---
tag: 58895A7T4H
kind: concept
name: Trajectory optimization
msc: [49M37, 49M05, 49J15]
specializes: [T3V6ZNAMDN]
sources:
  - url: https://en.wikipedia.org/wiki/Trajectory_optimization
same_as:
  wikidata: Q7833062
  wikipedia: Trajectory optimization
---

## Summary

Trajectory optimization is the computation of a state trajectory and an open-loop control that minimise a cost for a [[control system|ZPDTVZQN5K]], subject to constraints.
It is the numerical side of [[optimal control|T3V6ZNAMDN]]: one wants one good trajectory from a given initial state, not a feedback law for every state.
Indirect methods solve the necessary conditions, as in [[shooting|RDTESNZTKQ]], while direct methods discretise the problem into a nonlinear program, as in [[direct collocation|T58W34D60G]].
It is used to plan rocket ascents, satellite transfers, walking robots and vehicle manoeuvres.

## Explanations

### L1 human

Suppose a robot arm must move from one pose to another in two seconds while using as little energy as possible and not hitting an obstacle.
Trajectory optimization finds the whole motion in advance: the positions over time and the motor commands that produce them.
A small example is the double integrator $x'' = u$ that must go from rest at $0$ to rest at $1$ in unit time.
Minimising $\int_0^1 u^2 \, dt$ gives the smooth motion $x(t) = 3t^2 - 2t^3$ with $u(t) = 6 - 12t$.