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

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The paper deals with the generation of smooth paths for the inversion-based motion control of wheeled mobile robots. A new path primitive, called <formula formulatype="inline"><tex>$ { \mmb{\eta } }^{3}$</tex></formula>-spline, is proposed. It is a seventh order polynomial spline which permits the interpolation of an arbitrary sequence of points with associated arbitrary tangent directions, curvatures, and curvature derivatives, so that an overall <formula formulatype="inline"> <tex>$\mmb G^{3}$</tex></formula>-path is planned. A <formula formulatype="inline"> <tex>$\mmb G^{3}$</tex></formula>-path or path with third order geometric continuity has continuous tangent vector, curvature, and curvature derivative along the arc length. Adopting this planning scheme and a dynamic path inversion technique, the robot's command velocities are continuous with continuous accelerations. The new primitive depends on a vector (<formula formulatype="inline"><tex>$ { \mmb{\eta } }$</tex></formula>) of six parameters that can be used to finely shape the path. The <formula formulatype="inline"><tex>$ { \mmb{\eta } }^{3}$</tex> </formula>-spline can generate or approximate, in a unified framework, a variety of curve primitives such as circular arcs, clothoids, spirals, etc. The paper includes theoretical results, path planning examples, and a note on general <formula formulatype="inline"><tex>$ { \mmb{\eta } }^{k}$</tex></formula>-splines. </para>