Percutaneous dilatational tracheostomy (PDT) is frequently performed on patients in intensive care units for prolonged mechanical ventilation. The needle puncture, as the most critical step of PDT, could lead to adverse consequences such as major bleeding and posterior tracheal wall perforation if performed inaccurately. Current practices of PDT puncture are all performed manually with no navigation assistance, which leads to large position and angular errors (5 mm and 30 degree). To improve the accuracy and reduce the difficulty of the PDT procedure, we propose a system that automates the needle insertion using a velocity-controlled robotic manipulator. Guided using pose data from two electromagnetic sensors, one at the needle tip and the other inside the trachea, the robotic system uses an adaptive constrained controller to adapt the uncertain kinematic parameters online and avoid collisions with the patient's body and tissues near the target. Simulations were performed to validate the controller's implementation, and then four hundred PDT punctures were performed on a mannequin to evaluate the position and angular accuracy. The absolute median puncture position error was 1.7 mm (IQR: 1.9 mm) and midline deviation was 4.13 degree (IQR: 4.55 degree), measured by the sensor inside the trachea. The small deviations from the nominal puncture in a simulated experimental setup and formal guarantees of collision-free insertions suggest the feasibility of the robotic PDT puncture.