The configuration space of planar spidery linkages

The configuration space of the mechanism of a planar robot is studied. We consider a robot which has $n$ arms such that each arm is of length 1+1 and has a rotational joint in the middle, and that the endpoint of the $k$-th arm is fixed to $R{e}^{\frac{2(k-1)\pi }{n}i}$. Generically, the configuration space is diffeomorphic to an orientable closed surface. Its genus is given by a topological way and a Morse theoretical way. The homeomorphism types of it when it is singular is also given.