Cosserat Rod Modeling

Slender structures made of stimuli-responsive materials represent an emerging class of programmable material systems, with applications in soft robotics, minimally invasive surgery, wearable electronics, and deployable structures. In biomedical applications, the on-demand design of slender structures, encompassing both their material layout and actuation, is essential to harnessing their large deformation and multiphysics actuation to meet patient-specific design requirements. However, owing to geometric and material nonlinearities, their structure-actuation-response relationship is highly nonintuitive and computationally demanding. This often makes trial-and-error and conventional design approaches challenging for addressing patient-specific needs. To this end, we present a Cosserat-rod-based design optimization framework for the autonomous design of programmable slender structures. Leveraging the Cosserat rod representation, the end-to-end differentiable framework is demonstrated on two case studies: (i) design of swellable stents that morph to the shape of carotid vessels; (ii) co-design of the remnant magnetization and the applied field of magnetically-active catheters to reduce the force exerted on the patient-specific vessel wall during insertion. Both self-contact and external contact with surrounding structures are accounted for. We consider two actuation mechanism, namely the swelling of gel-elastomer composites and the magnetic actuation of hard-magnetic composites. The proposed framework presents an autonomous design platform of soft slender material systems that support large deformations and multiphysics actuation toward personalized biomedical design.

Cosserat rod models for soft robots usually construct sectional stiffness by summing material properties over a common cross-section. This assumption becomes inaccurate for trimmed helicoid arms, where load-bearing helix domains are separated and connected only through sparse fused crossings. This paper formulates a separated-section constitutive law that evaluates each helix domain in its local frame and pulls its constitutive response back to the backbone, yielding an effective backbone stiffness. Sparse-fusion mechanics captures the additional compliance caused by relative motion between neighboring domains and determines channel-wise reduction profiles $η_c(s/L)$ for bending, torsion, and extension. The resulting effective sectional stiffness is strongly anisotropic: bending and extension are reduced by about one order of magnitude, whereas torsion remains close to the effective backbone stiffness. The resulting sectional law is embedded in a geometrically exact dynamic Cosserat model with GVS discretization and routed-tendon actuation. Across 103 measured configurations, the three datasets give pooled normalized position errors of $7.7 \ \%$, $6.7 \ \%$, and $7.8 \ \%$, while each full-arm solve requires approximately $0.3 \ \mathrm{s}$ on one CPU core (Intel Xeon, Cascade Lake, $2.8 \mathrm{GHz}$), enabling rapid model-based planning, state and load estimation, and morphology--control co-design for architected soft robots.

Deep-learning models of anatomy can be numerically plausible yet anatomically impossible, and they generalize poorly when data are scarce. We introduce Anatomy-Informed Neural Networks (AINN), in which soft anatomic priors enter as penalty terms in the loss (e.g., a branching penalty that treats a renal transplant artery off the iliac instead of the aorta as unexpected rather than impossible), in direct analogy to a physics-informed neural network, and hard anatomic priors (e.g., continuity of the vessel) are built into the architecture and state representation, making such invalid predictions impossible by construction wherever the prior admits architectural enforcement. We develop it on a clinical test case with limited data: how the aortoiliac tree deforms when a stiff wire is introduced endoluminally. This is important to contemporary aortic surgery and will matter to autonomous endovascular navigation. We lift the vessel centerline and the wire path from R^3 to curves of frames in the Lie group SE(3), and couple a Cosserat-rod wire to a tortuosity-modulated, anatomically anchored vessel through a unilateral lumen-contact inequality. The prediction is a constrained minimizer of the coupled elastic energy, with contact forces as its Lagrange multipliers. Supervision is a Wasserstein-2 optimal-transport loss between the predicted projection through the C-arm geometry and the observed angiogram, so a 2D angiogram can train a 3D prediction. The kinematics, loss and projection are verified against known ground truth; the mechanics solver only against its own optimality conditions, and predicted displacement is not yet mesh-converged. Here, no network is trained. Future work will transfer this in silico model to real CT scans and test whether it improves predictive accuracy and reduces the training data required.

Precise control of soft manipulators remains challenging due to the difficulty of developing accurate yet computationally tractable models for model-based estimation and control. Reduced Cosserat-rod models provide a physics-based and control-oriented description of soft-robot dynamics, offering an explicit alternative to purely data-driven input-output representations. In this paper, we propose a moving-horizon estimation (MHE) and nonlinear model predictive control (NMPC) framework for cable-driven soft manipulators based on reduced Cosserat dynamics. A smooth cable-length-driven modeling formulation is developed by approximating the complementarity relationship between cable tension and cable slackness, enabling cable-length control without direct tension sensing. Based on this formulation, an MHE method is introduced to estimate the reduced state and reconstruct the manipulator configuration from end-effector pose measurements and cable-length information. An NMPC controller is then formulated to achieve task-space control under cable-length and cable-rate constraints. The proposed framework is validated through numerical simulations and experiments. Simulation results demonstrate the effectiveness of the estimator and controller for pose and strain-related regulation on a multi-cable soft manipulator. Experimental results on a four-cable prototype further show that the proposed MHE-NMPC scheme can be implemented in real time and enables accurate end-effector position tracking through cable-length control.

