It can be considered as continuation of these topics: “Determination of the angles of the manipulator with the help of its mathematical model. Inverse problem.” and “The use of manipulators as multi-axis CNC machines”.
There is a simple two-link manipulator with three degrees of freedom. We change its last link with a fixed length to a variable length link. By this action we added one degree of freedom to our manipulator. Now we have a two-link manipulator with 4 degrees of freedom.
In a particular mathematical model the variable length of the link is related to the x7, and in the figure the total length of the last link is displayed as the result of subtracting the fixed part of the link (L2) from x7 (equation f2). At the same time, the value of the variable x7 depends on the inclination of the first link (equation f4). x1, x2, x3 are the coordinates of the moving point of the first link, x4, x5, x6 are the coordinates of the operating point. The equation f2 defines the type of connection between the links, the equations f5 and f6 are the trajectory of the working point.