Time-Optimal Path Parameterization

Given a fixed path, find the fastest possible timing schedule that still respects velocity, acceleration, and torque limits.

TOPP is the ‘fastest possible’ version of time parameterization: given a fixed geometric path q(s), it finds the s(t) that minimizes total time T = ∫ds/ṡ (where ṡ is the speed of progress along the path) subject to joint velocity, acceleration, and torque limits. The classic approach, from a 1985 paper by James Bobrow, Steven Dubowsky, and John Gibson, is numerical integration: on the (s, ṡ) phase plane, it alternately integrates at maximum acceleration and maximum deceleration to find the switching points, producing a ‘bang-bang’ profile — full acceleration, then full deceleration. This is fast to compute but the switching points are hard to locate and prone to numerical trouble; a second family of convex-optimization methods is more stable but slower. In 2018, Hung Pham and Quang-Cuong Pham published TOPP-RA in IEEE Transactions on Robotics, which solves a small linear program at each discretized point along the path by computing reachable and controllable velocity sets, balancing speed with robustness; the open-source toppra library implementing it is widely used. Industrial arms that need to shrink cycle time or move material quickly commonly rely on it.

ExampleAn arm moving along a fixed weld-seam path has that path discretized into points, and TOPP-RA computes the maximum achievable speed at each one: straight segments hug the speed limit, while corners automatically slow down because of the acceleration constraints.

Also called

TOPP, Time-Optimal Path Tracking

Related

Time Parameterization · TOPP-RA · Trajectory Planning · Trapezoidal Velocity Profile · Convex Optimization · Joint Limits

Sources

Pham, Pham: A New Approach to Time-Optimal Path Parameterization based on Reachability Analysis (arXiv:1707.07239, IEEE T-RO 2018)
Bobrow, Dubowsky, Gibson: Time-Optimal Control of Robotic Manipulators Along Specified Paths (IJRR 1985)
toppra: Time-Optimal Path Parameterization library (GitHub)