This paper presents a novel framework integrating a robust wrench observer with a hybrid pose and force controller for physical Human-Robot Interaction (pHRI). In pHRI tasks, robots are expected to perform task-space motions robustly while remaining compliance with human disturbance, which proposes distinct control objectives. Most existing approaches focus mainly on either robustness or compliance and rely on force sensors for real-time tracking. To address these limitations, we propose a novel framework with the following key contributions. Firstly, a neural network (NN) is trained to improve the observer performance by compensating for system friction and unmodeled dynamics. Secondly, a task-null-space decomposition is applied to separate the joint velocity and reduce the effect of the human disturbance on the observer. Finally, a hybrid pose and force controller is introduced to achieve the end-effector pose and force tracking. Experimental results demonstrate that the proposed framework provides robust and accurate wrench estimation while maintaining reliable pose and force tracking performance. The proposed approach offers an effective solution and framework for pHRI without the need of force/torque sensor measurements which can be applied to many robotic manipulator platforms which do not have force/torque sensors.

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