The configuration space of the n-arms machine in the Euclidean space

As a typical kind of mechanical linkage, we consider the n-arms machine in Rd. The machine consists of n 2-legs with equal length such that the initial point of each 2-leg is fixed to a circle, while all tips of the 2-legs are combined to a central joint. We determine the homotopy type of the configuration space of the n-arms machine.