The robotic shape control of deformable linear objects has garnered increasing interest within the robotics community. Despite recent progress, the majority of shape control approaches can be classified into two main groups: open-loop control, which relies on physically realistic models to represent the object, and closed-loop control, which employs less precise models alongside visual data to compute commands. In this work, we present a novel 3D shape control approach that includes the physically realistic Cosserat model into a closed-loop control framework, using vision feedback to rectify errors in real-time. This approach capitalizes on the advantages of both groups: the realism and precision provided by physics-based models, and the rapid computation, therefore enabling real-time correction of model errors, and robustness to elastic parameter estimation inherent in vision-based approaches. This is achieved by computing a deformation Jacobian derived from both the Cosserat model and visual data. To demonstrate the effectiveness of the method, we conduct a series of shape control experiments where robots are tasked with deforming linear objects towards a desired shape.
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