Location tracking on a surface

A location tracking apparatus comprising:

• a primary shaft with a proximal end and a distal end;
• a tip movably connected to the distal end of the primary shaft, the tip comprising a lightweight tab configured to undergo angular displacement around the primary shaft with respect to an initial location of the tip within a limited range responsive to the tip being pushed to a surface of an object or the tip being moved away from the surface;
• a tracking instrument configured to provide location data by measuring the initial location of the tip while being moved along the surface of the object by an operator, the initial location comprising a zero-point located at a fixed distance from the tracking instrument;
• a rotary variable differential transformer (RVDT) connected to the proximal end of the primary shaft, the RVDT configured to generate displacement data by measuring the angular displacement of the tip;
• a secondary shaft with a proximal end and a distal end, the proximal end of the secondary shaft connected to the tracking instrument and the distal end of the secondary shaft connected to the RVDT;
• an electronic circuit disposed within the tracking instrument, the electronic circuit configured to:
• receive the location data from the tracking instrument;
• receive the displacement data from the RVDT;
• generate an alarm when the displacement data falls below a lower displacement threshold or exceeds a higher displacement threshold, the lower displacement threshold corresponding to the lower angular displacement limit and the higher displacement threshold corresponding to the higher angular displacement limit;
• and
• transmit the location data and the displacement data to a processing unit, the processing unit configured to generate position data comprising information regarding a position of the tip by executing a process, the process comprising:
• converting the displacement data to a transform vector;
• converting the location data to a two-dimensional transform matrix;
• calculating a three-dimensional transform matrix by combining the transform vector and the two-dimensional transform matrix;
• and
• extracting the position data from the three-dimensional transform matrix;
• wherein the limited range comprises zero displacement corresponding to the zero-point, a lower angular displacement limit, and a higher angular displacement limit.