Using different imaging techniques (primarily, magnetic resonance imaging or MRI) as guidance in robotic-assisted brain surgery becomes more and more wide-spread. One of important issues in this area is using these imaging technologies in online mode. For that, robotic devices that are compatible with MRI systems are required. In particular, conventional electric drives are not suitable; some alternatives should be used, such as piezoelectric drives (PED). In this paper, a robotic system is described intended to deliver a needle (cannula) to a given point by means of a PED. Dependence of the driving force generated by PED upon the needle speed is studied. In order to describe the contact interaction between the cannula and the soft tissue, a mathematical model of their interaction is developed based on modified Kelvin-Voigt approach. A phantom of porcine brain is manufactured. Experiments are carried out where the cannula was indented into the phantom. Based on the obtained experimental data, parameters of the mathematical model are identified. Numerical simulation of the insertion of the cannula into the soft tissue is performed, and the effect of parameters of the feedback control loop upon the cannula dynamics is analysed.