it difficult for them to adapt to unstructured and unpredictable surroundings.

Unlike in nature, where the biological bodies of animals or humans can adapt their structure and limb trajectories to overcome obstacles in unstructured areas, the morphological adaptability can be analyzed through the studies of bio-inspired principles (Naris, Szczecinski, & Quinn, 2020). Following the bio-inspired approach, the flexibility of the structure, functioning similarly to muscle fibers in biological systems, is one of the key mechanisms for enhancing the adaptability of constructed robots to better match the capabilities of living creatures.

By replicating the elastic behaviour, we propose the development of a compliant structure inspired by the logarithmic spiral found in nature (Jafarpour, Gorb, & Rajabi, 2023) to enhance the adaptability and flexibility of robotic legs. The proposed structure incorporates double-spiral properties to design a multidimensional spring joint, as shown in Figure 1. Experimental results demonstrate that the designed structure exhibits stiffness properties similar to linear and torsional springs for rotational motion while also allowing translation and rotation along other axes within the limits of the designed grooves.

## 2. Materials and Methods

Based on the proposed concept, a double-spiral shape was designed using a logarithmic equation (1), as described in Jafarpour, Gorb, & Rajabi (2023), to emulate the mechanics of natural muscle-tendon structures.

$$r = (r_0 + T)e^{-k\theta} \quad (1)$$

The equation utilizes four parameters to control the spiral shape: $r_0$ represents the outer radius of the spiral arm, $T$ is the thickness of the spiral from the position of $r_0$, $k$ defines the turning rate of the spiral, and $\theta$ denotes the number of spiral turns.

Each parameter was combined to generate a logarithmic spiral curve, which was then used to create the 3D CAD model of the double spiral. The dimensions
<figure><img src="image_2.png" /><figcaption>**Figure 1.** (a) Experimental setup for a rotational stiffness of the joint. (b) Experimental results of different compliant joint structures.</figcaption></figure>

of the system were determined based on the leg geometry of the bio-inspired legged robot and the actuator size, following the approach described in Billeschou et al. (2020). Additionally, the double-spiral structure was selected based on fabrication feasibility using the Ultimaker S5 3D printer, with materials such as PLA (Poly Lactic Acid) and Nylon. Based on these criteria, the structure was designed to prioritize compliance over stiffness within the actuator's load range, enabling the leg to bend or deform as needed to absorb perturbation from the environment. However, the single-disc double-spiral structure exhibited varying stiffness depending on the direction of rotation due to the arrangement of the spiral arms. To address this issue, we implemented a double-disc structure with identical parameters to counterbalance the stiffness in opposing directions. Furthermore, a compliant joint structure was designed using conventional methods, integrating linear and torsional springs, as a baseline for comparison with the proposed method. The prototype, a bio-inspired compliant joint for legged robot was then assembled accordingly, as shown in Figure 1.

### 3. Experimental Setup and Results

The experiment was designed to measure the current feedback from the Dynamixel (XM-430) actuator when the joint rotates from 0 to 30 degrees, using a force gauge to verify the resistance trend as the proposed structure began to rotate. Measurements were taken 10 times for each combination: PLA and Nylon in both directions. However, for the assembled linear spring and torsion

spring, measurements were taken only on one side due to their mechanical properties. The setup is shown in Figure 1(a).

Results from Figure 1(b) indicate that the printed double-spiral structure exhibited a linear current slope throughout the designed range (10 to 30 degrees), similar to both linear and torsion springs. Additionally, although the proposed PLA and Nylon structures were designed using the same parameters from Equation (1), differences in material properties resulted in distinct trend behaviors. These regression trends, with slopes of 6.34 (PLA) and 3.02 (Nylon) and $R^2$ values of 0.95 and 0.95, respectively, indicate that the double-spiral structures exhibit consistent, predictable stiffness comparable to conventional torsional (slope = 4.59, $R^2$ = 0.96) and linear (slope = 1.60, $R^2$ = 0.80) springs. This consistency is critical for achieving reliable, passive compliance in legged robotic joints under varying load conditions.

While the results demonstrate the effectiveness of the double-spiral structure, certain limitations were observed during testing. Between 0 and 10 degrees, the slope of each configuration was not clearly separated as it was beyond 10 degrees. This uncertainty occurred due to the limited resolution of the actuator's internal current sensor, which may not be precise enough to detect small variations in current within this range. As a result, the internal sensor may not be suitable for precise current measurement in the proposed system, making it more challenging to accurately control the movement path of the link.
## 4. Conclusions

The 3D printed double-spiral spring successfully achieved linear stiffness properties within a limited range of rotation, demonstrating predictable and consistent mechanical behavior. Moreover, the structure exhibited the ability to move along three axes of rotation and three axes of translation, indicating potential for greater adaptability in dynamic environments. However, this study presents preliminary results obtained through rapid prototyping and has several limitations. Most notably, fatigue analysis was not conducted due to time and fabrication constraints. Material properties also constrained the rotational range before structural failure, and stiffness remained fixed. For these reasons, future work will explore alternative materials and design modifications to enhance flexibility and durability, as well as perform comprehensive fatigue testing to evaluate long-term performance across different manufacturing processes and material types. In conclusion, the findings highlight the potential of the double-spiral structure for applications requiring compliant and adaptable robotic components.

## Acknowledgements

This research was supported by the postdoc fellowship program of Vidyasirimedi Institute of Science and Technology, Siam Commercial Bank (SCB), the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation (grant number B13F680131).

## References

Billeschou, P., Bijma, N. N., Larsen, L.B., Gorb, S. N., Larsen, J. C., Manoonpong, P. (2020). Framework for developing bio-inspired morphologies for walking robots. *Appl. Sci. 2020, 10*, 6986. https://doi.org/10.3390/app10196986

Bjelonic, M., Kottege, N., & Beckerle, P. (2016). Proprioceptive control of an over-actuated hexapod robot in unstructured terrain. *2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)*, 2042–2049. https://doi.org/10.1109/IROS.2016.7759321

Jafarpour, M., Gorb, S., & Rajabi, H. (2023). Double-spiral: A bioinspired pre-programmable compliant joint with multiple degrees of freedom. *Journal of the Royal Society Interface, 20*, 220757. https://doi.org/10.1098/rsif.2022.0757

Naris, M., Szczecinski, N. S., & Quinn, R. D. (2020). A neuromechanical model exploring the role of the common inhibitor motor neuron in insect locomotion. *Biological Cybernetics*, 114, 23–41. https://doi.org/10.1007/s00422-019-00811-y