When Adaptive Sampling Fails: A Controlled Evaluation of Graph-Based Path Planning for a Cuspidal 3R Manipulator

Cuspidal manipulators admit nonsingular transitions between distinct inverse-kinematics solutions of the same task-space pose, so reliable path planning can require reasoning over complete IK branches rather than selecting one solution independently at each waypoint. This study evaluates whether nonuniform waypoint sampling can reduce the computational burden of an all-solution layered-graph formulation while preserving feasible-path recovery. A canonical cuspidal 3R model was reconstructed from published kinematic parameters, and an independent numerical IK solver, graph planner, greedy baseline, and adaptive refinement method were implemented. The final controlled experiments do not support an adaptive advantage on the tested straight-line 3R benchmark. At comparable native-grid budgets, uniform sampling approximates the dense discrete objective more accurately. More importantly, when returned routes are reevaluated on a common dense predictor-corrector continuation, most reference-feasible paths are numerically identical across adaptive, same-count uniform, uniform-29, and the dense reference. The study therefore identifies an important methodological issue: planner comparisons performed on different discretization grids can confuse sampling error with genuine trajectory differences. Supporting reproducibility materials include the implementation, benchmark paths, validation data, timing results, and scripts used to reproduce the reported summary statistics