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Elasto-plastic metal wire curving task is commonly seen in manufacturing and medical fields. This paper presents a combined task and motion planner (TAMP) for a robot arm to work aside a bending machine and carry out 3D metal wire curving tasks. We assume a collaborative robot that is safe for humans but has a weak payload and develop the combined planner for the robot to use the bending machine. The contributions of the study are three-fold. First, we propose a coarse-to-fine optimization-based method to convert a 3D curve to a structured bending set. Second, we build a planner to generate the feasible bending sequence, machine operation, robotic grasp poses, and arm motion while considering constraints from the bending machine and the robot. Third, we use visual feedback to build and dynamically update the springback model of a metal wire and use the model to predict and compensate for bending errors caused by springback. Compared with previous work, the proposed planner does not require the robot arm to have a large payload, making it suitable for lightweight collaborative robots. We evaluate the system using both simulated and real-world 3D curving tasks. The results show that the proposed planner can solve robotic 3D curving problems with satisfying time efficiency and precision. It is flexible and applicable to different robots and metal wire materials without a significant change. The method is expected to accelerate the high-variation low-volume manufacture of 3D metal wire curves. Note to Practitioners—Using robots to bend metal wires has been an old topic in robotics and automation. In previous robotic metal wire bending systems, the robot motion was usually pre-programmed to feed parts to bending machines. It was not easy to be extended to multiple goal shapes. Also, the bending was limited to a few discrete action points instead of an arbitrary curve. The method developed in this paper solved the problem by adding up optimized goal shape parameterization, combined task and motion planning, and springback compensation. It helps to auto-program robot actions and ensures satisfying precision by correcting bending results online with visual feedback. Practitioners are encouraged to use the planner for either offline programming or online motion generation. The springback estimation and compensation are independent of motion generation and can be connected to both motions pre-programmed offline or generated online. However, it is advisable to employ robots with higher DoFs (Degree of Freedom) and avoid the online planning of curves with many bending actions.