Path planning and path following are core components in safe autonomous driving. Typically, a path planner provides a path with some tolerance on how tightly the path should be followed. Based on that, and other path characteristics, for example, sharpness of curves, a speed profile needs to be assigned so that the vehicle can stay within the given tolerance without going unnecessarily slow. Here, such trajectory planning is based on optimal control formulations where critical cases arise as on-the-limit solutions. The study focuses on heavy commercial vehicles, causing rollover to be of a major concern, due to the relatively high centre of gravity. Several results are obtained on required model complexity depending on path characteristics, for example, quantification of required path tolerance for a simple model to be sufficient, quantification of when yaw inertia needs to be considered in more detail, and how the curvature rate of change interplays with available friction. Overall, in situations where the vehicle is subject to a wide range of driving conditions, from good transport roads to more tricky avoidance manoeuvres, the requirements on the path following will vary. For this, the provided results form a basis for real-time path following.