Geometric Approach to Robot Deceleration Control: Nonlinear Distance-Based Control Using a Parabolic Cylinder (PCV-NA)

We present the Parabolic Cylinder Velocity-Nonlinear Approach (PCV-NA), a distance-driven deceleration method derived from a 3-D parabolic-cylinder constraint that directly couples remaining distance and target speed. In the ideal setting (no noise, delay, or friction), the model yields finite-time convergence of both distance and velocity to zero. For deployment, we provide a lightweight constant-time discrete implementation with robust sensing (Hampel filtering with α-β prediction and gating), delay handling, friction compensation, soft-start ramping, and an adaptive jerk limiter. In 100 Hz simulations, the approach achieves mean jerk 0.48 m/s³, peak jerk 4.09 m/s³, and responsiveness to the 5% band median 0.06 s (p95 0.09 s, worst 0.13 s), while preserving instant distance-driven adaptation without time-profile replanning. When measured under a dwell definition (5% band with hysteresis and 0.02 s dwell time), the lag was effectively reported as 0 s because the trajectory remained within tolerance from the start. The method is efficient for embedded hardware and released with code-level mappings for reproducibility.