Trajectory Optimization 101

Finding the best possible path through motion, not just any path that technically works.

Trajectory optimization computes a full motion path over time — every joint's position, velocity, and often torque at each moment — that satisfies physical constraints (joint limits, contact forces, avoiding obstacles) while minimizing some cost, like energy use, time, or jerkiness.

Why it's more than just connecting two points

Unlike simple point-to-point motion planning, trajectory optimization accounts for the robot's full dynamics throughout the motion, not just its start and end configuration — which matters enormously for a legged robot, where the entire trajectory (not just the endpoint) needs to keep it balanced the whole time.

Where it's used

It underlies both offline motion planning (computing a good walking gait ahead of time) and, in a shorter-horizon, faster-solving form, real-time model predictive control — the same underlying math applied at very different timescales and computational budgets.