Continuum and soft robots can positively impact diverse sectors, from biomedical applications to marine and space exploration, thanks to their potential to adaptively interact with unstructured environments. However, the complex mechanics exhibited by these robots pose diverse challenges in modeling and control. Reduced order continuum mechanical models based on rod theories have emerged as a promising framework, striking a balance between accurately capturing deformations of slender bodies and computational efficiency. This review paper explores rod-based models and control strategies for continuum and soft robots. In particular, it summarizes the mathematical background underlying the four main rod theories applied in soft robotics. Then, it categorizes the literature on rod models applied to continuum and soft robots based on deformation classes, actuation technology, or robot type. Finally, it reviews recent model-based and learning-based control strategies leveraging rod models. The comprehensive review includes a critical discussion of the trends, advantages, limits, and possible future developments of rod models. This paper could guide researchers intending to simulate and control new soft robots and provide feedback to the design and manufacturing community.

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