Constraint-Free Static Modeling of Continuum Parallel Robot

TL;DR

This paper introduces a novel, constraint-free static modeling approach for continuum parallel robots that simplifies analysis and control by eliminating kinematic constraints, enabling accurate simulation under large deformations.

Contribution

It presents a geometric, configuration-based model that avoids constraints through kinematic embedding, applicable to large deformations and rotations, with an explicit Newton solver for equilibrium.

Findings

Model shows good agreement with experimental data.

Enables efficient simulation of large deformations.

Simplifies static analysis of CPRs.

Abstract

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.