A boring tool moves having a pitch orientation, a yaw orientation and a roll orientation and is steerable underground using the roll orientation. A maximum drill string curvature is established for steering. The boring tool is advanced over a path segment. An averaged roll characteristic is determined for movement of the boring tool along the path segment. A path segment pitch orientation is established based on at least one measured pitch orientation along the path segment. Using the maximum drill string curvature in combination with the averaged roll characteristic and the path segment pitch orientation, the yaw orientation is determined. The averaged roll characteristic is determined based on a series of incremental roll measurements that are spaced across the path segment. A set of coupled ordinary differential equations is used to characterize movement of the boring tool.