${ \mmb{\eta } }^{3}$-Splines for the Smooth Path Generation of Wheeled Mobile Robots

The paper deals with the generation of smooth paths for the inversion-based motion control of wheeled mobile robots. A new path primitive, called eta3-spline, is proposed. It is a seventh order polynomial spline which permits the interpolation of an arbitrary sequence of points with associated arbitrary tangent directions, curvatures, and curvature derivatives, so that an overall G3-path is planned. A G3-path or path with third order geometric continuity has continuous tangent vector, curvature, and curvature derivative along the arc length. Adopting this planning scheme and a dynamic path inversion technique, the robot's command velocities are continuous with continuous accelerations. The new primitive depends on a vector (eta) of six parameters that can be used to finely shape the path. The eta3-spline can generate or approximate, in a unified framework, a variety of curve primitives such as circular arcs, clothoids, spirals, etc. The paper includes theoretical results, path planning examples, and a note on general etak-splines.