Soft robots offer remarkable adaptability and safety advantages over rigid robots, but modeling their complex, nonlinear dynamics remains challenging. Strain-based models have recently emerged as a promising candidate to describe such systems, however, they tend to be high-dimensional and time-consuming. This article presents a novel model order reduction approach for soft and hybrid robots by combining strain-based modeling with proper orthogonal decomposition (POD). The method identifies optimal coupled strain basis functions—or mechanical synergies—from simulation data, enabling the description of soft robot configurations with a minimal number of generalized coordinates. The reduced order model (ROM) achieves substantial dimensionality reduction in the configuration space while preserving accuracy. Rigorous testing demonstrates the interpolation and extrapolation capabilities of the ROM for soft manipulators under static and dynamic conditions. The approach is further validated on a snake-like hyper-redundant rigid manipulator and a closed-chain system with soft and rigid components, illustrating its broad applicability. Moreover, the approach is leveraged for shape estimation of a real six-actuator soft manipulator using only two position markers, showcasing its practical utility. Finally, the ROM's dynamic and static behavior is validated experimentally against a parallel hybrid soft-rigid system, highlighting its effectiveness in representing the high-order model and the real system. This POD-based ROM offers significant computational speed-ups, paving the way for real-time simulation and control of complex soft and hybrid robots.

The robotics community has seen an exponential growth in the level of complexity of the theoretical tools presented for the modeling of soft robotics devices. Different solutions have been presented to overcome the difficulties related to the modeling of soft robots, often leveraging on other scientific disciplines, such as continuum mechanics, computational mechanics, and computer graphics. These theoretical and computational foundations are often taken for granted and this leads to an intricate literature that, consequently, has rarely been the subject of a complete review. For the first time, we present here a structured overview of all the approaches proposed so far to model soft robots. The chosen classification, which is based on their theoretical and numerical grounds, allows us to provide a critical analysis about their uses and applicability. This will enable robotics researchers to learn the basics of these modeling techniques and their associated numerical methods, but also to have a critical perspective on their uses.