Intra-cardiac echocardiography (ICE) catheters are of crucial importance in cardiac ablation and structural heart procedures. However, achieving optimal anatomical views by precisely positioning these devices can be challenging for the cardiologist. To address this issue, a 6-degree-of-freedom robot has been designed and fabricated for actuating catheters with up to 4 tendons, as investigated in this research. The robot comprises four backdriveable tendon actuation units and a rotation and insertion unit. A pair of against and antagonist tendons were actuated on a sinusoidal motion while the position of the tip of the catheter was captured using an electromagnetic sensor. A static model based on the Cosserat slender rod theory was fitted to the recorded data through an optimization process, resulting in an estimation mean absolute error of 4.29 with a standard deviation of 2.76. The module of elasticity, Poisson's ratio, tendon compliance, the inner radius of the catheter, and tendon offset were determined to be 3.32 GPa, 0.41, 0.29N/mm, 1.01 mm, and 1.12 mm, respectively. The model demonstrated good compliance with the experimental recordings; however, the lack of hysteresis due to friction has resulted in the model's inability to differentiate between the loading and the unloading phases.