Trajectory control for an autonomous ground vehicle typically utilizes the error from the desired path or trajectory (i.e., crosstrack error) to produce velocity and steering commands. If an obstacle is in the path, previous techniques have synthesized a new trajectory that avoids the obstacles, and the vehicle directly follows this new path. This approach has drawbacks at high velocity, because the synthesized trajectory must satisfy the stability criteria of the vehicle. This paper introduces a technique which we call epsilon dragging. The approach modifies the existing trajectory by some value e in order to avoid an obstacle at high speeds, while preserving the original trajectory as the desired path. Epsilon dragging is performed by inducing an additional error to the crosstrack error of the vehicle; this induced error can be bounded in order to stay within the velocity/turnrate profile that governs safe behavior at high speeds. The paper provides a method to construct epsilon such that a vehicle can avoid an obstacle at high speeds without the need to verify the trajectory's curvature before it is synthesized. The technique is demonstrated in completing a lane-change maneuver at different velocities, and verifying that the velocity/turnrate profiles are not exceeded.