*3.3. 3D Path-Tracking Tests*

Three tests were executed using the helix path for comparison to show the behavior of the proposed algorithm in both ends of the assistance range spectrum, namely, fully assisted motion or free motion (FM), and not assisted motion, i.e., force-applied motion (FAM). The combined motion with virtual forces (CVF) is the smooth transition between FM and FAM. Note that for the FAM test we defined *fv* = 0 N to test the case of no assistance individually. This causes that at any instant when there is no external force applied, the robot stops and maintains the last position, meaning that only the user can make the robot move in this case. Moreover, the proposed virtual force scheme provides the means for a smooth transition between FM and FAM. The detailed data of the tests are shown in Figure 7a–i, including the external forces in the TNB frame, the *s* parameter variation throughout the task, and the desired velocity *r*˙*<sup>d</sup>* in the TNB frame obtained from the constant virtual force *fv*.
