The most challenging procedure for lumbar puncture (LP) is accurately puncturing the spinal membrane (dura mater) based on an automatic needle insertion device (NID). Piezoactuated NID has shown its advantages in robotic-assisted LP with high precision and compact structure. Soft control of the NID is important for insertion safety; however, for stick-slip piezo-actuated NID, there are few studies due to the complex mechanism of stick-slip motion. Here, a modeling and admittance control method for a proposed stick-slip piezoactuated NID is proposed for safe puncture of the spinal membrane. To analytically model the NID, the compliant mechanism (CM) in the NID is reduced to a second-order system. The stick-slip friction and the spinal membrane are modeled based on the LuGre model and the Hunt-Crossley model, respectively. Based on these models, an admittance controller (AC) for the proposed NID is established to realize the precise control of the position and the safety protection against puncture errors. Simulations and preliminary experiments based on a prototype of the NID and a phantom of the spinal membrane were carried out to test the proposed modeling and control method. Results show that the proposed NID with AC has a maximum insertion error of 0.62 mm and the insertion depth decays by 80% when an unexpected force is applied. Therefore, the proposed model and control method have the potential to be used in real LP procedures by further development.