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In traditional trajectory planning it is usually assumed that the mean path of an ensemble of open-loop trajectories is the same as would be obtained if no noise were present. However, even zero-mean noise tends to cause the mean trajectory to deviate from the nominal one. This paper introduces a stochastic model-based motion planning method to compensate for this bias. An implementation for nonholonomic mobile robots based on the kinematic cart model is provided. The examples show that the proposed method takes full advantage of the results of existing optimal trajectory planning methods, and makes the resulting mean trajectory conform to the pre-selected nominal trajectory. As a result, the average amount of online trajectory correction required of a controller is minimized.