Time-Optimal Spiral Trajectories With Closed-Form Solutions

The Archimedean spiral is space-filling plane curve that is found in applications ranging from coverage path planning for robot exploration to scan pattern generation for medical imaging. The constant linear velocity (CLV) parameterization of this spiral is of particular interest due to its fixed path velocity and isotropic sampling capability, but the high accelerations near its origin singularity yield poor trajectory tracking that limit its utility. Here, I derive a closed-form time-optimal time scaling for CLV spirals with large path velocities that mitigates the singularity by inspecting the CLV spiral's acceleration envelope. When applied to two degree-of-freedom Cartesian scanner, I demonstrate that this approach reduces trajectory tracking error by up to 97.1 % as compared to naïve CLV spirals with low computational overhead. I further show that this time scaling eliminates the central image distortion near the origin for scanning applications that rely on CLV spirals.