We have a link which is given in its undeformed configuration in dotted lines. And we have the actual link which is in its deformed configuration in the solid line. So, we can use the existing theory of rigid linkages by finding a transformation of this portion of the link with respect to this portion of the link. This is well known, well understood rigid body transformations using what are called Denavit-Harteneberg (or DH) parameters.
Now, instead of this, in the case of flexibility, what we have is the link has taken a shape which is deformed from this undeformed configuration. Remember we have assumed that our deformations are small. Now, based on that we can go from some fixed origin to one of the links, from here using this to the rigid, assuming is was rigid, and from here we assume that the flexible part of the transformation, and then we can multiply all these transformations and find out the effective transformation.
