Robot control
Trajectory optimization
Definition
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.
Also known as: Trajectory optimisation
Updated
Choosing motion by an objective
A trajectory specifies how state changes with time. An optimizer can search for a trajectory that reaches a target while minimizing a cost such as elapsed time or control effort and respecting limits.
MIT's trajectory-optimization notes formulate this as a finite-horizon problem from a specified initial condition. Dynamic formulations include equations of motion; kinematic formulations can focus on geometry and motion limits.
Turning a trajectory into decision variables
Direct shooting optimizes inputs and obtains states by simulating the dynamics. Direct transcription represents sampled states and inputs as decision variables, imposing the dynamics as constraints. Direct collocation represents motion between samples with functions such as polynomials and constrains their consistency with the dynamics.
These choices affect numerical behavior and how a solver can use an initial guess. They do not guarantee the same solution for every problem.
Optimal depends on the formulation
Nonlinear problems can have local minima, and a solver may fail to find a feasible trajectory even when one exists. MIT discusses the importance of initialization and warns against confusing solver failure with proven infeasibility.
A computed trajectory is also distinct from a feedback policy. Executing it requires control that handles tracking errors. Model predictive control repeatedly solves a finite-horizon problem as new state estimates arrive.
Sources
Related terms
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.
Model predictive control
Model predictive control repeatedly optimizes future actions using a system model, applies the next part of the solution, and replans from updated state information. It can account for objectives and constraints over a finite prediction horizon.
Contact-implicit optimization
Contact-implicit optimization plans motion while allowing contact events and forces to emerge from contact constraints in the optimization. It avoids requiring every contact transition to be fixed in a predefined mode sequence.