Constraint-Aware Physics-Informed Neural Networks for Static Shape Estimation of Co-Manipulative Continuum Robots

Modeling concentric tube robots (CTRs) involves complex nonlinear continuum mechanics, and despite recent progress, physics-based models often lack an accurate representation of the experimental setups. To overcome these limitations, deep neural network-based models have been explored as alternatives with superior accuracy; however, they often overlook known mechanics, require large training datasets, and typically discard shape estimation of the robot. We present a physics-informed neural network (PINN) for kinematic modeling of a 6-DoF CTR with three pre-curved tubes that embeds the Cosserat rod differential equations and learns from few-shot observational data, balancing physics priors with data-driven fitting. PINN enables full-state estimation of shape, twist angle, torsional strain, bending moment, and orientation. Benchmark tests show a mean shape error below 1% of the robot length and accurately recovered other kinematic states, outperforming a purely physics-based Cosserat rod model baseline while using a minimal training set. The resulting model is also computationally efficient and robust, making it well-suited for real-time control applications.

In co-manipulative continuum robots (CCRs), multiple continuum arms cooperate by grasping a common flexible object, forming a closed-chain deformable mechanical system. The closed-chain coupling induces strong dynamic interactions and internal reaction forces. Moreover, in practical tasks, the flexible object's physical parameters are often unknown and vary between operations, rendering nominal model-based controllers inadequate. This paper presents a projected adaptive control framework for CCRs formulated at the dynamic level. The coupled dynamics are expressed using the Geometric Variable Strain (GVS) representation, yielding a finite-dimensional model that accurately represents the system, preserves the linear-in-parameters structure required for adaptive control, and is suitable for real-time implementation. Closed-chain interactions are enforced through Pfaffian velocity constraints, and an orthogonal projection is used to express the dynamics in the constraint-consistent motion subspace. Based on the projected dynamics, an adaptive control law is developed to compensate online for uncertain dynamic parameters of both the continuum robots and the manipulated flexible object. Lyapunov analysis establishes closed-loop stability and convergence of the task-space tracking errors to zero. Simulation and experiments on a tendon-driven CCR platform validate the proposed framework in task-space regulation and trajectory tracking.

Continuum parallel robots (CPR) combine rigid actuation mechanisms with multiple elastic rods in a closed-loop topology, making forward statics challenging when rigid--continuum junctions are enforced by explicit kinematic constraints. Such constraint-based formulations typically introduce additional algebraic variables and complicate both numerical solution and downstream control. This paper presents a geometric exact, configuration-based and constraint-free static model of CPR that remains valid under geometrically nonlinear, large-deformation and large-rotation conditions. Connectivity constraints are eliminated by kinematic embedding, yielding a reduced unconstrained problem. Each rod of CPR is discretized by nodal poses on SE(3), while the element-wise strain field is reconstructed through a linear strain parameterization. A fourth-order Magnus approximation yields an explicit and geometrically consistent mapping between element end poses and the strain. Rigid attachments at the motor-driven base and the end-effector platforms are handled through kinematic embeddings. Based on total potential energy and virtual work, we derive assembly-ready residuals and explicit Newton tangents, and solve the resulting nonlinear equilibrium equations using a Riemannian Newton iteration on the product manifold. Experiments on a three-servomotor, six-rod prototype validate the model by showing good agreement between simulation and measurements for both unloaded motions and externally loaded cases.