Posture-to-posture trajectory planning for an autonomous car using triple clothoid segments

We consider a steering plan for an autonomous car as a function of path length. The initial state of a car is composed of its position, direction, and curvature corresponding to its front-wheel direction. In the case that the final state of the steering plan is taken as one point on a curved road, not only the position but also the direction of the car and curvature corresponding to the curved road are specified. Defining posture as a state that consists of the car's position, direction, and curvature, we consider this steering plan as a trajectory plan from the start posture to the end posture. In cars, the upper and lower limits of allowable curvature exist due to the minimum turning radius of the cars. Lateral acceleration and jerk, which affect ride quality, are respectively proportional to curvature and sharpness, which is the rate of change of curvature. We chose clothoid segments, for which curvature and sharpness are easily adjusted, as components of a trajectory. Single or double clothoid segments are not flexible enough to match the end posture. We propose a trajectory planning method that uses triple clothoid segments whose lengths are variable and introduce a four-parameter model for this method. We moreover show a concrete numerical solution method based on an optimization problem framework.