A framework for joint position and orientation estimation of a rigid body using range measurements is proposed. We consider a setup in which a few sensors are mounted on a rigid body. The absolute position of the rigid body is not known. However, we know how the sensors are mounted on the rigid body, i.e., the sensor topology is known. The rigid body is localized using noisy range measurements between the sensors and a few anchors (nodes with known absolute positions), and without using any inertial measurements. We propose a least-squares (LS), and a number of constrained LS estimators, where the constrained estimators solve an optimization problem on the Stiefel manifold. As a benchmark, we derive a unitarily constrained Cramér–Rao bound. Finally, the known topology of the sensors can be perturbed during fabrication or if the body is not entirely rigid. To take these perturbations into account, constrained total-least-squares estimators are also proposed.