FIG. 3.

Dipole models of the helical robot (shown magnified) and the two rotating permanent magnets are used to localize the helical robot. The helical robot (with magnetization m) is contained inside a catheter segment between the two rotating permanent magnets with magnetization M₁ and M₂. ${\mathbf{p}}_{\mathrm{s}}^i$ is the position vector to the ith sensor from a reference frame and ${\mathbf{p}}_{\mathrm{s}-\mathrm{r}}^i$ is position vector to the ith sensor from the robot's frame of reference. p_d1 and p_d2 are position vectors to the first and second rotating permanent magnets from the reference frame, respectively. The axis of rotation of the dipole fields is parallel to the axis of the helical robot and the catheter segment. B₁ and B₂ are the fields of the permanent magnets and exert magnetic torque T_m to overcome the drag torque (T_d) and fretting torque (T_f), with the fluid and clot, respectively.