${ \mmb{\eta } }^{3}$-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 $ { \mmb{\eta } }^{3}$-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 $\mmb G^{3}$-path is planned. A $\mmb G^{3}$-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 ($ { \mmb{\eta } }$) of six parameters that can be used to finely shape the path. The $ { \mmb{\eta } }^{3}$ -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 $ { \mmb{\eta } }^{k}$-splines.