Probably the simplest way to do this is not to define exactly what distance is between two linkage cofigurations, but to require that
whever two positions of the linkage are connected by a continuous motion of the linkage, the two corresponding points of the configuration space are connected by a continuous path and this path has points which correspond to positions of the continuous motion, all in a continuous way.
That's a mouthful! Let's see what it means.
Here is a very simple linkage:
What kind of space might parameterize this linkage's positions?
