The physics of flipping

[ewhitman]

This problem can be addressed by the concept of the Zero Moment Point. The ZMP is most often used in the context of determining tipping for humanoid robots, but it applies here.

The ZMP is the point on the ground through which a single force would act to achieve a given Center of Mass acceleration without any change in angular momentum.

If the ZMP leaves your wheel base (convex hull of your points of support), your robot will begin to tip.

The relevant formula is: Z = -h/g*a

where:
Z = the location of the ZMP relative to the CoM (in m)
h = the height of the CoM (in m)
g = the acceleration due to gravity (9.8 m/s/s)
a = the acceleration of the CoM (in m/s/s)

This means that while sitting still or traveling at a constant speed (a = 0), the ZMP is directly under the CoM (Z=0), which is hopefully inside your wheel base.

However, if your CoM is 50 cm high, and you accelerate forward at 5 m/s/s, your ZMP will move backwards 25.5 cm. (0.255 = -0.5/9.8*5) If your CoM is less than 25.5 cm in front of your rear wheels, you will begin to tip because the ZMP will have left your wheel base.

If I haven't explained the math well enough, the moral is:
1) Keep your CoM low
2) Keep your wheel base big and your CoM near the center of it

[Aren Siekmeier]

The negative acceleration pointed out by Don is definitely the one we're worried about too. We're trying to keep the angle between the floor and the line between our rear contact and the CG less than 45 degrees so that we can experience as much as 1g laterally and be just fine. And even lower is even better. This includes when we are crossing the bump/bridge, so depending on wheel profile, the angle on flat ground needs to be potentially as small as 22 degrees, making a short wheel base harder and harder.