Worked solutionOne step at a time
Split the weight along and into the slope
On a ramp tilted by θ, gravity does two jobs: it pulls the robot down the slope, and it presses the robot into the surface.
Friction has a ceiling
The surface pushes back along the slope with at most μ times the force pressing into it. It makes no difference how many feet share the load: the total is the same.
It slides when the pull beats the ceiling
Set the two equal. The mass and gravity cancel, and so does everything about the motors.
Answer: B, about 31°
θ = atan 0.6 = 31°. The ramp in the reel is at 35°, four degrees past the limit, so the robot slides however strong it is. Torque decides how hard a foot can push. Friction decides how much of that push the ground will return sideways.
What the simulation showedWhy it is called a cone
The reel stands a Unitree Go2 on a ramp in MuJoCo and tilts it.
- At each foot, the forces the ground can supply fill a cone around the surface normal. Its half-angle is atan μ = 31°, the same angle as the answer.
- To stand still, the ground force at each foot must point straight up against gravity. Tilt the ramp by θ and that force sits θ away from the normal.
- Past 31° the force the robot needs is outside the cone. There is no joint torque that produces it, and the feet slide.
Wrong answersWhy each one is tempting
| Pick | What went wrong |
|---|---|
| A · 17° | Halved the friction coefficient. |
| C · 45° | Assumed friction can match the normal force, which needs μ = 1. |
| D · 60° | Far outside what μ = 0.6 allows; it needs μ = 1.73. |
Follow-ups an interviewer asks nextTry these before opening them
Does a heavier robot slide sooner?
No. Weight raises the pull down the slope and the friction ceiling by the same factor, so the angle depends on μ alone.
It now tries to walk up a 25° slope. Is 0.6 still enough?
Standing still needs tan 25° = 0.47, so yes. But speeding up the slope needs extra force along the surface, and so does each push-off. The margin of 0.13 is all it has, so hard accelerations will slip.
How do you climb something steeper than the friction angle?
Raise μ with softer rubber or spikes, or stop relying on friction: grip an edge, or press against a second surface so the normal force is not limited by your weight.
Where does a controller use this?
A legged robot's controller chooses foot forces every few milliseconds, with a rule that each one stays inside its friction cone. That is why the cone is drawn at every foot in the reel.
