The configuration space of the \(n\)-arms machine in the Euclidean space (Q876540)

The authors determine the configuration space of a specific spider-like robotic machine in \({\mathbb R}^d\) for \(d\geq 3\). A configuration of their machine is given by specifying \(n+1\) points \(C, N_1, N_2, \dots, N_n\in {\mathbb R}^d\) satisfying the following conditions \[ | | N_i-C| | =| | N_i-B_i| | =r/2, \quad i=1, \dots, n. \] Here \(B_1, \dots, B_n\) are fixed points situated at vertices of a regular planar \(n\)-gon, \[ B_i=(\cos \frac{2\pi i}{n}, \sin \frac{2\pi i}{n}, 0, \dots, 0), \quad i=1, \dots, n. \] Hence, the machine has its ``body'' at point \(C\) and \(n\) ``legs''\, of length \(r\), each having two parts with a ``knee''\ \(N_i\) in the middle. The endpoints of the legs are anchored at points \(B_i\). The length \(r\) is supposed to satisfy the inequalities \(1<r<2\) for \(n\) even and \(1<r<2\cos \frac{2\pi}{n}\) for \(n\) odd. The main result of the paper, Corollary C, states that the configuration space of this mechanism is a wedge of spheres; more specifically the authors prove that the configuration space is homeomorphic to \[ \Sigma^{d-2}\left(\left(\bigvee\limits_{i=2}^{n-2}X_i\right) \vee S^{n(d-2)+2}\right), \] where \(X_i\) denotes the wedge of \(n\binom{n-2}{i-1} - \binom{n}{i}\) copies of the sphere \(S^{i(d-2)+1}\).