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. 2025 Aug 23;15:31041. doi: 10.1038/s41598-025-15247-7

Table 1.

Inverse solutions at separate points from a predefined 3-D spiral trajectory with calculating the desired field-rotation axis (Inline graphic), desired dipole-rotation axis (Inline graphic) and desired vector of joint variables (Inline graphic) at 17 representative waypoints.

Predefined 3-D trajectory    Point Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic
graphic file with name 41598_2025_15247_Figb_HTML.gif Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic
Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic
Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic
Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic
Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic
Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic
Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic