Robot control
Time-optimal path parameterization
Definition
Time-optimal path parameterization assigns timing to a fixed geometric robot path so the path is traversed as quickly as possible while satisfying specified velocity, acceleration, torque, or other constraints.
Also known as: TOPP, Time-optimal path parameterisation
Updated
Separate the path from its clock
A geometric path can be written as robot configuration q(s), where the scalar path coordinate s moves from start to finish. Time-optimal path parameterization finds a time law s(t). The path shape stays fixed, but the robot speeds up and slows down along it.
The constraints are transformed into bounds on path velocity and acceleration. Depending on the formulation, these can represent joint-speed limits, acceleration limits, torque limits, or contact-related inequalities. The objective is minimum traversal time within those stated bounds.
Reachability can propagate feasible speeds
Pham and Pham formulate TOPP through reachable and controllable sets at discretised positions on the path. Their TOPP-RA algorithm solves small linear programs recursively and also supports admissible-velocity propagation and a treatment of parametric uncertainty.
This differs from motion planning, which must choose a collision-free path, and from general trajectory optimization, which may change the path and timing together. TOPP starts after a path has already been chosen.
The optimum inherits every modelling assumption
A poor geometric path remains poor after optimal timing. Constraints that omit motor heating, flexible transmission behaviour, contact uncertainty, or controller bandwidth can produce a schedule that is mathematically feasible but unsuitable for hardware.
Minimum time can also drive motion onto active constraint boundaries and create aggressive changes in acceleration. Practical systems may add jerk limits or a time margin, giving up the strict optimum for smoother tracking and tolerance to modelling error.
Sources
Related terms
Trajectory optimization
Trajectory optimization finds a time-varying motion, and often control inputs, that minimizes an objective while satisfying specified constraints. Robot applications can include geometric, kinematic, and dynamic constraints.
Motion planning
Motion planning finds a robot movement from an initial state to a goal while satisfying constraints such as collision avoidance. A planner may produce a geometric path, a timed trajectory, or a sequence of controls.
Reachability analysis
Reachability analysis computes sets of states that a dynamical system can reach, or from which it can reach a target, under specified controls, time limits, and disturbances. Robots use it to reason about feasibility, collision avoidance, and conditional safety guarantees.
Jerk
Jerk is the rate at which acceleration changes with time, or the third time derivative of position. Robotics uses jerk limits to constrain how abruptly a commanded motion changes acceleration.