@Article{frizza:machines:2025,
  author    = {Frizza, Irene and Kaminaga, Hiroshi and Fraisse, Philippe and Venture, Gentiane},
  title     = {Design and Validation of a Soft Pneumatic Submodule for Adaptive Humanoid Foot Compliance},
  journal   = {Machines},
  year      = {2025},
  volume    = {13},
  number    = {12},
  month     = {December},
  doi       = {10.3390/machines13121142},
  abstract  = {Achieving stable contact on uneven terrain remains a key challenge in humanoid robotics, as most feet rely on rigid or passively compliant structures with fixed stiffness. This work presents the design, fabrication, and analytical modeling of a compact soft pneumatic submodule capable of tunable longitudinal stiffness, developed as a proof-of-concept unit for adaptive humanoid feet. The submodule features a tri-layer architecture with two antagonistic pneumatic chambers separated by an inextensible layer and reinforced by rigid inserts. A single-step wax-core casting process integrates all materials into a monolithic soft\textendash rigid structure, ensuring precise geometry and repeatable performance. An analytical model relating internal pressure to equivalent stiffness was derived and experimentally validated, showing a linear stiffness\textendash pressure relation with mean error below 10\% across 0\textendash 30 kPa. Static and dynamic tests confirmed tunable stiffness between 0.18 and 0.43 N\cdot m/rad, a rapid symmetric response (2.9\textendash 3.4 ms), and stable stiffness under cyclic loading at gait-relevant frequencies. These results demonstrate the submodule’s suitability as a scalable building block for distributed, real-time stiffness modulation in next-generation humanoid feet.},
  url       = {https://www.mdpi.com/2075-1702/13/12/1142},
  publisher = {MDPI}
}