Smooth Trajectory Generation for Robotic Manipulators Using Adaptive Tanh-Based S-Curve Profiles

This work presents an innovative approach for smooth trajectory generation in robotic manipulators, using a hyperbolic tangent function whose slope is dynamically modulated by an inverted Gaussian profile. This formulation enables the construction of S-type motion profiles with third-order kinematic continuity (C3), ensuring that position, velocity, acceleration, and jerk are continuous. In particular, the jerk starts and ends at zero, a condition that eliminates residual vibrations and prevents abrupt dynamic excitations when initiating or completing a movement. This guarantees smooth boundary conditions while optimizing both agility and smoothness, without the complexity of high-degree polynomials or conventional segmentations. Validation through simulations in MATLAB/Simulink on a 3-degree-of-freedom manipulator demonstrated that the strategy effectively mitigates jerk. Key results include a JRMS value of 0.548, indicating a significant reduction in vibrations, with a balanced dynamic response. Compared to classical profiles such as the sigmoid or standard tanh, this adaptive approach exhibited superior control and enhanced dynamic continuity, even in multipoint trajectories in Cartesian space, establishing itself as a robust and efficient alternative.