Hopping and the Raibert Controller
A one-legged robot cannot stand still at all: it can only stay up by hopping. Marc Raibert showed in the 1980s that a surprisingly simple rule works. Control the hopping height with the leg's thrust, the body angle with the hip during stance, and the forward speed by where the foot lands during flight. That last part is the key.
The foot placement rule
During stance, which lasts about \(T_s\), the body passes over the foot. Placing the foot halfway along that path, the neutral point, keeps the speed the same. Placing it further ahead slows the hopper down, and behind speeds it up:
What to notice
With the rule on, the forward speed settles at the target within a few hops, and recovers after a shove. Turn the rule off and the foot lands straight under the hip every time, so any speed it has persists and it cannot slow down or turn around. This page also measures the real stance time on every hop and regulates hop height by pushing harder after a low hop, as Raibert's machines did. A gain that is too high makes the speed oscillate from hop to hop. The same idea, choosing where to step to control speed, is at the core of today's legged robots, including those from Boston Dynamics, which Raibert founded.