The problem of manipulation by low-complexity robot hands is a key issue since many years. The performance of simplified hardware manipulators relies on the exploitation of nonholonomic effects that occur in rolling. Beside this issue, more recently, the attention of the scientific community has been devoted to the problems of finite capacity communication channels and of constraints on the the complexity of computation. Quantization of controls proved to be efficient for dealing with such kind of limitations. With this in mind, we consider the rolling of a pair of smooth convex objects, one on top of the other, under quantized control. The analysis of the reachable set is performed by exploiting the geometric nature of the system which helps in reducing to the case of a group acting on a manifold. The cases of a plane and a sphere rolling on an arbitrary surface are treated in detail.