Rod models in continuum and soft robot control: a review

TL;DR

This review unifies the four primary rod theories—Cosserat, Kirchhoff, Euler-Bernoulli, and Timoshenko—under a common formulation to model slender continuum and soft robots. It classifies rod models by deformation classes (e.g., Bend, Bend & Twist, Bend & Stretch) and surveys actuation mappings, discretization choices, and validation practices. It then surveys model-based and learning-based control strategies that leverage rod models, highlighting where MB methods excel and where learning-based approaches are gaining traction through simulation, real-world validation, and sim-to-real transfer. The work emphasizes practical design implications, current gaps (notably hysteresis, manufacturing variability, and limited experimental validation of controllers), and future directions such as physics-informed learning and improved actuator-geometry mappings. Overall, rod-based modeling provides a compelling, computationally efficient pathway to simulate, control, and design soft robots operating in unstructured environments.

Abstract

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.

Figures (8)

• Figure 1: Overview of modeling techniques for continuum and soft robots. Data-driven models employ artificial neural networks to map actuation space to task space. Discrete methods discretize the continuum body a priori (e.g., pseudo-rigid models treat it as a rigid robot). Geometric approaches describe the robot's shape with parameterized curves. Continuum mechanical models are physics-based and use continuous configuration spaces in 3D (FEM) or 1D (rod theories). This paper investigates rod models.
• Figure 2: Overview of rod theories. a) Euler-Bernoulli considers an elastic rod that can only bend in one plane. The rod is supposed unstretchable and unshearable; b) Kirchhoff introduces the concept of directors, modeling bend and torsion modes; c) Timoshenko extends the Euler-Bernoulli formulation considering shear and elongation strain modes; d) From the directors' idea, Cosserat Rod Theory expands the Kirchhoff Rod Theory, including also linear deformations, such as shear and elongation.
• Figure 3: Cross section of a soft robot with pneumatic and cable-driven actuators. The vector $\bm{d}_i(s)$ represents the position of the $i$-th actuator w.r.t the local frame $\{S_s\}$.
• Figure 4: Deformation classes of rod models for continuum and soft robots. Classes are defined as combination of the principal deformation modes: Bend, Twist, Stretch, Shear.
• Figure 5: Deformations exhibited by common continuum and soft manipulators and locomotors. Cable-driven manipulators stretch (contraction), bend, and twist depending on the actuator disposition along the centerline and the cross-section. Similarly, pneumatic manipulators mostly stretch (elongate) and bend. Locomotors move following different strategies (e.g., undulation, peristalsis) that require stretching or bending. Moreover, they undergo shear due to a prominent plane interaction.