Consider a planar robot consisting of n rigid segments of length $l_i, 1 < i < n$, that are connected by revolute joints. The hand or tool is attached, by means of a freely orientable joint, to the last segment, while the first segment is attached to the origin of the plane $\Bbb R^2$, again by means of a revolute joint. An illustration of the case n = 3 is shown below:
Let the end effector space be the set of all points reachable by the hand. Find a mathematical description of this space for a general n.
For n = 3 the space is an annulus with outer radius equal to the sum of the segment lengths, and inner radius given by $r = max\{|l_1-l_2| - l_3,|l_2 - l_3| - l_1, |l_1-l_3| - l_2,0 \}$