The Impact of Geometric Continuities (C1, C2, and C3) on the Trajectories of Industrial Robots

This article presents a mathematical analysis of the geometric continuity of industrial robot trajectories, highlighting the influence of continuity conditions on velocity, acceleration, and vibration profiles. This study proposes an original approach: isolating geometric continuity as an independent parameter and the comparative evaluation of the continuity levels C1, C2, and C3 using a series of pre-computed kinematic metrics: vibration energy, acceleration variability, and trajectory curvature stability. The methodology is validated by numerical simulations performed with RoboDK for trajectory generation and by post-processing in Python for metric evaluation. The results indicate that C1 trajectories exhibit discontinuities in the higher-order derivatives, which lead to undesirable kinematic behaviors, while C2 continuity represents the minimum requirement for an acceptable, stable motion in industrial applications. Higher-order continuity, C3, brings greater regularity to the trajectories, but the practical advantages are limited and relatively insignificant for standard industrial applications.