Collision-free spatial motion of rigid bodies via topological complexity

The Topological complexity a la Farber $\text{TC}(-)$ is a homotopy invariant which have interesting applications in Robotics, specifically, in the robot motion planning problem. In this work we calculate the topological complexity of the configuration space of $k$ distinct rigid bodies without collisions in ${\mathbb{R}}^d$, for $d=2,3$. Furthermore, we present optimal algorithms which can be used in designing practical systems controlling motion of many rigid bodies moving in space without collisions. The motion planning algorithms we present in this work are easily implementable in practice.