Worked solutionOne step at a time
Find the pivot
A robot standing still on flat feet can only push on the ground inside its support polygon, the convex hull of its foot contacts. If it tips forward, it rotates about the front edge of that polygon: the toe line.
Measure every lever from the toes
The robot's own weight acts at its centre of mass, 10 cm behind the toes. The box is 50 cm ahead of the centre of mass, so it is 50 − 10 = 40 cm ahead of the toes.
Balance the moments
The robot's weight turns it backwards about the toes, the box turns it forwards. It stays up while the backward moment wins. Gravity cancels from both sides.
Answer: D, 10 kg
m = 40 × 0.10 / 0.40 = 10 kg. Put another way: the combined centre of mass, (40 × −0.10 + 10 × 0.40) / 50 = 0, sits exactly on the toe edge. One gram more and it is outside the polygon.
What the simulation showedThe same rule on a Unitree H1
The reel poses a Unitree H1 from MuJoCo Menagerie, soft knees and a slight forward lean, and reads every number from the model.
- Robot mass 51.4 kg. Support polygon from the six foot capsules: heel contacts 17.5 cm behind the toe edge, feet 41 cm apart.
- Box hugged at the chest: robot centre of mass 11.4 cm behind the toes, box 12.1 cm ahead. It would take a 48.5 kg box to tip it, so 20 kg is fine.
- Box at arm's length: reaching forward moves the arms too, so the robot's own centre of mass is now 9.3 cm behind the toes, and the box is 42.1 cm ahead. Limit: 51.4 × 0.093 / 0.421 = 11.3 kg.
- Then the posed robot and box are simulated as one rigid body standing on its feet. With 11.0 kg it stands; with 12 kg it rotates over its toes and falls onto the box in about 2.2 s.
Wrong answersWhy each one is tempting
| Pick | What went wrong |
|---|---|
| A · 8 kg | Measured the box's lever from the centre of mass, not from the toe edge it pivots about. |
| B · 20 kg | Used a fixed fraction of body weight; the limit depends on the two lever arms. |
| C · 4 kg | Stopped at the robot's moment (4 kg·m) and forgot to divide by the box's lever. |
Follow-ups an interviewer asks nextTry these before opening them
How can it hold more without new motors?
Move its own centre of mass back: squat and lean the hips back, or step one foot forward to push the toe edge out. Each centimetre added to a or taken off b raises the limit. Humans do exactly this when they carry something heavy in front of them.
What changes if it walks instead of standing still?
Then the centre of mass accelerates, and the point that must stay inside the polygon is the zero moment point (ZMP), which includes the inertial forces. Accelerating forwards moves the ZMP back; braking moves it forwards, so stopping with the box out front is the dangerous moment.
Does a heavier robot always hold more?
Only if the extra mass sits behind the toes. Mass added to the arms or the front of the torso can lower the limit, because it moves the robot's own centre of mass towards the toe edge.
Why does the H1 in the reel lean forward a little?
Standing bolt upright, its centre of mass is only about a centimetre in front of the heel contacts, so the smallest push backwards tips it. A slight lean puts the centre of mass over mid-foot, the way a balance controller would.
