An Investigation of the Clothoid Steering Model for Autonomous Vehicles

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Virginia Tech

The clothoid, also known as the Cornu spiral, is a curve generated by linearly increasing or decreasing curvature as a function of arc length. The clothoid has been widely accepted as a logical curve for transitioning from straight segments to circle arcs in roads and railways, because a vehicle following the curve at constant speed will have a constant change of centripetal acceleration. Clothoids have also been widely adopted in planning potential paths for autonomous vehicle navigation. They have been viewed as useful representations of possible trajectories that are dynamically feasible. Surprisingly, the assumptions that underlie this choice appear to be lightly treated or ignored in past literature.

This thesis will examine three key assumptions that are implicitly made when assuming that a vehicle will follow a clothoid path. The first assumption is that the vehicle's steering mechanism will produce a linear change in turning radius for a constant rate input. This assumption is loosely referred to as the "bicycle model" and it relates directly to the kinematic parameters of the steering mechanism. The second assumption is that the steering actuator can provide a constant steering velocity. In other words, the actuator controlling the steering motion can instantaneously change from one rate to another. The third assumption is that the vehicle is traveling at a constant velocity. By definition, the clothoid is a perfect representation of a vehicle traveling at constant velocity with a constant rate of change in steering curvature. The goal of this research was to examine the accuracy of these assumptions for a typical Ackermann-steered ground vehicle. Both theoretical and experimental results are presented.

The vehicle that was used as an example in this study was a modified Club Car Pioneer XRT 1500. This Ackermann-steered vehicle was modified for autonomous navigation and was one of Virginia Tech's entries in the DARPA 2005 Grand Challenge. As in typical operation, path planning was conducted using the classic clothoid curve model. The vehicle was then commanded to drive a selected path, but with variations in speed and steering rate that are inherent to the real system. The validity of the three assumptions discussed above were examined by comparing the actual vehicle response to the planned clothoid.

This study determined that the actual paths driven by the vehicle were generally a close match to the originally planned theoretical clothoid path. In this study, the actual kinematics of the Ackermann vehicle steering system had only a small effect on the driven path. This indicates that the bicycle model is a reasonable simplification, at least for the case studied. The assumption of constant velocity actuation of the steering system also proved to be reasonably accurate. The greatest deviation from the planned clothoid path resulted from the nonlinear velocity of the vehicle along the path, especially when accelerating from a stop. Nevertheless, the clothoid path plan generally seems to be a good representation of actual vehicle motion, especially when the planned path is updated frequently.

Clothoid, Autonomous Navigation, Clothoid Navigation, Path Planning, Bicycle Model