Sigmoid angle-arc curves: Enhancing robot time-optimal path parameterization for high-order smooth motion

Trajectory planning is crucial in the motion planning of robots, where finding the time-optimal path parameterization (TOPP) of a given path subject to kinodynamic constraints is an important component of trajectory planning. The tangential discontinuity at the intersection of continuous line segments limits the speed of trajectory planning and can easily cause jitter and over-constraint phenomena. Smooth transitions at corners can be achieved by inserting parameter spline curves. However, due to the insensitivity of parameter spline curves to arc length, their performance in the application of the TOPP algorithm, which relies on the higher-order robot kinematics smoothness (i.e., the function $q\left(s\right)$ of the configuration space to the Cartesian space), fails to meet expectations.

A smoothing method suitable for the TOPP algorithm is proposed: Sigmoid Angle-Arc Curve (SAAC). This curve exhibits excellent performance in smooth corner transitions of the TOPP algorithm and is parameterized using arc length. The curvature and geometry of its curves can be expressed analytically in terms of arc lengths. Compared with the traditional B-spline method and the symmetric Euler spiral blending (SE-spiral), SAAC can provide smoother $C^2$ robot kinematics characteristics. Using the TOPP algorithm based on SAAC can significantly enhance the robustness of the TOPP algorithm, significantly reduce jerks, and reduce the time required for movement.