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.

Soft continuum robots, characterized by their inherent compliance and dexterity, are increasingly pivotal in applications requiring delicate interactions with the environment such as the medical field. Despite their advantages, challenges persist in accurately modeling and controlling their shape during interactions with surrounding objects. This is because of the difficulty in modeling the large degrees of freedom in soft-bodied objects that become more active during interactions. In this study, we present a deep visual model to predict the interaction shapes of a soft continuum robot in contact with surrounding objects. By formulating this task as a forward-statics problem, the model uses the initial state images containing the object configuration and future actuation values to predict interactive state images of the robot under this actuation condition. We developed and tested the model in both simulated and physical environments, explored the model’s predictive capabilities using monocular and binocular views, and tested the model’s generalization ability on different datasets. Our results show that deep learning methods are a promising tool for solving the complex problem of predicting the shape of a soft continuum robot interacting with the environment, requiring no prior knowledge about the system dynamics and explicit mapping of the environment. This study paves the way for future explorations in robot-environment interaction modeling and the development of more adaptable interaction shape control strategies.