We present a novel method to simulate bending and torsion of elastic rods within the position-based dynamics (PBD) framework. The main challenge is that torsion effects of Cosserat rods are described in terms of material frames which are attached to the centerline of the rod. But frames or orientations do not fit into the classical position-based dynamics formulation. To solve this problem we introduce new types of constraints to couple orientations which are represented by unit quaternions. For constraint projection quaternions are treated in the exact same way as positions. Unit length is enforced with an additional constraint. This allows us to use the strain measures form Cosserat theory directly as constraints in PBD. It leads to very simple algebraic expressions for the correction displacements which only contain quaternion products and additions. Our results show that our method is very robust and accurately produces the complex bending and torsion effects of rods. Due to its simplicity our method is very efficient and more than one order of magnitude faster than existing position-based rod simulation methods. It even achieves the same performance as position-based simulations without torsion effects.