6.9 Trajectories

Instances of Trajectory are used to define an intended path for Entity motion in precise mathematical terms. The motion path may be specified using different mathematical shapes:

Instances of Trajectory may be specified using just the three-positional dimensions along the X, Y, and Z axes (see Section 6.3, "Coordinate systems" for coordinate system definitions). Alternatively, instances of Trajectory may also be specified using the three-positional dimensions and the three-rotational dimensions (heading, pitch and roll) for six total dimensions. In the second case, the path not only specifies the movement of the entity along the path, but also the orientation of the corresponding entity during that movement.

When a Trajectory is specified relative to the position of a moving entity, its absolute position cannot be calculated before the scenario runtime. In this case, the trajectory must be calculated during the scenario runtime when the corresponding FollowTrajectoryAction is triggered.

Additionally, an instance of Trajectory may be specified with or without a time dimension, allowing for the combined or separate specification of the entities longitudinal domain: A trajectory incorporating the time dimension completely specifies the motion of the entity, including its speed. A trajectory without the time dimension does not specify the speed along the path. This approach allows for the separate control of speed.

A Trajectory provides a mathematically precise definition of a motion path. The corresponding entities behavior, however, depends on the actions employing the trajectory: Either an Entity follows this path exactly, or it uses the path as a guidance for the controller to follow the trajectory as best as the entities rules of motion allow.

Polyline trajectories may be used to represent a sampled version of any arbitrary curve shape. When using polyline trajectories, it is advised to use a sufficiently high number of points.

Clothoid and clothoid spline trajectories can be used to ensure a constant change of curvature along the curve shape. This can be useful to ensure that a trajectory is driveable. Circular arc and straight line segments can be expressed as special cases of clothoids by setting the curvature to a constant value and zero respectively. For clothoid spline trajectories it is possible to define the connecting points between the segments. This may be used to reset numerical errors, which might occur in very large trajectories. If the intermediate positions are not provided, the end position of the last segment will be used. Clothoid spline trajectories shall always be gapless between any two segments. However, it is permitted to intentionally model discontinuities in the tangent and curvature between the segments. This may be used to directly connect a circular arc and a line segment or create a sharp kink to induce a strong jerk in an entities' motion.

By using NURBS trajectories, most relevant trajectory curves may be expressed either directly or with arbitrary approximation: NURBS curves form a strict superset of the curves expressible as Bézier curves, piecewise Bézier curves, or non-rational B-Splines, which can be mapped to corresponding NURBS curves. Because NURBS curves directly support the expression of conic sections (such as circles and ellipses), approximation of clothoids using arc spline approaches is feasible. Note that although NURBS are able to approximate most relevant types of curves, it is non-trivial to understand if a certain type of curve has been approximated using a NURBS when reading a scenario file using NURBS. Therefore, conversion of other curve types to NURBS should only be used if later editing of the trajectory in the file, based on the initial curve type, is not required.

Another useful property of NURBS is that they allow continuity up to a given derivative: A NURBS curve of degree k, meaning order k+1, is infinitely continuously differentiable in the interior of each knot span and k-M-1 times continuously differentiable at a knot. M is the multiplicity of the knot, meaning the number of consecutive knot vector elements with the same value.

Commonly used NURBS curves are curves of quadratic (order = 3) and cubic (order = 4) degree. Curves of higher order are usually only required if continuity in higher derivatives needs to be ensured. The effort for evaluating curves increases with rising orders. Therefore, it is recommended to restrict instances of Trajectory to lower orders.

The following sections describe edge cases in the definition of trajectories.