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This paper studies the relative position based node-localization problem for a sensor network without all nodes sharing a common reference frame in the presence of both measurement and communication noises. To solve the problem, a robust distributed orientation estimate algorithm and a robust distributed node-localization algorithm are designed, where unbiased estimators are constructed based on the historical measurement information to inhibit the measurement noise and the stochastic approximation method is adopted to inhibit the communication noise. Under the zero-mean and independence assumption on the measurement/communication noise, we show that all sensor nodes can asymptotically determine their own orientation angles and positions almost surely under the designed algorithms, if and only if the network contains at least one anchor node, and its communication and distance-sensing topology is 1-rooted at the anchor node set and the corresponding bearing sensing topology is connected. Moreover, the convergence rate is quantified if only the measurement noise or communication noise is involved. Simulation experiments are conducted to validate the effectiveness of the proposed algorithms.