Center of Mass and the Support Polygon

A robot stands up as long as its center of mass sits above the area enclosed by its contact points with the ground, called the support polygon. Reach out too far, carry too much, or park on too steep a slope, and the center of mass moves outside it.

Arm angle Arm reach Payload Slope

The combined center of mass

For parts of mass $m_i$ at positions $\mathbf{r}_i$:

$$\mathbf{r}_{\text{CoM}} = \frac{\sum_i m_i\,\mathbf{r}_i}{\sum_i m_i}$$

A robot standing still is stable when the vertical line through its center of mass passes inside the support polygon. In this side view the polygon is the strip between the front and rear wheels. The stability margin is the distance from that line to the nearest edge.

What to notice

Stretch the arm forward with a heavy payload and the center of mass slides toward the front wheel. Past it, the robot tips. Tilt it on a slope and the center of mass shifts downhill relative to the wheels, so a robot that is fine on flat ground can tip on a ramp. A low, heavy base keeps the center of mass low and inside, which is why mobile manipulators put their batteries at the bottom.

This rule covers a robot standing still or moving slowly. Fast motion adds inertial forces, and for walking robots the zero moment point extends the same idea.