This paper studies the relative-position-based reference-frame-free formation control in the plane for discrete-time multiagent systems under both measurement and communication noises. To achieve the desired formation, a robust distributed orientation estimate algorithm and a robust distributed formation control law are designed. The main idea of our method is to use the historical measurement information to get unbiased estimates and use the stochastic approximation method to inhibit the noise. Under mild assumptions on the measurement/communication noises and agents' initial orientation angles, we show that a common reference frame can be asymptotically shared by all agents and the desired formation can be asymptotically stabilized almost surely under the designed distributed control law, if and only if the magnitude of the bearing measurement noise is less than ( π / 4) , the communication and distance sensing topology is rooted and the corresponding bearing sensing topology is connected. Simulations are conducted to verify the correctness and effectiveness of our orientation estimate algorithm and formation control law.