Origami-inspired robotic grippers have shown promising potential for object manipulation tasks due to their compact volume and mechanical flexibility. However, robust capture of objects with random shapes in dynamic working environments often comes at the cost of additional actuation channels and control complexity. Here, we introduce a tendon-driven origami tentacle gripper capable of universal object gripping by exploiting a synergy between local, deterministic deformation programming and global, stochastic entanglements. Each origami tentacle is made by cutting thin Mylar sheets; It features carefully placed holes for routing an actuation tendon, origami creases for controlling the deformation, and a tapered shape. By tailoring these design features, one can prescribe the shrinking, bending, and twisting deformation, eventually creating deterministic coiling with a simple tendon pull. Then, when multiple coiling tentacles are placed in proximity, stochastic entanglement emerges, allowing the tentacles to braid, knot, and grip objects with random shapes. We derived a simulation model by integrating origami mechanics with Cosserat rods to correlate origami design, tendon deformation, and their collective gripping performance. Then, we experimentally tested how these coiling and entangling origami tentacles can grasp objects under gravity and in water. A stow-and-release deployment mechanism was also tested to simulate in-orbit grasping. Overall, the entertaining origami tentacle gripper presents a new strategy for robust object grasping with simple design and actuation.

The Lie group SE(3) of isometric orientation preserving transformation is used for modeling multibody systems, robots, and Cosserat continua. The use of these models in numerical simulation and optimization schemes necessitates the exponential map, its right-trivialized differential (often referred to as tangent operator), as well as higher derivatives in closed form. The $6\times 6$ matrix representation of the differential, $\mathbf{dexp}_{\mathbf{X}}:se\left( 3\right) \rightarrow se\left( 3\right)$ , and its first derivative were reported using a $3\times 3$ block partitioning. In this paper, the differential, its first and second derivative, as well as the Jacobian and Hessian of the evaluation maps, $\mathbf{dexp}_{\mathbf{X}}\mathbf{Z}$ and $\mathbf{dexp}_{\mathbf{X}}^{T}$, are reported avoiding the block partitioning. For all of them, higher-order approximations are derived. Besides the compactness, the advantage of the presented closed form relations is their numerical robustness when combined with the local approximation. The formulations are demonstrated for computation of the deformation field and the strain rates of an elastic Cosserat-Simo-Reissner rod.

Coordinated robotic manipulation of deformable linear objects (DLOs), such as ropes and cables, has been widely studied; however, handling hybrid assemblies composed of both deformable and rigid elements in constrained environments remains challenging. This work presents a quasi-static optimization-based manipulation planner that employs a strain-based Cosserat rod model, extending rigid-body formulations to hybrid deformable linear objects (hDLO). The proposed planner exploits the compliance of deformable links to maneuver through constraints while achieving task-space objectives for the object that are unreachable with rigid tools. By leveraging a differentiable model with analytically derived gradients, the method achieves up to a 33x speedup over finite-difference baselines for inverse kinetostatic(IKS) problems. Furthermore, the subsequent trajectory optimization problem, warm-started using the IKS solution, is only practically realizable via analytical derivatives. The proposed algorithm is validated in simulation on various hDLO systems and experimentally on a three-link hDLO manipulated in a constrained environment using a dual-arm robotic system. Experimental results confirm the planner's accuracy, yielding an average deformation error of approximately 3 cm (5% of the deformable link length) between the desired and measured marker positions. Finally, the proposed optimal planner is compared against a sampling-based feasibility planner adapted to the strain-based formulation. The results demonstrate the effectiveness and applicability of the proposed approach for robotic manipulation of hybrid assemblies in constrained environments.

Robotic cochlear-implant (CI) insertion requires precise prediction and regulation of contact forces to minimize intracochlear trauma and prevent failure modes such as locking and buckling. Aligned with the integration of advanced medical imaging and robotics for autonomous, precision interventions, this paper presents a unified CT-to-simulation pipeline for contact-aware insertion planning and validation. We develop a low-dimensional, differentiable Cosserat-rod model of the electrode array coupled with frictional contact and pseudo-dynamics regularization to ensure continuous stick-slip transitions. Patient-specific cochlear anatomy is reconstructed from CT imaging and encoded via an analytic parametrization of the scala-tympani lumen, enabling efficient and differentiable contact queries through closest-point projection. Based on a differentiated equilibrium-constraint formulation, we derive an online direction-update law under an RCM-like constraint that suppresses lateral insertion forces while maintaining axial advancement. Simulations and benchtop experiments validate deformation and force trends, demonstrating reduced locking/buckling risk and improved insertion depth. The study highlights how CT-based imaging enhances modeling, planning, and safety capabilities in robot-assisted inner-ear procedures.