Trajectory optimization and MPC

Level SpecializationDifficulty ★★★★★Application⌖ Open in the map

What is it?

Choose the whole motion that minimizes a cost (time, energy, jerk) subject to dynamics, limits and obstacles — a constrained optimization over functions, discretized into a large nonlinear program. Model predictive control re-solves it every few milliseconds over a short horizon.

Formulas

\min_{x(\cdot),\,u(\cdot)}\int_0^T \ell(x, u)\,\dd t, \qquad \dot x = f(x, u),\ \ g(x, u) \le 0

The mathematics behind it

Constrained optimization★★★★★fundamental
A robot trajectory minimizes effort subject to dynamics, joint limits and obstacles.
Higher-order derivatives★★★★★frequent
Minimum-jerk trajectories minimize $\int \dddot x^{\,2}\,\dd t$ for smooth robot motion.
KKT conditions★★★★★frequent
Nonlinear programming solvers used in model predictive control (IPOPT, SQP) iterate towards a KKT point.
Curvature (basic differential geometry)★★★★★frequent
Vehicle paths must respect a maximum curvature (minimum turning radius).

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.