In this work, we propose a framework to address the autonomous impedance regulation problem of robots in a class of constrained manipulation tasks. In this framework, a human arm endpoint stiffness model is used to extract the task stiffness geometry along the constrained trajectory, which is then encoded offline and reproduced online by a Gaussian Mixture Model (GMM) and the Gaussian Mixture Regression (GMR), respectively. Furthermore, the full Cartesian impedance of the robot is formulated through an optimal control problem, i.e., the Linear-Quadratic Regulator (LQR), in which the task stiffness geometry (extracted from human demonstrations) is considered as the time-varying weighting matrix Q. The optimal impedance is eventually realised by the robot through a task geometry consistent Cartesian impedance controller. A tank-based passivity observer is implemented to give evidence on the stability of the system during online impedance variations. To evaluate the performance of the framework, a comparative experiment with three different impedance settings (i.e., the proposed framework, the framework without LQR and the framework without GMM/GMR) for Franka Emika Panda to perform a door opening task was conducted. The results reveal that our framework outperforms the other two, in terms of tracking error and the interaction forces.